Nuclear physics

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Methods of Experimental Physics VOLUME 5

NUCLEAR PHYSICS PART B

METHODS OF

EXPERIMENTAL PHYSICS: 1. Marton, Editor-in-Chief Claire Marton, Assistant Editor

1. Classical Methods, 1959 Edited by lmmanuel Estermann

2. Electronic Methods, in preparation Edited by E. Bleuler and R. 0. Haxby 3. Molecular Physics, 1961 Edited by Dudley Williams

4. Atomic and Electron Physics, in preparation Edited by Vernon W. Hughes and Howard L. Schultz 5. Nuclear Physics (in two parts), 1961 and 1963 Edited by Luke C. L. Yuan and Chien-Shiung Wu

6. Solid State Physics (in fwo parts), 1959 Edited by K. Lark-Horovitz and Vivian A. Johnson

Volume 5

Nuclear Physics Edited by

LUKE C. 1. YUAN Brookhaven National Laboratory Upfon, New York

CHIEN-SHIUNG WU Columbia University New York, New York

PART B

1963

@

ACADEMIC PRESS New York and London

Copyright @ 1963, by

ACADEMIC PRESS INC. ALL RIGHTS RESERVED NO P A R T OF THIS BOOK MAY B E REPRODUCED I N ANY FORM B Y PHOTOSTAT, MICROFILM, OR A N Y O T H E R MEANS, WITHOUT WRITTEN PERMISSION FROM T H E PUBLISHERS

ACADEMIC PRESS INC. 111 FIFTHAVENUE NEW YORK3, N. Y.

United Kingdom Edilon Published by ACADEMIC PRESS INC. (LONDON) LTD.

BERKELEY SQUARE HOUSE,LONDON W. 1

Library of Congress Catalog Card Number 61-17860 PRINTED I N T H E UNITED STATES O F AMERICA

CONTRIBUTORS TO VOLUME 5, PART B E. AMBLER,Low Temperature Section, National Bureau of Standards, Washington, D. C.I Page 162. F. AJZENBERG-SELOVE, Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania. Page 339. M. BLAU,Institut f u r Radiumforschung, Vienna, Austria. Page 37. M. H. BLEWETT,Brookhaven National Laboratory, Upton, New York. Pages 580 and 623. 0. CHAMBERLAIN, Department of Physics, University of California, Berkeley, California. Page 485. B. CORK,Lawrence Radiation Laboratory, University of California, Berkeley, California. Page 747. H. DANIEL, M a x Planck Institute for Nuclear Physics, Heidelberg, Germany. Page 275. M. DEUTSCH, Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts. Page 303. H. E. DUCKWORTH, Department of Physics, McMaster University, Hamilton, Ontario, Canada. Page 1. R. D. EVANS,Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts. Page 761. G. FEHER,Department of Solid State Physics, University of California, L a Jolla, California. Page 127. H. FRAUENFELDER, Department of Physics and Coordinated Science Laboratory, University of Illinois, Urbana, Illinois. Pages 129 and 214. W. B. FRETTER, Department ifPhysics, University of California, Berkeley, California. Page 35. W. GENTNER, M a x Planck Institute for Nuclear Physics, Heidelberg, Germany. Page 275. S. GESCHWIND, Bell Laboratories, Murray Hill, New Jersey. Page 20. J. G. HIRSCHBERG, Project Matterhorn, Princeton University, Princeton, New Jersey. Page 44. J. C. HUBBS,E-H Research Laboratories, Inc., Oakland, California. Page 58. 1

Present address: 1504 Summit Road, Berkeley, California. V

vi

CONTRIBUTORS TO VOLUME

5,

PART B

C. D. JEFFRIES,Department of Physics, University of California, Berkeley, California. Page 104. C. K. JEN, Applied Physics Laboratory, The Johns Hopkins University, Silver Spring, Maryland. Page 85. H. W. KOCH,Betatron Laboratories, National Bureau of Standards, Washington, D. C. Page 508. H. KOUTS,Brookhaven National Laboratory, Upton, New York. Page 590. D. W. MILLER,Department of Physics, Indiana University, Bloomington, Indiana. Page 366. W. A. NIERENBERG, Department of Physics, University of California, Berkeley, California. Page 58. G. D. O'KELLEY,Oak Ridge National Laboratory, Oak Ridge, Tennessee. Page 555. J. S. PRUITT, Betatron Laboratories, National Bureau of Standards, Washington, D. C. Page 508. S . REED,O$ce of Naval Research, Washington, D. C . Page 807. L. ROSEN,Department of Physics, Los Alamos Scientific Laboratory, Los Alamos, New Mexico. Page 366. A. ROSSI,Department of Physics, University of Illinois, Urbana, IElinois.2 Page 214. R. M. STERNHEIMER, Brookhaven National Laboratory, Upton, New York. Pages 691 and 821. A. H. WAPSTRA,Instztute voor Kernphysisch Onderzoek, Amsterdam, Holland. Page 152. W. WHALING, Department of Physics, California Institute of Technology, Pasadena, California. Page 9.

* Present address: Sorin, Saluggia (Vercelli), Italy.

CONTENTS, VOLUME 5, PART B CONTRIBUTORS TO VOLUME 5, PARTB . . . . . . . . . . . . .

v

CONTRIBUTORS TO VOLUME 5, PART A . . . . . . . . . . . . . xiii 5, PARTA . . . . . . . . . . . . . . . . xv CONTENTS, VOLUME

2. Methods for the Determination of Fundamental Physical Quantities (Continued from Vol. 5, Part A )

2.3. Determination of Mass of Nuclei and of Individual Particles. . . . . . . . . . . . . . . . . . . . . 2.3.1. Mass Spectroscopy. . . . . . . . . . . . . . . by H. E. DUCKWORTH

1 1

2.3.2. Determination of Atomic Masses from Nuclear Reaction and Nuclear Disintegration Energies . . . by WARDWHALING

9

2.3.3. Atomic Mass Determination from Microwave Spectra. . . . . . . . . . . . . . . . . . . 20 by S. GESCHWIND 2.3.4. Measurement of Mass with Cloud Chambers and Bubble Chambers . . . . . . . . . . . . . . 35 by W. B. FRETTER 2.3.5. Determination of Mass of Nucleons in Emulsions by MARIETTA BLAU 2.4. Determination of Spin, Parity, and Nuclear Moments . 2.4.1. Spectroscopic Methods . . . . . . . . . . . . 2.4.1.1. Optical and Ultraviolet Spectroscopy . . by JOSEPH G. HIRSCHBERG

.

37

.

44

. 44 . 44

2.4.1.2. The Investigation of Short-Lived Radionuclei by Atomic Beam Methods. . . . by JOHN C. HUBBS and WILLIAMA. NIERENBERG

58

2.4.1.3. Microwave Method. . . . . . . . . . . 85 by C. K. JEN 2.4.1.4. Nuclear Magnetic and Quadrupole Resonance . . . . . . . . . . . . . . . . 104 by C. D. JEFFRIES vii

...

CONTENTS. VOLUME

Vlll

5.

PART B

2.4.1.4.7.3. The Electron Nuclear Double Resonance (ENDOR) Technique . . . . . . . 127 by G . FEHER 2.4.2. Indirect Methods . . . . . . . . . . 2.4.2.1. Angular Correlation . . . . . by HANSFRAUENFELDER

. . . . . 129

. . . . . 129

2.4.2.2. Conversion Coefficients . . . . . . . . . 152 by A. H . WAPSTRA 2.4.2.3. Nuclear Orientat,ion. . . . . . . . . . . 162 by E . AMBLER 2.5. Determination of the Polarization of Electrons and Photons 214 by H . FRAUENFELDER and A. ROSSI 2.5.1. Introduction . . . . . . . . . . . . . . . . . 214 2.5.2. Description of Polarized Beams . . . . . . . . . 216 2.5.3. Motion of Electrons in Electromagnetic Fields . . 223 2.5.4. Polarization Transfer . . . . . . . . . . . . . . 233 2.5.5. Detection of Electron Polarization . . . . . . . . 239 2.5.6. Detection of Photon Polarization . . . . . . . . 264 2.6. Determination of Life-Time . . . . . . . . . . . . . . 275 2.6.1. Long Life-Time . . . . . . . . . . . . . . . . 275 by H . DANIEL and W . GENTNER 2.6.2. Short Lifetimes . . . . . . . . . . . . . . . . 303 by M . DEUTSCH 2.7. Determination of Nuclear Reactions . . . . . . . . . . 339 2.7.1. Determination of the Q Value for Nuclear Reactions 339 2.7.2. Determination of Nuclear Energy Levels from Reaction Energies . . . . . . . . . . . . . . . 352 by FAYAJZENBERB-SELOVE 2.7.3. 2.7.4. 2.7.5. 2.7.6. 2.7.7.

Total Interaction Cross Sections . . . . . Nonelastic Neutron Cross Sections . . . . . Differential Interaction Cross Sections . . . Differential Elastic-Scattering Cross Sections Supplementary Remarks . . . . . . . . . by LOUISROSENand DANW . MILLER

2.8. Determination of Flux and Densities . . . . . . . 2.8.1. Determination of Flux of Charged Particles . by 0. CHAMBERLAIN

. . . . . . . . .

. . . . . .

366 397 411 472 482

. . . 485 . . . 485

CONTENTSJ VOLUME

5,

ix

PART B

2.8.2. Determination of Differential X-ray Photon Flux and Total Beam Energy . . . . . . . . , . . 508 by J. S. PRUITT and H. W. KOCH

3. Sources of Nuclear Particles and Radiations 3.1. Radioactive Sources . . . . . . . . . . . . .

.

.

. .

by G. D. O J K ~ ~ ~ ~ ~ 3.2. Artificial Sources . . . . . . . . . . . . . . . . . . 3.2.1. Low-Energy Sources . . . . . . . . . . . . . . 3.2.1.1. Cascade Rectifiers . . , . . . . . . . . 3.2.1.2. The Electrostatic (Van de Graaff) Generator by M. H. BLEWETT 3.2.1.3. Nuclear Reactors. . . . . . . . . . . . by H. KOUTS 3.2.2. Medium- and High-Energy Sources . . . . . . . by M. H. BLEWETT

4. Beam Transport Systems 4.1. Introduction , . . . . . . 4.2.

. . . . . . . . . . . . . Beam Bending and Focusing Systems . . , . . . . . .

by R. M. STERNHEIMER 4.2.1. The Trajectories of Particles in a Strong Focusing (Quadrupole) Magnet . . . . . . . . . . . . 4.2.2. The Lens Equations for a Single Quadrupole Magnet and for a Two-Magnet System. Conditions of Double Focusing by a Two-Magnet System . . 4.2.3. The Matrix Method of Calculation for Strong Focusing Magnet Systems . . , . . . . . . . 4.2.4. The Focusing Equations for Deflecting WedgeShaped Magnets with Finite Field Index n . . . 4.3. Beam Separators . . . . . . . . . . . . . by BRUCECORK 4.3.1. Degrader Type Separation . . . . . . 4.3.2. Electromagnetic Separators . . . . . . 4.3.3. Radiofrequency Separators . . . . . . 4.3.4. Separation by Nuclear Interactions . .

555 580 580 581 584 590 623

691 692

694

696 719 731

. . .

747

. . . . . . . . . . . . . . . . . . . . . . . .

747 748 750 751

. . . .

4.4. Some Examples of Beam Transport Systems 752 by BRUCECORK 4.4.1. High Momentum Beams Using Counters as Detectors . . . . . . . . . . . . . . . . , . 752

.

CONTENTS. VOLUME

X

5.

PART B

4.4.2. Separated Beams for Bubble Chambers . 4.4.3. Special Beams . . . . . . . . . . . .

. . . . . 755 . . . . . 759

.

5 Statistical Fluctuations in Nuclear Processes by ROBLEY D . EVANS 5.1. Frequency Distributions . . . . . . 5.1.1. The Binomial Distribution . . 5.1.2. The Multinomial Distribution . 5.1.3. The Normal Distribution . . . 5.1.4. The Poisson Distribution . . . 5.1.5. The Interval Distribution . . . 5.2. Statistical Characterization of Data 5.2.1. Mean Value . . . . . . . . 5.2.2. Variance . . . . . . . . . 5.2.3. Sample Variance . . . . . . 5.2.4. Standard Deviation . . . . 5.2.5. Standard Error . . . . . . 5.2.6. Probable Error . . . . . . 5.2.7. Dimensions . . . . . . . . 5.2.8. Precision vs . Accuracy . . .

. . . . . . . . . 761 . . . . . . . . . 761 . . . . . . . . . 762 . . . . . . . . . 763 . . . . . . . . . 764 . . . . . . . . . 767

. . . . . . . . . . 771 . . . . . . . . . . 772 . . . . . . . . . . 773 . . . . . . . . . . 774 . . . . . . . . . . 774 . . . . . . . . . . 776 . . . . . . . . . . 778 . . . . . . . . . . 778 . . . . . . . . . . 779

5.3. Composite Distributions . . . . . . . . . . . . . . . 780 5.3.1. Combined Probabilities . . . . . . . . . . . . . 780 5.3.2, Superposition of Several Independent Random Processes . . . . . . . . . . . . . . . . . . 781 5.3.3. Propagation of Errors . . . . . . . . . . . . . 783 5.3.4. Difference of Two Mean Values . . . . . . . . . 784 5.3.5. Significance Levels and Confidence Intervals . . . 785 5.4. Tests for Goodness of Fit . . . . . . . . . . . . . . . 786 5.4.1. Pearson’s Chi-square Test . . . . . . . . . . . 786 5.4.2. Poisson Index of Dispersion . . . . . . . . . . . 789 5.4.3. Confidence Interval for the Standard Deviation of a Normal Distribution . . . . . . . . . . . . 791 5.5. Applications of Poisson Statistics to Some Instruments Used in Nuclear Physics . . . . . . . . . . . . . . 792 5.5.1. Effects of Resolving Time in Single-Channel Counting . . . . . . . . . . . . . . . . . . . . . 792 5.5.2. Effects of Resolving Time in Coincidence and Anticoincidence Circuits . . . . . . . . . . . . . 794 5.5.3. Scaling Circuits . . . . . . . . . . . . . . . . 795 5.5.4. Counting-Rate Meters . . . . . . . . . . . . . 799

CONTENTS. VOLUME

5.

xi

PART B

5.6. Useful Inefficient Statistics . . . . . . . . . 5.6.1. Estimate of the Mean Value . . . . 5.6.2. Estimate of Standard Deviation . . . . 5.6.3. Estimate of Standard Error . . . . . . 5.6.4. Estimate of x2. . . . . . . . . . . . 5.6.5. Examples . . . . . . . . . . . . . .

. . . . . 803 . . . . . 803 . . . . . 803 . . . . . 803 . . . . . 804 . . . . . 806

Appendix 1 by SIDNEYREED 1 . Evaluation of Measurement . . . . . . . . . . . . . . . 807 1.1. General Rules . . . . . . . . . . . . . . . . . . . 807 2 . Errors . . . . . . . . . . . . . . . . . . . . . . . . . 808 2.1. Systematic Errors, Accuracy . . . . . . . . . . . . . 808 2.2. Accidental Errors, Precision . . . . . . . . . . . . . 808 3. Statistical Methods . . . . . . . . . . . . . . . . . . . 809 3.1. Mean Value and Variance . . . . . . . . . . . . . . 809 3.2. Statistical Control of Measurements . . . . . . . . . . 810 4 . Direct Measurements . . . . . . 4.1. Errors of Direct Measurements 4.2. Rejection of Data . . . . . . 4.3. Significance of Results . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

811 812 813 814

5 . Indirect Measurement . . . . . . . . . . . . . . . . . . 814 5.1. Propagation of Errors . . . . . . . . . . . . . . . . 814 6. Prcliminary Estimation . . . . . . . . . . . . . . . . . 819

7. Errors of Computation . . . . . . . . . . . . . . . . . . 819

.

Appendix 2 Kinematics by R . M . STERNHEIMER 1 . Equations for the Lorentz Transformation from the Laboratory System to Center-of-Mass System; The Jacobian for Particle Reactions . . . . . . . . . . . . . . . . . . . . . . 821 2 . Maximum Angle Criterion for the Identification of Outcoming

Particles in Fundamental Collisions . . . . . . . . . . . 828 3 . Kinematics of Two-Body Decay . . . . . . . . . . . . . 831 4 . Threshold Energies for Associated Production of Unstable

Particles . . . . . . . . . . . . . . . . . . . . . . .

833

5 . Phase Space Factors for Two- and Three-Particle Final States 838

xii

CONTENTS. VOLUME

5.

PART B

.

Appendix 3 Properties of Elementary Particles and Particle Resonance States Table 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 845

Table 2 . . . . . . . . . . . . . . . . . . . . . . . . . .

846

Table3 . . . . . . . . . . . . . . . . . . . . . . . . . .

847

AUTHORINDEX. . . . . . . . . . . . . . . . . . . . . . .

849

SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . .

874

CONTRIBUTORS TO VOLUME 5, PART A D. E. ALBURGER, Brookhaven National Laboratory, Upton, New York M. BLAU,Institut f u r Radiumforschung, Vienna, Austria J. E. BROLLEY, JR., Los Alamos Scientijc Laboratory, Los Alamos, New

Mexico

B. CORK, Lawrence Radiation Laboratory, University of California, Berkeley, California J. W. M. DUMOND,Department of Physics, California Institute of Technology, Pasadena, California

R. D. EVANS,Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts H. FRAUENFELDER, Department of Physics, University of Illinois, Urbana, Illinois

W. B. FRETTER, Department of Physics, University of California, Berkeley, California S. S. FRIEDLAND, Solid State Radiations, Inc., Culver City, California

T. R. GERHOLM, Institute of Physics, University of Uppsala, Uppsala, Sweden

W. W. HAVENS,P u p i n Physics Laboratory, Columbia University, New York, New York R. HOFSTADTER, Physics Department, Stanford University, Stanford, California D. J. HUGHES,* Brookhaven National Laboratory, Upton, New York S. J. LINDENBAUM, Brookhaven National Laboratory, Upton, New York G. C. MORRISON, Atomic Energy Establishment, Harwelt, Berkshire, England G. D. 0’KELLEY,Oak Ridge National Laboratory, Oak Ridge, Tennessee F. REINES,Department of Physics, Case Institute of Technology, Cleveland, Ohio G. T. REYNOLDS, Princeton University, Princeton, New Jersey A. SILVERMAN, Department of Physics, Cornell University, Ithaca, New York

* Deceased. xiii

xi v

CONTRIBUTORS TO VOLUME

5,

PART A

R. M. STERNHEIMER, Brookhaven National Laboratory, Upton, New York R. W. WILLIAMS,Department of Physics, University of Washington, Seattle, Washington L. C. L. YUAN,Brookhaven National Laboratory, Upton, New York F. P. ZIEMBA, Solid State Radiations, Inc., Culver City, California

CONTENTS. VOLUME 5. PART A

.

1 Fundamental Principles and Methods of Particle Detection 1.1. Interaction of Radiation with Matter . . . . . . . by R . M . STERNHEIMER 1.1.1. Introduction . . . . . . . . . . . . . . . 1.1.2. The Ionization Loss d R / d x of Charged Particles 1.1.3. Range-Energy Relations . . . . . . . . . . 1.1.4. Scattering of Heavy Particles by Atoms . . . 1.1.5. Passage of Electrons through Matter . . . . . 1.1.6. Multiple Scattering of Charged Particles . . . 1.1.7. Penetration of Gamma Rays . . . . . . . .

. .

1

. . .

1 4 . . 44 . . 55 . . 56 . . 73 . . 76 .

I .2. Ionization Chambers . . . . . . . . . . . . . . . . . 89 by ROBERT W . WILLIAMS 1.2.1. General Considerations . . . . . . . . . . . . . 89 1.2.2. Pulse Formation . . . . . . . . . . . . . . . . 95 1.2.3. Quantitative Operation and Some Practical Considerations . . . . . . . . . . . . . . . . . 100 1.2.4. Amount of Ionization Liberated . . . . . . . . . 103 1.2.5. Noise: Practical Limit of Energy Loss Measurable . 105 1.2.6. Some Types of Pulse Ionization Chambers . . . . 107 1.2.7. Current Ionization Chambers and Integrating Chambers . . . . . . . . . . . . . . . . . 109 1.3. Gas-Filled Counters . . . . . . . . . . . . . . . . . 110 by ROBERTW. WILLIAMS 1.3.1. Gas Multiplication; Proportional Counters . . . . 110 1.3.2. Geiger Counters and Other Breakdown Counters . 118 I .4. Scintillation Counters and Luminescent Chambers . . . . 120 by GEORGE T. REYNOLDS and F. REINES 1.4.1. Scintillation Counters . . . . . . . . . . . . . 120 1.4.2. Solid Luminescent Chambers . . . . . . . . . . 159

1.5. Cerenkov Counters . . . . . . . . . . . . by S. J . LINDENBAUM and LUKEC . L. YUAN 1.5.1. Introduction . . . . . . . . . . . . 1.5.2. Focusing Cerenkov Counters . . . . . 1.5.3. Nonfocusing Counters . . . . . . . . XV

. . . . . 162

. . . . . 162 . . . . . 168 . . . . . 186

xvi

CONTENTS. VOLUME

5.

PART A

1.5.4. Total Shower Absorption eerenkov Counters for Photons and Electrons . . . . . . . . . . . . 189 1.5.5. Other Applications . . . . . . . . . . . . . . . 191 1.6. Cloud Chambers and Bubble Chambers . . . . . . . . 194 by W . B. FRETTER 1.6.1. Cloud Chambers . . . . . . . . . . . . . . . . 194 1.6.2. Bubble Chambers . . . . . . . . . . . . . . . 203 1.7. Photographic Emulsions . . . . . . . . . . . . . . . 208 by M . BLAU 1.7.1, Introduction . . . . . . . . . . . . . . . . . 208 1.7.2. Sensitivity of Nuclear Emulsions . . . . . . . . 210 1.7.3. Processing of Nuclear Emulsions . . . . . . . . . 216 1.7.4. Optical Equipment and Microscopes . . . . . . . 224 1.7.5. Range of Particles in Nuclear Emulsions . . . . . 226 1.7.6. Ionization Measurements in Emulsions . . . . . . 240 1.7.7. Ionization Parameters . . . . . . . . . . . . . 245 1.7.8. Photoelectric Method . . . . . . . . . . . . . 264 1.8. Special Detectors . . . . . . . . . . . . . 1.8.1. The Semiconductor Detector . . . . . and F. P. ZIEMBA by S. S. FRIEDLAND 1.8.2. Spark Chambers . . . . . . . . . . . by BRUCECORK

. . . . . 265 . . . . . 265

. . . . . 281

.

2 Methods for the Determination of Fundamental Physical Quantities 2.1. Determination of Charge and Size . . . . . . . . . . . 2.1.1. Charge of Atomic Nuclei and Particles . . . . . . 2.1.1.1 Rutherford Scattering . . . . . . . . . . . . . 2.1.1.2. Characteristic X-ray Spectra . . . . . . . . . by ROBLEY D . EVANS 2.1.1.3. Charge Determination of Particles in Photographic Emulsions . . . . . . . . . . . . . by M . BLAU 2.1.2. Principal Methods of Measuring Nuclear Size . . . by ROBERTHOFSTADTER

289 289 289 293

298 307

2.2. Determination of Momentum and Energy . . . . . . . 341 2.2.1. Charged Particles . . . . . . . . . . . . . . . 341 2.2.1.1. Measurement of Momentum . Electric and Magnetic Analysis . . . . . . . . . . . . . . . 341 by T. R . GERHOLM

CONTENTS, VOLUME

5,

PART A

xvii

2.2.1.1.4. Measurement of Momentum with Cloud Chambers or Bubble Chambers . . . . . . . . . 375 by W. B. FRETTER 2.2.1.1.5. Momentum Measurement in Nuclear Emulsions 388 by M. BLAU 2.2.1.2. Determination of Energy . . . . . . . . . . . 409 2.2.1.2.1. Energy Measurement with Ionization Chambers 409 by R. W. WXLLIAMS 2.2.1.2.2, Scintillation Spectrometry of Charged Particles 411 by G. D. O'KELLEY 2.2.1.2.3. Measurement of Range and Energy with Cloud Chambersand Bubble Chambers. . . . . . 436 by W. B. FRETTER 2.2.1.3. Determination of Velocity. . . . . . . . . . . 438 2.2.1.3.1. Time-of-Flight Method . . . . . . . . . . . 438 by LUKEG. L. YUANand S. J. LINDENBAUM 2.2.1.3.2. Measurement of Velocity . . . . . . . . . . 444 by W. B. FRETTER 2.2.1.3.3. Measurement of Velocity Using cerenkov Counters . . . . . . . . . . . . . . . . 454 by LUKEC. L. YUANand S. J. LINDENBAUM 2.2.2. Neutrons . . . . . . . . . . . . . . . . . . . 461 2.2.2.1. Recoil Techniques for the Measurement of Neutron Flux, Energy, Linear and Spin Angular Momentum . . . . . . . . . . . . . . . . 461 E. BROLLEY, JR. by JOHN 2.2.2.2. Time-of-Flight Method . . . . . . . . . . . . 495 by W. W. HAVENS,JR. 2.2.2.3. Crystal Diffraction. . . . . . . . . . . . . . 566 by D. J. HUGHES 2.2.2.4. Determination of Momentum and Energy of Neutrons with Her Neutron Spectrometer. . . . . 570 by G. C. MORRISON 2.2.3. Gamma-Rays . . . . . . . . . . . . . . . . . 582 2.2.3.1. Internal and External Conversion Lines. . . . . 582 by T. R. GERHOLM 2.2.3.2. Determination of Momentum and Energy of Gamma Rays with the Curved Crystal Spectrometer . . . . . . . . . . . . . . . 599 by J. W. M. DUMOND

xviii

CONTENTS, VOLUME

5,

PART A

2.2.3.3. Gamma-Ray Scintillation Spectrometry. . . . . 616 by G. D. O’KELLEY 2.2.3.4. Determination of the Momentum and Energy of Gamma Rays with Pair Spectrometers . . . . 641 by D. E. ALBURGER 2.2.3.5. Shower Detectors . . . . . . . . . . . . . . 652 by R. HOFSTADTER 2.2.3.6. Gamma-Ray Telescopes. . . . . . . . . . . . 668 by A. SILVERMAN 2.2.3.7. Measurement of ?-Ray Energy by Absorption. . 671 by ROBLEY D. EVANS 2.2.3.8. Detection and Measurement of Gamma Rays in Photographic Emulsions . . . . . . . . . . 676 by M. BLAU 2.2.4. Neutrino . . . . . . . . . . . . . . . . . . . 682 2.2.4.1. Neutrino Reactions. . . . . . . . . . . . . . 682 by F. REINES

AUTHORINDEX . . . . . . . . . . . . . . . . . . . . . . .

699

SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . .

718

2. METHODS FOR THE DETERMINATION OF FUNDAMENTAL PHYSICAL QUANTITIES (Continued) 2.3. Determination of Mass of Nuclei and of Individual Particles 2.3.1. Mass Spectroscopy1 *

t

2.3.1 .l. Introduction. The first mass spectroscopes were invented by Thomsoq2 A ~ t o nand , ~ Dempster4 in the second decade of this century for the purpose of investigating isotopy among stable atoms. In these instruments, and in most of the mass spectroscopes of the present day, the specific charge ( e / M ) of a positive ion is deduced by observing the deflection experienced by the ion as it passes through electric and/or magnetic fields. The mass of the ion is readily calculated from this as the charge carried by the ion is generally known. Within the precision of the first experiments the masses of all atoms appeared to be integral multiples of that of HI. Small divergences from this “whole number rule” were discovered, however, in 1923 and these have since been the subject of intensive investigation by mass spectroscopists because of the information they yield concerning nuclear stability. The reader will recall in this connection that the mass of an atom is less than the combined masses of its constituent nucleons and electrons, and that it is this difference between deduced and observed masses, known as the binding energy, which accounts for the stability of the atom. Prior to September, 1960, atomic masses were expressed in atomic mass units (OIE = 16 amu), but since September, 1960 they have been expressed in relative nuclidic mass units, abbreviated u, so defined that C12 = 12u. On these scales the masses of all atoms are nearly integral, the nearest integer being in each case the mass number (number of

t See also Vol. 4, A, Section 4.1.1. K. T. Bainbridge, Charged particle dynamics and optics, relative isotopic abundances of the elements, atomic masses. I n “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 1. Wiley, New York, 1953;H. Ewald and H. Hintenberger, ‘‘ Methoden und Anwendungen der Massenspectroskopie.” Verlag Chemie, Weinheim, Germany, 1953; M. G. lnghram and R. J. Hayden, “ A Handbook on Mass Spectroscopy,” Nuclear Energy Series, Report No. 14. National Academy of Sciences, Washington, D.C., 1954;and H. E. Duckworth, “Mass Spectroscopy.” Cambridge Univ. Press, London and New York, 1958. * J. J. Thomson, “Rays of Positive Electricity.” Longmans, Green, London, 1913. F. W. Aston, Phil. Mag. [6]38, 709 (1919). ‘A. J. Dempster, P h y . Rev. 11, 316 (1918). 1

* Section 2.3.1 is by H. E.

Duckworth. 1

2

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

nucleons) of the atom in question, Aston and many since him have expressed the exact mass of an atom as zM*= A ( l f), where f is the “packing fraction” and A and Z are the mass number and atomic number (number of protons), respectively. Thus

+

packing fraction, f = ( M

- A)/A.

An algebraically small packing fraction indicates an atom whose mass is numerically small with respect to its mass number and whose stability is consequently high. It is more common to express this stability in terms of the “binding energy per nucleon,” defined as follows: b.e./nucleon =

ZH’

+ ( A - 2)n - zMA A

where H1and n are the masses of the hydrogen atom and neutron, respectively. The term b.e./nucleon implies that the binding energy is associated with the nucleus, as indeed moat of it is. The binding energies of the orbital electrons, here practically neglected, are not only small but increase with 2 in a gradual manner so that the b.e./nucleon gives a significant picture of the variations and trends in nuclear stability. Modern precision mass spectroscopy is concerned with the study of these variations and trends. 2.3.1.2. Mass Spectroscopes for the Determination of Atomic Mass. The mass spectroscopes employed in the determination of atomic masses are necessarily high resolution instruments. With a single exception those under construction or now in use are instruments in which the ions are deflected by a linear combination of magnetic and electric fields as, for example, in the instrument shown in Fig. 1. The essential parts of such an apparatus are (a) a source of positive ions, (b) a region in which the ions are accelerated, (c) a system of slits for collimating the ion beam, (d) an electromagnetic analyzer which separates the beam into its several specific charge groups, and (e) a device for detecting the ions after analysis. Depending upon whether photographic or electrical detection is employed, “lines” or “peaks” are recorded by the detector, each representing a particular group of ions present in the initial ion beam. The electrical detector of Fig. 1 could be replaced, if desired, by a photographic plate located at Saand lying in the direction SaS4. These instruments possess the property of focusing at the final detector ions which emerge from the collimating system with an angular spread in the plane of the paper and/or with a velocity spread. They are, therefore, said to be “double focusing,’’ a term which has unfortunately been

2.3.

MASS O F NUCLEI AND INDIVIDUAL PARTICLES

3

abrogated by beta-ray spectroscopists to describe instruments which achieve angular focusing in two directions. The original double focusing instruments of Dempster, Bainbridge and Jordan,B and Mattauch,7 and the majority of their lineal descendants, are based on first-order focusing theory. In these the final image

90" ELECTROSTATIC ELECTROS 90"

ELECTRON MULTIPLIER

FIG. 1. Schematic diagram of the Nier double focusing mass spectrometer (from reference 10).

breadth contains aberration terms involving second-order and higher powers of the angular and velocity spread in the transmitted ion beam. Notable exceptions are the mass spectroscope of Nier and his collaborators,*-1° shown in Fig. 1, which possesses second-order direction A. J. Dempster, Proc. Am. Phil. SOC.76, 755 (1935). K. T. Bainbridge and E. B. Jordan, Phys. Rev. 60,282 (1936). 7 J . Mattauch, Phys. Rev. 60, 617 (1936). * A. 0. Nier and T. R. Roberts, Phys. Rev. 81, 507 (1951). E. G. Johnson and A . 0. Nier, Phys. Rev. 91, 10 (1953). l o K. S. Quisenberry, T. T. Scolman, and A . 0. Nier, Phys. Rev. 102, 1071 (1956). 6

6

4

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

focusing plus first-order double focusing, and the new instrument a t the Max Planck Institute for Chemistry11v12which will possess first-order double focusing along the entire photographic plate and second-order direction focusing in the middle of the plate. The resolution of a mass spectroscope is written as A M / M , which implies that two ions of mass M and M AM, respectively, can be just resolved. In instruments of the type we have been discussing, the resolution improves linearly with the radius of curvature of the electrostatic analyzer and also as the widths of the defining slits (Sz, for photographic detection; Sz and 54 for electrical detection) are reduced. With mass spectroscopes employing photographic detection the grain size of the cm) sets the lower limit to the actual line photographic plate width with the result that the resolution cannot be reduced indefinitely by reducing the width of the object slit Sz. Because of this grain size limitation in the case of photographic detection instruments and also because of the difficulty, whatever the method of detection] of adjusting and aligning the diminutive slits required to secure high resolution in any small scale apparatus, the instruments currently (1960) under coiistruction for the purpose of precision mass determinations exceed their predecessors in size by about an order of magnitude. That is, high resolution is being sought by increasing the radius of curvature a, of the electrostatic analyzer (and, of course, the size of the magnetic analyzer, which is related to it). Included in this category are new instruments at Osaka University13*14(a,= 1.09 m), Harvard Universityl6 (ae = 2.14 m), the Max Planck Institute for Chemistry11v12(ae = 5.43m), and McMaster University16 (a,= 2.75 m). Whereas the first double focusing instruments6-’ possessed resolutions ranging from 1/2500 - 1/10,000, the values for these new instruments should be considerably better than 1/100,000.

+

11 H. H. Hintenberger, H. Wende, and L. A. Konig, 2.Naturfursch. lOa, 605 (1 055) ; laa, 88 (1957). H. Hintenberger, J. Mattauch, W. Miiller-Warmuth, H. Voshage, and H. Wende, in “Proceedings of the International Conference on Nuclidic Masses” (H. E. Duckworth, ed.). Univ. of Toronto Press, Toronto, 1960. 18 K. Ogata and H. Matsuda, 2.Naturforsch. 10a, 843 (1955). 14 K. Ogata and H. Matsuda, in “Nuclear Masses and Their Determination” (H. Hintenberger, ed.). Pergamon, London, 1957. 16X. T. Bainbridge, R. C. Barber, H. E. Duckworth, and P. E. Moreland, Jr., in “Proceedings of the International Conference on Nuclidic Masses” (H. E. Duckworth, ed.). Univ. of Toronto Press, Toronto, 1960. 16 N. R. Isenor, R. C. Barber, and H. E. Duckworth, in “Proceedings of the International Conference on Nuclidic Masses” (H. E. Duckworth, ed.). Univ. of Toronto Press, Toronto, 1960.

2.3.

MASS OF NUCLEI AND INDIVIDUAL

PARTICLES

5

Smith, 17-19 a t the Brookhaven National Laboratory, has employed a completely different type of mass spectroscope known as the “mass synchrometer.” Here the ions describe circular paths in a homogeneous magnetic field, the frequency of the motion being f = Be/27rM. This cyclotron frequency is independent of the velocity of the ion but linearly dependent upon its specific charge. Measurements are therefore made of the cyclotron frequencies of the ions under study. The resolution is adjustable electrically between 1/10,000 and 1/25,000. 2.3.1.3. T h e D o u b l e t M e t h o d of Mass Comparison. In theory, in a typical mass spectroscope, one could measure the potential through which an ion is accelerated, ascertain the deflection produced by an analyzer of known characteristics, and from these data calculate the ion mass. This procedure, however, involves a knowledge of certain absolute values which it is not yet possible to determine to the desired precision. Instead, in practice, the mass of an unknown atom is found by comparing it to one whose mass is known. As a rule the known and unknown ions have specific charges which are nearly equal, with the result that the two peaks or lines appear on the mass spectrum at the same mass number. For example, singly charged CL2 ( =12u) and doubly charged Mg24 (=23.9850446 f 19u) both appear a t mass number 12, where they are slightly displaced with respect to one another by virtue of the one part in 1600 mass difference between them. These two constitute a “doublet” and it is the concern of the mass spectroscopist to ascertain, in terms of mass, this doublet spacing. In the main, three factors determine the precision with which mass differences can be determined. These are (a) the accuracy with which the dispersion of the mass spectroscope is known, and (b) the accuracy with which the lines or peaks belonging to the atoms under study can be located, and (c) the extent to which the lack of perfect velocity and angular focusing in the mass spectroscope introduces aberrations in the image. OF DISPERSION LAW,The theoretical dispersion 2.3.1.3.1. UNIFORMITY law governing a mass spectroscope is never realized experimentally because of lack of homogeneity in the analyzer fields. Thus, although it is possible in principle to compare the masses of atoms even if their mass spectral lines are wide apart on a photographic plate, it is difficult in practice to secure a uniform dispersion law over the large region involved. It is much less difficult to do so over the region occupied by a doublet. This consideration has led to the almost exclusive use of the doublet method in mass spectroscopes that employ photographic detection. G. Smith, Phys. Rev. 81, 295 (1951). G. Smith and C . C . Damm, Phys. Rev. 90, 324 (1953). 19L.G. Smith and C. C . Demm, Rev. Sci. Instr. 27,638 (1956). 17L.

18L.

2.

6

DETERMINATION O F FUNDAMENTAL QUANTITIES

In the case of electrical detection the fact that the ions, a t the time of detection, have each traveled identical paths makes the achievement of a uniform dispersion law much easier than in the case of photographic detection. The doublet members are brought in turn to the collector by varying some circuit parameter as, for example, in the case of the mass spectroscope of Fig. 1, a resistance that controls the voltage across the electrostatic analyzer. The mass change is expected to be proportional to the resistance change and is so in practice to a very close approximation. This has been verified by direct determination of the H* mass as, for example, from the CaHs-CaH, mass difference. This, it will be realized, is an extraordinarily stringent test, as these two ions do not constitute a doublet but are spaced one mass unit apart. Twenty-four such determinationsZ0of the H1 mass yielded a mean of H’ = 1.0078239 13u, as compared with the best value from the same laboratory, based on doublet comparisons, of H’ = 1.0078247 jz 2u. The extensive linearity of this dispersion law makes feasible, therefore, the determination of mass-unit differences with a precision which is not much inferior to that associated with doublet comparisons. 2.3.1.3.2. LOCATION OF MASSSPECTRAL LINESOR PEAKS. The precision with which a given mass difference can be determined is dependent upon the accuracy with which the mass spectral lines or peaks may be located, Let us designate the mass width of a line or peak by AM, a quantity which is directly proportional to the resolution of the mass spectroscope. I n the case of photographic recording, the position of a mass spectral line may be determined to some fraction of its width, say, for an observer who is neither unduly optimistic nor unduly conservative, Thus, if a resolution of 1/20,000 be available, a mass difference can be determined with a precision of one part in a million. With electrical recording it has been demonstrated that it is possible to locate a peak with a precision of & of its width, and perhaps even more closely. At least a tenfold improvement over the photographic case is thus obtained. This is done by a “peak-matching” technique introduced by Smith and Damm17 and also employed, in a modified form, by Nierlo and several others. In this technique, by taking advantage of rapidly responding detector systems, the two doublet peaks are made to appear on an oscilloscope screen on alternate sweeps. These two peaks are then brought into coincidence by adjustment of some variable (in Smith’s case, a frequency; in Nier’s case, a resistance) whose value gives the doublet mass difference. The peaks are thus “matched” by the human eye, an organ which can discern lack of coincidence with exceptionally keen discrimination. Two peaks which have almost been brought into coincidence by this method are shown in Fig. 2.

*

10

K. S. Quisenberry, C. F. Giese, and J. L. Benson, Phys. Rev. 107, 1664 (1957).

2.3.

MASS

OF NUCLEI A N D INDIVIDUAL

PARTICLES

7

2.3.1.3.3. IMAGE ABERRATIONS RESULTING FROM IMPERFECT FOCUSING. It has been mentioned that at least two of the modern precision mass

spectroscopes7-~*provide second-order angular focusing. Actually, even for first-order angular focusing instruments, the lack of perfect angular focusing is often not serious for the reason that the ions are strongly collimated during their acceleration (25-50 kev). The lack of perfect velocity focusing, on the other hand, had led in the past t o serious systematic errors in cases where the velocity spread associated with one of the doublet members differs from that associated with the other. This will be true if one of the ions and not the other arises from the dissociation of a molecule. The one may then possess significant initial kinetic energy as, for example, the S32-ion (from HZS or SO,) in the

FIQ. 2. Peak matching of two doublet members. The degree of mismatch shown (as indicated by the lateral displacement of one peak with respect to the other) corresponds to a mass difference of approximately one part in 500,000. Resolution is 1/40,000. (Courtesy of Professor A. 0. C. Nier and Dr. K. S. Quisenberry.) 02-S32 doublet. As i t possesses no velocity focusing feature, the mass synchrometer of Smith1?-l9 is particularly vulnerable in this respect and Smith has, therefore, taken pains to study doublets in which both members are nondissociation or “parent” ions. I n this way, for example, masses for OX6and D2 can be obtainedz1from the two doublets C8Ds-C4D2 and CDd-D20. I n the usual deflection-type mass spectroscope the slit S3 in Fig. 1, located between the electrostatic and magnetic analyzers, serves the purpose of limiting the velocity band which enters the magnetic analyzer. This slit should be narrow enough to ensure that it is completely illuminated by both doublet members. 2.3.1.4. Accuracy of Mass Spectroscopic Mass Values. Although “best” values have a habit of changing, and any listed here will in all probability be out-of-date a t the time of publication of this text, it is 21

L. G. Smith, Bull. Am. Phys. SOC.[2] 2, 223 (1957).

8

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

nevertheless instructive to list a few, if for no other reason than to indicate the degree of precision of modern mass spectroscopy. The values for three key light atoms, shown in the tabulation, are the best available in the autumn of 1960. Value

Reference 20

H’ DZ

1.0078247 k 2u

0’8

15.9949142 -t 5u

Reference 21 2.01409964 f 3u 15.99491550 i 15u

Thus, the error associated with H1 is 200 ev, with D2, 30 ev and with 0 1 6 , 130-500 ev. In the case of D2 the accuracy approaches the point where one must take into account differences in the ionization potential of the two ions forming a doublet. The errors quoted above are statistical ones, based on the internal consistency of the experimental data, and therefore represent the actual errors only if systematic errors be absent. Systematic errors will manifest themselves as discrepancies between the values obtained in laboratories that employ different experimental methods; otherwise there is no basis for suspecting their existence. As mentioned above, the mass of 0l6has been determined to a high degree of precision both a t the University of Minnesota20 and a t Brookhaven National Laboratory, 2 1 using radically different types of mass spectroscopes. The discrepancy between the two values is -1.3 5- 0.6 micro-u, that is, somewhat greater than twice the probable error, suggesting that systematic errors are of more importance than statistical ones. This is a fairly typical example; if anything, the discrepancy is rather larger here than usual. On the whole it appears that the actual probable errors associated with mass spectroscopic mass values are 1.5-2 times the stated statistical errors, with the limits of error being larger again by an additional factor of two. Finally, a word will be said concerning the accuracy as a function of mass. At the time of writing, the actual probable errors in the various mass regions that have been studied with modern techniques are as follows: A = 32, -1.5 kev;20 A = 48, -2 kev;22 A = 64, -4 k e ~ ; ~ ~ A = 132, -5 kev;Z4 A = 150, -0.14 and A = 200, -25 kev.26 22

C. F. Giese and J. L. Benson, Bull. Am. Phys. SOC.[2] 2, 223 (1957).

K. S. Quisenberry, T. T. Scolman, and A. 0. Nier, Phys. Rev. 104, 461 (1956). R. R. Ries, R. A. Damerow, and W. H. Johnson, Jr., in “Proceedings of the International Conference on Nuclidic Masses” (H. E. Duckworth, ed.). Univ. of Toronto Press, Toronto, 1960. 26 V. B. Bhanot, W. H. Johnson, Jr., and A. 0. Nier, Phys. Rev. 120, 235 (1960). *a

24

2.3. MASS

OF NUCLEI AND INDIVIDUAL PARTICLES

9

Further details concerning these and other mass regions can be found in published tab~lations2~,~~,27 of mass spectroscopic atomic mass data.

2.3.2. Determination of Atomic Masses from Nuclear Reaction and Nuclear Disintegration Energies*

The energy balance in a nuclear reaction is expressed in terms of the

Q value of the reaction, the difference between the mass of the target and

+ + +

projectile atoms before the reaction Ma M1, and the mass of the atoms that are produced in the reaction M z M,. The Q value is customarily expressed in energy units, Q = c2(Mo M 1- M z- Ma). For decay processes the Q value is the difference between the mass of the radioactive atom and the daughter atom, in the case of @-decay, or the daughter atom plus a helium atom in the case of a-decay. The electron mass does not appear explicitly as an emitted particle in 6-decay because the daughter atom has just one more, or less, electron than the radioactive atom, and the use of atomic masses rather than nuclear masses automatically includes the emitted electron. However, in order that the Q value for positron decay include all of the energy released in the transition, one must add 2m0c2,the energy which eventually appears as annihilation radiation, to the beta endpoint energy. The atomic masses are customarily expressed in atomic mass units. Prior to 1960, the mass unit universally used in precise studies of atomic masses was OL6/16.In September 1960, the International Union of Pure and Applied Physics adopted C12 as the mass standard and C12/12 as the atomic mass unit in order to take advantage of the fact that mass spectroscopic doublets containing carbon provide the most convenient measurement of the mass of heavy atoms. The new unit CI2/12 is larger than 016/16 by a factor 1.000 317 917 k 17,' and all atomic masses expressed in the new units are thereby decreased, but the mass to energy conversion factor is increased to (C12/12) c2 = 931.441 Mev so that any mass difference expressed in energy units is independent of the mass unit chosen. * B H. E. Duckworth, B. G . Hogg, and E. M. Pennington, Revs. Modern Phys. 26, 463 (1954). 27 H. E. Duckworth, Revs. Modern Phys. 29, 767 (1957). F. Everling, L. A. Konig, J. H. E. Mattauch, and A. H. Wapstra, Nuclear Phys. 16, 342 (1960).

* Section 2.3.2 is by Ward Whaling.

10

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

The following discussion will deal with the mass of atoms in their ground state, and the Q values will always refer to ground-state transitions. The experimental measurement of a Q value is based on the fact that the kinetic energy of the particles produced in the reaction exceeds the kinetic energy of the reacting atoms by just the amount of the Q value, and the experimental measurement consists of the determination of the kinetic energy of all of the atoms involved.* If the particles produced in the reaction carry away some of the energy released in the reaction as internal energy of excitation, either nuclear or atomic, the kinetic energy of the outgoing particles is thereby reduced, and the measured Q value does not represent the true mass difference. Nuclear excitation energy is usually so large that it is not easily overlooked, but it is much more difficult to detect, or otherwise make allowance for, atomic excitation of the residual atom. An estimate2 of the amount of energy that might be expected to appear as electronic excitation indicates that it is of the order of a kilovolt for reactions of atoms with atomic number in the neighborhood of ten. For reactions among nuclei with large atomic number, this atomic excitation energy may constitute a significant correction to the measured Q value3 and must be taken into account. From the mathematical point of view the experimental Q value is equivalent to a linear equation relating two or more masses. By proper choice of the &-value equations, one obtains a set of simultaneous equations which may be solved to find the unknown masses. It will be recognized that this procedure is identical with that used to determine masses from mass-spectroscopic doublet measurements. Indeed, a mass spectroscopic doublet is also a linear relation between unknown masses and hence is exactly equivalent to a Q-value measurement. I n selecting the set of simultaneous equations for solution, one may include mass spectroscopic doublets as well as reaction Q values. The mass tables published in the past few years (1957) have been based either on mass spectrographic doublets or Q values, and the two types of measurements have not been combined. The purpose behind this artificial separation of the data from the two sources is to discover possible inconsistencies between the different methods of measurement. As more accurate measurements remove the sources of disagreement, the information provided by both * A wide variety of techniques have been developed for measuring Q values, of which the most precise are based on electric and magnetic analysis (see Sections 2.2.1.1.1 and 2.7.1). ¶ A . B. Brown,C. W. Snyder, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 82, 159 (1951).

'R. Serber and H. Snyder, Phys. Rev. 87, 152 (1952).

2.3.

MASS OF NUCLEI AND INDIVIDUAL

PARTICLES

11

types of experiment will be combined to provide the most accurate information about atomic masses. The procedure used in determining masses from Q values differs from that used with mass doublets only in that a system of equations involving Q values is necessarily large, much larger than the minimum system of equations that can be used when treating mass doublets. 2.3.2.1. Formulation of Input Data. The large number of equations introduces difficulties, in addition to the obvious mathematical complexity, since it becomes necessary to combine values measured in different laboratories by different methods, values which may be based on inconsistent energy calibration standards. All of the input data must first be examined and, if necessary, corrected to correspond to best values of the energy calibration standards. It is convenient to consider the mass excess M , expressed in energy units, rather than the atomic mass M. If A is the mass number, then M = (M - A)$, and the Q value of a reaction is a simple combination of the mass excesses of the atoms taking part in the reaction. The set of input Q value equations would have the form

2

AikMk =

Qi

AQt

i

=

1, 2, 3,

. . . ,n

(2.3.2.1)

k=l

where A i k is a n integer, usually i 1 or 0, the Jfk are the unknown mass excesses which one wishes to find, Qi is an experimental Q value, and AQ, is the standard deviation in the experimental measurement. It will be assumed throughout the following discussion that AQiare standard errors.* This set of equations, called conditional equations in the mathematical literature, may be solved to find Mk if the following conditions are satisfied. (1) The determinant of the Aik must not be zero. If the set of equations (2.3.2.1)is large, evaluation of the determinant may be quite tedious, and the following graphical method determines immediately whether this condition is satisfied. Let us consider, for example, the set of Q value equations listed in Table I. This set of equations is represented graphically in Fig. 1, in which the nuclides are located as a function of A - 2 and 2. 4 link connecting two nuclides represents a mass difference equation relating the two nuclides. The mass difference equation given by a

* In practice, the errors quoted by various experimental physicists are sometimes standard errors, sometimes probable errors, and are frequently not identified at all. Statistical analyses of experimental data from many laboratories have indicated that the internal consistency of the measurements is such that, on the average, the quoted error is usually greater than the probable error and smaller than the standard error.

2.

12

DETERMINATION O F FUNDAMENTAL QUANTITIES

TABLEI. A Set of 40 Experimental Q Values Expressed as Linear Equations Which Relate the Masses of 20 Nuclides between n and N16 In these equations the symbol for a nuclide represents the mass excess of the atom, expressed in units of MeV.&

(1) n - HI = 0.783 f 0.013 Mev (2) n HI - HZ = 2.227 L- 0.002 = 6.251 f 0.008 (3) n t Ha - H3 (4) 2Ha - n - He3 = 3.267 f 0.007 = 0.0185 f 0.0002 (5) H 3 - He3 (6) He3 n - H1 - H3 = 0.7637 f 0.001 (7) Lie H' - He4 - He3 = 4.023 f 0.002 (8) Lie Ha - Li7 - H1 = 5.027 f 0.003 (9) 2 H a - H' - HS = 4.038 f 0.005 = 1.6449 f 0.0004 (10) Be7 n - H1 - Li7 HI - 2He4 = 17.346 0.010 (11) Li7 (12) Be8 - 2He4 = 0.0941 rt 0.007 (13) Be8 n - Be9 = 1.665 f 0.0014 = 6.816 f 0.006 (14) Be9 n - Bela (15) Be9 H' - HZ - Be8 = 0.559 f 0.001 (16) Be9 H' - He4 - Lie = 2.126 f 0.002 (17) Be9 HZ - H1 - Belo = 4.587 f 0.005 (18) Be9 H Z - H a - Be8 = 4.598 f 0.012 (19) Be9 Ha - He4 - Li7 = 7.153 f 0.003 = 0.557 f 0.004 (20) Belo - BIO (21) BO ' n - He4 - Li7 = 2.786 f 0.008 (22) BIO H' - He4 - Be7 = 1.148 f 0.002 (23) B'O Ha - H' - B" = 9.229 f 0.005 (24) BI1 H1 - He4 - Be8 = 8.585 rt 0.006 (25) B11 Ha - He4 - Be9 = 8.024 rt 0.004 = 4.949 f 0.006 (26) CI2 n - CI3 (27) N13 + n - H2 - C" = 0.281 f 0.003 (28) C'a H* - H' - CIa = 2.721 f 0.002 (29) NIS + n - HI - CI3 = 3.003 f 0.003 (30) C" + H a - H' - CI4 = 5.943 f 0.003 (31) C" Ha - H3 - CIa = 1.310 f 0.003 (32) C13 + H a - He4 - B1' = 5.164 f 0.004 (33) C'4 - N'4 = 0.156 0.001 0.005 (34) N14 + n - HI - CI4 = 0.6264 = 2.23 k 0.010 (35) N'3 - C'* (36) N14 n - Nx6 = 10.832 f 0.008 (37) N" + H a - H' - N" = 8.614 f 0.007 (38) N16 H1 - He4 - CI2 = 4.961 f 0.003 He - He4 - CIa = 7.681 f 0.009 (39) N16 = 3.115 f 0.0025 (40) 0 1 6 + H e - He4 - "4

+

+ + +

+

+

*

+ + + + + + + + + + + + + + +

+

*

+ + +

o This particular set of equations is taken from reference 8. The experimental Q values are t a b n from D. M. Van Patter and W. Whaling, Revs. Modem Phys. 26,

402 (1954)'.

2.3.

MASS OF NUCLEI AND INDIVIDUAL

13

PARTICLES

nuclear reaction Q value is represented by a link connecting the target and residual nuclei in the reaction; e.g., the line connecting Li6 and Li7 might represent the Li6(d,p)Li7reaction, or the equation Li6

+ H* - Li7 - H1 = &.

The legend in the corner of the figure indicates the links established by several common types of reactions.

a 7

6 5 z

4

3 2

I

0

0

I

2

3

4

5

6

7

8

A-Z FIG.1. Graphical representation of the set of &-value equations in Table I. A link connecting two nuclides represents a nuclear reaction or decay process in which one of the nuclides is the target, the other is the residual nucleus. The links established by common reactions are indicated in the lower corner of t h e figure.

In order that the determinant of the set of equations represented by the figure have a nonzero determinant, it is necessary that the links in the figure form a continuous path connecting C12, or some other atom of known mass, with every other nuclide that appears in the set of equations. From the figure it is apparent that the set of equations in Table I satisfies this requirement. However, if the C13(d,a)B" reaction were removed

14

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

from Table I, the remaining set could not be solved. Furthermore, the subset formed by reactions (28)-(40) in Table I can be solved for this group of masses connected to CI2 only if the p , n, d, I, and a masses, the bombarding and emitted particles in these reactions, were furnished as auxiliary information. (2) The mass standard C12,or some other atom of known mass, must appear in a t least one of the equations in the set. This requirement follows from the fact that the conditional equations are restricted to those in which the mass number, had it been included in the expression for the mass, would cancel out of the equation. For example, suppose the first equation in set (2.3.2.1) were n - H’ = QI (we may neglect the +AQi in this discussion). Since the difference n - H’ is the only combination of these masses which cancels the mass numbers, any other equation containing n and H1 must also contain at least one additional mass (e.g., n H1 - H2 = Q 2 ) . Each additional equation adds at least one additional unknown mass, and any system of s equations must contain therefore at least s 1 unknown masses. However, if to this system we add the (a 1)th equation, in which the only new mass appearing is C12,the set of equations now becomes soluble. If our second equation had been 6n 6 H - C12= Q2, the two equations would determine the n and H1 masses. Mattauch has shown that certain linear combinations of the masses, such as the nuclear binding energy, may be determined from a set of equations which does not contain the mass standard.4 2.3.2.2. Reaction Cycles. Since the set of equations (2.3.2.1) usually contains many more equations than unknowns, n > rn, there exist n - rn independent equations relating the Q values in the set. These equations, usually called reaction cycles, are of the form Z, a,&, = 0, where a. is a n integer, usually ? 1. The reaction cycles are useful in screening the input data for experimental Q values that contain gross errors. From the experimental values Q, k AQ, one computes the value of the cycle sum, and the uncertainty in the sum. If the value differs from zero by several times the standard deviation of the sum, it is likely that a t least one of the experimental Q values in the cycle is wrong. One then examines one at a time each Q value in the original cycle by summing other cycles which contain one of the original Q values. I n this way, it is possible to uncover experimental Q values which appear to be in error and eliminate them from the original set before proceeding with the solution.

+

+ +

+

J. Mattauch, The masses of light nuclides according to an adjustment of nuclear data and a comparison with masses based on mass spectroscopic doublets. I n “NUdear Masses and Their Determination’’ (H. Hintenberger, ed.), p. 123. Pergamon, New York, 1957.

2.3.

MASS OF NUCLEI AND INDIVIDUAL PARTICLES

15

TABLE 11. Reaction Cycles These are linear relationships which must be satisfied by the Q values in Table I by virtue of the fact that there are more equations in Table I than unknown masses.'

Table I1 contains 20 reaction cycles that relate the Q values of Table I. The first cycle, Q 1 6 - Q 2 4 &26 may be written out and evaluated

+

+ p - Be8 - d -& - p + Be8 + a Qza B11 + d - Be9 - a QI6 = Be9

-B11

24

=

0

= = =

0.559 k 0.002 -8.570 k 0.009 8.018 k 0.007 0.007 k 0.012

I n this example the experimental Q values are consistent with a cycle sum of zero. Cycles (6)-(11) all contain Q1= n - H1. If the cycle sum is set equal to zero, each cycle (6)-(11) determines a value of the mass difference n - HI, and the values determined in this way may have errors smaller than the direct determinations of the mass difference. For example, cycle (6) gives the value n - H1 = 0.7822 5 0.001 MeV, whereas the direct measurement of the beta-decay endpoint energy gives the experimental value listed in Table I, 0.783 jz 0.013 MeV. The weighted average of the several values of n - H1 obtained from the cycles and the directly measured value, should be substituted in the cycles (6)-(11) to reduce

16

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

the error in the cycle sum. Similarly, cycles (12)-(16) determine values of n H1 - H2 = Q2, cycles (18)-(21) contain the mass difference 2 H Z - He4. In principle, the reaction cycles could be discovered by eliminating the unknown masses from the original set of n equations. However, in practice, it is much easier to construct the cycles by examining the closed figures formed by the links in Fig. 1 . Since each mass must appear at least twice in a reaction cycle, the links representing the reactions in the cycle form a closed polygon in the graphical representation. The sum or difference of any pair of cycles is also a cycle, and the choice of the cycles is therefore somewhat arbitrary. For the purpose of screening input data, the shortest cycles and those with the smallest errors in the sum are most useful. Mattauchs discusses limitations that should be observed in order to obtain independent cycles; he also considers the significance of the deviation from zero relative to the probable error of the sum. 2.3.2.3. Solution of the Equations. The method of least squares is the common method of solving the set of equations (2.3.2.1) to find the most probable values of t'he mass excesses, defined as those values M k for which the sum

+

m

has its minimum value, or those masses M k for which d S / d M k = 0. It is evident from the form of the sum S that an experimental observation Qi f AQi is to be given a weight ( l / A Q i ) 2 .This weighting of the observations may be accomplished conveniently by multiplying the ith equation of the set (2.3.2.1) by a factor (l/AQi). If aik = (l/A&i)Aik, and qi = (l/AQi)Qi, after multiplication the set of equations (2.3.2.1) becomes

and the least squares condition is n

m

for all Mk. Setting the derivative with respect to each of the m d u e s 6

J. Mattauch, L. Waldmann, R. Bieri, and F. Everling, Ann. Rev. Nuclear Sci. 6,

179 (1956).

2.3.

of

Mk

17

MASS OF NUCLEI AND INDIVIDUAL PARTICLES

equal to zero leads to the m equations of the form n

m

C2

n

astaaiMi

s=l i

-

C

astq8= 0,

t = 1, 2, 3,

. . . , m.

(2.3.2.3)

8

The equations (2.3.2.3) are called the normal equations of the least squares solutions. I n vector form Eqs. (2.3.2.3)may be written aaa =

aq

(2.3.2.4)

where a is the matrix whose elements are aik, and 6 is the transpose of 6,&k = ski. The solution of Eq. (2.3.2.4) is =

(2.3.2.5)

(6a)-l&j = bQ

where (iia)-' is the inverse of the matrix &a,and the matrix b is defined by Eq. (2.3.2.5).Inversion of the square matrix 6a may be a very extensive computation if the number of unknown masses is large. Table 111 TABLE 111. Atomic Masses of the First Few Nuclides Appearing in Table I These are most probable values obtained by a least squmes solution of the set of equations in Table I. The values below were computed using the 0l6mass standard; the results are also expressed in terms of the C1*/12 mass unit.= 0 1 8 Mass standard atomic mass unit = 016/16

Nuclide

Mass excess (Mev)

n H1 H* Ha Hea He4

8.3622 f 11 7.5811 f 14 13.7189 f 20 15.8228 f 37 15.8044 f 37 3.6047 f 15

Atomic mass (am4 1.008 1.008 2.014 3.016 3.016 4.003

9806 1417 7334 993 973 8713

f 12 f 15 f 21 f 4

f 4 f 16

C12 Mass standard atomic mass unit = CL*/12

Atomic-mass (&mu) 1.008 1.007 2.014 3.016 3.016 4.002

6599 8213 0931 034 014 5988

Mass excess (MeV)

f 11 8.0662 f 12 f 14 7.2851 f 15 f 20 13.1269 L- 21 f 37 14.935 f 4 f 37 14.916 L- 4 5~ 15 2.4206 f I6

~~

For mass values computed from more recent and accurate Q values, see reference 1.

lists the first few mass values obtained by the least squares method of solution of the observational equations listed in Table I. 2.3.2.4. Accuracy of the Solution. According to Eq. (2.3.2.5), M i is a linear combination of the experimental Q values,

2.

18

DETERMINATION O F FUNDAMENTAL QUANTITIES

and the standard deviation AMi may be computed from the experimental AQ's n

(AMi)2 =

2 n

[(5) AQ.]'

=

aQ8

b:8 =

(bb)ii =

(6~);~.

(2.3.2.6)

8

8

Equation (2.3.2.6) follows from the assumption that all of the AQ's are random errors, independent of each other. Although this assumption is not true in practice, since many Q-value measurements contain a common source of error in the energy calibration standards used, no one has yet attempted to trace the truly independent errors though the vast amount of data involved in the input equations, and the assumption leading to Eq. (2.3.2.6) is usually made. One attractive feature of the least-squares method of solution is that the matrix (&a)-l which contains the AIMi information is computed during the course of the solution. Table IV lists the first few elements of this matrix (&a)-' obtained in the solution of the observational equations in Table I. The square root of the diagonal elements are the errors which appear in Table 111. The off-diagonal elements of the matrix (&a)-l are used to find the error AC in a linear combination of masses, C = ZlclMl, such as a Q value that one might compute from the masses: (AC)z

=

(2.3.2.7)

c~c~(&u),'.

Since the sum in Eq. (2.3.2.7) may contain negative terms, the value of AC may be much smaller than any of the AMt appearing in the expression for C. For example, from Table I11 we find that the value of H3 - He3 is 15.8228 - 15.8044 = 0.0184 MeV. The error in this value of the beta decay energy can be found by Eq. (2.3.2.7) with the matrix elements in Table IV.

A2(H3 - HeS) = aEs,H8

+

U E ~ S , H~ S2

=

13.76

~ ~ 8 , ~ ~ :

- 2(13.77)

+ 13.82

= 0.04; AC = 0.2 kev.

The error in this mass difference is far smaller than the error in H8 or Hea, and reflects the small error in the fourth observational equation in Table I. In the least squares solutions that have been published, Mattauchs and Drummonda have also published the matrix from which the errors can be computed. 8

J. E. Drummond, Phys. Rev. 97,

1004 (1955).

2.3.

19

MASS OF NUCLEI AND INDIVIDUAL PARTICLES

TABLE IV. The Matrix (&z)-~ in Units of (kev)2 The standard error in the neutron mass is ( ( E U ) ; ; ) ~ / * = (1.277 kev*)”a = 1.13 kev. These matrix elements were computed using the 0 1 6 ma68 standard.

n H‘ HZ

Hs Hea He4

n

H’

H2

Ha

Hea

He

1.277 1.191 1.860 2.927 2.924 1.170

1.191 1.903 2.381 3.845 3.874 1.549

1.860 2.381 3.960 5.943 5.964 2.431

2.927 3.845 5.943 13.76 13.77 3.597

2.924 3.874 5.964 13.77 13.82 3.612

1.170 1.549 2.431 3.597 3.612 2.152

2.3.2.5. Other Methods of Solution. From the original set of n equations (2.3.2.1) containing m unknowns, one may choose a particular subset of m equations and solve them to find the m unknown masses. The mass values obtained in this way will, of course, depend on the particular subset chosen, unless the original set of equations (2.3.2.1)is self-consistent, i.e., unless the original equations satisfy the auxiliary conditions imposed by the reaction cycles. This follows from the fact that any particular subset can be converted into any other subset by adding or subtracting reaction cycles; if the cycle sums are zero, the solutions to the modified subset will be the same as the solutions to the initial subset. Hence another method of solution, employed by Tollestrup, Fowler, and Lauritsen17is to adjust the experimental Q values so that one obtains a set of Q values which satisfy all of the reaction cycles. The increment SQi = Q:di - Qf””, the amount which is added to the ith experimental Q value to produce a Q;dj in the self-consistent set, is made directly proportional to the weight of the experimental Q value, SQi a (l/AQi)z. Li et aZ.* describe a solution of this type for the equations listed in Table I in which the adjustment process was iterated until each cycle sum was within 1 kev of zero. Once the set of self-consistent Q values is determined, the solution of any m equations of the original set for the m unknowns is straightforward, as is the calculation of AM from the errors in the adjusted Q values. Whittaker and Robinsong describe the method of computing the error in the adjusted Q values. I n order to compute the error in a linear combination of the masses, it is necessary to express the desired combination as a sum of the adjusted Q values, then compute A. V. Tollestrup, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 78, 372 (1950). 8 C . W. Li, W. Whaling, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 83, 512 (1951). 9 E. Whittaker and G. Robinson, “The Calculus of Observt~tions,”p. 252. Blackie, Glmgow, 1924.

2.

20

DETERMINATION OF FUNDAMENTAL QUANTITIES

the error in the sum from the errors in the adjusted Q values. This method of solutions has the advantage, a t least for hand computation, that in the iterative procedure of adjusting the Q values, one sees a t each stage of the solution which of the input experimental values requires the largest adjustment in order to bring it into consistency with the remaining data. The examination of the input data by means of the reaction cycles is thus continued throughout the solution, and one has the opportunity of altering the choice of input data if that should appear to be desirable.

2.3.3. Atomic Mass Determination from Microwave Spectra* 2.3.3.1. Introduction. In microwave spectroscopy, t information on relative atomic masses is obtained by correlating the shift in frequency of a pure rotational absorption line of a molecule with isotopic mass substitution for one of the atoms in the molecule. The rotational frequencies of molecules fall in the microwave range (1 to 300 mm in wavelength), a region of the electromagnetic spectrum which only became accessible in the last two decades. Prior to the advent of microwave techniques the rotational spectra of molecules were observed as a fine structure on their vibrational spectra which fall in the infrared region. Historically there was one very notable infrared mass measurement of this type at its time. In 1932 Hardy et al.‘ measured the deuterium mass t o an accuracy of 0.1 mmu by comparing the spacing of the vibrational lines in the DCl vibrational spectrum with those of HC1. The most general class of molecule suitable for mass measurements is the symmetric-top molecule shown in Fig. 1 whose rotational energy levels are given byz-* Wg where B

=

==

hBJ(J

+ 1) + h ( A - B ) K z

(2.3.3.1)

h/87r21B and A = h/8?r21A.Here I B and I A are the moments

t See also Vol. 2, Chapter 10.6.1; Vol. 3, Chapter 2.1; and Vol. 4, Chapter 4.1.2. J. D . Hardy, E. F. Barker, and D. M. Dennison, Phys. Rev. 42, 279 (1932). Diatomic Molecules.” Van Nostrand, Princeton, New Jersey, 1950. * G . Herzberg, “Infrared and Raman Spectra.” Van Nostrand, Princeton, New Jersey, 1945. 4 C. H. Townes and A. L. Schawlow, “Microwave Spectroscopy.” McGraw-Hill, New York, 1955. 1

* G. Hemberg, “Spectra of

* Section 2.3.3 is by S. Geschwind.

2.3.

MASS OF NUCLEI AND INDIVIDUAL PARTICLES

21

of inertria of the molecule perpendicular and parallel respectively to the symmetry axis of the molecule; [ J ( J 1)]*% is the total angular momentum of the molecule and K h is its projection along the symmetry axis. If the molecule possesses a permanent electric dipole moment it will have a pure rotational absorption spectrum corresponding to the transition J - 1 -+J with AK = 0. The pure rotational absorption frequencies are then v = 2BJ. (2.3.3.2)

+

In many cases the rotational energy levels of a molecule are split by interactions of the rotation with the nuclear electric quadrupole moment

L-SYMMETRY AXIS 1

OF MOLECULE

I

H

FIQ.1. Symmetric-top molecule.

and nuclear magnetic moment. The theory of these effects are thoroughly explored in reference 4,for example. It has been worked out in sufficient detail so that the value of the rotational frequency as it would be in the absence of these effects can be precisely determined, and will be assumed to have been corrected for in all that follows. For a diatomic molecule B in Eq. (2.3.3.2) is inversely proportional t o the reduced mass which for DC1 is about twice that of HC1 so that the isotopic shift in this case is quite large, i.e., a factor of two in the rotational spacings. However, for heavier atoms such as C1 these shifts are, of course, considerably smaller. For example, in IC1 the shift in rotational frequency for the two isotopes C136and C13’ is only about 5% and, because of the

22

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

limited resolution of infrared techniques, mass measurements in the infrared region are completely unfavorable. However, in the microwave range, due t o the monochromatic oscillator sources available along with refined techniques of frequency measurement, the situation for accurate mass determination becomes extremely favorable. For example, the above-mentioned shift of 5% in the IC1 rotational spectrum corresponds t o a frequency shift of 1200 Mc in the 24,000 Mc region which in turn was measured to an accuracy of 0.01 Mc so that this change of two mass units in C1 can be experimentally measured to an accuracy of 0.05 mmu, and even more accurately (to 0.02 mmu) in other alkali halides. Even more striking is the relative measurement of the carbon and oxygen isotopes in the CO molecule to 2.5 micromass units by Rosenblum et aLs I n general, isotopic frequency shifts may vary from a few Mc to a few thousand Mc per mass unit in the 24 kMc region. In the middle mass region around Se, typical shifts are 100 Mc/mass unit in the 24 kMc region, which measured to an accuracy of 0.002 Mc correspond to a relative mass measurement 0.02 mmu, which compares favorably with other techniques of measurement. The major contributions t o this technique of mass determination have been made by Professor C. H. Townes and his students and reference to their work may be found in the review articles cited which contain tables of all masses measured by this technique.&* The accuracy of experimental measurement of isotopic frequency shift will be determined by absorption signal-to-noise ratio combined with absorption linewidth. These considerations will be outlined in the next section dealing with the various spectrometers. Secondly, the accuracy of mass determination will depend on small molecular corrections needed t o correlate experimentally determined isotopic frequency shifts of rotational spectra with mass change and this will be treated in Section 2.3.3.3. The general requirements for relative mass determination of isotopes of the same atom from microwave spectra may be summarized as follows. (1) The atom should be contained in a molecule with a permanent electric dipole moment (requirement for pure rotational spectrum). (2) The molecule should be in a gaseous state (lo-$ to mm of Hg), or in a molecular beam. B. Rosenblum, A. H. Nethercot, Jr., and C. H. Townes, Phys. Rev. 109,400 (1958). 5. Geschwind, G. R. Gunther-Mohr, and C. H. Townes, Revs. Modern Phys. 26, 444 (1954). 7 S. Geschwind, in “Handbuch der Physik-Encyclopedia of Physics” (S. Flilgge, ed.), Vol. 38, Part 1, p. 38. Springer, Berlin, 1958. 8B. Rosenblum, C. H. Townes, and 5. Geschwind, Revs. Modern Phys. SO, 409 (1958). 5

f)

2.3.

MASS OF NUCLEI A N D INDIVIDUAL PARTICLES

23

(3) The molecule should have a reasonable isotopic shift of its rotational frequency for the atom in question (generally realizable and almost characteristic of microwave spectra). (4) The molecule should have a symmetry no lower than a symmetric top, as the spectra of asymmetric rotors are too complicated for mass determination.

Requirement 2 implies the need for only a very small amount of substance and only gm of S3swas present in the form of OCS in the gas absorption cell in the S3‘jmass determination by W. A. Hardy (private communication). In the case of radioactive isotopes much larger amounts may be needed, however, only to do the chemistry. The list of radioactive isotopes whose masses have been measured by microwave spectra may be found in earlier 2.3.3.2. Experimental Methods for Measuring Microwave Absorption Lines. Detailed accounts of the techniques of microwave spectroscopy may be found in the l i t e r a t ~ r e , ~ so s ~that - ~ ~only the salient features of the different types of spectrometers which have been used in mass determinations will be outlined here, It is characteristic of microwave gas absorption spectra that over many decades of pressure peak absorption intensity is constant, as long as the linewidths are determined by collision broadening. Thus, while the total number of molecules is proportional to pressure the peak absorption of an individual molecule is inversely proportional to the mean time between collisions or pressure, so that the peak absorption intensity of the gas is constant. As narrower lines are obviously desirable for more accurate frequency measurement, one therefore tries to work at as low a pressure as possible to lo-* mm of Hg) before other sources of line broadening set in, after which any further reduction in pressure would mean a loss in intensity. The different sources of line broadening are listed in Table I and are evaluated for the case of the linear molecule OCS. This table is pertinent to an understanding of the factors that lead to the selection of a particular spectrometer as described below. The factors contributing to the peak absorption coefficient, a! (the fractional absorption per centimeter), such as the dipole moment, matrix element for the transition, etc., are fully discussed in Chapter 13 of the work by Townes and S c h a ~ l o wThese . ~ authors also include various tables in the appendices which are helpful in calculating a! for the different molecules. The value of a! may range from to cm-’ or less in the M. W. P. Strandberg, “Microwave Spectroscopy.” Wiley, New York, 1954. W. Gordy, W. V. Smith, and R. F. Trambarulo, “Microwave Spectroscopy.” Wiley, New York, 1953. l1 “Microwave Spectroscopy.” Ann. N . Y . Acad. Sci. 66 (November, 1952). 9

lo

24

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

TABLE I. Contributions to Line Width of Microwave Gas Spectra Evaluated 2(Av) for OCS at room temperature

Half line width, AV (half-half width)d Doppler effect

A V / V = 3.58 X l O - T z / T l i i ?

Collision broadening (10-1 mm Hg)

Avo0ll

Power saturation and collision broadening.

120 kc

= 1 / 2 ~

” ‘v AvcO1l

38 kc at 24 kMc

-

4’

~?FPI~.&,,, 280 kc for P = 200pw -I- hic A ~ ; ~ , , and Avcall = 120 kc

Stark-modulation broadeningb

120 kc for f = 120 kc

dT

-

Wall collisionsc

Av

‘v

Av ,

0.6 X lo9

M

17 kc for 24 kMc wavcguide

R. Karplus and J. Schwinger, Phys. Rev. 73, 1020 (1948). R. Karplus, Phys. Rev. 73, 1027 (1948). M. Danos and 5. Geschwind, Phys. Rev. 91, 1159 (1953). KEY: T = temperature in degrees Kelvin; M = molecular weight; T = mean time between collisions; P = microwave power flow in ergs/sec cma; Ip.t,lmsx = largest dipole moment matrix element between the rotational levels a and b; f = Stark modulation frequency; A / V = wall area to volume ratio of gas absorption cell in cm-1. a

b

24 kMc region. A discussion of the limiting sensitivity of gas absorption cell spectrometers may be found in many place^.^*^^^^ 2.3.3.2.1. GAS ABSORPTION CELLS. 2.3.3.2.1.1 Basic Gas Absorption Cell (Crystal Video Detection). An outline of the waveguide gas absorption cell is shown in Fig. 2. The molecule under examination is contained in the hollow waveguide cell, in gaseous form at a pressure of approximately to mm. The microwave klystron oscillator is swept in frequency by application of a sawtooth voltage t o the repeller electrode. This same sawtJoothvoltage is simultaneously applied t o the horizontal plates of an oscilloscope to provide a time base proportional to frequency. As the frequency of the oscillator sweeps through a molecular absorption in the gas cell, the resultant reduction in the power reaching the microwave crystal detector, after passage through a video amplifier, is displayed on the vertical plates of the oscilloscope. The strength of these absorptions expressed in terms of the fractional absorption per centimeter, a, ranges from lo-* t o cm-l or less in the 24,000 Mc region. These absorption coefficients vary with the third power of frequency so 12

C. H. Townes and 6. Geschwind, J . Appl. Phys. 19, 795L.

2.3. MASS

25

OF NUCLEI AND INDIVIDUAL PARTICLES

that for very light molecules such as CO whose first rotational transition is in the 100 kMc region, the absorption becomes as large as 50% in a 6-foot length of waveguide. The frequency of an absorption line is measured very precisely by comparing the signal oscillator frequency with the harmonics of a standard quartz crystal oscillator which have been obtained by multiplication up to microwave f r e q ~ e n c i e s . ~ * ' ~ WAVEGUIDE GAS. ABSORPTION CELL

GAS INLET

TO VACUUM PUMP

\

\

M SAW TOOTH VOLTAGE GENERATOR

AMPLIFIER CROSS SECTION

FIG.2. Basic gas absorption cell microwave spectrometer.

2.3.3.2.1.2. Stark-Modulation Spectrometer. In practice these very small absorptions are masked by spurious signals from frequency sensitive guide reflections and discontinuities in the mode pattern of the microwave source klystron. To separate the true absorption from these spurious effects, the characteristic gas absorption frequencies are Stark modulated by applying a time varying (100 kc) square wave of voltage between the guide proper and a metal septum insulated from the guide as shown in Fig. 3. This technique was first introduced by Hughes and Wilson.14 The 100 kc component of the absorption is detected by a crystal rectifier, amplified in a tuned 100 kc amplifier and then detected in a phasesensitive detector. This scheme of detection greatly reduces spurious signals, since the amplitude of noise or spurious signals which fall in the pass band of the amplifier is small. Widths of absorption lines, as measl*S. Geschwind, Ann. N . Y . Acad. Sci. 66, 751 (1952). I4 R. H, Hughes and E. B. Wilson, Jr., Phys. Rev. 71, 562 (1947).

26

2.

DETERMINATION OR FUNDAMENTAL QUANTITIES

ured between points of half-maximum intensity, will vary from about 150 to 300 kc/sec in this type of spectrometer. This width results from the combined effects of Doppler broadening, pressure broadening, collisions with walls, Stark modulation, and power saturation (see Table I). Power saturation and collision broadening are major contributors to this width. A minimum of several hundred microwatts of power must impinge upon the silicon crystal detector for optimum detection sensitivity. However, this amount of power will saturate the absorption lines

SIGNAL OSCl LLATOR

I d l

-i

I

- VARIABLE -ATTENUATOR

I 1

I

1

INSULATING ,*STRIPS STARK WAVE GU IDE SECT I ON

i00 KC SQUARE WAVE GENERATOR

'I '

CRYSTAL DETECTOR

TUNED 100 KC AMPLIFIER

I

1 SAWTOOTH GEN ERATOR

-

FIG.3. Stark-Modulation spectrometer.

if the gas pressure is reduced too far, and so one is constrained by this power requirement for good detection sensitivity to work at pressures where collision broadening is a factor. 2.3.3.2.1.3. Balanced Microwave Bridge with Superheterodyne Detection.'* If one could use lower power levels without a reduction in sensitivity and consequently reduce the gas pressure, then linewidths would be limited by Doppler effect and wall collisions. This can be done by using a balanced microwave bridge with superheterodyne detection (shown in Fig. 4). The optimum power necessary for sensitive detection is furnished by the beat oscillator rather than the signal oscillator whose power can now be kept low enough (a few microwatts) t o avoid saturation with

2.3.

MASS OF NUCLEI AND INDIVIDUAL PARTICLES

27

the reduced gas sample pressure of 10-3 to 10-4 mm. In the initial work with this spectrometer, the Stark modulation was dispensed with t o avoid distortion or displacement of the absorption line by Stark effects. Even without Stark modulation, amplitude fluctuations and mode discontinuities in the signal klystron and reflections in the arms of the bridge were canceled in first order by use of a symmetrical bridge. Linewidth of 60 kc (for OCS) and a sensitivity of a 3X cm-' were achieved with this spectrometer. The limit in sensitivity was here imposed by uncancelled second-order effects of reflections in the bridge arms.

-

SAW TOOTH

FIG.4. Balanced microwave bridge spectrometer with superheterodyne detection.

An improvement of sensitivity of a factor of ten in the balanced bridge scheme was realized by adding low frequency, 1 kc Stark modulation.16 However, great care must be exercised to accurately clamp the Stark voltage on zero to avoid displacement of the absorption line. 2.3.3.2.1.4. High Temperature Spectrometer. A noteworthy achievement was the extension of microwave gas absorption spectroscopy t o very Such a high temperahigh temperatures (25Oo-930"C) by Stitch et aZ.16v17 ture is necessary to obtain a sufficient vapor pressure for the alkali halide diatomic molecules, many of whose pure rotational absorption spectrum were studied by this technique yielding accurate mass information for the C1, Br, Li, K, and Rb isotope^.'^ A standard 100 kc Stark modulation spectrometer was used in which formidable difficulties were overcome in extending it to this high temperature range. l a G . R. Gunther-Mob, R. L. White, A. L. Schawlow, W. E. Good, and D. K. Coles, Phys. Rev. 94, 1184 (1954). 16 M. L. Stitch, A. Honig, and C. H. Townes, Rev. Sci. Instr. 26, 759 (1954). A. Honig, M. Mmdel, M. L. Stitch, and C. H. Townes, 86, 629 (1954).

28

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

2.3.3.2.1.5. Millimeter-Wave Spectrometers. Since the absorption coefficient, a,varies as the third power of the frequency, a in the wavelength region of 1 to 10 mm is often high enough so that absorptions can be seen directly without the need for Stark-modulation or superheterodyne techniques. Furthermore, Stark modulation cells are quite lossy a t millimeter wavelengths so that instead one uses frequency modulation of the signal klystron. This source modulation was used in a millimeter-wave spectrometer by Rosenblum et al.6,1sin their very high precision determination of the carbon, oxygen, and hydrogen mass ratios. 2.3.3.2.2. MOLECULAR BEAMELECTRIC RESONANCE.19s2o The ultimate limit upon linewidth imposed by the Doppler broadening in the gas absorption cell spectrometers, described in the preceding section, is '

MICROWAVE RADIATION

0

A-FIELD

C-FIELD

0-FIELD

\

FLOPPED-, BEAM

HOT WIRE DETECTOR

>

FIG.5. Molecular beam electric resonance apparatus.

circumvented in the electric resonance molecular beam type spectrometer shown in Fig. 5. The beam of molecules traverses successive regions of oppositely directed inhomogeneous electric fields separated by an intermediate region (the c-field) which for the study of rotational transitions has no static field. The effective electric dipole moment, peff,of a molecule is, among other things, a function of the rotational state, J. In addition, the force on a molecule is proportional to p e f f ( ~ E / d 2 Thus, ). by a suitable arrangement of stops a particular rotational state is focused at the detector. By irradiating the molecules in the c-region with microwaves of the proper frequency, transitions are induced from the J to J 1 rotational state, The resultant change in the J for some of the molecules changes the magnitude of their deflection in the B-field so that they are not refocused at the detector, causing a decrease in beam intensity.

+

B. Rosenblum, A. H. Nethercot, Jr., Phya. Rev. 97, 84 (1955). H. K. Hughes, Phye. Rev. 72, 614 (1948). zo J. W. Trischka, Phys. Rev. 74, 718 (1948). l*

lo

2.3.

MASS OF NUCLEI AND INDIVIDUAL PARTICLES

29

Doppler broadening is absent as the molecules travel in essentially one direction relative to the microwave field. The linewidth is determined by the amount of time T , the molecules spend in the transition region 1. Typical linewidths that have been as is given by the relation Av * T achieved in mass measurements with this type of spectrometer range from 10 to 20 kc.21 By use of split RF field techniques,22linewidths of 1 kc are possible although no mass measurements have been made by this technique. 2.3.3.2.3. MASER-TYPESPECTROMETER.23A third general type Of microwave gas spectrometer is the gas-type maser (Fig. 6). While no mass measurements have been made with this device, it represents such an important conceptual and practical advance in the state of the art that it is worthy of mention here. I t is a combination of the above methods in that a molecular beam is used but a change in microwave

-

! U -d OUTPUT GUIDE

INPUT GUIDE

I,=_=L,_,_,_,---'----------_--_______________--__----J

GAS SOURCE

CAVITY

FOCUSER

FIG. 6. Gas maser spectrometer.

power absorbed by the beam is measured. The beam of molecules passes through a system of electrodes which produce a field gradient which focuses only those molecules in a particular energy state above the ground state into a microwave cavity. Microwave radiation of the proper frequency will stimulate emission to the ground state so that the output radiation is enhanced. A linewidth of 4 kc has been obtained for NHI at room t e m p e r a t ~ r e . ~ ~ , ~ ~ The maser-type spectrometer should also be capable of far greater sensitivity than the conventional ones.26 2.3.3.3. Correlation between Isotopic Shift of Rotational Frequency MOLECULES. The correlation and Mass Change. 2.3.3.3.1. DIATOMIC between frequency shift in the rotational spectrum and mass change is C. A. Lee, B. P. Fabricand, R. 0. Carlson, and I. I. Rabi, Phys. Rev. 91,1395 (1953) . N. F. Ramsey, Phys. Rev. 76, 996 (1949). 23 J. P. Gordon, H. Zeiger, and C. H . Townes, Phys. Rev. 95, 282 (1954). Z 4 J. P . Gordon, Phys. Rev. 99, 1253 (1955). M. W. P. Strandberg and H . Dreicer, Phys. Rev. 94, 1393 (1954). Z 8 C. H. Townes, Phys. Rev. Letters 6, 428 (1960). *I

22

30

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

most easily made for a diatomic molecule. Equation (2.3.3.2)is only a first approximation which assumes a rigid rotator. The effect of the vibration of the molecule is in a first approximation represented by v = 2J[B, -

(Y(V

+ 4)]

(2.3.3.4)

where Be = h/8r21, and I , is the equilibrium moment of inertia assuming the nuclei are stationary at their equilibrium separation, r,; v is the vibration quantum number and a is a correction which measures the rotation-vibration interaction; I , = pr.% where p is the reduced mass of the molecule. By measuring v for two different vibrational states, a can be found and hence Be. The equilibrium internuclear distance, r,, is determined to a very high approximation only by the electronic structure of a molecule, so that it is the same for different isotopic species. Thus by measuring the equilibrium B values first with masses M and mo and then with M and ml, where ml and mo are isotopes of the same atom, the relative masses of ml and mo will be given by (2.3.3.5)

Note that knowledge of re or constants such as h is not required. Assuming (ml - mo) is small, an uncertainty in M/mo of A produces an uncertainty in ml/mowhich is given by (2.3.3.6)

In practice this is usually negligible. In RbC1, for example, an uncertainty of 1 mmp in both Rb and C1 produces errors of only 1 part in lo7in the C1 mass ratio or Rb mass ratio. There are a set of approximations or uncertainties connected with Eq. (2.3.3.5)most of which can usually be very accurately corrected for or are negligible, which will now be briefly described. (a) Centrifugal distortion of the molecule due to rotation. This correction to Be is of the order of B,3/u82where ws is the vibrational frequency of the molecule. In all but the very lightest molecules this correction is usually negligible, When it is important, however, we can usually be determined with sufficient accuracy from optical band spectra to make this correction. 4sb (b) Anharmonicity of the potential function.27 Here corrections are also of the order of B,3/wz and are usually negligible except in lighter molecules, where they can also be corrected for as in *'J. 4. Duaham, Phys. Rev. 41, 721 (1932).

2.3.

MASS OF NUCLEI A N D INDIVIDUAL PARTICLES

31

(c) Contribution of molecular electrons to moment of inertia. Implicit in the writing of Eq. (2.3.3.5) is that the mass of an entire neutral atom, nucleus plus electrons is concentrated at a point. In reality, of course, the electrons have an extension in space about the nuclei and one must consider their extra contribution to the moment of inertia of the molecule due to this spatial extension. If the electrons are assumed to be almost spherically distributed about their respective nuclei and if indeed they rotated rigidly with the molecule, then one might expect the moment of inertia to be greater than the point mass assumption by approximately an amount equal to the moment of inertia of the electrons about their respective nuclei. If really present, this contribution to the moment VALENCE ELECTRONS NUCLEUS I

/’ ./

CLOSED

FIG.7. Illustration of the slipping of closed shell electrons in a rotating molecule.

of inertia would be quite large; but fortunately it is absent because the orientation of a closed shell of electrons remains fixed in space as the molecule r0tates.28,2~This slipping of the closed shells as the molecule rotates is illustrated in Fig. 7 and is akin t o the motion of a chair on a ferris wheel. On the other hand, the valence electrons may be visualized as rotating rigidly with the molecule and will give an extra contribution t o the moment of inertia, A single rigidly jotating valence electron at an average distance of approximately 1 A from its nucleus would give an error of less than one part per million in the C136/C137ratio in the alkali halides and is in general negligible in the heavier molecules. However, a similar valence electron in a lighter molecule such as CO results in an uncertainty of one part in lo6in the C12/C1aratio, which is one hundred times greater than the experimental error in this case. 2 8 G. C. Wick, Phys. Rev. 75, 51 (1948). 29See C. H. Townes and A. L. Schawlow, “Microwave Spectroscopy,” p. 213. McGraw-Ell, New York, 1955.

32

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

However, the nonslipping valence electrons contribute proportionately t o the magnetic moment of the molecule, so that by measuring the magnetic moment of the molecule, one can very accurately correct for this extra contribution, A I of the valence electrons t o the moment of inertia above the idealized situation. This connection between the two derives from nothing more than Larmor’s theorem for rotating charges and is given by4J-g (2.3.3.7) where B:‘Om is the idealized Be value assuming all the mass for the neutral atom concentrated at a point. Here M p is the proton mass; m, the electron mass; p2&,the nuclear magneton; pJ, the magnetic moment of the molecule in the state J . As magnetic moments of molecules are of the order of a nuclear magneton for lighter molecules, this fractional correction t o the B value will be of the order of a part in several thousand and corresponded t o a correction of one part in loEin the C13/C12ratio in CO, which was one hundred times greater than the experimental error.6 (d) “Wobble” stretching: Rosenblum et aL6 have pointed out that in a Z molecule, even though the projection of the electronic orbital angular momentum along the internuclear axis is zero, it may still have perpendicular components which precess about the internuclear axis causing it t o “wobble” and hence produce a type of centrifugal distortion. The effect of wobble stretching can be taken into account only by using one accurately known mass ratio t o compare with the microwave value of the same mass ratio. In this sense it represents a basic limit on the accuracy of mass determination by the microwave method. This correction in CO (which was ten times the experimental error) amounted t o about 1 part in lo6 in the C and 0 mass ratios and was applied to the C12/C1sand 018/01E mass ratio determination using the known C12/C14 ratio to calibrate for this correction. The very detailed examination of the afore-mentioned electronic corrections t o mass determinations in the molecule CO by Rosenblum et aLs has confirmed the validity of the molecular theory involved t o very high accuracy. This is illustrated in Table I1 by noting the consistency of their results for the Cla/C12ratio in the molecules C 0 l Eand CO1*and the good agreement of these results with other determinations. In the higher mass region these electronic corrections are far less important. This is borne out by R b mass ratios in Table I1 which have not been corrected for elect,ronic effects. The microwave result in RbCl which was obtained with a molecular beam apparatus is seen t o be consistent with the determinations in RbI and RbBr which were made in a

2.3.

33

MASS O F NUCLEI AND INDIVIDUAL PARTICLES

TABLE11. Illustrative Mass Determinations in Diatomic Molecules Mass ratio Molecule

From microwave spectra

Rbsa/Rb87 RbC136 RbI RbBr C13/C1* C016 COls

0.9770163 k 45" 0.9770177 f 45d 0.9770146 f 55d 1.08361283 f 27" 1.08361290 f 28"

From nuclear reactions ~~~

~~~~

From mass spectra ~~

~

0.9770191 f 20 1.08361308 f 20a 1,08361278 k 200

1.08361309

3*

See reference 5. T. T. Scolman, K. S. Quisenberry, and A. 0. Nier, Phys. Rev. 102, 1076 (1956). J. W. Trischka and R. Braunstein, Phys. Rev. 06, 968 (1954). See reference 17. a T. L. Collins, W. H. Johnson, Jr., and A. 0. Nier, Phys. Rev. 94, 398 (1954).

0

high temperature gas absorption cell, and all three in good agreement with the mass spectrographic result. 2.3.3.3.2. MOLECULES MORECOMPLEX THANDIATOMIC MOLECULES. In molecules other than diatomic, zero point vibrations prevent the determination of accurate mas6 ratios but nonetheless an accurate determination of atomic masses can be made in terms of mass difference ratios. In analogy to the representation of the rotation-vibration interaction for a diatomic molecule, the effective B value, Bo for a polyatomic molecule may be written as2-4

B O = Ba -

2

ai(Vi

+ di/2)

(2.3.3.8)

i

where Be = equilibrium B value, ai is the rotation-vibration interaction, vi is the vibrational quantum number, and di the degree of degeneracy of the ith vibrational mode. In order to determine Be, one must measure BOfor the same rotational transition for each vibrational mode in both the ground vibrational state, vi = 0, and the first excited state, vi = 1. This is, however, exceedingly difficult as the Boltzmann population of the excited vibrational states of the less abundant isotopic species will be too low so that to measure all the a's is in general beyond the reach of the sensitivity of present-day microwave techniques. In addition, the a's in a polyatomic molecule depend in a rather complex way upon anharmonic potential constants and mass, so that knowing their values for a more abundant isotopic species does not allow one to theoretically predict their values for the less abundant species as one can do in a simple diatomic molecule. By a straightforward application of the parallel axis theorem of mechanics, it can be shown that the following relationship obtains for

34

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

isotopic substitutions of an atom lying on the symmetry axis of a symmetric-top or linear molecule : 4 * 6

m l - mo - BLl)) - M1B!2)(BLo) M~B:~)(B:o’ -~12))‘ m2 - mo

(2.3.3.9)

Here mo,m land m2 are the isotopic masses, M1 and MO are the molecular masses with isotopes ml and m2, respectively, and the B,‘s are the equilibrium B values corresponding t o the three different isotopic species. While the equilibrium B values appear in Eq. (2.3.3.8), it can be shown that the error introduced by using instead the effective or measured B value, Bo (determined from the measured rotational frequency by vrot = 2BoJ), is small. For example, a detailed estimate* of the error introduced by this neglect of a in the (P3- S a 2 ) / ( S 3 *- S12) mass ratio as determined from the linear molecule OCS shows that it is less than 1 part in 15,000. This corresponds to an uncertainty of 0.03 mmp in the determination of S33mass taking the and V4masses as known, and is representative of the accuracy obtainable in the middle mass region. As a matter of fact the best evidence for the smallness of the error introduced by neglecting zero point vibrations is furnished by the very good agreement between mass difference ratios determined by the microwave method, as outlined, neglecting zero point vibrations, and other methods. Such comparisons have been tabulated in several p l a ~ e s For . ~ exam~ ~ ~ ~ ~ ple, in OCS the oxygen atom, being light, undergoes large displacements from its equilibrium position so that one might expect the effect of zero point vibrations to show up in this case; yet the microwave value of (0’’ - 016)/(018 - OI6) = 0.501042 f 8 6 is in good agreement with the most recent mass spectroscopic values of 0.501045 & 3 3 1 and 0.5010446 jz 9 . 3 2 In addition, it can be shown that the typical uncertainty in M1/Mz is also generally unimportant.6 Thus if one measures the rotational frequencies corresponding to three isotopic substitutions for an atom on the symmetry axis of a symmetric top molecule and takes the masses of two of the isotopes as known, one can accurately determine the mass of the third. Taking the masses of two of the isotopes as known can be regarded as a calibration of the isotope shift and the effect of zero point vibrations. Note that no information concerning the structural parameters of the molecule, such as bond angles and internuclear distances is required. 30 S.Geschwind, in “Nuclear Masses and Their Determination,,’ p. 163. Pergamon, New York, 1947. 31 H. Ewald, Z. Naturjorsch. 6a, 293 (1951). 31 T. T. Scolman, K. S.Quisenberry, and A. 0. Nier, Phys. Rev. 102, 107 (1956).

2.3.

MASS OF NUCLEI AND INDIVIDUAL

PARTICLES

35

Further increase of the experimental accuracy, would bring one into the region where electronic effects and zero point vibrations would undoubtedly become important. It would be very difficult to correct for these effects in polyatomic molecules as one is able to do for diatomic molecules and so one must regard the present state of the theory of the electronic structure of polyatomic molecules as setting the limit of accuracy attainable in mass determinations in polyatomic molecules.

2.3.4. Measurement of Mass with Cloud Chambers and Bubble Chambers*

Cloud chambers have proved very useful in the determination of the mass of the unstable nuclear particles produced in nuclear disintegrations, and with the rapid development of bubble chamber technique, there is every hope that mass measurements will be commonly made with those instruments. The mass of a charged particle detected in a cloud or bubble chamber may be determined in various ways. The most important of these are the following: 1. 2. 3. 4. 5.

Momentum-ionization Momentum-range (energy) Momentum-change in momentum Ionization-range Scattering-range

Momentum-ionization. If the momentum p = mopyc and the velocity @c are measured, the mass mo can be found, since y depends only on p. The momentum can be measured either prior to the entry of the particle into the chamber, or in the chamber, if the chamber is immersed in a magnetic field. Likewise the velocity may be determined either within the chamber by measurement of ionization, or outside the chamber by time-of-flight, cerenkov, or ionization techniques. Since these two measurements are independent, the error in the mass is obtained from

where A p and AI are the errors in p and I , and m ois taken to be a function of p and I . It is found that the error in ionization usually contributes

* Section 2.3.4 is by W.

6. Fretter.

36

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

more to the error in mass than does the error in momentum, when both are measured' in the same chamber. Under favorable circumstances the error in mass for an individual particle may be as small as 5%. I n the relativistic region, where the slope of the rise is rather slow, the error in mass determination is much larger, but it is still possible to distinguish mesons from nucleons in the momentum range of 5-20 Bevlc. Momentum-range. If the particle can be stopped and its range determined, and a range-energy relation is available, the mass can be found from a combination of the equations p = moPycand kinetic energy

= moc2(y -

1).

There are two general procedures used, depending on the energy of the particle: either the particle is stopped in the medium of the chamber, or in heavy material as in a multiplate chamber. In either case a separate measurement of the momentum is usually necessary, since momentum measurements are virtually impossible in a multiplate chamber and inaccurate in a chamber when a particle approaches the end of its range. In a large bubble chamber, however, momentum and range measurements are simultaneously possible. Where the range-energy relation is accurately established, and accurate momentum measurements can be made, the momentum-range method is the most accurate for mass measurements. The range straggling error on an individual measurement is only one or two per cent, and momentum measurements can usually be made to a per cent or so under good conditions. Depending on the region in which the measurement is done, the error in mass on an individual determination is also of the order of one or two per cent. Momentum-change in momentum. When it is inconvenient to bring a particle to rest, but momentum measurements may be made, the momentum loss of a particle in passing through a given thickness of material may be used together with the momentum itself, to determine the mass. This method was discussed by Wheeler and Ladenburg.* It is not very accurate because it involves differences in momentum measurements which are ordinarily fairly small, and is not often used. Ionization-range. If momentum measurements cannot be made, the mass can be determined if the velocity is found from the ionization, and the energy from the range of the particle. Because of the inaccuracies in the ionization measurements, and the limited region over which they can be made, this method has not been used very much, except for fairly rough measurements. In a large bubble chamber, where simultaneous range and velocity measurements should be possible, this should provide

* W. B. Fretter and E. W. Friesen, Rev. Sci. Innstr. 26, 703 (1965). * J. A. Wheeler and R. Ladenburg, Phys. Rev.

60, 754 (1941).

2.3.

MASS

OF NUCLEI A N D INDIVIDUAL

PARTICLES

37

a method for mass measurements in the absence of a magnetic field. If the medium is so heavy as to give large multiple scattering making an ordinary magnetic field useless, this method should give useful results, as i t does in emulsions. Scattering-range. Rarely applied to chamber conditions, this method gives mo-yp2cand moc2(r- 1) which can be combined to determine the mass. It has been applied to multiplate chambers3 and to high-pressure cloud chamber^.^

2.3.5. Determination of Mass of Nucleons in Emulsions* I n previous sections (2.1.1.3, 2.2.1.1, an d 2.2.1.2) the method of charge, momentum, and energy determination of particles have been discussed. I n all calculations, connected with these problems, the knowledge of the particle mass or the simultaneous mass determination was implicitly involved. Therefore we will discuss below only a few cases of mass determination which are interesting from a methodical standpoint. The discussion will be limited to particles of single charge and mainly treat particles ending in the emulsion. 2.3.5.1. Constant Sagitta M e t h ~ d . l J .I~n the case of singly charged, nonrelativistic particles (pat = 2E), ending in the emulsion, the rangeenergy relation together with the measured mean scattering angle give sufficient information for the determination of the particle mass. The range-energy relation can be written in the form

E

= aRnm1-n

(2.3.5.1)

where a is a constant and n the power factor which actually is not a constant, but varies slightly with particle ~ e l o c i t yThe . ~ mean scattering angle is given by6

(2.3.5.2)

K being the scattering constant and t the cell length. Using relation C. Peyrou, Nuovo cimento [9]11, Suppl. No. 2, 322 (1954). P. Baxter and F. R. Stannard, Proc. Phys. Soc. (London) A70, 19 (1957). 1 S. Biswas, E. C. George, and B. Peters, Proc. Indian Acad. Sci. S A Y418 (1953). 2 C. Dilworth, S. J. Goldsack, and L. Hirschberg, Nuovo cimento 191 11, 113 (1954). 3 R. G . Glasser, Phys. Rev. 98, 174 (1955). 4 See Section 1.7.5.2. 6 See Section 2.2.1.2. 3 4

~

* Section 2.3.5 is by

Marietta Blau.

38

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

(2.3.5.1) with the approximation that the product momentum times velocity is equal to twice the particle energy, (2.3.5.2) can be written in the form 8 = -tllZR-nmn-l K

2a

(2.3.5.3)

where m is the only unknown magnitude, which therefore can be calculated. The method of constant sagitta, which has been briefly mentioned in connection with other scattering methods, was developed independently by the laboratories of Brussels and Bombay and was presented in a joint communication at the Congress on Cosmic Rays at Bagnere de Bigore, July, 1953. The most common procedure in scattering measurements of fast particles is the use of constant cell lengths along the whole track or limited track segments within which the particle energy can be considered constant. The rate of energy loss per unit length, however, increases with decreasing residual range and toward the end of a track the sections of nearly constant energy become exceedingly small. I n this region, measurements made with cell lengths, appropriate for the more energetic part of the track, lead by necessity to results of low statistical accuracy. In the constant sagitta method variable cell lengths are chosen in such a way that the mean sagitta remains constant along the whole track. With the aid of the range-energy relation one can calculate for particles of given mass the cell length required such that at a given residual range the mean value of the second difference I) has a predetermined value A, (2.3.5.4)

The relation between t, R, and m for a given value A can be written: t3/2

-

ml-nRn.

(2.3.5.5)

The general procedure in constructing a “scattering scheme” is the following. One starts at a point of the track with the residual range Ro and chooses for the first cell a value t o ; the subsequent cells tl, t2, . . . tn can be easily found from a log-log plot, describing the relation (2.3.5.5) for a particle of given mass m. A particle of mass m2 # ml, measured with the same scattering scheme will again lead to a constant second difference D,, which, however, is different from D,. The ratio of the two second differences is related to the ratio of the masses by (2.3.5.6)

2.3.

MASS OF NUCLEI AND INDIVIDUAL PARTICLES

39

Equations (2.3.5.5) and (2.3.5.6) are not strictly correct for various reasons. (1) The power factor n is not constant, but varies with particle energy. (2) The relation p@c = 2E is not valid for higher energies. (3) The scattering constant varies slightly with particle velocity and cell length. (4) Corrections for “noise,” so far neglected, have to be introduced. The effects due to factors 1-3 can be accounted for in various ways. Dilworth et aL2 have shown that an adequate procedure consists in first calculating the scattering scheme with a single power factor (the authors use n = 0.58), assuming a constant value for the scattering constant and neglecting deviations due to the assumption ppc = 2E, and afterward using correction terms to adjust the calculated value D,,,‘to Bobs,the value of the mean second difference observed. Biswas et al.’ use values for the power factor n which were found in experiments with protons with energies up to 35 MeV, and, for higher energies, values derived from a n empirical formula; for the scattering constant theoretical values, depending on cell length and particle energy, were used. Glasser3 determines the constants b and n

ID1 = bR-n

(2.3.5.4a)

from measurements on proton tracks of 57 Mev from a cyclotron beam by fitting the experimental data to a power law and found a power factor n, = 0.607. Equation (2.3.5.4a) is derived from (2.3.5.4) and the measurements were started a t a point RO= to of the track, where to is the initial cell length: the following cell lengths ti are then related to the first by ti = t o ( R / t ~ ) 2 n ’ [compare 3 (2.3.5.5)]. According to Glasser’s experiments the mean absolute second differences as a function of particle mass are then given

ID1 = 0.053

e)””” (to)0.893.

(2.3.5.7)

where m, is the proton mass. The noise problem (point 4) is less complicated in the constant sagitta than in the constant cell method. The cells are smaller and therefore the troublesome stage noise is either completely absent or has a t least a constant value. The most important source of noise is the grain noise, arising from the fact that the grain centers are not completely aligned along the trajectories: the error is independent of cell length and of the order of half a grain diameter. The independence of the error from cell length is a great advantage of the constant sagitta method, because of the possibility of choosing cell lengths in such a way that the signal-tonoise ratio assumes an optimum value for the whole trajectory.

2.

40

DETERMINATION OF FUNDAMENTAL QUANTITIES

According to Biswas et al.’ the optimum conditions for the signal-tonoise ratio can be expressed by the relation A = 2.42

(2.3.5.8)

where A, the mean absolute value of the true second differences, is given by A = dDZ - p2, and a the mean value of the noise. Relation (2.3.5.8) suggests that the scattering scheme should be chosen in accordance with particle mass, since A is a function of mass [Eq. (2.3.5.6)]. RESIDUAL RANGE R in mm,SCALE

IT

-

500

200 -SCALE

II

i

I

100

I I-

(3

z w

-1

50-

-I -I W

0

2 0 - SCALE

10

-

I

20

50 100 200 RESIDUAL RANGE IN p ,SCALE

500

1000

I

FIQ.1. Plot of cell length versus residual range for protons and pions

Figure I, taken from the paper by Biswas et aLli is a plot of cell length versus residual range for protons and pions, chosen in such a way that the mean absolute second differences after noise subtraction assume the constant values of 1, and 1.6 p, respectively. Glasser’sa experiments indicate that the optimum signal-to-noise ratio is obtained for D = 2.4Z where D is the observed value due to scattering noise. He finds that the optimum initial cell t o for R = t o is 4.4 p for protons, 1.9 for pions, and 3.9 p for a particle of mass 1000 me, where m, is the electron mms.

+

41

2 . 3 . MASS OF NUCLEI AND INDIVIDUAL PARTICLES

The error in mass measurements is relatively large, even if optimum conditions are used since: am 1 aA - = -_. m 1-nA [See Eqs. (2.3.5.4) and (2.3.5.5).] In Table I (Glasser) the percentage statistical error in mass measurements for particles of 1000 m, measured a t optimum conditions is given for tracks of various lengths. The error becomes considerably less severe, of course, if many tracks are available. TABLE I. Statistical Error in Mass Measurements Available length,

(I0 1000 3000 6000 10,000 50,000

Number of cells

45 81 131 178 350

Errors on 1 track

(%I

Errors on 25 tracks (%)

45 33 26 23 14

8.9 6.7 5.3 4.5 3

Tables with scattering schemes for various particles were published by different authors. 2.3.5.2. Photometric Method. Mass measurements on singly charged particles, ending in the emulsion, can be made with any one of the ionization methods described in earlier sections. However, for dense tracks near the end of the range, the blob counting or mean gap length methods are not suitable. In this case the method of gap or blob length per unit path length should be used, or a photometric method based on opacity or profile measurements of the track. In the following the Lund photometric method, the so-called mean track width method, will be discussed. The general measuring arrangement and the measuring procedure has already been described in the section on charge determination. Details of the multiplier equipment, the electronic part, and the stabilized high-voltage supply of the measuring instrument are published by von Friesen and Kristiansson.' The objective used in the measurements is a K.S. X 100, and the dimensions of the slit referred to the objective plane are 2 . 4 X 30 1.1. The optimum slit dimensions depend on the density of the track, the track inclination, and the background conditions. Each measurement with the track H. Fay, K. Gottstein, and K. Hain, Nuovo cimento Suppl. [9]11, 234 (1954). S.von Friesen and K. Kristiansson, Arkiv Fysik 4, 505 (1952).

42

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

segment below the slit was followed by two background measurements to each side of the track; the three measurements are used for the determination of the true opacity, due only to grains activated by the particle along its path through the emulsion. The method of “mass measurement” is based upon the following considerations, The mean track width, or MTW(Rm), of a particle of mass m a t a point with residual range R is for singly charged particles a function of the particle velocity, v , and hence of the ratio R/m. (2.3.5.9)

Because MTW depends also on development and emulsion conditions in a complicated way, it was found impossible to obtain a simple analytic expression for f(R/m). found it expedient to write the function MTW (Rm) The in the form of a power series: MTW(Z2m) = MTWo

+

(2.3.5.10)

where A, B . . . in the expansion depend on emulsion and development condition, but are independent of particle mass; MTWo, the mean track width a t zero range, is normalized to the value 100. The number of terms in the expansion depends on the length of the track. The constants A, B . . . can be determined for a special emulsion stack and development conditions if particles of known mass, ending in the emulsion are available. The MTW of two particles with masses ml and mz a t the same residual range can be written in the form MTW,

=

100

+ A I R + BIRZ+ -

. and MTWZ = 100

+ AzR + BzR2 +

*

.

*

where (Al/Az) = (mz/ml) and (Bl/B2) = (mz/m1)2. It is advantageous to check this ratio on tracks of particles with known mass in order to ascertain the absence of systematic errors. The mass of an unknown particle is determined by the method of least squares with the condition, that Eq. (2.3.5.10)should fit the experimental results as well as possible, or that

0=

2

{MTWL- MTW(Rm)) aMTW(Rm) am

n 8

S.von Friesen and K. Kristiansson, Arkiv Fysik 8, 121 (1954). K. Kristiansson, Arkiu Fysik 8, 311 (1954).

(2.3.5.11)

2.3.

MASS OF NUCLEI AND INDIVIDUAL

PARTICLES

43

where MTW1, MTWz . . . MTWL represent measured values of the unknown track a t residual ranges R I , Rz, . . . RL. The mass of the unknown particle m, is related to the mass m of the known particle by (m/m,) = (RL/RL,) if R I , R2, . . . RL and R1,,R2, . . . RL, are the residual ranges where the ionization and hence the MTW of the two particles are equal. Particles of unknown mass should be compared with protons or other known particle tracks found a t nearly the same part of the emulsion. The value of the MTW of a track depends on the depth in the emulsion; dept,h corrections can be applied, but may introduce additional errors. Furthermore the inclination of tracks used for comparison should be the same, and the best results can be obtained only if the dip angle is small. The error in mass measurements depends on the total track length measured of the particle used for comparison, on the residual range of the unknown particle, on the mass ratio, and finally on the resolving power of the emulsion. The errors are relatively small if optimum conditions prevail, because the method is based on a great number of single measurements. Kristianssons estimates the error in mass to 5 3 % ’ if tracks of a t least 1 cm length are available. 2.3.5.3. M a s s Measurement of Particles Which Do Not End in the Emulsion. I n the case of particles not ending in the emulsion, the conditions are more complicated, because in addition to the mass the particle velocity must be determined. It is difficult to treat this problem in a general way, since it depends strongly on the velocity of the particle, length of track, dip angle, and emulsion conditions. For particle tracks near minimum ionization, where the grain density does not change perceptibly over an appreciable length in the emulsion, the only possible method in determining the mass consists in a combination of ionization and scattering measurements (constant cell method). In this case it is helpful to use sets of curves-scattering angle or p p versus blob density-for particles of various masses. The accuracy of the method depends nearly entirely on the error made in the scattering measurement; in general this error will be large unless the tracks are very long and flat and the emulsion is free from distortion. I n gray tracks with more than 1.5 minimum ionization, the gradient of the blob density along the track gives additional information on the particle mass, provided that the track length is sufficient. Johansson and Kristianssonlo have applied the photometric method to gray tracks which do not end in the emulsion. They approximate the MTW versus range relation by a straight line, and the slope of this line represents the rate lo

F. Johansson and K. Kristiansson, Arkiv Fysik 11, 467 (1956).

44

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

of change, d(MTW),/dR, which is a measure of the particle mass. They again use normalization procedures and comparison with tracks of particles of known mass and velocity. The MTW and d(MTW)/dR, the latter in units of MTW, of the unknown track, are calculated by the least square method. They find that identification is possible even without additional scattering measurements if very long and flat tracks are available; the best results were obtained with blob densities between 2-5 times minimum value, but even with tracks as light as 1.75 times, minimum ionization mass measurements can be made if the track length exceeds 10 mm. Yekutielli and Rosendorf" have recommended blob density measurements along the track, plotted as a function of the distance from the entrance of the track to its final disappearance. They discuss the problem in a mathematical way and give the solution of the problem in an analytical form.

2.4. Determination of Spin, Parity, and Nuclear Moments 2.4.1. Spectroscopic Methods 2.4.1.1. Optical and Ultraviolet Spectroscopy.* t 2.4.1.1.1. HISTORICAL BACKGROUND. Hyperfine structure in optical spectra was discovered in 1891 by A. A. Michelson using his celebrated interferometer. Fabry and Perot, and Lummer and Gehreke also brought their interferometers to bear, and by 1910 a considerable number of observations had been made. At length it became clear that two causes of this structure were present, one due to the different isotopes of an element, now usually called isotope shift, and the other due to some effect within a single isotope. I t was not until 1924 that Pauli' suggested that the hfs in the case of a single isotope was due to nuclear moments. Subsequent experiment rapidly disclosed that many nuclei possess spin, and a number of magnetic and electric quadrupole moments were 11

R. Rosendorf and C. Yekutielli, Nuovo eimento Suppt. [9] 12, 416 (1954).

t See also Vol. 3, Part 2; Vol. 4, A, Chapter 4.2. W. Pauli, Naiurwissenschaften 12, 741 (1924).

J. E. Mack, Revs. Modern Phys. 22, 64 (1950). G. Laukien, Handbuch der Physik, XXXVIII/l, 338 (1958). G. H. Fuller, 1960 Nuclear Data Tables, National Academy of Sciences, National Research Council Part IV, Table 4, 72 (1961). J

4

~

* Section 2.4.1.1is by Joseph G.

Hirschberg.

2.4.

DETERMINATION OF SPIN,

PARITY,

A N D NUCLEAR MOMENTS

45

In the case of elements with several isotopes the measurement of nuclear moments is complicated by the fact that the structure of each of the various isotopes is displaced by the isotope shift. Tin, for example, has ten stable isotopes, several of which have additional hfs caused by nuclear spin. With the advent, since the Second World War, of the availability of separated isotopes, the evaluation of isotope shifts and nuclear moments has been greatly facilitated. 2.4.1.1.2. ISOTOPE SHIFT.A short description of isotope shift is included here for two reasons. In any evaluation of nuclear moments of an element with several isotopes, the isotope shift structure must be disentangled from the structure due to the moments of the nucleus in order to measure the latter. Also, the isotope shifts themselves, in the case of heavy elements may directly shed light on the structure of the nucleus. The isotope shift can be divided into three categories: that for light elements, that for heavy elements, and that for intermediate elements. For light elements, the effect is due to the difference in mass of the nucleus for the different isotopes, In its simplest form this occurs in the spectrum of hydrogen. In one-electron spectra the effect of a finite nuclear mass is taken into consideration by the use of the reduced mass p :

Mm ' " W m

(2.4.1.1.1)

where M is the nuclear mass and m is the electron mass. The energy T , of a particular level will depend on M , the nuclear mass according to (2.4.1.1.2) where T , is the value which would obtain if M were infinite. The energy levels of a lighter isotope are shifted more toward the series limit than those of a heavier one. This simple situation obtains only with the two-body case; hydrogen and hydrogen-like spectra. When more than one electron is involved, corrections to the simple reduced mass solution must be made. In this case, detailed knowledge of the wave functions is necessary to apply the corrections, and the measured shifts do not agree well with calculation, except in the simplest cases. Beyond mass 50 or 60, the mass effect becomes very small, and is finally obscured by the nuclear volume isotope shift which may be thought of as being caused by the finite size of the nucleus. Since the nucleus is not an infinitely small point, but has a radius of the order of All3 X 10-13 cm, the

46

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Coulomb field breaks down in its neighborhood. Orbital electrons whose wave functions have an appreciable value at the nucleus, such as configurations involving s and to a smaller extent p electrons, are most affected. This effect becomes important for 2 > 20 and increases toward heavier nuclei approximately as Z2.The effect on the energy levels is just opposite to the mass effect, the energy levels of the lighter isotopes lying lowest in the energy level diagram. For details in the evaluation of both shifts the reader is referred to references 5, 6, and especially 7. 2.4.1.1.3. EFFECTSOF NUCLEAR MOMENTSAND SPIN. Besides the isotope shift another cause of hyperfine structure was soon required, since such structure was often found to occur in nuclei with only a single isotope. As indicated above, this was ascribed in 1924 by Pauli to the existence of a nuclear spin, I, and to the interaction between the resulting nuclear moments and those of the optical electrons. The most important of the nuclear moments is the magnetic moment pr. The change in the energy of the atom due to the magnetic energy between this moment and the orbital electrons is given by: AW,

=

-

-prH(O)cos prH(0)

(2.4.1.1.3)

where H(O)is the average value of the magnetic field of the orbital electrons in the neighborhood of the nucleus; p~ is the magnitude of the nuclear magnetic moment; H(0) and pr are the vector field and nuclear moment respectively. This expression becomes : AW,

=

A I J cos(1,J)

(2.4.1.1.4)

when it is remembered that H(0) and J are antiparallel (due to the negative sign of the electronic charge), and pr is parallel to I (because the proton has a positive magneticlmoment). A = -prH(O)/IJ and is evaluated by a method considered below. We may form a vector diagram for the composition of two angular moments (see references 6 and 7), as in Fig. 1, where I, the spin of the nucleus, and J, the total angular momentum of the electrons, are combined to form F. We can apply the law of cosines and write cos(1,J) =

FZ - 1 2 - J 2 21J

(2.4.1.1.5)

5E. U. Condon and G . H. Shortley, “Atomic Spectra.” Cambridge Univ. Press, London and New York (1935). 6 H. E. White, “Introduction to Atomic Spectra.‘’ McGraw-Hill, New York, 1934. 7 H. Kopfermann, “Nuclear Moments.” Academic Press, New York, 1958.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

47

Thus :

W,

=

A - (F2 - I 2 - J')

(2.4.1.1.6)

2

A rigorous solution on the quantum mechanics6 necessitates the following changes: F2, 12,and J 2 are replaced by F ( F l), 1(1 l), and J ( J 1) respectively. The magnitudes pr and H ( 0 ) are replaced by the z components of and H(0) if I and J are taken parallel to the z axis. Thus we obtain: A (2.4.1.1.7) W , = 2, F ( P 1) - 1 ( 1 1) - J ( J 1)

+

+

+

+

+

+

where F can assume the values

I+J,

I + J - l ,

I + J - 2 * . . ( I - J ) . (2.4.1.1.8)

This expression for the splitting AW, results in the characteristic hyperfine flag patterns, in which the same interval rule occurs as in

FIG. 1 . Vector diagram for combining I and J.

FIG.2. Hyperhe structure levels showing interval rule.

ordinary fine structure: The energy difference between two adjacent hfs levels, for instance one of F and one of F - 1, is just proportional to F. For example, if I = and J = 2 then F can have the values 4, $, +, 4 by Eq. (2.4.1.1.8)and the levels will appear as in Fig. 2. In many cases, the nuclear spin can be determined simply from the number of levels, or from the intervals, but in more difficult cases where some of the structure cannot be resolved, intensities must be determined. Detailed methods are discussed by Kopfermann.7 Sometimes it is noticed that the interval rule is not exactly followed. This effect is attributed to an electrostatic interaction with the orbital electrons by the nuclear quadrupole moment. The only other nuclear moment which normally can be measured by

2.

48

DETERMINATION OF FUNDAMENTAL QUANTITIES

means of hyperfine structure, is the electric quadrupole moment, Q. The additional energy due t o the quadrupole moment AWQ is given by: (2.4.1.1.9)

where e is the elect,ronic charge, the electric gradient a t the nucleus in the z direction and 0 is the angle between the direction of the quadrupole moment and the electric field gradient. The quadrupole moment may be thought of as a measure of the departure of the nucleus from spherical shape and has positive or negative sign according t o whether the nucleus is prolate or oblate, respectively. Casimirs has expressed the formula for AWQ as follows:

where

c = F ( F + 1) - 1(1+ 1) - J(J + 1)

and

B

= eQ@JJ(O)

where ipJJ(0)is the average gradient in the J direction of the electric field due t o the orbital electrons at the nucleus. We have now obtained expressions for the effect on the atomic energy levels of the magnetic dipole and electric quadrupole moments in terms of two constants A and B. If these formulas are combined we obtain for the total effect: AWF = AWm AWQ. (2.4.1.1.11)

+

As has been suggested above, much the largest effect is due t o th e magnetic moment ; the AWQ is often negligible and when present can be treated as a perturbation on AW, resulting in a departure from the intervals predicted. The hyperfine structure levels are affected by external magnetic fields in a similar way t o the ordinary fine-structure levels, resulting in Zeeman and Paschen-Back effects. A discussion of these will be found in Kopfermann’ and White,6 and is beyond the scope of this treatment. I n the study of spectra, transitions between levels are observed, not the levels themselves. This requires a consideration of the combination rules, telling us which levels combine with each other. The general rule is that values of a given F will combine so that AF = 0 or k 1. The intensities depend on I and J as well as F , and tables may be found in standard texts. 8

H. B. G. Casimir, Teyler’s Tweede Genootshap, Verhandel. (Haadem) (1936).

2.4.

DETERMINATION

OF SPIN, PARITY,

AND NUCLEAR MOMENTS

49

As an example of the transitions to be expected, consider a simple example: I = $, and J = 0 and 1. Here (see Fig. 3) there are just as many lines as F levels in the lower J state since there is only one hyperfine upper state. Other simple cases arise when one of the states arises from a configuration not split by the nuclear moments, the result then being similar t o the example.

1/2

FIG.3. Hyperfine structure transitions.

2.4.1.1.4. EVALUATION OF MOMENTS. The nuclear moments which have been determined from the study of hfs are the mechanical or spin moment I , the magnetic dipole moment p, and the electric quadrupole moment Q. As mentioned above, F , the total angular momentum, can assume J - 2 * - ( I - J ) , (or if J is the values I J , 1 J - 1, I larger than I , two I 1 levels) and for each value of F there is a hyperfine structure level. I can be determined if the spectrum is sufficiently well resolved by counting the maximum number of hfs components. If this cannot be done, for example because of insufficient resolution, the interval rule, intensities, or Paschen-Back effect may be used. By the use of Eqs. (2.4.1.1.7) and (2.4.1.1.10), the constants A and B can be deduced from the measured structure. As indicated above, A is related t o the nuclear magnetic dipole moment and B t o the electric quadrupole moment, and these latter could be determined exactly if the values of the mean magnitude field H ( 0 ) and the mean electric gradient in the J direction in the neighborhood of the nucleus were known. Goudsmitg has deduced formulae for A in the single electron case in terms of the nuclear g-factor gr, where pr = (e/2MP)hIgr, e is the electron charge, h is the Planck-Dirac constant, M , is the proton mass, c is the velocity of light, and I is the nuclear spin; for electrons with I > 0:

+

+ +

A. =

+

MPhcR,a2Z3 mn3(Z

9

-

+ + ) j ( j + 1) gz

S. Goudsmit, Phys. Rev. 48, 636 (1933).

(2.4.1.1.12)

50

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

and for s electrons

8 M,hcR,a2ZS A, = 3

mn3

BI

(2.4.1.1.13)

+,

Here R, is the Rydberg, a the fine structure constant, 1 and j the orbital and total electronic momentum quantum numbers respectively. For alkali-like and alkaline earth-like spectra similar formulas exist, semiempirical in nature, and indeed these calculations can be extrapolated to complex spectra with good accuracy.I Casimirs has obtained an expression for the B factor involving the quadrupole moment Q : 1 2j B = -@&(2.4.1.1.14) < l / r 3 > Rr(l,j) H. 2j+ 2 Unfortunately, the factor < l / r > , the average reciprocal of the magnitude of the electron radius vector, appears in this equation, but. this can be eliminated by consideration of the A factor in the derivation of which < l / r > also appears. The following relation7 can then be obtained:

Q

+

B PB' 41(1 1) P I m 1 F, A e2 j(2j - 1) pn M, I R,

= EOPO- -

(2.4.1.1.15)

where eo and P O are the electric and magnetic fundamental constants, 8.85 X 10-l2 farads per meter and 41 X lo-' henrys per meter respectively, p n is the nuclear magneton, pB the Bohr magneton, p r the nuclear magnetic moment and P, and R, are relativity corrections tabulated by Kopfermann. 'I 2.4.1.1.5. LIGHTSOURCES. The hfs splitting is smaller than the ordinary fine structure of atomic spectra by about the ratio of the nuclear magneton p n t o the electron magneton, or roughly %&. Thus, the evaluation of nuclear properties by optical spectroscopic methods ordinarily involves the measurement of very small energy differences, often of the order of thousandths of a wave number unit, To measure energy differences of the order of a few thousandths of a wave number unit, only a few light sources can be used. The principal causes of broadening that are encountered are: Doppler effect, Zeeman effect, and Stark broadening. Doppler broadening resulting from random thermal motion is given by:

where r D is the half-intensity breadth of the Doppler-broadened line, T is the absolute temperature of the source, and M the molecular weight

2.4.

DETERMINATION

OF SPIN, PARITY, AND NUCLEAR MOMENTS

61

of the radiating atom. From this we see that for narrow lines a cooled source is necessary; for light elements cooling by liquid air is commonly used. An atomic beam, where the effective temperature may be very small indeed, has been used t o obtain some of the narrowest spectral lines yet attained. The Zeeman effect is proportional t o the magnetic field strength, and in most sources is not a troublesome source of broadening. The Stark effect is of two types. A relatively large effect proportional t o the electric field occurs in hydrogen, and hydrogen-like atoms. In other cases, a much smaller quadratic effect obtains, proportional to the square of the field. Because of random intermolecular and interatomic electric fields, the linear Stark effect can contribute t o the broadening of spectral lines, the Stark broadening, r8,being approximately proportional to the density. For hydrogen and hydrogen-like spectra subject to the linear Stark effect, rdis especially large and the spectral lines are greatly broadened, especially at high pressures or if large externally applied electric fields are present. Because of the factors discussed above, we see that sources for high resolution must exhibit low temperature and low density, and should have as small magnetic and electric fields as possible. Other requirements, in some cases, are at least moderate intensity and sometimes high excitation. The accompanying tabulation summarizes the most commonly used sources with their advantages. ~~~~~~~~~

Type of source Arc Spark Vacuum spark Hollow cathode

~

Excitation energy Medium High Very high Medium

Electrodeless Low (radio frequency) Atomic beam in Medium emission Atomic beam in Resonance absorption lines only

Doppler temperature High High High May easily be cooled May be cooled May be very low May be lowest

Brightness

Pressure

Fields

High High High Medium

High High Low Low

Medium High High Very low

Medium

Very low

Low

Low

Lowest

Very low

Low

Lowest

Very low

From this tabulation we see that three sources appear to be of greatest use for the study of hfs: the hollow cathode, the electrodeless discharge tube, and the atomic beam. The hollow cathode (see Fig. 4), first used by Paschen, has been brought to a high state of development by many workers, especially

52

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Schiiler, Arroe, and Mack. It combines simplicity, ease of cooling, and very low electric fields with relatively low pressures, & to 10 torr. The electrodeless tube, excited by radio frequency, has the great advantage t ha t no electrodes are necessary, so that the tube can simply be of glass or quartz. It can also be cooled, but not quite so easily, and has the complication that suitable radio-frequency excitation must be provided. It also operates at low pressure, in the neighborhood of torr. An example is the well-known Meggers lamp, containing Hg198.

FIG.4. Hollow cathode discharge tube.

FIG.5. Atomic beam.

For ultimate resolution, where the line breadth must be reduced as far as possible, the atomic beam (Fig. 5 ) is used. Here the effective temperature is dependent only on the component of the atomic velocity in the line of observation. If the average velocity of the atoms emerging from the oven at temperature To is Do, where mijo2/2 = kTo, then the component V8 in the line of sight is 2fio tan 8, so that T,, the effective temperature, is given by T , = 4To tan2 0. For a well-collimated beam, 0 may be as small Thus even for a n oven a t 1000"K, the as lo, so that tan2 0 = 3 X effective temperature in the beam can well be reduced t o the order of l0K! The atomic beam is, however, much more complicated than the other light sources mentioned, requiring an oven and some means, for example an electron gun, for exciting the spectrum (in the case of the

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

53

emission beam) : it is usually not especially bright. A discussion of atomic beams has been given by Meissner.10 2.4.1.1.6. SPECTROGRAPHIC APPARATUS.* As noted above, the problem of determining nuclear properties by measuring the hyperfine structure of atomic spectra entails high resolving power. Of the many possible high-resolution spectroscopic instruments, only a few can be selected for discussion here and we will confine our remarks to the two most commonly used-the diffraction grating and the Fabry-Perot interferometer. As developed by Rowland in the nineteenth century and especially by Harrison in recent years, the diffraction grating (plane and concave) is a practical instrument of high resolution, suitable from the vacuum ultraviolet t o the far infrared. The theoretical resolving power R of the grating is given by the simple relation : (2.4.1.1.17) R = A/Ah = nN where n is the order number of the spectrum, and N is the total number of grooves in the grating. With modern ruling techniques it has been possible t o produce highly efficient gratings 25 cm wide with upwards of 15,000 grooves per centimeter, or more than 375,000 grooves. This affords a resolving power of over a million in the third order; such diffraction gratings can be very useful in the measurement of hfs, especially in the ultraviolet where high orders can be employed. The Fabry-Perot interferometer, devised toward the close of the nineteenth century, has emerged during the last decade with the advent of mnltilayer coatings and the photomultiplier as a n instrument t o be used for highest resolution. The resolving power of the Fabry-Perot is virtually unlimited, since it is just proportional to the separation bet ween the plates. Separations of more than a meter have been achieved, corresponding t o resolution widths smaller than any presently-available spectral lines. The Fabry-Perot, operating as it does at extremely high order, ordinarily requires a predispersion monochromator to eliminate unwantrd orders. Despite this complicating factor, the use of the Fabry-Perot, both photographically and photoelectrically, has become very widespread. This interferometer, as is well known, consists of a pair of partially reflecting mirrors, accurately flat and parallel (see Fig. 6). The light passed by the Fabry-Perot is a series of ring-shaped fringes a t infinity, the intensity of which, for perfect plates, is given by the Airy forrnula:

I * See also lo

=

T,,,[l

+ F Z sin2 6/2]-'

Vol. I , Part 7.

K. W. Meissner, Revs. Modern Phys. 14, 68 (1942).

X IO

(2.4.1.1.18)

54

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

where T,,, is the maximum transmittance, T,,, = T 2 / ( 1- R)2, where T and R are the transmittance and reflectance of the partially reflecting mirrors; F is proportional to the sharpness of the fringes and is given F = 2 di?/(l- R ) ; 6 is the phase difference of adjacent rays, and 10is the incident intensity. When the absorption of the partially-reflecting films is very low, T,,, is very nearly unity, and as Jacquinot” pointed out, the Fabry-Perot is, under these conditions, the most luminous of spectrometers ; its transmittance is highest for a given spectral range. Multilayer films achieve extremely high reflection, upwards of 95 %, with very low absorption, so that the sharpness of the fringes is limited not by the reflectivity through the function F, but by the available flatness of the plates.

Fro. 6. The Fabry-Perot interferometer.

The traditional method of using the Fabry-Perot is photographic and fully described elsewhere.10p12The photographic method has a number of disadvantages, however-among them nonlinearity in both wavelength and intensity, the first being quadratic and the latter logarithmic. A photoelectric method, developed by Meissner,I3 Jacquinot, and others, eliminates the two difficulties mentioned, and in addition has a considerable advantage in sensitivity. Here a diaphragm is placed at the center of the fringe pattern, and the light which passes through falls on a photomultiplier or other suitable detector. It can be shown that 6 = (%/X)2pt cos 8, so that I can be made to vary by changing 1/X, p , t, or cos 8. In the photoelectric method p is usually changed by varying the gas pressure between the interferometer plates, although sometimes t and 8 are varied. As 6 is changed, by whatever method, the spectrum is spread out repeatedly, and a monochromatic line appears as in Fig. 7. The condition P. Jacquinot, J . Opt. Sot. Am. 44, 761 (1954). S. Tolansky, “High Resolution Spectroscopy.” Pitman, New York, 1947. 18 K. W. Meissner, J . Opt. SOC.Am. 31, 405 (1941); ibid. 32, 185 (1942).

11

12

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTG

55

for the maxima is evidently 6 = 2 m , whence

nX = 2pt cos

e

(2.4.1.1.19)

often quoted as the fundamental formula for the Fabry-Perot. The halfintensity breadth, A61/z, for the perfect (exactly plane plates of infinite extent) interferometer is given to a first approximation by Meissner as 2F/r or 0.64F. Figure 8, which shows the logarithm of the transmittance TO,where To = Z/Zo [see Eq. (2.4.1.1.18)lof the interferometer plotted against the logarithm of the phase angle 6, indicates the strong dependance of the half-intensity width on the reflectance of the mirrors. In practical interferometers, a flatness of about & wavelength (green) can be expected, which would allow a reflectance of about 97% to be used. A 63

I I

2

6-

I

'

-I

FIG.7. Schematic appearance of a monochromatic line in a photoelectric Fabry-Perot.

higher R than this would merely lead to loss of light; for poorer plates, correspondingly lower reflectances must be used. As can be seen from Eq. (2.4.1.1.19), the wavelength free spectral range, @A, between successive peaks (n changing by l ) , is just X2/2pt, or, in terms of wave number, 6iv = l / 2 ~ t . In general, the predispersion instrument must have a wavelength band pass less than or equal to 26i to prevent the confusion of orders alluded to earlier. 2.4.1.1.7. RESULTSFROM SPECTROSCOPIC OBSERVATIONS. Of the many possibilities, two examples are given of the phenomena and methods described above. The first is a measurement of the spin of U 2 3 5 by Blaise et al.14Figure 9 shows two orders of a tracing by a photoelectric FabryPerot interferometer of a uranium hollow-cathode discharge cooled by is separated from the liquid nitrogen. The very strong line due to UZ3* U236and U233structure by the isotope shift. The U236components are not l4

J. Blaise, S. Gerstenkorn, and M. Louvegnies, J . phys. radium 18, 318 (1957).

8

FIG.8. The Airy formula.

-1

235

i- 233-l

FIQ.9. Hyperfine structure of Ua36. 56

235 :-233-1

.

100

50

0

Tb159 I 3/2 FIG.10. Hyperfine structure of

TbI59.

2.

58

DETERMINATION OF FUNDAMENTAL QUANTITIES

completely resolved after the first three, a, b, and c, but by the use of the intensity rules, the spin, I, of 4 was confirmed. The other example,16 Fig. 10, very clearly shows the hfs flag pattern in Tb169.Two orders of a photoelectric Fabry-Perot trace are presented, clearly showing I = $ and exemplifying the excellent resolution and intensity measurement possible with this method. 2.4.1.1.8. SUMMARY. The optical spectroscopic method provided the first knowledge of nuclear moments, and still is the source for much of present-day knowledge. Although superseded in some cases by more accurate methods, such as molecular beams and nuclear resonance, it is still indispensable in many cases where the other methods are not applicable or are applicable only with difficulty. Two examples are: (i) shortlived or rare isotopes where the quantities required for a beam are out of the question; and (ii) elements whose electronic ground state has no magnetic moment such as the rare gases. Several lists of moments have been p u b l i ~ h e d . ~ - ~ ACKNO w LEDGMENTS It is a pleasure for the author to acknowledge the kind help of many in preparing this section, especially of Dr. Jean Blaise, Bellevue, France, who put original photographs of hfs at the author's disposal, and of Professor Julian Ellis Mack of the University of Wisconsin, who read the section and made several valuable comments.

2.4.1.2. The Investigation of Short-lived Radionuclei by Atomic Beam Methods.* 2.4.1.2.1. INTRODUCTION.Atomic and molecular beams pro-

vide one of the oldest methods of experimental physics. Their earliest applications were, as to be expected, to the kinetic theory of gases, because study of the effusion of gas molecules from a source was of fundamental importance in understanding the Maxwellian theory of gases. The technique was soon recognized, however, as especially valuable for the study of molecular, atomic, and, later, nuclear structure. I n a beam, the atomic or molecular system is in effect isolated from any other molecule or atom. Thus, in a study of atomic, molecular, or nuclear properties, all experimental complications arising from the effects of the solid, liquid, or gaseous state can be avoided. If the experiment utilizes a resonance method, the lower limit to the width of the resonance line depends only upon the radiation lifetime and the duration of 0bservations.t lr

J. Blaise and F. Tomkins, private communication (1961).

t For further details of the atomic beam method see also Vol. 4, A, Chapters and 2.2; Vol. 4, B, Sections 7.1.1, 7.2.1, and 9.1.2.

* Section 2.4.1.2 is by John C.

Hubbs and William A. Nierenberg.

1.3

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

59

Advantages of the reaction-free state become apparent from consideration of the numbers involved in a typical experiment with a nonradioactive beam. Such a beam consists of 1O1O or lo1' atoms per second at the detector in an apparatus whose length is of the order of 1 meter. The distribution of velocities is the Maxwellian distribution multiplied by the velocity v. The mean velocity may be 50,000 cm/sec, giving a lineal density of -loB atoms/cm; for a beam 0.01 cm wide and 1 cm high, this is a volume density of -lo8 atoms/cm3. The result is a much smaller collision rate than in a standard gas density of 10lg atoms/cm3. Furthermore, if the cross section for collision of the beam atoms with themselves is spherically symmetric, the probability that products of collisions will remain in the beam is only a little greater than the apparatus transmission. Since this is about or less, the effect of collisions is negligible. Scattering due to residual gas in the apparatus is scarcely significant: the pressure of the residual gas is 5 X lo-' mm of Hg, or less, which corresponds to a mean free path ten times the length of the apparatus. The molecular (or atomic) beam became an important device for modern physical research when 0. Stern and his collaborators used it for studying such diverse phenomena as molecular scattering and space quanti~ation.l-~I n these experiments he studied the deflection of the atomic system in very strong inhomogeneous magnetic f i e l d ~ . ~The J specific conformation of magnet pole tips required to produce this inhomogeneity still goes by the name Stern-Gerlach field. The next major step in the technique, based on an interesting theoretical observation by Breit and Rabi,6 was put into experimental practice by Rabi and his collaborators.~-9 This concerned the strong magnetic interaction between the currents produced by the electron angular momentum of the atom and the nuclear magnetic moment. The nuclear spin is rather strongly coupled to the electronic angular momentum, and the two systems would not be decoupled in a magnetic field of less than 1 L. Dunoyer, Compt. rend. acad. sci. 162, 594 (1911); 167, 1068 (1913); Le radium 8, 142 (1911); 10, 400 (1913). 9 W. Gerlach and 0. Stern, Ann. Physik [4] 74, 673 (1924); [4] 76, 163 (1925).

0. Stern, 2.Physik 7, 249 (1921). R. D. Frisch and 0. Stern, 2.Physik 86, 4 (1933). I. Estermann and 0. Stern, Z. Physik 86, 17 (1933); 86, 132 (1933); I. Estermann, 0. C. Simpson and 0. Stern, Phys. Rev. 62, 555 (1937). 'G. Breit and I. I. Rabi, Phys. Rev. 38, 2082 (1931). I. I. Rabi, Phys. Rev. 49, 324 (1936). *I. I. Rabi and V. W. Cohen, Phys. Rev. 48, 582 (1933); D. R. Hamilton, Phys. Rev. 66, 30 (1939). SS. Millman, I. I. Rabi, and J. R. Zacharias, Phys. Rev. 63, 384 (1938); N. A. Renzetti, Phys. Rev. 67, 753 (1940). 4

60

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

several hundred gauss.* I n a weak field, therefore, the two systems behave as one, with an angular momentum that is the vector sum of the two angular momenta, but with a magnetic moment that, to a very good approximation, is due to the electronic structure alone. Thus the behavior of the system as a whole in a n external field is primarily influenced by the spin of the nucleus, and not by its magnetic moment, except through the hyperfine coupling. This was the basis for the zero-moment method used by Rabi and his collaborators for the measurement of the spins, magnetic moments, and quadrupole moments of the stable isotopes of the alkalies, gallium, and indium. (The Breit-Rabi diagram is later explained in some detail.) The next step in the development of the subject was the discovery and application by Rabi of the nuclear radiofrequency resonance method to molecular beams. This was an advance of many orders of magnitude in the precision of measurement of nuclear moments, and the application of the method to HD and D2led to the discovery and measurement of the deuteron quadrupole moment. lo Immediately afterward, the method was applied to the radiofrequency spectroscopy of alkali atoms by Millman et al.,” and because of its enormous precision, virtually supplanted the zero-moment method. Zacharias measured the spin of K40, the rare radioactive isotope of potassium, by an ingenious application of the method, and thereby demonstrated its applicability to trace amounts of material in the presence of large numbers of background atoms. l2 The resonance method was extended to measurements

* Throughout this paper all magnetic moments are in units of

the Bohr magneton,

PO, about 0.927 X lO-*O erg/gauss. The vector magnetic moment of the electron

system is gJpoJ, where J is the vector angular momentum in units of h, and gJ is known as the gyromagnetic ratio of the system and depends upon the details of the e‘ectronic motion. For a single electron in an s state, we have gJ E - 2, and, in general, g J - 1. For a nucleus, however, the magnetic moment is gIp& where I is the vector nuclear mlm. As a n example, for a single s spin in units of h, and, what is important, gr electron, the magnetic field at the nucleus due t o the spin oi’ the electron is roughly 2p0/r3. If r is taken as ~ 0 . X 5 10-8 cm, this is equivalent to about lo6 gauss. A unit nuclear moment in this field of 106 gauss is, therefore, equivalent to about 5 X erg. When this is divided by Planck’s constant, one gets 108 cps or 100 Mc/ sec, if frequency is used as a unit of energy. This represents a rather strong coupling of the nuclear spin t o the electronic angular momentum, and, since the electron precession frequency in a magnetic field, if uncoupled, is gJpoH/h, or about 1.4 Mc/ sec/gauss, for gr = 1, it would take a field of several hundred gauss to completely decouple the two systems. l o J. M. B. Kellogg, I. I. Rabi, N. F. Ramsey, and J. R. Zacharias, Phys. Rev. 66, 728 (1939); H. G. Koski, T. E. Phipps, N. F. Ramsey, and H. B. Silsbee, Phys. Rev. 87, 395 (1952); N. J. Harrick, R. G. Barnes, P. J. Bray, and N. F . Ramsey, Phys. Rev. 90, 260 (1953). I I S . Millman, P. Kusch, and I. I. Rabi, Phys. Rev.66, 165 (1939). l2 J. R. Zacharias, Phys. Rev. 60, 168 (1941); 61, 270 (1942).

-

-

2.4.

DETERMINATION

OF SPIN,

PARITY,

AND NUCLEAR MOMENTS

61

of quadrupole interaction in nuclei, including what is known as “pure quadrupole resonance.”13 The Kusch-Foley measurements established the anomalous magnetic moment of the electron14 and set the stage for the rapid development of the renormalized-electron-field theory and Lamb’s experiment leading to the discovery of the “Lamb shift.”16 For a broad discussion of procedures and techniques of atomic beams, the reader should refer to several articles that cover all the above applications. l 6 * This discussion is primarily concerned with the applications to the measurement of the spins, magnetic moments, and quadrupole moments of unstable nuclei of short half-lives. These quantities are of primary importance in the study of nuclear structure, but until recently there has been very little done in the field, considering the large number of known isotopes. As a result, although much has been known about the nuclei t ha t lie along the “stabie line” of the Seer&chart and longlived isotopes that may be produced in abundance from these in reactors, very little was known about those isotopes that are neutron-deficient by two, three, four, or more neutrons. The primary reason is that these are cyclotron-produced isotopes and, in general, are not available in quantities greater than about 1013 atoms. By a n amalgamation of the techniques of nuclear physics with atomic beams, measurements have been successfully carried out on 1 O l o atoms and on species whose halflives are as short as 10 minutes. Within the last four years, the nuclear spins of about one hundred isotopes have been determined by this extension of the method of atomic beams. The half-lives of these isotopes have ranged from 10 minutes to 24,000 years. It should be emphasized that these measurements have been made with no loss of the great resolution available and thus have a very high reliability compared with other methods. The usefulness of the results can be appreciated from the fact that nuclear spins are now known for ten isotopes of Cs, ranging from a magic number of neutrons for 65Csi23’ to ten less than the magic number of neutrons for 5sCs:227. I n addition, long-lived isomeric states have been investigated. Much useful information has been obtained on Au and on Ag isotopes. Because these atoms are in an electronic state with J = $, it is not feasible to measure the quadrupole moments of these nuclei.

* See also the Vol. 4 references cited on page

58. W. A. Nierenherg and N. F. Ramsey, Phys. Rev. 72, 1075 (1947). 1 4 P. Kusch and H. H. Foley, Phys. Rev. 74, 250 (1948); 72, 1256 (1947). 16 W. E. Lamb, Jr., R. C. Retherford, S. Treibwasser, and E. S. Dayhoff, Phys. Rev. 73, 241 (1947); 79, 549 (1950); 81,222 (1951); 86, 259 (1952); 86, 1014 (1952); 89, 98 (1953); 89, 106 (1953). 16 J. G. King and J. R. Zacharias, Advances in Electronics 8 , l (1956); N. F. Ramsey, “Molecular Beams.” Oxford Univ. Press, London and New York, 1956. 13

62

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Instead, the hyperfine-structure anomalies for these nuclei are observed. This term is also nuclear-shape-dependent. The advantage to be gained is that measurements can be made on a J = $ state, and a quadrupole (as well as an octupole) term can be measured. If this is possible, it will result in the knowledge of the quadrupole moments for eight or nine isotopes of the same element, and should be extremely useful in the interpretation of the unified nuclear model and similar theories. 2.4.1.2.2. GENERAL PRINCIPLE.* The fundamental method for the spin and magnetic moment measurements is based on space quantization. Because of its spin 1, the nucleus alone has a spatial degeneracy of 21 1. Because of the electronic angular momentum J , the electronic system has a degeneracy 2J 1 . In the absence of any external fields or interactions between the electrons and the nucleus, the total degeneracy is (21 1)(2J 1 ) . Because of the magnetic interaction between the nucleus and the electrons, the degeneracy between levels of the total angular momentum, F = I J, is removed. There are either 21 1 or 2J 1 of these hyperfine levels, whichever number is the smaller. F takes on the values I J, I J - 1 , . . . \I - JI,and each of these levels is 2F 1 degenerate. On the application of a weak magnetic field H , this degeneracy is removed, and the energy dependence of the levels is a constant plus a term proportional to H . The difference between any two energy levels is a possible transition frequency under the influence of an applied radiofrequency magnetic field, and this transition is observed, provided that it is allowed by the selection rules, and provided that the apparatus is designed to detect the particular transition involved. There are many different types of transitions, and a t least three different ways of designing the atomic beam magnets to observe them. As an example, the class of transitions that involve AF = _ + I is particularly suitable for precision measurements of hyperfine intervals. If no previous value of this separation is available, however, a tedious search is involved because of the high resolution of an atomic beam apparatus. A procedure that operates in stages and requires only a minimum amount of isotopes has been devised for determination of the various nuclear constants of interest. The quantities to be determined are, in order, the spin, the magnetic dipole term in the hyperfine interaction, the electric quadrupole term, the sign of each, and, if possible, the magnetic octupole term and the hyperfine-structure anomaly. From these terms, the values of the nuclear magnetic moment and electric quadrupole moment can be determined by ratios. The sequence as described requires monotonically increasing precision. As an example, the alkalies have a 2S1,2ground state; therefore, the quadrupole and octupole terms vanish identically, and

+

+

+

+

+

+

+

* See also Vol. 3, Part 6.

+

+

+

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

63

there is no hope of measuring these moments in this atomic state. Since the hyperfine-structure anomaly may be as high as several per cent, there is no point in measuring the dipole interaction with extreme accuracy for the purpose of determining the nuclear magnetic moment b y comparison with a known isotope. However, in order to determine the sign of the interaction] it is generally necessary to measure the hyperfine-structure constant to a greater accuracy to check the consistency of the assignment. As a n example of some aspects of this technique, we consider C S ' ~ ~ . The spin of this isotope is 2. Figure 1 is the plot of the energy levels versus the magnetic field. The energy is in units of a, the hyperfinestructure constant, and the abscissa is the dimensionless quantity,

+

X = (-gJ gr)poHha(I $1 +

(9.7

< 0 for electrons)

(2.4.1.2.1)

The Hamiltonian representing the interaction between nuclear spin and electronic angular momentum, and the interaction with the external applied field, is X

=

a1

- J - gJpoJzH - ____ grpoIpH h h ~

(2.4.1.2.2)

where X and a are in frequency units. The Land6 factor g.7 is known very accurately from atomic beam experiments on the stable Cs isotopes and is very nearly equal to -2; g I is small, mrnlm; a is approximately several kilomegacycles per second. Initially, in the weak field or Zeeman region, the levels vary linearly with H . The corresponding frequency differences between adjacent upper levels, F = 8, AF = 0, and AM = f1, are equal to one another and to approximately Q poH/h, or about 0.5 Mc/sec/gauss. The field is too weak to decouple the nucleus from the electrons, but it does cause a precession of the entire system due to the torque on the total dipole, which is why all the upper frequencies are the same for low field. For a free electron ( I = 0), this frequency is gJpoH/h or 2.8 Mc/sec/ gauss. The nucleus adds inertia to the system because of its angular momentum, but makes a negligible contribution to the torque; therefore the precession frequency is reduced. The exact expression for 2S1/2states to first order in the field for the upper levels is y o = - -

g.rpoH/h 21 1

+

2IgrpoH/h. 21 1

+

(2.4.1.2.3)

The last term is negligible for initial measurements in the determination of the spin. If a resonance can be observed a t low enough values of H , the spin is determined by the factor (21 l)-l. The advantage is the discreteness of the search. The resonance of the carrier or known isotope

+

64

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

2 .O

M

1.0

F.5

2

w -

AW

F

=% -1.c

-2.0

I

I

I

.

1.0

2.0

30 3.0

X

ENERGY LEVELS

J . 12

1.2,

FIQ.1. The Breit-Rabi diagram for I

=

2.

is used to calibrate the field, and the position of possible resonances for different spins can, in general, be discretely predicted and compared. Once a spin resonance is located and verified, an estimate can easily be made of the hyperfine interval Av, the difference in energy between the upper and lower F levels: A v = [(2I 1)/2]a. This is done by following the resonance to a field that will introduce a small deviation from the linearity expressed by Eq. (2.4.1.2.3).This term is, in general, quadratic, and is due to the incipient deeoupling of I and J by the external field,

+

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

65

and is 2Iv02 Av

(2.4.1.2.4)

V ~ V O + - -

to second order in the field. The shift 21v02/Av gives a rough estimate of Av, and this shift can be increased by increasing the frequency until Av is determined to any reasonable desired accuracy. Since this procedure is capable of determining Av to 0.1% or better if necessary, one must use the exact solutions for the Hamiltonian, including the terms in g r ; gr itself is estimated from Av by comparison with a known isotope and the relation” Avi/Avz = (211 l ) g r 1 / ( 2 I 2 l)grE. (2.4.1.2.5)

+

+

For alkalies and other 2S1/2 states, this formula is good to about 1%, which is more than enough for the small correction it affords. The deviation of this relation from equality is a measure of the finite distribution of the nuclear magnetism and is known as the hyperfine anomaly. However, the sign of gr is not determined, and the usual method is to check the constancy of the calculated A v versus H for assumed gr > 0 or gr < 0. If the resolution of the apparatus is sufficiently good, there is enough discrimination to tell the sign. So far, nothing has been said about the apparatus needed to observe this resonance. A particular arrangement of magnets in an atomic beam apparatus is used that results in what is inelegantly called a “flop-in)’ trajectory rather than the original “flop-out” design. Reference to Fig. 1 shows the rather interesting phenomenon that the particular level corresponding to the most negative value of M = - I - J (e.g., the level -$ in the diagram) varies exactly linearly with H and “crosses the diagram” in the Paschen-Back region. to take its place with the levels WLJ = The level immediately above, M = - I - J 1 (e.g., in the diagroup. The slope of gram) eventually winds up with the WLJ = these lines at a particular magnetic field is a measure of the force on an atom in that state in an inhomogeneous magnetic field. The two levels, M = - I - J and M = - I - J 1, therefore have equal but opposite forces exerted on them in the same inhomogeneous field, providing only that the field is sufficient to decouple the angular momenta completely as in the Paschen-Back region. Figure 2 is a schematic of an actual apparatus. The atoms leave the source and pass between the pole tips of two successive magnets that supply the field necessary to cause the decoupling and the inhomogeneity to deflect the atoms. These magnets are called the “A” and “B” magnets respectively, and are so arranged that the fields of each and the field gradients of each are in the same

-+

+

17E.Fermi,

Z . Physik 60, 320 (1930).

+ +&

-+

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

I

FIG.2. A cross section of an atomic beam apparatus.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

67

direction. The apparatus then acts like a Stern-Gerlach system in the first instance. The atoms are deflected either to the left or to the right, depending upon the sign of mJ, and there is no signal a t the detector. Between the two deflecting magnets there is a uniform field, the “C” field, which is normally at just a few gauss for spin determinations. The resonance takes place in this field via a small perturbing rf field induced by a pair of wires, generally called a “hairpin,” running parallel to and near the beam. If the resonance condition is met, transitions are caused between the two states,

F=I+J -I - J

MF =

and M F =

-I - J

+ 1.

The result is that the moments of the atoms initially in these states may be reversed in sign in the B field with respect to the A field, and these atoms will go in a sigmoid path and be refocused onto the detector yielding a signal. The result is a positive signal compared to no signal, which gives far better statistics than the flop-out method in which the indication is a reduction in beam intensity. This is an essential feature in experiments involving trace amounts of material. I t is important to note that the matching and stability of the A and B fields can be quite crude without too much effect on the efficiency of the apparatus. The C field must be stable within a fraction of the line width, and since this may be as low as 20 kc/sec, this is often a stringent requirement. Since only two 1) are involved, the best fractional inlevels out of a total of 2 ( 2 1 tensity obtainable is ( 2 1 l ) - I , and because of various losses, about one-quarter of this is actually realized. The final identification of the radioisotope is usually made certain by a determination of the half-life of a collected resonance sample. Figure 3 shows the results of the decay of a Cs beam produced by a-particles on Xe. The undeflected-beam decay curve is composite, with a t least three components. Three of the spin buttons collected showed counts that were above background and each decayed with a half-life ~ ~ and , CsI3* characteristic of a known species. Thus the spins of C S ~CsI3l, were measured. Cs129 and Cs131 had been measured earlier after having been produced by different methods. The procedure for atomic and nuclear levels where both I and J are greater than one-half is more complicated because of the existence of the quadrupole term. However the spirit of the method is the same, but two different transitions must be observed for sufficient accuracy, this is

+

+

68

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

always possible because there are always two or more “flop-in” transitions in this case. Further accuracy gives a direct measurement of gr. 2.4.1.2.3. CONSIDERATIONS IN APPARATUS DESIGN. The design of an atomic beam apparatus for the study of radioactive species should clearly involve the optimization of the transmission which is the total number of atoms from the source that reach the detector. This is a different condition from the many that are usually employed, so a brief description is required here. The apparatus designed and constructed a t

-

0

5

I5

10

20

25

DAYS

FIQ.3. The decay of three resonances corresponding to three different Cs isotopes.

Berkeley will be discussed as an example. I t is more fully described in the thesis of R. A. Sunderland, and the calculations that follow are abstracted from more complete accounts in his thesis. The apparatus being considered is to be designed so that the largest possible fraction of the manufactured atoms is available a t the detector. The C field will, therefore, be made as short as possible, thus sacrificing ultimate precision, and dead spaces between the source, the magnets, and the detector will be held to a minimum. In order to obtain a first idea, it will be assumed that the C-field length and the dead spaces are negligible. The deflection

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

69

of an atom from a straight-line path is then (2.4.1.2.6)

where 6y is the deflection perpendicular to the axis of the apparatus, p is the effective moment of the atom, a H / a y is the perpendicular magnetic field gradient, 1 is the length of the deflecting field, M is the mass of the atom, and v is the velocity of the atom. For l 2 = 1000 cm2,

aH/ay

=

6000 gauss/cm

T = 1000"K, and p = 10Vo erg/gauss, the deflection is 0.3 cm. In what follows, numerical factors of the order of unity will be discarded. If T is the source temperature, Mu2 = ICT for a n average atom. The field inhomogeneities are developed by pole tips which are cylindrical arcs. If the radius of one of the pole tips is a, we can assume ( l / H ) ( a H / a Y ) = U/a> therefore

6y = PHp/kaT.

(2.4.1.2.7)

If the exit slit is of height h and width w, we may assume, dimensionally, that the source slit and width are proportional to h and w (and, in fact, h, w, and 6y must be proportional to a in a n analysis of this type). The total number of atoms that reach the detector is then (remembering that 1 a h a w 0~ 6y a a) (2.4.1.2.8)

Here p is the number of atoms per cm3 in the vapor phase in the oven. For an experiment with an unlimited supply of material, the maximum permissible p 00 l / w by the Knudsen condition. This is the condition that the slit width should not exceed the mean free path in the oven, so that the condition for pure molecular effusion can be maintained. I n this case, N varies as the length of the apparatus. The only limit to the increase in 1 is the scattering of the beam atoms by the residual gas and the expense. In practice, it has seldom been necessary to consider an apparatus of length greater than 2 meters, and most of this length is due to the C field needed for high precision. In the situation in which trace amounts of material are involved, the total amount of material is fixed and all of it is needed. This is the case a t hand. The proper condition is that phw equal a constant. Then we find

N

a

Hp/T.

(2.4.1.2.9)

70

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

This means that the efficiency of the apparatus is independent of its length. The apparatus can be made quite short until the dead spaces and the finite C field introduce other limiting factors. A convenient over-all length is about 75 cm. Of course, the half-life of the material also enters these considerations. As the apparatus is shortened, the rate of efflux is decreased and the carrier needed must be decreased. This can reach the point a t which the amount of carrier is too small to be successfully handled. The foregoing analysis was carried through on the assumption that the ratios of ail lengths were held constant. There are other considerations, however. Provision must be made for a collimating slit and for a stop wire. The latter is an obstacle that just blocks the direct beam from the detector. It prevents the fast, relatively undeflected tail of the Maxwellian distribution from reaching the detector. Furthermore, the ratio of the A to B magnet lengths need not be unity. I n fact, if the A magnet has all its dimensions reduced, it remains equivalent in focusing power and transmission, as the above analysis showed (including the necessary reduction in source dimensions). If carried to the limit, this means a factor of two in length for the total apparatus and, therefore, a factor of four in transmission. In practice, a factor of three can be obtained, A factor of this size is not trivial when viewed against cyclotron time and counting time. One pays for this improvement with a higher source pressure. For the alkalies, this represents a reasonable temperature rise. For low-vapor-pressure substances, such as U, Pu, Am, Np, Th, etc., the necessary rise in oven temperature is barely tolerable, and the improvement in intensity probably is not worth it. Under these conditions, the stop wire allows only about one-half the atoms through that undergo a change of effective moment of 1 Bohr magneton. The transmission of such an apparatus is about when a cosine distribution is assumed. Channeling the source never seems to produce the improvements expected from the simple theory and, further, is less effective for material of short half-life. The exit slit is 0.1 by 0.5 in., and the source is 0.002 by 0.080 in. The one remaining question of importance is the length and value of the rf field. A typical beam velocity is 50,000 cm/sec, and if the hairpin is 1 cm in length, the transit time is 1/50,000 sec, given a resonance band with 50,000 cps. At a field of 10 gauss, a nucleus of spin I = 0 in an alkali atom resonates at 28 Mc/sec, and this bandwidth will just resolve spins in the neighborhood of I = 50. Such resolution is far more than necessary, but usually the C field is not sufficiently uniform to attain ideal resolution, and resolutions of 200 kc/sec are more common. This is still more than adequate. The strength of the rf field is determined as follows.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

71

The system precession a t resonance about the C-field direction can be viewed from a rotating coordinate system that has the same velocity and sense of rotation. In the rotating system, the angular-momentum vector appears stationary, and the applied magnetic field is taken as zero. Such is the essence of Larmor’s theorem. The rf field, which appeared as a rotating vector perpendicular to the C field in the laboratory system, is now stationary if the oscillator is tuned to resonance. In this coordinate system, the spin will precess about the perturbing field H’ with the Larmor frequency corresponding to H’. If the strength is just right, it will make just one-half revolution in the time the atom spends in the rf field. Denoting this time by At, the condition for a precession of ?r is (2.4.1.2.10)

For spin 0, At = 1/50,000 sec-I, this yields a field of 20 mgauss, a field that is very easily obtained up to quite high frequencies. For a monoenergetic beam, the resonance will decrease if this field is exceeded. I n practice, the Maxwellian velocity distribution will smooth out the oscillations of the resonance versus perturbing field. 2.4.1.2.4. P R O B L E M S INVOLVED I N P A R T I C L E DETECTION OF ATOMIC BEAMS.2.4.1.2.4.1.General Considerations. Only experimental methods that take advantage of particle detection can now give the high sensitivities required for the investigation of short-lived radioisotopes. Two essentially complementary techniques now being used meet this requirement; one is the detection of individual beam atoms by ionization, massspectrometric analysis, and subsequent detection with electron multipliers; the other is radioactive collection and detection. For the first scheme, the following limiting situation is typical: at least lo3 positive ions must be observed a t the electron multiplier to obtain a statistically significant sample of the beam; mass-spectrometer transmission is about 10%; and the fraction of beam atoms that is ionized ranges from a lower limit of (with the universal electron-bombardment detector first introduced by Wessel and Lewls) to unity for special elements, such as the alkalis and Group IIIb series, in which Langmuir-Taylor surface ionixationlg is part,icularly applicable. For radioactive collection and detection this situation is typical; a beam sample is required to have a decay rate in excess of 1 cpm and is observed in detectors with a n efficiency of 25%. Under these simple assumptions the crossover point, where one method becomes comparable to the other in sensitivity, occurs G. Wessel and H. Lew, Phys. Rev. 92, 641 (1953). I. Langmuir and K. H. Kingdon, Proc. Roy. SOC.A107, 61 (1925); J. B. Taylor, 2.Physik 67, 242 (1929); Phgs. Rev. 86, 375 (1930). ‘8

l9

72

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

for materials with half-lives between one week and 100 years, depending upon the ionization efficiency attainable. There is, however, another consideration, difficult to evaluate, but nonetheless important. Even when there is no common stable isotope having the same mass as a nuclide to be investigated by the ion method, there is inevitably some spillover of carrier, beam contamination, and apparatus background, especially oils, into the mass channel of interest. The effect is to make the ion method somewhat less sensitive than would first appear. The present cross-over, therefore, occurs somewhere in the neighborhood of 1- to 100-year activities. An excellent r6sum6 of the ion system is contained in an article by King and Zacharias16 and the publication of Lew,20therefore, this discussion is generally limited to research on materials with lives of one year or less. The central procedures of the radioactive method are the collection of beam material for a suitable integration time and the detection of the sample decay by a n appropriate counter. Let us first consider the conditions under which the material available may be most effectively utilized. The central fact is that if reliable counters are available, the relative uncertainty t in a beam sampling is (I, t =

+

c)l'2 ct"2

+

(2.4.1.2.11)

where c is the sample decay rate, b G is the observed decay rate, and t the counting time. It is assumed that the background b is well known. We dispense with the b term by noting that in most instances an efficient counting system with background of about 1 cpm or less can be found (more on this point later), and that 1 cpm is also about the lower limit of decay rate set by the time required t o count down the results of a run and by the investigator's patience. Thus any discussion of radioactive detection rather naturally separates into two categories; into the first class fall all materials with half-lives of several days or more, and into the second go isotopes with shorter lifetimes. For investigations in the first class, the rate of progression of research is strictly proportional to the rate a t which the beam exposures can be counted, since the exposures themselves can be taken a t essentially any rate, and since the investigator will have sufficient time to analyze the data and decide on a n optimum search procedure. The rate a t which exposures can be counted is, from Eq. (2.4.1.2.11), ne2c2 (2.4.1.2.12) d N / d t = -= nt2c c+b za

H. Lew, Phys. Rev. 74, 1550 (1948); 76, 1086 (1949).

2.4.

DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS

73

where n is the number of counters used, c is a prescribed relative uncertainty in the beam sampling, c is the net counting rate of beam samples, and b the counter background. The total activity available for the experiment will invariably be limited by availability, expense, or health precautions, and the counting rate c will be some appropriate apparatus constant times the total activity divided by the number N of exposures required to execute the experiment. Thus perhaps a more illustrative form of Eq. (2.4.1.2.12) is 1/r = 2Qne2/N2 (2.4.1.2.13) where r is the counting time required for the entire research project, and Q is proportional to the total activity. Note that there is a large premium connected with optimum search procedures, which reduce N, and with increase of the allowable relative uncertainty in the individual resonance points. In addition, the counter effect occurs as the first power of the number of counters. We now consider investigations falling into the second class, where one deals with materials in which the lifetime itself is a major factor in the control of exposure and counting procedures. Here the progress of the research is taken to be proportional to the number of resonance points that can be obtained per run, although other considerations, such as the fact that no extensive data analysis is possible during the course of the run, also play an important role. The maximum counting time per resonance point is T n / N , where N is now the number of resonance points taken during the run, T is a characteristic time which is essentially the half-life, and n is the number of counters. But, as before, the individual counting rates are & / N ; so for this case, assuming b 10-yr) species. Perhaps the most significant contribution to knowledge of nuclear structure has come from the enumeration of ground-state spins and magnetic moments for a large number of isotopes of the same element; most theoretical treatments of the nuclear system are considerably simplified by the fact that the “core” remains the same across such a series. I n addition to this general feature of the research, results are obtained having particular and specific interest, such a s the spin of Cs127 and Cs129, which showed conclusively the breakdown of the shell model in this region,2‘ and the spin 0 of Gass that, besides being interesting in its own right, has had some import in parity experiment^.^^

+

2.4.1.3. Microwave Method.* t During the last decade, the microwave method of measuring nuclear spin and moments has offered a n additional tool for determining nuclear properties. The principal feature of this method lies in the observation of the hyperfine structure which may be present in the microwave spectra of atoms or molecules. The existence of a n observable hyperfine interaction is the necessary condition for the derivation of any nuclear information from microwave spectroscopy. 3 4 H. Frauenfelder, A. 0. Hanson, N. Levine, A. Rossi, and G. DePasqualli, Phys. Rev. 107, 910 (1957). t See also Vol. 2, Part 10; Vol. 3, Chapter 3.1; Vol. 4, A, Sections 4.1.2 and 4.2.3.

* Section 2.4.1.3 is by C.

K. Jen.

86

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Since the energy splitting due to nuclear interactions is in general rather small, the high resolving power inherent in microwave and low-frequency techniques proves to be particularly effective in such investigations. There are essentially two areas where the microwave method has been used in the measurement of nuclear properties. One is the microwave spectroscopy in gases, and the other is the paramagnetic resonance in solids. The former, dealing principally with free molecules in electronically nonparamagnetic states, has led to a number of new determinations on nuclear spins and electric quadrupole moments and, to a lesser extent, on nuclear magnetic moments. The latter, dealing principally with paramagnetic ions in solids under the influence of external magnetic fields, has yielded another set of new determinations on nuclear spins and moments. Although the two approaches just mentioned differ somewhat in their detailed experimental execution, they are basically similar in applying microwave techniques to the observation of absorption spectral lines. 2.4.1.3.1. EXPERIMENTAL METHODSI N MICROWAVE SPECTROSCOPY. * 2.4.1.3.1.1. General Considerations for Microwave Spectroscopic Observation. a. General Description. When reduced to its minimum essentials, a spectroscopic system of the absorption type should contain a monochromatic source of radiation with a continuously variable frequency, a material whose absorption of radiation is under investigation and a detector of the radiation following absorption. These conditions are readily met by a microwave system. A monochromatic microwave source for the spectroscopic work is usually supplied by a klystron tube whose frequency can be varied continuously over a range. The wave propagates from the source through an absorption cell containing the material to be studied. The absorption cell may be a section of the waveguide which when containing a gaseous material can be blocked off at the two ends by sheets of material (usually mica) transparent to the microwave, or it may be a microwave resonant cavity which contains a small solid substance for the study of paramagnetic resonance. In any case, the outgoing wave is allowed to reach a detector for which a semiconductor (usually silicon) crystal is commonly used. Thus it is, in principle, possible to observe an absorption spectrum by noting the change of the crystal

* Most of the material presented in Section 2.4.1.3.1 can be found in the general references given below.laJb,lo la C. H. Townes and A. L. Schawlow, “Microwave Spectroscopy.” McGraw-Hill, New York, 1955. lb N. F. Ramsey, “Molecular Beams.” Oxford Univ. Press, London and New York, 1956. lo B. Bleaney and K. W. H. Stevens, Paramagnetic resonance. Repts. Progr. in Phya. 10, 108 (1953).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

87

current with the frequency (or alternatively the magnetic field as in the case of paramagnetic resonance). The absorption of a material is usually described by the imaginary part (x”) of the complex quantity x denoting the susceptibility of the material as

x = XI - i f .

(2.4.1.3.1)

The susceptibility may be due to electric dipoles as in most cases in gaseous microwave spectroscopy, or it may be due to magnetic dipoles as in paramagnetic resonance. I n practice it is seldom necessary to consider the absorption due to both kinds of susceptibility at the same time. b. Absorption in a Gas-Filled Waveguide. It is customary to use the absorption coefficient a instead of XI‘ for the description of the gaseous absorption. The relationship between the two quantities is (2.4.1.3.2)

where X is the free space wavelength and X I ‘ is the susceptibility per unit volume. Since X I ’ is dimensionless, a has the dimension of cm-l. If we let y denote the absorption of an empty waveguide, then we have

pl

=

Poe-(~+Y)~

(2.4.1.3.3)

where 1 is the length of the waveguide absorption cell, PO and PI are, respectively, the powers a t the input and output ends of the cell. Since in general a .. = H l 2 V . For maximum sensitivity, QO should be made twice as large as the “loaded” Q , leading to the result P, = P0/2,where Po is the input power. Using the above relations, we have for the power absorbed by the sample (2.4.1.3.8) The expression for xi’ of a paramagnetic substance is often rather similar to that in Eq. (2.4.1.3.5) and can be generally written as

xi’

= CN*J(u)

(2.4.1.3.9)

where C is a constant of the substance and varies inversely as the absolute temperature, N , is the number of paramagnetic’ions in the sample, and f(v) is the spectral form factor. At the center of the resonance line, f(v) = v o / A v , where Au is the half-width.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

89

d. Noise Power. We shall not give an analysis of the various sources of the noise power except to mention that the noise generated at the crystal is often the largest among them. The crystal noise power can be described by the expression

P (crystal noise)

=

(k T + Y2) ~

Af

(2.4.1.3.10)

where k = Boltzmann’s constant, I’ = absolute temperature, C = constant of a crystal, I = crystal current, f = output frequency, Af = output frequency bandwidth. The first term in Eq. (2.4.1.3.10) denotes the thermal agitation or the “Johnson” noise, and the second term the specific noise of a crystal. Equation (2.4.1.3.10) suggests that, for minimization of the crystal noise, f should be as high and Af as low as possible. The lowest limit of the noise power is set by the thermal term kT At. It is a common practice to describe the over-all noise power as N k T Af, with N designated as a noise figure not only for the crystal, but for the whole system. e. Sensitivity of Detection. The sensitivity of detection in a microwave spectroscopic system can be expressed by the minimum detectable a in gaseous spectroscopy or x:’ in paramagnetic resonance. This can be done by equating the signal voltage to the effective thermal noise voltage. For a gas-filled waveguide, the result is (2.4.1.3.11)

where e = Naperian base, I, = 2/7 (“optimized” length of waveguide). Similarly, for the case of a cavity, the result is (2.4.1.3.12)

Equations (2.4.1.3.11) and (2.4.1.3.12) point to the disadvantage of observing a signal at a very low output frequency, because the noise figure N with reference to Eq. (2.4.1.3.10) would be an enormously large number. Thus a spectrograph operating on the principle of detecting dc or very low frequency signals is apt to be extremely insensitive. The situation is somewhat improved if the microwave frequency is periodically swept (e.g., by a saw-tooth voltage) at a chosen rate across the frequency of a spectral line so that a band-pass receiver at a high frequency can be used. But the method that really gives a major improvement in sensitivity lies in the use of a modulation frequency such that the inverse frequency noise contribution in Eq. (2.4.1.3.10) is greatly reduced and the bandwidth Af can be made very narrow.

90

2.

DETERMINATION OF FUNDAMENTAL QUANTlTIES

2.4.1.3.1.2. Methods of Modulation and Detection. a. Stark Modulation. The method of modulation by employing the Stark effect is most extensively used in microwave spectroscopy of gases.* The modulation frequencies used by various workers are in the range of 6-100 kc. Consider first that the microwave frequency is tuned to an absorption line. If we apply an electric field, the absorption at this frequency can decrease considerably and often to zero because of the displacements in frequency for the Stark components of the original line. In the case of a periodically varying electric field, the fluctuating absorption would give a signal a t the modulation frequency. The variation of the amplitude of this signal with a slowly swept microwave frequency would yield a spectral presentation of the absorption line. The use of a square-wave (so biased that the field is zero half of the time) is especially rewarding in that the resulting spectrum will not only show the zero-field line, but also the Stark components at various positions as determined by the constant field during half of the cycle. However, there are also cases where we rather want to avoid seeing the Stark components when, for instance, we observe the Zeeman splitting of a spectral line. This is exactly our situation in the study of nuclear magnetic moments. Under these circumstances, we can use, for example, a coaxial waveguide for the absorption cell in which the electric field is very inhomogeneous, and hence will cause the Stark components to smear out. Such an arrangement is actually used in the experimental layout shown in Fig. 1. b. Zeeman Modu1ation.t Zeeman or magnetic field modulation is extensively used in paramagnetic resonance experiments. The modulation frequencies used by various workers are in the range of 10-1000 cps. In a few cases the modulation frequency has been raised to 100 kc or higher. A small sinusoidally varying magnetic field is usually placed along the direction of a high steady magnetic field. If the amplitude of the modulation field is much smaller than the spectral line width, then a signal at the modulation frequency would appear after the detection of the microwaves with a magnitude proportional to the derivative of the absorption coefficient (i.e., da/dH). c. Detection of Modulation Signals. Where a modulation scheme is used, only the information carried by the modulation frequency component in the detector output needs to be preserved. This signal is usually fed in succession to a preamplifier, a narrow-band amplifier, and a phase detector. In the phase detector, the signal is mixed with a modulation frequency reference voltage to produce essentially a dc output which can be delivered to an oscilloscope or a recorder. If it happens that there

* See for example, Vol. 3, Section 2.1.4.

t See also Vol. 4, A, Section 4.2.3.3.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

91

is more than one modulation frequency being used for special reasons, then each frequency component can be separately handled through its own channel as in the above description. The spectroscopic sensitivity of a relatively high frequency modulation system has been experimentally found to be considerably higher than that of a nearly dc system. But, even here, the noise figure N in typical setups is far greater than the ideal ( N = 1).Referring to Eq. (2.4.1.3.11), under the conditions of a good waveguide spectrometer, e.g., I,, lo3cm, T 300"K, A! 30 cps, P o 1 mw, the experimental value for amin is

-

-

MICA WINDOW ROTATABLE POLARIZER

1

-

-

/,,\ \ LONGITUDINAL FIELD COIL

TRANSVERSE FIELD ELECTROMAGNET STARK ELECTRODE

I

ROTATABLE lDETECTOR

AMPLIFIER

T147] OSClLLOSCOPE

1OOkC

PHASE

Fro. 1. Schematic diagram of a microwave Zeeman spectrograph.

roughly in the range lo-* to cm-l. The corresponding value for N is of the order of lo3. The application of field modulation method to paramagnetic resonance in solids calls for some special consideration. Here the modulation frequency for the magnetic field is, with a few exceptions, usually restricted below 1 kc because of the eddy current limitations of the cavity used as the absorption cell. I n order that full advantage can be taken for the lower noise figure a t higher frequencies, a superheterodyne scheme can be adopted with increased sensitivity. A local microwave oscillator is set a t a frequency differing by a constant amount, for example, 30 Mc/sec, from the signal microwave frequency. The difference frequency due to the mixing of the local oscillator output and the signal wave after absorption can be handled by an I.F. amplifier, and eventually by a phase-sensitive detector. If a field modulation is used, then the output contains the

92

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

signal at the modulation frequency which can be handled as discussed before. The sensitivity of a paramagnetic resonance spectrometer can be conveniently rated according to the minimum number of unpaired electron spins N , in Eq. (2.4.1.3.9) that the apparatus can detect when the sample is a t a given temperature. By the use of Eq. (2.4.1.3.12),it can be shown that the ultimate sensitivity of a cavity spectrometer is of the order of 1011 spins at room temperature. Practical spectrometers are considered to be good if their capability of detecting spins covers the range 1 0 1 2 to 1013. 2.4.1.3.1.3. High-Resolution Spectroscopic Methods.* High-resolution spectrometers are occasionally needed to resolve very closely spaced lines (hyperfine structure or Zeeman splitting) , or to enable more accurate determination of line frequencies. Increase in resolution is mostly accomplished by seeking ways to reduce the linewidth. When a modulation method is used for the sake of sensitivity, it also broadens the line in the same process. This means that when high resolution is important one must choose a sufficiently low modulation frequency and/or low degree of modulation at the expense of sensitivity. In gaseous spectroscopy, the linewidth can often be reduced by using lower pressures when pressure broadening is the dominating feature. For further reduction of linewidth, methods for decreasing Doppler broadening have been successfully worked out. But the systems used are often rather complex. For paramagnetic resonance in solids, the linewidth can be reduced by decreasing the concentration of the paramagnetic species, or sometimes by lowering the temperature. A rather different form of high-resolution spectroscopy has been introduced by FeherId with his double resonance method. Here, a resonance condition is first established for the electrons in a paramagnetic species. The resonance is actually between the hyperfine levels of two magnetic states of the electron. A second resonance is then introduced between one of the participating hyperfine levels and another hyperfine level of the same electronic state. The second resonance is essentially of the nuclear magnetic type, being in the range of radio frequencies. If the electronic resonance is initially intensity saturated with a marked reduction of signal, then the application of the nuclear resonance can cause a visible change in the electronic resonance. There is thus a natural selection of only such nuclear transitions that can affect a given electronic transition. This method has the advantage of combining the high resolution of a nuclear resonance experiment with the high sensitivity of an

* See also Vol. 4, A, Section 4.2.3.2. Id

G. Feher, C. S. Fuller, and E. A. Gere, Phgx. Rev. 107, 1462 (1957).

2.4.

DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS

93

electron resonance experiment. But, more important for our present subject, the method has been demonstrated to give a direct determination of the nuclear g factor from the separation between two lines in the radiofrequency spectrum. 2.4.1.3.1.4. Measurement of Frequency, Magnetic Field, and Spectral Intensity. 2.4.1.3.1.4.1. Frequency measurement.* Measurement of the signal oscillator frequency is mostly done with reference to an absolute frequency standard like the transmissions of the station WWV. In gaseous spectroscopy, the frequencies a t the centers of all the spectral lines are to be measured. However, for the measurements involving the hyperfine structure or Zeeman spectrum, only frequency differences between lines are essential. This less stringent requirement can often be met by a simpler operation. In paramagnetic resonance, the oscillator frequency is usually held at a fixed value while the resonance condition is fulfilled by the variation of the magnetic field. It is comparatively a simple matter to make an absolute measurement of a constant frequency. The accuracy of frequency determination depends upon the stability of the signal frequency and the spectral linewidth. Under typical conditions, it is possible to determine the line frequency within about one-tenth of the linewidth. Since a typical linewidth in gaseous spectroscopy is of the order of 1 Mc/sec, the accuracy of the line frequency determinations is usually within the range 0.01 to 0.1 Mc/sec. 2.4.1.3.1.4.2.Magnetic jield measurement.t The measurement of magnetic field is mostly done through the use of a proton resonance magnetometer. Under typical conditions, it is usually practicable to measure the field with an accuracy within 0.1 oersted for a field of several thousand oersteds. This accuracy is more than adequate for the Zeeman measurements in gaseous spectroscopy and is quite satisfactory for most of the paramagnetic resonance work. 2.4.1.3.1.4.3.Intensity measurement.$ Unless extreme care is taken, intensity measurement is subject to many types of variability. Fortunately, only relative intensities are of interest for all measurements concerning the hyperfine structure from which nuclear information is derived. The average accuracy of relative intensity determination is of the order of 1%. 2.4.1.3.1.5. Illustrative Microwave Spectrographs and Spectral Presentations. 2.4.1.3.1.5.1. Spectroscopy in gases. As an illustrative example, the schematic diagram of a microwave spectrographs specially designed

* See also Vol. 2, Chapter 9.2.

t See also Vol. 1, Chapter 9.3.

1See Vol. 3, Section 2.1.7.5. 5 The specific description given here refers to

a spectrograph of this type at the Applied Physics Laboratory of The Johns Hopkins University.

94

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

for measuring the Zeeman effect in gaseous spectroscopy is given in Fig. 1. The absorption cell is formed by a cylindrical waveguide 0.62 in. inside diameter and 53 in. long, with mica windows a t the two ends. A center conductor of 0.125 in. outside diameter serves as the Stark electrode, being supported by perforated mica discs in the waveguide. A zero-biased square-wave voltage a t 500 volts maximum and 100 kc is used for the Stark modulation. The absorption cell is placed in the gap of a 4-fOOt long electromagnet, which can supply a transverse magnetic field up to about 8000 oersteds. An auxiliary coil wound around the absorption cell can supply a longitudinal magnetic field up to about

24,019.6 Mc/ssc

FIG.2. Observed hyperfine spectrum of O I @ C W 3for J

= 1 + 2.

1000 oersteds. The longitudinal field is used only for the determination of the sign of the magnetic moment. For the latter operation, a circular polarization of the microwaves is produced by a “polarizer” (mica phase changer) and detected by an “analyzer” (rotable rectangular wave with crystal detector). Using a spectrograph as described in Fig. 1, we can observe the hyperfine spectrum of a molecule in the absence of a magnetic field. Such a spectrum for 016C12S33 is shown in Fig. 2, for the rotational transition J = 1 4 2. A very good fit between the observed pattern and the theoretical predictions can be obtained only when the following assumptions are made: I ( S 3 3 ) = +, en& = -29.1 Mc/sec [see (Eq. 2.4.1.3.13)l. When a magnetic field is applied, each line in the hyperfine spectrum is split in manner characteristic of the magnetic properties of the par-

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

95

ticipating states. The Zeeman pattern2 is the simplest for the transition F = + 4, where the magnetic splitting gives a doublet shown in Fig. 3. The nuclear g factor for S33can be calculated with the known value for the molecular g factor of 0cs3*( -0.026).

+

F=

v2

-bZ

I

t

24,019.6 Mc/sec

H = 3912 oc. -2

-I

0

I

2

Mc/sec

FIG. 3. Zeeman splitting (after Esbach et ~ 1 . 2 ) .

(r components)

for F =

-

8 4 (J

= 1 + 2) of O‘~C”Sa8

CAVITY AND SAMPLE

OSCl LLATOR

1

FIG.4. Schematic diagram of a microwave paramagnetic resonance spectrograph. 2.4.1.3.1.5.2. Paramagnetic resonance. A moderately simple paramagnetic resonance spectrograph is given schematically in Fig. 4. A field modulation of several hundred cycles per second can be used, even with a thick cavity wall. If a thin metallic wall for the cavity is used, 2 J. R. Esbach, R. E. Hillger, and M. W. P, Strandberg, Phys. Rev. 86, 532 (1952).

96

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

then the modulation frequency can be raised to the range of several hundred kilocycles with the attendant advantage of high sensitivity. The source modulation and its related circuitry are used to stabilize the klystron frequency to the sample cavity resonance frequency. This frequency should be kept at least several times higher than the field modulation frequency. There should be a provision, not indicated in the figure, for rotating the sample in the cavity if a single crystal is used. A sample spectrum of paramagnetic resonance is given in Fig. 5 which shows the magnetic hyperfine structure of Ce141 obtained by Kedzie et aLs with an irradiated fission product of U236 in a L a ~ M g ~ ( N 0 , ) ~ ~ - 2 4 H ~ 0 single crystal. * The eight-line spectrum is clearly associated with a

t---

1060 o e r s t e d s

t

3725 oersteds

FIQ.5. Hyperfine structure due to Ce"' in the paramagnetic resonance of Ce3+ ions in LazMgs(N03)~ 2 4 H ~ This 0 . spectrum was observed at 4.2"K and 9380 Mc/sec (after Kedzie et aZ.3).

nuclear spin of 4. The magnetic moment of Ce141is evaluated from the measured hyperfine coupling constant to be about 0.89 nm. BACKGROUND AND RESULTS OF MEASUREMENT 2.4.1.3.2. THEORETICAL FOR SPIN AND NUCLEARMOMENTS.2.4.1.3.2.1. Spins and Nuclear Moments by Microwave Spectroscopy in Gases. Microwave spectroscopy in gases has been largely concerned with the study of the rotational transitions in various molecules with negligible mutual interaction except for the effect of molecular collisions on line broadening. For the most part, the molecules studied do not possess a resultant electron spin or orbital angular momentum. This is the natural situation for almost all stable gaseous molecules in their ground electronic states with an extremely small number of exceptions. To be eligible for the present consideration of nuclear interaction, the molecule in question must have a t

* In their original work, the authors got several other types of spectral lines including those for Nd"7 from the same sample. We have reproduced only the part for Ce141for the benefit of a simple presentation. R. W. Kedzie, M. Abraham, and C. D. Jeffries, Phys. Rev. 108, 54 (1957).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

97

least one nucleus with a spin. Generally speaking, both the electric and magnetic hyperfine interactions are possible under these conditions. It turns out, however, that magnetic hyperfine interactions are usually negligible in view of the fact that the magnetic fields, due to paired electrons, tend to cancel out almost completely. The few molecules which do show a detectable magnetic hyperfine structure have not yet figured significantly in giving new nuclear moments. On the contrary, the electric quadrupole hyperfine interaction in a molecule is often very prominent except for a nuclear spin of +, which has no quadrupole moment, or, in some cases, a hyperfine coupling much too small for detection. Assume now a molecule answering the above description with just one nuclear spin and also being, for simplicity, a symmetric top variety (which includes a linear molecule as a special case). The restriction to a single nuclear interaction and a simple molecular configuration is fortunately not too stringent because most of the pertinent information to be presented has been obtained in just this way. Also assume that no magnetic field is being applied externally for the moment. Then the nuclear electric quadrupole interaction energy is, by a standard formula (reference la, page 154)

where

+

+

+

c = F ( F 1) - 1(1 1) - J ( J 1) F = J + I , J + I - 1 , . . . [J-I1 eqQ = quadrupole coupling constant K = component of J on molecular symmetry axis in a symmetric top, being zero for a linear molecule in the ground vibrational state. The quadrupole coupling constant is the product of the proton charge e and the nuclear quadrupole moment Q and the second derivative of V , q = r32V/r3z2,evaluated along the molecular axis z for the potential V a t the nucleus due to all charges outside of the nucleus. The quantity WQ is the hyperfine energy for each state characterized by the quantum number F , with all the F states associated with the same rotational state characterized by J and K . The rotational energy for a symmetric top is W R= B J ( J 1) - ( A - B ) K 2 , where A and B are the usual rotational constants. Any given transition is a result of the relation

+

A(WR

+ WQ)

=

hv

where Y is the microwave frequency and h is Planck’s constant. The selection rules in F , J, and K are: AF = 0, 1; A J = 0 , +_ 1; AK = 0 .

+

98

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

The intensity distribution for the transitions is well known and will not be given here. It is clear from the above discussion that, with the knowledge of J and K ,the nuclear spin I can be determined by matching the pattern of a calculated hyperfine spectrum (including intensity distribution) based upon an assumed I with that of the observed spectrum. The adjustable scale factor for perfect matching gives directly the constant eqQ. As a result of such measurements over a large number of molecules, the known spin values of very many nuclei have been confirmed; but, more significantly, more than ten new nuclear spins have been established by the microwave spectroscopic method for the first time. Table I gives a listing TABLE11 Atom Mass number I

B

0 S

c1 Ge Se

I

10 11 17 33 35 36 73 75 77 79 125 129 131

dPN)

3

Q

$ +0.633(10)

8 2

i

+l.OO(lO) +1.32(8)

H

8 8 4

-1.018(15) 2.74(14) 2.56(12)

cma) Reference

+

1.80081 (49)] [+2.68852(4)1 [-1.89370(9)] [O.64342(13)]

+0,06(4) +0.0355(2) -0.005(2) -0,050 +O ,035 [ + I .28538(6)] 0.0172(4) [ -0.8791 4 (12)] -0.21 (10) 1.1 [+0.534058(14)] I < 2 x 10-31 0.7 -0.66 [+2.617266(12)] - 0 . 4 3 0 5 )

[

i3

&(lo-24

Ib lb lb lb,2,4 lb lb,5 lb 6 lb lb 7,8 lb,9 8

This table lists only those atoms whose nuclear spins Z were first determined by microwave spectroscopy in gases. The other quantities p and & were also first obtained from the same method, but, where they are improved by another method, the more accurate results are given in square brackets. The numbers in parentheses indicate experimental errors in the last figures of the associated values.

of the nuclei and their new spin values, together with other items to be d i s c ~ s s e d . ~ ~Notable ~ ~ ~ 4 -among ~ these new spins are the ones such as I(B1O) and I(C136)which ran counter to the then current predictions. It is just this type of information that is most helpful to the theories of nuclear structure. We come now to the other result of the hyperfine structure observation, J. R. Esbach, R. E. Hillger, and C. K. Jen, Phys. Rev. 80, 1106 (1950). C. Aamodt and P. C. Fletcher, Phys. Rev. 98, 1317 (1955). EL. C. Aamodt and P. C. Fletcher, Phys. Rev. 98, 1224 (1955). 7 P. C. Fletcher and E. Amble, Bull. Am. Phys. SOC.2, 30 (1957). * P. C. Fletcher and E. Amble, Phys. Rev. 110, 536 (1958). W. Gordy, 0. R. Gilliam, and R. Livingston, Phye. Reu. 76, 443 (1949). 6L.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

99

namely, eqQ. Unlike the direct determination of nuclear spin, the nuclear quadrupole moment Q is as yet concealed in its product with the quantity q which needs to be ascertained. Of course, if the electronic wavefunction is accurately known, there should be no difficulty in calculating q. As this is practically not the case, some methods of approximation have to be used. The problem is greatly simplified by the observation of Townes and Schawlow (see reference la, page 232) that, for a heavy atom participating in a covalent bond in a molecule through a p orbital, the contribution to q is nearly the same as in an atom. In the case of an atom, q can be calculated readily from the fine structure separations or the magnetic hyperfine structure. However, measurements of eqQ for molecules containing the same atom show rather large variations, which can only mean that q varies from molecule to molecule on account of local conditions. The local conditions are construed to consist mainly of the ionic character of the bond and the degree of s or d hybridization. I n spite of some inherent uncertainties in this manner of analysis, it has been demonstrated that reasonable values for the nuclear quadrupole moments (with a specification of their signs) can be obtained. Such values for Q are listed in Table I only for those nuclei whose spins were first determined by the microwave method. There have been, in fact, many other nuclear quadrupole moments estimated from microwave measurements even though the corresponding nuclear spins were established by other methods earlier. We shall now take up the subject of nuclear magnetic moments in microwave spectra of gas molecules. I n the absence of an external magnetic field, nuclear magnetic moment can only reveal itself in a magnetic hyperfine interaction. However, except for an exceedingly few molecules, each of which has an electronic angular momentum, most molecules very rarely show any detectable magnetic hyperfine structure, which is rather difficult for interpretation in any case. Therefore, the information on nuclear magnetic moment has to be sought in the Zeeman splitting of the quadrupolar hyperfine lines previously discussed. Assume now we apply an external magnetic field H . For an electronically nonparamagnetic molecule, the only quantities which could be influenced by the magnetic field are the nuclear magnetic moment and the rotational moment of the molecule. These moments are usually of the same order of magnitude and are sufficiently small that the magnetic energy involved is often considerably smaller than the quadrupolar interaction for the fields commonly applied. Under these conditions, the first order magnetic energy (reference l a , page 290)

WH = -MpArH(ar.rg,

+ wgz)

(2.4.1.3.14)

2.

100

DETERMINATION O F FUNDAMENTAL QUANTITIES

where

M

magnetic quantum number associated with F nuclear magneton g J = rotational g factor = p(rotation)/J qr = nuclear g factor = ~ I / I (YJ = [F(F 1) J(J 1) - I ( I 1)1/2F(F 1) ffI = 1 - ( Y J . As a consequence of this situation, and subject to the well-known selection rule of AM = 0 for a ?r transition and AM = 1 for a u transition, there results a number of Zeeman components, with a defined intensity distribution, for each of the hyperfine lines. From the measured splittings for two or more hyperfine lines, it is possible to determine 1gJJ and 1911 and their relative sign, assuming that F , J , and I values are all known. The absolute sign of the g-factors can be experimentally determined by using circularly polarized microwaves. The aforementioned method has offered a few new determinations of nuclear magnetic moments before many of these moments were redetermined by some other methods (chiefly nuclear resonance) with much higher accuracy. In the interest of the present discussion, the first determinations of nuclear magnetic moments by the microwave spectroscopic method are listed in Table I, with the more accurate values by some other methods placed in square brackets. 2.4.1.3.2.2. Spins and Nuclear Moments by Paramagnetic Resonance. Paramagnetic resonance means, by usage, the electron spin magnetic resonance rather than any other paramagnetic resonance such as nuclear magnetic resonance. The presence of an unpaired electron spin, then, is an essential feature of this study. When a spinning electron, as a part of an atom, is placed in a magnetic field, a resonant absorption of radiation energy occurs when the radiation frequency is related to the magnetic field in a specific manner. It so happens that, with a convenient laboratory field of a few kilogauss, the resonant frequency falls directly in the microwave region. This is the only visible connection that paramagnetic resonance has with the microwaves. Paramagnetic resonance can be and has been done in gases, liquids, or solids. More productive work, however, has been done with the solid phase than any other, partly because most paramagnetic compounds are solids. The most extensively studied paramagnetic atoms or ions are those in the transition groups, the iron and the rare earth groups of elements. Paramagnetic resonance studies in recent years, mainly led by Bleaney at Oxford, have supplied the much needed experimental basis for many of the deductions and speculations in the quantum theory of paramagnetic solids. Among the significant contributions on the subject is a clear =

~ L N=

+ +

+

+

+

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

101

demonstration of the effect of the crystalline field on the splitting of the orbital degenerate energy levels, “quenching” of the orbital angular momentum, and magnetic anisotropy. Though somewhat disparaging to our present discussion, it must be said that the nuclear effects come in somewhat as by-products of the principal development in paramagnetic resonance in much the same way as they do in gaseous microwave spectroscopy. The nucleus of a paramagnetic ion can make its presence felt only through the channel of hyperfine interaction. Whenever the nucleus possesses a resultant angular momentum, and hence a nuclear magnetic moment, it can have a prominent magnetic hyperfine interaction with the electronic magnetic moment. Also, there may be a nuclear electric quadrupolar interaction, although generally very much smaller. The dominating role of the crystalline field in paramagnetic resonance of solids, and the variety of field symmetries involved, make it difficult to describe the situation in most general terms. For simplicity of presentation as well as to cover practically the most important case, let us assume that the crystalline field has an axial symmetry and the “spin Hamiltonian” takes the formga X = DSz2

+ P[gllHzSz + gI(HzSz + HuSg)l + AIzSz + B(1zSz(2.4.1.3.15) +

where the coordinate system is chosen with z along the symmetry axis and Sz,Su,S,= components of the electron spin S H,,H,,H, = components of magnetic field H 911 = electronic g factor parallel to symmetry axis gI = electronic g factor perpendicular to symmetry axis p = Bohr magneton and A , B, and D are constants. I n Eq. (2.4.1.3.15),it is assumed th a t the lowest orbital state is a singlet and the nuclear electric quadrupole energies are negligible. The complete solution of Eq. (2.4.1.3.15) is too complicated to be given here, but a simple description of the results can be made in sufficiently strong magnetic fields. If the field-independent terms are all negligible (i.e., A B D A 0), then there is just one resonance with the relation hv = gPH witheb

--

92 =

gI12COS*

8

+ gL2 sin2 8

((2.4.1.3.16)

where e = angle between magnetic field and the symmetry axis. If the term DSz2 is small but not negligible, there are 2 s equally spaced lines with sa

B. Bleaney and K. W. H. Stevens, Repts. ProgT. ,in Phps. 16,134 (1953). reference 9a, page 139.

@hSee

102

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

the intensity decreasing from the center. This spectrum is known as the fine structure, since it is completely independent of the nuclear effects. If, in addition, A and B are also not negligible, there is a hyperfine structure of 21 1 equally spaced lines for each fine-structure line present. All the hyperfine lines in one group are of equal intensity, since all the discrete nuclear orientations relative to the magnetic field are equally probable under normal temperature conditions. The spacing K between any two adjacent hyperfine lines is given assb

+

K2g2= A2g$ cos2 9

+ B2gL2sin2 0.

(2.4.1.3.17)

It is clear from the preceding discussion that the nuclear spin I can be directly determined from the number of hyperfine components. The nuclear magnetic moment can be easily determined if the moment of an isotopic nucleus is known from a previous direct measurement (e.g., nuclear magnetic resonance). Then the ratio of the nuclear magnetic moments between the two isotopes is simply equated to that of the total splittings of the two corresponding hyperfine spectra observed in the paramagnetic resonance experiment. When such information is not available, then one has to calculate A or B from the observed spacing between the hyperfine lines. Either A or B is directly proportional to the nuclear magnetic moment p with a proportionality factor which, for a non-s electron, depends principally upon the factor < F ~ >which , is the inverse cubic distance between the electron and the nucleus averaged over the spatial function of the electron. Without going into the detailed processes of evaluation, it suffices to say for our present purpose that the nuclear magnetic moments have been estimated for a number of nuclei from results of paramagnetic resonance. Table I1 tabulates the nuclear spins and magnetic moments, along with some rough values for the nuclear quadrupole moments, which represent new results first obtained by paramagnetic r e s o n a n ~ e . ~ ~ ~ l ~As , ~in ~ ~Table 0 - 1 9I, more prelo

M. M. Weiss, R. 1. Walter, 0. R. Gilliam, and V. W. Cohen, Bull. Am,. Phys.

SOC.2, 31 (1957).

Dobrowolski, R. V. Jones, and C. D. Jeffries, Phys. Rev. 104, 1378 (1956). R. V. Jones, W. Dobrowolski, and C. D. Jeffries, Phys. Rev. 102, 738 (1956). 13 W. Dobrowolski, R. V. Jones, and C. D. Jeffries, Phys. Rev. 101, 1001 (1956). l4 J. Owen and I. M. Ward, Phys. Rev. 102, 591 (1956). 16 M. Abraham, R. Kedsie, and C. D . Jeffries, Phys. Rev. 108, 58 (1957). 16 J. G . Park, PTOC. Roy. SOC.A246, 118 (1958). P. B. Dorain, C. A. Hutchison, Jr., and E. Wong, Phys. Rev. 105, 1307 (1957). B. Bleaney, P. M. Llewelyn, M. H. L. Pryce, and G . R. Hall, Phil. Mag. [7] 46, l1 W. l*

991 (1954). J. C. Hubbs, R. Marrus, W. A. Nierenberg, and J. L. Worcester, Phys. Rev. 109, 390 (1958).

2.4.

DETERMINATION

OF SPIN, PARITY,

AND NUCLEAR MOMENTS

103

cise values later derived from some other methods are placed in square brackets. Not listed in Table 11, but which can be found elsewhere, are the cases where paramagnetic resonance confirmed previous spin assignments but offered new values for nuclear magnetic and, sometimes, quadrupolar moments. TABLEI I e Atom Mass number I

-P

v Mn

co Mo Ru Ce Nd Sm

Eu DY

Er U P

PbN)

&(lo-24 cm*) Reference

-

32 49 50 53 56 57 60 95 97 99 101 141 143 145 147 147 149 152 154 161 163 167 233 235 239 24 1

1

5

-0.2523(3) 4.46(5)

6

4

4

4

5

[+3.34702(94)] 5.050(7) 3.855(7) 4.65(20) 3.800(7)

4

[ -0.93270(18)] [ -0.95229(10))

Q

= 1.09(3)

3 & 4 4 4 4

0.89(9) --1.0(2) -0.62(9) 0.56(6) -0.78(37) -0.65(23)

3

2.1 -0.37(4) +0.51(6) 0.50(12) 0.48-0.54 0.32-0.35 0.4(2) 1.4(6)

4 4 4 8 4

$

.’”” = 0.965(10) &I64

+

1.1(4) +1.3(4) 10.2(30) 3.3-3.5 3.9-4.1

[O ,021

Id 10 lb 11 12 lb 13 lb, 4 lb,14 Ib lb 3 Ib Ib 3 lb lb 15 15 16 16 lb 17 17 18,19 18

This table lists only those atoms whose nuclear spins Z were f i s t determined by paramagnetic resonance in solids. The other quantities p and Q were also first obtained from t h e same method, but, where they are improved by another method, the more accurate results are given in square brackets. The numbers in parentheses indicate experimental errors in the last figures of the associated values.

In concluding the subject on the microwave method for measuring spins and nuclear moments, it may be said that this method has offered a number of new assignments for the spins and many reliable values for the magnetic and quadrupolar moments of nuclei. This type of measurement has the advantage of requiring only small samples, a factor of great importance, particularly in connection with radioactive nuclei.

104

2.

DETERMINATION O F FUNDA4MENTAL QUANTITIES

2.4.1.4. Nuclear Magnetic and Quadrupole Resonance. 2.4.1.4.1.*

t

PRINCIPLES OF THE METHOD. 2.4.1.4.1.1. Introduction. Since their introduction by Purcell, et al.1 and by Bloch et a1.,2the techniques of nuclear magnetic resonance and nuclear induction have been successfully used in the precision determination of the magnetic moments of over 100 nuclei and the spins of about 25. The subsequent observation of nuclear quadrupole resonance absorption in solids by Dehmelta and by Pound4 has resulted in the measurement of nuclear electric quadrupole moments for some 20 nuclei. These experiments are all so closely interrelated that they will be collectively treated. The method is a modified extension of Rabi's principle of magnetic resonance (see Section 2.4.1.2) to nuclei contained in matter of normal density; here the resonance condition is observed by a macroscopic electromagnetic absorption or dispersion in contrast to the deflection and subsequent detection of free atoms. It should also be noted that the method is the nuclear analog of microwave paramagnetic resonance absorption (see Section 2.4.1.3). Consider a nucleus characterized by its angular momentum Ih and total magnetic moment p = yhI. Here p represents the component along I of the resultant magnetic moment of the individual nucleons, which are so tightly coupled by nuclear forces in comparison to the energies involved in magnetic resonance that the assumed proportionality between p and I is completely justified. The maximum possible components of I and t) in any given direction are called the spin I and the magnetic moment The proportionality factor y = p / I h is here called the gyromagnetic ratio; for protons, y E 2.67 X lo4 sec-l oersted-'. The interaction of the nucleus with a n applied dc magnetic field H will be given by the usual magnetic dipole term X M = - y . H = -yhI * H. The 21 1 equally spaced energy levels (see Fig. 1) are given by Em = -ykHom where I 2 rn 2 - I , if we assume H is directed along t See also Vol. 2, Chapter 9.7;Vol. 3, Chapter 4.1;Vol. 4, A, Chapter 4.4. E. M. Purcell, H. C. Torrey, and R. V. Pound, Phys. Rev. 69,37 (1946);N.Bloem-

+

bergen, E. M. Purcell and R. V. Pound, Phys. Rev. 73, 679 (1948). *F. Bloch, W. W. Hansen, and M. Packard, Phys. Rev. 89, 127 (1946); F. Bloch, Phys. Rev. 70, 460 (1946);70, 474 (1946). a H. G. Dehmelt, Am. J . Phys. 22, 110 (1954),which see for earlier references. 4R.V. Pound, Phys. Rev. 79, 685 (1950). 6 I.( is in Gaussian units, which are used throughout this section; the unit of magnetic field H i s the oersted. Sometimes nuclear magnetic moments are given in units of t h e nuclear magneton 10 = eh/2mc = 5.05 X erg/oersted. Usually experiments are performed in magnet air gaps in which H (oersteds) = B (gauss) to high precision; for this reason, the literature of magnetic resonance often specifies t h e magnetic field in units of t h e gauss.

* Sections 2.4.1.4.1 through 2.4.1.4.7.2 are by C. D. Jeffries. Section 2.4.1.4.7.3 is by G. Feher.

2.4.

DETERMINATION OF SPIN,

PARITY,

AND NUCLEAR MOMENTS

105

the z axis with magnitude Ha. Consider now a large number of such nuclei which are immersed in a lattice, i.e., a reservoir of thermal energy, such as the protons in a sample of water or a crystal of ice. Because of thermal agitation, there will be relaxation processes which tend to bring the system of proton spins into thermal equilibrium with the lattice.

m =+A 2

(b)

(a)

FIQ.1. (a) Energy levels of a magnetic dipole with spin I = in a magnetic field. (b) Additional splitting due to nuclear electrical quadrupole interaction with electric field gradient in a crystal.

We can describe the system by assigning Boltzmann population factors to the energy levels. Between an adjacent pair of levels there will exist the population ratio: N(m)/N(m

+ 1)

=

exp[-E,/kT

+ E,+l/lcT]

E 1 - rhHo/kT

1-

lo-'

(typical value a t room temperature). Now if electromagnetic radiation at the Larmor frequency wo = [Em- Em+J/h = ~ H radians/sec o is applied to the system, stimulated magnetic dipole transitions may occur according t o the selection rule Am = fl. However, there will be more stimulated absorption (Am = - 1) than stimulated emission (Am = 1) because of the slight excess population in the lower state of any pair of levels. Thus there will be a net resonant absorption of electromagnetic energy by the system, resulting in a slight heating of the lattice. For HD fields of a few thousand oersteds, Y O = w 8 / 2 r is of the order of 10 Mc/sec, and the resonance absorption can be detected by straightforward radio frequency techniques, e.g., b y wrapping a coil around the sample and observing by means of a bridge the loss factor of the coil for frequencies in the neighborhood of va. As in related phenomena, the absorption is accompanied by a dispersions which can be observed by adjusting the

+

R. d e L. Nronig, J . Opt. SOC.Am. 12, 547 (1926); H. A. Kramers, Atti del congr. J. Gorter and R. de L. Kronig, Physica 3,1009 (1936); A. R. Von Hipple, "Dielectrics and Waves." Wiley, New York, 1954. 6

inst. jk., Como 2, 515 (1927); C.

106

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

bridge to detect the change in reactance of the coil. Thus by observing the resonance frequency v o in a known magnetic field, one obtains in a relatively simple experimental arrangement the nuclear gyromagnetic ratio p / I h , from which the magnetic moment p may be determined if the spin I is known. Furthermore, as shown later, the spin may be determined from the amplitude of the resonance absorption. If I > 1, the nucleus may be further characterized by an electric quadrupole moment and there may be in addition to an interaction of the form XQ = Q * AE, which represents the dyadic product of the nuclear quadrupole moment operator Q with the electric field gradient tensor AE a t the nuclear site. The nuclear electric quadrupole moment, defined explicitly in reference 44, is a measure of the deviation from spherical symmetry of the nuclear charge distribution. Although the quadrupole interaction may be averaged to zero in the rapid tumbling of liquids, we find in single crystals that the magnetic energy levels are shifted unequally (see Fig. 1) by this additional term, so that resonance absorption between pairs of levels may be observed a t several different frequencies. I n the case of zero applied magnetic field, resonance absorption between pure quadrupole levels may be observed. The measurement of the resonance frequencies determines the product qQ of a principal value q of the electric field gradient tensor with the scalar quadrupole moment Q . Calculated values of q must be used to obtain Q from the data; consequently the precision in the determination of Q, in this or any other method, is about two or more orders of magnitudes lower than for p. However the precise ratio of the quadrupole moments for two isotopes can be easily measured. Furthermore the spin I is explicitly given by the number of resonance lines. From the broad field of nuclear magnetic and quadrupole resonance absorption, we discuss here only those aspects of direct interest to nuclear physics. For details we refer to more general and extensive treatments.’-’l 2.4.1.4.1.2. The Bloch Phenomenological Theory. The simple optical model of magnetic resonance absorption given above does not take into consideration the interactions of the spin system with itself or with the lattice. Detailed treatments of these relaxation processes have been N. F. Ramsey, “Nuclear Moments.” McGraw-Hill, New York, 1953. G. E. Pake, Solid State Phys. 2, 1 (1956). E. R. Andrew, “Nuclear Magnetic Resonance.” Cambridge Univ. Press, London and New York, 1955. lo A. Abragam, “Principles of Nuclear Magnetism.’’ Oxford Univ. Press, London and New York, 1961. 11 T. P. Das and E. L. Hahn, Nuclear quadrupole resonance spectroscopy. Solid Stale Phys. Suppl. 1 (1958).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

107

given,lJ2-14 but from the viewpoint of nuclear physics it is sufficient to consider the original phenomenological treatment of Bloch in which these effects are represented by two empirical relaxation times.2 Consider n identical nuclei per unit volume of spin 1 and magnetic moment p under the influence of an applied field H with the components: H , = 2H1 COB wt, H , = 0, H , = H ; here H is a large and uniform field; HI is a small oscillating field applied by means of a coil around the sample to induce magnetic dipole transitions. Following Ehrenfest’s theorem one assumes that M, the resultant magnetization per unit volume of all the nuclei, obeys the classical equation of motion dM/dt = M = rM X H in the absence of relaxation effects. These are next taken into account as follows: owing to thermal processes in which the spin system exchanges energy with the lattice, it is assumed that the longitudinal component of the magnetization M , approaches its thermal equilibrium value Mo in a characteristic thermal relaxation time T1 according to the relation: k?,= - ( M z - Mo)/T1. Here M ais given by the familiar Curie formula

+

M~ = I--HO1 npz I 3kT

= (I -I- 1 ) l - nyh2 wo

3kT

(2.4.1.4.1)

where T is the absolute temperature of the sample and k is Boltzmann’s constant. It is also assumed that the transverse components M,, M , will vanish exponentially in a characteristic time T 2according to the relations M , = -M,/T2; a, = -M,/TZ. The time T, 5 TI and is called the transverse or total relaxation time, and is essentially the characteristic time for incoherence of M,, M , due to processes in which the total energy of the spin system does not change; more specifically, there may be mutual spin flips, or variations of the locally effective value of H throughout the sample due to the dipole fields of neighboring nuclei. By the Fourier theorem the resonance linewidth Aw 0 1/T2, unless there is appreciable inhomogeneous broadeningl4” due, e.g., to magnetic field nonuniformities. I n the following we assume that there is no appreciable inhomogeneous broadening, as is often the case for liquids but not for solids. The total effect of both applied fields and internal interactions is then approximately given by the phenomenological equations :

M,

=

kf,= Ma =

-MZ/TS - r M , H -MM,/T2 r(M,2HI cos w t - M,H) - ( M z - Mo)/TI - rM,2H1 cos wt,

+

R. K.Wangsness and F. Bloeh, Phys. Rev. 89, 728 (1953). F. Bloch, Phys. Rev. 102, 104 (1956); 106, 1206 (1957). l 4 A. G. Redfield, ZBM J . Research Develop. 1, 1 (1957). 148 A. M. Portis, Phys. Rev. 91, 1071 (1953). 12

(2.4.1.4.2)

108

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

In the Purcell technique’ the experimentally observed quantity is essentiaIIy a bridge unbalance voltage V , proportional to A$z. I n the Bloch technique2 of nuclear induction, an additional coil is wound around the sample along the y axis (see Fig. 3), and one observes the induced voltage V , a Mu. I n either technique, resonance is usually observed by noting the periodic change in V , or V , as the applied field H is periodically swept through the resonance value H O = wo/y by superposing on the dc field a small modulation field H , cos Qt. From the nature of Eqs. (2.4.1.4.2), it is seen that the exact shape of the resonance curves V,(H) and V,(H) will depend upon the rate a t which the resonance is swept through, i.e., upon whether or not the passage through the resonance line in a time T is adiabatic in the Ehrenfest sense (7 >> l/yHl), and also upon the ratios r/T1 and ~ / T zPerhaps . the simplest and most common case is that of adiabatic slow passage: resonance is swept through so slowly that M , continuously reaches its steady-state value under the influence of HI and the relaxation processes. This requires T >> T l , T z , l/yHl. Another interesting case is that of adiabatic fast passage in which T >> l/yH1, T 1. Thus the total receiver coil voltage V , V , can be made essentially proportional to either v or u;i.e., one may observe either the absorption or the dispersion (or any combination). Similar considerations hold for the Purcell single coil technique : by adjusting the balance of an rf bridge containing the coil as a n element, one can V e at nuclear resonance make the total bridge unbalance voltage V , proportional to either the imaginary or the real part of the complex nuclear susceptibility corresponding to the absorption v or the dispersion u,respectively. Note that whereas the algebraic sign of the nuclear induction signals are dependent upon the sign of the gyromagnetic ratio y, those of t>hesingle coil signals are not. This is due to the circumstance that two (crossed) coils are needed to establish the sense of a rotating field. The shape of experimentally observed homogeneously broadened resonance curves for protons in HzO under adiabatic slow passage conditions are shown in Figs. 2a and 2b. They follow closely the form expected from Eqs. (2.4.1.4.3): the absorption v has a maximum at resonance ( A H = 0) and a half-width at half-maximum H1,2 = l/yTz, provided the rf field H 1 is small enough so that ( ~ H I ) * T I Tr in one channel, or by using two independent sources and separating the counters. ( 5 ) Scattering from one counter into the other, or from material close to source or counters, is checked. (6) The over-all performance of the directional correlation system is tested by measuring V(6)for well-known cascades (Nieo,Pd106). Most of these tests apply also to polarization-direction correlations. I n addition, however, the asymmetry ratio R of the polarimeter must be determined by one or more of the following experiments. (7) R can be found by measuring the linear polarization of gamma rays which were scattered by a n angle of about 90°.32--34,40 (8) The polarization-direction correlation of a well-known cascade (NiSo,Pdlo6)can serve to fix R.33 P. Lehmann and J. Miller, J . phys. radium 17, 526 (1956). S. M.Harris, Nuclear Phys. 11, 387 (1959). M.E.Rose, Phys. Rev. 91, 610 (1953). 82 J. S. Lawson, Jr., and H. Frauenfelder, Phys. Rev. 91, 649 (1953).

2.4.

DETERMINATION OF SPIN, PARITY, .4ND NUCLEAR MOMENTS

149

(9) The linear polarization of y rays following Coulomb excitation in even-even nuclei is well known and hence can also be used to find R.37 (10) The linear polarization of annihilation quanta yields a value for R a t the energy 0.51 MeV. This experiment, however, requires two polarimeter. 39 2.4.2.1.7.2.Procedure. It is simplest to discuss a y-y directional correlation. Other measurements are obvious generalizations of this basic experiment. Before a run, and at intervals during runs, resolving time and back, are determined. N a n d C denote single counts ground counts [ N i 0 ( 8 ) Co(9)] and coincidences, respectively, per unit time. 8 denotes the angle subtended a t the source by the counters (Fig. 1). A source of proper strength is prepared. The maximum source strength is determined by the ratio of accidental to true coincidences. For accurate measurements, this ratio should be smaller than about 0.2;under unfavorable conditions, one may be forced to make it unity or even higher. The source is then centered to about 1 %; i.e., the counting rate N ( 9 ) of the movable counter as a function of the angle 6 should be constant to about 1 %. After the source is centered, the coincidences and the single counts are recorded for various angles. The angle should be changed frequently, e.g., about every 20 min. For short-lived radioisotopes, more frequent changes are advisable. The choice of angles depends on the form of the correlation function. I n general, it is best, to use a few angles only (e.g., go", 120", . . . 270") and to obtain a t each angle a large number of coincidences. For extended experiments, automatic equipment for changing the angles and recording the data is very convenient. From the measured data, one obtains the "true" values Ct(S) and N t ( 8 ) by subtracting background, accidental coincidences, and contributions due to disturbing radiation^;^^ e.g. :

Cf(fi)= C m e a s (9) - Co(8) - Carc(9) @(9) = 2TrNr;1""(9)Ny""(8).

- Cd(9) (2.4.2.1.18) (2.4.2.1.19)

The contribution Cd(8),due to other coincident y rays present in the source must be found in a separate measurement, with the energy selection adjusted in such a way th at the various contributions can be singled out. The true number of single counts and coincidences can now be written as : Nit(&) = Mpifli~i (2.4.2.1.20) Ct(8) = ~ P , P , Q l E l Q 2 ~ 2 E , ~ ( i ) . (2.4.2.1.21) 63

N. Levine, H. Frauenfelder, and A. Rossi, Z. Physik 161, 241 (1958).

150

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

M is the number of nuclear disintegrations per unit time, pi the probability per disintegration that the radiation selected in counter i is emitted, Qi the solid angle in units of 4n, ei the efficiency of channel i, and ec the efficiency of the coincidence circuit. K (8) denotes the directional correlation function, as measured with the equipment under discussion. Comparison of Eq. (2.4.2.1.21) with Eq. (2.4.2.1.19) shows that the accidental coincidences increase with the square of the source strength, the true coincidences only linearly. As mentioned above, an optimum source strength exists which can be calculated from Eqs. (2.4.2.1.19)(2.4.2.1.21). Decreasing the resolving time T , permits the use of correspondingly stronger sources. Equations (2.4.2.1.19)-(2.4.2.1.21) also show that the ratio of true to accidental coincidences is unaffected by the solid angle. Very large solid angles hence are often advantageous. (Optimum solid angles are discussed in Devons and Goldfarb.e) For the further evaluation, the coincidence ratio C f ( 8 ) / N 1 ' ( 8is ) formed. From Eqs. (2.4.2.1.20) and (2.4.2.1.21), one then finds for the desired uncorrected experimental correlation function K ( 8 ): (2.4.2.1.22)

If one uses movable counters, it is seen from this equation that one best divides through by the counting rate in the niovable counter, i.e., that one identifies counter 1 with the movable one. The ratio K ( 8 ) then becomes independent of the solid angle of the movable counter. Furthermore, it is independent of the decay of the source, of the branching probability pZ and of the efficiency of the movable counter. Small errors in the centering of the source are corrected in first order. The numbers K ( 8 ) are now fitted by

+ A;~'~(COS8) +

K ( 8 ) = Ka(1

*

*

.

+ A~maPkmar(COS 8))

(2.4.2.1.23)

using the method of least square^.^^^^^-^' It usually suffices to take = 4. Details concerning the determination of the coefficients A k ' and their errors are given in RoselB1 Breitenbergerl64 and Price;6s applications of the method are treated in Klema and McGowanBsand Breitenberger.67 2.4.2.1.7.3. Corrections. K ( 8 ) corresponds to the correlation function W(0)only under the assumption of centered point sources, point detectors,

,k

E. Breitenberger, Proc. Phys. Soc. (London) A69, 489 (1956). 86

P.C.Price, PhiE. Mug.[7]46,237 (1954);Proc. Cumbridge Phil. Soc. 60,491 (1954).

s8E. D.Klema and F. K. McGowan, Phys. Rev. 91, 616 (19531. E. Breitenberger, Proc. Phys. SOC.(London) A69, 453 (1956).

2.4.

DETERMINATION

OF SPIN, PARITY,

AND NUCLEAR MOMENTS

151

TABLE11. Corrections in Angular Correlation Experiments [The references are not complete; in some cases, only a few out of a large number of papers are given as examples.] Correction for

References

Perturbing influence of extra nuclear fields Finite size of counters Finite sire of source Decentering of source Scattering in the source Count er-to-counter scattering Disturbing radiations

a, b, c, d b, eel, r b, c, e, f, i, k

m

f,

c, n c, e, 0 j , P, q

A. Abragam and R. V . Pound, Phys. Rev. 92, 943 (1953). Devons and L. J. B. Goldfarb, in “Handbuch der Physik-Encyclopedia of Physics” (5.Flugge, ed.), Vol. 42, p. 362. Springer, Berlin, 1957. H. Frauenfelder, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 531. Interscience, New York, 1955. R. M. Steffen, Advances in Phys. 4, 293 (1955). M. Walter, 0. Huber, and W. Ziinti, Helv. Phys. Acta 23, 697 (1950). f S. Frankel, Phys. Rev. 83, 673 (1951). 0 M. E. Rose, Phys. Rev. 91, 610 (1953). E. Breitenberger, Proc. Phys. Soc. (London) A66, 846 (1953). i A. M. Feingold and S. Frankel, Phys. Rev. 97, 1025 (1955). i E. L. Church and J. J. Kraushaar, Phys. Rev. 88,419 (1952). J. S. Lawson, Jr., and H. Frauenfelder, Phys. Rev. 91, 649 (1953). E. D. Klema and F. K. McGowan, Phys. Rev. 92, 1469 (1953). E. Breitenberger, Phil. Mug.171 46, 497 (1954). E. Breitenberger, Proc. Phys. SOC.(London) A67, 1108 (1954). H. Frauenfelder, Ann. Revs. Nuclear Sci. 2, 129 (1953). p E. D. Klema, Phys. Rev. 100, 66 (1955). q N. Levine, H. Frauenfelder, and A. Rossi, 2.Physik 161, 241 (1958). H. I. West, Jr., Univ. of California Report UCRL 5451 (1959). a

* S.

no scattering, no disturbing radiations, and no extranuclear effects. In order to compare experimental and theoretical results, K(6) must be corrected for all deviations from such an ideal arrangement. All of the necessary corrections have been discussed in the literature ; relevant references are collected in Table 11. After applying the corrections, the final form of the experimental correlation function becomes P P ( 6 )=

K O (1 + AYP~(cos6) + . . .

+ A ~ s x P k , , ( ~6)). ~~ (2.4.2.1.24)

One now makes the assumption 6 = 0 and compares theoretical coefficients A k with AYp in order to extract the desired physical information.

152

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

2.4.2.2. Conversion Coefficients.* 2.4.2.2.1. INTRODUCTION; THEOCONSIDERATIONS. The ratio of the probabilities for emitting Y-shell conversion electrons (see Section 2.2.3.1; Y = K , LI * * Mv * Ltotal *) to that of nuclear gamma rays is called the Y-shell conversion coefficient ay. This quantity, as well as the ratio of conversion coefficients in different shells depend on the character of the transition under consideration, i.e., on whether the parities of the initial and the final state are the same or not, and on the distribution of the angular momenta of the emitted gamma rays. As discussed in Section 2.4.2.1, the last datum is connected with the difference between the spins of the initial and the final levels. Information about spins and parities of nuclear levels can, therefore, be obtained by comparing measured conversion coefficients, or ratios of conversion coefficients for the same transition in different shells, with the theoretically computed values. We will therefore first discuss some results of conversion theory and then some experimental procedures for measuring conversion coefficients. Conversion coefficients also depend on the transition energy and on the nuclear charge. Since these quantities are mostly known with sufficient precision, their influence will not be discussed here (except for remarking that conversion coefficients mostly increase strongly with increasing Z and with decreasing E ) . Moreover, only theoretical results for pure multipole transitions will be considered; the probabilities for emitting gamma rays and those for emitting conversion electrons of different multipole orders have to be added incoherently. Conversion coefficients depend on the nuclear structure, both in a static as well as in a dynamic way.' The static effect arises through the action of the average nuclear charge distribution on the atomic electrons. Practically, it is dependent on the nuclear radius only. The dynamic effect originates in the circumstance that nuclear matrix elements due t o penetration of the electron wave function inside the nucleus allow ejection of conversion electrons but no emission of nuclear gamma rays. These matrix elements are normally much smaller than the gamma ray matrix elements which then determine the conversion coefficients (almost independent of nuclear structure). But in cases where the gamma matrix elements become very small, either by I-forbiddenness* or through violation of asymptotic

RETICAL

- -

--

1 L. A. Sliv and M. Listengarten, Zhur. Ekuptl. i Teoret. F i z . 29, 29 (1952); T. A. Green and M. E. Rose, Phys. Rev. 110, 105 (1958); E. L. Church and J. Weneser, Ann. Rev. Nuclear Sci. 10, 193 (1960). 2 E. L. Church and J. Weneser, Phys. Rev. 104, 1382 (1956).

-

* Section 2.4.2.2 is by A. H. Wapstra.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

153

selection rules13the penetration matrix elements become prominent, thus giving rise t o relatively large conversion coefficients. Evidently, large deviations can occur only for delayed transitions. K selection rules apply to both kinds of matrix elements, so that transitions delayed by K forbiddenness alone should have normal conversion coefficients.3 I n only slightly delayed transitions, interference between the two kinds of matrix elements may cause conversion coefficients t o be even somewhat smaller than those for undelayed transition^.^ The influence of the nuclear size effect on K, LI, and ,511 conversion coefficients is approximately the same; that on LIIIconversion coefficients is very much less, even in a case6 where LI and LII conversion coefficients are large by a factor 20. Theoretical conversion coefficients for the K , LI, LII, and LIII shells have been computed by RoselBtaking into account the influence of the static correction for the finite nuclear size and of the screening of the electric field of the nucleus by the electron shells around it. The same book gives M-conversion coefficients uncorrected for the last two effects. As a result, these conversion Coefficients are somewhat large (e.g., about 40% for E2 and 2 = 80); the ratios of the conversion in the M subshells are more nearly correct. Sliv and Band’ published K and L conversion coefficients computed from a model in which the nuclear currents are located on the surface of the nucleus, thus including the static correction and part of the dynamic correction. Their results differ from those of Rose by less than 5% in almost all cases of physical importance. Data permitting the calculation of conversion coefficients for any nuclear model have been published by Green and R0se.l The precision in measuring conversion coefficients is mostly limited due t o the difficulties in measuring gamma ray intensities. Ratios of Conversion coefficients for one gamma transition, which are identical with ratios of conversion line intensities, can be determined much more accurately. Since these ratios too depend on multipolarity and character, they can also be used to get information about these data. The K to L S. G. Nilsson and J. 0. Rasmussen, Nuclear Phys. 6, 617 (1958). L. S. Kisslinger, Bull. Am. Phys. SOC.2,358 (1957); A. H. Wapstra and C. J. Nijgh, Nuclear Phys. 1, 245 (1955); K. 0. Nielsen, 0. B. Nielsen, and M. A. Waggoner, ibid. 3

4

2, 476 (1956). 5 F. Asaro, F. S. Stephens, J. M. Hollander, and I. Perlman, Phys. Reu. 117,492 (1960). 6 M. E. Rose, “Internal Conversion Coefficients.’’ Interscience, New York, 1958. ‘L. A. Sliv and I. M. Band, “Coefficients of Internal Conversion.’’ Acad. 8ci. U.S.S.R., Moscow, 1956 and 1958; translated as Reports 57-ICC-K1 and 59-ICC-L, Department of Physics, University of Illinois, Urbana.

2.

164

DETERMINATION OF FUNDAMENTAL QUANTITIES

conversion line ratios can be measured most easily, but are often not very sensitive to the multipolarity (see Fig. 1). The L subshell ratios give much more information (see Fig. 2), but they can only be measured with instruments of high resolving power. IOC

2-

K / L ratios

Gamma Energy

FIQ.1. Ratio of K and L conversion lines for different pure multipole transitions, for Z = 79. At other nuclear charges, the same K / L ratios appear at approximately the same values for E / Z z .

2.4.2.2.2. DETERMINATION OF BETA AND GAMMA RAY INTENSITIES, The most natural way for obtaining conversion coefficients is to measure electron intensities in a beta ray spectrometer and gamma rays in a gamma ray spectrometer. Measurements of relative electron intensities in a magnetic or electric beta ray spectrometer with moderate precision offers no difficulties, since in a well designed instrument of this type the efficiency is nearly independent of the energy (see Section 2.2.1.1). It is, however, necessary t o take several precautionss if high precision is desired. 8 See, e.g., A. H. Wapstra, “Report on the Israel Conference on Nuclear Physics.” North-Holland Publ., Amsterdam, 1957.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

155

2-

I

0.2

1.0E

,

I

0.4

0.3

.

I

I

I

I

l

0.5 0.7 Gamma Energy

l

I

I MeV

2

3-

#/-

0.2

I

I

0.3

0.4

t

1

I

I

I

l

0.7 Gamma Energy 0.5

l

I MeV

I

1

2

2-

=--_-_ ---------------

0.01

I

I

I

I

I

I

-------_

MI I

I

l

l

I

156

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

For measuring absolute intensities, beta ray spectrometers can be calibrated using sources with simple spectra, the strength of which has been determined using 4~ counters or coincidence techniques. In cases with few, well separated electron lines, electron intensities may be measured in proportional or scintillation counter spectrometers, or with the new solid state detectors (see Sections 1.3.1, 1.4.1, and 1.8.1). Efficiencies of gamma spectrometers vary with energy in a way which, in many cases, cannot be computed with the necessary accuracy. These instruments have, therefore, t o be calibrated even for measurements of ratios of intensities of gamma ray lines with different energies. The most accurate measurements can be obtained, in principle, with crystal diffraction spectrometers.9 Such instrhments demand very strong sources. The more popular scintillation spectrometer needs much weaker sources, but has a much lower resolving power. Even for sources with few, well separated gamma lines, intensity measurements in a scintillation spectrometer with a precision of 5 % or better are mostly hard to make due to the difficulty of separating the lines from the Compton continua, caused by lines of higher energies in an unambiguous way. Secondary electrons emitted by a thin “converter” placed at the source position of a beta ray spectrometer and irradiated by the gamma ray source can be measured as much sharper lines than those found in scintillation spectrometer (see Section 2.2.3.1.2). By choosing a converter of material with high atomic number, the ratio of the intensities of the photo lines and the Compton continua can, moreover, in principle be made larger than is possible in scintillation spectra. Uranium converters are most suitable, but for many purposes lead converters, which can be obtained more easily, are sufficient. Sources for this type of measurement should be about a hundred times stronger than those used in scintillation spectrometry. In the computation of gamma ray intensities from the measured photopeak areas, one needs to know the photo cross sections as a function of the direction at which the photo electron is emitted. Habitually, the influence of this function is expressed as the product of the total photo cross section T Y and an angular distribution factor fY, so that the photopeak area A Y (if plotted as number of electrons per momentum interval versus momentum) becomes A y = 1 7 T y f y db Q the quantity d being the thickness of the converter and b a dimensional factor, and Q being the solid angle of acceptance of the spectrometer used. Values for the total cross sections can be derived from the tables of White 9

See Vol. 5, A, Section 2.2.3.2.

2.4.

DETERMINATION O F S P I N , PARITY, .4ND NUCLEAR MOMENTS

157

Grodstein or from older ones by Davisson.’O The recent work of Hultberg and his collaborators on the computation of the angular distribution factor fK is extremely important for the use of this method in intensity measurements. Yet stronger sources are necessary if one wants t o measure Compton lines,12 since then the gamma rays have to be collimated before they strike the converter, which in this case is preferably of low 2 material. The last method has the advantage that it produces only one line for every gamma ray. 2.4.2.2.3. DETERMINATION O F CONVERSION COEFFICIENTS; SPECIAL METHODS. Accurate measurements of conversion coefficients can be made in special cases, e.g., a single y transition preceded by a single /3- transition. The conversion coefficients can then be found by measurement of the intensities of the conversion lines and the beta continuum in a beta spectrometer: 1/aK

=

/3/K

- (K +L + M

+

. . .)/K

(2.4.2.2.1)

The symbols K , L , M , . . . stand for the intensities of the K,L,M, . . . conversion electrons. The intensity /3 of the beta continuum is mostly obtained by extrapolating the high energy part of the spectrum t o lower energies with help of the theoretical spectrum shape, since the measured spectrum is always distorted at low energies by counter window absorption and by the occurrence of scattered electrons. This extrapolation can, in allowed spectra, be made most easily with help of a Fermi plot. Forbidden spectra in the so-called “unique” class have also well-known shapes.I3 The other forbidden beta spectra have shapes depending on the accidental values of some nuclear matrix elements which are unknown. This fact introduces an additional uncertainty in the measurement of the lo G. White Grodstein, Natl. Bur. Standards (U.S.) Circ. No. 683 (1957); C. M. Davisson, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 857. Interscience, New York, 1955. 1 1 S. Hultberg, Nuclear Instr. & Methods 10, 24 (1960); S. Hultberg, D. Horen, and J. Hollander, Nuclear P h p . 28, 471 (1961); W. F. Frey, J. H. Hamilton, and S. Hultberg, Arkiv Fysik 21, 383 (1962). 12 M. Mladjenovic and A. Hedgran, Arkiv Fysik 8, 49 (1954); B. S. Dzhelepov et al., Nuclear Phys. 2, 408 (1956). 13 N. Dismuke, M. E. Rose, C. L. Perry, and P. R. Bell, U.S. Atomic Energy Comm. Rept. 0.R.N.L.-1222 (1952); M. E. Rose, C. L. Perry, and M. M. Dismuke, U.S. Atomic Energy Comm. Rept. 0.R.N.L.-1459 (1953); K. Siegbahn, ed., “Beta- and Gamma-Ray Spectroscopy,” cf. p. 875. Interscience, New York, 1955; A. H. Wapstra, G. J. Nijgh, and R. van Lieshout, “Nuclear Spectroscopy Tables.” Interscience, New York, 1959.

158

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

intensity of the continuous beta spectrum, though in the most frequent case of first forbidden transitions, the deviations are mostly small.14 The influence of this uncertainty on the conversion coefficients can in principle be avoided by measuring the electron spectrum in a beta ray spectrometer in coincidence with K X-rays due to internal conversion; the X-rays may be detected by a scintillation counter. In the special case discussed above we now find 1 / a ~=

TL,-K/TLSK

- (K

+L + M +

*

*

.)/K

(2.4.2.2.2)

in which npg indicates the number of coincidences per nuclear electron selected by the beta spectrometer (of course corrected for backgrounds and accidental coincidences), and n,-K the same per K conversion electron. Equations (2.4.2.2.1) and (2.4.2.2.2) are rather similar; the difference is that, using Eq. (2.4.2.2.2),n g K can be measured at places in the beta spectrum where the influence of disturbing effects and backgrounds is lowest. In the same simple cases, the K conversion coefficient may be obtained by comparing the gamma ray intensity with that of the K X-rays caused by the conversion process: a~ =

(E,/EK)~K/TWK

where X K and y represent the photopeak arcas in the scintillation spectrum, W K is the fluorescent yield and E, and EK are the respective efficiencies for detecting the gamma and the X-rays in the photopeak of the scintillation spectra. Sources used in this method should be thin enough to avoid absorption or fluorescent excitation of X-rays. The ratio E J E K can be computed accurately if the gamma energy is so low that the mean free path for absorption is smali compared with the dimensions of the counting crystal. I n this case, this method is capable of high precision. A somewhat more involved coincidence method was applied by Petterson et aZ.16on Aulg8.They measured coincidences between the 412 kev gamma rays and the nuclear electrons in Aul9s, and between 134 kev gamma rays and the L conversion line of the 165 kev transition in Hglg7 (decay schemes see Fig, 3). In this case,

in which n are again the number of coincidences per detected gamma ray; N ~ ~and Z KNp are the count rates of nuclear and conversion electrons, at the top position of the conversion line and at the point where the Py coinl4

l6

A. H. Wapstra, Nuclear Phya. 9, 519 (1958). B. G.Petterson, J. E, Thun, and T,R. Gerholm, NucZeur Phys. 24,243 (1961).

2.4.

DETERMINATION

OF SPIN, PARITY, A N D NUCLEAR MOMENTS

159

cidences were taken respectively. The factor c is the L conversion probability of the 165 kev line except for a 4 % correction for the difference between the line shapes of the 412K line and the complex 134L line. Another met,hod was used by Lewin et a1.16 in a study of TIzoo.They compared coincidences between the group of 1200 kev lines (see Fig. 3) preceding the 368 kev transition and the K conversion line of the last TLOO

ffl

Au 198

7

E @ I97

h

' .

O+

O+ ~ 1 9 8

YZ+

HqKX>

p" ~1203

FIQ. 3. Decay schemes of Hg197, Aulgg, Tlzo'J, and HgZoa,the middle two partly simplified. The values give transition energies in kev.

transition with coincidences between the 412 kev gamma rays and the nuclear electrons in Aulg8.Then, l / a ~= c(nsr41z/n,-,izoo)(n,/ns>

-

(K

+L + M + .

*

.)/K.

Here, ne is the ratio of top counting rate and area of a monoenergetic electron line, and ng is the ratio of the counting rate of nuclear electrons in AulS8a t the point where nar is measured and the area of the continuous beta spectrum. The quantities ng, and ne-,are the numbers of coincidences per recorded gamma ray; the factor c, nearly unity, corrects for the dependence of the coincidence and counting efficiencies on th e electron energy, and for the fact that the actual decay schemes are somewhat more involved than depicted in Fig. 3. I n complicated cases, some of the above methods cannot be applied t o yield accurate results. Then, the method of comparison of external and internal conversion lines may become very useful. If these lines are measured in the same spectrometer, both are proportional t o the same solid angle, which therefore does not occur in the final result allowing higher precision. l6

W. H. G. Lewin, B. van Nooijen, and A. H. Wapstra, Nuclear Phys. 27, 681

(1961).

160

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

If no strong enough sources are available to allow external conversion measurements, one still has not necessarily to know the absolute efficiencies of electron and gamma spectrometers. Often, a conversion coefficient of one of the transitions in the nuclide studied may be known; it can then be used to calibrate the electron and gamma counters relative to one another. Or, they may be relatively calibrated by measuring a standard nuclide in both. In nuclides decaying by negaton decay alone, the total amount of K X-radiation should be equal to the number of K conversion electrons multiplied by the fluorescent yield.17 Then, the ratio of K Auger electrons to that of K X-rays is known with considerable accuracy. And in nuclides emitting positrons, two annihilation quanta of 511 kev are emitted for every positon. Thus, one of the last three possibilities may be used in calibrating beta and gamma spectrometers relative to one another in order to determine conversion coefficients. In cases where the above methods fail, one may adopt a theoretical conversion coefficient for one of the transitions studied. Thus, the transition between the first excited 2+ state and the ground state in even-even nuclei is necessarily a pure E2 transition; and its conversion coefficient is often found to agree well with theory (though a few real exceptions may exist la). 2.4.2.2.4. EXAMPLE OF USEOF CONVERSION DATA.As a rather typical example for the use of conversion coefficient measurements we consider the case of HgZo3. This nuclide decays as depicted in Fig. 3 to a 279.12 kev excited level in TlZo3 decaying to the ground state by a single transition; the conversion coefficients can therefore be measured by the special methods outlined above. Accurate meas~rernents’~ yielded the results given in Table I; all experimental data have errors of only a few per cent. Theoretical values for the conversion coefficients of pure multipole transitions of a 279.12 kev gamma transition in TI are also given in Table I. Since parity is a good quantum number for electromagnetic transitions, only those multipole transitions can occur mixed in which the change in parity is the same (the upper and lower parts in Table I, respectively). The conversion coefficient for a mixture of two multipolarities is a! = aa1 (1 - a)a!,

+

1’ A. H. Wapstra, G. J. Nijgh, and R. van Lieshout, “Nuele&rSpectroscopy TabIes.” Interscience, New York, 1959. 18 J. F. W. Jansen, S. Hultberg, P. F. A. Goudsmit, and A. H. Wapstra, Nucleaf Phys. (to be published). 1 9 G. J. Nijgh, A. H. Wapstra, L. Th. M. Omstein, N. Salomons-Grobben, J. R. Huizenga, and 0. Almh, Nuclear Phys. 9,528 (1958), and other work discussed there.

2.4.

DETERMINATION O F SPIN, PARITY, AND NUCLEAR MOMENTS

161

and LYZ being the theoretical conversion coefficients for these multipolarities and a the percentage gamma rays of the first type (a is therefore the same for conversion in different shells). Comparison of the measured K / L ratio (the most easily obtained datum) with the theoretical values shows that the transition is a combiLYI

TABLEI. Conversion Coefficients ( x 104) and Ratios for the 279 kev Transitions in T I 2 0 3 (Theoretical data for transitions in which the parity does not change are printed above the experimental data printed in bold-face type, the other below. All theoretical data are obtained from Rose's tables6 except those printed in italics which are due to Sliv.') Conversion ratio

fif 1

41 10 3920

E2 M3

Exp. El M2 E3

764 770 41200 1620 278 14650 2060

646 684 479

586

55

616

64

108

618

110

19700 480 46 3890 4530

12700 242 34 3170 499

243 271 2400 166 7 434 2890

5 5 127 137 4550 82 6 291 1135

6.37 6.76 1.60 1.49

2.10 3.37 6.08 3.76 0.45

0.907

0.086

0.900

0,093

0.226 0.212 0.647

0.508 0.623 0.122

0.604 0.732 0.814 0.110

0.326 0.145 0.111 0.639

0.007 0.007 0.266 0.265 0.231 0.171 0.123 0.075 0.251

TABLE 11. K Conversion Coefficients and L Subshell Ratios Giving the Correct K / L Ratio for the 279 kev Transition in T l * ' J 3

+ + M2 + E3, Rose

E2, Rose MI E2, Sliv Experimenta1 MI

OIK

Lr / L

LllIL -

LIIIIL

0.179

0.479 0.517 0.604 0.732

0.351 0.332

0.170 0.151

0.326 0.173

0.171 0.095

0.196

0.162 1.33

+

+

nation of either M 1 E2, or M 2 M3. The theoretical K conversion coefficients and the LI :LII :L I I Iratios for the mixtures yielding the correct K / L ratio are given in Table 11. Comparison with the experimental values E 2 mixture. The ground state has a spin points very strong t,o a M 1 and positive parity; our result therefore proves that the 279 kev level in TI2O3has a spin and positive parity. The theoretical values in Table I1 do not agree completely with the experimental result, probably due to the dynamic finite size effect discussed in the beginning of this section. According t o an estimate by Kiss-

+

+

162

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

linger,20the influence of this correction would decrease the theoretical value of LYK by about 10%. Unfortunately, this arbitrariness in the theoretical conversion coefficients for pure multipoles hampers an accurate determination of the multipole mixing ratios a from conversion coefficients alone: the values for a as computed from Rose’s theoretical values and the experimental K , LI, LII, and LIII conversion coefficients are 0.256 0.012, 0.280 k 0.013, 0.46 f 0.03, and 0.37 k 0.03, respectively. One would be inclined to reason that the LIII conversion coefficients are least affected by the dynamic finite size effects since the LrII electron wave function has a minimum at the nucleus. The value for a as obtained above from the LIII result agrees indeed fairly well with the result a = 0.31 5 0.025 found by McGowan and StelsonZ1from angular distribution measurements. The theoretical results of Sliv lead to a value a = 0.42 f 0.03, in somewhat worse agreement with McGowan and Stelson’s result.

2.4.2.3. Nuclear Orientation.* 2.4.2.3.1. INTRODUCTION. A nucleus of total angular momentum I is oriented when the ( 2 1 1) magnetic substates have unequal occupational probabilities. Such a nucleus defines a certain spatial anisotropy and, when subsequent emission or absorption processes occur, this anisotropy is carried over into the final state as a consequence of the conservation laws involved. The detailed theories of these effects are closely related to those for angular correlations, and both can be treated by the same very general and refined

+

L. S. Kisslinger, Bull. Am. Phys. SOC.2, 358 (1958). F. K. McGowan and P. H. Stelson, Phys. Rev. 107, 1647 (1957);and private communication. 1 L. C. Biedenharn and M. E. Rose, Revs. Modern Phys. 26, 729 (1953). F. Coester and J. M. Jauch, Helv. Phys. Acta 40, 3 (1953). a S. Devons and L. J. B. Goldfarb, i n “Handbuch der Physik-Encyclopedia of Physics” (S. Flugge, ed.), Vol. 43, p. 362. Springer, Berlin, 1957. M. E. Rose, “Elementary Theory of Angular Momentum.” Wiley, New York, 1957. 5 U. Fano and G. Raoah, “Irreducible Tensorial Sets.” Academic Press, New York, 1959. * M. Jacob and G. C.Wick, Ann. Phys. (N.Y.) 7,404 (1959). 7 L. J. B. Goldfarb, i n “Nuclear Reactions” (P. M. Endt and M. Demeur, eds.). North-Holland Publ., Amsterdam, 1959. 8 L. C. Biedenharn, in “Nuclear Spectroscopy” (F. Ajzenberg-Selove, ed.), Part B, Chapter V.C. Academic Press, New York, 1960. 9 S. R. de Groot and H. A. Tolhoek, in “Beta and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 613. Interscience, New York, 1955. $0

$1

-

* Section 2.4.2.3is by E. Ambler.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

163

Indeed, it is appropriate to regard nuclear orientation and angular correlation as special cases of the most general observation, where a complete set of observations of the angular distribution and polarization of particles and radiation is made. Of course, for any particular purpose, e.g., for extracting given quantities of physical interest, many of these observations would be irrelevant or redundant, and one would have to select those which are appropriate. In particular, one would need to determine whether it is a n advantage to know the orientation of the initial nucleus. At this point we should note that there are two types of nuclear orientation: nuclear polarization, where there is a net nuclear moment; and nuclear alignment, where orientations parallel and antiparallel to a given axis are preferred over other orientations although no net moment exists. The experiments that have been carried out to date involve mainly the orientation of radioactive nuclei and the observation of the angular distribution of the decay products, although work in other fields, e.g., that of nuclear reactions, has begun. It should be remembered, however, that the method is an indirect one, and quantities such as spins and parities are seldom determined in a straightforward and unambiguous way starting from a completely unknown situation. The experimental method separates naturally into the following parts: (I) the production of oriented nuclei; (2) the determination of the degree of nuclear orientation produced; (3) the measurement of the relevant angular distributions, including in some cases polarization or correlation measurements; (4) the computation of theoretical curves and their subsequent comparison with experiments.

As a general rule, the measurement of spins and parities is possible simply because, for a given degree of nuclear orientation, angular' distributions for certain spin sequences can be fit to the data and others cannot. The measurement of nuclear moments, on the other hand, is possible when a decay scheme is known so that the degree of nuclear orientation, which is directly related to a nuclear moment, can be determined. The method is entirely reliable for the measurement of spins provided sufficient initial information concerning the decay scheme is available. The same can hardly be claimed for the measurement of nuclear moments, where the sensitivity is seldom such as to give accuracies better than lo%, and where agreement with more direct determinations is often not very good. The method has the advantage that it can be carried out with radioactive nuclei of which only a small quantity is available.

2.

164

DETERMINATION O F FUNDAMENTAL QUANTITIES

TABLE I. Experiments Using Oriental Nucleia Nucleus

Ref.

So46 Cr61 Mn 62 Mn MnKZ MnKZ Mn 6zm MnK4 Mn6* MnK4 MnK6 Mn66 Mn 68 Mn 68

1 2 3 4

co

C066

co

co CO67 co co co Cob8 C068

co

c o 80 coeo

co

ED

COSO CoEO

co co 80 go

Co60 As16 In114m

In116 SblZ2 Sblz2 I131

Cela7 ce1a7m

Cela9 Cela9 CeI4l Cel4I

Method S6

5 6

53 53 53 s3 53

7

53

8 9 7

53 53 s3 53

10 11

12 13

14 13 15,16 17 18 19-21 22 23 24 6 17 25

20,26 22 27 28 29-31 1 24,32,33 34 35 36,37 35 38,39 40 41 41 42 43 43 44

52

53 53 s3

s3 82 53 52 s2 s2 s2

53

s3 53 s3 s2 52 53 53 s5 S6 53 D5 S6

s1 S6

D 4 & D5 54 52 s2 52 52 s2 s2

Substance

Observation

Iron alloy “Brine holes” in CMN CMN CMN CMN CMN CMN CMN CMN CMN Mn (ND4)2(S04)2.6D~0 NFS CMN CMN CMN CMN CU(NHdz(S04)~.6HzO CMN CU(NHJ z (SO,) 2.6H20 CU(NHdz(S04)~6HzO CU(NHAz(S04)2.6H20 NFS CMN CMN CMN CMN CU(NH~Z(SO~)Z.~HZ~ CU(NH,)z(SOdz.6HzO CMN CMN Co metal Iron alloy CMN As in silicon Iron alloy Metal Iron alloy Sb in silicon Co (Zn)p-t,p-i NES NES CES CMN CMN CES

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

165

TABLE I (Continued) Nucleus

Ref.

Method

Substance

45 46 47 13 48 49 43 50 51 52 53 54 55,59 56 56 57 58 59 60 35 61 61 61,62

s2 & s 3 s2 & 53 s2 s3 s2 S2,S3 s2 52 s2 & S3 s3 Sl S2,S3 S2,S3 s2 s2 s3 52 s3 52 S6 s4

NES NES CES CMN N ES NES CMN NES NES SES Metal NES NES NES NES HES NES NES YES Iron alloy URN URN URN

54 S2$4

Observation

a Column one refers to the nucleus oriented, column two to the original publication, column three to the method used (cf. Table 11), column four to the substance used, and column five to the observations made. In the fourth column we have used the following abbreviations: CMN = cerous magnesium nitrate, NFS = nickel fluosilicate, Co (Zn)p-t,p-i = cobalt (zinc) p-toluenesulfonate, p-iodobenzenesulfonate, NES, CES, SES, HES, YES = neodymium, cerium, samarium, holmium, ytterbium ethyl sulfates respectively, and URN = uranyl rubidium nitrate. I n the fifth column the angular distribution of alpha, beta, and gamma rays is denoted by w(a,q), w(p,q), and w(k,v) respectively; w(k,v)& and w(k,q).5 refer to plane polarization and circular polarization measurements of gamma rays respectively; an refers to a measurement of slow neutron capture cross section; w(p,k,v) refers to beta-gamma angular correlation; w(kl,k2,q)to gamma-gamma angular correlation, and w(f,q) to the angular distribution of fission fragments.

REFERENCES TO TABLE I 1. A. V. Kogan, V. D. Kul'kov, L. P. Nikitin, N. M. Reinov, I. A. Sokolov, and

M. F. Stel'makh, Zhur. Eksptl. i Teoret. Fiz. 99, 47 (1960); Soviet Phys. JETP 12, 34 (1961). 2. M. Kaplan and D. A. Shirley, Phys. Rev. Letters 6, 361 (1961). 3. W. J. Huiskamp, M. J. Steenland, A. R. Miedema, H. A. Tolhoek, and C. J. Gorter, Physica 22, 587 (1956).

166

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

REFERENCES TO TABLE I (Continued) 4. W. J. Huiskamp, A. N. Diddens, J. C. Severiens, A. R. Miedema, and M. J. Steenland, Physica 23, 605 (1957). 5. E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Rev. 110,787 (1958). . . 6. H. Postma, W. J. Huiskamp, A. R. Miedema, H. A. Tolhoek, and C. J. Gorter, Physim 24, 157 (1958); H. Postma, Thesis, University of Leiden, Netherlands,

unpublished. 7. R. W. Bauer, M. Deutsch, G. 8. Mutchler, and D. G. Simons, Phys. Rev. 120, 946 (1960). 8. M. A. Grace, C. E. Johnson, N. Kurti, H. R. Lemmer, and F. N. H. Robinson, Phil. Mag.[71 46,1192 (1954). 9. G. R. Bishop, J. M. Daniels, H. Durand, C. E. Johnson, and J. Perez y Jorba, Phil. Mag.L71 46,1197 (1954). 10. S. Bernstein, L. D. Roberts, C. P. Stanford, J. W. T. Dabbs, and T. E. Stephenson, Phys. Rev. 94, 1243 (1954); J. W. T. Dabbs and L. D. Roberts, ibid. 96, 970 (1954). 11. P. Dagley, M. A. Grace, M. Gregory, J. S. Hill, and C. V. Sowter, reported by N. Kurti, Physica Suppl. S164 (September, 1958). 12. R. W. Bauer and M. Deutsch, Phys. Rev. 117, 519 (1960). 13. H. Postma and W. J. Huiskamp, PTOC.7th Intern. Conf. Low Temp. Phys. p . 183 (1961). 14. R. W. Bauer and M . Deutsch, Nudear Phys. 16, 264 (1960). 15. L. J. Gallaher, C. Whittle, J. A. Beun, A. N. Diddens, C. J. Gorter, and M. J. Steenland, Physica 21, 117 (1955). 16. 0. J. Poppema, J. E. Siekman, and R. van Wageningen, Physica 21, 223 (1955). 17. E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Rev. 108,503 (1957). 18. G. R. Bishop, M. A. Grace, C. E. Johnson, A. C. Knipper, H. R. Lemmer, J. Perez y Jorba, and R. G. Scurlock, Phil. Mag. [7] 46, 951 (1951). 19. J. M. Daniels, M. A. Grace, H. Halban, N. Kurti, and F. N. H. Robinson, Phil. Mag.[7] 43, 1297 (1952). 20. J. C. Wheatley, D. F. Griffing, and R. D. Hill, Phys. Rev. 99,334 (1955). 21. D. F. Griffig and J. C. Wheatley, Phys. Rev. 104, 389 (1956). 22. G. R. Bishop, J. M. Daniels, G. Goldschmidt, H. Halban, N. Kurti, and F. N. H. Robinson, Phys. Rev. 88, 1432 (1952). 23. P. Dagley, M. A. Grace, J. S. Hill, and C. V. Sowter, Phil. Mag. [S] 5, 489 (1958). 24. E. Ambler, R. W. Hayward, D. D. Hoppes, R. P. Hudson, and C. 6. Wu, Phys. Rev. 106, 1361 (1957). 25. E. Ambler, M. A. Grace, H. Halban, N. Kurti, H. Durand, C. E. Johnson, and H. R. Lemmer, Phil. Mag. [7] 44, 216 (1953). 26. B. Bleaney, J. M. Daniek, M. A. Grace, H. Halbaa, N. Kurti, F. N. H. Robinson, and F. E. Simon, Proc. Roy. Soc. A221, 170 (1954). 27. P. S. Jastram, R. C. Sapp, and J. G. Daunt, Phye. Rev. 101, 1381 (1956). 28. J. C. Wheatley, W. J. Huiskamp, A. N. Diddens, M. J. Steenland, and H. A. Tolhoek, Physica 21, 841 (1955). 29. M. A. Grace, C. E. Johnson, N. Kurti, R. G. Scurlock, and R. T. Taylor, in “ConfBrence de physique des basses temphratures,” p. 263. Centre National de la Recherche Scientifique and UNESCO, Paris, 1956; BuZl. Am. Phys. Soc. 2, 136 (1957).

2.4.

DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS

167

REFERENCES TO TABLEI (Continued) 30. G. R. Khutsishvili, Zhur. Eksptl. i Teoret. Fiz. 29, 894 (1955); Soviet Phys. J E T P 2, 744 (1956). 31. J. M . Daniels and M. A. R. LeBlanc, Can. J . Phys. 37, 1321 (1959). 32. C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Rev. 106, 1413 (1957). 33. C. E . Johnson, M. A. Grace, R. G. Scurlock, and C. V . Sowter, Phil. Mag. [8] 2, 1050 (1957). 34. F. M. Pipkin and J. W. Culvahouse, Phys. Rev. 109, 1423 (1958). 35. B. N. Samoilov, V. V. Sklyarevskii, and E. P. Stepanov, Zhur. Eksptl. i Teoret. Fiz. 38, 359 (1960); Soviet Phys. J E T P 11, 261 (1960). 36. J. W. T. Dabbs, L. D. Roberts, and S. Bernstein, Phys. Rev. 98, 1512 (1955). 37. A. Stolovy, Phys. Rev. 118, 211 (1960). 38. F. M. Pipkin, Phys. Rev. 112, 935 (1958). 39. G. E. Bradley, F. M. Pipkin, and R. E. Simpson, Phys. Rev. 123, 1824 (1961). 40. C. E. Johnson, J. F. Schooley, and D. A . Shirley, Phys. Rev. 120, 1777 (1960). 41. J. N. Haag, C . E. Johnson, D, A. Shirley, and D. H. Templeton, Phys. Rev. 1% 591 (1961). 42. M. A. Grace, C. E. Johnson, R. G. Scurlock, and R. T. Taylor, Phil. Mag.,to be published. 43. E. Ambler, R. P. Hudson, and G . M . Temmer, Phys. Rev. 101, 196 (1956). 44. C. F. M. Cacho, M. A. Grace, C. E. Johnson, A. C. Knipper, R. G. Scurlock, and R. T. Taylor, Phil. Mag. 171 46, 1287 (1955). 45. D. D. Hoppes, E. Ambler, R. W, Hayward, and R. S. Kaeser, Phys. Rev. Letters 6, 115 (1961). 46. D. D. Hoppes, Natl. Bur. Standards (US.)Rept. 93 (unpublished). 47. M. A. Grace, C. E. Johnson, R. G. Scurlock, and R. T. Taylor, Phil. Mag. [81 3, 456 (1958). 48. C.A. Lovejoy, J. 0. Rasmussen, and D. A . Shirley, Phys. Rev. 123, 954 (1961). 49. D. A. Shirley, J. F. Schooley, and J. 0. Rasmussen, Phys. Rev. 121,558 (1961). 50. G. R. Bishop, M. A. Grace, C. E. Johnson, H. R. Lemmer, and J. Perez y Jorba, Phil. Mag. [Sl 2, 534 (1957). 51. G. A. Westenbarger and D. A. Shirley, Phys. Rev. 123, 1812 (1961). 52. L. D. Roberts, S. Bernstein, J. W. T. Dabbs, and C. P. Stanford, Phys. Rev. 95, 105 (1954). 53. A. Stolovy, Bull. Am. Phys. SOC.[II] 6 (3), 275 (1961). 54. C. A. Lovejoy:andLD. A . Shirley, Proc. 7th Intern. Cmf. Low Temp. Phys. p. 164 (1961). 55. C. E. Johnson, J. F. Schooley, and D, A. Shirley, Phys. Rev. 120, 2108 (1960). 56. Q. 0. Navarro and D. A . Shirley, Phys. Rev. 123, 186 (1961). 57. H. Postma, H. Marshak, V. L. Sailor, F. J. Shore, and C. A . Reynolds, Bull. Ant. Phys. SOC.[I11 6 (3), 275 (1961). 58. H. Postma, A. R. Miedema, M. C. Eversdijk Smulders, Physiea 26,671 (1959). 59. H. Postma, M. C. Eversdijk Smulders, and W . J. Huiskamp, Physica 27, 245 (1961). 60. M. A. Grace, C . E. Johnson, G. R. Scurlock, and R.:T. Taylor, Phil. Mag. [81 2, 1079 (1957). 61. J. W. T. Dabbs, L. D. Roberts, and G. W. Parker, Physica Suppl. 569 (September 1958); Proe. 7th Intern. Cmf. Low Temp. Phys. p. 174 (1961). 62. S. H. Hanauer, J. W. T. Dabbs, L. D. Roberts, and G. W. Parker, ORNL Report 2919 (unpublished).

168

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

It is clear that the analysis of the data of each experiment will be very different in detail, and that the results of such analysis may be complicated. It would be neither possible nor appropriate, therefore, to redescribe all the experiments that have been done to date. We have tried to list them concisely in Table I, however, so that the reader may refer to the original papers for important and special details. The aim of the greater part of the article is to give an account of the general physical principles, some of the special techniques, and some of the mathematical methods of the theory that are fast becoming developed to the point where the experimentalist might be expected to use them. The reader will find useful a number of review articles.1°-17 2.4.2.3.2. THEPRODUCTION O F O R I E N T E D NUCLEI-STATICMETHODS. We may divide into three groups the methods proposed for orienting nuclei, and call them the static, dynamic, and transient methods. We have listed these methods together with certain information about them in Table I1 (a), (b), and (c) and have numbered them S1, S2, etc., for easy reference. The basic ideas of the static methods are as follows. The degeneracy of the ( 2 1 1) spatial orientations of the nucleus can be removed either by a coupling of the nuclear magnetic moment to a magnetic field or by a coupling of the nuclear quadrupole moment t o an electric field gradient. If the nuclei are then cooled to a temperature, T , such that kT ( k is Boltzmann’s constant) is of the order of the hyperfine splitting, the lower levels, which correspond to preferred orientations of the nuclear spin, become preferentially populated. The use of external electric fields is not possible since the magnitudes required exceed the dielectric strength of the substances to which they must be applied. The use of a n external

+

10 R. J. Blin-Stoyle and M. A. Grace, i n “Handbuch der Physik-Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 42, p. 555. Springer, Berlin, 1957. 1lR. J. Blin-Stoyle, M. A. Grace, and H. Halban, i n “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 600. Interscience, New York, 1955. 12 M. J. Steenland and H. A. Tolhoek, i n “Progress in Low Temperature Physics” (C. J. Gorter, ed.), Vol. 11, p. 293. North-Holland Publ., Amsterdam, 1957. 13E. Ambler, in “Progress in Cryogenics” (K. Mendelssohn, ed.), Vol. 2, p. 235. Heywood, London, 1960. 14 C. D. Jeffries, in “Progress in Cryogenics” (K. Mendelssohn, ed.), Vol. 3, p. 129. Heywood, London, 1961. 14& G. R. Khutsishvili, U s p e k h i Piz. Nauk 71, 9 (1960); Soviet P h y s . - U s p e k h i 3, 285 (1960). R. P. Hudson, in “Progress in Cryogenics” (K. Mendelssohn, ed.), Vol. 3, p. 99. Heywood, London, 1961. 16 N. Kurti, Nuouo cimenfo Suppl. [lo] 6, 1101 (1957). 17 W. J. Huiskamp and H. A. Tolhoek, in “Progress in Low Temperature Physics” (C. J. Gorter, ed.), Vol. 111, p. 333. North-Holland Publ., Amsterdam, 1961.

2.4.

DETERMINATION

OF SPIN,

PARITY,

AND NUCLEAR MOMENTS

169

magnetic field, on the other hand, provides a most direct and general way of orientating nuclei. Although it has been used successfully to a certain extent, rapid development of this so-called “brute-force” method has been hindered largely by the slow technical development of reliable steady high magnetic fields. With the continued development of high power solenoids, and the further possibility of high field superconducting solenoids, however, this situation is likely to change, and since the method has such a wide applicability we shall discuss it in some detail. We shall then discuss the other methods that have been proposed, which rely upon the presence in certain compounds of high internal fields, i.e., large hyperfine splitting. Some of the methods in this last group have met with considerable success. 2.4.2.3.2.1. External Field Polarization-El. The feasibility of the method in principle,’* and the appreciation of the practical problems in detai1,lg have long been known. More recent and extensive studies have allowed the method to be used in a significant although limited way.20-22 If an assembly of nuclei of magnetic moment p is coupled to an external field H a t temperature T , the nuclear polarization is given by: fl

= &(X)

=

21

+

1 ___ coth 21

(T + 21

1

1

x) - 2f coth

($x )

(2.4.2.3.1)

where &(x) is the Brillouin function and x = p H / k T . The value of f l is the same as the ratio of the nuclear magnetization to that a t saturation, and this quantity has been tabulated already in connection with electronspin m a g n e t i ~ m . 2 ~If~we 2 ~ adopt the figure of f l = 0.2 as being a reasonably large nuclear polarization for carrying out experiments, we find that for nuclear magnetic moments normally encountered, we require H I T = 106-107 gauss/”K. Now, as we shall discuss later, a value of T = 0.05”K can be obtained fairly easily, whereas a value of T = 0.01”K should be possible; therefore, magnetic fields in the range 104-106 gauss are suitable. As an illustration, we take what we believe to be a reasonable figure, viz., H / T = 4 X lo6 gauss/”K (e.g., 100 kilogauss a t 0.025”K) F. E. Simon, Conrpt. rend. ccungr. sur le magnt%isme, Strasbourg, 1939,p. 1 (1939). N. Kurti, “Les phknombnes cryomagn&iques,” p. 29. Coll&gede France. Paris, 1948. 20 J. W. T. Dabbs, L. D. Roberts, and S. Bernstein, ORNL Report, central files number 55-5-126(un-published). 2 1 J. W. T. Dabbs, L. D. Roberts, and S. Bernstein, Phys. Rev. 98, 1512 (1955). 22 A. Stolovy, Phys. Rev. 118, 211 (1960). 2s J. R. Hull and R. A. Hull, J. Chem. Phys. 9, 465 (1941). 24 L.P. Schmid and J. S. Smart, LLTables of Some Thermodynamic Functions which Occur in the Theory of Magnetism.” Navard Report 3640, U.S. Naval Ordnance Laboratory, White Oak, Maryland. 1s 19

TABLE 11. List of Methods Proposed for Orienting Nuclei (a). Static Methods of Orienting Nuclei Method

No.

Ref.

External field polarization

s1

1-3

Magnetic hfs alignment

52

4

w

l

0

Magnetic hfs polarization Electric hfs alignment

S3

5,6

s4

7

Ferromagnetics and anti-ferromagnetics

55

8-10

Dilute solid solutions in above

S6

10-13

Criteria

-

&upling p"H.1 HIT lohi07 gauss per degree K Coupling AS,Z,or

+

B(SJ, S J d S = 4 and A >> B or B >> A S>&andA,B#O Z > $ As above, but S = $ and A = B included P#O

Large enough hfs; coupling for alignment as in S2 and for polarization as in 53 As above

Applicability

All nuclei with pn. Long nuclear relaxation times may limit method to metals and alloys Paramagnetic compounds, e.g.9 Ce2Mg,(N03)1~24H20 Nd(C,H,SO,)a.SH,O Rare earth metals (See

Special Requirements Adiabatic demagnetization and contact cooling. Poor thermal conductivity is a severe problem Adiabatic demagnetization, usually

55) As above

Covalently bonded complexes, e.g., UOZ in (U02)Rb(N03)r High density of relevant nuclei is a very important advantage for work on nuclear reactions Whenever mechanism (e.g. diamagnetic core polarization in iron alloys) producing large hfs is operative

As above, but external field

102-103 gauss needed Temperatures < 1"K, usually

As above

As above

(b). Dynamic Methods of Orienting Nuclei Method Optical pumping

No.

Ref.

Criteria

D1

1

Resonance with low-field Zeeman splitting; F = I S, is oriented by transitions induced by a polarized light beam As D1 but orientation is passed o n t o other atoms in vapor by collisions. Large cross section for spin-flip exchange with atoms of added vapor. Depolarization at walls must be small Z relaxes via A S . I coupling; saturate electron spin resonance

Applicability

+

Optical pumping with ‘spin transfer’

D2

2

rf and microwave pumping in metals

D3

3

Extension t o nonmetals and semiconductors

D4

4-6

Use of ‘forbidden’ transitions

D5

Atoms in vapor

Metals with unpaired s-electrons

Nuclear relaxation via Paramagnetic centers with resolved hfs AS . I; or A S J , B(SzZz &ZV) with H along z-axis hfa with some anisotropy _ _ in order to give reasonable transition probabilities with rf and external fields parallel. Relaxation transitions, which produce orientation in D3, are ‘driven’ by rf field. Differential effect with inhomogeneously broadened line

+

7

Atoms in vapor, e.g., alkali metals, mercury. Possible extension t o solids

+

Special Requirements Vapor at ordinary temperaturea. When radioactive species used, large background due t o wall absorption is troublesome A s above

T

‘V 1°K preferably; relatively high rf powers needed, which can be a disadvantage T ‘V 1°K preferably

As above

TABLE I1 (Continued) (c). Transient Methods of Orienting Nuclei Method

No.

Ref.

Double resonance

T1

1

hfs must be resolved, but Paramagnetic centers with detailed form not so resolved hfs important. Necessary to make adiabatic rapid passage through electron spin resonance and nuclear spin resonance successively. Nuclear spin relaxation must not be too short

Three other transient effects

T2

2

Z and S relax via A S . I coupling, i.e., main relaxation process is that where S and Z flip together

c. -3 t 9

T3

T4

3

Criteria

Hyper6ne coupling is mainly of the type AS * I

Applicability

Special Requirements

T

‘v

1°K preferably

Only necessary to turn on external field Saturate two electron resonances in turn where Z does not flip After thermal equilibrium, decrease external field from Paschen-Back to Zeeman region (a) so that change is at all times adiabatic, and (b) so that the time taken is +. For S = 4 there are two extreme cases. When A >> B the energy levels are as shown in Fig. 3a, with the nuclear magnetic substates I , = + I lowest. I n this case the nuclear spin aligns parallel and antiparallel to the z-axis. When B >> A the energy levels are as shown in Fig. 3b, the nuclear magnetic (for I half-integral) or I , = 0 (for I integral) lie substates I , = lowest, and nuclear spin “aligns,” or rather precesses, in a plane perpendicular to the z-axis. When S > p there are again two cases, viz., D negative and D positive. We have shown the energy levels in the former

*+

s4 J.

M. Baker and B. Bleaney, Proc. Roy. Soc. A245, 156 (1958). B. Bleaney and K. W. H. Stevens, Repfs. Progr. in Phys. 16, 108 (1953). K. D. Bowers and J. Owen, Repts. Progr. in Phys. 18, 304 (1955). J . w. Orton, Repts. Progr. in Phys. 22, 204 (1959). M. H. L. Pryce, Nuovo cimento Suppl. [lo] 6, 817 (1957). 39 A. Simon, M. E. Rose, and J. M. Jauch, Phys. Rev. 84, 1155 (1951).

36

180

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES rn

m f 1/2

=T

( i 3 / 2 ,i1/2) f f 3/2

if51/2 /2

f 5/2

i 5/2

f 3/2 f 1/2

f 1/2 f3/2

f 5/2

f 3/2

L2P

+1/2

L-

(C)

FIG. 3. Energy levels for various values of the parameters A , B, D and P in Eq. 2.4.2.3.8;in (a) D = 0, P = 0, and A >>B, in (b) D = 0, P = 0, and B >> A , in (c) P = 0, D >> A , B and positive, in (d) D = 0, A = 0, B = 0 and P positive. The energy levels are given assuming I = $, with S = 6in (a) and (b),and S = Q in ( c ) . The quantity m labels the eigenstates of I..

case in Fig. 3c, and have assumed for simplicity that A 2 B and D >> A . The case of D negative corresponds to the case where the electron spin levels 8,= + S are lowest, and is more favorable for producing nuclear orientation than the case of D positive. We have listed in Table I11 the paramagnetic salts most frequently used

2.4.

181

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

TABLE 111. List of Paramagnetic Salts Used for Nuclear Orientation by Methods 52 and 538

~~

8

Mn55 (X)(5) (Y) C O (X)(5) ~ ~ (Y) M+++

1.997 1.997 4.108 7.29

f g4 -_I

S

911

4+ 3i

CeI4l(6) Pr141 NdI4'(6)

4

1.997 - 80 1.997 -215 4.385 2.338 -1.84 2.72

0.25 1.55 0.45

+

D

91 ___-

90 90 85 283

- 90 - 90 103 5 1

A

B

P

A

126 237

-

-

-

-

-

-

(-20) 770 (-40)

-

-

-

-

-

-

Ethylsulfates: M + + + ( C ~ H S S O ~ ) ~ . ~ H ~ O Cooling substances: Ce(C2HSS04)3.9H20; gll(7) = 3.80, 91 = 0.22, T-T*correl.(8) Nd(C2HoS04)3.9HzO;911 = 3.535, g 1 = 2.072, T-T* correL(9)

I Ce(l0) ~r141(11) Nd14a ~m147(12) Sm147 ~b169(11 Dy167 Ho l E 616) ( Yb17s(17)

IS/

-

911

i i

D 91 50.4

3.80 $ 1.525 3 3.535 2.072 4 0.432 0.604 4 0.596 8 i 17.72 < 0 . 3 (#)i 10.76(14: 0 4 15.36 0 C4)+ 3 . 4

-

+

+

-

-

A

I

B l

755 380 165 60 2090 691(15 3340 02

A

-

(288) ( 3) -

3870 -

650 -

P

-

.

Cooling substance: (Ni 15%, Zn 85%) SiFr6HzO; S = 1, g = 2.29, D = -1400, T-T* correL(l8)

_ -s

4$

co59

4

911

2.000 5.82

_

91 _

_

D

_

2.000 -134 3.44 -

~

-

A

91 184

A

P

A

47

-

-

A

P

A

B

- +150(19) -1654(21) +178 +301 862

-

-

-

a For the cooling substances references are given t o g-factors and t o T-T* correlation data. For other ions, constants in spin Hamiltonian are given for one isotope;

182

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

those for other isotopes differ only in the values of A , B[ a r / Z ] and P [ a Q / I ( 2 I - l)]. Except where explicit references are given (in parentheses), the data are taken from references 35-37 of text. The units for A , B, P, and A are lo-' cm-'. It should be noted that the constants have not usually been measured directly in the cooling substances, but rather in an komorphous diamagnetic salt (see references 35-37), and in some cases differences have been noted. REFERENCES A N D FOOTNOTES TO TABLE I11 1. M+++, M++, and Mf refer to trivalent, divalent and monovalent ions respectively. 2. A. H. Cooke, H. J. Duffus, and W. P. Wolf, Phil. Mag. [7] 44, 623 (1953). 3. R. P. Hudson, R. 5. Kaeser, and H. E. Radford, Proc. 7th Intern. Conf. Low Temp. Phys. p. 100 (1961). 4. J. M. Daniels and F. N. H. Robinson, Phil. Mag. [71 44, 630 (1953). 5. There are two sets of nonequivalent ions in unit cell; the ratio of X: Y appear to be about 1 6 : l [B. Bleaney, K. D. Bowers, and R. S. Trenam, Proc. Roy. Soc. A228, 157 (1955)l. 6. R. W. Kedzie, M. Abraham, and C. D. Jeffries,Phys. Rev. 108,54 (1957). 7. A. H. Cooke, 5. Whitley, and W. P. Wolf, Proc. Phys. Soc. (London) B68, 415 (1955). (See also reference 14 of text for discussion.) 8. C. E. Johnson and H. Meyer, Proc. Roy. Soc. A263, 199 (1959). 9. H. Meyer, Phil. Mag. [8] 2, 521 (1957). 10. No direct measurement of hyperfine splitting; see reference 41 of Table I. 11. J. M. Baker and B. Bleaney, Proc. Roy. Soc. A246, 156 (1958). 12. H. J. Stapleton, C. D. Jeffries, and D. A. Shirley, Phys. Rev. 124, 1455 (1961.) 13. Not directly determined, but see references 48 and 49 of Table I. 14. A. H. Cooke, D. T. Edmonds, F. R. McKim, and W. P. Wolf, Proc. Roy. SOC. A262, 246 (1959). 15. See reference 56 of Table I. 16. J. M. Baker and B. Bleaney, Proc. Phys. Soc. (London) A68, 1090 (1955). (See also reference 11.) 17. See reference 60 of Table I. 18. See reference 14 of text. 19. See reference 61 of Table I. 20. M. H. L. Pryce, Phys. Rev. Letters 3, 375 (1959). 21. See reference 62 of Table I. 9

for nuclear orientation by methods S2 and 53, together with the appropriate spin Hamiltonian constants. It will be noticed that almost all the substances used are those met with in work on magnetic cooling, and the nuclei to be oriented (usually radioactive isotopes) are incorporated substitutionally. This method, using a crystal of mixed constitution, works particularly well since the nuclei are in excellent thermal contact with the coolant. It is important to note, however, that spin Hamiltonian constants are usually obtained from paramagnetic resonance measurements made on a salt considerably diluted with isomorphous diamagnetic ions, and it is not invariably the case that they are the same as for the magnetically concentrated salt.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

183

To produce nuclear polarization we must, of course, apply an external magnetic field large enough to cause appreciable polarization of the electron moment. At the temperatures reached by magnetic cooling a few hundred gauss will usually suffice, provided the direction is suitably chosen with respect to the crystal field axes. Thus if we have a large D term in Eq. (2.4.2.3.8),a very anisotropic g-factor with 911 >> gl, or a very anisotropic hfs with A >> B , a magnetic field applied perpendicular to the z-axis will produce only a second order Zeeman effect. I n this case the spins remain oriented with respect to the z-axis, which is determined by the crystal field symmetry and not the external magnetic field. It is clear that the desired effect can be achieved by applying the magnetic field along the z-axis. There is an obvious advantage, therefore, in using single crystals with one kind of magnetic ion in unit cell instead of either polycrystalline specimens or single crystals with ions with different magnetic axes, e.g., the alums. The methods described in Section 2.4.2.3.8may be used to obtain the following “high” temperature approximations for the polarization and alignment : fl

= g,PHzAS(S

fz = -P(I

+ 1)(1 + 1)/9k2T2

+ 1)(21 + 3)(21 - 1)/451kT

(2.4.2.3.9)

+ (A2 - B2)S(S+ 1 ) ( 1 + 1)(21 - 1)(21 + 3)/2701k2T2 + P2(1 + 1)(21 + 5)(21+ 3)(21 - 1)(21 - 3)/18901k2T2. (2.4.2.3.10) 2.4.2.3.3. Electric hfs Alignrnent4O-S4. The coefficient P in Eq. (2.4.2.3.8)is given by41 3eQ d2V P= (2.4.2.3.11) 41(21 - 1)

where e is the absolute value of the electronic charge and Q the nuclear electric quadrupole moment. V is the electrostatic potential a t the nucleus, and therefore d2V/dz2 is the electric field gradient along the symmetry axis. Now we require P to be a t least 10V ev to obtain an appreciable nuclear alignment, and since values of Q may be of the order of one barn in favorable cases, we can see that electric field gradients of the order of 1Ol6 esu are required. Such intense field gradients are found locally in certain compounds, where the electron cloud surrounding the nucleus is very asymmetrical, for example, of the type encountered in certain covalent bonds. A notable series of compounds where this R. V. Pound, Phys. Rau. 76, 1410 (1949). See, e.g., A. R. Edmonds, “Angular Momentum in Quantum Mechanics,” p. 115. Princeton Univ. Press, Princeton, New Jersey, 1957. 40

41

184

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

situation obtains is the actinyl double nitrate^,^^^^^ where the hyperfine splittings are very large and where alignment can be achieved without having recourse to magnetic cooling. I n order to give a n idea of the pattern of the energy levels when the hfs is due to quadrupole splitting only, we have drawn the case of P negative in Fig. 3d. It will be noted from Eq. (2.4.2.3.10) that whereas with magnetic hfs alignment f 2 a 1/T2 a t high temperatures, in quadrupole alignment fz cc 1/T. 2.4.2.3.2.4. The Use of Ferromagnetics and Anti-ferromagnetics. Several of the transition and rare earth metals and alloys, as well as numerous intermetallic compounds, are ferromagnetic, and some are known to have a sufficiently large hfs so th at nuclear orientation becomes practicable (see Table I). The hfs is produced through magnetic coupling, although with transition metals, unlike the situation that obtains in paramagnetic salts, the electrons that give rise to the magnetic moment are not tightly bound. It is also a remarkable fact44that normally diamagnetic ions, when dissolved in small quantities in a ferromagnetic material such as iron, show sufficiently large hfs to make nuclear orientation possible. The mechanism that brings this about seems to be due to a polarization of the inner electron shells, which, in the case of S-shells is particularly potent in producing hfs. The method promises to have very wide application, especially for the investigation of radioactive nuclei that are normally nonmagnetic in the soIid state. Before going on to discuss the dynamic and transient methods for nuclear orientation, we shall now discuss some points of technique which pertain especially to the static methods. TECHNIQUES.2.4.2.3.3.1. Specimen 2.4.2.3.3. SOMEEXPERIMENTAL Preparation.* Most of the salts shown in Table I11 may be obtained as single crystals b y growing them from an aqueous solution. Small seed crystals are obtained, which are placed one in each of a number of small beakers of a few cm3 capacity. The beakers are filled with saturated solution which is allowed to evaporate very slowly. As the seed crystal grows, a layer of faults often occurs a t the place where the original surface of the seed was located. This can be avoided very often by warming the solution slightly (about +"C) before pouring it over the seed. The seed tends to dissolve slightly and if the warming is overdone may dissolve

* See also Vol. VI, A, Part 41

2.

J. C. Eisenstein and M. H. L. Pryce, Proc. Roy. SOC.A229, 20 (1955); A238, 31

(1956). 43 B. Bleaney, Discussions Faraday Sac. No. 19 (1955). 44 B. N. Samoilov, V. V. Sklyarevskii, and E. P. Stepanov, Zhur. Eksptl. i Teoret. Fit. 38, 359 (1960); So& Phys. J E T P 11, 261 (1960).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

185

completely! A slight dissolving, however, seems to be of value in avoiding faults at the surface of the seed. It is preferable to start with the highest grade of materials available, and sometimes a number of further purifications by recrystallization are beneficial. The rate of growth must be kept very slow and regular (one or two cm3 per week) in order to obtain clear single crystals, and even then all the crystals obtained are unlikely to be entirely free from occlusions of solution. To achieve the best results the ambient temperature should be controlled to better than i-1”C,and it is preferable to control the humidity also. The beaker should be covered to avoid dust and other contaminants entering the solution, which might cause fresh seeds to form. In practice it is not always possible to prevent the formation of new seeds, although these can be kept from interfering with the growth of the main crystal by a suitable arrangement, e.g., by suspending the main crystal in the middle of the solution away from the newly formed seeds which generally fall to the bottom of the beaker. It is always possible to stop t,he growth temporarily and to filter the solution, but there is a chance of new faults appearing when growth is restarted. Special care must be exercised in heating some solutions, particularly the ethyl sulfates since there is a tendency for the ethyl sulfate to break down into acid and alcohol. This is particularly serious because further deterioration is rapid. In growing ethyl sulfates the acidity of the solution should be checked a t regular intervals. It is also possible to grow the crystals from alcohol solutions, and often an advantage to grow them a t temperatures below normal room temperatures. The anhydrous uranyl rubidium nitrate crystals may be grown satisfactorily from solution in concentrated nitric acid. Controlled evaporation of the solvent may be effected by slowly passing dry nitrogen gas over the solution. If it is necessary to include radioactive nuclei in the crystal for gammaray measurements, the activity may be placed in the solution. Allowance should be made for the fact that in this case, as in the general case of growing crystals of mixed composition, the relative concentration of ions in the solution may not be the same as that in the crystal. When beta radioactivity is to be investigated, however, or when only a small amount of activity is available or when the half-life is very short, the activity can be mixed with a drop of solution which is then placed upon the surface of an inert crystal and allowed to crystallize. Similar remarks apply to specimens which are alpha radioactive although, because of the greatly differing scattering properties of alpha particles and electrons in matter, the need for a thin source is not so critical. The obtaining of an “ideal” beta-particle source under the circumstances is of course not possible. The preparation of a “good” beta-particle source is a t best somewhat of an

186

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

art, although it is possible in most cases to check the quality by comparing beta and gamma counting rates. In the case of the preparation of metal specimens, a number of alternative methods of preparation have been employed. It is possible to irradiate metal single crystals in a pile, for example, and subsequently anneal them.44s46 It is also possible to plate the activity on the surface and then heat treat so as to allow a slight diffusion to take place.44 2.4.2.3.3.2. Magnetic Cooling. Although there are a few useful experiments that can be done at about 1°K (the practical low temperature limit which can be reached by causing liquid He4 to boil under reduced pressure), and a t about 0.25”K (the practical low temperature limit reached by similarly using liquid He9, in most cases temperatures of about 0.01”K are required and the method of magnetic cooling is used. The principle of the method of magnetic cooling is to apply a magnetic field H O(typically in the range 10-30 kilogauss) to a suitable paramagnetic salt a t a low temperature T O (usually about 1°K). After thermal equilibrium has been established the salt is isolated thermally, usually by pumping away exchange gas, and the magnetic field is removed. The temperature then falls to a value T, as the field falls to a value H such that T, = -To *H. (2.4.2.3.12) Ho When the external field becomes very small, Eq. (2.4.2.3.12) ceases to hold and the temperature becomes dependent upon internal rather than external constraining forces, such as crystal fields and hfs. When we approach a cooperative region, as is often the case in practice, the relation is more complicated, and in fact at temperatures below a Nee1 point, the application of an external field may actually cause a slight cooling of the sample. For a detailed discussion of these and other points we refer the reader to more detailed treatments of the subject of magnetic co0ling,4~-~~ and also to Section 2.4.2.3.8. We shall refer here only to a few points of technique that are somewhat special to nuclear orientation. 41 M. A. Grace, C. E. Johnson, N. Kurti, R. G. 8curlock, and R. T. Taylor, in “Conference de physique des basses temptktures,” p 263. Centre National de la Recherche Scientifique and UNESCO, Paris, 1956. 48 C. G. B. Garrett, “Magnetic Cooling.” Harvard Univ. Press, Cambridge, Massachusetts, 1954. 47 E. Ambler and R. P. Hudson, Repts. P r o p . in Phys. 18, 251 (1955). 48 D. de Klerk and M. J. Steenland, in “Progress in Low Temperature Physics” (C. J. Gorter, ed.), Vol. 1, p. 273. North-Holland Publ., Amsterdam, 1955. 49 D. de Klerk, in “Handbuch der Physik-Encyclopedia of Physics” (S. Flugge, ed.), Vol. 15, p. 38. Springer, Berlin, 1956.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

187

In experiments with radioactive nuclei the activity is included in a host crystal essentially as an impurity. The type of paramagnetic ion in the host crystal can then be chosen to produce efficient cooling. Among the considerations that affect this choice is the condition that the coolant ion have little or no hfs. The reason for this can best be illustrated by considering the entropies involved. For a given type of ion the maximum entropy of the electron and nuclear spin systems is R ln[(2S 1)(2Z l)]. In order to produce considerable orientation, it is necessary to remove as much of this entropy as possible from the nuclear spin system. By magnetizing at 1”K, however, the maximum amount of entropy that can be l), i.e., the electron removed from the system as a whole is R ln(2S spins are completely ordered, whereas the nuclear spins are completely disordered. Since the demagnetization process is isentropic this is also the final entropy, although it is now shared between the two systems. For I appreciably greater than S, it is clear that the final state will not be a highly ordered one, i.e., the nuclear orientation will be small. When a second type of ion, of the kind mentioned above, is incorporated in small concentration, however, the entropy balance is very favorable for producing a large nuclear orientation. The best example of this is the case of cerous magnesium nitrate,60*61 which may be cooled by demagnetization to a temperature of 0.003”K1partly because it is a very dilute salt magnetically and partly because there is no hfs. This salt has the further advantage that it is a double salt into which both trivalent rare earth ions and divalent transition metal ions may be incorporated. Another point to be noted about magnetic cooling is that it is a “single shot” process, i.e., after demagnetization the specimen begins to warm up again, and measurements are taken during this warming period. Since the heat capacities of the salts are often very small, the question of thermal insulation becomes most important. Particular attention must be paid to technique in this connection, and sources of heat influx such as arise from radiation, and more particularly from vibration of the sample and conduction by residual exchange gas desorbing from the walls of the cryostat, must be minimized. For a carefully designed apparatus, a heat influx of 10 ergs/min into a sample weighing a few grams would be considered very satisfactory. For details of specimen support inside the cryostat, the use of exchange gas, thermal “guard-rings,” etc., the reader is referred to references 10-22, 25, and 4 6 4 9 . When a radioactive substance is being investigated, the heating due to absorption of some of the energy released during decay may become noticeable. For the more commonly used

+

+

+

J. M. Daniels and F. N. H. Robinson, Phil. Mag. [7] 44, 630 (1953). R. P. Hudson, R. S. Kaeser, and H. E. Radford, Proc. 7th Intern. Conf. Low Temp. Phys. p. 100 (1961). Kt

188

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

isotopes with beta-gamma cascades, a useful average working figure is 1 erg/min per microcurie of activity. The amount of activity required is usually in the range 1-100 microcurie. Thus even with cooling substances such as cerous magnesium nitrate, warm up times of one hour can be achieved, although since much larger effects are observed at the lowest temperature, not all of this time is equally useful. I n fact, the more usual procedure is to take data only at the lowest temperatures, and then deliberately to warm up the sample and take the “warm” data needed for normalization purposes. Besides requiring less stringent conditions for counter stability, this method overcomes inaccuracies due to inhomogenous heating in the sample, which arises on account of the rather low thermal diffusivity of the salts a t low temperatures. In fact, for the most accurate work, the taking of “cold” data is often confined to the first few minutes after demagnetization. There is a further point to note when experiments are being performed with polarized nuclei, I n this case, there is an external field present which affects both the specific heat, C , and the temperature of the sample. Now we have c = [b CH2J/T2 (2.4.2.3.13)

+

where c is the Curie constant and b is the “specific heat constant.” For work on radioactivity, or on nuclear reactions where the incident particles are uncharged, no serious trouble due to small heat capacities should be encountered. For charged particle reactions, however, the situation is likely to be very different, and a change of technique would be needed. One could consider producing continuous refrigeration by the use of He3 as ref1igerant~2-~4or by the development of ways of producing cyclical magnetic as well as resorting to experiments with a rather poor duty cycle, as illustrated for a somewhat extreme case in Fig. 2. In the case considered above for producing nuclear polarization, the residual external magnetic field limits the extent of the cooling and hence the polarization. An alternative procedure is to use two different specimens separated sufficiently in space but connected by a thermal link, the first specimen being cooled directly by demagnetization and the second by contact with the first, in the manner outlined in the previous section. This method has the further advantage that the substances in the V. P. Peshkov and K. N. Zinov’era, Repts. Progress in Phys. 22, 504 (1959). W. Taconis, in “Progress in LOW Temperature Physics” (C. J. Gorter, ed.), Vol. 111, p. 153. North-Holland Publ., Amsterdam, 1961. 54 E. Ambler and R. B. Dove, Rev. Sci. ZnstT. 32, 737 (1961). 56 C. V. Heer, C. B. Barnes, and J. G . Daunt, Rev. Sci. Instr. 26, 1088 (1954). 52

6s K.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

189

second stage need not necessarily be a paramagnetic salt. There are a number of points of technique involved, as we have already indicated, the most important one being that of obtaining good thermal contact between the thermal link and the salt. I n certain cases, demagnetization to a nonzero field can be achieved most advantageously without sacrificing the degree of cooling by using single crystals which are magnetically very anisotropic and by essentially rotating the sample with respect to the magnetic field.66Equation (2.4.2.3.12)is then modified to read: (2.4.2.3.14) where go and g are the g-factors in the directions of H o and H respectively. The advantage of using this method when g o >> g , as is the case with cerous magnesium nitrate, is evident. When the g-factor along H is not small, i t is necessary to allow for saturation effects. This is the case, for example, with neodymium ethyl sulfate where very often a polarizing field is applied along the z-axis while susceptibility measurements are made perpendicular. I n this case a useful relation is i

T*

=

( T H *tanh y)/y

(2.4.2.3.15)

where y = gllpH/2kT, TH* is the measured magnetic temperature, and T* is the one to be substituted in the T-T* correlation (see below). Finally, we should point out th at when we are in the range of cooperative effects in paramagnetic salts, changes in temperature may be complicated but, can be found by a procedure based upon the thermodynamic relati~n~’*~~*~~ (2.4.2.3.16) where M is the magnetic moment of the sample. I n order t o determine the degree of nuclear orientation (see Section 2.4.2.3.5),it is usually necessary to know the temperature of the specimen. When working with liquid He4, 68 or liquid He3, 69 temperatures can be obtained simply from the vapor pressure, although corrections for thermomolecular pressure differences may be significant.6oI n the magnetic cooling 56A.

H. Cooke, S. Whitley, and W. P. Wolf, Proc. Phys. SOC.(London) B68, 415

(1955).

W. F. Giauque, Phys. Rev. 92, 1339 (1953). F. G. Brickwedde, H. van Dijk, M. Durieux, J. R. Clement, and J. K. Logan, J. Research Natl. Bur. Standards A64 1 (1960). 59 S. G. Sydoriak and T. R. Roberts, Phys. Rev. 106, 175 (1957). 60 T. R. Roberts and S. G. Sydoriak, Phys. Rev. 102, 304 (1956). 57 68

190

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

region the absolute temperature is usually correlated with the susceptibility of the paramagnetic salt and published as a T-To correlation. The procedure is to measure T*, compute T o , and then use the published data to find T46-61[cf. refs. in Table 1111. The following relation defines T*:

c/T* = M / H ,

(2.4.2.3.17)

where c is the Curie constant and M the magnetic moment caused by a small external measuring field, He. T* is thus related only to measurable quantities, although it will be different for differently shaped specimens. The symbol To refers to a spherically shaped sample, and it is customary to relate all T* values to To through the relationship

+ /3

(2.4.2.3.18)

(n,- n)c

(2.4.2.3.19)

T o = T* where

/3

=

where n, is the demagnetizing coefficient for a sphere (= 47r/3), and n that for the specimen used. The following points are worth noting: at higher temperatures, T o 3 T; a t lower temperatures the determination of the T-Tb correlation is a lengthy, and also somewhat inaccurate, procedure; for magnetically anisotropic substances, the T-To correlation will depend upon the direction of measurement with respect to the crystallographic axes; and finally the value of T o measured in the presence of a magnetic field, is not necessarily equal to that which would be measured in zero field at the same temperature, since saturation effects can occur at low fields in the magnetic cooling range. Once the T-TD correlation is known, secondary thermometers, such as carbon thermometers-or, more particularly, y-ray anisotropies of radioactive nuclei, can be calibrated and used. This latter method, the “nuclear thermometer” as it has come to be called, has many advantages. It can be used, for example, to give the temperature of a very specific region (e.g., for work on beta decay of the surface layer of a paramagnetic salt). It is also convenient to use in experiments on radioactivity, since the “thermometric” isotope can be simply included in the crystal and the gamma ray intensity, and hence the temperature, recorded simultaneously with the other data. 2.4.2.3.4. THE PRODUCTION OF ORIENTED NUCLEI-DYNAMICAND TRANSIENT METHODS. The relevant properties of substances used in this class of experiment can usually be described by a spin Hamiltonian. Preferential nuclear population distributions are not caused primarily by lowering temperature; however, they are caused by driving transitions between different energy levels by means of oscillating fields a t a fre-

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

191

quency chosen to satisfy some resonance condition. The oscillating field is usually an electromagnetic field (although the use of acoustic vibrations for the purpose has been proposed) and the frequency ranges from the optical region down through the microwave to the rf region. The basic idea is to induce transitions a t rates comparable to the natural relaxation rates of the system. Saturation effects then occur, and the system is driven away from thermal equilibrium into a state of dynamic equilibrium where certain levels, not necessarily the lowest in energy, are preferentially populated. The new situation can then be thought of as being maintained by a pumping action, where the applied field excites the system in one way and the internal processes relax the system in another way. The optical method, which was the first of its kind to be suggested,61 is in many ways different from the others, For example, it is carried out a t or near room temperature and resonance is performed on free atoms in the vapor phase. The orientation is with respect to the total angular momentum of the atom, F = J I, where J refers to the electronic angular momentum of the atom. Relaxation occurs through spontaneous emission, rather than by taking place through thermal processes as in other methods, although often unwanted relaxation mechanisms such as wall collisions are present and prove to be troublesome. Relaxation effects may be induced also by collision with impurity atoms, a phenomenon that can in fact be used to orient the impurity nuclei themselves.s2 In general, for work with radioactive nuclei, these methods have the disadvantage that the activity tends to stick to the walls of the containing vessel, and since nuclei in such a position are not oriented, they merely give unwanted background counts. Other proposals which have been made refer generally to solid state systems, and t o a large extent stem from a suggestion by O v e r h a u ~ e r . ~ ~ His suggestion was based upon a theoretical study of the relaxation mechanisms for nuclear spins and the spins of conduction electrons in metals. Given hyperfine coupling of the type AS * I, it was shown that upon saturation of the electron spin resonance signal, there accompanied the disappearance of the electron polarization an enhanced nuclear polarization of approximately equal magnitude (in angular momentum units) to that which originally existed among the electron spins. The method is not confined to metals, but it is applicable to a much wider variety of substances; in fact, to most paramagnetic s u b ~ t a n c e s . ~ ~

+

61 A. Xastler, Proc. Phys. SOC.(London) A67, 853 (1954); J . Opt . Soc. Am. 47, 460 (1957). ' 62 H. G . Dehmelt, Phys. Rev. 106, 1487 (1957); 109, 381 (1958). a* A . W . Overhauser, Phys. Rev. 92, 411 (1953). 6' A. Abragam, Phys. Rev. 98, 1729 (1955).

192

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

I n paramagnetic salts we can study the various dynamic effects by simply adding to Eq. (2.4.2.3.8) the appropriate term ho * S cos wt for the applied rf field of amplitude ho, and a random time perturbation X ( t ) for the thermal relaxation effects,l4 The detailed b e h a v i ~ r ’of~the ,~~ system will depend upon matrix elements of these perturbing terms between the various energy levels of the unperturbed spin Hamiltonian. The form of X ( t ) may be quite complicatede6but in some cases, e.g., dielectrics at low temperatures, it gives rise to only one kind of relaxation process, e.g., where only the electron spin is flipped by interaction with the lattice waves (phonons) with no change in direction of the nuclear spin. These processes by themselves do not affect the nuclear polarization directly, and the required effects are brought about by second order processes where the phonons modulate the hyperfine coupling. Consider the simplest case of S = +, I = +, with coupling AS I, and with relaxation processes as described above. I n thermal equilibrium in an external field the energy levels and populations are as shown in Fig. 4a, where it is assumed for convenience that x level will become enhanced at the expense of the - - > level, and a nuclear polarization f l = +s is obtained. In the case that only one of the two allowed transitions is saturated the polarization will be x/2. I n the case of a metal, where the electrons taking part in the processes belong to a highly degenerate Fermi-Dirac system, the details are somewhat more involved but the basic idea is the same. Indeed it has been showne4 that the method should work to a varying degree for a wide variety of substances possessing different types of hyperfine coupling. A variation of these ideas which has proved to be most successful was proposed by Jeffries. The method relies on the fact that, as shown in Fig. 4c, all the energy levels are not necessarily exact eigenstates of S , and I,, and a so-called weak “forbidden” line can be excited. The idea is to saturate this line with an external field and thereby drive the transition which, in the previous method, is caused by second order thermal relaxation processes. In the steady state, shown in the figure, the same degree of polarization but of opposite sign is obtained. In discussing the dynamic methods it has been assumed that the different hyperfine components of a resonance line are resolved. It has been 86 See, e.g., C. H. Townes, ed., “Quantum Electronics.” Columbia Univ. Press,

I

New York, 1960.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

193

shown recently that this criterion is by no means necessary, and that in fact polarization effects can occur in surrounding nuclei, e.g., protons in the water molecules of crystallization, or in plastics where the protons can be coupled to F-centers. This occurs in spite of the fact that the nuclei are only weakly coupled by dipolar forces to the electron spins. It

I-

x

--

1+I -

I - 2 x I++>

Main relaxation relaxation processes

I i ASJ,

”forbidden”

Saturate

t hrf

S to lotticc

I+X

I -x

ItX ---.-

Itx I

a Thermal Equlllbrlum 1, = o

1t2x

I-->

--I

I b. Overhauser Effect f,.+X

c. Jeffries Effect f,=-X

FIG.4.Energy levels and relative populations of an electron and nuclear system with S = and Z = & in an external field H , and coupled through the hyperfine interaction, are shown. The quantity z = p H / k T is assumed to be . The symbols a,a’represent those quantum numbers that refer to quantities other than nuclear spin. The statistical tensors, or rather their complex conjugates, are also the expectation values, , of a set of tensor operators T,k..3973&69 We shall need to consider only systems with sharp angular moment (I’ = I ) and with axial symmetry ( q = 0; m = m’). In this case we may also drop a, a’ and obtain the following simple relations. The only nonzero elements of the density matrix are = am (2.4.2.3.21) where am is the occupational probability of the mth nuclear magnetic substate, and [ ]hk)

=

1(

-)r-m


a,.

(2.4.2.3.22)

m

6e A.

Simon, Phys. Rev. 92, 1050 (1953).

A. Simon and T. A. Welton, Phys. Rtc. 90, 1036 (1953);93, 1435 (1954). 68 M.E.Rose, Phys. Rev. 108, 362 (1957);G.F.Koster and H. Statz, ibid. 113,445

67

(1959). 89 E. Ambler, J. C. Eisenstein, and J. F. Schooley, J . Math. Phys. (January, 1962), 3, 118 (1962);ibid. 3, 760 (1962).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

195

Other notations that appear in the literature and are identical to that defined in Eq. (2.4.2.3.22) are fk(I)’OJ1 and Rk.72Other notations that which are related as follows: differ somewhat are Bk73and fk(I),70~74

Rk

=

Bk/l^ =

(y)

Ik[(2h

+ 1)(21 - k)!/(21 + k f 1)!]”2fk(I) (2.4.2.3.23)

2 = (21 + 1)1‘2.

where

The first four nuclear orientation parameters are given explicitly by (2.4.2.3.24a) (2.4.2.3.24b) f3

f4

[ m3am- ( 3 P + 31 - 1) ma,/5] = I-4[ 2 m4am - (612 + 6 I - 5) 1m2am/7

=

I-3

m

m

m

m

+ 3(1 - 1)(1)(I + 1 ) ( I + 2)/35].

(2.4.2.3.24d)

For nuclear alignment, as opposed to nuclear polarization, the odd orientation parameters, f l , f3, etc., are zero. It should be noted that in many cases we are dealing with a nuclear spin coupled to an electron spin, as displayed in Eq. (2.4.2.3.8). The two systems in physical states a and s, say, are c ~ r r e l a t e d ,so~ ~we must consider their joint density matrix pas. We then have

= TrE(

}

(2.4.2.3.25)

where the quantum numbers M refer to the eigenstates of Sz, and Trl stands for the trace taken over the electron states. I t will be noted also from Eq. (2.4.2.3.8) that it is not possible in general to diagonalize the H. A. Tolhoek and J. A. M. COX, Physica 19, 101 (1953). M. Morita and R. S. Morita, Phys. Rev. 109, 2048 (1958). 7* R. B. Curtk and R. R. Lewis, Phys. Rev. 107, 1381 (1957). 7 3 T. P. Gray and G. R. Satchler, Proc. Phys. SOC.(London) A68, 349 (1955). 74 5. R. de Groot and H. A. Tolhoek, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 613. Interscience, New York, 1955. ‘&See,e.g., U. Fano, Revs. Modern Phys. 29, 74 (1957). 70

71

196

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

energy and I , simultaneously. In computing the a,a it is then preferable to proceed by solving the secular determinant, as we illustrate later. In addition to describing the initial nuclear orientation, the statistical tensors can be used to describe the orientation of any other particle or radiation involved. I n general, for a particle with spin s, only tensor moments up to rank 2s can be nonzero, Thus for particles of spin +,only the tensor of rank 1 exists, and the orientation is completely specified by the polarization. The photon is in a special category. For present purposes it may be regarded simply as a particle of spin 1 with zero rest mass and only two independent polarization states. The polarization states may be chosen in a number of ways, e.g., two orthogonal states of plane polarization, or the two states of circular polarization, i.e., angular momentum projections along the wave vector Ic of + 1 (optically left circular polarization) or - 1 (optically right circular polarization). I n the former case we choose directions of plane polarization, Al and A2, for the vector potential such that (A1,A2,k) forms a right-handed set of axes.7ev77The polarization of any photon can then be written

+

A = aiAi a z A 2 where a1 and a2 are complex numbers such that la112

+

(2.4.2.3.27a)

1

la212 =

(2.4.2.3.26)

and A1 and Az are unit vectors. We now write

-

la212

=

41

aza1*

=

62

i(a1az" - aza1*) =

53

la112 a1az*

+

(2.4.2.3.27b) (2.4.2.3.27~) (2.4.2.3.27d)

so that El measures the difference in intensity of radiation polarized along A1 over that polarized along A2, t2measures a similar quantity for axes rotated through n/4 in a direction A1 + A2, and E3 measures the intensity of optically left-circularly polarized radiation over optically right-circularly polarized radiation. We have for the density matrix

1 P = t 2

+2

El

P = 77

31

- its 2

+2 i t 8

which can be written 76

52

1

- 61 2

+t

'

4.

H. A. Tolhoek, Revs. Modern Phys. 28, 277 (1956). F. W. Lipps and H. A. Tolhoek, Physica 20, 85, 395 (1954).

(2.4.2.3.28)

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

197

For further discussion of the quantities El, t2, t3, which are closely related to the Stokes’ parameters, we refer the reader e l~ e w h e re .~ O-~ ~ If we choose the more appropriate spherical representation for vector quantities

eo = e, (2.4.2.3.29)

and write the following equation to describe the photon polarization

A

+ LYA-

=

the density matrix is given b y 1 P =

+

53

-(El

2

-(51

(2.4.2.3.30)

+itz)

2

- it21

12

2

(2.4.2.3.31) F3

Finally, the statistical tensors are given by

Roo

=

1/43,

Rlo = &/d2, R2o

=

1/46,

Rzkz =

i.5)/2 (2.4.2.3.32a)

-(I1

with

R1*1

=

Rz*l

=

0.

(2.4.2.3.32b)

I n the description of polarization so far, we have considered a single particle in a pure state. The concepts need to be extended in order to describe a beam of radiation which may be unpolarized, partially polarized, or completely polarized. This may be done, for example, simply by taking the appropriate incoherent average over the large number of particles in the beam.7s-79 For photons, for example, a beam traveling along a direction described b y polar angles (0,4) with intensity W(O,4) can be described with the aid of Stokes’ parameters and written as a 4 X 1 matrix:

w e , + ) . (I,w.

w(e,4)- ( i , P t 1 , P ~ 2 , P t 3 )

(2.4.2.3.33)

The magnitude P measures the magnitude of the polarization and ranges from zero for a n unpolarized beam to 1 for a completely polarized beam The components of describe the type of polarization, a s described above. 78 7t1

U. Fano, J. O p t . SOC.Am. 39, 859 (1949). W. H. McMaster, Revs. Modern Phys. 33, 8 (1961).

198

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

2.4.2.3.5.2. Determination. We shall defer until the next section any discussion on the measurement of the degree of polarization of a beam. Here we discuss the determination of the degree of nuclear orientation of the initial state. For the static methods, if we know the Hamiltonian of the system and the temperature of the sample, we have the formal expression : = Tr(T,kp)/Tr p (2.4.2.3.34)

where, for thermal equilibrium

The trace is taken over all states, i.e., if X is the spin Hamiltonian over both electronic and nuclear states. It is useful to have expansions in powers of 1/T, which are obtained by the methods described in Section 2.4.2.3.8and have been given explicitly for a number of cases in Section 2.4.2.3.2. At the lowest temperatures, however, these expressions are not accurate enough, so that it is necessary to solve the eigenvalue problem and compute the am's.Fortunately this is often quite simple since the secular determinant factors and closed . ~ ~ ~ parameters ~ ~ ~ ~ ~ ~ ~ expressions can be obtained for the u , ' ~ Orientation for a few simple cases have been tabulated for a number of values of The secular determinant corresponding to the most frequently met with spin Hamiltonian, viz., A=, Av, and D all zero, and H along the z-axis, in the representation that makes M and m diagonal, factors into two linear equations and 2 1 quadratic equations. The latter are of the form: m,

m

-t

= 0.

- 1, +t

a22

(2.4.2.3.36)

-W

The eigenvalues are given by ‘w = ;[(a11

+ a d k ((a11-

+ 4 a l ~ a ~ ~ )(2.4.2.3.37)

a22)2

1’2]

and the transformation between energy and angular momentum eigenstates is IWr> = a r , m l m , -& > 4- a,,,-llm - 1, ++> (2.4.2.3.38) 80 81

N. R.Steenberg, Proc. Phys. Soc. (London) A66, 399 (1953). T.P. Gray and 0. R. Satchler, Pmc. Phys. Soc. (London) A08, 349 (1955).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

with a,,, and

ar,m-l

199

given by (a11 -

wr)ar,m

+ a i 2 ( ~ ~ , ~ -01 =

(2.4.2.3.39)

and the normalization condition

We then obtain am by means of the equation

When the terms in D, As and A,,, or H , and H,, are present the secular determinant may not factor so simply. Perturbation methods can be employed in many cases, however, while in the most complicated cases the dimensionality of the problem [i.e., (2s 1)(21 l ) ] is small enough to permit rapid solution with the aid of an electronic computer. 2.4.2.3.5.3. Perturbing Effects. Two effects will be considered that can alter the nuclear orientation from the values computed by the above methods, viz., the effect of interionic couplings and the effect of perturbations acting on intermediate states. Strictly speaking, Eq. (2.4.2.3.8) applies only to substances of high magnetic dilution, where the interaction between the paramagnetic ions themselves can be neglected. This coupling must certainly be taken into account in concentrated salts, e.g., in computing specific heat, and the question arises whether the coupling affects the nuclear orientation. The type of interionic coupling predominant in the salts we are considering is of the dipole-dipole type. This can be included formally by summing Eq. (2.4.2.3.8)over all ions, and adding the interaction

+

+

where the prime denotes summation over all pairs. The components of the magnetic moment p are (g.pSz,gypSy,gzpS,). It has been ~ h ~ ~ nbY~ ~ ~ ~ ~ applying the methods described in Section 2.4.2.3.8that these interactions do not appear in the leading terms of the orientation. Thus, for example, they do not appear in the expression for fz until the term in l/T4. There are a few cases found experimentally, however, where the interactions do have a significant effect on the orientation, viz., the cases of the 8z

N. R. Steenberg, Phys. Rev. 93, 678 (1954). Can. J . Phys. SB, 1133 (1957).

J. M. Daniels,

200

2.

DETERMINATION O F FUNDAMENTAL Q U A N T I T I E S

divalent ions Co++ 84 and Mn++ in cerous magnesium nitrate. It has been suggested that the effect is the result of antiferromagnetic ordering,86 and it has also been conjectured how a strong interaction between pairs of ions might be i m p ~ r t a n t . ~An ’ explanation is still lacking, however, although from the more practical point of view i t is worth noting that the disturbing effects can be overcome completely by applying an external In the special case of neomagnetic field of a few hundred dymium ethyl sulfate, interionic interactions are also important, but here the crystal structure is such that often a good approximation is obtained by considering only the two nearest neighbors.88 Reorientation effects in an intermediate state of a decay sequence have been investigated a great deal in connection with angular correlation s t u d i e ~ , ~and ~ - to ~ ~a lesser extent specifically for oriented n~clei.~~*94,96 The effects are divided into two groups, dynamic and static, depending on whether or not the interaction is described by a Hamiltonian depending explicitly on time. Static perturbations comprise effects due to external magnetic fields, hyperfine splitting, and so on. Dynamic perturbations comprise effects due to spin-lattice relaxation, nuclear recoil (e.g., as in a-decay), and rearrangement of electrons among the atomic orbits (e.g., following p-decay and, particularly, K-capture). The static effects can be estimated quite accurately, whereas only qualitative arguments are made concerning the dynamic effects. Generally it appears that reorientation effects could be important for mean nuclear lifetimes longer than about lo-” sec. The experimental evidence on the subject, which is meager, seems to indicate that for cascades initiated by p-decay, effects if they exist at all are small, but for K-capture they may be large. Thus in practically the same substance, there are no effects observableg6 a t all for CoflO, 84 E. Ambler, M. A. Grace, H. Halban, N. Kurti, H. Durand, C. E. Johnson, H. R. Lemmer, and F. N. H. Robinson, Phil. Mag. 171 44, 216 (1953). 85 M.A. Grace, C. E. Johnson, N. Kurti, H. R. Lemmer, and F. N. H. Robinson, Phil. Mag. [7]46, 1192 (1954). 86 M.W.Levi, R. C. Sapp, and J. W. Culvahouse, Phys. Rev. 121, 538 (1961). ST E. Ambler, R. P. Hudson, and G. M. Temmer, Phys. Rev. 101, 196 (1956). 88 C. E. Johnson, J. F. Schooley, and D. A. Shirley, Phys. Rev. 120, 2108 (1960). ED A. Abragam and R. V. Pound, Phys. Rev. 92, 943 (1953). V. Gillet, Nuclear Phys. 20, 561 (1960). 91 H. Frauenfelder, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 531. Interscience, New York, 1955. 92 R. M. Steffen, Phil. Mag. Suppl. [8]4, 293 (1955). 93 E. Heer and T. B. Novey, Solid State Phys. 9, 199 (1959). 94 N. R.Steenberg, Phys. Rev. 96, 982 (1954). s6 H. A. Tolhoek, C. D. Hartogh, and S. R. de Groot, J . phys. radium 16,615 (1955). S6 H.R. Lemmer and M. A. Grace, Proc. Phys. SOC.(London) A67, 1051 (1954).

2.4.

DETERMlNATION O F SPIN, PARITY, A N D NUCLEAR MOMENTS

201

whereas the 137 kev level in FeS7(lifetime -5 X sec) seems to be perturbed ~ t r o n g l yfollowing ~ ~ , ~ ~ K-capture in Cofi7.For further discussion on dynamic perturbations we refer the reader to references 91-93, There is one essential point of difference between static perturbations as they affect angular correlation as compared to nuclear orientation. In the former case one tries to choose substances where the effects of external fields at the nucleus are small, whereas in the latter case they must par excellence be large. This does not mean that attenuating effects should be larger in the latter case; on the contrary, they may be less. As is well known from angular correlation theory, an external field when suitably oriented, i.e., along the direction of the first radiation, serves to preserve or restore the angular correlation. Similarly, if the symmetry of the perturbing field is the same as the (assumed) axially symmetric field that produced orientation, angular distributions from oriented nuclei may be unaffected. The same conclusion would hold for angular correlations with oriented nuclei, provided the first radiation is observed along the axis of symmetry, but would not necessarily hold if the first radiation were observed in a perpendicular direction. This comes about firstly because the nuclear orientation remains axially symmetric, and secondly -and partly as a result of this-because the spin flipping terms are small. Thus if we suppose that the interaction in the intermediate state is given by a spin Hamiltonian, X', with constants g', A', B', etc., the terms that give rise to reorientation are off-diagonal, e.g., B', and if these are zero there is no reorientation a t all. 2.4.2.3.6. MEASUREMENTS OF ANGULAR DISTRIBUTIONS AND POLARIZATIONS. The majority of experiments have been carried out below 1°K and have involved magnetic cooling. The apparatus used has been developed using techniques already established in this field, with appropriate modifications for the different types of particle to be counted. Measurements have been made so far on the angular distribution of gamma rays,98,99 the plane polarization of gamma rays, l o O , l o l the circular polarthe angular distribution of beta parization of gamma ray~,~02,103 97 G. R. Bishop, M. A. Grace, C. E. Johnson, A. C. Knipper, H. R. Lemmer, J. Perez y Jorba, and R. G. Scurlock, Phil. Mag. [8]46,951 (1955). 98 B. Bleaney, J. M. Daniels, M. A. Grace, H. Halban, N. Kurti, F. N. H. Robinson, and F. E.Simon, Proc. Roy. SOC.A221, 170 (1954). s9 0. J. Poppema, M. J. Steenland, J. A. Beun, and C. J. Gorter, Physica 21,233 (1955). loo G. R. Bishop and J. Perez y Jorba, Phys. Rev. 98,89 (1955). lo* R. W. Bauer and M. Deutsch, Nuclear Phys. 16,264 (1960). lea J. C. Wheatley, W. J. Huiskamp, A. N. Diddens, M. J. Steenland, and H. A. Tolhoek, Physica 21, 841 (1955). 1°3 1%.W. Bauer and M. Deutsch, Phys. Rev. 117, 519 (1960).

202

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

t i c l e ~ , ’ ~the 4 ~ ~angular ~~ distribution of alpha particles, 106,107 and the angular distribution of fission fragments.los In addition to the original references given above to descriptions of apparatus, we also refer the reader to references 10-14. Experiments have been made also on the angular distribution of gamma rays using dynamic methods of nuclear orientation. 14,108 We cannot hope to describe all the above apparatus in detail, so we shall discuss those aspects of the design that are particular to the nuclear orientation method and are not normally encountered in nuclear spectroscopy on the one hand and magnetic cooling on the other. These stem largely from the fact that one has to deal with thick sources placed inside a cryostat. When one is dealing with penetrating particles, e.g., gamma rays of energy greater than about 10 kev, there is no need for any special technique. Conventional NaI (Tl) scintillators may be placed outside the cryostat, and only small corrections are required for absorption and scattering within the source and in the walls of the cryostat. The use of a multichannel a n a l y ~ e for r ~recording ~ ~ ~ ~data ~ ~ is useful, among other things, for assessing the amount of scattering, as well as for recording data for several gamma ray energies simultaneously. For short range particles, problems arise from the fact that the source cannot be made close to ideal, i.e., “massless,” and from the inability of such particles to penetrate the walls of the cryostat. The procedure adopted with alpha and beta particle sources is to take an inactive crystal weighing a few grams and place on one well-defined surface a drop of concentrated solution of the salt containing the activity. It is found that when the solution is allowed to crystallize it does so as a continuation of the base single crystal. I n the case of alpha particle sources the base crystals of uranyl rubidium nitrate are normally employed (it should be noted that there is in these samples a background heating due to alphaparticle activity of about 5 ergs per minute per gram). I n the case of beta-particle sources it is found that suitable sources can be made by confining the activity to a surface layer about 10011 thick. Although there is some multiple scattering and considerable backscattering preslo4 C. 8. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Rev. 106, 1413 (1957). IosH.Postma, W. J. Huiakamp, A. R. Miedema, M. J. Steenland, H. A. Tolhoek, and C. J . Gorter, Physica 24, 157 (1958). lo8J. W. T.Dabbs, L. D. Roberts, and G. W. Parker, Physica Suppl. S69 (September, 1958). lo’s.H. Hanauer, J. W. T. Dabbs, L. D. Roberts, and G. W. Parker, ORNL Report 2919 (unpublished). 108 F. M. Pipkin, Phys. Rev. 112, 935 (1958). loO D. A. Shirley, J. F. Schooley, and J. 0. Rasmussen, Phgs. Rev. 121, 558 (1961).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

203

ent, so that substantial corrections have to be applied, a check can generally be made on the consistency of this procedure. This involves comparing the value of f~ deduced from beta-particle asymmetries, the value of fi deduced from gamma-ray anisotropies, and the temperature as determined by a measurement of T* for a salt whose thermal and magnetic properties are known and for an isotope whose decay scheme is known. Some discussionllOolllhas been given of the more important corrections to be applied when using beta-particle sources. Of the electrons emitted in the backward direction, between and are returned to the forward hemisphere. Although there is some degradation in energy, this is not large for beta particles, so that it is hard to make an energy discrimination. The procedure adopted is to reject the very lowest part of the beta-particle spectrum since the corrections are too large to be allowed for accurately. The remaining part of the spectrum is then compared with a “reconstructed11spectrum formed by taking the spectrum of an ideally thin source and computing the thick source behavior, using experimental data on backscatteringl12,ll3of material of the same average Z as the crystal. This is also compared to the spectrum taken with a betaray spectrometer, first of the thin source alone, and then with the same source placed on a crystal of the paramagnetic salt. A correction must be applied also for Compton electrons generated in the surroundings. Since this is directly related to the geometry involved, it must be determined in relation to the particular apparatus used. It may be evaluated by making spectral measurements with and without an absorber placed over the source, just thick enough to stop the beta particles. Although in the particular case of positrons it is possible to measure the asymmetry by measuring the annihilation radiation with scintillators placed outside the cryostat,lo6 it is generally preferable to place the counters inside the cryostat. lo4 This immediately brings up the question of the behavior of light pipes, phototubes, scintillators, and other detectors, such as surface barrier counters a t low temperatures. The more usual photomultiplier tubes cease to operate a t low temperatures, principally because the photocathode does not remain electrically conducting as the temperature is lowered. Special types may be obtained, however, with photocathodes that can be operated down to liquid helium temperatures and which have remarkably low noise. An alternative

+

IlaR. W. Hayward and D. D . Hoppes, I R E Trans. on Nuclear Sci. NS-6, No. 3 (December, 1958). 111 D. D. Hoppes, Natl. Bur. Standards Tech. Note No. 99 (1961), unpublished. 114 W. Bothe, 2.Naturforsch. 49, 542 (1949). 113 H. H. Seliger, Phys. Rev. 88, 408 (1952).

204

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

arrangementlo4to that of operating a photomultiplier a t low temperature is to use a lucite light pipe, 50-100 cm long, between the scintillator in the cryostat and the photomultiplier a t room temperature. Since lucite is both a poor thermal conductor and opaque to infrared radiation, fairly large diameter light pipes can be used to extract light pulses with good efficiency, without introducing prohibitively large amounts of heat into the cryostat. As a scintillator for counting beta particles, anthracene works quite well when operated at low temperatures, although rapid cooling or warming will cause cracking and consequent loss of resolution. Under good conditions an overall resolution of about 15% can be obtained with the 624 kev conversion electrons from Cs’37. Anthracene crystals also work satisfactorily a t low temperatures as alpha-particle detectors, although the simplest arrangement in this case, and also for fission fragments, is to use surface barrier counters. We refer the reader elsewhere for further details.1l4~~~~ Up to the present time, measurement of the polarization of particles emitted from oriented nuclei has been confined to gamma radiation, for which detectors based upon Compton scattering have been used. The sensitivity of such a detector to polarization is obtained from the KleinNishina formula, and can be handled very simply by means of Stokes’ parameter^.^^,^^ Thus, in the notation of Eq. (2.4.2.3.33),if the incident and scattered photon beams are given by No(l,P0fo)and N,(l,P&) respectively, then (2.4.2.3.43)

where the matrix T is given by Eq. (2.4.2.3.44). T

= 2

(A)*(i)’ mac

1

+

x

OOS~

+ (ko - k ) ( l - cos x) sin2 x

sin2 x

1

0 -(1 - c o a x ) ( k c o s x + k o ) ’ S

COB

COB*

x

0

(1 - c o s x ) ( n a X n ) . ( k X S )

+ .

0 ‘0

2

+

x

-(1 - COB x)(ko coa x k) S (1 - 008 x)(n X no) (ko X S) (1 - cos x)(ko X n) S 2 coa x (ku k ) ( l COB X)COB x

. . -

1

.

(2.4.2.3,44)

- con x)(k X no) . S + In these expressions, ko and k are the momenta (in units of moc) of the incident and scattered photon respectively, and noand n are unit vectors (1

F. J. Walter, J. W. T. Dabbs, and L. D. Roberts, Rev. Sci. Instr. 81, 766 (1960). J. W. T. Dabbs and F. J. Walter, eds., “Semiconductor Nuclear Particle Detectom.” Publ. No. 871, National Academy of Sciences-National Research Council, Washington, D.C., 1961. 1~

116

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

205

along these directions respectively. Th e magnitudes k o and k are connected by the well-known relation

l / k - l / k o = 1 - cos x (2.4.2.3.45) where x is the scattering angle. S is a unit vector along the initial direction of the spin of the scattering electron, mo is the rest mass of the electron, c is the velocity of light, and T is in units of cm2/electron. The choice of axes is such that, with the notation of Section 2.4.2.3.5

A1” = A 1 = (noX n) (2.4.2.3.46) where A” and A are the vector potentials of incident and scattered photon, respectively. The axes to which the ko-photon is referred (z’y’z’, with ko along 02’) will be related to those of the nuclear orientation (zyz, with n along 02) by a right-handed rotation through angle 0 about Oy. Maximum . aligned nuclei the sensitivity is obtained by taking e equal to ~ / 2 For beam can be described by the intensities, N,. and Nu / 2 ! k 2 T 2+ < X > 2 / k Z T 2 )+

( a

*

a )

*

+

*

.][1

+ /kT

. . .]. (2.4.2.3.67)

This expression simplifies further since for all tensor operators, except for those of rank zero, the trace is identically zero. The trace of the spin Hamiltonian, moreover, is generally made zero by adding constant terms, so that Eq. (2.4.2.3.67) finally becomes, setting 0 equal to T,",

F,"

=

[- < T , " X > / k T ]

+ [ { 3 < T , " X > . (2.5.6)

If the angle between P and ii is given by 8, then the probabilities that a measurement of P, yields the value + l , respectively - 1, are given by prob (+1) prob (-1)

= =

cos2(8/2) sin2(8/2).

(2.5.7)

Equations (2.5.4) to (2.5.7) describe fully polarized electrons, as can be seen easily by calculating the degree of polarization P , Eq. (2.5.2), for Eq. (2.5.4) or (2.5.6): I’ = 1. (2.5.8) This result, a t first sight puzzling, is due to the fact that only pure spin states have been considered so far, i.e., states that can be described by a wave function $. In the general case, where the electrons can no longer be described by a single wave function, the degree of polarization can be between 0 and 1. Such a situation is called a mixed state. To find the resulting polarization, one must average the polarization of the various pure states: (2.5.9) (2.5.10) Equation (2.5.7) is replaced by prob (fl) = P cos2(8/2) prob (- 1) = P sin2(e/2).

(2.5.11)

Mixed states cannot be described by wave functions, since they consist of an incoherent superposition of pure states. The most convenient tool to

2.5.

POLARIZATION OF ELECTRONS A N D PHOTONS

219

handle such systems is the density matrix which will be briefly described in Section 2.5.2.3. 2.5.2.1.3. NONRELATIVISTIC ELECTRONS. The step from electrons a t rest to nonrelativistically moving particles leads to some useful definitions. Assume that the electron has a momentum p and that @ is the unit vector in the direction of the momentum. One then defines the longitudinal pofarization or helicity P, by

P,

=

(2.5.12)

P, = +1 means that the polarization vector is parallel to the momentum p; P, = - 1 indicates antiparallel orientation. In general, the polarization vector will be neither parallel nor antiparallel to the momentum and the absolute value of P , will be less than 1. One can also measure the expectation value of the polarization in a direction i perpendicular to p, i.e., the transverse polarization Pt :

P,

= .

(2.5.13)

In such a measurement, an azimuthal angle is implied. Usually, the conditions of an experiment dictate the choice of the direction i. Often i is selected in such a way that Pt is maximized. 2.5.2.1.4. RELATIVISTIC ELECTRONS. The situation for relativistic electrons is more complicated, because in the Dirac theory the spin and the momentum of a particle are coupled. The approach used for the nonrelativistic electrons therefore does not work. The description of the polarization of relativistic particles is nevertheless very important. This importance has actually increased in the last few years, because the high-energy accelerators not only produce relativistic electrons, but also relativistic muons, nucleons, and hyperons. The simplest description of the polarization of a relativistic electron was originally suggested by Darwin3P:one defines the polarization as the expectation value of the spin operator d in the rest frame of the electron. To find the polarization in any other coordinate system, one uses a Lorentz t r a n ~ f o r m a t i o n . ' ~ ~ ~ ~ - ~ ~ The polarization can, however, also be defined directly by selecting an appropriate operator. The various ways in which such operators can be introduced, their justification, their properties, and the relationships C. G. Darwin, Proc. Roy. SOC.Al'dO, 621 (1928). H. A. Tolhoek and S. R. deGroot, Physica 17, 1, 17, 81 (1951). 34 G. Ascoli, 2.Physik 180, 407 (1958). *& T. K. Khoe and L. C. Teng, Argonne National Laboratory Report ANLAD 69, 1962. Unpublished. 3*

*3

220

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

among different points of view have been the subject of a number of papers. 2 2,24-27,86-46 One of the most convenient generalizations of d has been introduced by S t e ~ h For . ~ ~electrons, the corresponding operator can be written as22,27,44,46 where

z = (d ' PIP + P x

(bJx

is the 4 X 4 generalization of Eq. (2.5.3) and

(:

P)

(2.5.14)

is the 4 X 4 Dirac matrix

:1)*

The operator B commutes with the Hamiltonian and with the momentum operator. In the nonrelativistic limit B is identical with the spin operator d.

From Eq. (2.5.14), using Dirac spinors, one can see that the expectation value of the helicity is the same in the laboratory system as in the rest system of the electron : (2.5.15) The nonrelativistic operator d 6 for the transverse polarization is replaced by pd 6. The matrix fi introduces a factor mc2/W, where W is the total energy of the electron. The transverse polarization, measured in the laboratory system, is hence smaller than the transverse polarization measured in the rest system: mc2 P,(lab) = = = -Pt(rest). (2.5.16)

-

W

2.5.2.2. The Polarization of Photons. Photons also possess only two polarization states and the formal description of the photons polarization seL. Michel and A. S. Wightman, Phys. Rev. 98, 1190 (1955). B. Stech, 2. Physik 144,214 (1956). C.Bouchiat and L. Michel, Nuclear Phys. 6, 416 (1958). J. Werle, Nuclear Phys. 6, 1 (1958). 40 C. Fronsdal and H. UberalI, Phys. Rev. 111,580 (1958). R. H. Good, Jr., and M . E. Rose, Nuouo cimento 13, 1182 (1959). 4-2 M. Froissart and R. Stora, Nuclear Znstr. 7,297 (1960). 4 * F. Calogero, Nuovo cimento 20, 280 (1961). 44 M.E.Rose and R. H. Good, Jr., Nuouo cimento 22, 565 (1961). 45 D.M.Fradkin and R. H. Good, Jr., Nuovo cimento 22, 643 (1961).

*'

2.5.

POLARIZATION

OF ELECTRONS AND PHOTONS

22 1

is hence very similar to the one for electrons. One main difference exists, however. The spin of the electron can point in a n arbitrary direction with respect to its momentum. The photon spin, however, lies always parallel or antiparallel t o the m ~ m e n t u m . A~ photon . ~ ~ ~ in ~ ~a ~plane-polarized ~ state is not in an eigenstate of the photon spin operator; its state is described by a linear superposition of the two eigenstates of the spin operator. Formally, one can describe the complete polarization state of a photon moving along the + z axis b y the expectation value of the Pauli spin vector Eq. (2.5.3): P(photon) = . (2.5.17) The physical interpretation of the spin vector is different, however, for electrons and photons. For electrons, P is a vector in physical space. P(photon), on the other hand, is a vector in a "polarization space" (PoincarB space)8 and its components are defined as follows: P I is the plane polarization with respect to two arbitrary orthogonal axes; Pz is the plane polarization with respect to two axes oriented a t 45" to the right of the previous set; and P o is the circular polarization (helicity, longitudinal polarization). 2.5.2.3. Density Matrix and Stokes Parameters. T h e complete description of an electron or photon beam, propagating in a given direcTABLE I. Stokes Parameters for Photon and Electron Beams Stokes parameter

I PI

Pz P8

Photons

Electrons

Intensity Plane polarization Plane polarization a t 45" to PI Helicity (circular polarization)

Intensity Transverse polarization Transverse polarization a t 45' to PI Helicity (longitudinal polarization)

tion, requires four experimentally determinable quantities : the intensity 1 and three components of the polarization vector P . (Methods for measuring beam intensities are treated in Chapter 2.8.) A particular set of these four quantities, known as Stokes parameters, is given in Table 1.8,46-61

U. Fano, J. Opt. Soc. Am. 39, 859 (1949). D. L. Falkoff and J. E. McDonald, J . O p t . SOC.Am. 41, 862 (1951). 48 W. H. McMaster, Am. J. Phys. 22, 351 (1954). U. Fano, Revs. Modern Phys. 29, 74 (1957). 6o G. Passatore, Nuovo cimento 6, 850 (1957). 61 W. H. McMaster, Revs. Modern Phys. 33, 8 (1961). 411 47

222

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Most experimental arrangements are designed without explicit reference to the Stokes parameters. Their value and convenience becomes apparent, however, in the theoretical treatment of the experiments. Hence, some of the properties of Stokes parameters are listed in the following : (1) The Stokes parameters can be written in the form of a column vector : (2.5.18)

(2) If several independent beams are superposed incoherently, the Stokes parameters of the resulting beam are the sum of the Stokes parameters of the individual beams. (3) From the Stokes parameters, the density matrix: p30s31948849 describing the polarization state can be constructed immediately: p =

+(I

+P

*

d)

(2.5.19)

where d is the Pauli spin vector (Eq. 2.5.3). (4) A polarization semitive detector can also be described by a set of Stokes parameters. Assume that the detector passes only particles or photons characterized by the Stokes parameters (1,Q). The set (1,Q) is then used to represent the detector. (5) The probability W of detecting with a detector (1,Q) a particle out of a beam characterized by (1,P) is given by (2.5.20) (6) If a particle undergoes a polarization-sensitive interaction, then the Stokes parameters of the scattered beam (I,P) are given by (2.5.21)

The interaction matrix 3 has been worked out explicitly for a number of cases.46-61 2.5.2.4. Sign Conventions. A particle or photon is called “righthanded” when its spin and its momentum are parallel; it is called “lefthanded” when they are antiparallel. From Eq. (2.5.12) it then follows that a right-handed particle possesses positive helicity, and a lefthanded particle negative helicity. A right-handed particle corresponds to a right-handed screw.

2.5.

POLARIZATION O F ELECTRONS AND PHOTONS

223

This definition is opposite to the one used in classical optics, where observers face the source of light. The magnetic moment of the electron is opposite to its spin; an electron with positive helicity thus has a magnetic moment which trails its momentum. The sign of the transverse polarization occurring in nuclear interactions of particles with spin is fixed by the Basel Convention: Positive polarization is taken in the direction of the vector product

+

ki X ko, where ki and koare the wave vectors of the incoming and outgoing particles respectively. [SUN Document, Physics Today 16, No. 6, 23 (1962).] 2.5.3. Motion of Electrons in Electromagnetic Fields* 2.5.3.1. Preliminary Remarks. A knowledge of the behavior of spin particles in electromagnetic fields is crucial for designing and evaluating experiments involving their polarization. I n homogeneous fields, the trajectory of a particle beam is independent of its polarization. The problem can therefore be separated into two parts, the motion of “spinless” particles in external electromagnetic fields and the influence of these fields on the polarization. The trajectories of particles in external electromagnetic fields have been calculated for many cases of interest. Discussions of this problem and further references can be found in references 52-54. The behavior of the polarization in electromagnetic fields is of major interest for the applications discussed in the present chapter. Some important cases and a list of references are given in Section 2.5.3.2. In Section 2.5.3.3, experimental arrangements for the transformation from longitudinal to transverse polarization are treated. 2.5.3.2. Theoretical Results. The motion of polarized particles in electromagnetic fields has been treated in a number of papers.14~26,21~33-36. In discussing a particular problem, one has to decide whether

+

42,43966--62

6zK. T. Bainbridge, in “Experimental Nuclear Physics” (E. SegrB, ed.), Voi. I, Part V. Wiley, New York, 1953. 63 0. Chamberlain, Ann. Rev. Nuclear Sn’. 10, 161 (1960). 6 4 M. S. Livingston and J. P. Blewett, “Particle Accelerators,” Chapter 5. McGrawHill, New York, 1962. b6 H. Mendlowitz and K. M. Case, Phys. Rev. 97, 33 (1955); 100, 1551 (1955). 6E K. M. Case, Phys. Rev. 106, 173 (1957). s7 L. M. Carassi, Nuovo cimento 6, 955 (1957); 7, 524 (1958). s8 H. Mendlowitz, Am. J . Phys. 26, 17 (1958). 69 V. Bargmann, L. Michel, and V. L. Telegdi, Phys. Rev. Letters 2 , 435 (1959). V. S. Popov, Soviet Physics J E T P 11, 1141 (1960). R I R H. . Good, Jr., Phys. Rev. 126, 2112 (1962). E2 H. J. Meister, 2. Physik 166, 468 (1962). * See also Vol. 4A, Part 3.

224

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

the treatment must be relativistic, whether one has to take the anomalous magnetic moments into account, and whether the fields can be considered to be homogeneous. These three criteria will become more apparent in the following. In Sections 2.5.3.2.1 to 2.5.3.2.5, typical situations will be treated. The notation is as follows: y = [ l - (v/c)”-”~

(2.5.22)

E and B denote the electric and magnetic field strengths; fi and are unit vectors in the direction of the momentum and the transverse polarization, respectively; v is the particle velocity. The rest mass and the charge are m and e; for an electron, e < 0. The dimensionless g factor is defined as the ratio of the magnetic moment I to the spin S:

I = SPOS

(2.5.23)

p o = eh/2mc.

(2.5.24)

with For positive Dirac particles, g and PO are both positive; the spin and the magnetic moment are parallel to each other. For negative Dirac particles, where the magnetic moment points opposite to the spin, Eqs. (2.5.23) and (2.5.24) can lead to some confusion. The negative sign can be introduced in g or in pa. In the definition used here, e and hence po are negative for the electron; g is positive, The g factor has a normal and an anomalous part, g = 2(1

+ a).

(2.5.25)

Numerical values for the anomalous part a are given in Table 11. TABLE 11. Anomalous Parts of the Particle Electron Muon Proton a

a = (g - 2)/2

(1160.9 f 2 . 4 ) X (1162 f 5 ) X 1.79276 0.00003

g

Factor Reference a b

Schupp, Pidd, and Crane.I6 Charpak et al.83,84

4 3 G. Charpak, F. J. M. Farley, R. L. Garwin, T. Muller, J. C. Sens, V. L. Telegdi, and A. Zichichi, Phys. Rev. Letters 8, 128 (1961). 04 G. Charpak, F. J. M. Farley, R. L. Garwin, T. Muller, J. C. Sens, and A. Zichichi, Phys. Letters 1, 16 (1962).

2.5.

POLARIZATION OF ELECTRONS A N D PHOTONS

225

The values in Table I1 and the equations in the following sections together determine whether the anomalous magnetic moment must be taken into account in the design and the evaluation of a given experiment. 2.5.3.2.1. HOMOGENEOUS EXTERNAL FIELDS.I n order to describe the behavior of the polarization in external electromagnetic fields, one introduces the angle # between the polarization vector P and the momentum unit vector 0 : (2.5.26) (P * p)/P = cos #.

d is called the polarization angle. I n an external electromagnetic field, 4 will, in general, be a function of time. The laboratory time rate of change, d#/dt, in a homogeneous field is given by59,36

1""

[

5 + i . (fi X B ) ( g - 2 )

fi=-=e " (9 - 2 ) dt 2mc ( u / c ) Y

Equation (2.5.27) is valid as long as the distances over which the fields vary are large compared to the dimensions of the wave packet describing the particle. Inhomogeneous fields are treated in reference 61. Equation (2.5.27) can be generalized to include the effect of external electromagnetic fields on particles with electric dipole rn0ments.b~t6~ The resulting equations have been used in the evaluation of experiments to determine the electric dipole moment of muons and electr0ns.6~-6* 2.5.3.2.2. LONGITUDINAL FIELDS.Assume that both the electric and the magnetic fields are parallel to the momentum:

EX$=O

BXP=O.

The longitudinal polarization (Eq. 2.5.15) remains constant. The transverse polarization (Eq. 2.5.16) is affected in two ways. I n a longitudinal magnetic field, the maximum transverse polarization also remains constant (# = const) but the transverse polarization processes around the field direction with an angular frequency WB

=

geB/2myc

=

gpoB/hy.

(2.5.28)

This rotation has been used for a direct determination of the g factor of the free e l e c t r ~ n . ' When ~ using lens spectrometers in experiments BB

R. L. Garwin and L. M. Lederman, Nuovo cimento 11, 776 (1959). D. F. Nelson, A. A . Schupp, R. W. Pidd, and H. R. Crane, Phys. Rev. Letters 2,

492 (1959). 67 G. CharpRk, L. M. Lederman, J. C. Sens, and A. Zichichi, Nuovo cimento 17,288 (1960). 68 G. Charpak, F. J. M. Farley, R. L. Garwin, T. Muller, J. C. Sens, and A. Zichichi, Nuovo cimento 22, 1043 (1961).

226

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

designed to determine a transverse polarization, one must either compensate or correct for the resulting rotation of the polarization vector (see Section 2.5.5). In a longitudinal electric field, the transverse polarization remains unchanged in the rest system of the particle. The electric field, however, changes the total energy W of the particle. According t o Eq. (2.5.16), the transverse polarization observed in the laboratory system will then change also. 2.5.3.2.3. TRANSVERSE ELECTRIC FIELD. Assume an electric field which is always perpendicular to the momentum:

E*f=E

B=O

E*fi=O.

The trajectory in such a field will be circular and the laboratory angular frequency of the particle will be given by: WE =

(2.5.29)

eE/myv.

The momentum will also rotate with a frequency angle 4 changes according to d$/di

= QE = & E [ ( g

- 2)y

WE.

- g/rl.

The polarization (2.5.30)

If 60is the polarization angle of a particle before entering the electric field, and if the momentum is turned by an angle A$ = w ~ by t the field, then the polarization angle becomes 4’ = $ J ~ A 4 , where

+

A4

=

4A$[(g - 2 ) ~ g/~l.

(2.5.31)

Equation (2.5.31) yields for the angle A#* required to transform longitudinal into transverse polarization (A4 = --.rr/2) : (2.5.32)

In the nonrelativistic limit, y = 1, the situation becomes very simple. From Eq. (2.5.30) it follows that QE(nonre1) = - W E . Seen from the laboratory, the polarization vector keeps its direction in space; the polarization direction is not affected by the electric field. The angIe A$* in Eq. (2.5.32) becomes a/2. 2.5.3.2.4. TRANSVERSE MAGNETIC FIELD. I n a transverse magnetic field, E=O B*QXfi=B the time rate of change (Eq. 2.5.27) of the polarization angle becomes

n,

eB 2mc

= - (g

- 2).

(2.5.33)

2.5.

POLARIZATION

OF ELECTRONS

AND PHOTONS

227

The orbital angular frequency of a charged particle in this field is given by the Larmor frequency: (2.5.34) (2.5.35) Equations (2.5.33) and (2.5.35) show that a particle with g = 2 will precess in a transverse magnetic field without change of the polarization angle 4. A particle with g # 2 , on the other hand, will suffer a change of the polarization angle and this change can be used to determine the anomalous part a = &(g - 2 ) directly. Such experiments have been performed with free electrons'6 and with free muon^.^^,^^ For high-energy polarized beams, the property expressed by Eq. (2.5.33) c8n be used to transform the polarization angle. 2.5.3.2.5. CROSSED FIELDS. Consider transverse crossed fields,

E.B=O

B*IX@=B

(2.5.36)

adjusted in such a way that the particle trajectory is straight:

E

= -V

X B/c.

(2.5.37)

Inserting Eqe. (2.5.36) and (2.5.37) into Eq. (2.5.27) yields QEB

=

W L ~ / ~ Y .

(2.5.38)

Arrangements other than the ones treated in Sections 2.5.3.2.2 to 2.5.3.2.5 can be discussed along similar lines, always starting from Eq. (2.5.27). 2.5.3.3. Polarization Transformers. In some experiments it is desirable to change the polarization of a beam. One example is the determination of the longitudinal polarization of beta particles emitted in nuclear beta decay. If Mott scattering is used to measure the polarization, the longitudinal polarization must first be changed to a transverse one, since Mott scattering is only sensitive to the transverse polarization. A second example is the determination of the anomalous part of the g factor of the electron or the muon. The transformation of the polarization state of a particle beam can be achieved in a number of ways, all based on applications of Eq. (2.5.27). In the following sections, the most important polarization transformers will be discussed. One general remark is in order. Most of the spin transformers act also as energy, momentum, or mass filters. It is therefore possible to use them simultaneously for the selection of a certain energy or momentum band and for the polarization transformation.

228

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Nevertheless it is often desirable to precede the transformer in the experimental set-up by an energy selector in order to prevent particles with undesired energies from reaching the detector after scattering from the walls of the polarization transformer. 2.5.3.3.1. CYLINDRICAL AND SPHERICAL CONDENSERS. It is clear from Section 2.5.3.2.3 that a radial electric field acts as a spin transformer. Equation (2.5.31) gives the change A$ in the polarization angle $I if the momentum is turned by an angle A$. Equation (2.5.32) describes the angle Afi* through which the momentum has to be turned in order to transform a longitudinal polarization in to a transverse one. A radial electric field can be realized with cylindrical and spherical condensers. Both types have been used extensively as velocity selectors6* and as polarization transformers. l r 3 References and data for designing velocity selectors are given in reference 52. The spherical condenser has been treated in references 69-73. Among the many publications in which the use of radial electric fields as polarization transformers is described, the following offer the most information for the experimental design and evaluation : cylindrical c o n d e n ~ e r sspherical ,~~ condensers.76-7s Figure 1 shows a setup in the cylindrical g e ~ m e t , r y Figure . ~ ~ 2 shows the schematic arrangement for a polarization transformer using a spherical ~ondenser.7~ The deflection voltage which must be applied to the plates of the condensers is given by62 (2.5.39)

for the cylindrical case, and by (2.5.40) M. Purcell, Phys. Rev. 04, 818 (1938). E. Persico and C. Geoffrian, Rev. Sci. Znstr. 21, 945 (1950). 71 F. T. Rogers, Rev. Sci. Znsfr. 22, 723 (1951). 72 N. Ashby, Nuclear Instr. 3, 90 (1956). 7aR. H. Ritchie, J. S. Cheka, and R. D. Birkhoff, Nuclear Znslr. 6, 157 (1960); 69E.

70

8, 313 (1960). 74 J. S. Greenberg, D. P. Malone, R. L. Gluckstern, and V. W. Hughes, Phys. Rev. 120, 1393 (1960). 76 H. Bienleh, K. Guthner, H. von Issendorff, and H. Wegener, Nuclear Instr. 4,

79 (1959). 76 A. Ladage, Ph.D. Thesis, University of Hamburg, 1961 (unpublished). 77 R. 0. Avakyan, G . L. Bayatyan, M. E. Vishnevskii, and E. V. Pushkin, Soviet Phys. JETP 14, 491 (1962). 78 A. R. Brosi, A. I. Galonsky, B. H. Ketelle, and H. B. Willard, Nuclear Phys. 33, 353 (1962).

2.5.

POLARIZATION OF ELECTRONS AND PHOTONS

229

Grounded

At. C o l l i m a t o r Plbstic Scintillator

S , ,S,

S7 -Collimating

Slits

FIG.1. Spin transformation with a cylindrical condenser. The figure also shows the arrangement to determine the transverse polarization with Mott scattering. The d i g r a m is not to scale. (From Greenberg, Malone, Gluckstern, and Hughes, reference 74.)

for the spherical case. Rz and R1 are the radii of the outer and the inner plates, respectively, m is the rest mass and v the velocity of the particle, and y is given by Eq. (2.5.22) (Gaussian units). Some typical design parameters and properties for two systems are given in Table 111. TABLE 111. Design Parameters for Electric Condensera Type

Cylindrical

Spherical

Reference Mean bending radius Gap width Deflection angle Azimuthal acceptance angle Maximum voltage Transmission Energy resolution

74 20 cm l.8cm 90"

78 25.4cm 1.Ocm 137" 45" 50 kv 5 x 10-4 45%

5

50 k v x 10-5 5%

230

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

The following additional remarks apply to the design and the evaluation of experiments involving cylindrical and spherical condensers : (1) It is difficult to use voltages higher than 50 kv without sparking, regardless of gap width. The systems, hence, must be so designed that the highest energy to be investigated can be passed without exceeding this limit. (2) It is preferable to apply the voltage symmetrically to the two plates so that the mean path is a t ground potential.

FIG. 2. Spin transformation with a spherical condenser. The indicated value for the polarization angle 6 applies to electrons of 616 kev. (From Brosi, Galonsky, Ketelle, and Willard, reference 78.)

(3) The transformation from longitudinal to transverse polarization in any given set-up is complete only for the one energy which satisfies Eq. (2.5.32). However, for energies within a very wide band, the deviations from a complete transformation are small and a corresponding correction is easy to apply. (4) The walls of the system should be lined with a material with a low backscattering coefficient (polyethylene) in order to reduce the background. (5) Corrections which must be applied to the data in order to find the final polarization angle are discussed in references 74-78. (6) As can be seen from TabIe 111, the spherical condenser has a t least one order of magnitude better transmission than the cylindrical con-

2.5.

POLARIZATION

OF ELECTRONS AND PHOTONS

23 1

denser. It is, however, much more difficult to construct and to adjust. The choice between the two hence depends on the particular problem to be solved. (7) As pointed out above, it is desirable to preselect the energy of the particles entering the polarization transformer, for instance with a magnetic spectrometer. Unfortunately it is impossible to match a magnetic spectrometer to a spherical analyzer without a considerable loss in intensity. One thus has two alternatives, either a preselection with a corresponding loss in intensity, or no preselection and a higher background and increased scattering. 2.5.3.3.2. CROSSEDFIELDS (WIEN FILTER).Consider crossed electric and magnetic fields, adjusted according to Eq. (2.5.37) :

E / B = lE[/lB[

= V/C.

(2.5.37’)

Particles with the velocity v then travel undeflected through this system. Their polarization vector, according to Eqs. (2.5.38) and (2.5.34), is turned by an angle At$ = egBL/2mcvy2 (2.5.41) where L is the length of the system. Such crossed fields have been used as polarization transformers in a number of investigation^.'^-*^ An example of a setup is shown in Fig. 3. To give some idea about the orders of magnitude involved in the design of a Wien filter for low-energy electrons, consider a n apparatus of length L = 30 cm and assume electrons of 200 kev energy ( v / c = 0.695). From Eq. (2.5.37) it follows that E (in kv/cm)/B (in oe) = 0.21. If a complete transformation from a longitudinal to a transverse polarization is to be obtained, A 4 in Eq. (2.5.41) is r / 2 and one finds for the required magnetic field strength B = 120 oe. The electric field strength then becomes E = 25 kv/cm. Crossed fields have a number of advantages over other types of polarization transformers: (1) The entire arrangement is axially symmetric. This symmetry reduces the appearance of experimental asymmetry effects. 78 P. E. Cavanagh, J. F. Turner, C. F. Coleman, G. A. Gard, and B. W. Ridley, Phil. Mag. 21, 1105 (1957). A. I. Alikhanov, G. P. Eliseiev, V. A. Lyubimov, and B. V. Ershler, Nuclear Phys. 6,588 (1958). P. E. Spivak and L. A. Mikaelyan, Nuclear Phys. 20, 475 (1960). 82 R. Sosnowski, Z. Wilhelmi, and J. Wojtkowske, Nuclear Phys. 26, 280 (1961).

232

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

(2) The magnetic and the electric field can be varied in such a way that the transformation of the polarization is complete, i.e., A 4 = n/2, for a wide range of energies. (3) If one reverses B and E simultaneously, the geometry of the apparatus remains unchanged but A 4 changes its sign. This property allows a simple elimination of asymmetry effects due to different counter efficiencies and other instrumental asymmetries.

O&O 2&- J

Ir

I

\

\

cm

I,

‘._I

IL-.---A

-

FIG. 3. Spin transformation with a crossed field. The upper part shows the horizontal, the lower the vertical cross section. The electron energy is preselected with a magnetic spectrometer (2); 1 indicates the source; 3 denotes the magnet pole and 4 the electrostatic plates of the crossed field system. Magnetic shields (5) are used to define the length of the crossed fields. The transverse polarization of the electrons leaving the transformer is determined with Mott scattering; 6 indicates the scattering foil and 7 the counter telescopes. (From Spivak and Mikaelian, reference 81.)

(4) Since the change in polarization can be chosen to be go”, small errors in spin rotation due to an error in the determination of the effective length L are small. ( 5 ) Crossed fields can be used as mass filters, even for particles with very high energy.s3@It may be possible to use the combined properties of mass filter and polarization transformer. One major disadvantage of a Wien filter is the fact that gamma rays from the source can pass straight through the system without encountering an absorber. This disadvantage can be removed, with some loss of intensity, by inserting a magnetic spectrometer (lens spectr~rneter’~ or transverse sector fields1)between source and polarization transformer. 83

A. Yokosawa, Argonne National Laboratory Report ANLAD-70, 1962. Unpub-

lished.

2.5.

POLARIZATION OF ELECTRONS AND PHOTONS

233

2.5.3.3.3. COULOMB SCATTERING. Coulomb scattering in materials of low atomic number 2 can also be used to transform the polarization state of electron^.^^.^^ The Coulomb field is essentially a radial electric field. For low 2, the spin-orbit coupling is small and all the arguments presented in Section 2.5.3.3.1 apply. Details about experiments can be found in references 86-90. A particularly elegant m e t h ~ d consists ~ ~ s ~ ~in using a semicircular aluminum foil as the polarization transformer. With properly placed source and second scattering foil, electrons scattered anywhere in the foil are changed from a longitudinal to a transverse polarization state. The semicircular scatterer results in an appreciable intensity gain over a flat scatterer. The main advantage of this type of polarization transformer is its simplicity; the main disadvantages are the necessity for a correction for depolarization and the fact that Coulomb scattering does not act as an energy selector. 2.5.4. Polarization Transfer* 2.5.4.1. interactions between Electrons and Photons. It was pointed out in Section 2.5.1 that the helicity is to a large extent conserved in the various transformations involving photons and electrons. This “persistence of helicity” forms the basis of some methods to determine the longitudinal polarization. It can, for example, be more convenient in a certain experiment to measure the helicity of photons than of electrons. In this case one forces the longitudinally polarized electrons to produce bremsstrahlung ; the bremsstrahlung is circularly polarized and one detects this circular polarization. In the present section, the various processes of interest for experimental applications will be outlined. Two of the processes of interest are shown in Fig. 4, both involving electrons of momentum p and energy E (including the rest energy), and photons of momentum k and energy E?. Only the observed particles are shown in Fig. 4.Figure 4a hence can represent bremsstrahlung or positron annihilation. Figure 4b can stand for pair production or photoeffect. To describe the particles and the interaction processes, it is best to use the Stokes parameters (Table I) and the interaction matrix 3, Eq. (2.5.21). L. J. Tassie, Phys. Rev. 107, 1452 (1957). F. Gursey, Phys. Rev. 107, 1734 (1957). 86 A. De-Shalit, S. Kuperman, H. J. Lipkin, and T. Rothem, Phys. Rev. 107, 1459

S4

86

(1957).

J. Heintze, 2. Physik 160, 134 (1958). A. I. Alikhanov, G . P. Eliseiev, and V. A. Liubimov, Nuclear Phys. 7, 655 (1958). 89 W. Bahring and J. Heintze, 2. Physik 163, 237 (1958). 9 0 S. Cuperman, Nuclear Phys. 28, 84 (1961). * See also Vol. 4R, Chapter 7.6.

87

88

234

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Explicit representations of the interaction matrix for various processes can be found in references 46-51. (Before using the Stokes parameters or interaction matrices given in a paper, it is advisable to check the definition of the components of the Stokes parameters since it is not the same in all papers.)

X

t

FIG. 4. Interaction between electrons and photons. (a) An incident electron creates one (or more) photons. (b) An incident photon gives rise to one (or more) electrons. Only one emerging particle is shown. The interaction takes place in the zz plane.

If the incident electron in Fig. 4a has a longitudinal polarization P, and the outgoing observed photon a circular polarization Pa, one defines a helicity transfer egiciency L by the relation Pt(photon)

=

LP,(electron).

(2.5.42)

L is a function of the parameters describing the process under consideration. Examples are discussed later. If the incident electron is transversely polarized, the outgoing photon can be circularly polarized; the corresponding transfer efficiency T is given by Pt(photon) = TPt(electron).

(2.5.43)

Similar transfer efficiencies can be defined for processes represented by Fig. 4b, for instance P,(electron) = L’Ps(photon).

(2.5.44)

In general, the calculation of the helicity transfer efficiencies L and L’ requires a detailed investigation of the processes under consideration. However, for incident particles of high energy (>>mcz)>,a general connection holds: I n a Coulomb field, almost complete momentum transfer implies that helicity is also transferred nearly completely, i.e., L -+ 1,

2.5.

POLARIZATION OF ELECTRONS AND PHOTONS

235

L’ -+ 1. This result can be interpreted as a consequence of conservation ~ ~the , ~following ~ sections, specific examof spin angular m o m e n t ~ m .In ples will be given. 2.5.4.2. Bremsstrahlung.* Of particular interest for experimental applications is bremsstrahlung. Regardless of the polarization state of the incident electron beam, bremsstrahlung is, in general, polari~ed.~3-~* If the incident electron beam is unpolarized, the bremsstrahlung is partially linearly p o l a r i ~ e d . ~At ~ -electron ~~ energies of a few Mev or less, the polarization vector (vector of the electric field) is perpendicular to the plane of interaction (plane of photon emission) for low photon energies. At the high energy tip of the bremsstrahl spectrum, the polarization vector lies in the plane of interaction. A typical plot of the linear polarization versus photon energy is shown in Fig. 5 . At high electron energies, the polarization vector is always perpendicular to the plane of interaction and the amount of polarization decreases with increasing photon energy (Fig. 6). The high energy tip of the bremsstrahl spectrum of high energy electrons shows very little or no linear polarization. The polarization of the bremsstrahlung due to polarized electrons has been treated in many papers.61~91~92~95~g7-107 The following facts emerge. ( 1 ) The linear polarization of the bremsstrahlung is independent of the polarization of the incident electrons if the spin of the final electrons is not observed. ( 2 ) L o n g i € u d i ~ a lpolarized l~ electrons give rise to circulady poIarized photons. The sign of the helicity remains unchanged; right handed U. Fano, K. W. McVoy, and J. R. Albers, Phys. Rev. 116, 1159 (1959). R. H. Pratt, Phys. Rev. 123, 1508 (1961). 98 A. Sommerfeld, “Atornbau und Spektrallinien,” Vol. 2, Chapter 7. Vieweg, Braunschweig, Germany, 1939. 9 4 H. A. Bethe and E. E. Salpeter, Quantum Mechanics ofthe One- and Two-Electron Atoms, Section IV, c. Springer, Berlin; Academic Press, New York, 1957. 96 R. L. Gluckstern, M. H. Hull, Jr., and G. Breit, Phys. Rev. 90, 1026 (1953). 96 R. L. Gluckstern and M. H. Hull, Jr., Phys. Rev. 90, 1030 (1953). 97 C. Fronsdahl and H. Uberall, Phys. Rev. 111, 580 (1958). 9s C. Fronsdahl and H. Uberall, Nuovo cimento 8, 163 (1958). 99 H. Olsen and L. C. Maximon, Phys. Rev. 114, 887 (1959). 1 0 0 1 . B. Zel’dovich, Doklady Akad. Nauk S.S.S.R. 83, 63 (1952). 101 K. W. McVoy, Phys. Rev. 106, 828 (1957); 111, 1333 (1958). 102 G. L. Visotskii, A. A. Kresnin, L. N. Rozentsveig, Zhur. Eksptl. i Teoret. Fiz. 82, 1078 (1957); Soviet Phys. J E T P 6, 883 (1957). 103 A. Claesson, Arkiv Fysik 12, 569 (1957). 104 G. Bobel, Nuovo cimento 6, 1241 (1957). 106 B. K. Kerimov, I. M. Nadzhafov, Izvest. Akad. Nauk S.S.S.R. ser. Fiz. 22, 886; Columbia Tech. Transl. 22, 879 (1958). H. Banerjee, Phys. Rev. 111, 532 (1958). 107 S.Sarkar, Nuovo &mento 21, 410 (1961). * See also Vol. 4A, Section 3.5.3. 91

92

236

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

electrons give rise to right-handed photons. The transfer eflciency L, i.e., the ratio of the photon helicity to the electron helicity [see Eq. (2.5.42)], depends on the energy of the electron, on the fractional energy E,/(E - mc2) of the photon, on the photon emission angle 8, and on the charge Z of the nucleus in whose field the bremsstrahl process occurs.

E ~ (E / - M C 2~

FIG. 5. Polarization effects in bremsstrahlung from electrons with initial energy = 2.5 MeV. The transfer efficiencies L (Eq. 2.5.42) aod T [(Eq. 2.5.43)]and the linear polarization Plin are plotted vs. the fractional photon energy E , / ( E - mcz). The curves are for 2 = 26 (iron) and for a photon emission angle e = 10'. (From Fronsdal and tZberall, reference 97.)

E

L reaches a maximum near the high energy tip of the bremsstrahl specthis maximum is unity for all trum. For high energy electrons ( E >> m2), elements. Typical curves for L are shown in Figs. 5 and 6. (3) Transversely polarized electrons also give rise to circularly polarized photons. The transfer efficiency T [Eq. (2.5.43)] is considerably smaller than unity. Two typical curves for T are given in Fig. 5 and Fig. 6. (4) For arbitrary polarization, characterized by a degree of polarization P [Eq. (2.5.2)] and a polarization angle 4 [Eq. (2.5.26)], the circular polarization P 3 of the bremsstrahlung is given by combining (2.5.42) and (2.5.43) as Pa(photon) = P ( L cos 4 T sin 4). (2.5.45)

+

2.5.

POLARIZATION OF ELECTRONS AND PHOTONS

237

The helicity of the bremsstrahlung is thus proportional to the degree of polarization of the electrons; unpolarized electrons cannot emit circularly polarized bremsstrahlung. (5) The intensity of the bremsstrahlung from transversely polarized electrons exhibits an azimuthal asymmetry about the direction of motion of the incident electron beam.108

0

0.2

0.4

0.6

0.0

I .o

Ey/ E FIG.6. Polarization effects in bremsstrahlung from electrons with initial energy E = 50 MeV. The transfer efficiencies L (Eq.2.5.42) and T (Eq.2.5.43) and the linear polarization Pli, are plotted vs. the fractional photon energy E,/(E - mez) = E,/E. The curves are for Z = 82 (lead) and for a photon emission angle e = 0.23". (From 01sen and Masimon, reference 99.)

(6) For longitudinally polarized electrons of very high energy, finite nuclear size effects can become noticeable in the transfer efficiency L.Io9 2.5.4.3. Pair Production. Pair production can be considered to be the inverse process of bremsstrahlung ; the physical observables of interest can be obtained simply from the ones calculated for bremsstrahlung (see references 8, page 161; references 51, 101). For experimental applications, the following results are important :61,99, 101,110,111

W. R. Johnson and J. D. Roeics, to be published. B. K. Kerimov and F. S. Sadykhov, Zhur. Eksptl. i Teoret. Fiz. 40,553 (1961); Soviet Phys. J E T P lS, 387 (1961). 110 K. W. McVoy and F. J. Dyson, Phys. Rev. 106, 1360 (1957). 111 I. G.Ivanter, Z h r . Ekspft. i Teoret. Fiz. S6, 1093 (1959); Soviet Phys. J E T P 9, 777 (1959). 108

109

238

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

(1) A photon beam with helicity PI produces an electron-positron beam that is longitudinally polarized. (2) The faster particle of the pair is always polarized in the same sense as the circularly polarized photon. (3) The longitudinal polarization P, of the faster particle of the pair is proportional to PI. At the upper end of the spectrum, where this particle takes all the available energy, the helicity transfer efficiency L' = P,/Pa reaches a maximum. (4) For high energy photons ( E , >> mc2), the maximum transfer efficiency for any element becomes unity. (5) I n a shower, created by a longitudinally polarized high energy electron, the initial helicity remains conserved to a very large extent."O This follows from the remarks here and in Section 2.5.4.2:in both bremsstrahlung and pair production, the helicity is conserved a t high energies. This preservation of helicity may be used to determine the helicity of the particle originating the shower. 2.5.4.4. Photoeffect.* The photoelectric effect is closely related to bremsstrahlung. In first order, it is the inverse of the high-energy limit of bremsstrahlung a t which the incident electron radiates all of its kinetic energy and is left with zero momentum.112 The calculation of the relevant cross sections how^:^^^^^^^^^*^^-^^^ (1) I n the high energy limit, photoelectrons produced by circularly polarized gamma rays are longitudinally polarized, regardless of the angle of emission. The helicity transfer efficiency P,(electron)/P3(photon) is unity. (2) At lower energy, circularly polarized photons also produce polarized electrons. However, the transfer efficiency is less than unity and the polarization vector of the electron is in general not parallel to the electron momentum. Curves and equations to find t'he magnitude and direction of the polarization vector are given in references 114-117. 2.5.4.5. Positron Anni hi I a tion-in- FIi g ht. The annihilation-in-fligh t of positrons is the basis for some experimental methods to determine the positron polarization by measuring the polarization of the annihilation quanta. Polarized positrons, when annihilating in flight, transfer a certain amount of their polarizarion to the resulting quanta:

t

K. W. McVoy and U. Fano, Phys. Rev. 116, 1168 (1959). llJK. W.McVoy, Phys. Rev. 108, 365 (1957). 114 H.Olsen, Kql. Norske Videnskab. Selskabs. Forh. 51, Nos. 11, Ila (1958). 116 H. Banerjee, Numo eimento 11, 220 (1959). U.Fano, K. W. McVoy, and J. R. Albers, Phys. Rev. 116, 1147 (1959). 11' B. Nagel, Arkiv Fysik 18, 1 (1960). 118 H.Kolbenstvedt and H. Olsen, Nuovo cimento 22, 610 (1961). * See also Vol. 4B, Chapter 8.2. t See also Vol. 4A, Section 2.1.9. Ila

2.5.

POLARIZATION OF ELECTRONS AND PHOTONS

239

(1) In single quantum annihilation of longitudinally polarized positrons, the resulting single gamma ray is circdarly polarized in the same sense as the positron. The helicity transfer efficiency increases with positron energy and becomes unity for high energies.llaJ16Single quantum annihilation is forbidden for free electron-positron pairs. However, if a third body takes up momentum, the process can take place. Usually, single quantum emission will only account for a small fraction of the observed annihilation quanta. 119 (2) The two quantum annihilation of longitudinally polarized positrons in an unpolarized target gives rise to circularly polarized photons.s.61J20-122 The higher energy gamma ray, emitted in the direction of the positron momentum, carries the larger helicity, with the same handedness as the positron. The helicity transfer efficiency is unity a t high energies; it is about 0.5 a t a positron energy of 100 kev. (3) Transversely polarized positrons also give rise to circularly polarized annihilation quanta.lZ3 (4) The annihilation quanta from transversely polarized positrons show a very small left-right asymmetry.124This asymmetry is, however, too small to be experimentally useful a t the present time. (5) The photon emission left-right asymmetry can be quite large in single quantum annihilation in high 2 atoms. 2.5.5. Detection of Electron Polarization* An experimental setup to determine the polarization of an electron beam will, in general, consist of a momentum selector and a polarizationmeasuring device. Information about momentum selectors can be found in Chapter 2.2 and in references 52, 125, and 126. The most important methods used to measure the polarization are listed in Table IV and discussed in the following sections. 2.5.5 1. Electron-Electron Scattering.t 2.5.5.1.1. THEORETICAL BACKGROUND. If two colliding (negative) electrons have parallel polarizations, the differential cross section for scattering is smaller than if they have 119

L. Sodickson, W. Bowman, J. Stephenson, and R. Wehatein, Phys. Rev. 124,

1851 (1961).

A. Page, Phys. Rev. 106, 394 (1957). W. R. Theis, 2.Physilc 160, 198 (1958). ‘22 W. H. McMaster, Nuovo cimento 17, 395 (1960). I2*L.A. Page, Phys. Rev. 109, 2215 (1958). 1 2 4 s . C. Miller and R. M. Wilcox, Phys. Rev. 126, 629 (1962). 120L. 121

l2s M. Deutsch and 0. Kofoed-Hansen, in “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 111, Part XI. Wiley, New York, 1959. 128 K. Siegbahn, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), Chapter 111. North Holland, Amsterdam, 1955. (Second ed. in preparation.) * See also Vol. 4A, Chapter 3.5. t See also Vol. 4B, Section 6.1.3.

TARLE IV. Methods to Determine the Polarization of Electrons Method

Used mainly for: PolarizaCharge tion Energy

Section

~

electronelectron scattering

~~

P, P,

>0.3Mev >0.3Mev

2.5.5.1 2.5.5.1

Remarks

~

~

Depends on the spin dependence of electron scattering. Electrons are scattered from thin magnetized iron foils. Effect is small (0.2Mev

2.5.5.3

Longitudinally polarized electrons produce circularly polarized bremsstrahlung. Circular polarization is determined by methods given in Section 2.5.6. Effects are small ( < l o % ) ; strong sources are required.

EC

Annihilation in flight (Helicity transfer)

e+

P,

>0.2Mev

2.5.5.3

The helicity of the high energy annihilation quanta is determined.

Annihilation in flight or a t rest (cross section)

ef

P,

2 0 . 0 3 MeV

2.5.5.4

The annihilation of positrons in magnetized iron depends on the direction of polarization.

Annihilation a t rest in magnetic fields

e+

PP

2.5.5.5

The energy term d - B permits the determination of the direction of d with respect to the applied field B.

2.5.

POLARIZATION OF ELECTRONS AND PHOTONS

24 1

antiparallel polarizations. This fact can be predicted by a naive application of the Pauli principle: If the electrons are in the same spin state, they cannot be in the same space state and hence cannot collide. The dependence on polarization is strongest for “head-on collisions,’’i.e., scattering by 90” in the c.m. system. For such collisions, the parallel polarization cross section vanishes entirely a t nonrelativistic energies and the polarization dependence remains strong at all energies. It weakens as the scattering angle departs from 90’ and vanishes a t 0” and 180”. Positronelect,ron collisions have the same polarization dependence as electronelectron collisions in the extreme relativistic limit, but the polarization dependence vanishes nonrelativistically. The polarization dependence of electron-electron (Mdler), electronpositron (Bhabha), and electron-muon collisions has been investigated theoretically by many author^.^^^'*^-'^^ These calculations do not contain radiative corrections. Such corrections are very small a t energies of interest in experiments on ordinary beta decay. In very high energy scattering experiments, they may have to be taken into a c c ~ u n t . ~ ~ ~ ~ ~ The electron-electron cross sections for parallel and antiparallel spin directions are given in Section 1.1.5.3. Of particular interest is the ratio of parallel to antiparallel differential cross section. As shown in Section 1.1.5.3, this ratio depends on the fractional energy transfer w (see Eq. 1.1.83); for a given electron energy, it is smallest for w = 6, i.e., for a head-on collision (see Fig. 9 in Chapter 1.1). The ratio up/u. for w = is shown in Fig. 7 for electron-electron and positron-electron scattering. (The designation parallel and antiparallel can lead to ambiguities. Here, parallel means that the two spins are parallel. In some theoretical papers, parallel means that each electron has its spin parallel to its momentum.) 2.5.5.1.2. THE EXPEMMENTAL METHOD.To use the polarization dependence of the Mgller cross section in measuring electron polarization, A. M. Bincer, Phys. Rev. 107, 1434 (1957). A. A. Kresnin and L. N. Rosentsveig, J. Exptl. Theoret. Phys. ( U . S . S . R . )32, 353 (1957); Soviet Phys. J E T P 6, 288 (1957). I29 G. W. Ford and C. J. Mullin, Phys. Rev. 108, 477 (1957). lS0J. M. C. Scott, Phil. Mug. [8] 2, 1472 (1957). A. I. Mukhtarov and I. S. Perov, Izvest. Akad. Nauk. S.S.S.R. 22, 883 (1958); Columbia Tech. Transl. 22, 876 (1958). I32 R. Raczka and A. Racska, Bull. acad. polon. sci. 6, 463 (1958). 133 K. Bockmann, G. Gamer, and W. R. Theis, 2. Physik 160, 201 (1958). la4 K. Nagy and I. Farkas, Nuovo ciniento 7, 570 (1958). Is6 P. Stehle, Phys. Rev. 110, 1458 (1958). S. Sarkar, Nuovo cimento 24, 139 (1962). l3’ G. Furlan and G . Peressutti, Nuovo cimento 15, 817 (1960); 19, 830 (1961). la* Y. S. Tsai, Phys. Rev. 120, 269 (1960). lZ7

128

242

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

electrons must be scattered by target electrons of known polarization, and some aspect of the scattering process must be measured. The troubles inherent in absolute measurements of cross sections can be avoided by measuring a quantity that depends on the ratio of cross sections for two polarization states of the target electrons. One such quantity can be developed as follows. Let Pi be the polarization vector of the electrons to be measured, while Pois the polarization 1.0

l

'

l

-

l

.

1

2

-

ELECTRON-POSITRON (Bhabha)

0

0.5

1.0

ENERGY (Mcv)

1.5

2 .o

FIQ.7. Cross section ratio (parallel to antiparallel spins) for Mprller and Bhabha scattering as a function of the incident electron energy in MeV. The curves are valid for symmetric scattering, where both emerging electrons have the same energy. The energy transfer in this case is w = i. vector of the target electrons. Let u+ be the scattering cross section with both electrons completely polarized in the direction of Pi and Po,while u- is the cross section with complete polarization but with the direction of POreversed. The cross section for scattering of electrons with uncorrelated polarization will be (u+ u-)/2. The magnitude of P is the fraction of particles that are fully polarized. Therefore a fraction PiPo of the collisions takes place between particles that are fully polarized, and have cross section u+ or a- depending on the direction of Po. The rest of the collisions have cross section (a+ u-)/2. With Poin one direction, the counting rate will be proportional to

+

+

(2.5.46)

2.5.

POLARIZATION OF ELECTRONS AND PHOTONS

243

With the direction of P o reversed, it will be proportional to (2.5.47) From these equations, one obtains (2.5.48) The polarization dependence of the cross section is often expressed in terms of u-/u+, and experimental results are given in terms of 8 , the relative change in counting rate on polarization reversal : 6=2

c+- c- = 2PtPo 1 - u-/u+* c++ c1 + .-/+

(2.5.49)

TARGET ELECTRONS. I n order to use the relation 2.5.5.1.3. POLARIZED Eq. (2.5.49), one must have target electrons of known polarization whose direction of polarization can be reversed. Such electrons exist in a magnetized ferromagnetic foil. About two of twenty-six orbital electrons in iron are aligned at saturation. The fraction f of electrons aligned can be obtained by measuring the magnetization of the foil. It is related to the magnetization through f = MJNP (2.5.50) where M, is the magnetic moment per unit volume in the material due to electron spin, N is the number of electrons per unit volume, and p is the Bohr magneton. M a is not equal to the total magnetic moment per unit volume in the material: in addition to Ma there is a small but significant orbital c o n t r i b u t i ~ n . ‘The ~ ~ relative magnitudes of spin and orbital contributions can be obtained from a measurement of the gyromagnetic ratio for the foil material by an Einstein-de Haas or Barnett experiment. M, is related to the total magnetization M by M,=2-

g’

-1 9’

M

(2.5.51)

where g‘ is the magnetomechanical f a ~ t 0 r . l ~ ~ Values of the magnetomechanical factor g’ (which is sometimes also called the gyromagnetic ratio) are collected in reference 140. The data in reference 140 also allow reasonable estimates for alloys. C. Kittel, “Introduction to Solid State Physica,” 2nd ed., pp. 408-414. Wiley, New York, 1956. 140 G . G. Scott, Revs. Modern Phys. 54, 102 (1962).

244

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

2.5.5.1.4. THE DETECTION OF ELECTRON-ELECTRON SCATTERING. The cross section for Mgller scattering from all electrons of an atom is smaller than the Mott cross section for scattering from the nucleus by a factor of about the atomic number of the scatterer. Therefore, Mgller-scattered electrons must be detected among a much larger number of Mott-scattered ones. Fortunately, there are differences between Mott-scattered and Mller-scattered electrons. The Mgiller-scattered ones come from collisions between particles of equal mass. In such collisions, as seen in the lab system, the incident electron gives up much of its kinetic energy to the target electron, and both emerge from the collision. This allows Mgller collisions to be distinguished from the Mott background by counting the two electrons in coincidence, Use of coincidence counting is helped by the fact that Mgller scattering can be treated as a two-body process as long as the bombarding energy is much greater than the binding energy of the target electrons. If the incident momentum is known, there is a fixed relation between the momenta of the two outgoing electrons. The counters can be placed so that if one electron reaches one counter, the other electron will almost certainly reach the other. This relation can be obscured by multiple scattering in the foil. Energy conservation in the collision can be used to discriminate against spurious coincidences by requiring the energies of the two electrons to add up to the incident energy. 2.5.5.1.5. EXPERIMENTAL ARRANQEMENT. The scattering method described above has been used to determine the helicity of electron^,'^^-^^^ of p o s i t r ~ n s , ' ~and ~ J ~of~ muons.147 A very simple experimental arrangement is shown in Fig. 10 of Chapter 1.1. An improved system is represented schematically in Fig. 8. A magnetized ferromagnetic foil provides polarized target electrons and coincidence counting is used to detect Mgller collisions. The angle 0 is the I 4 l H. Frauenfelder, A. 0. Hanson, N. Levine, A. Rossi, and G. DePasquali, Phys. Rev. 107, 643 (1957). 14* N. Bencaer-Koller, A. Schwarzschild, J. B. Vise, and C. S. Wu, Phys. Rev. 109, 85 (1958). 143 J. S. Geiger, G. T. Ewan, R. L. Graham, and R. D. MacKensie, Phys. Rev. 112, 1684 (1958). 144 J. D. Ullman, H. Frauenfelder, H. J. Lipkin, and A. Rossi, Phys. Rev. 122, 536 (1961). 146 D. M. Harmsen and K. Holm, 2.Physik 166, 227 (1962). * 4 6 J. C. Hopkins, J . B. Gerhart, F. H. Schmidt, and J. E. Stroth, Phys. Rev. 121, 1185 (1961). 147 G. Backenstoss, B. D. Hyams, G. Knop, P. C. Marin, and U. Stierlin, Phys. Rev. Letters 6, 415 (1961).

2.5.

POLARIZATION

OF ELECTRONS A N D PHOTONS

245

laboratory scattering angle corresponding to a 90" c.m. scattering angle. A beta monochromator selects electrons in the desired momentturn band. The Mdler-scattered electrons a t 90" c.m. scattering angle possess half the selected energy. Pulse-height analysis in the two counters thus distinguishes the Mplller-scattered electrons from those which have undergone Coulomb scattering. Energy selection and coincidence arrangement together form a very effective means for singling out the desired events. I n principle, the arrangement shown in Fig. 8 can be used to detect longitudinal as well as transverse polarization. The cross section ratio u+/u- is not very different from one for transverse polarization, h ~ w e v e r . ~

SOURCE Magnetized

MONOCHROMATOR

in direction of arrow

COUNTERS

FIG.8. Basic arrangement for the measurement of electron and positron helicity by means of Moller and Bhabha scattering.

In practice the method is therefore always used for the determination of longitudinal polarization. In this case the target electrons should be polarized in the direction of the incident electron beam. Unfortunately, extremely high fields are necessary to magnetize a thin foil in a direction normal to its surface. Therefore, it is necessary to magnetize the foil along its surface and incline i t a t an angle to the electron beam, as shown in Fig. 8. If the fraction of electrons aligned in the foil is f, and the angle of inclination of the foil to the electron beam is a, the target electrons will have longitudinal polarization f cos LY when looked at in the c.m. system. The relative change 6 of the coincidence counting rate then follows from Eq. (2.5.49) if one sets P O = f cos a and uJu+ = u p / u a :

(2.5.52)

The ratio up/ua is given in Chapter 1.1, Eq. (1.1.86). AND PROCEDURE. One realization of the arrange2.5.5.1.6. EQUIPMENT ment in Fig. 8 is shown in Fig. 9. A detailed drawing of a similar instrument is reproduced in reference 145. Additional experimental information

246

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

LEAD STOP,

SOURCE MQNOC;HAOMATOR COILS

n

HELMHOLTZ

\I

COILS

FIG.9. Experimental realiaation of the arrangement shown schematically in Fig. 8. (After Ullman, Frauenfelder, Lipkin, and Rossi, reference 144.)

can be found in references 141-147. A few remarks about equipment and experimental procedure follow here. (1) The angle 28 between the two counters must be adapted to the energy selected by the spectrometer. If the total energy of the incoming electron is W , and if both scattered electrons have equal energy (w = t), then sin2 e =

2mc2

w + 3mc2

(2.5.53)

At small energies, the two electrons emerge a t an angle 20 = 90"; a t higher energies, scattering is peaked forward and 8 decreases. (2) The design of the spectrometer is important. It is best to use one with small spherical aberration (intermediate image spectrometer, see Section 2.2.1.1). Otherwise only a small fraction of the electrons striking the scattering foil will reach the counters. (3) The scattering foil must be selected so that the multiple scattering in the foil is not too disturbing. The fraction of electrons aligned should be as large as possible and the external magnetic field necessary to polarize should be aa small as possible. The first requirement is satisfied if the foil

2.5.

POLARIZATION OF ELECTRONS AND PHOTONS

247

thickness is selected such that the root-mean-square scattering angle for the incident electrons equals the counter aperture.144To meet the other two conditions one usually selects Deltamax or Supermendur foils. Superm e n d ~ r with l ~ ~ a composition of 49% Fe, 49% Co, and 2 % V, has a fraction f of electrons aligned which varies between 5 and 9 % and can be kept near saturation with a holding field of 3 oe. Deltamax possesses an f around 5 % and requires a holding field of about 4 oe. These values apply to foils of thickness 2-8 mg/cm2. The determination off is straightforward, but somewhat difficult; details are discussed in references 143 and 144. Equations (2.5.50) and (2.5.51) are used to find f from the measured magnetization M . (4) Scintillation counters are usually selected as detectors. The thickness of the (plastic) scintillators is chosen to be about the range of the electrons to be counted. ( 5 ) The coincidence circuit should have a resolving time which is as short as possible in order to minimize accidental coincidences. A conventional slow-fast system permits a good pulse-height selection and a resolving time of the order of a few nanoseconds. (6) In order to determine 6, Eq. (2.5.49), the foil is magnetized in one direction and coincidences are recorded for a fixed period of time. The magnetization of the target foil is then reversed and the corresponding coincidence rate is measured. (7) The coincidence rates so determined must be corrected for accidental coincidences and for spurious coincidences due to background, to beta-gamma cascades, and to annihilation radiation.144 (8) Asymmetries in the equipment can be tested by replacing the ferromagnetic scattering foil with a nonmagnetic foil of about equal thickness (A1 or Cu). (9) Counter-to-counter scattering must be checked and eliminated. (10) The effect of reversing the magnetic field at the scattering foil on the counters and on the electron trajectories must be minimized. 2.5.5.1.7. CORRECTIONS. The polarization Pi, determined from 6 by Eq. (2.5.52), must be corrected for the following effects: 2.5.5.1.7.1. Finite Apertures and Finite Energy R e ~ o l u t i o n . ~The ~~J~~ relations (Eq. 2.5.52) are true for a single energy, scattering angle, and angle between the trajectory of electrons and the axis of the foil. I n an actual experiment, a fairly wide range of all three of these quantities has to be accepted to get a reasonable counting rate. The problem of averaging the expression on the right of Eq. (2.5.52) over all the accepted events is, in general, very complicated. 14*

H. L. B. Gould and D. H. Wenny, Elec. Eng. 76, 208 (1957).

248

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

The problem is simplified if the accepted energy range is reasonably narrow. Except a t low energy, where the Mprller scattering method is not at its best anyway, both up/ua and Pi vary slowly with energy. Pi can then be treated as a constant, and up/uacan be calculated over the mean accepted energy, The averaging of cos a over electron trajectories can be separated from the averaging of R, Eq. (2.5.52)) over scattering angles as long as the accepted distribution of scattering angles does not vary appreciably over the range of variation of a. This can be assured by making the counter faces large enough. The averaging of cos a over electron trajectories is a straightforward geometry problem. Averaging R over scattering angles is more complicated. First, one must know the distribution in scattering angle of the counted events. Controlling this distribution is helped by the fact that in Meiller scattering there is a definite reIation between the scattering angle and the energy of an electron after the collision. This correlation is important, because the energy will hardly be changed by multiple scattering that radically changes the direction of motion. It is possible to limit the accepted range of scattering angles by limiting the accepted range of energies. If the counters are large enough to intercept all the electrons allowed by the energy selection, multiple scattering will have little effect on the accepted distribution. The distribution of accepted events as a function of electron energy after scattering can be measured by sweeping a narrow channel across the energy range accepted by one counter. This procedure gives the distribution of Mplller-scattered electrons reaching the counter. The resulting curve, when multiplied by the energy-sensitivity curve of the counter as used in the polarization measurement, yields the accepted distribution of events. Once this distribution is known, the averaging process can be carried out by numerical integration. The averaging process is discussed in references 144 and 145. 2.5.5.1.7.2. D e p o E ~ r i z a t i o n Electrons . ~ ~ ~ ~ ~ ~can ~ be depolarized by multiple and single scattering in the source, and by multiple scattering in the foil and any material between the source and the foil. It is generally true that depolarization corrections are less important in experiments involving Mprller scattering than in those using Mott scattering (Section 2.5.5.2)) for the following reasons. The energies involved in MPler-scattering experiments are usually larger than those involved in Mott scattering. The depolarization in the source is hence smaller. Depolarization in the scattering foil is less important in Mprller scattering because the energy and angle conditions which an event must satisfy in order to be counted are stringent. Details of the corrections for depolarization in the source

2.5.

POLARIZATION OF ELECTRONS AND PHOTONS

249

and in the absorber are given in references 144 and 145 and in Section 2.5.5.2.6. 2.5.5.2. Mott Scattering.* 2.5.5.2.1. THEORETICAL BACKGROUND. The classical way to study electron polarization is by means of Mo tt scattering, i.e,, by the scattering of the electron in the Coulomb field of a nucleus. l3v14 The spin-orbit interaction leads to a difference in scattering a t a fixed scattering angle between electrons with spin up and electrons with spin down. This spin dependence is the basis of the method discussed in the present section. Mott scattering is best suited for the analysis of the transverse polarization of negative electrons of energies between about 30 and a few Lob.

System

Rest System of the Electron

FIG. 10. Mott scattering. In the lab system, the incident electron with spin d perpendicular to the plane of scattering is deflected by an angle e to the right. In the electron rest system, the nucleus orbits partially around the electron and creates a magnetic field B. I n the situation shown in the figure, this magnetic field and the spin d are antiparallel. The magnetic moment and the field B hence are parallel, the spin+rbit interaction adds to the Coulomb attraction, the scattering to the right is enhanced.

hundred kev. If the energy is lower than about 30 kev, the experimental problems of preparing thin enough scattering foils become formidable. If the electron energy is much higher than, say, 600 kev, the asymmetry due to the spin-orbit interaction becomes very small. The method is also not well suited for positrons. The Coulomb repulsion keeps the positron away from the nucleus; hence the spin-orbit interaction and the asymmetry in scattering remains small a t all energies. The basic physical idea underlying Mott scattering can be described b y reference to Fig. 10. Assume that an electron is scattered in a plane perpendicular to its spin. The existence of a spin-orbit interaction is then easiest to see in the rest system of the electron, where the nucleus moves around the electron. This motion gives rise to a magnetic field at the electron. If the magnetic field and the magnetic moment of the electron are parallel, the attractive potential is increased. I n the case of Fig. 10, this happens for electrons scattered to the right and one expects increased * See also Vol. 4A, Section 3.5.1.

250

2. DETERMINATION

OF FUNDAMENTAL QUANTITIES

scattering to this side. Hence one has the following rule, useful in discussing experimental arrangements : Electrons with spin up, i.e., with the magnetic moment pointing down, approaching a nucleus of positive cha,rge straight ahead, are predominantly scattered to the right.

(2.5.54)

2.5.5.2.2.THEASYMMETRY FUNCTION 8(e): THEORETICAL RESULTS."~'4 Mott scattering is discussed in Section 1.1.5.3. The two quantities important for experimental work are (du/dQ), the differential cross section for an unpolarized beam, and 8(8,WJZ), the asymmetry function. 8(e,W,Z)is the polarization produced if an unpolarized electron beam of energy W is scattered by a thin target of atomic number 2; 13 is the scattering angle. In the following, S(e,W,Z) will be written S(e). To describe the Mott-scattering process of a polarized beam, one can introduce spherical coordinates, with the z-axis along the momentum of the incident electron. The transverse polarization P of the incident electron shall lie along the azimuth Q O and the scattering shall be described by a polar angle 0 and an azimuthal angle Q. The cross section for Mott scattering then can be written as

Assume that the scattering occurs in the plane perpendicular to the polarization vector, so that Q - (DO = 3a/2 = -a/2 for scattering to the left and Q - Q O = +a/2 for scattering to the right. "Left" is thus defined by a unit vector such that

e

L=PXp

(2.5.56)

where $ and are unit vectors in the direction of the polarization and the momentum of the incident electrons, respectively. (This definition of left and right agrees with the one shown in Fig. 10.) The ratio of intensities of electrons scattered to the left to electrons scattered to the right then becomes

Hence one finds for the transverse polarization P

p

1L-R =--.

SL+R

(2.5.58)

2.5.

POLARIZATION

OF ELECTRONS AND PHOTONS

25 1

The generalization to the case where the polarization vector is not perpendicular to the scattering plane is trivial. The most complete recent calculation of du/dQ and s(0)has been performed for electrons with p = v/c = 0.2, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, for scattering on mercury (2 = 80), cadmium (2 = 48), and aluminum (2 = 13).149Partial results are shown in Figs. 11 and 12 of Chapter 1.1. Additional values for gold (2 = 79) and aluminum for v/c = 0.49 and 0.59 are given in reference 150. A number of additional remarks may be useful: (1) Figure 11 in Chapter 1.1 shows that S(e) for mercury is negative for most values of v / c and e. For positive P, Eq. (2.5.58) shows that scattering then occurs predominantly to the right, in agreement with rule (2.5.54). (2) In all calculations, the transverse polarization P is defined in the rest system of the electron. (3) The calculation149has been performed using unscreened Coulomb functions. A number of calculations using screened functions exist,151J62 and these indicate that screening is indeed important below about 150 kev. Unfortunately, these calculations are not accurate and extensive enough to be of much use in experiments. Experimental methods hence must be resorted to at the present time for estimates of the correct asymmetry function s(e)at low electron energies (see Section 2.5.5.2.3). (4) The calculations of s(e)in references 149 and 150 assume point nuclei. The influence of a finite nuclear size on s(0)has also been investigated153;a t the energies used in most experiments, the corresponding correction will be less than 0.1 %. (5) The contribution to s ( 0 ) due to inelastic electron scattering is less than 0.5% for gold scatterers16*and it can hence be neglected in most experiments. (6) The scattering of polarized muons from extended nuclei has also been c a l ~ u l a t e d . ~The ~ ~ resulting ~ ~ ~ 6 asymmetry may be useful to determine the polarization of muon beams. 2.5.5.2.3. THEASYMMETRY FUNCTION s(6): EXPERIMENTAL RESULTS. In the absence of a completely satisfactory calculation of s(0) for lowN . Sherman, Phys. Rev. 103, 1601 (1956). N.Sherman and D. F. Nelson, Phys. Rev. 114, 1541 (1959). J. Bartlett and T. Welton, Phys. Rev. 69, 281 (1941). 16* C.B. 0. Mohr and L. J. Tassie, Proc. Phys. Soe. (London) 67, 711 (1954). l K 3B. K.Kerimov and V. M. Arutyunyan, J . Exptl. Thewet. Phys. (U.S.S.R.) S8, 1798 (1960);Soviet Phys. J E T P 11, 1294 (1960). lK4 G. Felsner and M. E. Rose, Nuovo cimento 20, 509 (1961). l K 6J. Franklin and B. Margolk, Phys. Rev. 109, 525 (1958). lK6G.H.Rawitscher, Phys. Rev. 112, 1274 (1958). 14*

lKo

252

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

energy electrons, the existing experimental data give some useful information about the influence of screening. Two methods have been used to find s(e). The more straight-forward approach is to use double scattering of e l e c t r o n ~ . * ~ In~ ~the 4 first scattering, an unpolarized electron beam becomes partly polarized; the second scattering acts as an analyzer. The asymmetry in the double-scattering experiment yields the product S(0,) X S(0,). I n a second method, one assumes that electrons emitted in allowed beta transitions possess a polarization P , = v / c . The longitudinal polarization is first transformed to a transverse one (Section 2.5.3.3) and the asymmetry of the transversely polarized beam is then determined. With the assumption P = v / c , S(0) then follows from Eq. (2.5.57). The early experiments, restricted to double scattering, are summarized in references 1, 13, and 14. Recent investigations have used double A doubles ~ a t t e r i n g * ~ ~and - l ~polarized l electrons from beta decay.162*163 scattering experiment a t an electron energy of 1.5 kev has been performed with a mercury atomic beam.ls4 The most recent data are plotted in Fig. 11, which shows the ratio S,,,(0)/Sthe0,(O) as a function of the electron energy. Sexp(0) is the asymmetry function as determined by one of the two methods outlined above. &haor(e) is the value calculated without s ~ r e e n i n g . ' * ~ ~ ' ~ ~ Figure 11 shows that screening is very important a t energies below 150 kev. Even above 150 kev, there may still be some deviations from the "unscreened" value. Figure 11 also demonstrates the need for more and more accurate experiments and for improved theoretical calculations. At the present time, the data in Fig. 11 are useful for estimates of the correct asymmetry function S ( 6 ) for electron energies between 40 and 200 kev. 2.5.5.2.4. DETERMINATION OF THE TRANSVERSE POLARIZATION OF ELECTRONS. Mott scattering is best suited for the measurement of the W. G. Pettus, Phys. Rev. 109, 1458 (1958). D. F. Nelson and R. W. Pidd, Phys. Rev. 114,728 (1959). P. E. Spivak, L. A. Mikaelyan, I. E. Kutikov, and V. F. Apalin, Nuclear Phys. 23,

lS8

lS9

169 (1961). P. E. Spivak, L. A. Mikaelyan, I. E. Kutikov, V. F. Apalin, I. I. Lukashevich, and G . V. Smrinov, J . Exptl. Theoret. Phys. (U.S.S.R.)41, 1064 (1961); Soviet Phys. J E T P 14, 759 (1962). V. A. Apalin, I. Y. Kutikov, I. I. Lukashevich, L. A. Mikaelyan, G . V. Smirnov, and P. Y. Spivak, Nuclear Phys. 31, 657 (1962). 162 H. Bienlein, G. Felsner, K. Guthner, H. von Issendorff, and H. Wegener, Z. Physik 164,376 (1959). H. Bienlein, G. Felsner, R. Fleischmann, K. Giithner, H. von Issendorff, and H. Wegener, 2.Physik 166, 101 (1959). '64 H. Deichsel, 2.Physik 164, 156 (1961).

2.5.

253

POLARIZATION OF ELECTRONS A N D PHOTONS

transverse polarization of (negative) electrons with energies between about 40 and a few hundred kev. The basic setup for such a measurement is very simple. The beam of electrons strikes the scattering foil. A small fraction of the electrons (10-4-10-6) suffers a large angle scattering and enters one of the two symmetrically arranged counters C L or CR. The ratio L / R of the intensities in the two counters is a direct measure for the transverse polarization [Eq. (2.5.58)]. In theory this procedure is straightforward and simple. I n practice there exists a n enormous number of difficulties and the experimental ratio

"1" 1 I

REFERENCES

0.71 ,

0.60

,

,

,

100

,

,

,

,

200 ELECTRON ENERGY (kew I

,

,

300

FIG. 11. Ratio of the experimentally determined asymmetry function 5.,,(120") to the function St~,,,(120"), calculated without screening. All values apply for a gold scatterer (Z = 79).

( L / R ) e x pmust be corrected for many deviations from a n ideal setup before a reliable value for the polarization P can be obtained. I n the present section, some of the important points in designing a system for such an experiment will be sketched; the corrections which must be p discussed in Section 2.5.5.2.6. applied to ( L / R ) e x are Electrons can only be transversely polarized if a direction, different from that of the momentum, is given by some physical process or quantity. I n double scattering, the normal to the scattering plane in the first scattering constitutes such a quantity. I n radioactive decay, electrons can be transversely polarized if measured in coincidence with a preceding or following radiation. An example is the transverse polariza-

254

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

tion of conversion electrons following beta decay.16b169The theory predicts that the conversion electron will, in general, be emitted polarized, with the polarization vector parallel to the beta electron momentum. Thus, conversion electrons emitted at right angles to the beta particles are transversely polarized. I n order to select these electrons one must record coincidences between a beta counter and two counters which record the left-right asymmetry after scattering. Because this type of

FIQ. 12. Schematic representation of a setup to measure the transverse polarization of conversion electrons following beta decay. (From Blake, Bobone, Frauenfelder, and Lipkin, reference 172.)

experiment shows all the basic aspects of a transverse polarization measurement, a typical setup will be discussed here. The transverse polarization of conversion electrons following beta decay has been determined in a number of experiment~.17~-~7~ A typical setup is shown in Fig. lZ.172 The beta particles emitted by the source S are detected in the counters 0 and 00. The subsequent conversion electrons are focused by a two-lens magnetic spectrometer upon the gold foil T. 166

H. Frauenfelder, J. D. Jackson, and H. W. Wyld, Phys. Rev. 110,451 (1958).

186

€3. V. Geshkenbein, Nuovo cimento 10, 365 (1958).

V. B. Berestechij and A. P. Rudik, Nuovo cimento 10, 375 (1958). R. L. Becker and M. E. Rose, Nuovo cirnento 18, 1182 (1959). 169 H. R. Lewis and J. R. Albers, 2. Physik 168, 155 (1960). 170 J. E. Alberghini and R. M. Steffen, Nuclear Phys. 14, 199 (1959). 171M. E. Vishnevskii, V. A. Lyubimov, E. F. Tretihkov, and G . I. Grishuk, J. Ezptl. Theoret. Phya. (U.S.S.R.) 38, 1424 (1960); Soviet Phys. J E T P 11, 1029 (1960). 178 B. Blake, R. Bobone, H. Frauenfelder, and H. J. Lipkin, Nuovo cimento 26, 942 (1962). 167

168

2.5.

POLARIZATION OF ELECTRONS AND PHOTONS

255

The currents in the two spectrometer lenses are of equal magnitude but opposite sign in order to eliminate a rotation of the conversion electron spin. The transverse polarization of the conversion electrons is then determined by Mott scattering from the gold foil T a t an angle of 120". The counts L and R in the two detectors CL and CR, respectively, are recorded in coincidence with the events in the beta detectors p and Po. The coincidences with the detector Po, which counts beta particles emitted parallel to the spectrometer axis, serve to determine the instrumental asymmetry. The source in all experiments should be as thin as possible and as uniform as possible, in order to minimize depolarixation and backscattering. The energy or momentum selector and the slit systems should be completely symmetric and should not contribute spuriously scattered electrons. Lining the entire system with polystyrene and constructing the scattering chamber as large as possible reduces the amount of unwanted scattered e l e c t r o n ~ . ~ ~ ~ 7 ~ A thin gold foil is usually employed as the target. I n principle, uranium foils would be better, because the asymmetry function S(0) becomes larger with increasing 2. However, gold is the heaviest material which can be worked easily into thin uniform foils. The position of the scattering foil is crucial. In early scattering experiments, foils were placed a t 45" to the beam and the intensities L and R were then determined. It turned out, however, that such an arrangement is extremely inconvenient. Multiply scattered electrons may mask the polarization effect nearly c0mp1etely.l~The foils are now usually placed perpendicular to the beam and the scattered electrons are observed a t angles around 120" (Figs. 2, 3, and 12) or around 70" (Fig. 1). I n the first case, the intensity of the Mott-scattered electrons is small, but the asymmetry function S(l2OO) is large. In the second case, the intensity is much larger, but S(70") is quite small. The choice between these two solutions depends on the particular problem. Generally, the larger angle is to be preferred. Two methods allow the determination of some instrumental asymThe first, more common one, is to replace the gold scatterer by an aluminum foil. S(0) is small for A1 and the two counters CL and C R hence should give the same counting rates. In the second method, one uses the fact that even for gold S ( 0 ) is very small a t scattering angles around 30" (see Fig. 11, Chapter 1.1, Vol. 5 A). Two additional counters at small scattering angles are employed to get the data necessary for the corrections of the instrumental asymmetries.l* Unfortunately, these measurements eliminate only some of the instrumental asymmetries. Even small misalignments of the electron beam with respect to the symmetry axis of the system can introduce large errors. These errors and the

256

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

methods to correct for them are discussed in detail in references 74, 75, and 78. 2.5.5.2.5. DETERMINATION OF THE LONGITUDINAL POLARIZATION OF ELECTRONS. The basic idea involved in the measurement of the longitudinal polarization of electrons by Mott scattering is now clear. First, the longitudinal polarization is transformed to a transverse one (see Section 2.5.3.3). The polarization transformer usually acts also as a momentum selector. Second, the transverse polarization is then determined as outlined in Section 2.5.5.2.4.Three basic systems are shown in Figs. 1-3. References are also given in Section 2.5.3.3. One particular idea deserves to be mentioned here. The difficulty of the only approximately known asymmetry factor s(0) a t low electron energies can be partly avoided by accelerating the electrons to a fixed energy. Relative measurements can then be made easily and absolute ones are ~ i m p l i f i e d . ~ ~ 2.5.5.2.6. CORRECTIONS. I n order to find the polarization, the raw data L and R must be corrected for depolarization in the source, the target, and the polarization transformer, the background must be taken into account, and geometric factors resulting from the finite sizes of the source, the spectrometer apertures, the target, and the detectors must be determined. These corrections are treated in detail in references 1, 74-76, 78, 79-81, 87-90. Here a few remarks concerning the most important corrections are made. 2.5.5.2.6.1. Plural and Multiple Scattering in the Target. This contribution is the most important source of errors in Mott scattering.14 Plural scattering consists of two scattering events of about equal angle; multiple scattering consists of a succession of small-angle scattering. Both effects contribute to the counting rate in the detectors CL and CR. In either case, the individual scattering event will involve angles of less than, say, 60'. Figure 11, Chapter 1.1, Vol. 5 A, then shows that s(e < 60') is nearly zero. The contribution from plural and multiple scattering thus does not show an asymmetry, and it must be subtracted from the total counting rate in order to find the true asymmetry L / R . The contribution from multiple scattering can be calculated reasonably well173(see also the discussion and the references in Section 1.1.6). The foil can be made thin enough so that the contribution from multiple scattering is very small. It is much more difficult to get rid of plural scattering or to correct for it by calculation. Foils thin enough to eliminate the contribution due to double scattering yield only very low counting rates. Moreover, the ever-present wavyness of very thin foils considerably enhances the contribution of double scattering over the amount expected from theoretical estimates. H. Wegener, 2.Phyaik 161, 252 (1958).

2.5. POLARIZATION OF

ELECTRONS AND PHOTONS

257

The best method to correct for the contribution from plural and multiple scattering in the target consists in determining the intensities L and R as functions of the target thickness and then extrapolating to zero thickness. For thin targets (thickness t < 1 mg/cm2, even at electron s e d One . energies of 600 kev), a linear extrapolation is ~ plots either L / R or ( L - R ) / ( L R ) versus the foil thickness and extrapolates to zero thickness.

+

1.6

fl

LEAST-SQUARES STRAIGHT-LINE FIT TO THESE POINTS

FIG.13. Experimental correction for plural and multiple scattering in the target. extrapolated to zero target thickness, yields [PS(O)]-'l*.The kinetic energy of the electrons is 616 kev, the scattering angle 0 is 135". (From Brosi, Galonsky, Ketelle, and Willard, reference 78.)

An improved approach, which fits the data for a much larger range of target thicknesses, is described in reference 78: one plots the function

[PS(e>]-"~ =

d(J5 + R)/(L- R)

(2.5.59)

versus the target thickness. The extrapolation to zero thickness then yields the desired value of P . Figure 13 shows an example of such an extrapolation. 2.5.5.2.6.2. Depolarization. It was pointed out in Section 2.5.3.3.3 that the polarization vector of electrons remains approximately fixed with respect to the laboratory system during multiple scattering. More accurately, if the momentum is turned by an angle A# by multiple scattering in low-2 material, then the polarization vector turns only

~

2.

258

DETERMINATION OF FUNDAMENTAL QUANTITIES

through an angle A 4 , given by (2.5.60)

where v is the velocity of the electron. This effect leads to an energydependent depolarization of an electron beam. Electrons initially moving in the direction towards the analyzer, i.e., the target or the polarization transformer, suffer no appreciable depolarization. Electrons which initially moved in other directions and are scattered into the acceptance angle of the analyzer have their polarization vector pointing in different directions. Electrons originally moving backwards and being scattered forwards may even have polarization vectors pointing in the opposite direction from the vectors of the “true electrons.” The scattered electrons hence decrease the degree of longitudinal polarization of the electron beam. The multiple scattering of polarized electrons has been studied by various authors.129,174-1” Detailed formulas can be found in these references. In an actuaI experimental setup, depolarization can occur in the source, in the source-backing material, in all material between the source and the target foil, in the walls, and in the polarization transformer. The magnitude of the depolarization is usually estimated with the help of theoretical calculations. Moreover, the calculations are best checked by additional experiments, for instance varying the source or the sourcebacking thickness. Most important, theoretical estimates are used to find values for the maximum thickness of the various components; the system is then designed in such a way that all corrections remain small. In the souwe, depolarization occurs either by a large-angle single scattering or by small-angle multiple scattering, These two effects must be considered separately. Calculations have been performed for longit ~ d i n a l l y ”and ~ t r a n s v e r ~ e l y ’polarized ~~ electrons. It turns out that the two contributions, multiple and single scattering, contribute terms of the same order of magnitude. Under identical conditions, transversely polarized electrons are much less depolarized than longitudinally polarized ones. E. Rose and H. A. Bethe, Phys. Rev. 66, 277 (1939). B. Muhlschlegel and H. Koppe, 2. Physik 160, 474 (1958). I. N. Toptygin, J . Ezptl. Theoret. Phys. (U.S.S.R.) 80, 488 (1959). G. Passatore, Nuovo cimento 18, 532 (1960). B. Miihlschlegel, 2.Physik 166, 69 (1959). B. Blake and B. Miihlschlegel, 2.Physik 167, 584 (1962).

114 M.

ll6 ln

2.5.

POLARIZATION OF ELECTRONS AND PHOTONS

259

Figure 14 shows an experimental investigation of the influence of source thickness and backing thickness on the measured value of the polarization. The depolarization due to scattering from the walls, from the foil and source holders, and from the plates of the polarization transformer or the momentum selector is much more difficult to study. Theoretical calculations are nearly impossible. Experimentally, one can find some information by changing the various parts of a setup in a controllable manner.’8

P v/c

--

co60 Source 2 rng /cm2

0

Co60 Source

Thickness = I rng/crn2

0.8 I.6 2.4 3.2 Thickness of Ni Backing (mg/crn2)

4.0

FIQ.14. The effect of the thickness of the source backing and the source on the meaaured polarization. Electron energy 194 kev. (From Greenberg, Malone, Gluckstern, and Hughes, reference 74.)

2.5.5.2.6.3.Finite-Angle Corrections. The source, the polarization analyzer, the target, and the counters all subtend finite solid angles. Moreover, the momentum selector will also possess a finite momentum resolution. Corrections must be applied for all these effects.The solid angles will also introduce a spread in the angle & between the polarization vector and the scattering plane. Equation (2.5.57) thus must be used in the form in which the term sin(+ - &) is still retained. The function sin(+ - cb0) then must be averaged over the accepted angles.78 2.5.5.3. Methods Depending on Helicity Transfer. In the present section, experiments are discussed in which a longitudinally polarized electron gives rise to one (or more) circularly polarized gamma rays, and the circular polarization of the gamma rays is measured. The discussion can be very brief because the polarization transfer is already treated in Section 2.5.4 and the determination of the circular polarization of gamma rays is dealt with in Section 2.5.6. 2.5.5.3.1. EXTERNAL BREMSSTRAHLUNG. The basic idea underlying the determination of the electron helicity by using the bremsstrahlung is

2.

260

DETERMINATION OF FUNDAMENTAL QUANTITIES

extremely simple. Electrons are allowed to emit bremsstrahlung in a radiator. The polarization of the emitted radiation is then measured in a polarimeter. 180-18’ An experimental arrangement is shown schematically in Fig. 15. Before discussing Fig. 15, some differences between internal and external bremsstrahlung must be mentioned. The external bremsstrahlung is produced

FIG.15. Arrangement for measuring the helicity of internal and external bremsstrahlung. The main part of the external bremsstrahhng originates in the radiator of high atomic number 2. The absorber (low Z ) prevents electrons from entering the polarimeter and producing spurious bremsstrahlung.

by an electron which leaves the nucleus and then radiates in the field of another nucleus. The intensity of the external brernsstrahlung is approximately proportional to the atomic number 2 of the material in which the radiation occurs (see Section 1.1.5.1). The internal bremsstrahlung, which originates in the atom of the decaying nucleus, is much weaker than the external bremsstrahlung and its intensity is approximately independent of the atomic number of the source.18* In the arrangement shown in Fig. 15, the source will emit internal M. Goldhaber, L. Grodzins, and A. W. Sunyar, Phys. Rev. 106, 826 (1957). F. Boehm and A. H. Wapstra, Phys. Rev. 109, 456 (1958). 18* H. Schopper and S. Galster, Nuclear Phys. 6, 125 (1958). 188 E. G. Beltrametti and S. Vitale, Nuovo cimento Q, 289 (1958). 184 U. Amaldi, Jr., M. Bernardini, P. Brovetto, and S. Ferroni, Nuovo cimento 11,

180

181 ’

415 (1959). 18)

A. Bisi and L. Zappa, Phys. Rev. Letters 1, 332 (1958); Nuclear Phys. 10, 331

(1959). 186

G . Culligan, 5. G. F. Frank, and J. R. Holt, Proc. Phys. Isoc. (London) 78, 169

(1959). 187

P. Lipnik, J. P. Deutsch, L. Grenacs, and P. C. Macq, Nuclear Phys. SO, 312

(1962). 188 C. S. Wu, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 649. North Holland, Amsterdam, 1955.

2.5.

POLARIZATION OF ELECTRONS AND PHOTONS

26 1

bremsstrahlung and beta particles. The internal bremsstrahlung quanta are circularly polarized. 189-191 The beta particles give rise to circularly polarized external bremsstrahlung quanta in the radiator. The experimental setup must be so designed that two conditions are fulfilled: (1) No electrons should give rise to bremsstrahlung originating in the polarimeter

FIG.16. Total intensity of internal and external bremsstrahlung, after subtraction of the background, a8 a function of the atomic number 2 of the radiator. The int.ensity a t 2 = 0 is due to internal bremsstrahlung alone. Points are for photon energies between 0.6 and 1 Mev (x), a.nd between 1 and 1.4 Mev ( 0 ) . (After Schopper and Galster, reference 182.)

which could falsify the results. This condition is enforced by the absorber, which consists of a very low 2 substance, such as water, polyethylene, or paraffin; (2) A separation between internal and external bremsstrahlung must be possible. It is achieved by measuring the bremsstrahlung intensity as a function of the atomic number, 2, of the radiator. 182,188 The result of such an experiment is shown in Fig. 16. The main problem consists in finding the correct correspondence between the longitudinal polarization of the original electron and the G. W. Ford, Phys. Rev. 107, 321 (1957). R. E. Cutkosky, Phys. Rev. 107, 330 (1957). lv1 A. Pytte, Phys. Rev. 107, 1681 (1957). I89 I90

262

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

circular polarization of the bremsstrahlung. The helicity transfer has already been discussed in Section 2.5.4. I n addition, however, the following problems arise: (1) The depolarization in the source, the radiator, and the absorber must be calculated. Backscattering must be taken into account. (2) The energy distribution of the beta rays entering the radiator must be known. (3) The angular and the energy distribution of the bremsstrahlung must be found. The arrangement shown in Fig. 15 is not restricted to electrons from radioactive sources. Electrons emitted by short-lived reaction products can also be used. In this case, where the experiment is done at an accelerator, a pulsed beam and a gated detector can be employed to reduce the background. lE7 2.5.5.3.2. ANNIHILATION-IN-FLIGHT. As pointed out in Section 2.5.4.5, the higher energy gamma ray from positron annihilation-in-flight carries most of the helicity of the original positron. This property has been used to determine the positron helicity :lg2--1g4 positrons impinge on a converter, in which they annihilate in flight. The highest energy annihilation quanta, emitted in a relatively narrow cone forward, are then analyzed with a Compton polarimeter (Section 2.5.6). It is advantageous to use a spectrometer to select the energy of the incoming p o s i t r o n ~ . ~ ~ The 2 , ~ 9converter ~ can be made out of lucite. I n order to reduce the background, it is possible to use a plastic scintillator as converter and then measure coincidences between the analyzed gamma rays and the converter.lg4 2.5.5.4. Positron Annihilation with Polarized electron^.^^^ The spin dependence of positron annihilation has been used in two different ways to determine the helicity of positrons. Both methods are experimentally quite difficult and require a detailed knowledge of the depolarization during the slowing-down of positrons. As a result, very few experiments have been performed and i t seems unlikely that either method will be used often in the future. 2.5.5.4.1. ANNIHILATION-IN-FLIGHT IN A POLARIZED TARGET. Assume that a longitudinally polarized positron beam strikes a target which contains electrons polarized parallel or antiparallel to the momentum of the incoming positrons. The situation is thus superficially the same as in the 191 M. Deutsch, B. Gittelman, R. W. Bauer, L. Grodzins, and A. W. Sunyar, Phye. Rev. 107, 1733 (1957). 19a F. Boehm, T. B. Novey, C. A. Barnes, and B. Stech, Phys. Rev. 108,1497 (1957). lD4J. B. Gerhart, F. H. Schmidt, H. Bichsel, and J. C. Hopkins, Phye. Rev. 114,

1095 (1959).

2.5.

POLARIZATION

OF ELECTRONS AND PHOTONS

263

Bhabha-scattering experiments (Section 2.5.5.1). However, the magnetized target is chosen to be so thick that most of the incident positrons are stopped, and instead of observing the scattered electrons, one determines the number of annihilation quanta of a given energy. The gamma rays of 0.511-Mev energy are due to annihilation-at-rest. The gamma rays of higher energy are produced by annihilation-in-flight. Annihilation-inflight is strongly spin-dependent. 1,61,120--122 In the exact forward direction, angular momentum can only be conserved if the spins are antiparallel. In other directions, the spin dependence is a function of the positron energy. At low energies, annihilation with opposed spins is predominant; a t high energies, annihilation in the triplet state is more probable. To use this spin dependence for a determination of the positron helicity, positrons from a very strong source are focused by magnetic lenses onto a magnetic annihilator foil.l g 6The gamma-ray intensity for various gammaray energies is then observed for the two directions of magnetization. From the observed intensity difference, the polarization of the positrons can be calculated. The main difficulties of this method are: (1) the counting rate is small even for very strong sources and the background is large; (2) the observed intensity difference is small (1-3%) even for completely polarized positrons; (3) the depolarization in the thick annihilator can be large.129,1g6 2.5.5.4.2. ANNIHILATION OF SLOW POSITRONS. The annihilation of slow positrons in magnetized ferromagnets can be used for the determination of the helicity of the positron^.^^^^^^^ This method depends on the fact that at zero positron energy, two-quanta annihilation can occur only in the singlet state, i.e., for antiparallel spins. In iron, about two of the d-electrons can be polarized but the s-electrons cannot. Annihilation of slow positrons occurs mainly with s- and d-electrons. In order to favor the observation of the annihilation quanta from d-electrons, one uses the fact that the d-electrons are faster than the s-electrons. The angular distribution between the two annihilation quanta will hence be wider for annihilation occurring with d-electrons than for those occurring with s-electrons. The intensity of the wings of the angular distribution, determined as a function of the direction of magnetization of the annihilator, thus gives information about annihilation with d-electrons. The effects upon revers196s.

Franltel, P. G . Hansen, 0. Nathan, and G . M. Temmer, Phys. Rev. 108, 1099

(1957). 197

S. M. Neamtan, Phys. Rev. 110, 173 (1958). S. 5. Hanna and R . S. Preston, Phys. Rev. 106, 1363 (1957); 108, 160 (1957);

109, 716 (1958). lQ*

R. S. Preston and S. 5. Hanna, Phys. Rev. 110, 1406 (1958).

264

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

ing the magnetization can be quite large (-lo%), but the accurate calculation of the polarization of the incident positrons is extremely difficult. 2.5.5.5. Positron Decay in Magnetic Fields. The formation and the decay of positronium in a magnetic field can be used to determine the polarization of p o ~ i t r o n s . ~ , ~The ~ ~ 9theory 8 underlying this method is described in references 1, 3, and 199. The unavoidable depolarization effects which occur in thermalizing positrons make this method unsuitable for very precise experiments. 2.5.6. Detection of Photon Polarization

2.5.6.1. Introduction. In principle there exist a number of possibilities to detect the circular polarization of gamma rays200: (1) Gamma rays can interact with polarized matter; (2) The helicity of gamma rays can be transferred to secondary particles, for instance, electrons, and the polarization of these particles can be measured ; (3) The Mossbauer effect can be used to study the intensity of the various components of a nuclear Zeeman pattern.

The possibilities (1) and (2) are discussed in detail in reference 200. In the present section, only those methods are described that have been successfully applied to experiments, namely Compton scattering and Mossbauer effect. A t the present time, Compton scattering from polarized ferromagnets is the best method t o determine the circular polarization of gamma rays in the energy range from about 0.2 t o a few MeV. The method suffers from the same limitations that all experiments with polarized ferromagnets do: only about two out of twenty-six electrons can be polarized and the maximum effect to be expected from any polarimeter is thus of the order of 10%. The method using the Mossbauer effect can be applied only to the small class of gamma rays that can be emitted and absorbed without recoil and that can be split into individual Zeeman components by a magnetic field. If these conditions are fulfilled, the helicity can be determined to a much higher degree of accuracy than with the Compton effect. General discussion of the methods used to determine photon helicities is contained in references 1, 2, 3, 14, and 200. 2.5.6.2. Compton Polarimeter. 2.5.6.2.1. THEORETICAL BACKGROUND. The interaction of unpolarized gamma rays with matter is treated in l@Q

L. A. Page and M. Heinberg, Phys. Rev. 106, 1220 (1957). Inslr. 8, 158 (1958).

zoo H.Schopper, Nuclear

2.5.

POLARIZATION OF ELECTRONS AND PHOTONS

265

Section 1.1.7; the Compton effect, in particular, is dealt with in Section 1.1.7.2. The differential cross section for Compton scattering is described by the famous Klein-Nishina formula, Eq. (1.1.118). For the application to the determination of the polarization of gamma rays, the form of the Klein-Nishina formula shown in Eq. (1.1.118) is unsuitable since i t applies to unpolarized gamma rays and unpolarized electrons. The scattering of a gamma ray with circular polarization P3 from a n electron with spin d is given by the following expressions : 1 ~ 2 ~ 1 4 ~ 4 6 ~ 4 9 ~ 6 1 ~ 2 0 0 - 2 0 ~ (2.5.61) 9 0 @3

=

+

+

1 cosz 6 (Ico - k)(l -(I - cos 0)(ko cos 0

=

- cos 0)

+ k)

*

d.

(2.5.62) (2.5.63)

Here, T O = e2/mc2is the classical electron radius, ko the initial and k the final photon momentum. a0 denotes the ordinary (KIein-Nishina)

FIG. 17. Compton scattering. The incoming gamma ray with momentum ko is scattered by an electron with spin d; k is the scattered gamma ray. A reversal of the spin of the electron, d + -d, corresponds to a change $ + $ T.

+

Compton cross section; @ 3 is that part of the cross section th a t depends on the helicity Pa. 0 is the scattering angle; f denotes the fraction of polarized electrons. Introducing the angle between ko and d, and 4 between the (kod) and the (kok)planes (Fig. 17), one can rewrite Eq. (2.5.63) as

+

=

- (1 - cos 0)[(ko

+ k)cos 6 cos + + k sin 0 sin + cos 41.

(2.5.63')

Equations (2.5.61) to (2.5.63') show that Compton scattering can be a tool for determining the circular polarization Pp of gamma rays. (The Zo1 *OZ

W. Franz, Ann. Phys. 33, 689 (1938). F. W.Lipps and H. A, Tolhoek, Physicu 20, 85, 395 (1954).

266

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

dependence of the Compton cross section on the linear polarization, and the corresponding application to measurements of the linear polarization, are discussed in Section 2.4.2.1.4and in reference 2.) Circular polarization measurements using the Compton effect can be performed in two different ways. One either observes the attenuation of a gamma ray in a polarized absorber as a function of the direction of magnetization of the absorber (transmission method), or one determines the intensity of the scattering from polarized matter as a function of the magnetization (scattering method), For any given problem, one must decide which method will yield the largest effect, the highest intensity, and the best signal-to-noise ratio. This decision will depend on the gamma-ray energy, the intensity of the available sources, the background radiation, and on whether the gamma rays are observed in coincidence with other radiations. In each case, two directions of magnetization exist which can be denoted with and -. The change of a counting rate N on reversal of the polarizing magnetic field is observed. The relative change 6 in the counting rate N is given by

+

(2.5.64) where N+ and N - are the counting ratee with the magnetic field in one or the other direction respectively. Which direction is defined as depends on the arrangement. The factor 2 is convention and it is sometimes omitted. In Fig. 18, 6 is plotted versus the energy of the incoming photons for four different arrangements. Each curve is calculated under optimum conditions. In actual experiments, the effect to be expected is usually smaller. The problem of production of sources of circularly polarized gamma rays deserves a remark. The fact that electrons emitted in nuclear beta decay are longitudinally polarized is intimately connected with parity nonconservation in weak interactions. Gamma rays, however, are created in electromagnetic interactions and these always conserve parity. Gamma rays hence can only be circularly polarized if they are emitted by a nuclear system which is polarized. Such a polarization can be the result of the application of low temperatures and very high fields, or of the observation of a preceding or following radiation in coincidence with the gamma ray under investigation. I n both cases, extranuclear effects can destroy part of the circular polarization of the gamma ray. (Similar effects occur in angular correlation experiments; they are briefly discussed in Section 2.4.2.1. References to detailed investigations are also given there, par-

+

2.5.

267

POLARIZATION OF ELECTRONS AND PHOTONS I

I

I

I

I

C. BACKWARD SCATTERING

15

1

b. FORWARD SCATTERING

10

8

IN% 0 . TRANSMISSION

5

d. BEARD- ROSE 0

I

I

I

1

I

I

2

3

4

5

K O IN UNITS OF

4

MC2

FIG. 18. Relative change 8 in counting rate N as a function of the incident photon energy ko (in units of mcz) for various experimental arrangements. Curve a applies to transmission with optimum length. Curve b refers to forward scattering under optimum angle. Curve cis for backward scattering, and d for the Beard-Rose method. All curves are calculated under the assumption of saturated iron and completely polarized incident photons.

ticularly in Table 11.) A strong influence of extranuclear fields on the circular polarization of gamma rays has indeed been observed.203 2.5.6.2.2. THETRANSMISSION METHOD.The simplest way to measure the circular polarization of a gamma ray consists in determining the transmission through a polarized absorber, as shown schematically in Fig. 19. In this experiment, the total Compton cross section is used. By integrating the differential cross section (Eq. 2.5.61), one obtains the total cross section for Compton effect in the form Cr

101

=

6 0 +.fP363

F. Boehm and J. Rogers, Nudear Phys. 33, 118 (1962)

(2.5.65)

268

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

where the polarization sensitive part

US

is given by

The upper sign corresponds to the photon spin parallel to the electron spin and hence antiparallel to the magnetization.

SOURCE

SCINTILLATION CRYSTAL MAGNET

,

PHOTOMULTIPLIER I

FIG.19. Measurement of the helicity of gamma rays by transmission. The magnet is alternatively magnetized parallel and antiparallel to the direction of the gamma ray. The corresponding change in counting rate is proportional to the gamma-ray helicity .

The relative change 6 in counting rate (Eq. 2.5.64) on reversing the sign of the magnetization is then given by 6 = 2 tanh(NLfZu3-P3).

(2.5.67)

N is the number of iron atoms per cm3,L is the length of the iron absorber, and Zf is the number of oriented electrons per iron atom [Zf = 2.06 at saturation; see Section 2.5.5.1.3, particularly Eqs. (2.5.50) and (2.5.51)J. The sign in Eq. (2.5.64) applies if the electron spin is parallel to the direction of the incident photons. Equation (2.5.66) and Fig. 18 show that a change in sign of 6 occurs at 0.65 MeV. Around this energy, the transmission method is obviously not usable. At higher energies, the transmission is greatest when the photon spin is parallel to the electron spin. The relative change 6 increases with increasing length of the magnetized absorber; but the counting rate decreases. The optimum length Lo of the absorber can be calculated easily,200it is given by

+

NLOT, = 2

(2.5.68)

where T~ is the total absorption coefficient of the gamma ray. The curve a in Fig. 18 has been calculated for this length LO.

2.5.

POLARIZATION OF ELECTRONS AND PHOTONS

269

A number of experiments have been performed with the transmission method. 18,180,187,204--207 The disadvantages of the transmission method are its small counting rate and the small effect (see Fig. 18). Its advantages are the basic simplicity and the fact that energy discrimination is simpler than in the scattering methods. In the latter experiments, the gamma-ray lines are degraded in energy and widened ; in the transmission arrangement, the gamma-ray counter can be set on the energy of the incident gamma ray. One additional difficulty in the transmission method arises in the determination of the effective length L, because the ends of the iron absorber are not fully magnetized. A toroidal analyzer has been used to avoid this diffi~ulty,20~.207 but the results from these experiments indicate that some unsolved problems remain. 2.5.6.2.3. FORWARD SCATTERING. The majority of all experiments has A typical arrangebeen performed with forward scattering.17~1s1~184,208-212 ment is shown in Fig. 20. The gamma rays are scattered from magnetized iron and the counting rate is determined as a function of the direction of magnetization. The entire setup is cylindrically symmetric in order to increase the counting rate. A central absorber prevents the detection of unscattered gamma rays. The effect t o be expected from such an arrangement is given by Eqs. (2.5.61) to (2.5.63):

(2.5.69)

N+ is the number of scattered photons if the electron spin is approximately parallel to the incoming gamma ray (0 _< $ < a/2) and N - is the corresponding counting rate with the electron spin approximately antiparallel to ko. As shown in Fig. 17, reversal of the electron spin, d + -d, corresponds to a change $ --f $ T . “Approximately antiparallel” therefore is represented by angles $ such th at T I $ < 3 ~ / 2 ,and the corresponding cross section as- is given by

+

93-

+3

(T

5

$

< 3~/2).

(2.5.70)

a04 S. B. Gunst and L. A. Page, Phys. Rev. 02, 970 (1953). zosA. Lundby, A . P. Patro, and J. P. Stroot, Nuovo cimento 6, 745 (1957); 7, 891 (1958). *On L. A. Page, B. G. Peterson, and T. Lindqviat, Phys. Rev. 112, 893 (1958). an’ I. Marklund and L. A. Page, Nudear Phys. 9, 88 (1958/59). *08 H. Schopper, PhiE. Mag. 181 2, 710 (1957). R. M. Steffen,Phys. Rev. 116, 980 (1959). zla H. Appel, 2. Physik 166, 580 (1959). D. Bloom, L. G. Mann, and J. A. Miskel, Phys. Rev. 136, 2021 (1962). *laH. J. Behrend and D. Budnick, 2. Physik 168, 155 (1962).

ANALYZER MAGNET

8- DETECTOR

ANTHRACENE

MAGNETlC

MAGNETIC SHIELD

Y- DETECTOR

FIQ.20. Apparatus to determine the circular polarization of gamma rays by Compton scattering from magnetized iron. The gamma rays are detected in coincidence with the preceding beta particles. (From Steffen, reference 209.)

70'

I

I

I

60'

mE

50'

40'

30'

I

I

4

2 k0

I

6

I

FIQ. 21. Optimum scattering angle Om for forward scattering as a function of the incident photon energy k~ (in units of mc*). The angle fi between the incoming photon and the electron spin is optimized according to Eq. (2.5.71). (From Sohopper, reference 200.) 270

2.5.

POLARIZATION OF ELECTRONS AND PHOTONS

27 1

The difference in sign in the definitions of 6 (Eq. 2.5.64) and 6' (Eq. 2.5.69) is due to the fact that 6 is defined for transmission and 6' for scattering.200 In order to design a n optimum instrument, one must compute @s-/@o for various scattering angles and determine its maximum for various gamma-ray energies. The optimum angle $, between the direction of the incoming photon ko and the electron spin is given byzoo (2.5.71) where 8 is the scattering angle. The optimum scattering angle em must be calculated numerically. Figure 21 shows the optimum scattering angle for t,b = $m as a function of energy of the incident photon. The corresponding change 6' in the counting rate is given by curve b in Fig. 18. To determine the sign of the gamma ray helicity, the following rule, which follows from Eqs. (2.5.69) and (2.5.63), can be useful: More photons are scattered in the forward direction if the electron spin and the photon spin are antiparallel. Right-handed photons thus are scattered more strongly in the forward direction if the magnetization in the scatterer points away from the gamma-ray source.

I n the experimental realization of the ideas outlined above, a number of factors must be taken into account: (1) The magnetic scatterer should contain as high a fraction f of polarized electrons as possible. Usually, Armco iron is chosen as material. Hyperco alloy has been used for improved efficiency.203 The calculation off from the saturation magnetization is discussed in Sections 2.5.5.1.3 and 2.5.5.1.6. (2) The gamma rays observed in the counter (see Fig. 20) should come only from the saturated part of the magnetic scatterer, but not from other material. To satisfy this condition as completely as possible, lead collimators are introduced and the magnetic core is chosen to be so thick that scattering from the magnet coils is negligible. A second possibility of avoiding scattering from the coils, and at the same time minimizing multiple scattering in the magnetic material, is the use of a large air gap.211 (3) I n the cylindrical geometry shown in Fig. 20, only very few gamma rays are scattered at the optimum angles. It is possible to shape the magnetic scatterer in such a way that most gamma rays are scattered at about the optimum angle 0,,,.200~203~213 (4) The influence of the magnetic field on the gamma ray counter must *I*

€3. Behopper and S. Galster, Nuclear Phys. 6, 125 (1958).

272

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

be eliminated. Long light pipes and magnetic shielding can reduce changes in the counting rate on switching the field to less than 0.1 %. (5) In every system there exist drifts. The influence of such drifts can be minimized if the magnetic field is changed at very short intervals of time. One such system, with a cycle of less than 10 see, is described in reference 21 1. (6) The raw data from a polarimeter must be corrected for background, for multiple and plural scattering in the magnetic scatterer,200and for deviations from an ideal geometry. Calculations of the effects of the finite size of source, scatterer, and absorber can be found in references 184, 200, 209, and 210.

‘;;lo

SCATTERER

DIRECTION OF MAGNETIZATION

COLLIMATOR

FIG.22. Principle of the method of Beard and Rose for the determination of photon helicity. The iron scatterer is magnetized alternately in the two directions indicated. Two counters are used to speed up data collection and to reduce experimental asymmetries.

2.5.6.2.4. BACKWARD SCATTERING. Figure 18 shows that, especially for small energies, backscattering yields the highest changes in relative counting rate. Nevertheless, experiments in the backscattering geometry are hard to perform, because the cross section is small and the energy of the backscattered quanta is nearly independent of the incident energy. The selection of the desired photons is hence difficult and the background is high. Details about the determination of the photon helicity using backscattering can be found in references 214 and 215. 2.5.6.2.5. THE BEARD-ROSE METHOD. The geometry shown in Fig. 22 permits the simultaneous observation of the scattered gamma rays in two direction^.^^^^^^^ Data gathering hencecan be faster than inother methods. *I4

M. Bernardini, P. Brovetto, S. DeBenedetti, and S. Ferroni, Nuooo cimento 7,

416 (1958).

R. M. Steffen, Phys. Rev. 118, 763 (1960). D. B. Beard and M. E. Rose, Phys. Rev. 108, 164 (1957). 217 E. C . Beltrametti and S. Vitale, Nuovo cimento 9, 289 (1958). z16

218

2.5.

POLARIZATION OF ELECTRONS AND PHOTONS

273

Curve d of Fig. 18 shows th at this approach is best suited to gamma-ray energies around 1 MeV. 2.5.6.3. Determination of the Helicity of Gamma Rays by Mossbauer Effect. 2.5.6.3.1. THEMOSSBAUEREFFECT.^^*-^^^ A free nucleus N * of mass M that emits a gamma ray of energy E will recoil with an energy R given by R = E2/2Mc2 (2.5.73) and the gamma ray line will, in general, show Doppler broadening. If the decaying nucleus is embedded in a solid, a fraction f of all decays will show a remarkable behavior,21* namely : (1) the emitted gamma ray line will show the natural line width r ; and ( 2 ) the gamma ray will not show any measurable energy loss due to recoil.

The natural line width

r is given by rr = h

(2.5.74)

where T is the mean life of the excited nuclear state. The fraction f of recoilless emissions is determined by

f

=

exp(- /X2)

(2.5.75)

where < x 2 > is the mean square deviation of the nucleus from its equilibrium position in the lattice, and 2nX = 2nhc/E is the gamma ray wavelength. If the solid can be described by a Debye model, and is characterized by a Debye temperature 0, then

f

=

exp

[- a 6R

1

+

2):(

L””s]) (2.5.76)

where T is the ambient temperature of the solid. In practice, only lowenergy gamma rays ( E 5 100 kev) show an appreciable fraction of recoillessly emitted gamma rays. I n order to observe these gamma rays, a material containing the nucleus N in its ground state is used as an absorber. The gamma ray beam contains a fractionf of gamma rays with the natural line width. A fraction 2 1 8 R. L. Mossbauer, Z . Physik 161, 124 (1958); Naturwissenschajten 46, 538 (1958); 2.Naturforsch. 14a, 211 (1959). 219 R. L. Mossbauer, Ann. Rev. Nuclear Sci. 12, 123, 1962. 220 H. Frauenfelder, “The Miissbauer Effect.” W. A. Benjamin, New York, 1962. 221 A. Schoen and D. M. J. Compton, eds., “The Mosshauer Effect, Proceedings of the Second Mossbauer Conference.” Wiley, New York, 1962.

274

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

f’ of these gamma rays are then absorbed without recoil, and this excess absorption can be observed. If the source is moved towards the absorber with a velocity v, the emitted gamma ray will receive an additional Doppler energy AE: AE = (v/c)E. (2.5.77) By giving the source various velocities v and hence various additional amounts of energy, the absorption curve can be used to trace out the natural line shape, or, more accurately, the overlap of the natural lines of source and absorber. If the source or the absorber is placed in a strong magnetic field of magnitude B,either artificially created or produced by extranuclear fields in solids, the gamma-ray line splits into Zeeman components. Assume the very simplest case where the excited state N* has spin 1 and the ground state N spin 0. The emitted line then splits into three Zeeman components. These components can be resolved if

g*poB > 2 r

(2.5.78)

where g* is the g factor of the state N* and p o is the nuclear magneton. If the gamma ray is observed in the direction of the magnetic field, only two lines will be seen and these two components are completely circularly polarized (longitudinal Zeeman effect). 2.5.6.3.2. THEOBSERVATION OF THE H E L I C I T Y . The ~ ~ ~observation J~~ of the circular polarization is now straightforward. The gamma-ray beam emitted in the direction of the magnetic field is observed with an absorber possessing an unsplit absorption line. The absorption spectrum will show two lines, and the intensity of these two lines is proportional to the strength of the right and left circularly polarized component respectively. Alternately, the absorber can be placed in a strong magnetic field and it can then be used to find the helicity of an unsplit gamma ray. A detailed description of the theory underlying this method and of some experiments is contained in reference 222. It is essential to point out, however, that this method can only be applied to gamma rays that show an appreciable Miissbauer effect and that satisfy the condition expressed by Eq. (2.5.78).

**sH. Frauenfelder, D. E. Nagle, R. D. Taylor, D. R. F. Coohran, and W. M. Visscher, Phys. Rev. 126, 1065 (1962).

2.6. Determination of Life-Time 2.6.1. Long life-Time* 2.6.1.1. Radioactive Decay. For a (large) number N of identical radioactive nuclei, the decay rate is proportional to the number of the existing nuclei : - -dN =

AN.

dt

(2.6.1.1)

The constant h is called the decay constant. Integration of Eq. (2.6.1.1) leads to N = Noe-At (2.6.1.2) as the number of the nuclei still existing a t the time t, if this number was Noat t = 0. N and d N / d t are to be understood as statistical mean values. The mean life is found to be 7 =

i*N t

dt

- -.

(2.6.1.3)

h*Ndt A more common figure than 7 is the half-life T , defined as the time when N = N0/2. One has T = 7 log 2 = 7 * 0.693.

If there are two ways of decay, e.g., alpha and beta decay, the partial half-life TI is given by TI =

e2 T

(2.6.1.4)

P1

+

with p l / ( p l p z ) as the branching ratio for the decay via branch 1. An analogous expression holds for n ways of decay. For a constant production rate d N / d t , the number of active nuclei N increases as (2.6.1.5) N = N,(1 - e--ht) where N , is the saturation activity and X the decay constant defined in Eq. (2.6.1.1). I n the case of decay chains, Eq. (2.6.1.2) and Eq. (2.6.1.5) must be combined, N, being replaced by N of the mother substance. Figure 1 shows, as an example, the amount of RaA, RaB, and RaC (Po218, Pb214,and Bi214)in a source which contained only RaA a t the time t = 0. We speak about radioactive equilibrium if the ratio between

-

* Section 2.6.1 is by H.

Daniel and W. Gentner. 276

276

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

the activities does not change. The equilibrium is called secular if the decrease in time is negligible. Then the number of nuclei show the same ratio as the half-lives while the activities are equal:

N1:Nz: . . . N ,

=

T1:Tz:

*

*

*

T,.

(2 .6.1.6)

I n the case of a nonsecular equilibrium, the number of nuclei as well as the activities of the daughter substances N 2 through N , are larger than the values obtained from Eq. (2.6.1.6).

t-

FIG.1. Amounts of RaA, RaB, and RaC in a source which contained RaA only at the time t = 0.

The laws of radioactive decay have frequently been checked by experiment, especially in the beginning of radioactive research. Excellent review is given by Meyer and Schweidler,’ Rutherford el a1.,2and B ~ t h e . ~ The theory of radioactive decay is treated here and by Rasetti4 and Segr&.6In particular, Bothe3 and SegrP deal with the statistical fluctuSt. Meyer and E. Schweidler, “Radioaktivitiit,” 2nd ed. Teubner, Berlin, 1927. E. Rutherford, J. Chadwick, and C. D. Ellis, “Radiations from Radioactive Substances.” Cambridge Univ. Press, London and New York, 1930. * W. B o t h , in “Handbuch der Physik” (H. Geiger and K. Scheel, eds.), 2nd ed., Vol. XXII, Part 1. Springer, Berlin, 1933. F. Rasetti, “Elements of Nuclear Physics.” Prentice-Hall, New York, 1936. E. SegrB, in “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 111, pp. 1-53. Wiley, New York, 1959. 2

2.6.

DETERMINATION OF LIFE-TIME

277

ations in radioactive decay. The decay constants X have finally been found, in every case, to be, within very narrow limits, practically unchangeable numbers characteristic for the individual activity. Small changes, however, have definitely been observed. Leininger et a1.6 measured the difference in the decay constants between Be metal and BeFz to be X(Be) - X(BeF2) = (0.84 5 0.10)10-3X(Be); Kraushaar et ~ 1 . ~(0.74 7 5 0.05)10-3; and Bouchez et U Z . , ~ (1.2 5 0.1)10-3. Similar results were found for the difference between X(Be0) and X(BeF2).6s7The radioactive isotope Be7 decays by electron capture, and the small effects on the decay constant can be explained by the change of the electron density at the nucleus. In the case of Tcggm which decays by a strongly converted isomeric transition, Bainbridge et d 9found the following differences : and

X(KTc04) - X(TczS7) = (2.7 5 0.1) 10-3X(Tc&) X(TC)- X(TczS7) = (0.31 0.12)10-3X(T~zS7).

Again the effect is caused by a change of the electron density a t the nucleus. Byers and Stumplo measured the effect of low temperature and superconductivity on the decay constant of metallic Tcggm. They obtained X(4.2'K superconductive) - X(293'K) = (6.4

k 0.4)10-4X(2930K).

The effect was greatly reduced in the absence of superconductivity: X(4.2'K

normal) - X(293"K)

=

(1.3 f 0.4)10-4X(2930K).

Porter and McMillan" treated theoretically the effect of compression on the decay rate of metallic Tcggm. The question of a cosmological variation of the decay constants cannot definitely be answered a t present. The reader is referred to the review article of Dicke.I2 Before measuring unknown half-lives, it is useful to estimate the expected order of magnitude. In any type of radioactive decay, the decay rate increases with increasing energy available. Besides, the decay rate depends on spin and parity change of the transition, and on other factors such as the more or less overlapping of the initial and final state wave functions. Instead of discussing these questions in detail, we want to give some help for rapid orientation. R. F. Leininger, E. SegrB, and C. Wiegand, Phys. Rev. 81, 280 (1951). J. J. Kraushaar, E. D. Wilson, and K. T. Bainbridge, Phys. Rev. 90, 610 (1953). 8R. Bouchez, J. Tobailem, J. Robert, R . Muxart, R. Mellet, P. Daudel, and R . Daudel, J. phys. radium 17, 363 (1956). K. T. Bainhridge, M. Goldhaber, and E. D. Wilson, Phys. Rev. 90,430 (1953). lo D. H. Byers and R. Stump, Phys. Rev. 112, 77 (1958). l 1 R. A. Porter and W. G . McMillan, Phys. Rev. 117, 795 (1960). 12 R. H. Dicke, Revs. Modern Phys. 29, 355 (1957). 7

278

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Figure 2 shows the relationship between half-life and energy for alpha transitions between the ground states of even-even nuclei.1s For these transitions the half-lives can be calculated most accurately. Transitions between other nuclei are often strongly hindered (by up to more than a factor of thousand). Extensive information is contained in the review article of Perlman and Rasmussen.13 For beta transitions, with

FIQ.2. Plot of logarithms of partial alpha half-lives for ground state transitions versus the inverse square root of the effective total alpha-decay energy (Qea = alphaparticle energy recoil energy electron screening correction). Points used in the analysis are indicated by open circles, and points not used, by full circles. Nuclei with 126 or fewer neutrons are not shown on this plot. The last digit in the mass number of the alpha emitter is given beside each point.

+

+

the decay energy, spins, and parities known, one calculates the order of magnitude for the half-life from the expected ft-value. A method for rapid determination of ft-values is given in references 14 and 15. A rough figure for the expected gamma transition probability is obtained from the Weisskopf formula.16~” Here again, large deviations are known : enhancements by more than two orders of magnitude, and hindrance I. Perlman and J. D. Rasmussen, in “Handbuch der Physik-Encyclopedia of Physics” (S. Fliigge, ed.), Vol. XLII, pp. 109-204. Springer, Berlin, 1957. 14 S. A. Moszkowski, Phys. Rev. 88, 35 (1951). D. Strominger, J. M. Hollander, and G. T. Seaborg, Revs. Modern Phys. SO, 585 (1958). la V. F. Weisskopf, Phys. Rev. 83, 1073 (1951). 17 S. A. Moszkowski, in “Beta and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), pp. 373-395. North-Holland, Amsterdam, 1955.

2.6.

DETERMINATION O F LIFE-TIME

279

factors of up to 10’. Competing processes, especially internal conversion, speed up the total decay according to Eq. (2.6.1.4). 2.6.1.2. Survey on the Methods. There are a number of ways of measuring long half-lives. The most important ones are: (a) direct observation of the decay; (b) determination of activity and number of radioactive nuclei; and (c) determination of daughter product content, collected during a known time interval. These three methods will be treated below in some detail. The decay of the free neutron and double beta decay will be treated separately. Besides the methods (a) through (c), there are methods that are applicable in special cases, which shall not be discussed, e.g., half-life determination via the mole ratios for members of radioactive decay chains (cf. Section 2.6.1.1). Direct observation of the decay is useful with shorter half-lives and is applicable up to the order of ten years half-life. This method becomes more involved and less accurate with increasing half-life, a t least, for half-lives longer than about a month. Variations of the efficiency of detection probably are the main source of error. Zero methods are advantageous. Variations within the source, such as loss of activity or change in the self-absorption, can be avoided by careful source preparation. The activity can be followed either by counting methods or by integral methods with an ionization chamber or calorimeter. Counting methods are more suitable for shorter half-lives and less suitable for longer half-lives, as compared with integral methods. Ionization chambers or calorimeters are useful for half-lives longer than about an hour. Ionization chambers need lower activities than do the calorimeters and are therefore more common in use. Half-lives longer than about ten years are usually measured with the specific-activity method. This method is applicable down to about one year half-life. Errors arise mostly from uncertainty in the determination of the specific activity as well as from determination of the number of radioactive nuclei in the sample. In many cases dilution is necessary and gives rise to additional errors. Which of the three single processes contributes most to the over-all error depends on the individual activity. For example, determination of the number of radioactive nuclei is easy and no dilution is necessary in the case of Rb87, whereas measuring the specific activity is difficult. I n the case of C14,however, all three processes are not easy; determination of the specific activity is most difficult. For determining the specific activity 4r counters are preferred, particularly in the case of beta-active nuclei. The content of radioactive nuclei in the source is frequently measured with the help of a mass spectrometer. The dilution process is done mostly in the liquid phase. I n a number of cases, the gaseous phase is chosen.

2.

280

DETERMINATION OF FUNDAMENTAL QUANTITIES

For measuring very long half-lives the collection method (c) is preferred. More important is the inverse method: with the half-life known, one determines the daughter product and hence the age of a sample. The suitable age determination method is often uniquely determined by the sample itself. I n many cases, however, different methods are applicable, thus providing a check or indicating different types of ages. 2.6.1.3. Direct Observation of the Decay. I n this method the decay rate -dN/dt is measured as a function of the time t. According to Eqs.

I

-

\ \

I

I

I

I

I

I

\

I

I 1 \ 1

I

I

I

I

I

t (arbitrary units)

1

I

I

-

FIG.3. Counting rate from a source containing two activities with half-lives T I and Tz, respectively. A constant counter background is assumed.

(2.6.1.1) and (2.6.1.2) the logarithm of the decay rate, when plotted against the time, is a straight line if there is only one activity present. Figure 3 shows the decay of a source with two different activities, with a background assumed. I n practice it is advantageous to follow the decay as long as possible, e.g., for detecting radioactive contaminations. The classical instrument for measuring decay curves is the ionization chamber.* Here the current which arises between two electrodes as a consequence of the ionization and a voltage difference is measured directly. All the charge-transporting ions in the gas between the electrodes * See also Vol. 2, Chapter 4.1.

2.6.

DETERMINATION OF LIFE-TIME

28 1

are directly produced; the chamber is operated in such a way that ion multipIication does not take place. Gas pressure and voltage are such that one works in the region of saturation (no recombination). For more accurate measurements the differential ionization chamber, introduced by Rutherford,'* is preferred. This instrument consists of two single chambers, as identical as possible, operated with equal voltages of different sign. One chamber is irradiated by an activity of known half-life and, the other is irradiated by the activity as a rule, long half-life (e.g., RazZn); with unknown half-life (for measuring small differences, as in the Be7 and Tcggm experiments discussed in Section 2.6.1.1, one takes the same isotope

BRASS

SOURCE HOLDER

POLYSTYRENE LUCITE

4

B O O CERAMIC OR CRUSHED Bc METAL COVEREP WITH WAX

FIG.4. Schematic diagram of balanced ionization chambers.

for both chambers). The source strengths are chosen in such a way that a t t = 0 both chambers show about the same current, or the chamber with the shorter-lived activity a little more. The ionization current difference is taken as a function of time. Figure 4 shows schematically the differential ionization chamber of Segr& and Wiegand.l9 Here the sources are inside the chambers. The instrument is operated with argon having a small excess pressure (1356 mm Hg). The voltages are provided by batteries. The galvanometer, measuring the difference current after amplification by an electrometer circuit, is operated by the rate-of-drift method. Bainbridge et al.9 describe a high-pressure differential ionization chamber welded of steel. The sources are also inside the chambers, if possible at a position where the ionization current is maximum so that small source displacements have only a negligible effect. TobailemZ0gives a description of a differential ionization chamber filled with air at atmospheric pressure. The sources 18 E. Rutherford, Sitzber. Akad. Wiss. Wien, Math.-natunu. Kl., Abt. IIa 120, 303 (1911). 10 E.Segrh and C. E. Wiegand, Plays. Rev. 76, 39 (1949). *o J. Tabailem, Ann. phys. (12110, 783 (1955).

282

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

are outside the chambers. Robertzl reports a differential ionization chamber which can be operated a t 4 atm and 1600 volts per chamber. In all cases the chambers are connected by a tube for pressure balance, The chambers are shielded against each other with lead (5-10 cm Pb). The induced currents are very smalI and therefore need special means for measuring. Robert21reports for each chamber with the source a t the center a current of 7.5 x l O - l 4 amp per microcurie Ra including daughter products, with a filter of 2 mm lead. Practically, one measures the charge of a well insulating condenser (capacitance of the order of 5000 ppf) coupled,-e.g., to a vacuum tube electrometer. Care must be taken to insure a high input impedance of the system (1012-1014ohms) and a low zero point drift of the galvanometer. Tobailem20describes a setup which approximately compensates for the discharge of the heater battery for the electrometer tube. Mme. Curiez2describes a method which makes use of a piezoelectric crystal and a balance. In order to compensate for differences between the two chambers the sources may be exchanged: first, source 1 is measured in chamber 1 and source 2 in chamber 2; then source 1 in chamber 2 and source 2 in chamber 1 ; then again source 1 in chamber 1 and source 2 in chamber 2, etc. Exact source positioning is important with this method. This may lead to difficulties in the case of setups with the sources outside the chambers.2l If the difference current vanishes at time t = 0, the current i as a function of 2 is given by

i = IW[l- exp(-AX,t)]

(2.6.1.7)

I , = current a t t = 00, A, = decay constant of unknown activity. TobailemzOalso treats the following subjects : decay of standard not negligible; different background in the two chambers; effect of a source on the other chamber; and a method for defining t = 0. The general operation of ionization chambers is treated by S t a ~ b . ~ ~ The ionization chamber makes use of the ionization produced by the radioactive radiation. Instead, use can also be made of the calorimetric effect for determining d N / d t , particularly in the case of strong sources. The produced heat may either be measured directly or be compensated for by the Joule Thompson or the PeItier effect. There are objections against the use of the Peltier effect, because of the strong ionization present.a1 Differential calorimeters have been proved successful. Accord-

-

*‘J. Robert, Ann. phye. [13]4,89 (1959). 22 M.Curie, “RadioactivitB.” Hermann, Paris, 1935. H. H. Staub, in “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. I. Wiley, New York, 1953.

2.6.

DETERMINATION

OF LIFE-TIME

283

ing to Angstromz4 and von Schweidler et a1.,2Kone takes two single calorimeters as identical as possible, one of them heated by the radioactive source under investigation and the other heated by an electric current. Comparison is made by a thermocouple. The current is adjusted so as to give vanishing temperature difference. Measurements are best started after approximate temperature equilibrium has been reached. Robertz1describes a microcalorimeter he has recently used for precise half-life determinations (Fig. 5 ) . The radioactive source is inside a small calorimeter made of gold or lead, which hangs on silk strings within

FIG.5. Microcalorimeter (schematically). S source, C calorimeter, 5°Cthermocouple, E red copper vessel, Ch chimney, B water bath.

a copper vessel; good thermal insulation is necessary for a high sensitivity. The copper vessel is surrounded by a large water bath. A thermocouple compares the temperatures of calorimeter and copper vessel. The water temperature is measured by a precision mercury thermometer divided into hundredths or thousandths of a degree. The water temperature is kept at the calorimeter temperature while measuring, by adding warm water. The water temperature is taken as a function of time. The ionization chamber and the calorimeter are preferred because of their great stability. However, for short half-lives (a few minutes or less), they are too slow. Here (and up to several years half-life) counting methods are suitable. Moreover, counters usually are more handy. Proportional counters are more stable than Geiger counters. I n particular, age effects are much smaller. Another disadvantage of Geiger counters is the low counting rates permissible. Therefore, statistical errors are large K. Angstrom, Physik. 2. 6, 685 (1905). E. von Schweidler and V. F. Hess, Sitzber. Akad. Wiss. Wien, Math.-naturw. KZ.. Abt. I I a 117, 879 (1908); St. Meyer and V. F. Hess, ibid. 121, 603 (1912). 24

26

284

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

in the case of short half-lives. Hence Geiger counters will be useful only for survey measurements, e.g., for identifying the activity. Scintillation counters withstand high counting rates but are usually not stable enough for precise work. A severe source of error is the dependence of the photomultiplier gain on the counting rate.26 This effect makes a high discrimination level unrecommendable. Instead, the use of a suitable filter is indicated for eliminating soft unwanted radiation. I n order to reduce gain variations a stabilizer may be ~ s e d . ~ ’ , ~ ~ A combination of magnetic spectrometer and scintillation counter has proved to be very useful (see, e.g., reference 29). The magnetic spectrometer is set at the maximum of the intensity distribution of a beta spectrum while the scintillation counter threshold is set about halfway between the line and zero. I n this way even large variations of the spectrometer setting, the photomultiplier or amplifier gain, or the discrimination threshold do not seriously change the counting rate. Of course, the resolving power of the magnetic spectrometer may also be used for discriminating against unwanted radiation. Use of a good mechanical clock or a quartz timer is recommended for all counting measurements. The power line frequency is not always constant enough to serve as a time standard. I n the case of very short half-lives, of the order of a second, the time is fixed by a quartz timer which automatically switches a scaler plus printer or two scalers with printers alternatively. Another method consists in operating two scalers with lighting tubes, one of them being driven by a quartz clock and the other by the radiation counter. The scalers which are not switched are photographed a t about equal time intervals. Another method uses the constant sweep speed of a cathode ray oscilloscope as time axis, the counts being indicated by vertical deflection pulses. The scope screen is again photographed. Other electrical means for measuring decay curves have been described.a0 After registration of a pulse, the counting setup, in most arrangements, will not be ready to register any pulse coming within a fixed time interval, the dead time TO.When T ohas elapsed the setup will be ready for the next pulse. Under these conditions the measured counting rate n has to be corrected to give the “true” counting rate no according to no = n / ( l

- nTo).

(2.6.1.8)

H. Jung, Ph. Panussi, and J. Jlnecke, Nuclear Znstr. 9, 121 (1960). B. Astrom, Arkiv Fyaik 12, 237 (1957). ** H. de Waard, Nucleonics 13, No. 7, 36 (1955). z9 H. Daniel, Z . Naturforsch. laa, 363 (1957); H. Daniel and U. Schmidt-Rohr, Nuclear Phys. 7, 516 (1958). 80 A. J. Bureau and C. L. Hammer, Phya. Rev. 106, 1006 (1957).

2.6.

DETERMINATION OF

LIFE-TIME

285

The conditions under which Eq. (2.6.1.8) holds are not always fulfilled. A counting setup may have several dead times, e.g., the dead time of the first scaling circuit and the dead time of the mechanical counter. For more . ~ ~ complicated cases, the reader is referred to the paper of J o ~ t There are several experimental methods for determining the dead time correction. A very direct method consists in taking the decay of a strong pure source. One can also work with several sources of known strength constant in time. However, it is not advisable to vary the counting rate by varying the distance between source and detector and to depend on the validity of the 1/74 law. This law does not hold accurately enough, because of absorption and backscattering from the walls. The dead time can also be measured electronically with a double pulse generator. It is good t o apply two independent methods in parallel. The counter background is measured before and after the measurement of the sample. The decay curve itself should be followed as long as possible, e.g., to notice radioactive contaminations. Weak long-lived contaminations can be corrected for fairly easily if one has measured long enough. If the decay curve falls off too rapidly a t the beginning, this may indicate a short-lived activity or may be caused by a counting-ratedependent gain. When higher precision is wanted a least square fit of the decay curve must be made. Standard least square methods are sufficient. P e i e r l ~ ~ ~ treats counting statistics including background problems and data evaluation. A graphical method, in the course of which the remaining number of pulses is plotted, is given by KuiEer et al.33 I n the methods described above, the counting rate is taken as a function of the time. In some cases it is practical t o measure directly the number of remaining nuclei N as a function of time. For this purpose, the mole ratio between the radioactive isotope and an isotope of the same element is determined with a mass spectrometer. In this way Wiles et aL3* measured the half-life of CsI37 with C S ~ ~ ' / C Sand ' ~ ~T, h ~ d the e ~half-life ~ of Kr*5 with Krs5/Kr*6. Another method, which makes use of the alpha radioactivity of Rn222, the gamma radioactivity of some daughter products, and of the very different ionizing effect of the two radiations, has been applied by Bothe36 in order to measure the RnZz2half-life with high precision. 31

R. Jost, Hetv. Phys. Acta 20, 173 (1947).

52

R. Peierls, PTOC.Roy. SOC.A149, 467(1935).

33

34

I. KuSEer, M. V. Mihailoid, and E. C. Park, Phil. Mag. [ S ] 2,998 (1957). D. R. Wiles, B. Smith, R. Horseley, and H. G . Thode, Can. J . Phys. 31, 419

(1953). 3 6 R .K. Wanless and H. G. Thode, Can. J . Phys. 31, 517 (1953). 36 W. Bothe, 2. Phyaik 16, 266 (1923).

286

2.

DETERMINATION OF FUNDAMENTAL QUANTITIEE.

2.6.1.4. Specific Activity Method. The specific activity method is favored in cases where the half-life is too long to measure directly the decrease of activity. Instead, one measures the specific activity and the number of radioactive nuclei per weight unit. If

A =

decay rate weight unit

=

specific activity

and @=

number of active nuclei weight unit

then it follows from Eq. (2.6.1.1) X = A/g.

(2.6.1.9)

For the direct decay rate measurement, counting methods are applied. However, the ionization chamber and calorimeter are also used. With these methods, the energies of the radiations must be known whereas with counting methods approximate knowledge of the decay scheme is often sufficient. The calorimetric method has the advantage of being free of self-absorption effects and the disadvantage of needing strong sources. The heat production rate .J is given by

J

= 5.93 X

10-TB watt

with C = source strength in curies and 2 = mean energy per disintegration in MeV. For counting alpha rays, counters with a defined small solid angle are suitable because of the small directional scattering of alpha rays. Figure 6 shows a proportional counter due to Hurst and Source holder and collimating baffles are a rigid unit. In this way constancy of the geometry factor, i.e., the reciprocal of the fractional solid angle, is guaranteed. This geometry factor amounts to 716.8 & 0.8, including a ball correction of 2.5%. The source is mounted on a platinum disk in the middle of the source-holding plate. Small dislocations on the plate are not critical. The counter is usually operated as a proportional counter filled with 60 mm Hg of methane. The degree of evacuation in the region between source and counter, necessary to ensure that all alpha particles emitted into the fractional solid angle reach the counting volume, can be calculated. It is advantageous to make all alpha particles reaching the counter travel through the entire counter, for in this way pulses of about the same height result. s7R.Hurst and G. R. Hall, Analyst 77, 790 (1962).

2.6.

DETERMINATION

OF LIFE-TIME

287

To Amplifier

t

FIG.6. Alpha proportional counter with defined small solid angle.

Counters with a small solid angle are not suitable for low specific a large proportional activities. Macfarlane and K ~ h m a ndescribe ~~ counter 011 the walls of which (1200 cmz) the active layer is deposited (27r geometry). The pulses are registered with a multichannel analyzer. With shielding and anticoincidence counters, the background between 1 and 3 Mev alpha energy amounts to only 9 pulses per hour. Four n counters are also suitable for alpha rays. I n a n y case, the sources must be thin enough to avoid self-absorption losses. The source backing, however, can be made thick. With thick backing a correction must be 38

R. D. Macfarlane and T. P. Kohman, Phys. Rev. 121, 1758 (1961).

288

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

applied which, in the case of 2a geometry, has been c a l ~ u l a t e d The .~~ detection probability is ( F = without correction) :

+

F ( B ) = +[l

+ 0.201@(B)].

@ ( B )is a tabulated function depending on material and range.39 With Pt as backing and U as source, 0.2013(B) = 0.016. I n the case of nonnegligible source thickness, the self-absorption must be corrected This correction depends also on the discrimination bias. For beta rays counters with defined small solid angle are also used. The method is recommended by its simplicity. The accuracy, however, is limited by a number of necessary corrections which are not very well known: Within the source the beta particles lose energy, are scattered, and are absorbed. The scattering leads to increased intensities in directions not too far away from the normal direction.40 Backscattering also increases the intensity.41The phenomenon of backscattering, which because of its complexity is hard to treat theoretically, has been subject to many experiments. The results are not in good agreement, especially with respect to the difference between electrons and positrons. The various measurements are not always comparable, however, because the results depend on the geometry chosen. Qualitatively, it can be said that the backscattering strongly increases with increasing atomic number for backings of equal surface density. 41-44 Therefore 8 thin film made from a low-atomic-number material is taken as source support. This film should be conductive in order to avoid charging the source, (e.g., Al, formvar, or collodion with an evaporated Au layer on the back). The most accurate absolute measurements of beta activities are performed in 4?r geometry, either with a gaseous source where the active gas is added to the counting gas or is the counting gas itself, or with a thin solid source. For gaseous sources, cf. Section 2.6.1.8. Figure 7 shows a 4n counter due to Pate and Yaffe.4KThe counter consists of two hemispherical brass cathodes 7 cm in diameter and two ring-shaped anodes of 1 mil tungsten wire. The source is supported by a plastic film of 5-10 pg/cm2, evaporated with 2 pg/cm2 gold. Methane of 39 B. B. Rossi and H. H. Staub, “Ionization Chambers and Counters.” McGrawHill, New York, 1949. 40 D. Fehrentz and H. Daniel, Nuclear fnstr. 10, 185 (1961). 4 1 W. Paul and H. Steinwedel, in “Beta and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), pp. 1-23. North-Holland, Amsterdam] 1955. 42 W. Bothe, 2. Naturjorsch. 4a, 542 (1949). 43 G . Marguin, Ann. phys. 1131 2, 318 (1957). 4 4 H. H. Seliger, Phys. Rev. 78, 491 (1950); 88, 408 (1952). 46E. D. Pate and L. Yaffe, Can. J . Chem. 33, 610 (1955).

2.6.

DETERMINATION

289

OF LIFE-TIME

atmospheric pressure is taken as counting gas; it is dried by silica gel. The counter is operated in the proportional region. The slope of the ' per 100 volts. The detection probability is unity plateau is less than 0.1% within 0.1 %. The finite thickness of the supporting film makes a correction necessary. Pate and Yaffe46discuss the various methods according to their own 0040' COPPER ANODE LEAD

TEFLON INSULATOR

,/

GA> n n

INLET (2001' )I* TUNGSTEN

AND MOUNT

GAS OUTLET

FIG.7. 4~ counting chamber.

experiments. (a) With the sandwich method as described b y Hawkings et a1.,47 the source is first measured in 47r geometry, then on the front side covered with a film identical to the backing and again measured in 4n geometry. The observed reduction in the counting rate is taken as correction for the first measurement. The result should be independent - ~ ~ a correction of the film thickness but is noL48 (b) Mann and S e l i g e ~give formula the result of which depends on two not well known quantities, the fractional backscattering factor and the fractional absorption factor. B. D. Pate and L. Yaffe, Can. J. Chem. 33, 929 (1955). R. C. Hawkings, W. F. Merrit, and J. H. Craven, i n Proceedings of Symposium on Maintenance of Standards, Natl. Phys. Lab., May, 1961. (Puhl. by H. M. Stationary Office, London, 1952.) 48 1).B. Smith, Brit. Atomic Energy Rept. A.E.R.E. I/R 1210 (1953). 4B W. B. Xlann and H. H. Seliger, J. Research Natl. BUT.Standards 60, 197 (1953). 46

47

290

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

(c) With the absorption curve the counting rate is taken as a function of the backing thickness for the same source. This curve, which is not a straight line, is extrapolated to zero. The error of the disintegration rate is of the order of 0.2% even for beta emitters of low maximum energy.42The preparation of thin films is described in references 50-52. Particular difficulties often arise from self-absorption within the source, sometimes even if the specific activity of the sample is sufficient to prepare a source of a low average thickness. If possible, sources should be prepared by evaporation in vacuo because this method gives the most uniform sources. Electrodeposition frequently gives good results, too. I n some cases drying by evaporation of colloidal suspensions is applicable. Evaporation of salt solutions, on the other hand, results in sources of a crystalline structure with an effective thickness generally much larger than the calculated average. I n order to keep the crystals small, one may try special drying methods such as freeze drying or the application of insulin.63Source preparation is treated in the handbook of Siegbahn.K4 The source quality is not easy to check. Autoradiographies help in detecting inhomogeneities. Inspection with a magnifying glass or a microscope is advisable in any case. The average source thickness can be determined with a balance or with an alpha gauge. Estimating the source thickness from activity, area, and specific activity of the material frequently leads to values too small. Pate and Yaffe66compare various methods for source preparation. In order to correct for self-absorption in homogeneous sources, the self-absorption curve is taken. This procedure is tedious because each point needs a source of its own, the preparation method being always the same. The self-absorption curve is fitted by the theoretical curve of Gora and Hickey6sor extrapolated directly to zero. The error is of the order of 1%. Bayhurst and Prestwood67give a method for transforming, within an error of 3%, results obtained with thick sources into values for negligible source thickness. Heintze and Fischbecks8 describe a setup for measuring absolute beta activities in 2 r geometry for sources of saturation B. D. Pate and L. Yaffe, Can. J. Chem. 33, 15 (1955). E. Huster and W. Rausch, Nuclear Instr. 3, 213 (1958). 61 H. Mahl, in “Technische Kunstgriffe bei physikalischen Untersuchungen,” 12th ed. (H. Ebert, ed.), pp. 115-122. Vieweg, Braunschweig, Germany, 1959. 68 L. M. Langer, Rev. Sci. Instr. 20, 216 (1949). 6 4 H. Slatis, in “Beta and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), pp. 259-272. North-Holland, Amsterdam, 1955. 66 B. D. Pate and L. Yaffe, Can. J. Chem. 34, 265 (1956). 66 E. K. Gora and F. C. Hickey, Anal. Chem. 26, 1158 (1954). 61 B. P. Bsyhurst and R. J. Prestwood, Nucleonics 17, No. 3, 82 (1959). SS.1. Heintze and H. Fischbeck, 2. Physik 147, 277 (1957). *L

2.6.

DETERMINATION OF LIFE-TIME

29 1

thickness which has an error of 5%. In the two cases the decisive step for eliminating the effect of the spectral shape is the use of the mean beta energy (instead of the maximum energy). Vincentb9gives a description of setup (spherical 4s proportional counter) and method (variation of gas pressure) for measuring absolutely electron capture activities. Instead of proportional or Geiger counters, use can also be made of scintillation counters for 47r counting. Nevertheless, difficulties arise from the higher background. The scintillator may be liquid or solid.60-62 If the activity is in solution or incorporated into the crystal, no difficulties with source thickness or backscattering are encountered. If there is a second electron in cascade a correction must be applied if the solid angle is less than 47,or if the electron emission is delayed, as in the case of CsLS7. In some cases specific activities can be determined with the help of coincidence methods. Consider a nucleus emitting one beta particle and one gamma quantum per disintegration only. The beta counter shall accept beta rays only, and the gamma counter gamma rays only. The efficiency of the coincidence circuit shall be unity. Then one has, as can easily be derived, for the source strength - d N / d t (disintegrations per unit time) - d N / d t = npn,/n,, (2.6.1.10) independent of the beta and gamma detection probabilities; np, ny, and neodenote the beta, gamma, and coincidence counting rates, respectively. Of course, the method is applicable also in the case of alpha-gamma coincidences and gamma-gamma coincidences, and can be extended to more complicated decay schemes (see reference 63). In practice there are many sources of error. The beta counter usually also accepts gamma radiation. The method frequently in use to measure the gamma sensitivity of the beta counter, by putting an absorber in front of the counter, leads in most cases to values which are too high, because the secondary electrons from the absorber are also counted. Coincidence circuits with a short resolving time often have efficiencies less than unity. Gamma quanta may be scattered from one counter into the other. To reduce this effect the counters are put a t right angles to the source. In this way the effects of anisotropic directional correlations are also eliminated. Because of the finite resolving time, there are also accidental coinciD. H. Vincent, Nukleonik 1, 332 (1959). C. F. G. Delaney and I. R. McAulay, Sci. Proc. Roy. Dublin Sac. A l , 1 (1959). 6 1 K. F. Flynn and L. E. Glendenin, Phys. Rev. 116, 744 (1959). 62 G. M. Lewis, Phil. Mag. [7] 43, 1070 (1952). G . Wolf, Nukleonik 2, 255 (1961).

69

292

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

dences. I n a twofold coincidence circuit the rate of accidental coincidences is given by n, = 2 m p , where 27 = (double) resolving time; r is determined by adding a delay into one of the pulse channels, by irradiating the two counters with independent sources, or with sources like which do not give true coincidences. I n the case of a modern fast-slow coincidence circuit, determining the accidental coincidences is not so simple. Mayer-Kuckuk and N i e r h a d 4 describe a method which consists in measuring all possible twofold coincidences (with elimination of the third coincidence requirement). For extremely long-lived activities no direct counting methods are applied. The photographic emulsion is a detector which allows accumulation of data for a long time. I n this way Riezler and co-workersR5investigated the alpha decay of a number of nuclei. I n many cases it is not possible to determine directly the number of radioactive nuclei in a source. Sometimes the composition of the source material must be found out first. If there is enough material available the number g of radioactive nuclei per weight unit can be determined with a balance.* One can also add a known amount of inactive material to the unknown amount of active material and, in this way, prepare a standard solution. The mole ratio between active and inactive nuclei can be determined mass spectroscopically. An aliquot is taken from this standard solution and its decay rate measured. When preparing aliquots, care must be taken that the activity is really shared according to the volume ratio. I n carrier-free solutions the activity tends to stay on the walls. It is therefore advantageous to add as much carrier as is compatible with the activity measurement. The minimum amount corresponds to a layer a few atoms thick on all surfaces which come in touch with the activity during the whole procedure. All surfaces should be rinsed before use. The conditions, e.g., the pH value, should be chosen in such a way that the radioactive ions really stay in solution. As an example of a mass spectrometric analysis (CI3*source produced in a reactor), the work of Boyd et ~ 1 . should 6 ~ be cited, and as an example of a decay rate determination including source preparation, etc., the work of Bartholomew et U Z . , ~ ’ performed with the same source.

* Sometimes this number is too small for direct weighing. T. Mayer-Kuckuk and R. Nierhaus, Nuclear Instr. 8,76 (1960). W. Porschen and W. Riezler, 2. Naturforsch. l l a , 143 (1956); W. Riezler and G. Kauw, ibid. laa, 665 (1957); 13a, 904 (1958); 14a, 196 (1959). 66 A. W. Boyd, F. Brown, and M. Lounsbury, Can. J. Phya. 33,35 (1955). 67R. M. Bartholoqen;, A. W. Boyd, F. Brown, R. C. Hawkings, M. Lounsbury, and W. F. Mqrrit, Cdn. J. Phys. SS,43 (1955). 64

6s

I

2.6.

DETERMINATION OF LIFE-TIME

293

2.6.1.5. Decay of the Free Neutron. The beta decay of the free neutron was presumed by Chadwick and Goldhaber,'j8 and established by Snell and Miller'j9 and by Robson.'O The free neutron is the simplest beta-unstable nucleus. Quantities usually not calculable can reliably be calculated in this case. One of the quantities most important for beta decay theory is the half-life of the free neutron. However, special experimental arrangements are necessary, because of the impossibility of keeping neutrons a t a given place. The high background resulting from neutron capture disturbs the measurements. All neutron half-life measurements done u p to now are of the specificactivity type with direct observation of the decay. There is a proposal, however, to collect the decay product hydrogen in a n evacuated vessel.71 Figure 8 shows the setup of Sosnovsky et al.,72which has led to the most accurate value T = 11.7 k 0.3 min. A well collimated neutron beam from a reactor runs from left to right through the vacuum chamber 1 (pressure 1-2 X 10-6 mm Hg) in which there is a high voltage electrode 6 (20 kv). Neutrons of the beam decaying off the baffle 2 are registered if the recoil protons are emitted in the direction of the proton counter 5 . Before reaching the counter the protons fall through the potential of 20 kv and are simultaneously focused. A proportional counter with thin window serves as a proton detector (collodion 0.07 p , grid supported). The decay rate of the neutrons is measured with a constant-power reactor. I n practice, difference measurements (reactor on-reactor off; accelerating voltage on-accelerating voltage off) are necessary for eliminating the background. The transmission of the system and the source volume were carefully calculated with regard to the fact that the neutron density is not constant over the beam cross section. As the recoil protons enter the accelerating region with a small divergence only, the method is almost independent of the proton energy spectrum which depends strongly on the beta decay interaction and is not known well enough. Besides the decay rate, the number of neutrons in the source volume is needed. This quantity is measured by activating N a or Au targets. R ~ b s o n ?puts ~ two magnetic lens spectrometers back to back and leads a neutron beam from a reactor through the spectrometers perpenJ. Chadwick and M. Goldhaber, Proc. Roy. SOC.A161, 479 (1935). A. H. Snell and L. C. Miller, Phys. Rev. 74, 1217 (1948). J. M. Robson, Phys. Rev. 78, 311 (1950). 71 B. T. Feld, in "Experimental Nuclear Physics" (E. SegrB, ed.), Vol. 11, p. 218. Wiley, New York, 1953. 7 1 A. N . Sosnovsky, P. E. Spivak, Yu. A. Prokofiev, I. E. Kutikov, and Yu. P. Dohrinin, Nuclear Phys. 10, 395 (1959); Zhur. Eksptl. i Teoret. Fiz. 36, 1012 (1959); 36, 1059 (1958). 7 a J. M. Robson, Phys. Rev. 88, 349 (1951). 88 69

FIG.8. Diagram of the apparatus for measuring the neutron half-life. KEY:(1) body of apparatus; (2) diaphragm; (3) diaphragm with grid; (4) spherical grid; (5) proportional counter; (6) high voltage electrode; (7) flange of dzusion pump; (8) monitors; (9) reactor shield.

2.6.

DETERMINATION OF LIFE-TIME

295

dicular to their axis. One of the spectrometers detects the electrostatically accelerated recoil protons, with an ion-sensitive multiplier tube as detector; the other spectrometer detects the electrons. The two spectrometers are operated in coincidence; the finite time of flight of the protons is compensated for by a delay line. Snell et ~ 1 . ’also ~ take coincidences between electrons and recoil protons, but without magnetic spectrometers. The electrons are detected by two proportional counters in coincidence, and the protons by an ion-sensitive multiplier tube. D’Ange10’~ works with a diffusion chamber for detecting the electrons. The recoil protons cannot be identified. In order to reduce the background] the neutron beam is cleaned of gamma radiation by reflexion on a nickel mirror, and the chamber is operated with an oxygen-helium mixture. 2.6.1.6. Double Beta Decay. Double beta decay is a process of second order in the beta interaction. As this interaction is weak, extremely long half-lives are expected. Reliable experimental results for the occurrence of double beta decay do not exist a t present. I n double beta decay the nuclear charge is changed by two units. Instead of positron emission, electron capture is possible. Accordingly, four cases are to be distinguished if one asks for the role of the charged particles only: Emission of two electrons] emission of two posit,rons, emission of a positron and capture of an electron, and capture of two electrons. Each of the four processes may occur without neutrino emission or with emission of two neutrinos (in the case of double 8- decay more precisely : antineutrinos). I n the neutrinoless case, the socalled Majorana case, the energy sum of the two emitted electrons is always constant and equal to the Q-value, whereas in the case of neutrino emission, the socalled Dirac case, the energy sum is a continuous distribution between zero and the Q-value. This difference has important consequences for the experimental technique. The expected half-lives are much shorter in the Majorana case than in the Dirac case (of the order of 10l6 years and of the order of loz3years, respectively). For the connection with the regular beta decay, and generally for more information, the reader is referred to the excellent article by Primakoff and R ~ s e n . ’ ~ Experiments for detecting double beta decay can essentially be done in three different ways: (a) by counting the two electrons in coincidence; (b) by observing the two electrons with a cloud chamber or with a photographic emulsion; and (c) by determining the amount of daughter substance grown up in a mother substance of known age. Method (a) is the A. H. Snell, F. Pleasonton, and R. V. McCord, Phys. Rev. 78, 310 (1950). N. D’Angelo, Phys. Rev. 114, 285 (1959). 76 H. Primakoff and S. P. Rosen, Repts. Progr. in Phys. 17, 121 (1969).

296

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

most direct. In principle, method (a) and method (b) give information concerning not only the half-life but also the energy spectrum. A counting experiment of the type (a) was performed by Dobrokhotov et ~ 1 on. Ca48. ~ The ~ source consisted of 423 mg Ca4*,enriched to 76%; the inactive sample for background determination consisted of enriched Ca44. The two samples were alternatively put between two scintillation counters with a diameter of 50 mm. The solid angle subtended by the source at each counter was nearly 2 ~ A. 70 liter liquid scintillation counter viewed by 21 photomultiplier tubes served as an anticoincidence counter. Coincident pulses were displayed on a scope screen and registered with a camera. The whole arrangement was located in a lead shield at a depth of 65 meters water equivalent. Two different runs were made in different seasons, of 430 hours and of 300 hours, respectively. The over-all result was a lower limit of 7 X 10l8yearsfor the double beta decay half-life of Ca48.Other counting experiments were performed by Cowan et U Z . , ~ ~ A w s c h a l ~ mMcCarthy,so ,~~ Detoeuf and Moch,81Fulbright,82Pearce and D a r b ~Kalkstein ,~~ and LibbylE4and Fireman.86Berthelot et aLE6searched for double K capture. As an example of an experiment of the type (b) we sketch the work of Winter.87 Winter used a cloud chamber of 24 cm in diameter. The foils under investigation were stretched across the chamber. Stereoscopic pictures were taken at a random expansion rate of 2/min, in a magnetic field of 790 gauss. Pictures with two electrons, according to a first rough inspection, coming from the same place were stereoscopically projected back for further inspection. The last criterion was the energy balance. The following results were obtained: Cdlle, Tb-8- > 1 X 1017 years; > 6 X 10l6 years; Mo'OO, To-8- > 3 X 1017years; and MoQ2, Cd1O6,TB+B+ TB+B+> 4 X 10l8years. The values for Tp+p+are partial half-lives because of competing electron capture. Other experiments of this type were 77

E. I. Dobrokhotov, V. R. Laearenko, and S. Yu. Lukyanov, Zhur. Eksptl. i Teoret.

Fiz. 36, 76 (1959); English translation. Soviet Phys. J E T P 36, 54 (1959). 78

C. L. Cowan, Jr., F. B. Harrison, L. M. Langer, and F. Reines, Nuovo cimento

[lo] 3, 649 (1956).

M. Awschaloni, Phys. Rev. 101, 1041 (1956). A. McCarthy, Phys. Rev. 87, 1234 (1955); DO, 853 (1953). 8 1 J.-F. Detoeuf and R. Moch, J. phys. radium 16, 897 (1955); Compt. rend. acad. sci. 941,393 (1955). 8* H. W. Fulbright, Physica 18, 1026 (1955). 8 3 R. M. Pearce and E. K. Darby, Phys. Rev. 86, 1049 (1952). S4 M. I. Kalkstein and W. F. Libby, Phys. Rev. 86, 368 (1952). SsE. L. Fireman, Phys. Rev. 76, 323 (1949). 8 6 A. Berthelot, R. Chaminade, C. Levi, and L. Papineau, Compt. rend. acad. sci. 236, 1769 (1953). 87R.G. Winter, Phys. Rev. 88, 88 (1955). 79

805.

2.6.

DETERMINATION

OF LIFE-TIME

297

performed by Fremlin and Walters,*8 Fireman and S c h ~ a r z e r ,and ~~ Lawson. Inghram and R e y n o l d ~performed ~ ~ ~ ~ ~an experiment of the type (c). They investigated mineral samples of BizTe3which, according to geologic information, were 1500 k 500 million years old, and searched for Xe'3n from double beta decay of Te13n.The whole Xe content of the samples and the mass spectroscopical composition of the Xe were determined. In the second i n v e ~ t i g a t i o n 2.4 , ~ ~X 61113 S.T.P. argon plus 2.,, X lo-' cm3 S.T.P. xenon were extracted out of 371 gm Bi2Te3with 124 gm T e by heating in vacuo. A large excess of XelZg, Xe130, and Xe131was found as compared with atmospheric xenon. The excess of Xe129 and Xe131can be accounted for by (n,r) reactions with TeI28 and TeI3O and additional (T = 1.7 X 107 years). There was uranium XeIz9from the decay of in the neighborhood which can have provided the neutrons. A similar explanat,ion is excluded for Xe13"because 1 1 2 9 is unstable. The author^^^^^^ ascribe the XeL30excess to double beta decay of Te130 and obtain a halflife of 1.4 X loz1years. This investigation was later redone by Hayden and Inghram.93If all the Xe13nfound by these authors in their mineral sample is ascribed to double beta of TeI3O, a half-life of 3.3 X lo2' years is found. A similar experiment was performed by Levine et aLg4 The question of the stability of the free proton and of the nucleon bound in a beta-stable nucleus has also been investigated experimentaIly.96~*6 The free proton, e.g., might disintegrate into two positrons and an electron. Therefore the experimental arrangement to detect the decay rate by a counting method is not too different from the arrangements used in double beta decay work. The best results were obtained b y the CERN groupg6for the lower limits of the half-lives: 1.5-2.8 X years for bound nucleons and 2 .2 4 .7 x 1024 years for protons in hydrogen, the variations being due to postulates made about the decay mode. 2.6.1.7. Determination of Branching Ratios. I n many cases the partial half-lives are interesting (cf. Section 2.6.1.1). We do not want to treat 88 J. H. Fremlin and M. C. Walters, Proc. Phys. Soe. (London) A66, 911 (1952). E. L. Fireman and D. Schwarzer, Phys. Rev. 86, 451 (1952). J. S. Lawson, Jr., Phys. Rev. 81, 299 (1951). 9 l M. G. Inghram and J. H. Reynolds, Phys. Rev. 76, 1265 (1949). 92 M. G. Inghram and J. H. Reynolds, Phys. Rev. 78, 822 (1950). 93 R. J. Hayden and M. G. Inghram, i n "Mass Spectroscopy in Physics Research." Natl. Bur. Standards (U.S.) Circ. No. 622 (1953). 9 4 C. A. Levine, A. Ghiorso, and G. T. Seaborg, Phys. Rev. 77, 296 (1950). 96F. Reines, C. L. Cowan, Jr., and M. Goldhaber, Phys. Rev. 96, 1157 (1954); F. Reines, C. L. Cowan, Jr., and H. W. Kruse, ibid. 109,609 (1957); G. N. Flerov, D. S. Klochov, V. S. Skobkin, and V. V. Terentiev, Soviet Phys. Doklady S, 78 (1958). 96 G. K. Backenstoss, H. Frauenfelder, B. D. Hyams, L. J. Koester, Jr., and P. C. Marin, Nuovo cimento [lo] 16, 749 (1960). 89

90

2.

298

DETERMINATION OF FUNDAMENTAL QUANTITIES

the determination of partial half-lives of competing but equal processes, e.g., the decomposition of a beta continuum into its components, but restrict ourselves to the branching-ratio determination in the case of different processes, e.g., alpha and beta decay. Determination of branching ratios is particularly easy if the measurements themselves can be done with alpha particles only. This happens often with radioactive decay chains.07 In this way Schupp et aLg8determined the branching ratio ./(a p ) of Bi2'2 by comparison of the alpha lines of and Po212with a CsI scintillation counter. The error was small because there were two alpha groups with energies and intensities not too different but which could well be resolved, and because the angular scattering and backscattering of alpha particles is small in any case. Eastwood el U Z . ~ measured ~ the very small branching ratio a/(a p) of Bk248by separating chemically the daughter of beta decay Cf24aimmediately before the measurement of the alpha activity and following the increase of the over-all alpha activity. Extrapolation to the time of separation gave the alpha activity of Bk249.The specific beta activity was determined with a 47r counter (Section 2.6.1.4). Aliquots were used for both alpha and beta counting. For measuring electron-positron branching ratios, use can be made of a magnetic spectrometer, which is able to distinguish between electrons and positrons. Most "flat" spectrometers, such as the double focusing or the 180" spectrometer, do this; also, lens spectrometers which have a helical baffle. Weak positron branches are detected with the help of the annihilation radiation. In contrast to nuclear gamma radiation, radiation from annihilation in rest shows a strong directional correlation (angle between the two quanta, 180") which can be used to distinguish between the two. The branching ratio K / p + of K capture to positron emission can often be measured by comparing the intensities of K X-ray and Auger line on the one hand and positron radiation on the other. Difficulties arise in the case of light elements, because of the low K X-ray energy. Scobie and Lewis'o' measured the K / p t ratio of C1l with a wall-less proportional counter.101 Such a counter is also suitable for measuring the ratio between K and L capture.'O' 2.6.1.8. Age Determinations. Age determinations are the inverse of the problem of measuring half-lives: With the half-life known, one asks for the "age," i.e., the time which was needed to produce a certain experimentally determined amount of daughter product, or to reduce the

+

+

97

H. Daniel, Ergeb. exakt. Naturw. 82, 118 (1959). Schupp, H. Daniel, G. W. Eakins, and E. N. Jensen, Phys. Rev. 120, 189

98G.

(1960). 98 T. A. Eastwood, J. P. Butler, M. J. Cabell, H. G. Jackson, R. P. Schumann, F. M. Rourke, and T. L. Collins, Phys. Rev. 107, 1635 (1957). 100 J. Scobie and G . M. Lewis, Phil. Mag. [8] 2, 1089 (1967). 101 R. W. P. Drever, A. Moljk, and S. C. Curran, Nuclear Indr. 1,41 (1957).

2.6.

DETERMINATION OF LIFE-TIME

299

initial activity by a certain experimentally determined factor. Age determinations are of great importance for the study of the history of the earth. According to Eq. (2.6.1.2) the age t is given by (2.6.1.11)

If there is only one daughter product consisting of D nuclei, Eq. (2.6.1.11) can be rewritten t = -log 1

x

(I

+);

(2.6.1.12)

P = N being the number of parent nuclei still existing a t present, i.e., at the time t. In the case of the potassium-argon method with its two daughter products Ca40and A40 one has instead of Eq. (2.6.1.12)

( ;,g:)

t=Xlog 1 + - - ,

(2.6.1.13)

A,/X is the branching ratio for electron capture to A40, A, being the partial decay constant for this branch. A40/K40 means the mole ratio between A40and K40at present. Equations (2.6.1.11) through (2.6.1.13) hold only if there was no loss of material. Such a loss is particularly likely if the daughter product is a noble gas. If necessary, a correction must be applied.lo2 In many cases not the whole amount of daughter substance contained in a sample has been produced by radioactive decay of the mother substance. Help is given by the mass-spectrometric analysis. For example, Pb206is the final product of UZ38decay but is also contained in primordial lead. From the total PbZo6 amount determined in a Uz98-Pbzo6age determination, the fraction corresponding to the primordial lead must be subtracted. This fraction is found with the help of the PbZo4,PbZo7,and Pb208abundances. It is assumed here that these isotopes are of primordial origin. This is not quite right: Pb207is also produced by the decay of U236(abundance 0.72%) and its daughter products. If the sample also contains thorium it will contain as well the final product of the ThZS2 decay, PbZo8. Of the various age determination methods, only two shall be treated here, the potassium-argon method for dating large ages and the radiocarbon method for dating small ages. Table I lists the more common methods for age determination. 102 H. Fechtig, W. Gentner, and J. Zahringer, Geochim. el Cosmochim. Acta 19, 70 (1960); E. G. Wrage, ibid. 28, 61 (1962);H.Fechtig, W. Gentner, and S. Kalbitzer, ibid. 86, 297 (1961).

300

2.

DETERMINATION. OF FUNDAMENTAL QUANTITIES

TABLE I. Methods for Age Determinations" Parent

End product

U23B

6rB7 Pb*"a

K40

A40

U236

PbZ07

c136 C'4 H3

NI4 He 3

RbB7

2'112

(5.3 (4.51 (1.25 (7.14 (3.1 5568 12.28

+ Ca40

A36

of parent

Determined age

f O.2)1O1Oyears f 0.02)108 years f 0.05)108 years f 0.11)108 years f 0.3)106 years

Formation age Formation age Formation age Formation age Exposure age

f 30years f 0.03 years

Exposure age

0 The half-life values quoted are mean values computed from the information The stated errors are standard available in the literature (except in the case of 0"). deviations only, computed from the deviations from the mean (except CI4).Formation age, under ideal conditions, means the time elapsed since the last separation of parent and end product. Exposure age (for meteorites) means the time the sample (or its surface) has been exposed t o cosmic rays. The CI4method determines the time that the organic material has been dead.

Figure 9 shows a potassium-argon setup in use at the Heidelberg laboratory; in practice there is a second melting and purifying part which is also connected with the mass spectrometer. Before being placed in the melting oven, the sample is mechanically crushed to grains of 0.6 to 1 mm in diameter. The argon determination proceeds in three steps: extraction, purification of the extracted gases, and isotopic analysis. The chamber of the static high-vacuum mass spectrometer is made of glass. It can be heated by a sinkable oven. Transportation of the argon into the purifying part is done by adsorption on charcoal. The potassium content of the sample is measured by isotopic dilution with the same setup, or with a spectrophotometer, or by activation analysis. copillary

oven

-=+=7r - - - - - - - - - - -1

cal. gas

Ca

t-

melting

--i

Cu-CuO

purifying

Ca

1-

I

ion source

-

mass spectrometer

I

i-

FIG.9. Potassium-argon setup. rf radio frequency, M o Mo-crucible, T P Toepler pump.

2.6.

DETERMINATION OF LIFE-TIME

30 1

I n radiocarbon dating, the low activity still present in the sample is to be measured [Eq. (2.6.1.11)]. This is usually done with the help of a proportional counter but Geiger counters and, less frequently, scintillation counters are also used. In order to reduce the background, the counter is shielded b y absorbing material and anti-coincidence counters.

El

75 O h PARAFFIN 2 5 O/o BORIC ACID

GEIGER COUNTER

FIG.10. Proportional counter for radiocarbon dating. A vertical section of the whole shield perpendicular t o the axis of the proportional counter. Wire: stainless steel, 0.05 mm in diameter, sensitive length 300 mm. Inner diameter of the counter: 45 mm. Background: 0.98 cpm a t 20°C and 3 a t m Co2. Modern carbon: 9.0 cpm a t 20°C and 3 atm COZ. Maximum age: 38,000 years a t 3 atm COZwith t h e usual criterion of net counting rate = 4u, where u is the statistical deviation a t 48 hours of counting.

Figure 10 shows a counter arrangement. I n most cases, the C14 is added as a gas to the counting gas or forms the counting gas itself, notwithstanding the fact that Libby, who introduced and developed radiocarbon dating, worked with a solid source covering the walls of a Geiger counter. An explicit description of a C14 dating station using the COz proportionally counting technique is given by Olsson. lo3 Delaney and McAulayeodescribe a liquid-scintillator setup for radiocarbon dating. A general outline of the method, together with his Geiger counter technique, is given by Libby.104 The techniques used for sample preparation 1 03 104

I. Olsson, Arkiv Fysik 13, 37 (1958). W. F. Libby, “Radiocarbon Dating,” 2nd ed., Univ. Chicago Press, Chicago, 1955.

302

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

vary depending on the method chosen for determining the C14 activity but, as a general ruie, all the measurements are performed in comparison to a standard. As standard, wood about a hundred years old is preferred because it is easy to date with the help of the tree rings and there is no danger of a contamination. Such contaminations are found in very recent organic material as a consequence of industrial coal consumption (contamination with inactive carbon) and atomic bomb tests (contamination with C14). By determining the C14 content in comparison with a standard, all the uncertainties arising from wall effects in the counter, from counting losses because of the discrimination bias, etc. are eliminated. A serious source of error consists in contaminating the sample with recent organic carbon, An uncertainty arises also from possible variations, locally as well as in time, of the C14 content of the atmosphere.lo6 TABLE11. Some Important Half-Lives (Other Than Those Given in Table I)" n 014

Na2* NaZ4 Pal Srao Iiai

11.7 71.0 2.59 14.98 14.29 26.3 8.09

f 0 . 3 min rt 0 . 2 s e c f 0 . 0 3 years f 0 . 0 2 hours f 0 . 0 6 days 1.7years -fr 0 . 0 2 dayys

*

CS"'

PbS10 Bia1O((RaE) PbPl3

Rn"$ Rez2% Th232

28.5 f 1 . 2 years 21.3 f 0 . 8 years 5.01 f 0.01 days 10.64 f 0.01 hours 3.8242 f 0.0007 days 1607 f 11 years (1.41 f O.O1)lO1Oyears

For the computation cf. Table I footnote 8. Exceptions are here n and Na22.

For routine measurements, samples containing about 2-3 gm C are usually taken. It is, however, possible to work with smaller amounts, e.g., of the order of 0.5 gm C. Extended information on age determinations is found in references 104 and 106. When the C14 half-life itself is measured, corrections for the counter efficiency are to be applied. I n the case of a gaseous activity, the end effect is eliminated by taking two counters of the same diameter but of different length, and the wall effect can be found similarly with two counters of the same length but of different diameter or by a computation.104 Table I1 lists some important half-lives. 106 E. H. Willis, H. Tauber, and K. 0. Miinnich, Am. J . Sci. Radiocarbon. Suppl. 2, 141 (1960). 106L. T. Aldrich and G. W. Wetherill, Ann. Rev. Nucl. Sci. 8, 257 (1958); L. H. Ahrens, Repts. P r o p . in Phys. 19, 80 (1956); T. P . Kohman and N. Saito, Ann. Rev. Nuclear Sci. 4, 401 (1954); A. A. Smales and L. R. Wagner, eds., "Methods in Geochemistry." Interscience, New York, 1960; H. E. Guess, Ann. Rev. Nuclear Sci. 8,243 (1958).

2.6.

DETERMINATION OF LIFE-TIME

303

2.6.2. Short Lifetimes* 2.6.2.1. Introduction. An excited state of a nucleus which is not subject to strong external perturbations will in general decay according to an exponential law N = Noe-‘/r where T is the mean life of the state. The mean life is a n unambiguous property of the state and can be measured even when not all modes of decay are known or detectable. If there is more than one mode of decay, we can write 1/r =

x

=

xi a

where the A, are the rates of decay by each of the modes. The measured mean life is the same, regardless of the mode detected, provided the observed decay is exponential. In this section we deal with the meassec and primarily urement of short lifetimes, i.e., generally 7 < T < sec. Most of the states with lifetimes in this range, and still long enough t o be measured, decay by electromagnetic processes, i.e., gamma-ray emission or internal conversion and pair conversion. There are a few beta decays with lifetimes in the millisecond range among the light nuclei, e.g. BX2.Shorter beta-decay periods cannot be expected because the decay energies required would in general imply instability against nucleon emission. Alpha-particle decay of heavy nuclei can be expected to occur with periods in the range of interest. In fact, the lifesec) and ThC’ (3.0 X lO-’sec) were the first time of RaC’ (1.64 X very short lifetimes measured. Most direct measurements of r proceed by measuring the individual lifetimes T of a large number of nuclei and determining the mean value from their exponential distribution. Sometimes the mean of the measured values is determined without observing the exponential distribution itself. Such a direct measurement of the mean is the most obvious procedure, and in principle the most efficient, b ut in many cases it increases the magnitude of systematic errors. In some cases the measuring device itself takes the average and the individual values of T are not observed but only r. The lifetime T of a n individual nuclear state is the interval between its “birth” or formation and its “death” or decay. The second of these is always marked by the emission of some radiation which can be detected. The formation may be marked by emission or absorption of a particle which can also be timed or detected. Sometimes this radiation is not the one immediately leading to the state in question but one which precedes the formation by a negligible

* Sect,ion 2.6.2 is by M. Oeutsch.

304

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DETERMINATION OF FUNDAMENTAL QUANTITIES

time interval. This situation is fairly common because usually the states of interest lie quite low in the nuclear level scheme. For example the 0.50 Mev state in F17may be formed in the reaction Ol6(p,y)Fl7through a cascade of y-rays. The mean life of this state has been measured : (1) by detecting the 1.5 Mev gamma ray leading to the formation of the 0.5 Mev level and the 0.5 Mev gamma ray emitted in its decay and measuring the time delay between them; ( 2 ) by timing the bombarding proton and detecting the decay gamma ray. The hold-up in the gamma ray cascade is negligible compared with the 4 X 10-lo sec mean life of the 0.5 Mev state. Nuclear states formed after radioactive beta decay are frequently reached by a cascade of gamma rays through very short-lived states. For example the first excited state of deformed nuclei in the rare-earth region generally has an excitation energy below 0.1 Mev and a mean life of the order of sec. When it is formed by beta decay to a highly excited state followed by several gamma rays, the mean life is usually measured by detecting and timing the beta ray and a conversion electron from the decay transition, disregarding the delay due to the intervening gamma transitions. Of course i t must be proved in each case that this procedure is justified. Sometimes the detected radiation is not of nuclear but of secondary atomic origin. For example K X-rays following orbital electron capture may be used to time the formation of a state. The lifetime of the atomic levels, of the order of lO-'4 sec, may be neglected. Sometimes the 0.51 Mev annihilation radiation is detected to time the emission of positrons. This procedure must be used with caution since the mean time for annihilation ranges from about 2 X sec in some metals to about sec in air. Another possible source of error may arise when the radiations forming and depopulating the excited state are observed in a definite angular relationship with each other, as is frequently the case. The two radiations usually show an anisotropic angular correlation if the nuclear orientation is preserved during the intervening lifetime (cf. Section 2.4.2). The ambient molecular fields in the source material cause nuclear reorientation with a relaxation time which may be between sec and several minutes, depending on the nuclei and surroundings. The angular correlation of the radiations changes from its initial anisotropy to isotropy in a, time of the order of the nuclear spin relaxation time. If this time is comparable with the nuclear lifetime, the efficiency for decay varies significantly, distorting the apparent decay rate. Similar distortion can also occur in some circumstances' due to nuclear resonance absorption in the source or surrounding material. 1

F. J. Lynch, R. E. Holland, and M. Hamermesh, Phys. Rev. 120, 513 (1960).

2.6.

DETERMINATION OF LIFE-TIME

305

All of these interfering effects connected with the nuclear processes appear only rarely and under special circumstances. They usually cause only minor errors compared with those introduced by the technical difficulties of timing and detection. Indirect measurements of specific decay rates frequently also permit a determination of T, e.g., by measuring the cross section of an inverse process which is closely related to the decay. The most important example of this is Coulomb excitation of states which decay by y-ray emission and internal conversion. The same is true for determination of the width of the state which is connected with the mean life by the relation

r

=

These and similar indirect means of deducing lifetimes lie outside our subject. Any direct measurement of the lifetime T depends basically on a clock which starts at the moment of formation and stops a t the moment of decay. The phase or reading of the clock is examined to find T. Sometimes two clocks are involved, one starting at formation, the other a t decay. The phase difference between the two clocks is then measured. The rate of some clocks, such 8s oscillators, may be known precisely. Other clocks depending on individual atomic processes may be subject to statistical fluctuations. In practice even “cl&ssical”clocks have statistical uncertainties. A great variety of processes have been used as clocks. Most of them can be classified among a few general types. One distinction is provided by the manner in which the clock is started and stopped. Broadly speaking there are two types of clock: those attached to the nucleus itself and those under control of the experimenter. Of the first type we consider primarily the position or velocity of the nucleus after formation of the excited state. These can be used as clocks when there is sufficient information concerning nuclear recoil in the formation process and if the change of the recoil velocity due to interaction with the surrounding medium is sufficiently well understood. The second type of clock is usually an electronic device although there are a few examples of mechanical clocks. The starting and reading of the clock, especially an electronic clock, may be accomplished in two principal manners. Either a signal is derived from the process of formation or decay -a “birth” or “death” announcement-or else the clock is used to control the physical process by which the state is formed or its decay detected. We may call the first type the detector method, the second the shutter method. The greatest volume of results has been obtained by means of electronic clocks, using the detector method. When both the

306

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DETERMINATION OF FUNDAMENTAL QUANTITIES

formation and the decay are detected it is usuaIly called a coincidence method. When formation is gated by an electronic shutter we usually speak of a pulsed-beam experiment. A1though the distinction between the two procedures is not always perfectly sharp, the characteristics of the various schemes are sufficiently distinct to make this division useful for discussion. The classification of the main methods of lifetime measurement according to this scheme is shown in Table I.* TABLE I. Classification of Timing Methods “External” clocks controlled by the experimenter

_____-

Mechanical clocks

Electronic clocks

Detection of for- “Delayed coincidence” methods mation, and Time-to-pulse height converters Oscilloscope trace photography decay Chronotron Gating of forma- Pulsed accelerator beame t ion ,detect ion of decay Gating of forma- Electromagnetic deflection of decay particles with pulsed tion and deaccelerator or with coincicay dence detection

Measurement of recoil distance with coincidence detection Recoil distance measurement using Doppler shift t o detect stopping Recoil distance measurement, using collimators Mechanical shutters, rotating wheels, etc.

“Atomic” clocks depending on rate of extranuclear processes Doppler shift from recoil nuclei in various media Competition of nuclear decay with electronic transitions: Double-vacancy X-ray lines Positron “lines” from pair formation in ionized atoms

2.6.2.2. Electronic Timing. 2.6.2.2.1. DETECTORS FOR ELECTRONIC TIMING. The detector used with an electronic clock must produce a signal

capable of turning the clock on or off, maintaining the timing of the arrival of a ray from the source with a precision which is as least comparable with the lifetime to be measured. A constant time delay between the event and the signal does not cause any inconvenience. In principle it is also not necessary that the signal be very “fast,” i.e., that it contain very high frequency components, since the phase of a slow signal can still be determined with good precision. It is only necessary that this phase bear a fixed relation to the arrival time, not varying from event to event. In practice a fast signal is usually desirable to assure stable operation. * See also Vol. 2, Section 10.6.2 on atomic clocks, and various other chapters on electronic clocks, oscilloscopes, and other devices.

2.6.

DETERMINATION OF LIFE-TIME

307

Virtually all detectors depend on the collection of charge produced, directly or indirectly, by the passage of an ionizing particle through matter. Since this charge usually represents only a small number of electrons, fluctuations in this number are important and better time resolution can sometimes be obtained with a detector in which a larger amount of charge is collected relatively slowly. Apart from its timing characteristics, a detector must have good sensitivity and, in many applications, good selectivity for the relevant radiations. Other important considerations may be the ability to function a t high counting rates, suitable size, and reasonable simplicity. By far the most widely used detectors for lifetime measurements are scintillation counters employing photomultipliers. The most important, properties of these detectors are described in Chapter 1.4 of this volume.* The following is a brief summary of these. The scintillator transforms part of the energy loss of an ionizing particle into light; the photocathode converts part of the photons into electrons; the electron multiplier collects and accelerates the photoelectrons and amplifies the resulting current by repeated secondary emission. In each of these steps, except the photoelectric conversion, there occurs a time delay, subject to statistical fluctuations. Therefore timing information is lost. The statistical fluctuations are of two kinds: time fluctuations and efficiency fluctuations. The latter indirectly cause timing errors because of the manner in which the information is presented to the clock. The following are the main sources of time fluctuations: (1) Duration of the scintillation. After excitation, the scintillator begins to emit light without measurable delay, with an intensity which, initially, decays approximately according to a n exponential law. The period is of the order of a few nanoseconds for the fast organic phosphors, about 300 nsec for activated NaI, and about 30 nsec for pure N a I a t liquid nitrogen temperature. (2) Light collection. The time required for the scintillation light to reach the photocathode depends on the distance traversed. Along the line between the source of nuclear radiation and the photocathode the time At spread introduced in the distance d is (2.6.2.1)

where V L is the light velocity in the phosphor and V R the velocity of the radiation. At right angles to this line the spread is d/vL. With the usual detector dimensions and a reasonable geometric arrangement the spread sec. This may be increased by is rarely more than about 2 X multiple reflections in the phosphor and photomultiplier. * Ser also Vol. 2, Chapter 11.1.

308

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DETERMINATION OF FUNDAMENTAL QUANTITIES

(3) Transit time. In most of the widely used end-window photomultipliers the average time between the formation of a photoelectron and the arrival of the electron avalanche at the anode is about 50 nsec. Only the fluctuations about this mean transit time cause loss of timing information. Such fluctuations arise from differences in path length and, to a lesser degree, in electron velocity. By far the most important fluctuations occur in the region between the cathode and the first multiplying dynode because of the relatively long distance and weak electric field. Typical variations of transit time as a function of the point of origin of the photoelectron are about 3 nsec. This is reduced by as much as a factor ten when only the central portion of the photocathode is used. Ionizing particles, especially heavy ions, produce a sufficient number of secondary electrons from a solid surface to permit detection in an electron multiplier without the intervention of a phosphor and photocathode. Such devices eliminate most of the time fluctuations described. They have been used in time-of-flight applications but not in lifetime measurements. Gas ionization detectors were used universally before scintillation detectors became available. They are hardly useful for the measurement of lifetimes below about 100 nsec. Parallel-plate chambers yield a current pulse which starts immediately upon passage of the ionizing radiation. The current is however only of the order of 10-lo to lo-* amp and timing information is lost in the amplifier noise. Proportional counters and G-M counters avoid this difficulty. The geometry of such counters is necessarily such that transit time fluctuations in the migration of the primary electrons to the multiplying region are of the order of 0.1 psec. Parallel-plate spark counters are free of this difficulty and can, indeed, be used for nanosecond timing. Their construction and operation are rather awkward and they suffer from a long recovery time. Solid semiconductor ionization detectors are analogous to the parallelplate ionization chambers. Because of their smaller dimensions they yield currents a t least one hundred times larger than gas chambers. The noise problem has, nevertheless, still prevented their application to very fast timing. Devices of this type using collision multiplication may become useful in the future. 2.6.2.2.2. ELECTRONIC CLOCKS.The electronic device which most nearly resembles the usual idea of a clock consists of a gated oscillator* which is turned on by the formation signal and turned off by the decay (Fig. la). The number of oscillator periods can be counted by a scaling circuit which is examined after each event. Modern high-speed scalers could, in principle, allow this method to be used for periods of the order * On oscillators in general, see Vol. 2, Chapter 6.3.

2.6.

DETERMINATION

OF LIFE-TIME

309

of sec. It has been used, in this form, only for times in the microsecond range or longerl2ssprimarily in neutron time-of-flight measurements. When the resolution obtained by counting the nearest full cycle is insufficient, the phase of a single cycle may be examined. Most electronic

/---

c

T

-,‘

b

I

/

on

.. \

off

.’..-,

,

I

-T-

C

4 off

on

d

-T-

On

off

FIQ.1. Modes of operation of various electronic clocks.

clock schemes can be interpreted in this manner.4 Other discussions, from different points of view, have been given, e.g., by Bay,6 Baldinger and Franzen16 and deBenedetti and F i n d l e ~ .In ~ the simplest scheme, an R. A. Swanson, Rev. Sci. Znstr. 31, 149 (1960). * R . A. Reiter, T. A. Romanowski, R. B. Sutton, and B . G. Chidley, Phys. Rev. Letters 6, 22 (1960). E . Gatti and V. Svelto, Nuclear Znstr. 4 , 189 (1959). 6 2 . Bay, Nucleonics 14, No. 4, 56 (1956); Z. Bay, V. P. Henri, and H. Kanner, Phys. Rev. 100, 1197 (1955). E. Baldinger and W. Franzen, Advances in Electronics and Electron Physics 8, 289 (1956). S . deBenedctti and R. W. Findley, i n “Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 45. Springer, Berlin, 1958.

‘

310

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

“on” signal starts the oscillator which is turned off, before completing even one quarter cycle, by an “off” signal (Fig. Ib). The oscillator phase may be measured by the voltage reached. If the oscillator period is sufficiently long compared with the time interval measured, this output signal is nearly linear with T . For infinite period the oscillator becomes a constant current generator (Pig. Ic). This is the idealized mode of operation of most coincidence circuits and time-to-height converters. The distribution in output pulse heights can be conveniently sorted by a pulse-height analyzer. The triggered sweep circuit of a fast oscilloscope is such a constant current generator. If the sweep is started by the “on” signal, the “off” signal may be applied to the vertical plates or to the intensifier. Its position on the screen measures the time interval T.8s9This scheme is of value primarily when other characteristics of the signals are to be recorded together with T , e.g. the pulse height, and when the total number of events is relatively small. In practice, the clock scheme is usually somewhat modified. If the signal is derived from a detector which starts the clock unconditionally, the output will present a large number of meaningless long time intervals ocurring a t the rate of single counts instead of the rate of relevant time intervals. This is sometimes acceptable with shutter schemes in which the turn-on signal is provided by an oscillator, but not in coincidence schemes in which the signals occur a t random in time. Therefore, a gating signal is provided which allows the clock to be turned on only when an “off’) signal occurs within the interesting time interval. This can be achieved, e.g., by providing a relatively slow coincidence circuit as a gate and delaying both fast signals to the clock until the gate has time to operate. For example, in the oscilloscope presentation described above, the sweep trigger is derived from a gate circuit and both the “on” and “off” signals are displayed after a suitable delay. The distance between them is then used to measure T. By far the most widely used method to select only intervals in the desired range is the “overlap” method (Fig. Id). Here both the formation and decay signals must have a sharply defined duration. The clock current flows only during the interval when both signals are present. Therefore, the clock indicates the interval between the onset of the second pulse and the end of the first, i.e., the overlap time of the two pulses. The two pulses of well-defined duration are usually achieved by delay-line shaping of a step-function signal. We note also that the clock is now symmetric in the sense that either signal may be considered as 8 9

R. Hofstadter and J. A. McIntyre, Phys. Rev. 78, 24 (1950). K. G. Malmfors, J. Kjellman, and A. Nilsson, Nuclear Instr. I, 186 (1957).

2.6.

DETERMIXATION OF LIFE-TIME

311

the “on” or “off” signal. The symmetric arrangement is unable to distinguish intervals T from - T. In many experimental situations the nature of the detectors assures that in an event of interest the earlier signal always appears a t the same input. Accidental intervals may, however, appear in either order. When this constitutes a disadvantage, the same pair of signals may be presented to a second, similar circuit with a relative time delay such that only events occurring in the desired order will produce timing signals in both c i r c ~ i t s . ’ ~ Figure 2 illustrates an asymmetric clock arrangement of the type used in accelerator-chopper experiments’O in the manner illustrated in Fig. lc. The constant current generator is the tube V-4. It is turned on by a negative signal a t input 1 and turned off a t input 2. Similar circuits have been built with transist>orsreplacing the vacuum tubes.” Symmetric “overlap” circuits are frequently divided into two types,6 nonlinear addition or parallel-switch circuits and multiplication or series-switch circuits. The first type is basically derived from the Rossi coincidence circuit. The two pulses are added and the added signal is presented to a discriminator-integrat,or circuit. Many combinations of circuit elements have been used for this operntion.6,6A widely used scheme is addition in the common plate circuit of a pair of pentodes followed by a series12 or parallel13 diode discriminator or a pentode dis~ r i m i n a t o r .The ’ ~ pentodes may be replaced by transistors or by currentbiased diodes.16 Addition can also be performed a t the common cathode or common emitter.16 The commonest form of series switch is represented by the type 6BN6 beam switching tube in which current can pass to a target anode only when b0t.h of two control grids receive a positive pulse. Series switches can also be made of two transistors in series. Figure 3 illustrates a circuit due t’o Green and Bell.17 The two gate pulses are formed by shorted cables a t the two pentode plates and present>edto the series-switch circuit labeled “converter” which is the basic clock c,ircuit. Through a pair of taps on the cables, added pulses also appear at. the discriminator diode labeled “coincidence.” This sccond

+

10

L. E. Beghian, G. H. R. Kegel, and R. P. Scharenberg, Rev. Sci. fn,str. 29, 753

( 1 958).

G . Culligan and N . H. Lipman, Rev. Sci. Instr. 31, 1309 (1960). R. E:. Bell, R. L. Graham, and H. E. Petch, Cuan. J . I’liys. 30, 35 (1952). I 3 R. Garwin, Rev. Sci. Znstr. 24, 618 (1953). 1 4 S. Gorodetzky, R. Richert, R. Manquenouille, and A . Knipper, Nuclear fnstr. 7, 50 (1960). 15 S. deBenedetti and H. J. Richings, Rev. Sci. Znstr. 23, 37 (1952). 16 P. C. Simms, Rev. Sci. Znstr. 32, 894 (1961). l i R. E. Green and R. E. Bell, N,uclea,r Instr. 8, 126 (1958). 11

l2

2.6.

DETERMINATION OF LIFE-TIME

313

+

clock circuit serves to distinguish T from - T intervals as discussed above. Figure 4 shows the complete arrangement with a slow triplecoincidence gate circuit which permits the time interval to be recorded only when both signals have appropriate pulse heights as selected by the

R

Fro. 3. An overlap-type series-switch time-to-height converter with an auxiliary nonlinear-adding coincidence circuit to distinguish intervals +T from - T (from ref. 17).

circuits labeled A D A and A D s and when the second clock signal is present in ADc. Measurement of the accumulated change is equivalent to a phase measurement only if the amplitude of the oscillation (Fig. lb) is known. This can be achieved only if the start (and stop) signals are completely

314

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

I

I

1

I 7

I

I

I

I

- 1

\I/

i T I AA I- -I q J . I

I

Triple

I

Gate

Gating pulse

I

I

Kicksorter

I

Coinc.

I

I

FIG.4. A complete time interval sortcr. The 6BN6 converter and the fast coincidence circuit are shown in Fig. 3. A D A and A D B are single-channel pulse-height selectors (from ref. 17).

uniform. When detectors are used to produce these signals this is never true.6v1sThe (re1at)ivelysmall) loss in timing precision is accepted because of the simplicity of the procedure. A very elegant scheme for measuring the true phase of a gated or shock-excited oscillator has been introduced by Gatti (Fig. 5).18,1eThe “on” signal starts an oscillator with frequency v 1 or is known to occur a t a definite phase of this oscillator. The “off” signal does not block this oscillator as in the schemes described before but instead starts Av. The two oscillator anot,her oscillator with a frequency v 2 = v1

+

I*

C. Cottini and E. Gatti, Nuovo ciniento 4, 156, 1550 (1956).

19

R.L. Chase and W. A. Higginbotham, Rev. Sci. Inslr. 28, 448 (1957).

't

316

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

signals are added. The phase of the resulting beat signal with frequency v 2 - vl, is the same as the phase difference between the oscillators. The phase of the much slower beat signal can be measured, for example, by determining the time of the first zero crossing, that is the first half period. This then measures the initial time interval T modulo l / v l . This scheme is applicable particularly in the case of lifetime measurements with pulsed accelerators (shutter scheme) where the frequency v1 can be derived directly from the oscillator of the accelerator. The phase of this continuous wave train is more easily fixed than that of a signal derived from a particle detector. Most recent measurements utilize electronic clock schemes of the types described with a multichannel recording device so that the entire time interval spectrum of interest is recorded simultaneously. Many older measurements utilize a circuit only as null indicators, i.e., as coincidence circuits of short revolving time. Schemes of this type are called delayed-coincidence measurements. They usually use a circuit of the overlap type followed by a discriminator which accepts a signal only when the output pulse has nearly its maximum possible value, e.g., T is nearly zero. The time interval spectrum is then obtained by delaying one of the signals by accurately known amounts T’ and measuring the frequency of coincidences as a function of T’. The device used to introduce the delayed T’ is effectively the clock for this type of measurement. For T‘ larger than sec, active circuits such as univibrators are sometimes used. For shorter times almost invariably measured lengths of coaxial cables provide the delay. The disadvantage of this sequential method of obtaining data for a decay curve is obvious. One advantage of the method is the fact that the coincidence circuit null-indicating clock functions in exactly the same manner for all intervals T , so that nonlinearity of response does not affect the precision of the experiment. A rather elaborate and very successful scheme is represented by the so-called chronotronZ0which does not quite fit our classification. In this circuit a series of overlap or nonlinear addition circuits are placed along a pair of delay lines such that the two signals arrive a t the various circuits with different relative delays. The circuit giving the largest signal is the one a t which the two signals arrived most nearly simultaneously. If all the pulse heights from the various meeting points are recorded, e.g., on an oscilloscope, it is possible to interpolate between the possible discrete time intervals.21Another modification which extends the number of measur2o I. A. D. Lewis and F. H. Wells, “Millimicrosecond Pulse Techniques,” p. 269. McGraw-Hill, New York, 1954. W. Grismore and W. C. Parkinson, Rev. Sci. Znstr. 28, 245 (1957).

2.6.

DETERMINATION O F LIFE-TIME

317

able time intervals is the “vernier chronotron”22 in which the pulses circulate repeatedly through the two delay Iines. The manner in which the signal from the detector is used to turn the clock on or off is decisive for the resolution of the devices. The full timing information is contained in the entire pulse derived from the detector however deteriorated its frequency response, although the later part contains usually less information than the initial part. Ideally, then, the clock should examine the entire pulse and analyze it in such a manner that all of the information is used with appropriate statistical weight. Of the methods in use at present, only photography of oscilloscope traces could lend itself to such a procedure, since it preserves the information long enough to permit examination and evaluation by an electronic computer. The method has not yet been attempted in the measurement of lifetimes of nuclear states. Most of the electronic devices of the time-to-height converter or coincidence overlap types used with scintillation detectors utilize only one or two pieces of information. The most widely used scheme, which may be termed the limiter method, utilizes the time a t which the pulse reaches a certain level. In practice this is achieved by the use of a “limiter” circuit which reaches its maximum current when the signal reaches the desired IeveI. The most widespread limiter circuit is a conducting pentode which is cut off by the signal from the photomultiplier. Other limiters employ transistorsz3 which may be turned on to saturation or turned off. Recently, negative resistance elements, especially tunnel diode@ and avalanche transistors, have come into use for this purpose. Post and Schiff26 have analyzed the rise of an idealized charge pulse which would be derived from a photomultiplier without transit time or gain fluctuations. The charge collected a t the anode would be an irregular staircase, each step corresponding to the arrival of the charge due to a single photoelectron at the cathode. For pulses yielding, on the average, R photoelectrons, the time at which the level corresponding to the Q’th photoelectron is reached, is given by (2.6.2.2)

where r p is the period of the phosphor decay, assumed to be exponential.

** H. W.

Lefevre and J. T. Russell, I R E Trans. on Nuclear Sn’. “3,146 (1958). R. M. Sugarman a n d W . A. Higginbotham, Intern. Conf. Instr. for High Energy Phys. Berkefey I960 p. 54. Interscience, New York, 1961. a4 G . C . Bret and E. F. Shrader, Nuclear Znstr. 13, 177 (1961). *I R. F. Post and L. I. Schiff, Phys. Rev. 80, 1113 (1950). *a

318

2.

DETERMINhTION OF FUNDAMENTAL QUANTITIES

The standard deviation of this time, At, is given by

where R >> 1, R >> Q. It would, therefore, appear in this approximation that it is always advantageous to set the limit8erlevel to correspond to the lowest practical value of collected charge, ideally to the first photoelectron. The same article gives similar formulas for other phosphor decay laws and for less restrictive conditions. Equation (2.6.2.3) shows that the time resolution is an equally strong function of the light yield and photoelectric efficiency as of the decay time of the scintillator. The statistical fluctuations in light yield and photoelectric yield are included in Eq. (2.6.2.3) but not the pulse height spread due to different particle energies and varying efficiency of light collection from different parts of the scintillator. The signal time spread introduced by these effectsz6is

It is, therefore, customary to determine the height of each pulsc as a second parameter besides the time of t,he initial rise. I n the great majority of actual experiments this is done b y seIecting in a separate discriminator circuit only pulses falling within a narrow pulse height range and accepting through a gating arrangement only the corresponding time pulse (slow-fast scheme) (Fig. 4). This procedure is satisfactory if a large fraction of the relevant pulses falls into a narrow pulse height window. This is frequently true, for example, when counting internal conversion electrons or a-rays. It is also true for low-energy 7-rays counted with a NaI scintillator. In some other cases, for example with a continuous p spectrum or with a complex y-ray spectrum, the loss in counting rate may be too severe and it may be desirable to measure the time spectrum for a wide range of pulse heights. One procedure is to record for each pulse the measured time interval and the pulse height (or both heights if two detectors in coincidence are used). This can be done in a suitable digital memory device. The scheme involves rather formidable circuitry even with digital storage on magnetic tape.27 It has been used in connection with neutron time-of-flight experiments but, to the author’s knowledge, not for lifetime measurements. An alternative scheme is represented by analog rather than digital storage of the 26 27

E. Bashandy, Nuclear Znstr. 6, 289 (1960). C. C. Rockwood and N. G. Straws, Rev. Sci. Znsfr. S2, 1211 (1961).

2.6.

DETERMINATION OF LIFE-TIME

319

information, and this can be done in oscilloscope trace photography. In many cases it is sufficient to reduce the additional time uncertainty due to pulse height spreads only in first approximation. This can be done by a simple analog method. The time interval spectrum is presented as a pulse spectrum by a time-to-height converter. By adding to (or subtracting from) this converted pulse another pulse proportional to the total pulse height, a linear dependence of the signal time on pulse height can be compensated. Sometimes i t is possible to reduce the pulse height spread b y appropriate design of the detector. For example, a fairly uniform pulse height can he obtained even with a continuous beta-ray spectrum by using a scintillator much thinner than the range of the relevant beta rays. Gatti4s28and otherse have pointed out that, in fact, the transit time spreads and other factors make the representation of the signal as a staircase quite unrealistic. I n the case of scintillation detectors, these factors make the pulse start more slowly than corresponds to the exponential decay of the scintillator. It then becomes advantageous to derive the signal from a larger number of initial photoelectrons than would be indicated by the formula of Post and Schiff. I n all electronic clocks which may be considered gated oscillators, the effective turn-on or turn-off time (“machine time”) is determined by the centroid of the signal pulse used. This is also true for the linearly charging time-to-height converters. For example, when a limiter is used i t is the centroid of th a t part of the current which derives the limiter to saturation. When the entire scintillation pulse is used the standard deviation of this machine time is At = T = / R ~assuming /~ the entire spread to be due to the statistical fluctuations of the scintillations and the photoelectric yield. This spread is larger roughly by a factor R1/2than th at of the first photoelectron in the idealized case of Post and Schiff. On the other hand, the time of the centroid is independent of pulse height or number of photoelectrons. Thus, when the dominant factor in the resolution time is the variation in pulse height, the best procedure is to use the centroid of the entire pulse t o yield the machine time. The radio-frequency vernier circuit of Cottini and Gatti18 is specially suited to this. In many cases it has been found preferable to use a method which makes use of the fact that the mode or peak of the current pulse from a fast organic scintillator is not very far from the mean or centroid. The current pulse is differentiated and the time of zero crossing is used as clock signal. When the dominant factor in the resolution time is the decay time of the phosphor, the first photoelectron remains the best signal for the clock. This situation exists for small but uniform pulses detected with a 2*

S. Colombo, E. Gatti, and M. Pignatello, Nuovo cimenlo 6, 1738 (1957).

320

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

relatively slow scintillator such as NaI (Tl). I n most practical cases the three factors of scintillator decay, transit time spread, and pulse height variation make comparable contributions to the resolution time. A detailed analysis shows that, when only the initial part of the pulse is used, the variance of its centroid shows a flat minimum for a number of accepted photoelectrons, which depends on the specific conditions of the experiment but is in most practical cases of the order 0.1-0.2 of the total number R for fast organic scintillators and much smaller for NaI (TI), provided the pulse height does not vary over a large factor from one pulse to the other. The problems associated with other types of detectors are similar to those discussed above. In gas ionization counters the dominant factor is the transit time fluctuations. Therefore, small dimensions and strong electric fields in all parts of the counter are required for short response time unless the particle of interest always traverses the same small region of the sensitive volume.* 2.6.2.2.3. ANALYSISOF COINCIDENCE MEASUREMENTS. The deduction of the mean life r from the data obtained with a device which has a time resolution comparable with r requires an analysis similar to other measurements with devices of finite resolutions. We consider first the case of a constant source, i.e., coincidence experiments or devices gating the detector rather than the source. Figure 6 shows the counting rate versus clock time for three different resolution curves. The counting rate for sufficiently long time delays always approaches an exponential with logarithmic slope X = 1/r provided the slope of the resolution curve itself is steeper than A. The mean life can be calculated29 from the exponential part of the curve by the expression I

r = Ni(ti

- f)(log Ni - log Ni)’

(2.6.2.5)

i

The standard error in this value is (2.6.2.6)

It follows from this that a t least under these idealized conditions it is not the width of the resolution curve but its “steepness” which determines the shortest lifetimes which can be measured. If this condition is not

* Various chapters of Vol. 2 give further information on electronic problems connected with the type of measurements described in this section. 39R. H. Bacon, Am. J . Phys. 21, 428 (1953).

2.6.

DETERMINATION OF LIFE-TIME

321

satisfied or if it is satisfied only for times so long that the number of counts is prohibitively small, the mean life can still be measured moderately well under favorable conditions. It follows from the definition of the mean life that the mean shift of the solid curves in Fig. 6, with respect to the dotted curves, is exactly T . This “centroid shift” or first moment a

I,I

,____+--I

b

FIG.6. Time interval distributions due t o an exponential decay (solid curves) for three different resolution curves (dotted lines).

method6 is very attractive and has been widely applied to measure very short lifetimes. In the expression for the centroid given by

t = -i -

2 Ni

(2.6.2.7)

i

Ni is the observed number of counts in the time interval ti f At where 2 At is the separation of successive points on the time distribution curve.

322

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

The standard error in t is given by (2.6.2.S)

I t is applicable only when all counts in the solid curves are due to a sequence of radiation with the mean life 1/X, i.e., in the absence of a background of “prompt” events. Also, points with long time delays, subject to uncertainties due to accidental coincidences and low counting rates, enter with large statistical weight. The latter difficulty is avoided in a method of analysis proposed by Newton.30 The most serious limitation of the centroid shift method is the requirement of a very precise knowledge of the shape of the time resolution curve (or “prompt” curve) including the position of “0” time under the conditions of the experiment. This requirement may be partly relaxed by the use of higher moments of the two curves.b~31 A number of ingenious methods have been developed to calibrate the time resolution curve. The difficulty of this problem arises primarily from the fact that the position and shape of this curve depend on the pulse heights of the process in the “on” and “off” channels. In addition many circuits show a dependence of their behavior on counting rate and, used a t the limit of their resolution, show drifts which may occur during the time between calibration and measurement. The simplest method, frequently very adequate, is to observe the time spectrum with a source of radiation known to exhibit no nuclear time delays and producing pulses of the height encountered in the experiment. This pulse height may be selected by appropriate pulse height selector circuits. In some cases i t is possible to use a so-called “self-comparison” method. l 2 This method reverses the roles of the detectors used for the “on” and “off” signals in the lifetime measurement and compares the resulting curves with each other rather than with the “prompt” curve. It is applicable primarily when both radiations are selected in a magnetic spectrometer.26It suffers from the same limitations as the centroid shift method. Figure 7 shows an experimental result which is typical of the best measurements obtainable a t present with coincidence methods. The apparatus used was similar to that illustrated in Figs. 3 and 4, with the addition of a circuit to compensate for pulse height variation. Conditions in this case32 are very favorable because of the relatively high energy Newton, Phys. Rev. 78, 490 (1950). R. S. Weaver and R. E. Bell, Nuclear Znstr. 9, 149 (1960). 32 R. E. Bell and M. H. JPlrgeiisen, Nuclear Phys. 12, 413 (1959).

30T.D. 81

2.6.

323

DETERMINATION OF LIFE-TIME

of the gamma rays involved. The figure shows the centroid shift and the property of the normalized resolution curve to intersect the decay curve a t the point corresponding to the mean life. The decay curve has a horizontal slope a t this p ~ i n t . ~ , ~ ~ A very elegant modification of the “self-comparison” method has . scheme ~ ~ is applicable whenever recently been used b y Simms et ~ 1 This

DELAY FROM CENTER OF PROMPT CURVE

( I d l o s a c UNITS)

FIG. 7. Time distribution of counts due t o gamma rays of abcut 1 Mev energy following t h e 0.536 Mev gamma ray in the decay of Tb16e.The “prompt” curve was obtained with a Coeosource (from ref. 32).

the cascade of radiations passing through the state of interest is also accompanied by another radiation which can be detected in triple coincidence with the “on” and “off” signals. A strong calibrating source giving “prompt” radiations of the appropriate energies and of considerably greater intensity than the cascade of interest is placed in the apparatus at the same time as the source investigated. The triple coincidence signal is used to steer the clock readings from the delayed cascades to one set of channels of the time analyzer. Those readings without triple coincidence are overwhelmingly due to the calibrating source and are 33

P. C. Simms, N. Benczer-Koller, C. S. Wu, Phys. Rev. 121,

1169 (1961).

324

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

steered to another set of channels. Thus, calibration and measurement are performed simultaneously in the same apparatus. The two sources are actually physically mixed to assure identical geometrical conditions. Different cascades from the same nuclide may even be used. The problem of choosing truly identical pulse height spectra for the two sources remains, of course, in general, the same. The calibration of the clock for coincidence experiments, although straightforward in principle, presents a surprising amount of difficulty, and is the limiting factor in the precision of otherwise favorable cases. The usual time standard is the signal velocity in measured lengths of coaxial cable introduced into one of the detector signals. In practice, variations of the order of several per cent may occur, especially in cables with air as dielectric. Cables deteriorate with time and with rough handling. Additional uncertainties are introduced by connectors and by attenuation and dispersion of the signal in the cable and by connectors and terminations, Unless special precautions are taken, lifetimes based on such calibrations are uncertain by about 5 %. A more reliable standard is provided by the velocity of light (7-rays) in air. If a source of coincident ?-rays located between the detectors giving the time signals is displaced by a distance 2 along the line joining the detectors, the time interval spectrum is shifted by an amount 22/c. Detector separations much larger than detector size are usually required for suitable precision. Coincidences between the annihilation quanta emitted from a positron source are very useful for this procedure because they yield conveniently high coincidence rates even for large counter distances. For greatest precision, compensation must be made for changing counting rates if the source is displaced. It is frequently most convenient to calibrate a set of coaxial cable delay lines in terms of the displacement of a positron source, and then in turn use these cables to calibrate the time scale for radiations of different energy and for different counter arrangements. If oscilloscope display of the pulses is used as a clock, an oscillator may serve as calibration. The shortest lifetimes which can be measured observing decay curves with present coincidence methods are of the order lo-’” sec. The various methods for determining the mean delay of the entire sample of events discussed in the previous section can reduce this lower limit by about a factor 3 under favorable conditions, although a t the risk of systematic errors. Limits to lifetimes can be set as low as 5 X lo-’* sec by the centroid-shift method since a null result is much less affected by background effects due to the “prompt” radiations. Discussion of the time resolution of various experimental devices is somewhat impeded by the confusion-mostly semantic-attached to the

2.6.

325

DETERMINATION OF LIFE-TIME

term “resolving time,” which is used in a variety of ways to measure two quantities which are not very closely connected: (1) the range of clock time intervals corresponding to a single clock reading or channel, i.e., the clock channel width; we shall call this the coincidence resolving time 7 , ; (2) the range of true time intervals between two events which will correspond to a single clock channel; this we call the resolution time 7x. Although the two descriptions seem very similar, they describe two entirely different quantities. The point may best be illustrated by considering a time analyzer, e.g., a multichannel time-to-height converter with symmetric inputs. Each output channel registers all cases in which the time interval between the “start” and “stop” signals is between t and t r,. This width 7 cis usually completely under the control of the experimenter and may be made arbitrarily short. The limits of 7, are completely sharp since nothing is said about the time of occurrence of the events which caused the clock signal to occur. If the signals in the two channels occur completely a t random in time a t rates N 1 and Nz sec-’, and if N1 l ) , ~2

= .ImtUor2

(2.7.3.19)

while for a very thin sample measurement (mot 10 MeV. Generally speaking, therefore, the equations for relating nonelastic cross sections t o transmission measurements are considerably more complicated than Eq. (2.7.4.1). These equations have been worked out in great detail and with high-order corrections by Bethe, Beyster, and Carter.*O We can hardly hope to improve upon their treatment and must content ourselves with a brief description of the general approach to this type of problem. represent the transmission of neutrons which suffer no Let To = collision whatsoever, T1 the transmission of neutrons which undergo one elastic collision, T z the transmission of neutrons which undergo two elastic collisions, etc. Let us also assume 1 source neutron. Then the total transmission of neutrons which have not suffered nonelastic collisions is T = T O T I T 2 . . . Alternatively, if 10= 1 - To represents the number of first collisions, Il the number of nonelastic first collisions, Iz the number of nonelastic collisions following one elastic scattering (second nonelastic collision), I 3 the number of third nonelastic collisions, etc., we have ... (2.7.4.2) T=l-11-I2

+ + +

I1 ne - ulo

=

ne (1 - To) u-

and where

1 2

=

(2.7.4.3)

bt

66

EI(1 - Pl) ; une

(2.7.4.4) (2.7.4.5)

80 H. A. Bethe, J. R. Beyster, and R. E. Carter, J. Nuclear Energy 3 (3), 207 (1956); 8 (4), 273 (1956); 4 ( l ) , 3 (1957); 4 (2), 147 (1957).

400

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

and PI is the probability for escape (without further collision) of neutrons which have suffered 1 elastic collision. Similarly, (2.7.4.6)

E2

where

=

El(1

- PI)

(2.7.4.7)

and P 2 is the escape probability for neutrons which have suffered 2 elastic collisions. 14,16,etc. are given by similar expressions. \

\

\

\

FIG.12. Geometry for evaluating multiple-scattering effects in a spherical shell.80 For shells whose thickness is less than several simplifying assumptions are justifiable. First of all we may assume that a negligible fraction of nonelastic interactions occurs after two elastic scatterings. Secondly, in calculating the number of neutrons undergoing first elastic collisions we may assume that all the source neutrons are available for elastic scattering at every point in the shell, i.e., the probability for a n elastic collision occurring between r and r dr is redr (assuming as usual 1 source neutron). Since the second assumption implies that we are considering a larger number of first elastic and hence second nonelastic collisions than actually take place, while the first assumption eliminates high order elastic collisions and hence the nonelastic collisions which follow these, the errors introduced by the two assumptions tend to cancel.

+

2.7.

DETERMINATION OF NUCLEAR REACTIONS

40 1

Referring to Fig. 12 and invoking our first assumption, we see that the probability of nonelastic scattering following one elastic collision is, to first approximation, aney(r,e).We therefore have, (2.7.4.8) For the special case of isotropic scattering and t rz a(0,E) = (2.7.5.53) d E p an,p (E)Q’nH V P( E)Pn, The application of the time-of-flight method171 to the measurement of cross sections of the type under discussion is illustrated by Fig. 34. The scatterer is a hollow cylinder (to reduce multiple scattering) placed on the axis of the charged-particle beam. The cylinder axis is vertical and coincides with the axis of rotation of detector and shield. Different scattering angles are realized by rotating detector and shield around the axis of the scatterer. The desired resolution and the available yield, which are affected in reciprocal ways, determine the flight path. The detector can be a plastic scintillator cemented directly to the face of the photomultiplier, or a plastic phosphor optically coupled to the photomultiplier by means of a Lucite light pipe. As used in the geometry of Fig. 34, the scintillator has a more or less flat response above 0.5 MeV. I n any case, cross sections are always measured by direct comparison to the well-known n-p differential cross sections, This is accomplished by replacing the scatterer with an hydrogenous one (usually polyethylene) containing a known number of H nuclei and measuring the counting rate as a function of scattering angle and hence neutron energy. The neutrons scattered from the carbon in the polyethylene are easily resolved from those scattered by the hydrogen. If the n-p scattering angle is adjusted so that the energy of the neutrons scattered from the polyethylene is equal to the energy of the inelastic neutrons from the scatterer, a comparison of the detector response to neutrons from n-p scattering with the detector response corresponding to the unknown yield gives a direct comparison of the known to the unknown cross section. The cross section for reactions which result in the emission of neutrons of energy E a t angle 0 is then simply, (2.7.5.54) where CH and CsCrepresent, respectively, the counting rates due to H and scatterer of unknown cross section, n H and NBc are corresponding quantities for the number of nuclei in the two scatterers and u(E)]His the differential cross section for the scattering from hydrogen of the primary neutrons of energy Eo through the angle 4 where 4 is given by 17lL. Cranberg and J. S. Levin, Phys. Rev. 103, 343 (1956).

2.7.

DETERMINATION OF NUCLEAR REACTIONS

457

= EOcosz 9. Background corrections are made on the basis of counting rates with scatterer removed. To reduce background the detector is surrounded by 1 in. of lead which provides shielding from those gammas which do not originate in the scatterer. Enclosing the lead is 6 in. of paraffin loaded with LiC03 to moderate and capture neutrons.

E

POLYETHYLENE

MU METAL SHIELD RCA 581s PHOTOM

FIG. 34. Time-of-flight geometry for the investigation of the angular dependence of the cross section for elastic scattering of neutrons.17'

As already indicated, one of the basic problems in measuring differential cross sections of the type here discussed is the evaluation of multiplescattering effects, since for intensity reasons the dimensions of the scatterer must be a significant fraction of a mean free path for both the primary and secondary neutrons. The methods for making these corrections are basically similar to those described in Section 2.7.4. It is, however, usually necessary to make simplifying assumptions even when using fast electronic computers to evaluate the integrals. Some of the most useful of these, whose validity has been established, by Monte Carlo methods, for scattering dimensions not much longer than one mean free path are: (1) The angular distribution of neutrons leaving the sample is constant after the first collision.

458

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

+

(2) The fraction of n th collision neutrons suffering n 1 collisions is constant. (3) The number of neutrons suffering more than 3 collisions is negligible. Insofar as cross sections for monoenergetic neutrons are concerned, one must be cognizant of the departure from homogeniety of all neutron sources as the c.m. energy of the neutron-producing reaction is increased. Thus the Li'(p,n) reaction generates two neutron groups above a n 28 2,

FIG.35. Cross section as a function of energy and atomic weight for 14-Mev incident neutrons.

incident proton energy of 2.4 Mev as mentioned in Section 2.7.3.1.2. The p-T, d-D, and d-T reactions will produce a continuum of neutron energies above 8.3, 4.4, and 3.6 MeV, respectively. For many purposes the differential cross section with respect to energy, integrated over 4n solid angle is all that is required. A method which has been found extremely useful for carrying out this type of measurement makes use of the geometry of Fig. 11 with a detector which is capable of measuring the neutron energy as well as the neutron flux. The method of measuring the neutron spectrum is identical to th a t used for the evaluation of Eq. (2.7.5.51).The results of a series of such measu r e m e n t ~ using ~ ' ~ nuclear plate detection is illustrated in Fig. 35. 178 E. R. Graves and L. Rosen, Phys. Rev. 89, 343 (1953).

2.7.

DETERMINATION

OF NUCLEAR REACTIONS

459

The proton-recoil counter telescope, which is also very useful in measuring both the neutron energy and absolute flux in experiments of this type, has already been described in Section 2.7.5.2. 2.7.5.3.3. DIFFERENTIAL CROSSSECTIONS FOR PROCESSES INVOLVING THE EMISSION OF T-RAYS.The problem of measuring differential cross sections for processes involving the emission of y-rays from neutroninduced reactions is analogous and the methods quite similar to the problem, discussed in Section 2.7.5.3.2, of measuring cross sections for neutron emission. The chief differences occur in the type of detectors used and in the procedure for discriminating against background. SCATTERING RING GAS TARGET

SHIELDING CONE

Cn4 SCINTILLATION COUNTER

FIG. 36. Typical experimental arrangement, employing the ring geometry, for measuring the energy and angular dependence of the cross section for y-ray emission from neutron-induced reactions. (Day, ref. 173.)

The most precise experiments for determining y-ray energies have utilized the photopeaks from a NaI s ~ i n t i l l a t o r ~and ~ 7 ~this is, in fact, the most generally useful detector for y-radiation. A typical arrangement for measuring the energy and angular distribution of y-rays produced by the inelastic scattering of monoenergetic neutrons is illustrated in Fig. 36. The resemblance to the arrangement for the neutron case (Fig. 40) is immediately apparent. Although the detector can be shielded from the direct neutron flux, by the methods described in Section 2.7.5.3.2, these provide no relief from neutrons scattered by the sample into the detector. Of the techniques for discriminating against such background, the most powerful, suggested by L. Cranberg and developed by R. B. Day and D. A. Lind,174 utilizes a pulsed beam and time-of-flight to effect the separation of the y-rays to be observed not only from the neutrons scattered by the sample but also from the y-rays produced by room-scattered neutrons. A variation of this method avoids pulsing the beam by requiring coincidences between detector signals and signals from the associated particles generated in the neutron-producing reaction.176 The detector response to a scattered neutron will, of course, be delayed (by the neutron 1 7 3 R. B. Day, Phys. Rev. 102, 767 (1956) ; J. J. Van Loef and D. A. Lind, ibid. 101, 103 (1956). 174 R. B. Day, Proc. of t h e Conf. on Neutron Physics by Time-of-Flight, Gatlinburg, Tennessee, ORNL-2309, 90 (1956). 175 P. Shapiro and R. W. Higgs, Phys. Rev. 108, 760 (1957).

460

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

flight time from scatterer to detector) with respect to the anticipated y-ray signal. For measuring cross sections as a function of y-ray energy, Battat and Graves176 have employed a three-crystal pair spectrometer. Since such an instrument yields a single peak, in the pulse-height distribution, for each y-ray energy, the interpretation of the data is quite straightforward as compared, for example, to that from a single-crystal NaI spectrometer. However, the low detection efficiency constitutes a serious limitation on this method. Application of the three-crystal pair spectrometer to (charged, neutral) reactions has already been discussed in Section 2.7.5.2. One of the main problems in measuring differential cross sections for y-ray emission involves the determination of the detector efficiency as a function of y-ray energy. This has been discussed in the case of NaI counters in Section 2.7.5.2. Important corrections arise from self-absorption, multiple scattering, and angular resolution, all due to the finite size of scatterer and detector. DayITahas shown, by means of y-ray transmission experiments, that the first and most important of these corrections can be calculated on the assumption that the total Compton scattering cross section is effective in removing y-rays from the photopeak, and this assumption greatly simplifies the calculation of the correction due to self-absorption in the scatterer. Methods for evaluating the angular resolution corrections have been developed by a number of investigators. Under favorable circumstances it is possible to measure the counter efficiency and scatterer self-absorption a t one and the same time.178 This is accomplished by using a scatterer in the form of pellets which can be coated with a known amount of a radioactive substance. After first measuring the yield of y-rays produced as a result of a given irradiation, the scatterer is coated with a radio-nuclide, the y-rays from which have approximately the same energy spectrum as the ones originating from the neutron reaction under investigation. For energetic neutrons and scatterers of reasonable thickness, each neutron is capable of making more than one inelastic collision so that multiple-scattering effects must be evaluated if accurate cross-section data are to be obtained.l18 A summary of available data on the cross sections for the emission of y-rays from neutron-induced reactions, including angular distributions, is to be found in BNL-32546and BNL-400.”@ 1733177

M. E. Battat and E. R. Graves, Phys. Rev. 97, 1266 (1955). A. M. Feingold and Sherman Frankel, Phys. Rev. 97, 1025 (1955). 178 R. M. Kiehn and C. Goodman, Phys. Rev. 95, 989 (1954). 179 D. J. Hughes and R. S. Carter, Neutron cross sections and angular distributions. Brookhaven National Laboratory Report BNG400 (1956). 176

177

2.7.

DETERMINATION OF NUCLEAR REACTIONS

46 1

2.7.5.4. Cross Sections for y-Induced Reactions. Photodisintegration experiments are in general difficult to perform and in many cases difficult to interpret exactly. The first problem arises because the cross sections are in general small, so that intense y-sources are required. However, the most intense sources are not monoenergetic, for which case the interpretation of the results in terms of cross sections requires unfolding the energy dependence of the source. It is this process which leads to the difficulty in precise interpretation of the data. CONSIDERATIONS. There are two basic types 2.7.5.4.1. EXPERIMENTAL of sources for studies of y-induced reactions: monoenergetic sources and bremsstrahlung sources. The first class includes radioactive sources, which produce y-rays either as the result of natural or artificial radioactivity, and nuclear-reaction sources, which produce monoenergetic y-rays directly in a nuclear (ply) reaction. Included in the second class are electrostatic-generator sources, which produce either thin- or thicktarget bremsstrahlung with an accurately variable but low maximum energy, and betatron and synchrotron sources, which usually produce thick-target bremsstrahlung with a variable and high maximum energy. The advantage of a monoenergetic source is that the cross section under study may be determined quite unambiguously a t a known energy. However, these sources are generally characterized by very low intensity, which introduces the usual difficulties present in low yield measurements. Further, they often actually consist of several monoenergetic y-ray lines, which removes some of the advantage of their use. Radioactive y-ray sources are limited in maximum energy, but if a variety of them are used it is possible to obtain measurements a t a number of low energies. A typical investigation of this type was the determination of the photodisintegration cross section of beryllium in several experiments.180~181 The radioactive sources employed were all artificially produced with the following y-ray energies: ( S b 9 1.70 MeV, (MrP) 1.81 MeV, PI-'^^) 2.185 MeV, (La'4O) 2.50 MeV, and (NaZ4)2.76 MeV. The monoenergetic y-rays from nuclear-reaction sources are generally considerably higher in energy than those from radioactive sources. Some of the more important nuclear-reaction y-ray sources are the F19(p,ay) reaction (6.1-, 6.3-, and 7.1-Mev lines), the Li7(p,y) reaction (17.63- and 14.8-Mev lines), and the H3(p,y) reaction (20.4-Mev line, but very weak). The electrostatic generator producing the proton beam is usually adjusted 180 B. Russell, D. Sachs, A. Wattenberg, and R. Fields, Phys. Rev. 73, 545 (1948); A. Wattenberg, Photoneutron sources. Preliminary Report No. 6, Nuclear Science Series, Division of Mathematics and Physical Sciences, National Research Council, Washington, D.C. Unpublished. 1.91 B. Hamermesh and C. Kimball, Phys. Reo. 90, 1063 (1953).

462

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

to an energy above the appropriate capture-gamma resonance, and a thick target employed to guarantee maximum y-yield from the resonance. Bremsstrahlung sources have the advantage that large intensities are available, but unfortunately a continuous spectrum of X-ray energies up to the maximum electron energy is produced. This spectrum is in general not precisely known, which contributes seriously to the uncertainties of the cross-section determination. I n electrostatic-generator sources, an accurately homogeneous electron beam of known energy strikes a thick or thin target such as gold. By raising the electron energy in small and precise steps, the upper limit of the bremsstrahlung spectrum rises accordingly, allowing photodisintegration experiments to be performed under conditions of precision energy control. m This technique is quite limited in energy, however, being restricted so far to measurements in the range of a few MeV. Experiments with betatrons and electron synchrotrons can be performed with good intensity to much higher energies. However, the energy control is somewhat less precise, although the stability of betatrons has been improved to the point where the energy will remain constant to + 5 kev under favorable conditions.lXs Betatrons and synchrotrons have been used in the accumulation of a large amount of experimental photodisintegration data as a function of energy in the region of the “giant resonances’’ from 12 to 23 MeV. Detection of photodisintegration processes has usually been carried out either by observing the activities produced in the radioactive product nuclei, or by detecting the direct yield of the ejected particles. In the first case, one obtains direct information about a particular photodisintegration process in a particular isotope, provided it is not possible to form the same radioactive product nucleus by a different photodisintegration process involving a different isotope in the target. In experiments designed to observe the shape of the giant resonances, the residualactivity detection method yields only a lower limit to the width of the total resonance photon-absorption cross section, since processes leading to stable isotopes are ignored.**4The residual-activity method imposes certain requirements on the measurements due to the half-life of the product nucleus and the presence of unwanted activities if chemical separation is not employed. If the half-life is not too short, activation curves can be obtained by irradiating thin disks of the target material for some standard dose, and then removing them to a standard counter u*J. C. Noyes, J. E. Van Hoomiasen, W. C. Miller, and B. Waldman, Phys. Rev. 96, 396 (1954). 188 L. Katz, R. N. H. Haslam, R. J. Horsley, A. G. W. Cameron, and R. Montalbetti, Phys. Rev. 96,464 (1954). 184 Karl Strauch, Ann. Rev. Nuclear Sci.2, 105 (1953).

2.7.

DETERMINATION OF NUCLEAR REACTIONS

463

arrangement. If the radioactive product nuclei with disintegration constant X are produced at a constant rate B during the irradiation lasting for a time t l , and the counting begun a t a time 2 2 after the bombardment ceases, then the number of counts observed in the counting time t s is given by the familiar formula:

N,

=

(B/X)(l - e-XLl)e-XLz(l-

e-hL3).

(2.7.5.55)

From the experimental counting data, Eq. (2.7.5.55) allows one to determine B and, therefore, the saturated activity corresponding to the irradiation rate employed. This information, coupled with the number of nuclei in the sample and the photon flux corresponding to that irradiation rate, suffice to determine the cross section in principle, except for complications introduced by the photon-energy spectrum to be discussed later. I n these measurements corrections must be included for absorption in the sample, as well as for the presence of extraneous activities besides the one of interest. If the half-life is short, some method of rapid and automatic sample transfer to the counter is required (see, e.g., ref. 185). A variety of methods are employed for the detection of particles directly ejected in photonuclear processes. Most of these methods do not distinguish between possible processes producing the detected particle; for example, between neutrons from (-y,n), (y,np), or (y12n). As a consequence, detection of photoneutrons will ordinarily make the observed resonance appear wider than the true shape of the (7,n) cross-section curve, unless the thresholds for the unwanted reactions are sufficiently high.l84,186However, the practical advantages of direct particle detection generally outweigh any ambiguities in the source reactions of the particles.187 For the detection of photoneutrons, it is desirable that the detector have a constant efficiency for neutrons from thermal energies to about 10 MeV. For this reason, enriched BFs counters imbedded in a paraffin housing surrounding the neutron-producing target are usually employed (see, e.g., refs. 182, 188, 189). Arrays of several such counters may be used to increase yields near threshold and reduce exposure time.187~190A difficulty with this method is the pileup of secondary elec186L. Kats and A. G. W. Cameron, Phys. Rev. 84, 1115 (1951). 1saP. F. Yergin and B. P. Fabricand, Phys. Rev. 104, 1334 (1956). 187 R. Nathans and J. Halpern, Phys. Rev. 93, 437 (1954). 188 J. Halpern, A. K. Mann, and R. Nathans, Rev. Sci. Instr. 23, 678 (1952); L. E. Lazareva, B. I. Gavrilov, B. N . Valuev, G. N. Zatsepina, and V. S. Stavinsky, Conf. 1966. Part 1, p. 306. Acad. Sci. U.S.S.R. Peaceful Uses of Atomic Energy, MOSCOW, AECTR 2435, Supt. of Documents, U.S. Government Printing Office, Washington, D.C., 1955. 189 R. Montalbetti, L. Katz, and J. Goldemberg, Phys. Rev. 91, 659 (1953). 190 W. H. Hartley, W. E. St.ephens, and E. J. Winhold, Phys. Rev. 104, 178 (1956).

464

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

trons in the counter which occurs with betatrons during the X-ray pulse. This problem can usually be eliminated by gating the counter to count thermal neutrons only between pulses of the m a ~ h i n e . ~To ~ ~determine r~~l the true yield of photoneutrons in this method, the efficiency of the counting arrangement can be determined, for example, by using a calibrated Ra-Be source placed a t the sample p ~ ~ i t i obyn com, ~ ~ ~ ~ ~ paring with the residual activity method [for a sample in a region where the ( y , n p ) and (y,2n) reactions are energetically forbidden189],or in the case of nuclear-reaction sources by comparing with the neutron yields of known nuclear reactions.lgOFigure 37 shows a representative betatron experimental arrangement used a t the University of Pennsylvania for direct detection of photoneutrons with BFa counters (from Nathans and H a l ~ e r n ' ~ ~ ) . A second method for detection of photoneutrons is by means of induced activities due to neutron capture or (n,p) reactions in suitable substances placed in a paraffin block surrounding the target. The induced activity is then measured by Geiger counters located in intimate contact with the substance. Typical examples are rhodium (44second activity, slow neutrons), aluminum (9.6-minute activity, threshold about 3 Mev), and silicon (2.4-minute activity, threshold about 5 Mev). Such detectors with different neutron-activation thresholds give some information about the energies of the emitted neutrons. l g 3Another interesting photoneutroninduced activity method makes use of a NaI(T1) crystal placed in the vicinity of the target sample. Fast photoneutrons produced in the tiarget will then interact in the crystal, producing IlZ8with a half-life for P-emission of 25 minutes. Since this activity is actually produced within the crystal, the detection efficiency for the p r a y s is nearly In addition, the target geometry does not require thermalisation of the photoneutrons. One of the most important aspects of the direct detection of photonucleons is the possibility of obtaining the angular distributions of the emitted particles. A detection method useful for such studies is the nuclear emulsion, which is often used as a detector for yinduced reactions producing charged particles. A complete discussion of photodisintegration experiments with nuclear emulsions is given in a review article by Titterton. lg6 One method is to observe photodisintegrations within the J. Halpern, R. Nathans, and A. K. Mann, Phys. Rev. 88, 679 (1952). D. R. Connors, Ph.D. Dissertation, University of Notre Dame, Indiana, 1956. Unpublished 19sF. Ferrero, A. 0. Hanaon, R. Malvano, and C. Tribuno, Nuovo cimento [lo]4, 418 (1956). 194 K. G. McNeill, Phil. Mag. [7] 46, 321 (1955). 186 E. W. Titterton, Progr. in Nuclear Phys. 4, 1 (1955). 191 18)

Side View.

Betatron

B x.

Target

+

A- A , Ionization Chambers. Lucite Holder for ( r ) meter.

rtSl0t.

d'

12" 6" 0

End View.

FIG. 37. Experimental arrangement for the detection of photoneutrons produced by bremsstrahlung from a betatron.'*'

466

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

volume of the emulsion itself, which is “loaded” with the appropriate target substance. This is particularly useful for experiments with lowintensity sources, such as nuclear-reaction ( p , y ) sources. It has also been used a great deal for multiparticle disintegration measurements, such as C12(y,3a),for example. A second method, particularly useful for angular distributions, is to place the emulsions, shielded from the direct y-ray beam, a t a variety of angles around the target sample. This is particularly desirable in betatron and synchrotron studies, where highly collimated X-ray beams can be attained. The emulsion technique has the advantage over counter studies that the emulsions are relatively insensitive to background electrons, but the disadvantage that the counting techniques are more tedious. A complete discussion of the nuclear-emulsion technique applied to the detection of emitted neutrons is given in Section 2.7.5.3.2. A counter method for observing the angular distributions of photoneutrons190makes use of the Hornyak detector,s0 mentioned previously in Section 2.7.3.1.4. Grains of ZnS are molded in Lucite, which serves to produce recoil protons from the incident neutrons and transmit the emitted light. The higher specific ionization of the protons in the ZnS produces much larger proton pulses than electron pulses, unless severe pileup from the X-ray beam is present. Such a detector, suitably shielded from the main beam and arranged to rotate around the target sample, has been used successfully for photoneutron angular distribution measurements using 70-Mev X-rays from a s y n c h r ~ t r o n . ~ g ~ Another important experimental consideration concerning the determination of cross sections for y-induced reactions is the measurement of the y-ray flux. The method employed depends upon the type of measurement. In photodisintegration measurements with radioactive sources, the y-ray source strength may be determined by comparison with a known source, preferably emitting y-rays of about the same energy. For example, in the measurements of Hamermesh and Kimball,lsl the comparison was a reactor-produced NaZ4 source whose strength was determined from the neutron capture cross section and the known reactor flux employed. In nuclear-reaction source experiments, the y-flux may be determined by using a NaI(T1) scintillation counter whose counting efficiency has been determined. I n the Li7(p,y) work of Hartley et aZ.,19o for example, the efficiency of the NaI(T1) detector was obtained by calculation, by semiempirical comparisons, and by direct comparison with published absolute thick-target yields from lithium. Experiments with an electrostatic-generator bremsstrahlung source by Noyes et al. used aa an X-ray monitor a suitably shielded Geiger counter located behind and in line with the source. The measurements with betatron 196

W. R. Dixon, Cun, J . Phya. 38, 785 (1955).

2.7.

DETERMINATION OF NUCLEAR REACTIONS

467

bremsstrahlung sources normally determine the X-ray flux by a suitable “r-meter’’ placed in a cavity in a Lucite block. The “roentgen” recorded in this way does not correspond exactly to the usual definition of the roentgen, since the ionization is produced in the walls surrounding the cavity which do not have the same absorption coefficients as air.lS7 From the dosage recorded in the r-meter and the assumed shape of the bremstrahlung spectrum, the number of photons per cm2 per unit energy interval striking the target sample during that dose can be determined.’!’? 2.7.5.4.2. DETERMINATION OF CROSSSECTIONS FOR GAMMA-INDUCED REACTIONS. * The cross section for photodisintegration processes can be determined in a straightforward manner if monoenergetic sources are employed. For example, in a measurement of the photoneutron production cross section using a Li7(p,y) source, a BF3neutron counter imbedded in paraffin, and a NaI(T1) y-ray monitor, as described in the previous section, the cross section can be calculated fromlgo u =

(YeJnk,).

(2.7.5.56)

In this equation, Y is the observed number of neutron counts per y-ray count, corrected for background and y-ray attenuation in the sample, and n is the nuclear density. The y and neutron counter efficiencies, er and t,, are determined by methods described in the previous section. Because in such a measurement a cylindrical sample surrounding the source is usually used to increase the yield, the effective sample thickness f i s given by 190 1 t = (1 - e-z) dQ (2.7.5.57)

--/ 4rCL

4*

where p is the linear photon absorption coefficient for lithium y-rays in the sample. The cross section obtained in this way is actually a suitable average for the 17.6- and 14.8-Mev y-rays from lithium. For bremsstrahlung sources, the accurate determination of the cross section is very difficult, because it requires unfolding the bremsstrahlung spectrum from the experimental yield curve. The accuracy obtainable in this way is governed by184(a) the accuracy of the yield curve obtained as a function of peak bremsstrahlung energy, (b) the validity of the assumed bremsstrahlung spectrum used in the unfolding calculations, and (c) the validity of the assumed energy response of the incident flux monitor. The yield curve can be made quite accurate if enough points are taken with sufficient precision, assuming good machine stability.

* See Section 2.7.7 for more recent supplementary remarks. 19’

H. E. Johns, L. Kats, R. A. Douglas, and R. N. H. Haslam, Phys. Rev. 80, 1062

(1950).

2.

468

DETERMINATION O F FUNDAMENTAL QUANTITIES

However, the detailed character of the bremsstrahlung spectrum is not well known, so that this factor gives a major contribution to the uncertainty of any cross-section determination. The theoretical curves of Schifflgs are usually employed, as described in some detail by Katz and Cameron.199 The flux monitor problem is discussed by various authors (see review article by StrauchlS4for references). The Notre Dame group1s2,192~20°has made a number of measurements of photodisintegration processes using thin- and thick-target bremsstrahlung produced by an electron beam from an electrostatic generator. Figure 38 shows their target and counting arrangement used for the 7 BF, COUNTER LEAD PARAFFIN

TARGET ARRANGEMENT ( NOT TO SCALE 1

FIG. 38. Schematic diagram of target and counting arrangement for photodisintegration experiments near neutron threshold using bremsstrahlung produced by an electrostatic-generator electron beam.’$*

photodisintegration of beryllium. l g 2 One interesting result concerns the determination of thin- and thick-target “isochromats.” An isochromat is a curve representing the number of photons having energies between hv and hv dhv as a function of the electron beam energy, i.e., the energy of the upper limit of the bremsstrahlung spectrum. By observing line absorption into particular levels in indium, the thin-target isochromat was foundzooto be represented by a step function, with each new step appearing when a new line absorption process becomes energetically possible, The thick-target isochromat, on the other hand, was found to rise linearly with electron energy, but with sudden changes of slope a t the energies corresponding to new line absorption processes.zoo,201 A somewhat similar behavior has been observed by the group at the University

+

L. I. Schiff, Phys. Rev. 70, 87 (1946); 83, 252 (1951). L. Katz and A. G. W. Cameron, Can. J . Phys. 2Q, 518 (1951). 200 W. C. Miller and B. Waldman, Phys. Rev. 76, 425 (1949). 201 J. L. Burkhardt, E. J. Winhold, and T. H. Dupree, Phys. Rev. 100, 199 (1955) 19B

2.7.

DETERMINATION OF NUCLEAR REACTIONS

469

of Saskatchewan for betatron excitation functions in the light nuclei, la3 where the fine structure in the excitation function is taken to represent absorption into well-defined levels of the target nucleus. I n order to interpret the thick-target excitation functions obtained with electrostatic-generator techniques near photodisintegration thresholds, it is assumed that individual thick-target isochromats are each straight lines of positive slope, at least in the vicinity of threshold. Then the bremsstrahlung distribution can be represented by1g2

P(E,Eo) = (Eo - E ) m

(2.7.5.58)

where P(E,Eo) is the number of photons of energy E per cm2 per unit energy interval per microcoulomb of beam electrons of energy Eo, and m is the slope of the isochromat. If one assumes that the slope of succeeding isochromats is the same, i.e., that m is independent of photon energy E , then the yield Y ( E o )per microcoulomb of beam electrons of energy Eo is given by:lg2

Y(Eo) = Qm /E:: ( E o - E ) u ( E )dE

(2.7.5.59)

where Q is the number of nuclei exposed to the X-ray beam, a ( E ) is the disintegration cross section, and E,, is the threshold energy. The cross section is then obtained by taking the second derivative of the experimental yield curve :Ig2 1 d2Y u(E0) = --* (2.7.5.60) Q m dEo2 The assumption that m is independent of the photon energy E proves to be valid only over a limited energy regi0n.19~8~~~ The betatron and synehroton bremsstrahlung measurements have been interpreted in terms of the photodisintegration cross sections using two principal methods developed by the Saskatchewan group, the “total spectrum method”lg7 and the “photon difference method.”19gIn both of these methods, small changes in the experimental yield curve can introduce large variations in the calculated cross section. Therefore, in both cases “smoothing” processes are involved in evaluation of the data. The total spectrum method requires progressive smoothing of the crosssection curve during the computation, which introduces a certain amount of arbitrariness into the result.199 I n the photon difference method, the yield curves are also smoothed, but according to the criteria that the first and second derivatives should also be smooth and of an appropriate shape.Ig9Because the photon difference method is simpler to calculate and less arbitrary, it is more widely used. The photodisintegration yield Y (Eo) per roentgen of irradiation at

470

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

any betatron operating energy E o is given by:186

Y ( E o )= Q

IOEoa’(E)P’(E,Eo)dE

(2.7.5.61)

where P‘(E,Eo) is the number of photons at energy E per cm2 per unit energy interval per roentgen of irradiation, a’@) is the cross section for a photon of energy E to eject the particle detected, and Q is the number of nuclei exposed to the X-ray beam. Equation (2.7.5.61) actually represents a set of integral equations, one for each yield point obtained a t each betatron operating energy Eo. I n the photon difference method,1Bethe integral of Eq. (2.7.5.61) is replaced by a summation over a series of energy intervals of width E MeV: (2.7.5.62) where E represents the average energy of each interval, a(E) the cross section evaluated at this average energy in each interval, and P(E,Eo) the number of photons per cm2 per E Mev per roentgen of radiation. The difference between the observed yields for betatron energies E O and Eo - E is then

~(Eo - )Y(E0 - E ) = AY(E0) =

a(E) AP(8,Eo) (2.7.5.63) E

where

AP(E,Eo)

P(E,Eo) - P ( E , Eo - E )

(2.7.5.64)

and E takes on the values e/2, 3r/2, 5r/2, . . . , E o - 4 2 . Equation (2.7.5.63) can then be rewritten to obtain the cross section

B(EO- ~ / 2 )= AA(E0) - z a ( E ) A+(E,Eo)

(2.7.5.65)

E

where

(2.7.5.66)

and

(2.7.5.67)

From the observed difference AY in the yields a t successive energies, the cross sections a can be calculated as a function of energy using Eq. (2.7.5.65). Tabular details of the method, as well as a tabulation of the quantities Q AP(E0 - 4 2 , E0) and A+(E,Eo), are given by Xatz and Cameron.lg9Although the details of the cross section obtained by the total spectrum or photon difference methods are uncertain due to the form assumed for the bremsstrahlung spectrum, the peak positions of the “giant resonances,” and especially the cross sections integrated over energy, are relatively insensitive to the bremsstrahlung spectrum shape.lB7

2.7.

~

DETERMINATION OF NUCLEAR REACTIONS

MAX.

BETATRON

ENERQY

47 1

(Mov)

(a)

hv'

(Mrv)

(b)

FIQ.39 (a and b). Representative example of a photoneutron yield curve obtained with bremsstrahlung from a betatron, and the corresponding cross section obtained by a photon difference method.189

472

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

Figure 39 shows a typical photoneutron yield curve and the corresponding cross section extracted by the photon difference method (from Montalbetti et ~ 1 . 1 ~ 9 ) .

2.7.6. Differential Elastic-Scattering Cross Sections 2.7.6.1. Differential Cross Sections for Elastic Scattering of Neut r o n ~ . ~All ’ ~ of the methods described in Section 2.7.5.3 are equally germane to the measurement of differential neutron elastic-scattering cross sections. However, by thus limiting the objective of the measurements, certain simplifications can be realized since one can now utilize R I N G SCATTERER

T-=-

SOURCE

DETECTOR

I

. . ‘.

/’

\

‘.

/‘ I ’ \

Fro. 40. “Ring” geometry for measuring differential cross sections for the elastic scattering of fast neutrons.

the primary source to calibrate the detector response and thus avoid absolute flux measurements. This is especially convenient if the detector has a flat response in the energy region of the elastically scattered neutrons, or if the scattering nucleus is heavy enough not to alter significantly the energy of the scattered neutron, for then the detector may be biased sufficiently high to discriminate against nonelastic neutrons without the necessity for complicated corrections to take account of the energy dependence of detector efficiency. The general approach is to determine the counting rate of the detector when it is in its normal position and again when it is in the position usually occupied by the scatterer.zO2 Figure 40 shows a typical arrange2 0 1 R. N. Little, Jr., B. P. Leonard, Jr., J. T. Prud’Homme, and L. D. Vincent, Phys. Rev. 98,634 (1955);S. E. Darden, R. B. Perkins, and R. B. Walton, ibid. 100, 1315 (1955);H. S. Hans and S. C. Snowdon, ibid. 108, 1028 (1957);J. 0. Elliot, ibid 101, 684 (1956).

2.7.

DETERMINATION OF NUCLEAR REACTIONS

473

ment for measuring elastic-scattering cross sections without recourse to absolute flux determinations. The geometrical arrangement utilizes the scatterer in the form of a ring on whose axis lies the source and detector. This geometry maximizes the amount of scattering sample for a given attenuation and simplifies the problem of shielding the detector from the primary neutron source. Shielding of the detector from the primary neutron flux is accomplished by the introduction of a shadow cone, on the sample axis, between source and detector. The scattering angle e is varied by changing 11 and 1 2 . Ideally, the shadow bar is opaque for source neutrons. However a correction is readily applied for non-opacity and room scattering by determining the counting rate with shadow bar in place and scatterer out. If we denote by P,, the average Aux of primary neutrons in the scatterer, the flux FD scattered into the detector can be represented by (2.7.6.1) where n, is the number of scattering nuclei and u(0) is again the differential elastic-scattering cross section a t angle 0. The counting rate in the detector appropriately corrected for background, as above indicated, will be CI = CFDwhere e represents the detector efficiency. If we now replace the scatterer by the detector, and operate the neutron source a t the same intensity, the counting rate will be C2 = eF. and the ratio of counting rates is simply (2.7.6.2) where the last term represents the average attenuation of the incident neutron flux by the scatterer. The above result is independent of the absolute flux a t the detector and a t the scatterer. For reasonable geometrical configurations, Eq. (2.7.6.2) will give u(0) to an accuracy of approximately 10%. For higher precision it is necessary t o make the usual corrections for finite size of the scatterer, seIf-absorption and multiple scattering in the scatterer, energy dependence of the detector, and finite angular resolution. Disadvantages of the method are that the geometry changes with scattering angle, and one is usually obliged to take account of source and counter asymmetries. A geometrical arrangement which circumvents the above-mentioned disadvantages is illustrated in Fig. 41. Here one uses a small scattering sample around which the detector is rotated in order to vary the scattering angle. The detector is usually an organic scintillator or a proportional

474

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

counter containing hydrogen or propane. Shielding in the form of wedges is placed between source and detector.2°a Where room-scattered background is high, it often becomes necessary to enclose the detector in a shield containing a channel through which enter the scattered neutrons.204 In this geometry (Fig. 42), detector and shield are rotated around an axis passing through the center of the sample and perpendicular to the plane of the scatterer. CYLINDRICAL SCATTERING SAMPLE

NEUTRON SOURCE

-PROTON

amm

SCINTILLATION DETECTOR

FIG.41. “Wedge” geometry for measuring differential elastic-scattering cross sections for fast neutrons. (Walt and Beyster, ref. 203.)

A time-honored method for measuring neutron elastic-scattering cross sections is based on the detection of the recoil nucleus, while still another method utilizes the cloud chamber in the same way as for reaction studies.206The former method has found extensive application in elastic scattering from light nuclei. If the target material is in the form of a gas, and if it is known that only elastic scattering takes place, one can obtain the complete angular distribution of the scattered neutrons by measuring the pulse-height distribution in an ionization chamber. This comes about. as follows: the energy of the recoil nucleus is proportional to (1 - cos +no.m,) 80s H. B. Willard, J. K. Bair, and J. D. Kington, Phys.Rev. 98,669 (1955); M. Walt and J. R. Beyster, ibid. 98, 677 (1955); M. Walt and H. H. Barschall, ibid. 95, 1062

(1964). SorR.

C. Allen, R. B. Walton, R. B. Perkins, R. A. Olson, and R. F. Taschek,

Phys. Rev. 104, 731 (1956). 206 J. R. Smith, Phys. Rev. 96,

730 (1954).

2.7.

DETERMLNATION OP NUCLEAR REACTIONS

475

while the number of recoils in the energy interval AE is proportional to c( $J)A(COS$J~.~.). It is therefore apparent that the elastic-scattering

angular distribution has the same shape as the energy distribution of the recoil nuclei.206Although this method is simple in principle, one must use great care to insure that the pulse height is independent of the position a t which the ions are formed, that recombination is either negligible during the time of ion collection or proportional to the number of ions produced, and that wall effects are small, i.e., the counter dimensions are large compared to the range of recoils. Also, one must exercise particular

SAMPLE-

NEUTRONSOURCE

EXPERIMENTAL ARRANGEMENT

FIQ. 42. Experimental arrangement for measuring neutron elastic-scattering differential cross sections in a high background of neutrons and y - r a y ~ . ~ '

care to minimize and correct for the effects of neutrons scattered in the detector itself. I n deference to this consideration the amount of nonactive material in the detector should be kept as small as possible. A summary of neutron elastic-scattering cross sections with references to the original literature is given in ref. 179. 2.7.6.2. Differential Cross Sections for Elastic Scattering of Charged Particles. Most of the considerations required for the accurate determination of charged-particle elsstic-scattering cross sections, such as current integration, solid-angle determination, corrections for beam size, angular resolution, multiple scattering, etc., have already been discussed in detail in Section 2.7.5. In addition, Sections 2.7.5.1.1 and 2.7.5.1.2 include a discussion of most of the detectors used for such measurements. Therefore, only a brief review of the principal experimental arrangements *oeE. Baldinger, P. Huber, and H. Staub, Helv. Phys. Acta 11, 245 (1938); H. H. Barschall and M. H. Kanner, Phys. Rev. 68, 590 (1940); J. L. Fowler and C. H. Johnson, ibid. 98, 728 (1955).

476

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

employed and a few representative results of these methods will be given in this section. At low energies, the electrostatic generator is usually employed for elastic-scattering measurements. The beam from such a machine is normally passed through a magnetic analyzer to separate various beam components (e.g., protons from diatomic hydrogen ions), and then through a precision electrostatic anaIyzer.42 An electrostatic-generator beam is very useful for precision experiments because it can be made essentially parallel and its spread in energy can be made quite small (typically 0.1 % under routine operating conditions). Its energy is readily variable in arbitrary steps, so that it is extremely useful for the study of elastic-scattering experiments as a function of incident-particle energy. However, the electrostatic generator is limited a t present to proton energies of about 8 MeV* which, because of the Coulomb barrier, prevents its use for many types of experiments in heavier nuclei. The targets employed in electrostatic-generator measurements of absolute elastic-scattering cross sections are ordinarily gas targets (if avajlable), because of the precision with which the number of scattering nuclei may be determined. I n addition, target contamination then can be minimized, and no question of target uniformity exists. To prevent partial loss of the precision energy control available in an electrostaticgenerator beam, the scattering gas may be opened directly to the electrostatic analyzer through a suitable differential pumping column.207This presents several advantages over a foil separating the accelerator vacuum from the scattering (a) continuous flushing of the gas prevents buildup of contaminant gases and vapors; (b) energy degradation and straggling of the incident beam do not occur in an entrance foil (such a foil would have a variable thickness due to carbon buildup); and (c) small-angle scattering effects in an entrance foil are not present. The effects of the gas in the differential pumping column are negligible, since the pressure in the column is much less than in the scattering chamber itself .207 Detailed considerations of the calculation of the cross section from the geometry of gaseous targets have already been given in Section 2.7.5. Of course, many substances are not available in suitable gaseous form, so that solid targets must be employed. For elastic-scattering measurements, the thick-target technique employing a magnetic spectrometer described in Section 2.7.5.1.2 allows measurements to be taken under conditions such that target contamination can be accurately determined and allowed

* See Section 2.7.7 for more recent supplementary remarks. *o7H. L. Jackaon, A. I. Galonsky, F. J. Eppling, R. W. Hill, E. Goldberg, and J. R. Cameron, Phys. Reu. 89, 365 (1953).

2.7.

DETERMINATION

OF NUCLEAR REACTIONS

477

The detectors employed for elastic-scattering experiments with electrostatic generators are usually scintillation counters, proportional counters, or magnetic spectrometers or spectrographs.* A discussion of the application of scintillation counters and magnetic spectrometers and spectrographs to measurements of this type has already been given in Sections 2.7.5.1.1 and 2.7.5.1.2. Proportional counters will give larger pulse heights for a-particles than for protons of the same energy, so they are particularly useful in a-scattering experiments where singly charged reaction products may also be produced. However, the improved energy discrimination of scintillation counters and the fact that they may be mounted directly in the vacuum without the necessity for a thin gas-tight intermediate foil has led to increased use of scintillation counters for elastic-scattering measurements. Figure 43 shows a representative elastic cross section obtained with electrostatic-generator techniques, in this case the elastic scattering of protons by a thick F19 target using the magnetic-spectrometer method described in Section 2.7.5.1.2 (from Webb et ~ 1 . 9 . From the upper energy limit of electrostatic generators (about 8 MeV* a t present) to the upper energy limit included in the present discussion (50 Mev), cyclotrons or linear accelerators are normally employed for the measurement of charged-particle elastic-scattering cross sections. The external beam from a cyclotron normally diverges to some extent, so that one must employ auxiliary equipment with some care in order to focus the beam on the target in a small spot of usable intensity without too much angular divergence. Furthermore, a certain fraction of the beam is lost in the focusing system, which tends to cause background difficulties. I n these respects the cyclotron is at somewhat of a disadvantage to electrostatic generators and linear accelerators, whose beams are seldom far from parallel. However, the cyclotron provides the only method for making measurements in the energy region between electrostatic generators and linear accelerators. For a given number of sections, the linear accelerator (of nuclear particles) is basically a fixed-energy machine, whereas recent developments2°8 in cyclotron techniques permit variation of the beam energy over a wide range (although not as yet with the ease of a n electrostatic generator). I n order to produce a beam with the appropriate characteristics on a remote target (usually from 15 to 30 feet from the cyclotron snout) in an experimental area outside the main cyclotron shielding, a typical cyclotron experimental arrangement will generally include various steering magnets, a strong-focusing pair

* See Section

2.7.7 for more recent supplementary remarks.

R. L. Thornton, K. Boyer, and J. M:Peterson, Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1966 4, 87 (1956). 208

478 2. DETERMINATION OF FUNDAMENTAL QUANTITIES

0

1

’

600

I

700

&

I

9150

060

1100 I200 1300 PROTON ENERGY (KEV)

1 4 0

1500

I

1600

I

1700

1

lm

FIG.43. Typical elastic cross section obtained aa a function of incident proton energy using electrostaticgenerator techniques.13a

2.7.

DETERMINATION OF NUCLEAR REACTIONS

479

of magnetic quadrupole lenses, and an analyzing magnet. The energy spread in the beam spot on the target depends upon many factors, but under routine operating conditions will be around one-half per cent and under very favorable conditions may be reduced to less than O.2%.ll6A typical scattering chamber has already been described in Section 2.7.5.1.1 and shown in Fig. 24. Both gas targets and solid targets are employed in cyclotron elasticscattering measurements. In particular, solid targets have been employed in a rather extensive series of measurements performed to observe the breakdown of Rutherford scattering for a-particles on elements throughout the periodic table (see, e.g., refs. 209-211). In many measurements of this type, the target thickness is sufficient to be determined to good accuracy by weighing of the target foils. I n such a case one actually determines the areal density in mg/cm2, which may be converted into nt = number of nuclei/cm2 if the density is known. This allows the direct determination of the absolute cross section from Eq. (2.7.5.44) when the scattered-particle yield, the solid angle, and the number of incident particles are measured. Where such measurements have been made, the magnitude of the elastic-scattering cross section at forward angles checks with that predicted from Rutherford scattering.212I n many experiments the scattering at forward angles is checked to have the correct Rutherford energy or angle dependence, and then the magnitude assumed to be Rutherford for cross-section normalization. Another technique213 employed in angular-distribution measurements compares the elastic scattering from the target under investigation with that from a gold target inserted before and after each measurement. The ratio of the number of counts from the target under investigation (per monitor count in an auxiliary monitor counter) to the number of counts from the gold target (per monitor count) may then be related simply to the Rutherford cross section.213 The principal advantage to this scheme is its moderate insensitivity to slight systematic errors in the angle determination. The detectors employed in cyclotron elastic-scattering measurements are scintillation counters (discussed above and in Section 2.7.5.1.l), nuclear emulsions (discussed in Section 2.7.5.1.I), magnetic spectrometers or spectrographs (discussed in Section 2.7.5.1.2), proportional counters (e.g., 209

G . W. Farwell and H. E. Wegner, Phys. Rev. 96, 1212 (1954).

N.S. Wall, J. R. Rees, and K. W. Ford, Phys. Rev. 97, 726 (1955). 211 H.E. Wegner, R. M. Eisberg, and G. Igo, ibid. 99, 825 (1955);E. Bleuler and D. J. Tendam, ibid. 99, 1605 (1955); D.D.Kerlee, J. S. Blair, and G. W. Farwell, 210

ibid. 107, 1343 (1957). 212 113

H. E. Wegner, private communication, 1957. W. F. Waldorf and N. S. Wall, Phys. Rev. 107, 1602 (1957).

480

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

ref, 214) or ionization chambers (e.g., ref. 215), and activation foils.216 Gas counters are not often used currently for angular-distribution measurements, because of the greater convenience of scintillation counters; however, they are very useful in proportional counter-scintillation counter telescopes for mass determination (see Sections 2.7.5.2.1 and 2.7.5.3.1). The activation method has been used with internal cyclotron beams.216Representative of elastic-scattering measurements made with

looor 1

TARGET: Th SCATTERING ANGLE :SO"

CORRECTED loot 2

2

5 W

In In v)

0

a v

10-

W

2

G-I W

a

1110. . . .ISI . . . .20I , , . 25 .I....I....I....I.,.J 30 31 40 45 ALPHA-PARTICLE ENERGY(MEV)

FIG.44. Typical elastic cross section obtained as a function of incident a-energy using cyclotron techniques.200 The energy variation was obtained by inserting foils in the cyclotron beam.

cyclotron external beams using scintillation-counter detection are the curves shown in Figs. 44 and 45. Figure 44 shows the failure of Rutherford scattering for a-particles as a function of energy at a fixed angle obtained by Farwell and Wegner,Z09 and Fig. 45 a similar failure for both a-particles and deuterons as a function of angle at fixed energy obtained by Rees and S a m p ~ o n . ~ ~ ' A number of proton elastic-scattering angular distribution measureD. 0. Caldwell and J. G. Richardson, Phys. Rev. 98, 28 (1955). D. A. Bromley and N. S. Wall, Phys. Rev. 102, 1560 (1956). *laB. L. Cohen and R. V. Neidigh, Phys. Rev. 93, 282 (1954). a l r J. R. Rees and M. B. Sampson, Phys. Reu. 108, 1289 (1957). 214

216

2.7.

48 1

DETERMINATION OF NUCLEAR REACTIONS

ments have been carried out using linear accelerators at various fixed energies, for example, a t 9.8 Mev (Minnesota first section218),19 Mev (Physico-Technical Institute of the Academy of Sciences, USSR219), 31.5 Mev (Berkeley220),and 40 Mev (Minnesota second sectionzz1).In a representative experimental arrangement,221 the beam from the Minnesota linear accelerator second section passes through a deflecting magnet, a strong-focusing quadrupole magnet, and then some thirty feet into the

Gold

1.2

0

I

I

I

30

60

90

I

I 20

I

150

IE I

&m.

FIG.45. Representative angular distribution of elastically scattered deuterons and a-particles using cyclotron techniques.*17

target chamber in a shielded experimental area. The beam-energy spread on the target is less than 0.5%, the beam diameter about one-half inch, and the angular divergence less than 1.5 X radian. Most of these experiments have been performed with scintillation-counter detectors. Target and detector considerations have for the most part already been discussed. The accuracy of charged-particle elastic-scattering measurements varies with the type of experiment. The most accurate measurements have in general been determinations of the proton-proton scattering cross section with electrostatic generators. It might be informative to conclude with a brief summary of errors in the cross section obtained in a *1* 219 220

*21

R. H. Lovberg, Phys. Rev. 103, 1393 (1956); N. M. Hintr, ibid. 106, 1201 (1957). R. A. Vanetsian and E. D. Fedchenko, Soviet J. of Atomic Energy 2, 141 (1957). B. B. Kinsey and T. Stone, Phys. Rev. 103, 975 (1956). M. K. Brussel and J. H. Williams, Phys. Rev. 106, 286 (1957).

482

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

representative experiment of this type by Worthington et aLZz2carried out with a gas target and proportional-counter detection. The systematic uncertainties for a particular choice of defining slits were as follows: beam energy 0.1%, proton current 0.1%, integrator capacitance 0.04%, oil density (in measurement of gas pressure) 0.02%, gas contamination 0 to 0.3%, slit-edge scattering 0.06 % maximum, “G-factor” of analyzer slit system f0.04% and multiple scattering 0.01 %. The nonsystematic uncertainties were as follows: statistics k 0.22%, height of oil column k0.02%, gas temperature I f : 0.02%, and counting losses I f : 0.02%. The total error in the cross section resulting when these uncertainties were combined appropriately (except at forward angles) was k0.3%. Acknowledgment The authors are pleased to acknowledge their indebtedness to Leona Stewart for supervising the preparation of the figures, for checking many of the formulae and derivations, and for correcting, proofreading, and assembling the completed manuscript.

2.7.7. Supplementary Remarks The preceding sections (2.7.3 to 2.7.6 inclusive) were completed and submitted in April, 1958. The present Appendix was submitted in August, 1960. Although the general methods for the determination of nuclear cross sections have not been changing rapidly, it seems appropriate to mention briefly some of the more significant improvements in techniques which have developed since the previous sections were written. The energy ranges available for monoenergetic neutrons from various reactions have been considerably extended, from those indicated in Section 2.7.3.1.2, by the advent of the tandem electrostatic accelerator. It is a simple matter to calculate the new upper limits for neutron energies corresponding to the new accelerator energy limits. However, a t these higher energies extra caution is required to insure that low-energy neutron groups or breakup neutrons are biased out appropriately in the neutron detector. A major advance in charged-particle detection methods has been the solid-state detector.” Development of these detectors is proceeding so rapidly that definitive statements at this time about energy resolution, maximum energy capability, and general usefulness will undoubtedly be rapidly outdated. It seems appropriate to indicate briefly their characteristics, however, since they will certainly become one of the more important types of energy-sensitive detectors. The reader is referred to ((

222

H. R. Worthington, J. N. McGruer, and D. E. Findley, Ph98. Rev.90,899 (1953).

2.7.

DETERMINATION OF NUCLEAR REACTIONS

483

papers by Friedland et a1.22s and by McKenzie and BromleyZz4 for details and other references. Solid-state detectors are basically semiconductor junctions which function over a small region as solid-state ionization chambers. Examples of these junctions are Au-Ge junctions cooled to liquid nitrogen temp e r a t ~ r eand ~ ~silicon ~ p - n junctions operated at room temperature.223 A number of important advantages of solid-state counters over scintillation counters for the detection of heavy charged particles can be cited. If the particle range is not too great, solid-state counters produce voltage pulses proportional to the particle energy with an energy resolution which is considerably better than for scintillation counters (0.6% has already been attained). They are small and simple, requiring no high-voltage supplies, associated photomultipliers, gas flow systems, or windows. The only voltage requirement is a small bias, usually less than a hundred volts. They produce pulses with a rise time in the millimicrosecond range or less, making them particularly adaptable to coincidence measurements. There are, of course, certain limitations to the solid-state counters, at least at present. They are most useful for particles with short range because the sensitive “depletion region” of the junction is as yet rather small. Their upper limit for linear response is -35 Mev for a- and He3 particles and -10 Mev for protons. These limits will presumably be increased as junctions with thicker depletion regions are constructed. The resolution capabilities appear t o be limited primarily by noise, so that to attain best results, low-noise, capacity-compensated amplifiers are required. Counters of large area are difficult to construct, so that only about 1 cm by 1 cm counters retaining all of their desirable properties have been used up to 1960. Aside from counting rate considerations, very small counters are often a disadvantage for the determination of absolute cross sections because of difficulties with finite source sizes, multiplescattering when absorbers or d E / d x counters precede the solid-state counter, and other similar considerations. However, because of the rapid improvements being made in the field, the above limitations may soon be considerably reduced. An interesting application of solid-state counters is the construction of multichannel detection systems using a large number of small solidstate eounters placed side by side, as for example, in the focal surface of a magnetic spectrometer. When such systems are combined with small transistorized amplifiers, data accumulation can be greatly facilitated. An interesting new development in particle detection by scintillation 228

914

S. S. Friedland, J. W. Mayer, and J. S. Wiggins, Nucleonics 18, 54 (1960). J. M. McKenzie and D. A. Bromley, Phys. Rev. Letters 2, 303 (1959).

484

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

counters is the possibility of discriminating between electrons and heavily ionizing particles by the difference in decay time of the fluorescence. For example, in the few Mev range, the fluorescent decay time for electrons in Cs(T1) is about 0.70 psec as compared to 0.42 psec for a r - p a r t i ~ l e s . ~ ~ ~ ~ ~ ~ Such decay time differences are also teported for anthraceneZz7 and other organic scintillators.a2*This difference is sufficient to allow discrimination between alphas and electrons in C S I , ok~ ~between ~ fast neutrons (actually proton recoils) and y-rays in anthracene and some other organic scintillators.2ag A large fraction of the (charged particle, y) type of studies continue to concentrate on energy measurements and relative cross sections, without serious attempts at absolute cross-section measurements. One interesting method for determining absolute cross sections, not described in the previous sections, is by measurement of the /3-activity of the residual nucleus of the reaction, a procedure also frequently used in neutron capture cross-section measurements. The very small cross sections for the reactions 0le(p,y) and NeZ0(p,y),both of astrophysical interest, have recently been determinedZ3Oby counting the 66-sec F17 and 23-sec NaZ1positron activities produced in appropriate 0lEand Ne20 targets by a measured integrated proton flux. Unfortunately, the method is somewhat limited by the availability of appropriate activities produced by such capture reactions, and, as usual, accurate absolute cross sections require accurate knowledge of target thicknesses and counter efficiencies. Nevertheless, it is a very useful method for the determination of very small capture cross sections. A number of advances have been made in measurements of cross sections for y-induced reactions. In particular, much more emphasis has been placed on measurements using monoenergetic y-rays. Nuclear y-rays from the T(p,y) reaction have been used to give monoenergetic y-rays with variable energy from 20 to 25 Mev by varying the proton energy. Measurements of the total y-absorption in various elements have been made in this wayza1by attenuation experiments, as well as of the C12(y,n) 226R. S. Storey, W. Jack, and A. Ward, Proc. Phys. SOC.(London) A72, 1 (1958). 226 J. C.Robertson and A. Ward, PTOC. Phys. SOC.(London) A79, 523 (1959). 227 G.T.Wright, Proc. Phys. SOC.(London) B49, 358 (1956). 2~ F. D.Brooks, Progr. in Nuclear Phys. 6,284 (1956);also in “Liquid Scintillation Counting” (C. G . Bell and F. N. Hayes, eds.), p. 268,Pergamon, London, 1959. **o F. D. Brooks, Atomic Energy Research Establishment, Harwell, England, Unpublished Report AERE NP/GEN 8 (1959). 230R. E. Hester, R. E. Pixley, and W. A. S. Lamb, Phys. Rev. 111, 1604 (1958); N. Tanner, Gid. 114, 1060 (1959). *aM.M.Wolff and W. E. Stephens, Phys. Rev. 112, 890 (1958);E.E. Carroll, Jr. and W.E. Stephens, ibid. 118, 1256 (1960).

2.8.

DETERMINATION OF FLUX AND DENSITIES

485

cross section by use of the positron activity of C11.23z+z33 An example of the latter is the interesting experiment by R. B. Day233in which an annular ring of plastic phosphor is placed around the tritium y-source. After a bombardment for an appropriate interval of time, the plastic phosphor is placed directly on a photomultiplier tube and the C" positron activity in the phosphor itself counted. Perhaps the most important advance in the production of monoenergetic y-rays is that which utilizes the annihilation in flight of posit r o n ~ In . ~this ~ ~method, positrons produced a t the target of an electron linear accelerator are momentum-analyzed in a magnetic field and then allowed to annihilate in flight in a thin lithium target. By use of a suitable geometry, the transmitted positrons may be deflected from the photon beam and the photons converged to a focus a t the target of interest. At present, it appears that experiments performed with T(p,y) y-rays, although more limited in accessible energy range, are capable of higher resolution than positron-annihilation y-rays, but that the available intensities from the annihilation method will ultimately be considerably greater. Also, the T(p,y) source is subject to grave difficulties above the threshold for neutron production. Advances have also been made in the bremsstrahlung measurements of y-induced reactions. In particular, the method of Penfold and L e i ~ s ~ ~ ~ for the extraction of cross sections from these measurements is now widely used. Although not fundamentally different from some of the m e t h o d ~ l ~ described, ~ * ~ g ~ it has certain analytical and computational advantages.

2.8. Determination of Flux and Densities 2.8.1. Determination of Flux of Charged Particles* 2.8.1.1. Measurement of Parallel Beams of Charged Particles. We assume we are dealing with a beam of charged particles emanating from an accelerator. We wish to determine the number of particles nB that D . Cohen and W. E. Stephens, Phys. Rev. Letters 2, 263 (1959). R. B. Day, private communication, 1960. 234 J. Miller, C. Schuhl, G. Tamas, and C. Tzara, Cornpt. rend. acad. sci. 249, 2543 (1959); S. C. Fultz, C. P. Jupiter, C. R. Hatcher, and F. D. Seward; C. P. Jupiter, N. E. Hansen, H. W. Koch, and S. C. Fultz, Bull. Am. Phys. SOC.121 6, 36 (1960). 236 A. S. Penfold and J. E. Leiss, Phys. Rev. 114, 1332 (1959). 232L. 233

* Section 2.8.1 is by 0. Chamberlain.

486

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

have passed along the beam line and impinged on a target in a certain exposure. Obviously the beam must be well localized in space and its composition must be well known if it is to be of greatest usefulness. This implies collimator holes and focusing magnets designed to cut out scattered particles or particles whose energy may have been degraded by passage through material. We discuss first the primary monitors, capable in themselves of an absolute beam determination. I n a later section we discuss some useful secondary monitors-those which must be calibrated by comparison with some primary monitor. 2.8.1.2. Primary Monitors. 2.8.1.2.1. FARADAY CUP. The most common instrument for use as a primary standard for measuring beams of charged particles is the Faraday cup. The beam to be measured falls on an electrode sufficiently massive to stop the beam as well as almost all of the secondary particles resulting when the beam particles strike the material of the Faraday cup. The Faraday-cup electrode is electrically insulated from its surroundings, and the measurement of beam hitting the cup consists in measuring the electric charge carried to this electrode by the beam. If Q is the electric charge carried to the Faraday cup, and ze is the electric charge carried by each beam particle, then the total number of beam particles, is clearly nB

= &/ze.

I n this expression e is the magnitude of charge of one electron and z the number of electronic charges carried by each particle of the beam. For example, for a beam of alpha particles we would use z = 2. The basic electrical circuit of the Faraday cup is quite commonly the slide-back circuit shown in Fig. 1. At the end of a run the potentiometer is adjusted to bring the Faraday-cup potential back to its original value. Thus the electrometer amplifier and voltmeter serve only as a null instrument-to show there was no net change in potential of the Faraday-cup electrode. The magnitude of total beam charge must then be equal but opposite to the total charge supplied through C,, namely

Q = -C,AV where C, is the capacity of the condenser C. and AV is the difference in potential as shown by the two readings of the calibrated potentiometer, before and after the interval during which the beam was turned on. Many precautions must be taken if beam charges are to be correctly measured. These are discussed in the following paragraphs. (1) The drift of the instrument (with beam off) must be measured, preferably immediately before and immediately after the beam readings are made, and the drift corrected for, if necessary. Since the drift rate

2.8.

DETERMINATION

487

OF FLUX AND DENSITIES

may be different (before and after the run) the correction is usually based on the average. (2) To be sure the rate of drift used corresponds to conditions as they are during a beam measurement, the potentiometer should be continually adjusted during a run so as to maintain nearly zero reading of the voltmeter at all times. This is sometimes done electronically. The circuit

FARADAY-CUP ELECTRODE

r

.

INCIDENT BEAM

- - - - ---1 I

~

ELECTROMETER AMPLIFIER CALIBRATED POTENTIOMETER

VOLTMETER AT OUTPUT

FIG.1. Circuit diagram of Faraday cup with slide-back potentiometer.

shown in Fig. 2 gives continuous electronic slide back action through the condenser C . The circuit is used just as that of Fig. 1. Since the electrometer is used only as a null indicator when the final manual slideback is made, the condenser C need not be accurately known. The electrometer amplifier shown in Fig. 2 should be a high-gain highinput-impedance amplifier such as that shown in the article of Brown and Tautfestl or (for measurement of very weak beams) a vibrating reed electrometer.* (3) The condensers C, and C must be free from leakage and dielectric “soakage,” which in practice usually means a condenser with polystyrene dielectric,3 and C, should be calibrated on the occasion of its use. For dc capacitance comparison the circuit of Fig. 3 is very good. When the K. L. Brown and G. W. Tautfest, Rev. Sci. Instr. 27, 696 (1956). H. Palevsky, R. K. Swank, and R. Grenchik, Rev. Sci. Znstr. 18, 298 (1947) give an excellent description of the instrument. Good commercially made varieties are available. SAvailable from J. E. Fast Company, Chicago, Illinois, or as type L A “glass mike” of Condenser Products Corporation, Chicago. 1

2

488

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

FARADAY-CUP ELECTRODE

INCIDENT BEAM

- - - - -- - - - - - - - - - - - - - - - -- - - - - - - - - - I I I

I

I

r-----------I

I

I

I

I I

I

I

I

I I I

I

I I

I f

'

I I

'c

I

I

I I

I I

I I I

I

I

I I

I

ELECTROMETER AMPLIFIER (NEGATIVE GAIN)

I I I I

I

- 1 1

L-------_-

circuit is adjusted so that the output voltmeter I'is unaffected by opening or closing switch Sw2, then C z / C , = Rr/Rz. R1 and R z should be wellstandardized variable resistance boxes. Figure 4 shows more detail of the condensers with their containers, and indicates the stray capacities between each terminal and the container. Note that the magnitudes of the stray capacities do not affect the balance point: two of the stray capacities are strictly in the low-impedance part of the circuit and the other two suffer no potential change at the balance point. Figure 5 shows

7

ELECTROMETER

FIQ.3. Bridge circuit for dc comparison of capacitances.

2.8.

DETERMINATION OF FLUX AND DENSITIES

489

FIG.4. Detail of dc bridge, showing stray capacities in addition to the main capacities CI and C2.The balance point is independent of the stray capacities.

FIG. 5. Preferred electrometer circuit showing stray capacitances in connection with the standardized condenser CS.

490

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

the same detail of the condenser C, in the circuit of Fig. 2. Again one stray capacitance is in the low-impedance potentiometer circuit, and the other stray capacitance returns exactly to its original charge when the potentiometer is adjusted for zero reading (or the original reading) of voltmeter V . Comparison of Fig. 4 with Fig. 5 shows that in each case the effects of stray capacities are similarly eliminated, indicating the suitability of the bridge circuit for intercomparing condensers. Since absolutely calibrated condensers are usually available calibrated at finite frequency (often 1000 cps) one polystyrene condenser must be compared with a n absolute condenser a t this frequency. By choosing a polystyrene condenser very nearly equal in capacitance to the absolute condenser one can arrange to get an accurate comparison from a standard ac bridge such as the General Radio type 650-A impedance bridge. Effects of stray capacities can be minimized if a standard capacity a t least as large as 0.01 pf is chosen. (4) It is of course essential that, to the desired accuracy, all of the beam current and only the beam current reach the Faraday cup. The beam itself should be well-defined and free from any spray of scattered particles such as those which might emerge from a collimator. Ideally, the beam should be focused on a collimator and then the beam emerging from the collimator should again be magnetically analyzed and focused on the entrance to the Faraday cup, although it is frequently not convenient to use this procedure. More usually the beam emerging from the collimator falls directly on the Faraday cup. Carefully made exposures with pairs of X-ray films can be used in many cases to place limits on the fraction of the beam lying outside a certain geometrical area.4 Gas around the Faraday-cup electrode would give rise to ionization and hence to gain or loss of charge. The Faraday cup should be in vacuum, and experiments should be performed to make sure the charge collected is independent of pressure for the quality of vacuum used. A diffusion pump is usually required. Backscattering of beam particles can cause significant loss of beam charge, especially for low-energy electron beams. * This may be minimized by constructing the surface of the Faraday cup (where the beam strikes) of material of low atomic number and by allowing the beam to strike the Faraday cup a t the bottom of a deep hole. Backscattering may

* See also Vol. 4 B, Part 10. ‘The method is described in G . P. Thomson and W. Cochrane, “Theory and Practice of Electron Diffraction,” p. 244. Macmillan, London, 1939. The X-ray films determine accurate exposure ratios for different regions of the beam. Spatial integration then tells what fraction of the film exposure occurred outside any prescribed area.

2.8.

DETERMINATION OF FLUX AND DENSITIES

49 1

further be reduced if the region of the deep hole has a magnetic field, transverse to the beam direction, sufficient to deflect the incident beam appreciably. The magnetic field serves to trap charged particles, particularly if they are of lower energy than the beam. Secondary electron emission from the Faraday cup can cause a loss of negative charge, and secondary emission from the surroundings of the Faraday cup (especially the entrance foil) may cause an unwanted gain in negative charge by the Faraday cup. Most of this effect is eliminated if sufficient magnetic field is maintained in the region of the deep hole and in the region between the Faraday cup and any thin film where the beam may enter the vacuum region of the Faraday cup. A test for residual secondary emission may be made by biasing the Faraday cup (and all its electrometer circuit) with respect to the surrounding conductors, or by biasing the surroundings with respect to the Faraday cup. I n either case the bias voltage supply must be very stable, for any variations in biassupply voltage induce unwanted charges on the Faraday cup. Bias voltages typically used go up to a few hundred volts, and sometimes to more than 1000 volts. By comparing Faraday-cup charge with some secondary monitor a t various bias voltages (both positive and negative), it is possible to measure how much charge transfer is due to low-energy charged particles. With proper magnetic trapping it should in every case be possible to reduce this as low as desired, certainly always well below 1%. It is more difficult to test for high-energy charged particles backscattered off the cup or knock-on electrons from the entrance windows. A real test for backscattering may involve varying the depth of the hole into which the particles fall in the Faraday cup and varying the position of the entrance window. While proton beams with energies less than 20 Mev can be stopped in graphite without generating secondary particles, either electron beams or high-energy proton beams generate many secondary particles. The Faraday cup must be massive enough to stop a sufficient fraction of the secondary particles. For electron beams the reader is referred t o the articles of Tautfest and Panofskys and of Brown and Tautfest.'* Rest practice is to make a mock-up of the Faraday cup prior to actual construction, and measure with nuclear emulsions the numbers of charged particles entering and leaving the skin of the Faraday cup at various points over its surface. By making the conservative assumption that all entering particles are of one sign and all leaving particles are the opposite sign, it is possible to set a firm upper limit for the loss of charge due to secondary charged particles. This procedure has been described by * See also Vol. 4 B,Chapter 10. 6 G.W.Tautfest and W. K. H. Panofsky, Phys. Rev. 106, 1356 (1957).

-TO

ELECTROMETER

/ /-.ooe

>

SOLID BRASS

I J P

TO PUMP

I

I

II

FOILS

-

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BEAM

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FIG.6. Faraday cup of Peterson, as arranged for the calibration of an ionization chamber for 340-Mev protons. Permanent magnets, not shown, have been used outside the vacuum wall to provide a transverse magnetic field of 100 gauss across the face of the Faraday cup.

2.8.

DETERMINATION OF FLUX AND DENSITIES

493

rELECTRONlCS PLATE VACUUM

POLYSTYRENE BIAS RING

MOLDED RUBBER GASKET

-POLYSTl‘RENE

-12

INCHES+

FIQ.7. Faraday cup of Brown and Tautfest for electrons up to 300 MeV. Permanent magnets, not shown, were inserted in the cavity of the collector electrode to give a transverse field of 300 gauss.

INTEGRATOR BIAS VOLTAGE IN KV

FIG.8. Curve of response versus dc collector-electrode bias, as obtained with the Faraday cup of Brown and Tautfest. On the basis of this cwve the gain or loss of charge due to low-energy electrons could be guaranteed less than 0.2%.

494

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

Aamodt, Peterson, and Phillipsa in constructing a Faraday cup for 340Mev protons, and a similar procedure is described by Brown and Tautfest for a Faraday cup for electrons. Figure 6 shows the Faraday cup of Peterson, as set up for the calibration of a n ionization chamber for 340-Mev protons. Figure 7 shows the Faraday cup of Brown and Tautfest as used by Tautfest and Panofsky for electrons, up to 300 MeV. Figure 8 shows a curve of Faraday-cup response versus dc bias as given by Brown and Tautfest. 2.8.1.2.2. ABSOLUTE BEAMMEASUREMENT BY COUNTING. With the development of very fast scaling circuits7 it has become practical to measure charged-particle beam intensities by counting the individual beam

-1 U

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7-

LEAD SCATTERING

COUNTER TELESCOPE NUMBER 2

COUNTER CHAMBER TELESCOPE

NUMBER

I

FIG.9. Schematic diagram of experimental arrangement for absolute beam monitoring by counting, as used to calibrate an ionization chamber.

particles. Since in most experiments the primary beam is much more intense than any beam which could be counted directly] it will usually be necessary to calibrate a secondary beam monitor, such as an ionization chamber, using several intermediate steps. The method described here presupposes the availability of a t least two fast scaling circuits (resolving time less than 10-7 second) and a duty cycle of the accelerator which produces the beam of a t least 1%. Using these parameters one would estimate about 1 % counting loss at counting rates of about 1000/sec. Figure 9 shows a suitable experimental arrangement. One scaler would record counts from a telescope of two counters placed directly in the beam, to be used at beam rates near lo3 per second. The other scaler would be used to monitor a second telescope placed behind a lead foil (for multiple scattering) so as to count only about 1% of the beam, the second telescope to be used for beam rates of 1O3-1Ob/sec, approximately. Crudely speaking, the ratio of counting rates of the two telescopes would be determined a t lo3 particles per second, and the ratio of counting rate 6

R. L. Aamodt, V. Peterson, and R. Phillips, Phys. Rev. 88, 739 (1952).

Model 520A, high-speed decade scaler of the Hewlett-Packard Co.; V. Fitch, Rev. Sci. Instr. 20, 942 (1949); J. Marshall and J. Fischer, Proc. Nall. Electronics Conf. 9, 491-500 (1953). 7

2.8.

DETERMINATION OF FLUX AND DENSITIES

495

in the second telescope to the ionization-chamber current would be determined a t lo6 particles per second (a beam level well suited to the ionization chamber). Actually, in each comparison several different beam levels would be used, and a n extrapolation of the ratio to zero beam intensity would be performed. Somewhat better accuracy might be attainable if more intermediate steps were used. Calibrations to 1% or 2% should be obtainable with this method in counting times of the order of one day. 2.8.1.2.3. BEAMMONITORING BASED ON INDUCTION ELECTRODES. When a beam of charged particles passes near an electrode it induces an apparent charge on the electrode. If the electrode geometry is carefully chosen,

FIG. 10. Arrangement of induction electrodes for measuring total accelerated charge in a synchrotron.

the apparent charge is easily related to the beam density. Although this method of beam measurement is not a t present applicable to external beams unless they are extremely intense, the method has been used for some time in connection with large synchrotrons such as the Cosmotron and Bevatron to measure the circulating beam charge within the machine. The measurement is greatly facilitated by the bunching of a synchrotron beam during acceleration. Present practice allows beam charge measurements to 5 or 10% on an absolute basis, with indication that this accuracy can be improved. Beam charges of a t least lo7 electron charges are needed in most cases, this minimum being determined by rf pickup from the rf system of the synchrotron. A suitable arrangement of electrodes is shown in Fig. 10. Each of the three electrodes shown is a hollow rectangular conducting box without ends, so the beam can pass through all three electrodes. The two end electrodes are grounded guard electrodes, necessary to give a well-defined effectivelength to the center electrode. The effective length will be called L, and is the distance from center to center of the two gaps between the electrodes. These electrodes would most frequently be installed within a straight section of the synchrotron.

496

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

If the circumference of the synchrotron is called K , the average charge induced by the beam on the (center) electrode is qL/K, where p is the beam charge. The average potential of the electrode must be measured with respect to the potential when there is no beam in the vicinity of the electrode, that is, when the beam bunch is on the other side of the synchrotron. A block diagram of a suitable circuit arrangement is shown in Fig. 11. A pulse generator is also included. By calibrating with the pulse generator one may make measurements that are independent of many of the circuit parameters. The pulse generator is assumed to have a very low output impedance and the amplifier to have a high input impedance.

PULSE GENERATOR

C II

‘

LINEAR

AMPLl F I ER

-

FAST

A

DC

- INTEGRATOR -OUTPUT

L

FIG.11. Block diagram of induction-electrode circuit.

The amplifier might well be a triode amplifier of approximately unit gain, followed by distributed amplifiers. It is assumed that the rise time of the amplifier system is appreciably shorter than the time between pulses during which the induction electrode is not affected by beam charge, so that the potential between pulses may be well established at the amplifier output. Furthermore the amplifier must be linear to considerable precision.* We assume that there is no overshoot a t the amplifier output. (The whole pulse structure should be of one sign.) Any overshoot, if present, would be compensated by a passive linear network inserted in the amplifier input or output circuit. The fast integrator circuit must serve the function of determining the “area” under the induction-elec trode pulses. A number of techniques are currently in use, At the Bevatron, only the fundamental frequency component is measured, and one obtains the absolute charge by applying a correction factor based on the Fourier analysis of the pulse shape as photographed on an oscilloscope screen. Th at method has been described by Heard.0

* See also Vol. 2, Chapter 6.2.

* Harry G . Heard, University of

California Radiation Laboratory Reports UCRL3609 and UCRL-8092. Unpublished.

2.8.

DETERMINATION OF FLUX AND DENSITIES

497

A different method will be described here.g Within the limitations described above, the fast integrator circuit may be that of Fig. 12. I n this circuit t,he diode serves to establish the minimum or quiescent potential (between beam bunches) as ground. The output is then obviously proportional t o the time average of the pulse, averaged over a complete rf cycle. The constants in this circuit must be chosen to minimize the imperfections of the diode (e.g., type OA85). Reasonable values are R 1 = R z = R S = 200,000 ohms, C1 = Cz = C3, RICl = lOO/f, where f is the radiofrequency of the machine. However, for the higher frequency

FIG.12. Circuit of fast integrator.

synchrotrons the capacities given here may be too small, becoming comparable to stray capacities, or giving insufficient sensitivity. The slow integration-integration over many acceleration pulses of the synchrotron-may be accomplished by switching the condenser Cs out of the fast integrator circuit and discharging it into a dc electrometer such a s that of Fig. 2 in which the input is always very nearly a t ground potential. The switching operation is to be performed once each acceleration cycle, a t the time in the cycle when the circulating beam charge is to be measured. The switching circuit must be well shielded, and resistors may be required in series with switch contacts to avoid undesired charge changes as the contacts open. A vibrating-reed electrometer may be called for. The pulse generator to effect the calibration is shown here only in a form suitable for representing negative beams, although circuits can be devised for positive beams based on the same principles. The circuit is shown in its essential components only in Fig. 13. The vacuum tube must be operated so that it is guaranteed to be cut of between pulses. Care must also be used to avoid overloading the tube. The time average of the output voltage is t,hen - I R , when measured with respect to the voltage between pulses, where I is the current indicated by the (well-calibrated) ammeter in the plate circuit, and R is the (well-calibrated) resistance 9 The fast integrator described here is similar to that used at the Cosmotron. C. E. Swartz (private communication, 1957).

498

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

indicated in the plate circuit. When coupled to the induction electrode through the known capacitance C (see Fig. 11), it imitates a beam whose charge is just

Q

= -(K/L)IRC.

This is the equation through which the calibration is effected. Although the calibration of the system described here should be independent of the frequency of pulses from the pulse generator, errors will of course be minimized if the pulse generator is used very close to the radio frequency of the accelerator. To the extent that the fast integrator works properly, the response should be independent of the width of the beam bunch. The

R

VARIABLE BIAS NEGATIVE

FIG.13. Basic pulse-generator circuit for calibrating induction electrodes.

bunching of the beam represented by the pulse generator may be varied by changing the bias and rf amplitude, but care must be used to make sure the tube is always cut off between pulses. It may in fact be advisable to add a bit of circuitry that will allow continuous monitoring of the minimum grid voltage on a separate meter. More complicated arrangements of induction electrodes can be used to measure the radial or vertical beam position. One such apparatus, as used in the Cosmotron, has been described by Swarta,'* but will not be described here. 2.8.1.2.4. OTHERABSOLUTEBEAMMONITORS. Other primary methods of measuring fluxes of charged particles include calorimetric methods, and the counting of tracks in nuclear emulsion. A full discussion of these subjects is omitted because they are believed to be of less general usefulness. lo C. E. Swartz, Rev. Sci. Instr. 24, 851 (1953).

2.8.

DETERMINATION

OF FLUX AND DENSITIES

499

2.8.1.3. Secondary Monitors. 2.8.1.3.1. IONIZATION CHAMBERAS RELATIVE BEAMMONITOR. * I n most work with charged-particle beams it is important to have some secondary standard for monitoring beam intensities, chosen to interfere as little as possible with the beam and to be useful over wide ranges of beam intensity. By far the most common instrument of this kind is the parallel-plate ionization chamber. The ionization chamber can be constructed with sufficiently thin foils so that the beam is little affected by its presence, especially for proton beams above 30 Mev or electron beams above 2 MeV. Since each beam particle that traverses the ion chamber produces many ion pairs in the gas of the chamber, electrometer drift currents rarely cause trouble. A number of precautions should be taken to achieve reliable operation. To decrease the tendency of ions, once formed, to recombine, one fills the ion chamber with a gas, such as argon, in which the negative charges may be collected (quickly) as electrons rather than (slowly) as negative ions. Variations of barometric pressure may be prevented from affecting the internal pressure by using a pressure well above (or well below) atmospheric pressure. To prevent drift due to leakage currents along insulators, one should always arrange that the insulators responsible for the electrical insulation of the electrode connected to the electrometer input (sensitive electrode) should have negligible voltage drop across them-always less than one volt. The chamber should have a well-defined volume of gas from which ions are collected, determined by two highvoltage electrodes, one on each side of the sensitive electrode. This last measure is necessary since contact potentials of the electrodes will always give rise to some ion collection, even in a region between two electrodes a t the same electrical potential. (The author has experienced a considerable degree of nonlinear operation of an ion chamber which had a grounded foil on one side of the sensitive electrode-there was a partial collection of ions from this region, with very different fractions of ion recombination a t different beam levels.) Figure 6 shows a very satisfactory form of ion chamber used by Chamberlain, SegrB, and Wiegand for monitoring proton beams in the energy range 100-300 MeV. Whenever the ion chamber is used, a check should be made that the electric field in the ion chamber is sufficient to collect all the ion pairs formed by the beam. This is done by comparing the ion chamber with some other monitor while using various voltages on the high-voltage electrode of the ion chamber (various collection voltages). Voltages of 300-1000 are quite typical for the ion chamber of Fig. 6, filled with slightly over one atmosphere of argon. If the collection voltage is SUE* See also Vol. 4 A, Section 2.1.5.

500

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

cient, the ion chamber will give a response independent of collection voltage. In judging whether the collection voltage is adequate (which in principle means extrapolating to infinite voltage) the following ruleof-thumb is useful for parallel-plate ion chambers that are reasonably near the point of complete charge collection: when the collection voltage is doubled, the recombination is cut to one-half or less (at any given beam intensity). Typically the rule might be applied as follows: If raising the collection voltage from 400 volts to 800 volts causes 1% change in ion chamber response (charge collected by the dc electrometer) then there is no more than 1%of charge loss by recombination at the higher collection voltage. Jesse and Sadauskis” have found that impurities of the order of 0.1 % can quite appreciably affect the response of an ion chamber. From this we infer that unless special precautions are taken it will not be possible to compare beams in different laboratories by the use of ion chambers without transporting the chambers and gas-filling apparatus from one laboratory to another. I n practice, less trouble seems t o be experienced than might be expected from the work of Jesse and Sadauskis, in getting reproducible results with argon-filled ionization chambers. Probably the argon used (believed 99.5% argon and the remainder mostly nitrogen) is already “saturated” with impurities in the sense of Jesse and Sadauskis. Methane, ethane, and ethylene are other promising gases for ionization chambers. Although they have been used less often than argon, they may be more reproducible because they are less sensitive to small amounts of impurity. Nitrogen may also be less sensitive to impurities than is argon. Oxygen is to be avoided, even as a small impurity, for electrons attach to oxygen. Thus, there is much more ion recombination when oxygen is present. Effects due to radioactivity induced by the beam in the materials of the ion chamber are not usually noticeable unless the ion chamber is accidentally exposed to a very high beam level shortly before a very weak beam is to be measured. Even in these cases there will be little error if ion chamber drift (with beam off) is measured immediately before and immediately after a measurement. The use of the electrometer with an ionization chamber is just the same as with a Faraday cup-except possibly that the drift corrections are usually smaller, fractionally, than with the Faraday cup. Ionization chambers have proved most useful for average beam intensities between lo3 and lo9 particles/sec-cm2, for a 300-Mev proton beam with a duty cycle of somewhat less than 1%. At lower intensities elec‘1

W. P. Jesse and J. Sadauskis, Phys. Rev. 88, 417 (1952); ibid. 100, 1755 (1955).

2.8.

DETERMINATION

OF FLUX A N D DENSITIES

50 1

trometer drift becomes troublesome and a t higher intensities it is difficult to avoid recombination of ions within the gas. A number of other methods for secondary beam monitoring will be mentioned briefly. RADIOACTIVITY. Reactions such as C12(p,pn)C" 2.8.1.3.2. INDUCED have frequently been used'2 to give a measurement of charged-particle beams through measurement of the radioactivity induced. If, as in this example, the product measured is ,&radioactive, it is difficult to make a n absolute measure of the yield. Thus it is important to be able to calibrate such a monitor (against a primary monitor) in the same laboratory where i t is to be used. For proton beams of energy higher than 1 Bev the production of the alpha-active Tb149by bombardment of gold promises to be a useful m0nit0r.l~The 4-hour lifetime of the activity is convenient for many purposes, and i t is a n activity easily counted accurately. The reaction has a high threshold, about 600 MeV. 2.8.1.3.3. SECONDARY-EMISSION MONITOR. Tautfest and Fechter14have described a practical secondary-emission monitor consisting of 20 thin aluminum foils placed in the beam, in vacuum. It has the disadvantages t,hat it yields a very small current (comparable to that from a Faraday cup) and that it must be operated in a good vacuum (pressure less than 10-6 mm of mercury). However, it has the important advantage th a t it gives a linear response to very high peak beam intensities (as tested with high-energy electron beams, 16 ma/cm2, and probably much higher values). Surface effects may alter the sensitivity, so frequent calibration may be necessary. MONITORS. Any substance that scintillates 2.8.1.3.4. SCINTILLATION when the beam passes through it can be placed in the beam, and the yield of light measured with a phototube or photomultiplier connected to a dc electrometer integrator. KitchenI6 has described a secondary monitor based on the weak scintillation of air being detected by a photomultiplier tube, the monitor being proved useful in the range 10-lo10-7 amp for 30-Mev protons. Such a monitor is capable of measuring larger beam currents than an ionization chamber, but its calibration may need t o be remeasured frequently as a protection against changes in photomultiplier response. 12 See, for example, R. L. Aamodt, V. Peterson, and R. Phillips, Phys. Rev. 88, 739 (1952). 1s Private communication from B. J. Moyer and S. Parker (1958). Moyer and Parker have used this monitor, based on work of R. B. Duffield, G. Friedlander, and L. Winsberg. See Lester Winsberg, Bull. A m . Phys. SOC.3 (No. 8), 406, F5 (1958). l4 G. W. Tautfest and H. R. Fechter, Rev. Sei. Znstr. 26, 229 (1955). l 6 Summer W. Kitchen, Rev. Sci. Znstr. 26, 234 (1955).

502

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

2.8.1.3.5. CERENKOV MONITORS. Where relativistic particles compose the beam to be measured, eerenkov light may be used, in much the same way as scintillation light. The Cerenkov radiator may be a solid or liquid substance, or (for extremely relativistic beams) a gas. The comments about scintillation monitors apply also to Cerenkov monitors. 2.8.1.3.6. RELATIVEMONITORS BASEDON COUNTER TELESCOPES. A thin target may frequently be placed in the beam to be monitored, and a counter telescope arranged to count the charged particles ejected from the target in some convenient direction. The precautions applicable to any counter telescope are also applicable here. By judicious choice of target, angle of the telescope, size of telescope, etc., one can have a monitor capable of handling an arbitrary beam intensity. Unfortunately these monitors are sensitive to many parameters, so their calibration tends to be different each time they are placed in the beam. If left undisturbed, these monitors can be very useful for short periods, however. 2.8.1.3.7. DETERMINATION OF FLUXDENSITIESOF PARTICLES SCATTERED FROM A TARGET. To determine a differential scattering cross section we must measure the number of (charged) particles R scattered into a detector of solid angle w. These quantities will be used in the usual expression for the differential cross section

where n B is the number of beam particles traversing a thin target with target atoms per unit area, measured normal to the beam direction. 2.8.1.3.7.1. Determination of the Scattered Flux R. I . Counters.* Very similar considerations can be applied to a great variety of different types of counter. Pulse ionization chambers, proportional counters, and scintillation counters (either with gaseous, liquid, or solid scintillator) are intended to give as signals electrical pulses whose sizes are proportional (ideally) to the energies deposited in the active counter material by charged particles. Geiger counters are intended to give a uniform response to any ionization within their gas. Cerenkov counters*6 may be used where it is desired to count only very fast particles or where only particles within a certain velocity interval are to be counted.lBJ7Essentially the same considerations will apply to the use of all of these counters.

NT

*See also Vol. 9 A , Chapter 2.1. For general information on Cerenkov Counters aee John Marshall, Cerenkov counters. Ann. Rev. Nuclear Sci. 4, 141 (1954), and Sections 1.5 and 2.2.1.3.1 of Vol. 5 A. ' 1 0. Chamberlain and C. Wiegand, The velocity-selecting Cerenkov counter. Proc. CERN Symposium on High-Energy Accelerators and Pion Phys., Vol. 2, p. 82. CERN, Geneva, 1956. 16

2.8.

503

DETERMINATION OF FLUX AND DENSITIES

TABLE I. Approximate Characteristics of Various Types of Counter

Assumed active material

Type of counter Geiger counter Proportional counter Pulse ionization chamber Scintillation counter Scintillation counter Cerenkov counter

Uncertainty in determining the time of an event (set)

By volume 90% argon, 10% alcohol, 10 cm Hg total pressure By volume 96% argon, 4 % COS,1 a t m Argon, 1 atm

Dead time (see)

Type of particle for which counter is most useful

10-6

10-4

Any

10-8

2 X 10-6

Any

10'0

2

Stilbene or good plastic scintillator NaI (T1 activated)

10-9

10-8

Any

10-8

10-6

Any

Glass

2

4 x 10-9

Relativistic particles

x

10-9

x

10-6

a, heavy ion

Table I gives some very approximate information about the characteristics of a few counter varieties. The data given are intended only for orientation, and are made up on the assumption that the active regions of the counters have linear dimensions of the order of a few centimeters. Many special arrangements have been used to count only one kind of particle in the presence of a background of other particles. We will not attempt to include here these more special devices, but will restrict our attention to the case in which, apart from background effects, only one kind of particle emerges from the target in the direction of the detector. A typical counter arrangement is shown schematically in Fig. 14, PHOTOMULTIPLIER

LIGHT PIPE SCI NTlL LATOR OF AREA A TARGET

SCI NTILLATOR

LIGHT PIPEPHOTOMULTIPLIER-

FIG.14. Geometry of scintillation-counter telescope. By placing the photomultiplier tubes on opposite sides as shown, one may eliminate effects due t o pulses from particles that hit the photomultipliers rather than the scintillators.

504

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

which shows as a n example a telescope of two scintillation counters. We leave to another section the problem of determining the effective solid angle subtended a t the target by the counter arrangement and center our attention now on the measurement of the number of counts of both counters in coincidence. A typical coincidence system is assumed, such as that described by Garwin. l 8 This system measures the number of particles passing through both counters, subject to these restrict,ions. (1) Some accidental coincidences will be recorded, due to particles which traverse only one or other of these counters, two such events sometimes occurring within the coincidence circuit resolving time. For a given integrated beam passed through the target, the number of accidental coincidences is not constant with respect to beam intensity, but is proportional (in first approximation) to the beam intensity. One corrects for accidental coincidences by extrapolating the number of counts per unit integrated beam to zero beam intensity. (Plot number of counts per unit integrated beam versus beam intensity, and make a straight-line extrapolation to zero beam intensity.) The theory of accidental coincidences is included in an article by Feather. l9 Accidental coincidences become a very prominent problem in experiments involving pulsed accelerators. (2) Some true coincidence events will be lost because two scattered particles, each traversing both counters, pass through the counters at substantially the same time. This effect is also corrected for, a t the same time as (1) above, by performing the extrapolation to zero beam intensity. This correction is also accentuated when the beam is derived from a pulsed accelerator. (3) I n some experiments there may be a small background, a s from cosmic rays, which is not derived from the beam, hence the rate of these events must be measured with the beam off, and this background rate subtracted from all observed rates measured with the beam on. (4) Some coincidences are due to particles that do not come from the target, although they are derived from the beam. I n most experiments it is sufficient to measure the number of these counts per unit integrated beam, by counting with the target removed, and subtract this number from the corresponding number measured with the target in place. However', it is always possible that the presence or absence of the target may affect this background, hence the source of the background should be carefully investigated to establish whether a simple subtraction process is adequate. l8 '9

R. L. Garwin, Rev. Sci. Znstr. 21. 569 (1950). N. Feather, Proc. Cambridge Phil. SOC.46, 648 (1949).

2.8.

DETERMINATION

OF FLUX AND DENSITIES

505

(5) Some of the pulses from the counters will be too small to actuate the coincidence circuit in the usual way. These small pulses may arise from particles that pass only through the edge of one counter and hence do not lose much energy in the scintillator, or from many other causes. The understanding of these events and the correction for the counts missed usually depend to a large extent on the so-called “bias curves.” The “bias” originally referred to discriminator bias a t the output of a linear amplifier. At the present it, is customary to express the bias in terms of the input a t the counter necessary to actuate the whole circuit in the normal way. The bias is proportional to the reciprocal gain of the

a W

-0

BIAS OF ONE COUNTER

FIG. 15. Bias curve of scintillation counter. The straight line used t o extrapolate t o zero bias is shown dashed.

amplifier. A fairly typical bias curve is shown in Fig. 15. I n the region of the plateau the circuit is counting nearly all the coincidences. When the bias is adjusted to higher values (lower gain), a n appreciable fraction of counts are lost. When the bias is set very low, additional counts usually are seen, due largely to photomultiplier noise and overloading of parts of the circuits. Using the plateau region only, a straight line may be matched to the bias curve. Where this straight line intersects the vertical axis one may read the number of counts per unit beam “extrapolated to zero bias.” One method of correcting for small pulses missed b y the coincidence circuit is to use the extrapolat,ed value of counts per unit beam. This is not, however, the only method. I n practice, one must choose a method commensurate with the accuracy desired and the information available about the det,ailed nature of the bias curve. II. Photographic emulsions. Nuclear photographic emulsions are frequently used t o determine the flux of scattered particles. The emulsions are exposed in an accurately known geometrical arrangement at known

506

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

distance from the target while a known number of beam particles traverses the target. The emulsions are usually held in an accurately machined holder designed to minimize errors due to beam misalignment.20 The emulsions are oriented so the particles to be counted enter the emulsion surface at a small grazing angle because the tracks, after development, are more easily seen and counted if they run nearly parallel to the emulsion surface. The numbers of scattered particles are determined by counting the tracks entering a known area of emulsion. For many purposes the method is convenient because many pieces of emulsion, placed to measure the scattering at various angles, may be exposed simultaneously. However, up to the present time it is necessary t o scan the emulsions by handoperated microscope, so the scanning process becomes very tedious whenever many scattering events are to be recorded.* Apart from the statistical errors inherent in any counting process, the most troublesome source of error is in the determination of the scanning efficiency. Experience has shown that different individuals have different average scanning efficiencies, and that, furthermore, a given individual may have a different scanning efficiency at one time than another. The accurate determination of particle fluxes requires constant vigilance and imagination in repeatedly determining the average scanning efficiency. While emulsions rarely suffer from errors due to accidental coincidences, virtually the same corrections must be made for backgrounds as have already been discussed for counters. 2.8.1.3.7.2. Determination of Solid Angle. The solid angle is usually defined by a detector of effective area A located a t a distance 1 from the target. In many cases the greatest dimension of the area A is much smaller than the distance 1 and the solid angle may then be given by w =

A/P.

Two extreme cases will be discussed, the first is that in which the full sine of the detector determines the area A . The second is the case in which a slit or aperture determines the area A . I. Full detector area as sensitive area. For definiteness this case will be discussed using the example of high-energy charged particles detected by a telescope of two scintillation counters in the geometry of Fig. 14. Assuming that the second scintillator is sufficiently large to intercept all

* Refer to Vol. 5 A, Chapter 1.7. See, for example, J. C. Allred, L. Rosen, F. K. Tallmadege, and J. H. Williams, Rev. Sci. In&. 22, 191 (1961) as well a8 G . W. Tautfest and W. K. H. Panofpsky, Phya. Rev. 106, 1356 (1957). 10

2.8.

DETERMINATION

OF FLUX AND DENSITIES

507

particles that have traversed the first scintillator (even allowing for multiple scattering in the first scintillator), the solid angle is determined by the first scintillator. Strictly speaking, the solid angle w is a function of the sensitivity of the counter. One will use a slightly different figure for I (in the expression for solid angle), depending on how much of the scintillator must be traversed to cause a useful count. Knowledge of this is, in turn, dependent upon the bias curve, which has already been discussed. In general one may say that the solid angle determination is most accurate if the thickness t of the scintillator is very small compared to the distance 1 from target to scintillator.

FIQ. 16. Geometry of counters with aperture used to determine the solid angle, showing the trajectory of a particle reflected from the inside of the aperture. This arrangement is seldom satisfactory for a precise determination of solid angle.

I I . Solid angle determined by slit or aperture. It is sometimes more convenient, especially when counting low-energy particles, to define the solid angle by a slit or aperture in a shielding wall. The aperture is then followed by a larger counter system, as in Fig. 16. A persistent difficulty with this arrangement is that it is difficult to calculate accurately the corrections for particles that traverse the edges of the aperture or reflect from the inside faces of the aperture. This arrangement should be satisfactory only if the shield thickness t can be made so small compared with target-to-slit distance 1 that there is a negligible difference between A/Z2 and A / V 2 (see Fig. 16), and furthermore only if the dimensions (width or diameter) of the aperture can be made large compared to the range of the particles in the shield material. It is rarely practical to satisfy these criteria. Many other arrangements will be used in practice. I n all of them it is well to consider in detail the effects of scattering from dense materials and the proper analysis of the bias curves.

508

2.

DETERMINATION O F FUNDAMENTAL Q U A N T I T I E S

2.8.2. Determination of Differential X-ray Photon Flux and Total Beam Energy*

2.8.2.1. Introduction. The choice of a particular technique for X-ray photon measurements will depend on the nature of the source, the desired information, the range of photon energies and intensities, the required accuracy, and the investment of time and effort contemplated. This section consists of a description of the available photon sources, followed by descriptions of particular measurement techniques. t High-energy measurements are stressed, although some information which is useful at low energies is also given. For the present purposes the dividing line between high and low energies is arbitrarily drawn a t 3 MeV. The discussion is limited t o photon beams of small divergence. If the source emits photons of only one energy, flux determination is a simple counting problem, which is not discussed here explicitly. For a source which emits photons with several energies, the differential photon flux, N ( k ) , is defined as the number of photons/sec with energy k, per unit energy interval. The quantities which can be derived from N ( k ) , and the units in which they will be expressed, are: Photon flux

=

s

N ( k ) d k , photons/sec

Differential energy flux Beam power

=

Beam intensity

1

=

k N ( k ) , watts/Mev

k N ( k ) d k , watts

‘I

=-

A

k N ( k ) d k , watts/cm*

A is the area of a beam cross section. Another quantity which is often useful is the total beam energy ( T B E ) defined as the product of beam power and exposure time. The unit of total beam energy used here is joules, even though the use of Mev would be more convenient for converting t o “equivalent quanta,” a term occasionally used by high-energy physicists. The number of “equivalent quanta’’ is defined as the ratio of total beam energy to maximum photon energy. The determination of N ( k ) for a given X-ray beam can be separated into a determination of the relative dependence of N ( k ) on k , plus an absolute normalization. The dependence on k may be known from theory, but in most cases it must be determined with a photon spectrometer. If

t The techniques described are illustrative examples of methods which have proven useful; there has been no attempt to be comprehensive. ~

* Section 2.8.2 is by J.

S. Pruitt and H. W. Koch.

2.8.

DETERMINATION

OF FLUX AND DENSITIES

509

the absolute efficiency of the spectrometer is known, the measurements can also be used for normalization. In the case of thin-target high-energy bremsstrahlung beams, for which the shape of the differential flux distribution is assumed known from theory, absolute normalization, in principle, reduces to determination of the photon flux, which is a counting problem. However, the great preponderance of photons with energies less than 1 Mev makes a simple count extremely insensitive t o the flux of high-energy photons. The alternative customarily used when the bremsstrahlung spectral shape is known is t o measure the total beam energy, a quantity which is more sensitive to high-energy photons and can be evaluated by several techniques which do not require the use of a spectrometer. 2.8.2.2. Sources of Photon Beams. The most common photon beams are bremsstrahlung beams, discussed below. Their major characteristic is their broad energy spectra, which limit their value in physical research. For instance, use of a bremsstrahlung beam to determine a reaction cross section as a function of photon energy requires the untangling of the effects of photons of different energies, which greatly complicates the analysis. There are, however, several techniques which show promise of allowing direct experimentation with monoenergetic photons whose energy can be varied at will. One of these, positron annihilation in flight, is also discussed in this section. The last part of this section describes two examples of sources of monoenergetic photons of fixed energy. 2.8.2.2.1. BREMSSTRAHLUNG. Bremsstrahlung are produced when electrons bombard matter and are decelerated in the atomic fields. For electrons of kinetic energy T , photons will be generated with all energies between k = 0 and k = T . Theoretical calculations of the elementary process have been made by several authors, and their theories have been compared with experimental measurements of the differential bremsstrahlung cross section.' For T > 10 MeV, theory and experiment agree within the experimental errors. Theory should be accurate to within about 2% a t these energies, but the experiments are only good to about l o % , so the shape of a differential photon flux distribution can, in principle, be calculated relatively precisely and easily. For T < 10 MeV, however, the theoretical approximations lose their validity and th e differences between the approximate theory and experiment may be larger than 10%. The available sources of high-energy bremsstrahlung include betatrons, electron synchrotrons, and linear accelerators. They are usually sources of thin-target spectra, where the target is too thin t o significantly degrade the electron energies but thick enough for multiple scattering to occur. The experimental spectra in such a case can be approximated by those 1

H. W. Koch and J. W. Mots, Revs. Modern Phys. 31,920 (1959).

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

m

5

20

I

a

a

In ' v)

za m

O

.2

0

.4

.0

.6

1.0

k k,,

FIG.1. Product of theoretical integrated-over-angles thin-target bremsstrahlung cross section and photon energy.' The formula used is listed as 3BS(e) in reference 1.

i

2

5

10

20

50

100

200

500

k, Mev

FIG.2. Theoretical integrated-over-angles thin-target bremsstrahlung cross section.* The formula used is listed as 3BS(e) in reference 1.

2.8.

DETERMINATION

10 2.0

km=(Mev) Photons/sec X

19 1.3

511

OF FLUX AND DENSITIES

36 0.9

68 0.6

130 0.3

260 0.2

480 0.1

260 0.2

480 0.1

I

TABLE11. 15 Mev Photons/sec/Mev/watt 1

kmsr(Mev) Photons/sec X 10+

19 1.6

36 1.2

68 0.7

130 0.4

lists the number of photons/sec with energies k 2 0.9km,,, and Table I1 lists the photons/sec with k between 14.5 and 15.5 Mev, both for a bremsstrahlung beam power of 1 watt. The shape of thick-target differential flux distributions, where electron energy loss is no longer negligible, depend upon the target thickness and are much more difficult to evaluate. The available calculations are reviewed in reference 1. Specific examples based on simple assumptions can be found in references 4 and 5. 2.8.2.2.2. POSITRON ANNIHILATION IN FLIGHT. The annihilation of positrons with atomic electrons in a medium produces two photons. If it occurs when the positron is at rest, these photons are isotropic with opposite momenta and with energies of 0.511 MeV. If the annihilation occurs in flight, a less likely event in a thick medium, one of the photons has a larger energy, which depends only on the emission angle and positron energy. In recent experiments,s thick tantalum targets were L. I. Sohiff, Phys. Rev. 83, 252 (1951). 8 A . S. Penfold and J. E. Leiss, Phys. Rev. 114, 1332 (1959). E. Hisdal, Phys. Rev. 106, 1821 (1957); E. Hisdal, Arch. Math. Nuturvidenskub 54(3), 1 (1957). N. E. Hansen and S. C. Fultz, UCRL 6099 (1960). C. Tzara, Compt. rend. 246, 56 (1957);C.P. Jupiter, N. E. Hansen, R. E. Shafer, and-S. C. Fultz, Phys. Rev. 121, 866 (1961);F. D.Seward, C. R. Hatcher, and S. C. Fultz, ibid. 121,605 (1961);F. D.Seward, S. C. Fultz, C. P. Jupiter, and R. E. Shafer, UCRL 6177 (1960). a

512

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

bombarded with high-energy electrons from a linear accelerator. Positrons produced in the resultant shower were magnetically analyzed, and those of a given energy were allowed to annihilate in a thin beryllium foil. The photons observed at a given angle were monoenergetic within the limitations discussed below. This technique for producing monoenergetic photons of variable energy has a very low efficiency because it involves two processes, positron production and annihilation in flight. Each has an efficiency of the order of 10W when the width of the observed photon spectrum is just a few per cent of the peak energy, and only one photon is observed for lo1*electrons incident on the tantalum. Specifically, it has been shown’ that for a tantalum thickness, X , between 0.05 and 0.3 radiation lengths, with a n electron kinetic energy, T , between 10 and 100 MeV, the number of positrons observed in a solid angle, a, a8

I _15

-

-

1.0 -

-

U

0

5 Resolution

10

15

(%I

FIQ. 14. Experimental energy resolution versus photon energy for the anticoincidence scintillation spectrometer of reference 22. The resolution varies inversely as the square root of photon energy.

which together with the crystal at the rear acts as the anticoincidence shield. The great improvement in the response function shape can be inferred from the curves of Fig, 13, which show the pulse height distributions produced by 2.22-Mev gamma rays from neutron capture in hydro21 R. E. Connally, Rev. Sci. Instr. 24, 458 (1953); R. D. Albert, Rev. Sci. Instr. 24, 1096 (1953); P. R. Bell, Science 120, 625 (1954); K. I. Roulston and S. I. H. Naqvi, Rev. Sci. Znstr. 27,830 (1956); R. C. Davis, P. R. Bell, G. G. Kelley, and N. H. Lazar, IRE Trans. on Nuclear Sci. NS-S,82 (1956); C. C. Trail and S. Raboy, Rev. Sci. Instr. SO, 425 (1959); also see papers by R. W. Perkins, J. M. Nielsen, and R. N. Diebel, and by W. H. Ellett in Proc. Total Absorption Gamma-Ray Spectrometry Symposium, OTI-TID-7594 (1960). (2 C. 0. Bostrom and J. E. Draper, Rev. Sci. Insfr. 32, 1024 (1961).

2.8.

527

DETERMINATION OF FLUX AND DENSITIES

gen without anticoincidence (a) and with anticoincidence ( b ) . Curve c is background. Figure 14 is a summary of the energy resolutions obtained and Fig. 15 is a plot of the crystal efficiency. The efficiency drops rapidly with increasing energy, and is only 7% at 10 Mev. This decrease is caused by the increasing range of secondary electrons and photons, which make total absorption events increasingly rare. Figure 15 indicates that the value of the anticoincidence spectrometer is limited at energies above 10 MeV. 1.0

0.5 >.

0 Z

w - 0.2

0 LL LL

w 0.1

0.05 0.1

0.2

0.5

1.0

2.0

5.0

10.0

k, M e V FIG. 15. Experimental detection efficiency of the anticoincidence scintillation spectrometer of reference 22 as a function of photon energy.

2.8.2.3.1.2.1. Applications. There have been attempts to measure both relative and absolute gamma ray fluxes with single-crystal spectrometers.23 If photons of more than one energy were present, these determinations were greatly complicated by the mixing of the peak of one component of the pulse height distribution with the tails of other components. The pulse height distribution obtained with the spectrometer of Fig. 12 for chlorine is shown in Fig. 16, and indicates that the anticoincidence spectrometer is a valuable tool for sorting out complex spectra. A matrix solution of Eq. (2.8.2.4) is extremely impractical when the pulse height, spectrum is complex, as in Fig. 16. In such a case, it is more reliable t o successively subtract response functions fitted to the highest energy peak, the next highest, and so 0 n . ~ For ~ 3 ~the anticoincidence spectrometer, the response function tail (curve b minus curve c in Fig. 13) is so small that this peeling process will not reveal structure in the photon 28

N. H. Lazar, I R E Trans. on Nuclear Sci. NS-6, No. 3 (1958).

528

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

2000

I

I

I

I

I

PULSE HEIGHT

1

I

6,

I

I

1

Mev

FIG. 16. Anticoincidence scintillation spectrometer pulse height distribution produced by gamma rays from neutron capture in chlorine.2'

spectrum which was not apparent in the pulse height distribution, but this is not necessarily the case for a single-crystal spectrometer. 2 . 8 . 2 . 3 . 1 . 3 . Coincidence Scintillation Spectrometers. Figure 17 is a cutaway diagram of the two crystals used in a high-resolution coincidence spectrometer recently developed for use with high-energy X-rays.24 Incident photons pass through the axial hole in the front crystal and are absorbed by the 9-in. rear crystal. Only voltage pulses produced by pair production events which occur close to the entrance surface are recorded. Most of the volume of the big crystal is consequently available for absorbing the showers produced by these events, and the energy leakage is much smaller than with a single-crystal spectrometer which accepts pulses produced by events occurring at all depths. 2* B. Ziegler, H. W. Koch, and J. M. Wyekoff, Bull. Am. Phys. SOC. 6,11 (1961), and paper to be published.

2.8.

DETERMINATION

OF FLUX AND DENSITIES

529

The selection of pair production events is made by requiring coincidence of the rear-crystal pulses with pulses from the front crystal produced by 0.511-Mev gamma rays. These photons originate from annihilation of positrons at rest in the rear crystal (see Section 2.8.2.2.2). The total attenuation coefficient of sodium iodide for 0.511-Mev gamma rays is large enough (0.09 cm2/gm) to ensure that both the annihilation photons and the positrons were generated within a few centimeters of the entrance to the rear crystal.

FIG. 17. Cutawa,y diagram of the crystal arrangement in the coincidence scintillation spectrometer. Voltage pulses produced by absorption of photons in the 9-in. rear crystal are recorded only if a 0.511-Mev photon from positron annihilation simultaneously produces a pulse in t h e crystal with the axial hole.

Response functions for this spectrometer have been measured with 6.14- and 17.64-Mev gamma rays from F l g ( p , y )and Li7(p,y) respectively, and these are shown in Fig. 18. Comparison with the curves of Fig. 5 shows that the coincidence spectrometer has much better energy resolution than a single-crystal spectrometer. The detection efficiency of this instrument is only a few per cent for 10-Mev photons, but increases with increasing energy. 2.8.2.3.1.3.1. Applications. The coincidence spectrometer has been used to measure photon total attenuation coefficients in an experiment similar to the one described in Section 2.8.2.3.1.1.1 with the same experimental arrangement (Fig. 7). The data have not yet been analyzed, but

2.

530

DETERMINATION OF FUNDAMENTAL QUANTITIES

the accuracy of attenuation coefficient measurements will still be about 1 %. The anticipated improvement over the single-crystal measurements concerns the dips in the curves in Fig. 8, which are produced by nuclear absorption processes. The coincidence spectrometer has distinguished detail in the nuclear absorption cross section which the single-crystal instrument could not resolve.

-

1.0-

-

I

W

s

I

?L

-

+- .8I

2

-

W

I

W

.6

-

-

-

v)

J

-

3

n

w

-

-2.8 %

-2.0%

.4-

L a

-

2

.2-

k

-I

-

0

.

0

I

I

I

I

1

5

l

l

l

l

10

l

l

I

l

l

15

FIG.18. Coincidence scintillation spectrometer response functions measured with 6.14- and 17.64-Mev gamma rays from FIg(p,y) and LiT(p,y),respectively, normalized to unity at the peaks. The peaks correspond to capture by the crystal of all except 0.51 Mev of the incident photon energy.

When a high-resolution spectrometer is used to analyze a continuous photon flux distribution with little or no structure, there is a simpler method of solving Eq. (2.8.2.4) for N ( k ) than the matrix solution described in Section 2.8.2.3.1.1.1. The product N ( k ) e ( k ) in the integrand of Eq. (2.8.2.4) can be written as a Taylor series in the neighborhood of a particular photon energy, ko: P(k) = N(k)e(k) = P(ko)

1 dT(k0) Wko) +(k - ko)* dk (k - ko) + 2 7

+... Therefore, Eq. (2.8.2.4) can be rewritten :

(2.8.2.11)

2.8. For a given

E,

DETERMINATION OF FLUX AND DENSITIES

531

if ko is chosen so that :

ko(a)

(2.8.2.13)

=

the second term on the right of Eq. (2.8.2.12) vanishes, and the equation may be rearranged to yield: (2.8.2.14) where

+

higher order terms. (2.8.2.15) 6 = d2P’dk2(ko) J K ( k l r ) ( k - k o ) 2dk 2P(kO) J K ( k , 4 dL Therefore, the number of counts/sec in channel e of a pulse height distribution is approximately proportional to the differential photon flux at LO, the center-of-gravity energy of the response functions for E. If the differential photon flux distribution changes slowly with photon energy, as in the attenuation coefficient distributions of Fig. 11, 6 can be neglected in Eq. (2.8.2.14). In the region of the nuclear absorption dips, however, this technique could be used only if 6 was included in the calculations. The determination of N ( k ) would have to be made by successive approximations. 2.8.2.3.1.4. cerenkov Spectrometers. Large blocks of dense material like lead glass emit Cerenkov light when irradiated with X-rays. 26 The advantages of a Cerenkov spectrometer are that blocks large enough to absorb almost all of the photon energy can be manufactured with little difficulty, and faster response times are possible than with sodium iodide, since the Cerenkov light is emitted by the secondary electrons themselves rather than by the crystal atoms. The great disadvantage is that Cerenkov light is generated only by electrons moving with velocities greater than the velocity of light in the block. A Cerenkov spectrometer cannot detect electrons with energies below about 200 kev, and, since there are usually a large number in this category, the spectrometer resolution is relatively poor. Kantz and Hofstadter have made measurements which indicate that a large spectrometer will have an energy resolution of about 17% for 180-Mev electrons.26 2.8.2.3.2. MAGNETIC SPECTROMETERS. Magnetic spectrometers in general consist of a thin foil converter, where the secondary electrons and positrons are generated, a magnetic field to separate the particles in energy, and detectors t o count the number in a given energy bin. There are two main types, Compton and pair spectrometers. The former meas26 26

See Vol. 5, A, Section 1.5.4 by Lindenbaum and Yuan. A. Kante and R. Hofstadter, Nucleonics 13(3), 36 (1954).

532

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

ure the number of electrons generated in the foil in Compton scattering interactions, and the latter are coincidence instruments which measure the number of electron-positron pairs generated. There are two basic reasons for the low detection efficiency of highresolution magnetic spectrometers. In the first place, the converter foil must be very thin, so that multiple scattering of the particles is not significant, with the result that only a small fraction of the incident photons generate secondary particles. Second, most of the secondaries generated by photons of a given energy must be discarded in order to obtain a one-to-one correspondence between photon energy and electron positron) energy. (or electron

+

y BFAM I

FIG.19. Schematic diagram of a single-channel Compton spectrometer.27 F is a Be converter foil, P is the magnetic field outline, and B is a baffle which defines the spectrometer acceptance angle, 8.

2.8.

DETERMINATION OF FLUX AND DENSITIES

533

2.8.2.3.2.1. Compton Spectrometers. Compton scattering is a simple two-body process which is well understood theoretically, and the spatial and energy distributions of the electrons can be accurately predicted over a large energy range. This permits evaluation of the absolute detection efficiency and of its variation with photon energy, a prerequisite for accurate determination of the shape of a broad differential photon flux distribution. For example, it should be possible to predict the absolute efficiency of the simple single-channel spectrometer shown in Fig. 1g2' to within 10% for all photon energies between 1 and 30 MeV.

B e TARGET THICKNESS, m m

FIG. 20. Theoretical energy resolution (dashed curves) and detection efficiency (solid curves) of a Compton spectrometer with an acceptance angle of 1.4"and an exit slit width of 1 % of the electron energy.Z8

The electron counter cannot distinguish between Compton and pair electrons, and a correction must be made for the contribution of the latter. This is customarily done by repeating the measurements with the magnetic field reversed so that positrons are counted, and subtracting the positron count from the electron count. Detailed investigations of the characteristics of Compton spectrometers have been made below 30 Mev.28 The calculations of detection efficiency and energy resolution for a particular spectrometer are shown as a function of foil thickness and photon energy in Fig. 20. This indicates that 2% resolutions are possible, and also illustrates the rapid decrease of detection efficiency with decreasing photon energy. Z'J. W. Mots, W. Miller, H. 0. Wyckoff, H. F. Gibson, and F. S. Kirn, Rev. Sci. Znstr. 34, 929 (1953). z8 E. Keil and E. Zeitler, Nuclear Znslr. & Method8 10, 301 (1961).

534

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

A revealing comparison between a typical Compton spectrometerze and an anticoincidence scintillation spectrometer has been made by the designers of the instrument of Fig. 12.z2 They state that if the two spectrometers are placed as close to the photon source as possible without materially changing their resolutions, the counting rate in a pulse height peak will be larger for the crystal by five or six orders of magnitude. I n addition, the Compton spectrometer usually collects data at only one

FIQ.21. Experimental Compton spectrometer measurements of the relative bremsstrahlung differential energy flux distribution generated by 30-Mev electrons incident on a platinum target.81 The solid curve is predicted by Schiff*for thin-target bremsstrahlung at 0’ and the dashed curve has been corrected for electron energy loss in the target.

energy at a time and the crystal spectrometer measures the differential flux distribution over a wide energy range in one exposure. 2.8.2.3.2.1.1. Applications. Compton spectrometers have been used mainly to identify and measure the energies of nuclear gamma rays,29 but some measurements on bremsstrahlung spectra have also been reported.sOJ1Some relative bremsstrahlung measurements are shown in Fig. 21 as a function of relative photon energy for 30-Mev electrons incident on a platinum 2.8.2.3.2.2. Pair Spectrometers. Pair spectrometers are usually symmetrical instruments with identical electron and positron detectors which count only events where the photon energy (minus 2 m e ) is equally L. V. Groshev, A. M. Demidov, D. L. Lutsenko, and V. I. Pelekhov, “Atlas of Gamma-Ray Spectra from Thermal Neutron Capture.” Pergamon Press, New York, 1959. so J. W. Motz, W. Miller, and H. 0. Wyckoff, Phys. Rev. 86,968 (1953). H. Kulenkampff, M. Scheer, E. Schriifer, and J. Seyerlein, Naturwissenschaften 21, 1 (1958).

2.8.

DETERMINATION OF FLUX AND DENSITIES

535

divided between electron and positron. Coincident detection minimizes background and improves the energy resolution. The two main disadvantages of pair spectrometers are that it is difficult to calculate the absolute efficiency and its variation with photon energy, and that extremely long experiment times are required to obtain high-resolution data. This latter is best illustrated by the following simple arguments. Consider a spectrometer bombarded by N ( k ) dk photons of energy k per second. If the spectrometer only counts electrons and positrons with kinetic energies between T and T dT, the singles counting rates for electrons and positrons will be :

+

+

re = e(T)p dT JN(k)[u,(k,T) a,(k,T)I dk rP = e ( T ) p dT JN(k)a,(k,T) dk

(2.8.2.16)

where p is the foil thickness in atoms/cm2, and e ( T ) is the efficiency for detecting a particle of energy T ; a,(k,T) and o,(k,T) are respectively the differential cross sections per Mev for a photon of energy k to generate particles with energy T by pair production and by Compton scattering in the foil. The counting rate for true coincidences will be: rt where k’ mately :

=

2(T

=

e2(T)pd T 2 N(k’)a,(k’,T)

(2.8.2.17)

+ mc2). The random coincidence rate will be approxir r = 2rrerp

(2.8.2.18)

where 7 is the detector resolving time. The time required for n true coincidences is: time =

n Drt

__

(2.8.2.19)

where D is the duty cycle of the source (D = 1 for a steady source). With the help of Eqs. (2.8.2.16), (2.8.2.17), and (2.8.2.18), this time can be expressed as a function of the counting errors, random coihcidence errors, and energy resolution without knowing any of the spectrometer parameters except the resolving time, r , and the collection efficiency, e : time

=

87

De26,6,26,2f @‘I.

(2.8.2.20)

6, is the ratio of random coincidences t o true coincidences, 6, is the statistical counting error ( l / x h ) , and 6, is the spectrometer energy resolution (dk‘/k‘).f(k’) is the dimensionless factor:

536

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

f(k’) has been calculated for the case where N ( k ) varies as I l k for

which approximates a bremsstrahlung spectrum (Fig. 2). The approximation was made that u,(k,T) is independent of T , for T < k - 2mc2. The values of f(k’) listed in Table IV were obtained for a platinum foil and should be somewhat larger for lower atomic numbers. Table V lists the predicted experiment times as a function of energy resolution for 25-Mev photons in a beam with a peak energy kmax= 50 MeV. TABLE IV. f ( k ’ ) for Equation (2.8.2.20), with Bremsstrahlung Beams k’(Mev) k,,..(Mev) 25 50 100 200

25

50

100

200

0.8 2.4 3.7 4.6

0 0.3 2.4 3.5

0 0 0.1 2.4

0 0 0 0.02

TABLE V. Pair Spectrometer Experiment Time Versus Resolution &(%)

t(hours)

8 4 2 1 0.5

0.7 3 12 46 185

The other parameters used were 1% random coincidences, 1% counting error, 0.01 psec resolving time, e = 0.8, and D = 0.018, the latter correspond& t o 100-psec bursts of X-rays from an accelerator with a repetition rate of 180 bursts/sec. The numbers in this table indicat,e that high-resolution pair spectrometer measurements are impractical under normal conditions with a broad photon spectrum. The most obvious ways to decrease the experiment time are to reduce T , increase e, or increase D. 2.8.2.3.2.2.1. Applications. As with Compton spectrometers, pair spectrometers have been used principally with gamma rays below 20 Mev.32 A few pair spectrometer measurements with bremsstrahlung beams have s* R. L. Walker, acd B. D. McDaniel, Phys. Rev. 74, 315 (1948); see also the article by D. E. Alburger, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed ), pp. 767-794. Interscience, New York, 1955.

2.8.

DETERMINATION OF FLUX AND DENSITIES

537

been reported in the 1iteratu1-e.~~ In addition, a pair spectrometer with an energy resolution of +% at 20 Mev has recently been used t o determine The data from this latter the nuclear absorption cross section of 016.34 experiment are plotted in Fig. 22. They were obtained by studying the total attenuation coefficients of water, and did not require absolute knowledge of the spectrometer efficiency (Sections 2.8.2.3.1.1.1 and 2.8.2.3.1.3.1).This work was made possible by stretching the X-ray pulse until it was 3000 psec long, but it still required approximately 3 months of work at a rate of about 144 hours per week to obtain the data shown in Fig. 22. The long experiment time was justified in order to demonstrate and measure the structure in the tot(a1 photonuclear absorption cross section. 2.8.2.4. Total Beam Energy Measurements. The techniques for measuring total beam energy ( T B E ) differ for high- and low-energy X-rays partly because of differences in the available beam intensities. In units of pw/cm2 at a distance of 1 meter from the source, typical low-energy intensities range from about 0.5 a t 50 kev up to several hundred at 3 MeV. At high energies, corresponding numbers for betatrons and electron synchrotrons are 500 at 20 Mev up to 10,000 at a few hundred Mev. In both ranges, however, recently developed accelerators, the direct accelerators at low energies and linear accelerator at high energies, promise to increase these numbers by several orders of magnitude except near 50 kev. There were early attempts to use calorimetry to determine the total beam energy with low-energy X-rays, but the low beam intensities made accurate measurements impossible. The alternative which was adopted was to measure what is technically called expos~re-dose,~~ in roentgens. The roentgen is defined in terms of ionization in air, which can be accurately measured with low-intensity beams. The exposure-dose is proportional to the energy/cm2 in an X-ray beam, and the proportionality factor varies slowly with photon energy. One roentgen is equivalent to 3360 ergs/cm2 and one roentgen/min to 5.6pw/cm2 both to within 10% between 0.06 and 2 MeV. One advantage of the roentgen at low energies is that both the total beam energy and the energy locally absorbed in an irradiated medium (the absorbed-dose) can be calculated from the exposure-d~se.~~ This duality 8* J. W. DeWire and L. A. Beach, Phys. Rev. 83, 476 (1951); G. Diambrini, A. (3. Figuere, A. Serra, and B. Rkpoli, Nuovo cimento [lo] 16, 500 (1960). SIN. A. Burgov, G. V. Danilyan, B. S. Dolbilkin, L. E. Laeareva, and F. A. Nikolayev, to be published. *6 Report of the ICRU. Natl. Bur. Standards (U.S.) Handbook No. 78, 2 (1959). 86 Protection against Betatron-Synchrotron Radiations up to 100 MeV. Natl. Bur. Standards (U.S.) Handbook No. 66, 26 (1954).

50

40 L

O

e.E

..

0

30

I-

u W m u) m

g

20

0

2

0 I-

a U

::

10

m

a LT a W A

0

0

3

2

I 19

I 20

I

1

1

I

I

I

21

22

23

24

25

26

k, M e v FIG.22. Experimental pair spectrometer messurements of the cross section for photonuclear absorption in

0'' obtained from studies of the attenuation of a 200-Mev bremsstrahlung beam by 100 gm/cm2 of water.84

2.8.

DETERMINATION

OF FLUX AND DENSITIES

539

arises because at low energies X-rays essentially deposit their energy where they interact with the medium, At high energies, the roentgen is not a very useful unit of either exposure-dose or of absorbed-dose, because the ranges of secondary particles and primary photons are comparable. The exposure-dose in roentgens cannot be measured directly, and the absorbed-dose at a point depends on the incident energy at many points in the medium. These reasons, plus the higher intensities, have led to a return to exposure measurements in absolute energy units a t high energies. However, at least one of the techniques which has been used for this purpose at high energies, the calorimeter, has also been used for some low-energy measurements, as described below. 2.8.2.4.1. LOW-ENERGY FREE-AIRIONIZATION CHAMBERS. The roentgen is defined below 3 Mev as “the exposure-dose of X- or gammaradiation such that the associated corpuscular emission per 0.001293 gm of air produces, in air, ions carrying 1 electrostatic unit of quantity of electricity of either sign.JJ36For photons of energy k, the beam energy content in ergs/cm2 can be calculated from the exposure-dose in roentgens with the e q u a t i ~ n ~ ~ , ~ ~ : ergs/cm2 roentgen =

[g]

1 0.001293pa(k)

(2.8.2.22)

where W is the average energy in ev required to produce one ion pair in air, yn(k) is the energy absorption coefficient for air in cm2/gm, the sum of total attenuation coefficients describing photoelectric, Compton, and pair events, if each is multiplied by the fraction of the photon energy given to secondary electrons. Calculated values of the ratio of erg/cma t o roentgens are plotted in Fig. 23, using W = 33.6 ev.asThe experimental points at 0.66 and 1.25 Mev were measured with calorimeter^.^^ Exposure-dose in roentgens can be measured directly with a free-air ionization chamber, or more indirectly with a thimble chamber which has been calibrated with a free-air chamber. Figure 24 is a schematic diagram of a typical free-air chamber for photons with energies less than 500 k e ~ . ~ ~ It collects ionization equal t o the ionization produced by electrons gens’ G. J. Hine and G. L. Brownell, “Radiation Dosimetry,” p. 13. Academic Press, New York, 1956. ** Z. Bay, W. B. Mann, H. H. Seliger, and H. 0. Wyckoff, Radiation Research 7,567 (1957). 88 P. N. Goodwin, Radiation Research 10,6 (1959); I. T. Myers, W. H. LeBlanc, and D. M. Fleming, unpublished data (1961). 40 H. 0. Wyckoff and F. H. Attix, Design of Free-Air Ionization Chambers. Nail. Bur. Standards (U.S.) Handbook No. 64 (1957).

540

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

erat,ed in the volume V , which is defined by the aperture area and the Iength of the collecting electrode. The large dimensions of this chamber are necessary because the electrons generated in the beam must lose all of their energy in air before striking the plates, and because of the requirements of electronic equilibrium (the number of electrons and the electron energy entering the

0

0.01 0 0 2

0.05

0.1

0.2

0.5

I

2

5

10

k, Mev FIG. 23. Theoretical ergs/cm' per roentgen for photon energies between 0.01 and 10 MeV. The curve wm calculated for W = 33.6 ev required to produce an ion pair in air.asThe points at 0.66 and 1.25 Mev were measured calorimetrically with Cs1a7 and Coeo sources, respeotively.39

collecting region must equal the number ,and energy leaving). These requirements set an upper limit on the energy of photons which can be measured with a free-air chamber of practical size, but the same general type of chamber has been successfully used at energies as high as 2.6 Mev by filling it with air at high pressure to reduce the electron ranges.41 At still higher photon energies, electron ranges become comparable with 4 1 H. 0. Wyckoff, J . Research Natl. Bur. Standards 64C, 87 (1960); K. K. Aglintsev and G . P. Ostermukhova, Proc. Mendeleeu All-Union Sci. Research Znst. Metrology No. 66 (1961).

2.8.

DETERMINATION

OF FLUX AND DENSITIES

54 1

photon mean-free-paths, and it is difficult to correct for the lack of electronic equilibrium. 2.8.2.4.2. ABSOLUTEHIGH-ENERGY MEASUREMENTS. There are five techniques which have been used for measuring total beam energy in absolute units with errors of only a few per cent. Most of the measurements have been used to calibrate standard ionization chambers in lightly filtered bremsstrahlung beams with peak photon energies, kmax, between 6 and 1080 MeV.

FIQ.24. Schematic cross section of a free-air ionization chamber for measuring exposure-dose in roentgens at energies below 500 k e ~ . ~This O instrument is used t o determine the amount of ionization produced in the shaded volume V.

2.8.2.4.2.1. Calorimeters. X-ray calorimeters are designed t o absorb most of the energy in an incident photon beam and to detect the resultant temperature rise. The incident energy is determined by comparing this rise with the rise produced by dissipating an accurately measured quantity of electrical energy in the calorimeter. This technique has been used with low-energy bremsstrahlung beams with k,,, between 400 kev and and 1.4 Mev,42.43with CsI3’ and Co60 radioactive with high-energy bremsstrahlung beams with k,,, between 18 and 42 J. S. Laughlin, J. W. Beattie, W. J. Henderson, and R. A. Harvey, Am. J . Roentgenol. 70, 294 (1953). 4 s G. S. Dolphin and G. S. Innes, Phys. in Me& Bid. 1 , 161 (1956); J. MeElhinney, B. Zendle, and S. R. Domen, Radiation Research 6, 40 (1957).

542

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

300 Mev.42.4"41The advantage of this technique is that total beam energy can be measured without knowledge of either the photon spectral distribution or the atomic parameters of the calorimeter. It has the disadvantages of low sensitivity and the need for obtaining corrections for the amount of energy escaping from the calorimeter, which is sometimes as large as 10%.

?X-RAYS

FIG.25. Schematic diagram of a typical X-ray calorimeter, used for measurement of total beam energy.

Figure 25 is a generalized diagram of a typical X-ray calorimeter. It is designed to be adiabatic, with the absorbing elements, the lead cylinders, thermally isolated from their surroundings. The cylinder supports are made of a thermally insulating material (the use of threads is common) and the box is evacuated t o minimize exchange of heat by conduction. The surfaces of cylinder and box are normally plated with a metal of low thermal emissivity (such &g gold) to reduce the exchange of thermal radiation. Even with these precautions, it is necessary t o maintain the 44 P. D. Edwards and D. W. Kemt, Rev. Sci. Znstr. 24, 490 (1953). 4 6 s . P. Kruglov, Zhur. Tekh. Fiz. 28, 2310 (1958);Soviet Phys.-Tekh. Phys. 5, 2120 (1958). 48 J. S. Pruitt and S. R. Domen, J . Reeearch Natt. Bur. Standards 66A, 371 (1962). 47 J. S. Pruitt and W. Pohlit, 2. Nalurforsch. lLb, 617 (1960).

2.8.

DETERMINATION O F FLUX AND DENSITIES

543

calorimeter box at a constant temperature. Good thermal isolation and good temperature control are well worthwhile since they minimize the corrections for heat transfer and permit longer exposures at lower intensities. Extreme precautions were taken with the NBS ~ a l o r i m e t e r , ~ ~ for instance, with the result that 90-min exposures to an X-ray beam with an intensity of 2 pw/cm2, which used a 13.8-cm2beam, produced a temperature rise of 2 X 10-40C,and yielded determinations of total beam energy with a rms deviation less than 4%. A calorimeter of the type shown in Fig. 25 detects changes in the reIative temperature of the two cylinders when one of them is irradiated with X-rays. This is done with thermistors, resistors with large temperature coefficient, which are embedded in the cylinders. The thermistors are connected in a Wheatstone bridge so that the output voltage is proportional t o an increase in temperature of the bombarded cylinder. Calibration is performed by dissipating power in separate resistive elements embedded in the cylinders. 2.8.2.4.2.2. The Scintillation Spectrometer Method. Large-sodium-iodidecrystal spectrometers have been used to measure total beam energy with the same experimental arrangement used to determine total attenuation coefficients (Fig. 10).48849 As pointed out in Section 2.8.2.3.1.1, a large-crystal spectrometer counts the total number of incident photons efficiently and accurately, if a correction is made for photons which traverse it without interaction. On the other hand, a counter of this type would be swamped with counts from low-energy photons if it was used in a lightly filtered bremsstrahlung beam, as mentioned in the introduction. This experiment avoids jamming by filtering out low-energy photons with a long absorber. The total energy in a beam irradiating the crystaI with the absorber removed can be calculated from the number of spectrometer counts with the absorber in position, n, using the equation:

TBE =

nJN(k)kd k JN(k)e-p(k)zdk

(2.8.2.23)

where N ( k ) is the differential flux distribution of photons incident on the absorber, and x and p ( k ) are the length and total attenuation coefficient of the absorber, respectively. This technique has the advantage of high sensitivity and has been used at bremsstrahlung energies as low as 6 MeV. It does require knowledge of the incident photon spectra and the total attenuation coefficients. However, Eq. (2.8.2.23) is relatively insensitive t o errors in the shape of N ( k ) , 45

49

E. 0.Fuller and E. Haywsrd, J . Research Natl. Bur. Standards 66A,401 (1961). J. E.Leks, J. S. Pruitt, and R. A. Schrack, unpublished data (1958).

544

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

and the validity of the assumed attenuation coefficients can be tested by varying the absorber length. 2.8.2.4.2.3. The Balanced-Converter Method. Measurements of ionization behind thin foils of different atomic number in a high-energy bremsstrahlung beam can be used to determine total beam energy. Using foils of atomic number Z1 and Z z and identical thickness, measured in electrons per cm2, it has been shown50 that the total beam energy is given by:

where p is the ionization measured behind the foils in a thin-walled parallel-plate chamber with an air gap thickness t, p e is the foil thickness in electrons/cm2, and I,,, is the minimum specific ionization produced in air by an electron. R ( 2 ) is a slowly varying function of atomicnumber which has the units of a cross section. It is essentially the pair production cross section for hydrogen averaged over the photon flux distribution, but contains factors which describe the deviation of the pair-plus-triplet cross section from a Z 2 dependence on atomic number, and the deviation of the secondary electron-positron pairs from minimum ionization. The values of R ( 2 ) originally calculated for carbon, aluminum, copper, and lead for 330-Mev bremsstrahlung are listed in Table VI. TABLE VI. R(2)for Equation (2.8.2.24), at 330 Mev

z R ( Z ) , barns

0.0254

13 0.0255

29 0.0250

82 0.0220

The great advantage of this technique is its experimental simplicity, but this advantage is at least partially offset by the requirements of detailed knowledge of the differential photon flux distribution, the pair production cross sections, and the specific ionization of electrons. 2.8.2.4.2.4. The Shower Method and the Quantameter. Measurements of ionization at different depths in a large absorbing medium irradiated by a photon beam can also be used to determine the total beam energy. If q(x) is the ionization measured in a thin-walled parallel-plate ionization chamber with a gap thickness t at depth x in the medium, the incident energy can be expressed ass1:

TBE

W, @

q(x) dx. (2.8.2.25) dI3t W , is the energy required t o produce unit ionization in the gas of the =

6o

W. Blocker, R. W. Kenney, and W. K. H. Panofsky, Phys. Rev. 79, 419 (1950).

61

A. P.Komar and S. P.Kruglov, Zhur. Tekh. Fiz. SO, 1369 (1960);Soviet Phys.-

Tech. Phys. 6, 1299 (1961).

2.8.

DETERMINATION

OF FLUX AND DENSITIES

545

chamber, d, and dg are the densities of the medium and the gas, and S is the ratio of the mass electron stopping power of the medium (Mevcm2/gm) to that of the gas, averaged over the energy spectrum of electrons entering the chamber. S can be predicted theoretically if the electron spectrum is known. It is a slowly varying function of energy. There have been several measurements of bremsstrahlung beam energy with the shower technique in the energy range between 16 and 1080 HIGH VOLTAGE r l O N COLLECTOR

/-

f S T A I N L E S S STEEL 6 rnm INSULATION

2 mm X-RAYS

10 m m

%-dx+I mm

2 mm

QUANTAMETER

FIG.26. Schematic cross section of the quantameter, a copper ionization chamber with twelve ionization gaps arranged to evaluate in a single measurement the total ionization produced in B high-energy penetrating shower.54 Mev.60-53The most r e ~ e n have t ~ ~ been ~ ~made ~ with a quantameter, the copper ionization chamber shown in cross section in Fig. 26.54 The quantameter determines the integral in Eq. (2.8.2.25) in a self-contained copper medium with a single measurement, automatically summing the ionization at several depths to evaluate the integral by Simpson’s rule for parabolic interpolation. The quantameter is not large enough to absorb all of the incident energy but compensates for the energy leaking from the sides of the chamber with peripheral air cavities, and compen62

53

D. C. Oakley and R. L. Walker, Phys. Rev. 97, 1283 (1955). J. W. DeWire, private communication (1959). R. R. Wilson, Nuclear Znst. 1, 101 (1957).

2.

546

DETERMINATION OF FUNDAMENTAL QUANTITIES

sates for transmitted energy with an oversized ionization gap at the rear. These corrections are usually of the order of a few per cent. The quantameter constant, the ratio of incident enetgy t o measured ionization, has been theoretically evaluated for the normal gas filling of argon to a pressure of 800 mm of mercury at 2 0 T , and for air filling to a pressure of 760 mm of mercury at 20°C. The calculated values of this ,~~ 4.80 and constant for argon are 4.79 X 1018 k 3% M e v / ~o u l o rn b and 4.72 X 10'8 k 1.8% Mev/coulomb,6' and for air, 7.74 and 7.81 X lo1* k 2% Mev/coulomb.61 2.8.2.4.2.5. Pair Spectrometer Method, Pair spectrometers (Section 2.8.2.3.2.2) have been used t o determihe total beam energy from measurements of the differential photon flux distributions (Eq. 2.8.2.17).62*63 There is no need for a high-resolution spectrometer in this application, so the arguments about long experiment times given in Section 2.8.2.3.2.2 are not very pertinent. The accuracy is usually limited by uncertain knowledge of the absolute detection efficiency and energy resolution of the spectrometer. 2.8.2.4.3. RELATIVEHIGH-ENERGY MEASUREMENTS. There are two techniques which have been used to determine total beam energy in relative units, to investigate the variation of ionization chamber calibrations with the peak energy of a bremsstrahlung beam. 2.8.2.4.3.1. Pair Spectrometer. Pair spectrometers have been used to measure the ratio of the total beam energy at two different bremsstrahlung energies by comparing the numbers of pairs produced by If n ( f ) is the number of pairs counted, photons of a fixed energy, f.66,66 the ratio of total beam energies is simply:

T B E , - n1(k)C1(5) -where C ( f ) = TBEa n&G)Cs(f)

3

kdk.

(2.8.2.26)

N ( k ) is the differential photon flux distribution of the incident photon beam. C ( f ) can be calculated as a function of the peak energy of the bremsstrahlung beam for a given 5. This method yields accurate measurements of the energy ratio without knowledge of either the spectrometer efficiency or energy resolution, although it does depend upon knowledge of the shape of the differential photon flux distribution. 2.8.2.4.3.2. Copper Activation. A second method for relative measurements of total beam energy requires counting positrons emitted by Cuez produced by the reaction Cu6a(y,n)Cuazin a copper foil in the photon s 6 F .J. Loeffler, T. R. Palfrey, and G . W. Tautfest, Nuclear Instr. & Methods 6, 50 (1959).

w.P. swanson, private communication (1961).

2.8.

DETERMINATION OF FLUX AND DENSITIES

547

beam.66 The cross section for this reaction, acu(k>,has a peak at 18 Mev and is zero below about 10 Mev and above about 38 If Ac is the measured positron activity produced by a bremsstrahlung exposure, the ratio of total beam energies in beams with two different peak photon energies is:

and N ( k ) is again the differential photon flux distribution. This method requires knowledge of the shape of the reaction cross section, acU(k), but, at high bremsstrahlung energies, N ( k ) varies slowly over the region where acu is large, and 6 is insensitive to errors in the assumed values of ucu. However, 6 is sensitive to the assumed shape of the differential photon flux distribution. 2.8.2.4.4. HIGH-ENERGY IONIZATION CHAMBERCALIBRATIONS. The four standard ionization chambers shown in Fig. 27 have been calibrated with measurements of total beam energy which cover the energy range from 6 to 1080 MeV. The three copper chambers were designed at Cornell University,68 the University of Illinois,44 and the Leningrad Institute of Physics and Technology (LPTI).46 The chamber designed at the United States National Bureau of Standards (NBS)4shas walls of an aluminum alloy, 2024 Dural (nominally 93.4% Al, 4.5% Cu, 1.5% Mg, and 0.6% Mn). The chambers are all air-filled and open t o the atmosphere. The calibrations of these chambers in bremsstrahlung beams with different peak energies, k,,,, are listed in Tables VII-X, which also show the method used and the place where the data originated. In addition to the institutes where the chambers were designed, the data comes from the California Institute of Technology (Cal. Tech.),62s62the General Electric Company (GE),63the Max Planck Institute for Biophysics (MP1),47 and Purdue University.66 Some of the Purdue and Illinois measurements were relative, and these are marked with asterisks. A few of the calibrations were obtained by intercomparison of the chambers. Those obtained by quantameter measurements are all based on the original calculation of the quantameter constant, 4.79 X 10l8 Mev/coulomb (7.51 X lobjoules/coulomb). The calibrations are all listed in units of joules/coulomb at 2OoC and 760 mm of mercury, although most were reported in other units. The Illinois measurements with the Illinois chamber were originally reported in e r g - ~ m / e s uand ~ ~ have been converted to joules/coulomb for a chamber with the design air gap thickness of 0.104 in. The listed k,, in this case 57 68

A. I. Berman and K. L. Brown, Phye. Rev. 96, 83 (1954). D. R. Corson,J. W. De Wire, B. D. McDaniel, and R. R. Wilson, NP-4972 (1953).

HIGH VOLTAGE

3 8 m m A1

HIGH VOLTAGE ,

\

HIGH VOLTPGE

64mm

I

O I‘ N

COLLECTOR

ILLINOIS

CORNELL

P LPTl

COPPER

ALUMINUM A L L O Y

STAINLESS S T E E L

NBS INSULATION

FIG.27. Schematic cross sections of four standard ionization chambers which have been experimentally calibrated to determine total beam energy in high-energy bremsstrahlung beams. The name below each chamber is the institute where it waa d&gn4.4w6.aw

2.8.

DETERMINATION

OF FLUX AND DENSITIES

549

TABLEVII. Calibration of the Cornell Ionization Chamber in a Small-Diameter Bremsstrahlung Beam, a t 20°C and 760 mm of Hg kmar(Mev) Institute 60 90 90 120 125 130 144 150 170 175 197 200 200 220 225 250 250 250 270 275 300 308 315 315 318 320 325 497 500 500 600 780 789 1000 1080

NBS NBS Purdue Purdue Purdue NBS GE Purdue Purdue Purdue Cornell GE Purdue Purdue Purdue Purdue Cornell Purdue Purdue Purdue Purdue Illinois Cornell Cornell Purdue Purdue Purdue Cal. Tech. Cal. Tech. Cal. Tech. Cornell Cornell Cal. Tech. Cornell Cal. Tech.

Methoda

Calibration (joule/coulomb)

Reference

Comparison Comparison Cu activation* Spectrometer * * Cu activation* Comparison Spectrometer Cu activation* Spectrometer * * Cu activation* Spectrometer Spectrometer Cu activation* Spectrometer * * Cu activation* Balanced converter Spectrometer Cu activation* Spectrometer * * Cu activation * Cu activation* Comparison Spectrometer Quantameter Balanced converter Spectrometer* * Cu activation* Quantameter Spectrometer Shower Quanta meter Quantameter Quant ameter Quontameter Quantameter

6.01 X l o 6 f 2.7% 5.82 2.7 5.67 5.62 5.54 5.54 2.7 5.79 3 5.58 5.71 5.67 5.96 5 6.00 3 5.84 5.81 5.94 6.00 4.3 6.17 5 6.16 6.26 6.31 6.58 6.45 2.5 6.43 5 6.64 3 6.67 3.8 6.62 6.64 7.27 3 8.17 2.7 7.08 4.9 7.65 3 8.17 3 8.32 3 8.75 3 8.99 3

55 55 55 55 55 55 53 56 55 55 53 53 55 55 55 55 53 55 55 55 55 53 53 53 55 55 55 53 52 52 53 53 53 53 53

KEY: * Normalized to curve at kmax = 250 MeV; kmaX= 270 MeV.

** Normalized to curve at

2.

550

TABLE) VIII.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Calibration of the Illinois Ionization Chamber in a Small-Diameter Bremsstrahlung Beam, at 20°C and 760 mm of Hg

kmax(Mev)

Institute

Method

Calibration (joule/coulomb)

Reference

25 30 35 40 45 50 60 70 90 110 130 146 150 170 195 244 293

NBS NBS NBS NBS NBS NBS NBS NBS NBS NBS NBS Illinois NBS NBS Illinois Illinois Illinois

Comparison Comparison Comparison Comparison Comparison Comparison Comparison Comparison Comparison Comparison Comparison Calorimeter Comparison Comparison Calorimeter Calorimeter Calorimeter

39.8 X lo6 f 2.5% 40.0 2.5 39.8 2.5 39.9 2.5 40.7 2.5 41.4 2.5 43.1 2.5 44.6 2.5 48.6 2.5 52.8 2.5 56.7 2.5 60.9 3 60.5 2.5 63.8 2.5 64.9 3 72.3 3 83.3 3

46 46 46 46 46 46 46 46 46 46 46 44 46 46 44 44 44

T A B LIX. ~ Calibration of the LPTI Ionization Chamber in a 40-mm Diameter Bremsstrahlung Beam Filtered by 4.0 gm/cma of Aluminum, at 20°C and 760 mm of Hg kmax(Mev)

Institute

Method

Calibration (joule/coulomb)

Reference

16 22 25 29 40 53 53 68 72 a5 85 95

LPTI LPTI LPTI LPTI LPTI LPTI LPTI LPTI LPTI LPTI LPTI LPTI

Quantameter Quantameter Quantameter Quantameter Quantameter Calorimeter Qu ant amet er Qua ntameter Calorimeter Calorimeter Quantameter Quantameter

8.52 X lo6 f 3% 3 8.31 8.15 3 8.07 3 3 7.90 7.75 1.7 7.80 3 7.88 3 7.87 1.5 1.o 7.95 7.99 3 8.09 3

51 51

51 51 51 51 51 51 51 51 51 51

2.8.

551

DETERMINhTION O F FLUX AND DENSITIES

TABLE X. Calibration of the NBS Ionization Chamber in a 42-mm Diameter Bremsstrahlung Beam Filtered by 4.5 gm/cm* of Aluminum, at 20°C and 760 mm of Hg k,,(Mev) 6 8 10 13 16 19 20 20 25 26 30 31 35 35 36 40 41 45 45 50 50 60 60 70 70

90 90 110 110 130 130 146 150 150 150 170 170 170 200 230 260 a

CaIibration (joule/coulomb)

Institute

Methoda

NBS NBS NBS NBS NBS NBS NBS NBS NBS NBS NBS NBS MPI NBS NBS NBS NBS NBS N BS NBS NBS NBS NBS NBS NBS NBS NBS NBS NBS NBS NBS Illinois NBS NBS Illinois Illinois NBS NBS Illinois Illinois Illinois

Scintillator Scintillator Scintillator Scintillator Scintillator Scintillator Scintillator Calorimeter Scintillator Calorimeter Scintillat or Calorimeter Calorimeter Scint illator Calorimeter Scint illator Calorimeter Calorimeter Scintillator Scintillator Calorimeter Calorimeter Scintillator Scint illator Calorimeter Calorimeter Scintillator Scintillat or Calorimeter Calorimeter Scint illator Comparison Calorimeter Scintillator Pair spectrometer * Pair spectrometer* Calorimeter Scint illator Pair spectrometer * Pair spectrometer* Pair spectrometer*

KEY: * Normalized to curve a t k,,

=

170 MeV.

4.10 X lo6 4.17 3.99 4.07 4.09 4.17 4.04 4.16 4.14 4.11 4.10 4.12 4.04 4.10 4.11 4.07 4.09 4.02 3.96 3.86 3.99 3.94 3.82 3.73 3.86 3.80 3.71 3.60 3.84 3.81 3.64 3.84 3.82 3.69 3.75 3.81 3.87 3.71 3.87 3.98 4.09

+ 2% 2 2 2 2 2

3 2.1 3 2.5

3 1.6 1.5 3 1.3 3 1.7 1.3 3 3 1.5 1.5 3 3 2.4 2.3 3 3 2.4 1.3 3 3.6 1.3 3 1.6 3

Reference 48 48 48 48 48 48 49 46 49 46 49 46 47 49 46 49 46 46 49 49

46 46 49 49 46 46 49 49 46 46 49 46 46 49 56 56 46 49 56 56 56

2.

552

DETERMINATION O F FUNDAMENTAL QUANTITIES

are 2.5% lower than the values originally reported, because of an error in the original betatron energy c a l i b r a t i ~ n . ~ ~ If the calibrations of each chamber are plotted as a function of energy, it is possible to fit each with a smooth curve passing through most of the points to within the quoted error. The only large deviation is the 500-Mev pair spectrometer calibration of the Cornell chamber. The calibration

v; W

0

OUANTAMETER

l-

a [r

m a

/

0

W

m

a I 0

105

I II

1

11 kMAx,

Mev

FIG.28. Experimental calibration curves of four standard ionization chambers in air at 20°C and 760 mm of mercury as a function of the maximum photon energy of a bremsstrahluug beam (Tables VII, VIII, IX, X). The experimental calibration curve of the quantameter, filled with argon at 20°C and 800 mm of mercury, is also shown (Table XI).

curves are shown in Fig. 28, along with a curve labeled quantameter. The quantameter curve comes from Table XI, which was derived from Tables VII and IX by reversing the procedure and considering the quantameter calibration unknown. The slight decrease in quantameter calibration with increasing energy is well within the quoted errors, so that there is no real experimental evidence for variation of the calibration between 53 and 317 MeV. The calibrations of the four chambers of Fig. 27 must depend on the 6s

L. J. Koester, private communication (1958).

2.8.

DETERMINATION OF FLUX AND DENSITIES

553

shape of the spectrum of incident photons, since the calibrations vary with ksx. They will consequently also vary with beam filtJration,a variation which has been studied in detail for the NBS chamber.46It has been shown

TABLEXI. Calibration of the Quantameter in a Small-Diameter Bremsstrahlung Beam, for Argon Filling of 800 mm of Hg at 20°C.

kwx(Mev)

Institute

Method

53 72 85 308 315

LPTI LPTI LPTI Illinois Cornell

Calorimeter Calorimeter Calorimeter Calorimeter Spectrometer

Calibration (joule/coulomb) 7 . 6 3 X lo6 k 7.64 7.64 7.50 7.43

1.8% 1.6 1.1 2.5 5

Reference 51 51 51 53 53

for this chamber that the change of calibration with filtration is negligible at energies above 100 Mev for l o w 4 absorbers. The change becomes noticeable at lower energies, but is still only 2 % for a filtration change from 4.5 to 10 gm/crn2 of aluminum at 6 MeV.

This Page Intentionally Left Blank

3. SOURCES OF NUCLEAR PARTICLES AND RADIATIONS 3.1. Radioactive Sources* At some time in nearly all investigations of experimental nuclear physics, a need arises for a radioactive source. The source may be required for instrument tests, for the standardization of detector efficiency, or for the calibration of the energy scale of a spectrometer. It is desirable to combine, if possible, the functions of disintegration rate standard and energy standard in a single source; however, many of the nuclides suited for one purpose are unsuitable for another. The following discussion treats the preparation of sources of neufrons, a-particles, /3-particles, and y-rays, with some comments on the calibration of these sources for use in typical laboratory situations. 3.1.1. Radioactive Neutron Sources

The discovery of the neutron and the subsequent study of its properties and interactions are intimately connected with the development of radioactive neutron sources. Although the earliest work in neutron physics used neutrons from the bombardment of light elements with a-particles from polonium or radon, it was soon recognized that the photodisintegration of deuterium and beryllium could also be used to prepare neutron sources. Accelerators and nuclear chain reactors have now largely supplanted radioactive neutron sources in detailed studies of neutron reactions and in production of neutron-induced radioactivities. Nevertheless, radioactive neutron sources possess several advantages which make them very important in a nuclear laboratory. They are compact, and therefore portable; many have a constant output over many years; they require no maintenance, and are relatively inexpensive. Such features make them suited to the study of the slowing down and diffusion of neutrons in various media, and the determination of neutron scattering and absorption cross sections. They are also widely used as primary and secondary standards of neutron flux which, because of their small size, may be conveniently exchanged between laboratories. For more extensive data on the fundamental processes involved, the methods of determining absolute neutron yields, and results of inter-

* Chapter 3.1 is by G. D. 0 Kelley. 555

556

3.

SOURCES O F NUCLEAR PARTICLES AND RADIATIONS

national intercalibrations, the reader is referred to review articles by Anderson,’ Wattenberg,Z Feld,3 curt is^,^ and Hansoa6 3.1.1 .l.a-Particle Neutron Sources. Useful thick-target neutron yields have been observed when a number of light elements are irradiated with a-particles from the disintegration of heavy radionuclides. Yields obtained from thick targets, and yields obtained when the a-emitters were intimately mixed with different target materials, are summarized in Table I.6-EThe data are sufficiently accurate to show that the yield is highest for beryllium, with boron, fluorine, and lithium or sodium following in order. 3.1.1.1.1. THEB ~ ( a , n )REACTION.The high (a,n)yield of beryllium has been responsible for its extensive use as a target material. Naturally occurring a-particle sources such as Pozx0,RaZz6,AcZz7,and Em222 have long been popular as components of Be(cr,n)Cl2 neutron sources. More recently, Pu239,Am241,and Cm242 have been used successfully when alloyed with beryllium. The maximum neutron yields for these transuranium sources were measured by Runnalls and Boucher,’ whose data fit the following empirical expression :

nmax= 0.152E:66 neutrons/lOs alphas.

(3.1.1)

Table I shows also that Eq. (3.1.1) fits other cases well enough to be used for extrapolating to other a-particle energies for which no experimental yield data are available. The principal nuclear reaction in sources using a beryllium target is Be9

+ He4 = C12 + n + 5.71 MeV.

(3.1.2)

For PoZ1Oa-particles (5.30 MeV) on a thin beryllium target, the energy of the emergent neutrons would be expected to range from about 11 Mev (outgoing neutron and incoming a-particle in the same direction) to 6.7 Mev (outgoing neutron and incoming a-particle in opposite direcH. L. Anderson, Neutrons from alpha emitters. National Research Council, Nuclear Science Series, Preliminary Report No. 3, Washington, D.C. (1948). a A. Wattenberg, Phys. Rev. 71, 497 (1946). 8 B. T. Feld, in “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 11, Part VII. Wiley, New York, 1953. 4 L. F. Curtiss, “Introduction to Neutron Physics,” Chapter 111. Van Nostrand, Princeton, New Jersey, 1959. 6A. 0. Hanson, Radioactive neutron sources. I n “Fast Neutron Physics” (J. B. Marion and J. L. Fowler, eds.), Vol. I, Chapter IA. Interscience, New York, 1959. 6 R. J. Breen and M. R. Hertz, Phys. Rev. 98, 599 (1955). 7 0. J. C. Runnalls and R. R. Boucher, Can. J . Phys. 54, 949 (1956). 8 J. H. Roberts, Neutron yields of several light elements bombarded with polonium alpha particles. U.S. Atomic Energy Commission Report MDDC-731 (1944).

3.1.

557

RADIOACTIVE SOURCES

tions). In a practical source, where the Po*loand beryllium are intimately mixed, the spectrum is more complicated for a number of reasons. One complication is that the final nucleus can be left in its ground state, or in the excited states at 4.43 and 7.65 MeV. Less than 7 % of the Po-a-Re reactions lead to the 7.65-Mev state of C1*, but about 60% excite the 4.43-Mev state.g The 3.22-Mev cascade y-ray has an intensity of only TABLE I. Summary of Neutron Yields from a-Particle Neutron Sources Neutron yield per 106 a-particles

a-Source

Target

Measured thick targetsa

Measured actual sources

2.6 80 24

1 .o 60 16 2.8 1.1 0.80 0.53 46lC 6.gd 677 60 f 2 74 2 112 3

1.5 1.4 0.74

Ace27

Puzs9 Am2441 Cm242

Be Be Be Be

**

Reference 6 6 6 6 6 6 6 1 1 1

7 7 7

Calculatedb 66

485-552" 709 58 75 110

Roberts.* Runnalls and Boucher.7 c Measured for a source with no Poalopresent. d For a source in equilibrium with its decay products. Depends on PoZ1Ocontent of source. 0

b

about 3% that of the 4.43-Mev y-ray;IO y-ray transitions between the 7.65-Mev level of spin zero and even parity and the O+ ground state are, of course, strictly forbidden. Further, because the beryllium usually is thick with respect to the range of the incident Po2lo a-particles, a spread of neutron energies in the forward direction is expected to extend from the maximum of 11 Mev down to 5.3 Mev (for zero-energy a-particles) . An additional complication of the neutron-energy spectrum arises from the spread in a-particle energy introduced by the thick beryllium target. The variation of reaction cross section with incident a-particle 9K. G. Steffen, 0. Heinrichs, and H. Neuert, 2. Physik 146, 156 (1956); K. C. Steffen and H. Neuert, i b i d . 147, 125 (1957). 10L. E. Beheian, H. H. HaIban, T. Hussain, and L. C . Sanders, Phys. Rev. 90, 1129 (1953).

558

3.

SOURCES O F NUCLEAR PARTICLES AND RADIATIONS

energy is influenced by both the potential barrier of the Be9 nucleus (3.7 MeV) and the resonances caused by the level structure of the Clacompound nucleus. Low-energy neutrons have been observed , which may be due in part to the multibody breakup processes He4

+ Be9 = He4 + Be8 + n - 1.67 Mev = 3He4 + n - 1.57 MeV.

(3.1.3) (3.1.4)

Experimental determinations of the neutron energy spectrum from Po-a-Be sources have been made by measuring proton recoils in emulsionsl1I12 and counter teles~opes,'~~'4 and by measuring the energy released in the Li6(n,a)Ha reaction, using LiEI(Eu)as a scintillator.16The results of these experiments suggest two prominent peaks at about 3.5 and 5 MeV, with small "humpsJJ a t 8 and 10 MeV. Several workers also report considerabIe intensity below 2 MeV. A typical spectrum is shown in Fig. 1. Using thin-target excitation functions, Hessls calculated the spectrum expected for a Po-a-Be source. I n one case he assumed that all of the neutron yield not going to the levels at 0 and 4.43 Mev went via the 7.65-Mev state, and in another case he assumed that 80% of the excess yield went into the multibody breakup reaction. Although the experimental neutron-energy spectra are not in very good agreement with each other, none agrees very well with either spectrum calculated by Hess. Because of its long half-life (1622 years), Razzshas been used to prepare standard neutron sources. However, the Ra-a-Be neutron-energy spectrum is not as well studied as that of Po-a-Be, chiefly because of experimental difficulties associated with the intense y-radiation from radium. The energy spectrum from Ra-a-Be is very complicated, owing to the variety of a-particle energies emitted in the decay of radium and its products. After a few weeks, all of the a-emitting decay products have attained their steady-state concentrations except PoZ1O, whose growth is determined by 19.4-year Pb2'0. The neutron energy from Ra-a-Be peaks at about 4 MeV, and extends to a maximum of about 12 MeV. There seem to be more intermediate- and low-energy neutrons than in Po-a-Be sources, which may be due either to an enhancement of B. G. Whitmore and W. B. Baker, Phys. Rev. 78,799 (1950). L. L. Medveczky, Acta Phys. Acad. Sn'. Hungar. 6,261 (1956). 18 R. J. Breen, M. R. Hertz, and D. U. Wright, Jr., The spectrum of poloniuml1

l*

beryllium neutron sources. U.S. Atomic Energy Commission Report IVILM-1054 (1955). 14 R. C. Cochran and K. M. Henry, Rev. Sn'. Instr. 26,757 (1955). 16 R. B. Murray, Nuclear In&. 2,237 (1958). l6 W. N. Hess, Neutrons from (a,n)8ources. U.S. Atomic Energy Commission report UCRL-3839, revised (1957).

NEUTRON ENERGY (Mev)

2

3

I

I

Y

20

30

t 40

50

60

70

80

PULSE HEIGHT

FIG.1. Neutron spectrum of a l-curie Po-a-Be source. The measurement was made by using a LiBI(Eu) scintillator to measure the energy released in the Li6(n,a)Hsreaction (Murray16).

560

3.

SOURCES O F NUCLEAR PARTICLES AND RADIATIONS

the multibody reaction by the 7.68-Mev a-particles of or to (y,n) reactions from the large number of high-energy y-rays. The spectrum of Ac2Z7-a-Be is similar to that for radium sources, but has only about oneseventh the y-ray i n t e n ~ i t y . ' ~ Plutonium alloyed with beryllium makes a very convenient long-lived neutron source with a low y-ray intensity. The neutron-energy spectrum1* shows a considerable low-energy component, a prominent peak at about 4 MeV, followed by a much lower peak at about 7 MeV, and an upper limit of -11 MeV. IN OTHER ELEMENTS. As will be seen 3.1.1.1.2. THE(a,n) REACTION from Table I, several light elements beside beryllium can be used as targets in (a,n)neutron sources. The (a,n)reactions on boron, fluorine, and lithium appear most promising. The reactions in baron are:

+ He4 = + n + 0.5 Mev BI0 + He4 = N13 + n + 1.07 Mev

B" and

N14

(3.1.5) (3.1.6)

where most of the yield from natural boron comes from the reaction with B11 (81.2% abundance). The energy spectrum is simpler than that of Be (a,n)sources. Measurements of the Po-a-B spectrum1J4 indicate a single broad peak at -2.7 MeV, and a practical end point at -5 MeV. The spectrum of a Ra-a-B source is similar. The F(a,n) reaction is of interest because it produces a reasonable yield for Po a-particles, and, aIthough the spectrum from a pure fluorine source has not been measured, such a pure source is expected to exhibit a simple spectrum which peaks a t about 1.5 Mev.16 The Li7(a,n) reaction is useful as a source of lowenergy neutrons which is relatively free from y-rays. Because the threshold of the Li7(a,n) reaction is only 4.38 Mev and that for Lis(a,n) is 6.64 MeV, only the Li7 contributes when natural lithium is used with PoZl0 a-particles (5.30 Mev). The mean energy is probably about 0.4 Mev.16 OF a-PARTICLE NEUTRON SOURCES. A technique 3.1.1.1.3. FABRICATION for the preparation of Ra-a-Be sources developed for the Manhattan District has been described by Anderson.' Sources were made by evaporating to dryness a suspension of Be metal powder in a solution of RaBrz. The resulting dry, intimate mixture of RaBrz and Be was then pressed hydraulically into a pellet having a density of 1.75 grnjcmz by the conventional techniques of powder metallurgy. To prevent loss of radioactive gas, which would not only cause the neutron yield to vary, but which would also create a serious hazard, the pellets currently are sealed in an inner container, which is in turn sealed in another container. Such pressed sources are very convenient because of their small size, and their neutron 17

W. R. Dixon, A. Bielesch, and K. W. Geiger, Can. J . Phys. 86,699 (1957). Phye. Rev. 88,740 (1955).

18L. Stewart,

3.1.

RADIOACTIVE

561

SOURCES

emission rates are very constant with time because of the physical stability of the mixture. Transuranium elements are very useful constituents of a-particle neutron sources, as they can be readily alloyed with beryllium. These alloyed sources are very homogeneous and can be made quite reproducibly. Several reviews describe the fabrication methods applicable to P U ~ ~ ~ - ~ - B sources,18*and Runnalls and Boucher’ discuss extension of the alloy techniques to Am241and Cm242as well. TABLE 11. Y-Ray Yields from

(a,n) Sources

in Milliroentgens per Hour at

One Meter19

Source RazzB-a-Be PbZlo-a-Be PoZ’O-a-Be A~2~7-a-Be Pu zs9-~-Be Thzz*-,-Be

Tiiz

7-Yield per curie (mrhm)

Y-Yield per 106 neutrons /sec (mhm)

1622 yr 1 9 . 4 yr 138.4 day 21.6 yr 24 ,360 yr 1 . 9 1 yr

825 22 0.1 146 3.7 900

55-82 8.8 0.04 7.3-8.6 1.8 45-53 -

A 0.5-gram Ra-a--Be source typically may be 2 cm in diameter and 2 cm long; a 1-gram source would be about 25% larger. On a weight basis, polonium sources require less active material and so have much smaller dimensions. In practice a doubly sealed capsule of Po-a-Be may have over-all dimensions of 1 cm or more. High yield Pu-a Be sources are very bulky, because 16 grams of Pu are required per curie of a-radiation. 3.1.1.1.4. ?-RADIATION FROM (a,n) NEUTRON SOURCES. A recent survey of the y-radiation from a number of Po neutron sources has been made by Breen and Hertz,s who used a NaI(T1) scintillation spectrometer. The y-ray yields from all polonium sources are low, and are about one y-ray per neutron. Table 1119shows y-ray yields for a variety of sources, from which it is seen that most sources constitute a serious y-ray hazard, and that special care must be exercised if the average radiation level is to be kept below a tolerance dose. 188 M. J. F. Notley and J. Sheldon, Manufacture of Pu-Be Neutron Sources. British Atomic Energy Research Establishment Report AERE-R-3670 (1961) ; R. E. Tate and A. S. Coffinberry, “Plutonium-Beryllium Neutron Sources, Their Fabrication and Neutron Yield,” i n Proc. 2nd Intern. Conf. on the Peaceful Uses of Atomic Energy, P/700, 14, 427 (1958). Is Neutron Sources and Their Characteristics, a bulletin distributed by the Commercial Products Division, Atomic Energy of Canada Ltd., P. 0. Box 93, Ottawa, Canada; Private communication from M. R. Fleming, Atomic Energy of Canada Ltd., October 27, 1959.

562

3.

SOURCES OF NUCLEAR PARTICLES AND RADIATIONS

TABLE 111. Spontaneous Fission Neutron Sources Nuclide U232 U'as pu*aa Pusas

PU240 Pu94Z Pu944 Cmer2 Cmar4 (3262

Cf264 a

b

a-Decay Tina

Fission Ti/za

Fissions per lo1 alphas

74 yr 4.51 X 109 yr 2 . 8 5 yr 86.4 yr 6580 yr 3 . 7 9 X 1Obyr -7 .G X 107 yr 162.5 day 1 7 . 9 yr 2 . 2 yr

8 X l O I 3 yr 8 . 0 X l O l S yr 3 . 5 X 1 0 9 yr 3 . 8 X 10IOyr 1.22 X 10I1yr 7 . 1 X 10IOyr 2 . 5 X 1Oloyr 7 . 2 X 106yr 1 . 4 X 10Tyr 66 yr 56 day

9 . 2 X lo-' 5 . 6 X lo-' 8 . 1 X lo-' 2.3 X 5.4 X 5.3 3 . 0 X lo8 G.2 X 1.3 3 . 3 X lo4 Large

-

Neutrons per fissionb

1.89 2.04 2.09 2.32

i0.20 f 0.13 f 0.11 k 0.16

2.33 f 0.11 2.61 f 0 . 1 3 3.51 0 . 1 6

Strominger, Hollander, and Seaborg.ll Crane, Higgins, and Bowman.22

3.1.1.2. Fission Sources. In certain studies pertinent to nuclear reactor physics, it is desirable to employ sources with a neutron-energy spectrum similar to that of the fission neutrons. Such "mock fission" sources have been made using Po2lo a-particles on a mixture of light-element target materials.2O Now that heavy isotopes of the transuranium elements can be produced by multiple neutron capture in high-flux nuclear reactors, it has become possible to use heavy nuclides which decay by spontaneous fission as sources of fission neutrons. The use of spontaneous fission neutron sources should increase as reactors of higher flux become available for production of larger quantities of these heavy nuclides. A summary of pertinent data on several nuclides which may prove useful as spontaneous fission sources is given in Table III.21~22 Partial half-lives for a- and spontaneous-fission decay were taken from the nuclear data tabulation of Strominger, Hollander, and Seaborg.21The ratio of a-tofission half-life was multiplied by lo6 so that various nuclides could be compared with each other, and with the yields of Table I. It is important t o note that a high yield of neutrons per a-particle implies that the neutron spectrum will be relatively free from (a,n) reactions. 3.1.1.3. Photoneutron Sources. Photons of energy greater than the neutron binding energy can induce neutron emission from nuclei. The y-energies from nuclides of useful half-life are less than 3 MeV. Since the 1 0 E. Tochilin and R. V. Aves, Neutron spectra from mock fission sources. U.S. Report USNRDL-TR-201 (1958). *I D. Strominger, J. M. Hollander, and G. T. Seaborg, Revs. Modern Phys. SO (2), Part 11, 584-904 (1958). 59 W. W. T. Crane, G. H. Higgins, and H. R. Bowman, Phye. Rev. 101,1804 (1956).

3.1.

563

RADIOACTIVE SOURCES

(y,n) thresholds for deuterium (2.226 0.003 MeV) and beryllium (1.666 k 0.002 MeV) are the only ones known below about 6 MeV, radioactive photoneutron sources always use either deuterium or beryllium. Recent reviews of this subject have been made by Wattenberg,2s23 Hanson16and Curtiss.4 The energy of neutrons resulting from the bombardment of deuterium or beryllium by y-rays is, for a given laboratory angle e between the y-ray and neutron, given byg3

En =

[y]

]+

[E, - Q - E'2 2(A - 1)931 931A3

6 cos 0

(3.1.7a) (3.1.7b)

where En is the neutron energy, E, the y-ray energy, Q the neutron binding energy, all in MeV; A is the mass number of the target. To obtain the greatest intensity from practical sources, the y-ray source is usually surrounded by heavy water or beryllium, so the neutrons have an inherent spread AE, corresponding to an isotropic distribution in 0, of AEn = 26. ' (3.1.8) For example, a t En = 100 kev, AEn/Encli 4% for beryllium, and ~ 2 5 % for deuterium. I n most sources the considerable quantities of beryllium or heavy water introduce the most serious energy spread. Neutron scattering by the target materials may cause, in addition to a significant energy spread, a downward shift in the average energy of the emerging neutrons. The y-rays may lose energy by Compton scattering and then produce lowenergy neutrons, introducing a further uncertainty in the neutron spectrum. The characteristics of several radioactive photoneutron sources are summarized in Table IV.24-31 23 A. Wattenberg, Photo-neutron sources. National Research Council, Nuclear Science Series, Preliminary Report No. 6. Washington, D.C. (1949). 2 4 B. Russell, D. Sachs, A. Wattenberg, and R. Fields, Phys. Rev. 73, 545 (1948). 26 D. J. Hughes and C. Eggler, Phys. Rev. 7 2 , 902 (1947). 2 6 A. 0. Hanson, Phys. Rev. 76, 1795 (1949). 37 E. SegrB, unpublished work reported in reference 5, page 33. z8R. D. O'Neal, Phys. Rev. 70, 1 (1946). 29 L. G. Elliott, unpublished work reported in reference 5, page 33. 30 R. L. Walker, Los Alamos Scientific Laboratory Report MDDC-414 (1946); 0. R. Frisch and R. L. Walker, Los Alamos Scientific Laboratory Report LA-400 (1945). 3lL. F. Curtiss and A. Carson, Phys. Rev. 76, 1412 (1949).

3.

564

SOURCES OF NUCLEAR PARTICLES AND RADIATIONS

TABLE IV. Properties of Some Photoneutron Sources EW (Mev)

Source

y-Rays Per decay

(Mev)

Reference

En,*

Na24+Be Na24 f D

15.0hr 15.0hr

2.754 2.754

1.00 1.00

0.83 0.22

2 2

Mn66 + B e

2.58hr

1.81 2.13 2.65 2.98 2.65 2.98

0.28 0.20

0.15 0.30

25 25

Mn66+D

2.58hr

Gal3 + B e

14.3 hr

+D

14.3hr

Ga72

0.005 0.005

Standard yield, ReferX lo-' ence 29 14

24 24

2.9

24

0.31

24

5.9 3.7

24 23

6.9 4.6

24 23

(1-08)

0.22

2

1.87 2.21 2.51 2.51

0.08 0.33 0.26 0.26

(0.18) (0.48) (0.75) 0.13

2

0.99 0.01 0.01

0.16 0.22 (0.26)

26 28 -

+ Be Yaa + D

104day

1.83 2.73 2.73

In116m + B e

54.0min

2.09

0.25

0.30

25

Sb124 + B e

60.9 day

1.69 2.09

0'51} 0.07

0.024

26

La140+ B e La14'+D

40.2day 40.2day

2.52 2.52

0.01 0.01

0.62 0.13

2 2

0.23 0.68

24 24

(Th228)*+ B e

1.91 yr

(Th228)b + D

1.91 yr

1.7-2.4 1% ’ abundant. (2) Absence of intense internal conversion electrons which would tend to broaden a peak when using a pulse ionization chamber.

3.1.3. p- and y-Ray Sources Nuclides which decay by emission of nuclear p- or y-rays frequently must be counted in experimental situations which are far from ideal. Standard sources are then needed to determine either the counting yield or the energy scale of the apparatus, or both. 3.1.3.1. Source Preparation. The method used to prepare a source for beta counting depends on the purpose of the measurement. If the active material is in a carrier-free solution, an aliquot may be evaporated to dryness on a thin plastic film, and the source may then be counted using a 4xp counter or a counter with calibrated geometry. Pate and Yaffe40 have described a device for vacuum evaporation of aliquots of certain radioactive compounds onto thin films with almost a 100% yield. In some instances the source can be made very thin by using an electrodeposition procedure, but backscattering from the source mount becomes important. Frequently, the radioactivity is present in solution with large amounts of carrier. Sources are then prepared by precipitating the carrier in an appropriate chemical form, after which the precipitate is transferred onto either a filter paper, a watchglass, or a metal planchet. The chemical yield is determined by weighing and the counting geometry is calibrated for this type of source mount, taking into account the self-scattering and 6 1 F. A. White, F. M. %urke, J. C. Sheffield, R. P. Schuman, and J. R. Huizenga, Phys. Rev. 109, 437 (1958).

3.1.

RADIOACTIVE SOURCES

573

self-absorption of the thick source, and the scattering from the thick backing. An extensive literature has become available on the preparation of beta sources; some valuable suggestions may be found in references 52, 53, and 54. Although any of the methods of preparing beta sources can be used to prepare y-sources as well, liquid or other unusually thick sources can be used to advantage with y-counters because y-rays are absorbed and scattered to a lesser degree than ,&particles. A convenient mount for y-sources is made from a disc of blotting paper or a chemical “filter accelerator” taped to a card. An aliquot of the solution to be counted is simply transferred to the absorbent paper and allowed to saturate it. After the liquid is evaporated under an infrared heat lamp, the source may be covered by cellophane tape. Solutions contained in small, biological test tubes may be assayed very conveniently in a well-type NaI y-ray scintillation counter, or in a y-sensitive ionization chamber. 3.1.3.2. fi-Assay Methods. If an absolute determination of a 0-activity is to be made using a n end-window counter, the accuracy obtained will depend greatly on the details of the counting arrangement. The “counting yield” of an essentially weightless source on a thin backing can be standardized to 5% or better, provided that an absolute standard of the activity is available, or if the geometry, absorption, and scattering effects for the activity have been determined. An additional correction for y-ray background is necessary when counting 0-y sources. By determining an absorption curve for the sample, it is possible to obtain the y-counting rate with the p-rate excluded, and to correct the y-rate beyond the &range to zero absorber thickness. This method generally leads to a high value for the y-background, because all of the y-counting rate has been used, even though some coincident y-rays are detected simultaneously with the &particles, and hence would not increase the beta counting rate. Another objection is that y-scattering and production of secondary electrons by the @-absorberlead to a further uncertainty in the y-background. These corrections usually are small, except for nuclides such as CoB0,for which the 0-energy is low and the y-energy and intensity are high. Calibration standards must be used when assays of thick &sources are desired to better than 10%. A carrier-free source of the activity should be prepared and its absolute disintegration rate determined. Aliquots of this source are then processed in the same way as the unknown, taking B. P. Bayhurst and R. J. Prestwood, Nucleonics 17 (3), 82 (1959). G. Friedlander and J. W. Kennedy, “Nuclear and Radiochemistry” (revised ed.), pp. 278-282. Wiley, New York, 1955. 6 4 Oak Ridge National Laboratory, “Master Analytical Manual,” Sections 2, 3, and 5. U.S.Atomic Energy Commission Report TID-7015 (November, 1957). 62

63

574

3.

SOURCES OF NUCLEAR PARTICLES A N D RADIATIONS

care that the amounts of carrier, sample mounting procedures, and other experimental details reproduce the treatment of the unknown. An over-all counting yield may be obtained by comparing the counting rate of the calibration standard with the known disintegration rate of the activity added. Routine @-countingusing a 27r geometry is very convenient, especially if weak sources requiring high detection efficiency are to be measured. An additional advantage is that the anisotropic scattering effects are of less importance in this high geometry case. The internal sample (or “windowless”) counters are often necessary if low-energy 8-rays are to be determined. The general procedure for calibration of the counting yield is similar to that described above for end-window counters. The most widely used technique of primary standardization of 8emitters is the 4 ~ counter, 8 operating either in the Geiger or in the proportional regions. Since with the 4n/3 counter it is possible to record the total rate of emission of @-particles in all directions, coincident y-rays such as annihilation radiation are always detected with a 8-particle, and result in a single count. Likewise, the measured disintegration rate is not affected by scattering, because any discharges which are caused by repeated scattering of the primary particle, and which follow the main event, will occur within the resolving time of the counter. The same is true of secondary radiation produced within the counter. Besides the usual 47r counter using twin, facing 27r counters, various attempts have been made to achieve a 4?r geometry free of source absorption by introducing the source either as a gas in a gas-filled counter or as a liquid in a liquid scintillation counter. In applying either of these techniques, a counting efficiency correction is required because some of the 8-particles strike the walls of a gas counter without producing any primary ionization; an analogous “wall-loss” occurs in a scintillation counter with dissolved source, unless the scintillator dimensions are very large with respect to the 8-range. A t present, the most accurate and reliable 47rP counters operate in the proportional region. A carefully designed proportional counter, used with low-noise amplifiers having good overload characteristics, can yield counting rate-voltage plateaus 500 to 800 volts wide, with a slope of less than 0.1%/100 volts. Geiger counters, on the other hand, require less electronic instrumentation, but because they exhibit relatively narrower and steeper plateaus, 47rp Geiger counters are not generally considered to be as precise as the proportional type. A carefully designed 47rP proportional counter has a counting efficiency in excess of 99.5%, so the principal corrections arise from absorption in the source and in the mounting film. Indeed, self-absorption in the source

3.1.

RADIOACTIVE SOURCES

575

is and shall probably remain the factor which limits the accuracy of 47r counting. Seliger and SchwebelS6reported that even samples which gave the best 477 counting results were shown by microscopic examination to consist of lumps of varying size. They did not, however, make a systematic attempt to measure absorption losses. The source problem has also been treated by Pate and Yaffe140who measured the self-absorption of Nina,&radiation (67 kev) in nickel dimethylglyoxime of 0-250 pg/cm2 superficial density, and showed how quantitative corrections could be applied in other 4?r counting problems. Three techniques have been used to determine the source-mount absorption correction : (1) A so-called “sandwich” procedure used by Hawkings et a1.,66in which the counting rate of a source on a known thickness of backing is measured, followed by a determination made with an identical film covering the sample; (2) Seliger and co-workersS7proposed the calculation of the absorption correction using measurements of 27r and 4?r single film and sandwiched film counting rates, together with mathematical relationships developed using simplified assumptions of scattering and absorption effects; (3) A determination of the counting rate as a function of the actual mount thickness was first made by SmithlS8and this “absorption curve” approach has been the subject of an extensive study by Pate and Yaffe.69Any of these methods is satisfactory for counting /3-rays of energy greater than a few hundred kev, but a t the lower energies the absorption curve technique appears to be the most accurate. Pate and Yaffe estimate that the error in the disintegration rate arising from their source mount absorption correction does not exceed 0.2%, even a t 67 kev. Films to be used as source mounts should be rendered electrically conducting by a metallic coating a t least 2 pg/cm2 surface density applied by vacuum evaporation. This treatment will serve to prevent distortions in the electric field of the counter, which may arise either from source charging or from penetration of the electric field of one counter through the source aperture into the other, thus creating a region of low collecting field near the source. The general technique of 4?r@counting has been systematically studied H. H. Seliger and A. Schwebel, Nucleonics 12 (7),54 (1954). R. C. Hawkings, W. F. Merritt, and J. H. Craven, Proceedings of Symposium on Maintenance of Standards, National Physical Laboratory, 1951. H. M. S. O., London, 1952. 67 H. H. Seliger and L. Cavallo, J . Research Natl. Bur. Standards 47, 41 (1951); W. B. Mann and H. H. Seliger, J . Research Natl. Bur. Standards 60, 197 (1953). 6s D. B. Smith, 4 Pi Geiger Counters and Counting Technique. British Atomic Energy Research Establishment Report AERE-I/R-1210 (1953). B. D. Pate and L. Yaffe, Can. J . Chem. 33, 929 (1955). 66

SE

576

3.

SOURCES OF NUCLEAR PARTICLES AND RADIATIONS

by Pate and Yaffe.40.46J9060Other helpful information may be found in papers by Seliger and c o - w o r k e r ~ ~ ~ who ~~7 describe ~ ~ 1 measurements performed at the National Bureau of Standards and the results of international comparisons of samples. It is possible a t present to obtain a reproducibility of about 0.5%, and an accuracy nearly as good in favorable cases. In cases where a p-particle and a y-ray are emitted in sequence, 8-7 coincidence counting can be a very convenient and accurate method of absolute disintegration rate measurement. Consider a radioactive source which decays by emission of a single beta transition followed by a single unconverted y-ray, placed between a 8-counter and a y-counter. The counting rate of the p-counter Np is given by

Np = Noeg

(3.1.11)

where NOis the disintegration rate, and q is the counting yield of the 8-detector. Similarly, the y-counting rate N , is given by N y

=

NO!,

(3.1.12)

where e,. is the counting yield of the y-detector. The coincidence rate N , is No = Nocge, (3.1.13) which by use of Eqs. (3.1.11) and (3.1.12) reduces to

No

=

NgN,/N,.

(3.1.14)

Neither the source self-absorption nor the counting yields of the detectors need be known for this simple case, an attractive experimental simplification. The counting rates in the 8- and y-channels must be corrected for dead-time losses, if any, and for background rates. The background rate of the @-channelmust include the contribution arising from the y-sensitivity of the 8-detector, and is measured by use of a @-absorberas described above for conventional 8-counting. The coincidence counting rate must first be corrected for random coincidences in the customary way, i.e.,

N,

=

2rNgN,

(3.1.15)

where 27 is the resolving time of the coincidence circuit, and Ng and N , B. D. Pate and L. Yaffe, Can. J . Chem. 33, 1656 (1955). W. B. Mann and H. H. Seliger, Preparation, maintenance and application of standards of radioactivity. Natl. Bur. Standards (U.S.) Circ. NO.694 (1958). 60

61

3.1.

RADIOACTIVE SOURCES

577

are the uncorrected counting rates. Further, a coincidence background must be removed, which may be due solely to environmental radiation, or, in more complicated cases, to y-y coincidences from y-ray cascades in the source. A correction must also be applied for angular correlations, if they exist. Although this discussion used a simple example, the method may be adapted to other situations, such as complex decay schemes, 6-electron coincidence counting, X-y coincidence counting, and y-y coincidence counting. The P-y coincidence method was shown by P u tma n 6 z 6 3to be valid for nuclides with several @-groupsprovided th a t the sensitivity of either the P- or y-detector was the same for all branches of the decay scheme. The 4743 counter is the logical approach to this condition. Because 4 ~ 0 - y coincidence counting is nearly independent of adsorption in the source and its mount, the technique has been used to calibrate the counting . ~ ~ counting ~~~ nuclides with yield of thick sources in a 4 ~ cp o ~ n t e rWhen complex decay schemes by the 4rp-y coincidence technique, the corrections frequently turn out to be relatively insensitive to the details of the decay s ~ h e m e . So ~ ~ with * ~ ~ quantitative information about a decay scheme, it may be possible to standardize sources by 474-7 coincidence counting to a few tenths of a per cent.B4 3.1.3.3. y-Assay Methods. Gamma-ray counting by spectrometry a t a defined solid angle has been discussed in Section 2.2.3.3. Integral counting arrangements using y-counters a t arbitrary geometries or NaI wellcrystal counters without differential pulse-height analysis must be calibrated using sources of known disintegration rates. High-pressure ionization chamber^^^,^^ make excellent devices for the secondary standardization of y-emitters. Gamma-gamma coincidence counting has been applied a t the National Bureau of Standards to the calibration of Gos0 sources, and results which are repoducible to 0.1%have been obtained.Bs J. L. Putman, Brit. J . Radiol. 23, 46 (1950). L. Putman, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), Chapter 26. Interscience, New York, 1955. 6 4 P. J. Campion, in “Measurements and Standards of Radioactivity, Proceedings of an Informal Conference.” Natl. Acad. Sci.-Natl. Research Council Rept. No. 24,24 (1958);Intern. J . Appl. Radiation and Isotopes 4 , 232 (1959). 66 R. Gunnink, L. J. Colby, Jr., and J. W. Cobble, Anal. Chem. 31, 796 (1959). C. C. Smith and H. H. Seliger, Rev. Sci. Znstr. 24, 474 (1953). 67 J. L. Putman, in “Measurements and Standards of Radioactivity, Proceedings of a n Informal Conference.” Natl. Acad. Sci.-Natl. Research Council Rept. No. 24, 69 (1958). 68 R.W.Hayward, in “Measurements and Standards of Radioactivity, Proceedings of a n Informal Conference.” Natl. Acad. Sci.-Natl. Research Council Rept. No. 24,76 (1958). 82

O3J.

578

3.

SOURCES OF NUCLEAR PARTICLES AND RADIATIONS

3.1.3.4. 8- and y-Ray Standards, The calibration of the counting yield for a particular counter is generally accomplished using a disintegration rate standard. The National Bureau of Standards (NBS) produces such radioactivity standards for distribution to interested laboratories. A list of the NBS sources, and details of their standardization and use, were published by Mann and Seligersl in June, 1958; the latest information on available radioactivity standards can be obtained by requesting one of the periodic NBS bulletins on the subject. Most of the NBS standards TABLE VII. Standard Electron Lines

Source Th(B Th(B ‘I’h(B Th(B Th(B Th(B Th(B Th(R

+ C + C”)-A line + C + C”)-B line + C + C”)-F line + C + C”)-I line + C + C”)-Ia line + C + C”)-J line + C + C”)-Lline

+ C + C”)-X line

Electron energy, (kev) 24.510 36.151 148.08 222.22 222.90 234.61 422.84 2526.3

f 0.003 5 0.003 f 0.02

193.59 481.70 624.21 975.9 1164.5 1324.3

f 0.05

kO.02 rtO.02 50.02 rtO.06 f0.4

f0.15 f 0.10 f0.3 f0.3 f0.2

Magnetic rigidity of electron line, (gauss-cm) Reference 534.21 652.40 1388.45 1753.91 1757.07 1811.11 2607.18 9986.7

f0.03 ?c 0 . 0 4 f 0.10 f 0.14 f 0.14 f 0.15 f 0.35 f 1.5

69 69 69 69 69 69 70 71

1618.10 2838.9 3381.28 4657.9 5322.5 5879.4

f 0.20 k0.4

50 72 73 74 75 75

f 0.50

f 1.0 2 1.0 & 0.7

are solutions of long-lived a-, p-, and y-emitters; in addition, the Bureau can also supply standard a-sources mounted on metal disks and pointsource y-standards mounted between layers of polyester tape. Standardized samples of short-lived nuclides are now available commercially, and distribution of these materials has been discontinued by the NBS. The commercial standards are prepared in the same way as the NBS standards, and their accuracy is better than 3%. The KBS continues to maintain primary standards of both long- and short-lived nuclides and to compare them with samples from other national standardizing laboratories. Because these standards are at hand, the NBS also calibrates radioactive samples from laboratories requesting this service, provided that the samples conform to certain physical, chemical, and activity level specifications. Sources emitting internal conversion electron or y-ray lines of accii-

3.1.

RADIOACTIVE SOURCES

579

rately known energy are needed for the calibration of electron or y-ray spectrometers. The data on standard electron lines presented in Table VI169-76 may be used to calibrate various types of electron spectrometers. TABLEVIII. Standard 7-ray Energies" Source Am241 I131

Cd"39 ThB(Pb212) ThB(Pb212) Hg203 1"' I101

Au198 Th C" ( T l z O S ) Positron annihilation Biz07 Cs'37

Mn64 Biz07 c o 60 Na2z coao

Na24 Biz07 ThC"(TI2OS) Na24

7-ray energy, (kev) 59.568 80.164 87.7 115.03 238.62 279.12 284.31 364.47 411.77 510.84 510.976 569.7 661.65 837.9 1063.9 1172.8 1276 1332.6 1369 1771 2614.3 2754

f 0.017 f 0.009 f0.2 f 0.01 f 0.02 f 0.05* f0.05 f 0.05 zk 0.04 f 0.07 f 0.007 f0.1 fO.10 f 0.3" f0.3 f 0.5 f4 f 0.4 f1 f5 f 0.4 f1

Unless otherwise noted, the energy values were taken from data referred to in the Table of Isotopes by Strominger, Hollander, and Seaborg.21 6 From Wapstra, Nijh, and Van Lieshout.60 c From Katoh, Nozawa, and Yoshisawa.76 (1

For calibration of instruments which are linear with energy, such as scintillation spectrometers, the electron energies have been given. Magnetic spectrometers require a calibration of a momentum scale, and for such applications the magnetic rigidities of the lines in gauss-cm are included K. Siegbahn and K. Edvarson, Nuclear Phys. 1, 137 (1956). G. Lindstrom, Phys. Rev. 83, 465 (1951). 71 G. Lindstrom, Phys. Rev. 87, 678 (1952). 72 G. Backstrom, Arkiv Fysik 10, 393 (1956). 73 G. Lindstrom, K. Siegbahn, and A. H. Wapstra, Proc. Phys. SOC.(London) B66, 54 (1953). 74 D. Alburger, Phys. Rev. 92, 1257 (1953). 76 G. Lindstrorn, A. Hedgmn, and D. E. Alburger, Phys. Rea. 89, 1303 (1953). 69

70

580

3.

SOURCES OF NUCLEAR PARTICLES AND RADIATIONS

as well. A list of standard 7-ray energies is given in Table VIII,'" and will be found especially useful for calibrating scintillation spectrometers. No energy standards below 60 kev were included, because X-rays are the preferred standards in this energy region, and such energies are well known.

3.2. Artificial Sources* The first use of an artificial source in nuclear physics was in 1932 when Cockcroft and Walton successfully disintegrated lithium by protons of about 400 kev.' Further expansion in energy of direct-voltage accelerators and the development of the cyclotron provided sources for the study of the gross structure of the nucleus. For the study of nuclear level schemes and the structure of liquids and solids, the nuclear reactor has proved to be a prolific source of neutrons of energies ranging from a fraction of a n electron volt up to about 1 MeV. With the development of the synchronous type of accelerators, first a t energies of several hundred MeV, came the production of mesons and then, with the modern multi-Bev machines, the production of antiprotons and hyperons. 3.2.1. Low-Energy Sources

Charged particles and neutrons of energies up to a few Mev are usually obtained from dc (direct potential drop) accelerators which comprise about half of all existing accelerators.2 Although accelerators can serve usefully as thermal-energy neutron sources, neutrons of these energies are more frequently derived from nuclear reactors. The chief advantage of dc accelerators lies in their precision, the voltage spread being of the order of 0.1 % ' or less. They fall into two main classes: (1) those where high voltages are obtained by a cascade process such as the cascade rectifiers or voltage-multiplying circuits and (2) those where charge is actually transported to a terminal as in the belt-charged electrostatic generators. Many high-voltage devices were investigated around 1930 and voltages M. Nozawa, and Y. Yoshizawa, J . Phys. SOC.Japan 13, 1419 (1958). D. Cockcroft and E. T. S. Walton, Proc. Roy. SOC.(London) A137, 229 (1932). 2 G. A. Behman, Nuclear Instr. 3, 181 (1958);6, 129 (1959). 76T.Katoh,

* J.

* Sections 3.2.1.1 and 3.2.1.2 are by M. H. Blewett.

3.2.

ARTIFICIAL

581

SOURCES

of 2 or 3 million volts were reached but, in most cases, it was not possible to connect a discharge tube across the terminals without breakdown a t much lower voltages. Tesla coils and other types of resonant transformers have been found useful for producing X-rays for hospitals but have been supplanted by other machines in nuclear physics. Surge generators, wherein a stack of capacitors is charged in parallel and discharged in series, are not particularly useful since the pulses are so short and the voltage too irregular. For many years, cascade transformers were used for a considerable amount of research a t California Institute of Technology by Lauritsen and his co-workers3but, more recently, these have been replaced by electrostatic generators. 3.2.1.1. Cascade Rectifiers. Direct voltages up to 100-200 kv are rather easily obtained from a simple transformer-rectifier system such as

Fro. 1. Standard transformer-rectifier circuit. The capacitor serves as a filter to maintain constant voltage output.

that illustrated in Fig. 1. Higher voltages can be reached by voltage doubling, usually in several stages as in the Greinacher, or CockcroftWalton, type of circuit shown in Fig. 2. With the rectifiers acting as switches, the two capacitors in each stage are charged on alternate half cycles from the ac source and double the voltage appears across the two capacitors when they are connected in series. In modern installations, a separate stack of filter capacitors, charged through a series resistance, is usually included to reduce the voltage ripple. This circuit, also known as a cascade rectifier, together with an appropriate ion source and discharge tube, is usually termed a Cockcroft-Walton accelerator. Such accelerators can provide dc currents up to about 10 ma and, in air, have been operated up to about 1.5 MeV, each stage contributing about 200 kv. By using a pressure tank, somewhat higher voltages can be achieved but this is not very prevalent. The main advantage of this type of accelerator, compared H. R. Crane, C. C. Lauritsen, and A, Sdtan, Phys. Rev. 46, 507 (1934).

582

3.

SOURCES O F NUCLEAR PARTICLES AND RADIATIONS

with the belt-charged generator, is in the lack of moving parts and in the higher possible currents. Ripple and regulation in voltage multipliers have been analyzed by several authors, e.g., Lorrain et aL4 Ripple is inversely proportional to frequency of excitation and to capacity of the capacitors in the multiplier stack. It is directly proportional approximately to the square of the number of stages of multiplication. Hence, it is desirable to use capacitors of 6V

T

E

V sin w t

1

T Cf

FIQ.2. Voltage-multiplying circuit with R-C filter.

large value, high frequency of excitation, and as few stages as possible. Typical parameters for a 750-kv multiplier are: number of stages. . . . . . . . . . . . . 5, capacity of stack unit.. . . . . . . . 15,000 pF, frequency of excitation. . . . . . . . 600 cycles. Ripple at the terminal of this system is about 400 v, peak to peak, per milliampere of current drawn. This ripple can be reduced to negligible values by the resistance-capacity filter shown in Fig. 2. Typical values for the filter units would be R, = 5 megohms, Rf’= 20 megohms, and C, = 20,000 pF. Voltage measurement and information for voltage regu4 P. Lorrain, R. Beique, P. Gilmore, P. E. Girard, A. Breton, and P. PichB, Can. J . Phys. 58, 299 (1957).

3.2.

583

ARTIFICIAL SOURCES

lation systems is usually accomplished through a wire-wound resistor of high quality connected from the high-voltage terminal to ground. The most serious problems encountered in the use of these systems are associated with the design of the evacuated accelerating tube. Although many successful designs of equipment for generation of high voltage were available in 1932, it remained for Cockcroft and Walton to demonstrate that maintenance of high voltage across an evacuated accelerating tube Voltage dividing resistor

/

Axis

\ t/

Corona rings FIG.3. Section of accelerating tube.

required very careful voltage division and shielding of the insulating walls of the tube from discharges in the vacuum. Modern accelerating tubes follow the design of Cockcroft and Walton and generally have the form shown in Fig. 3. Voltage multipliers can be operated with either polarity and can be used for the acceleration of electrons or positive ions. However, they are most frequently used in nuclear physics laboratories as positive-ion sources. The ion source, which is housed in the terminal of the filter stack, can be of several forms,Kthe most prevalent being a radio-frequency source,6 a cold-cathode P.I.G. ~ o u r c e or , ~ a duo-plasmatron source.8 With any of 6 M. S. Livingston and J. P. Blewett, “Particle Accelerators,” Chapter 4. McGrawHill, New York, 1962. 6 R. N. Hall, Rev. Sci. Znstr. 19, 905 (1948). 7 J. Backus, “National Nuclear Energy Series,” Div. I, Vol. 5. McGraw-Hill, New York, 1949. * C. D. Moak, H. E. Banta, J. N. Thurston, J. W. Johnson, and R. F. King, Rev. Sci. Znstr. SO, 694 (1959).

584

3.

SOURCES O F NUCLEAR PARTICLES AND RADIATIONS

these sources, several milliamperes of protons, deuterons, helium, or other ions can be extracted by a suitable ion-optical s y s t e m g ~and ~ ~then carried through a n accelerating tube along the axis of the filter stack to the ground, or in a horizontal direction to a terminal connected to the ground terminal of the machine. Cockcroft-Walton accelerators of conventional types are commercially available from the Philips Company of Eindhoven, Holland, or from Haefely of Basel, Switzerland. Variants of the cascade-rectifier system have been evolved in the United States by Radiation Dynamics of Westbury, N.Y., and by the High Voltage Engineering Corp. of Burlington, Mass. The Radiation Dynamics “Dynamitron” operates a t a high frequency of about 300 kc and has been redesigned to use much smaller capacitors and a coupling system suited to the higher frequency. Units for various voltages up to 3 Mv are listed in the catalogues. The HVEC “Insulating-Core Transformer” consists of a stack of 30-kv three-phase transformer rectifiers with magnetic cores aligned and insulated from each other. Present ratings are for output voltages to 600 kv. I n 1962, both of these systems were in preliminary operation and showed promise of success. 3.2.1.2. The Electrostatic (Van de Graaff) Generator. Electrostatic or “Van de Graaff”” generators are very widely used for dc acceleration of charged particles and, with suitable targets, as neutron sources to energies in the range between 1 and 5 million volts with a precision of somewhat better than 0.1 %. Currents are available in this range up to about 500 pa for positive-ions and up t o about 1ma for electrons. I n a new I [ tandem” variation, positive-ion energies over 20 Mev are possible but at drastically reduced currents (20 pa or less). Electrostatic accelerators have also found wide industrial application and are available commercially from two firms. The Van de Graaff, belt-charged, machine can be purchased from the High Voltage Engineering Corporation of Burlington, Mass. A modern version of the Wimshurst machine, charged by a rotating, insulating cylinder, is manufactured by S.A.M.E.S. of Grenoble, France. The general principles of the belt-charged generator were known in the latter part of the last century but its practical value for nuclear physics was first demonstrated by R. J. Van de Graaff. l1 Since that time, many

* The name of Van de Graaff, often used as an adjective for these machines, is now used as a trademark by the High Voltage Engineering Corporation. R. W. Allison, el al., Rev. Sci. Instr. 32, 1331 (1961). 10 A. Yokosawa, “Proceedings, International Conference on High Energy Accelerators, Brookhaven, 1961,” p. 385. U.S.Government Printing Office, Washington 25, D.C., 1961. 11 R. J. Van de Graaff, Phys. Rev. 38, 1919A (1931).

3.2.

585

ARTIFICIAL SOURCES

Generating voltmeter

/

c

- Corona point for regulation

Charging belt

J

/ /Housing

Motorf

FIG.4. Schematic drawing of Van de Graaff generator.

developments and modifications have been made which have been summarized by Herb12 and by Van de Graaff, Trump, and Buechner.13 The method of operation is illustrated by Fig. 4.Charge from a 20- or 30-kv supply is sprayed from a row of corona points onto a moving belt which carries it to the field-free region inside a spherical metal terminal. Here, the charge is removed by another row of corona points and flows to the outer surface of the terminal. If this amount of charge is Q, and the la

R. G. Herb, “Handbuch der Physik,” Vol. XLIV, pp. 64-104. Springer, Berlin,

1959. l3

R. J. Van de Graaff, J. G. Trump, W. W. Buechner, Repts. Progr. in Phys. 11, 1-18

(1948).

586

3.

SOURCES OF NUCLEAR PARTICLES AND RADIATIONS

capacitance of the terminal to ground is C, the terminal is raised to a potential V = Q / C . As the charge-carrying process continues, the potential of the terminal rises until either the process is stopped or electrical breakdown occurs between the terminal and its surroundings. By means of an accelerating column, the terminal is connected to ground through a series of corona rings either by resistors or corona points. Inside the terminal is located an ion source (or filament if electrons are desired) and, after a small amount of preacceleration to pull the ions to the mouth of the accelerating column, they are accelerated down a beam pipe inside the column by the electric field therein. Early Van de Graaff generators were operated in air a t atmospheric pressure. This required a great deal of space around the terminal and the highest voltages attained were about 1 Mv. Today, the apparatus is enclosed in a pressure tank and operated at pressures of 10 or more atmospheres. Since electrical-breakdown voltage is proportional to pressure, this modification has made it possible to reach potential differences of 5 million volts, and more, in equipment of reasonable size. A modern 5-Mev accelerator is enclosed in a pressure tank about 20 ft long and 8 ft in diameter and the structure may have either a horizontal or vertical axis. I n either case, the terminal is supported from the ground end on insulating rods. As the insulating gas, ordinary air is unsatisfactory since it will support combustion in case of spark breakdown to a combustible component. Instead, a combination of nitrogen and carbon dioxide is frequently used. The superior insulating properties of the electronegative gases of the freon and sulfur-hexafluoride families are offset by the corrosive products they liberate in an electrical breakdown. In systems where breakdown is not expected, sulfur hexafluoride is now sometimes used. The voltage of the terminal is measured to a precision of 1% or better by a generating voltmeter mounted in the wall of the housing opposite the terminal. In this device a rotating vane alternately is exposed to and shielded from the electric field of the terminal. Charge induced on the vane provides an externally available signal whose amplitude is a measure of terminal potential. This signal can now be fed back to the charging supply to control the amount of charge carried to the terminal. For more precise control the energy of the accelerated beam is measured by deflecting it in a magnetic field to fall on an insulated slit system which gives a precise indication of beam deflection and hence of particle energy. Information from this system is fed back through amplifiers to corona points in the wall of the housing opposite the terminal. Charge from these points sprayed toward the terminal can be used to cancel part of its charge and so to drop its potential. With this arrangement terminal voltage can be controlled to better than one part in a thousand.

3.2.

ARTIFICIAL SOURCES

587

The current available from a Van de Graaff machine is determined by the amount of charge which can be carried by the belt. Originally made of paper, belts now in general use are of rubberized fabric, between 20 and 60 cm wide, and with speeds of the order of 1200 ft/min. The charge that can be sprayed on the belt is limited by gas breakdown and so increases with gas pressure. The charging rate for a 60-cm belt at 10 atm is usually about 500pa. The over-all charging rate can be doubled by using the downgoing half of the belt to carry charge from the terminal to ground. Belts also serve another function: to drive a generator in the terminal to provide power for the ion source and the extracting and focusing lens systems. For acceleration of electrons, the terminal contains a conventional electron gun. For positive ions, the more complicated ion-sources enumerated in the preceding s e c t i ~ n ~ must * ~ ~ be ~ . *mounted in the terminal. Control and monitoring of these components can be accomplished by light beams or by microwave links. However, it is more usual, and almost as effective, to control by pulling insulating strings and to monitor by watching meters in the terminal through a telescope. Sources of difficulty in this type of accelerator have been electrical breakdown across the insulating gas or along the support members, or discharge inside the evacuated accelerating tube, the latter being the major offender. This tube, which has the same design as that described in the preceding section, is mounted beside the charging belt between the terminal and the ground plane below the charging supply. Potential along the tube is divided either by a resistor string or by corona points between the corona rings that shield the accelerating tube. Another array of corona rings surrounds the belt and serves to distribute potential between the terminal and ground. It is important that all of these parts be carefully made, kept clean, and precisely placed. Small imperfections can cause loss of available voltage of as much as 50%, or even complete failure. A serious limitation in positive-ion machines has been electron loading from electrons emitted in the accelerating tube. These travel back to the terminal end to produce large quantities of X-rays which then ionize the insulating gas and cause leakage currents to the outside. By the addition of lead shielding at the ion source, TurnerL4has been able to effect considerable reduction in these effects. The positive ions that are usually accelerated are protons and deuterons; hydrogen-3 can also be accelerated but is more frequently used as a target; helium ions and alpha particles require a more elaborate ion source and preacceleration system. With a thick beryllium target, through the Beg(d,n)BLo reaction, the Van de Graaff generator provides a source 14

C. M. Turner, Phys. Rev. 96, 599 (1954).

588

3.

SOURCES OF NUCLEAR PARTICLES AND RADIhTIONS

of neutrons of either thermal or higher energies. For 3-Mev bombarding energy and a current of 200 pa, about 10’2 neutrons/sq cm are obtained. To slow these neutrons down to thermal energies (about 0.02 ev) the target is enclosed in moderating material which may be paraffin, water, or some other hydrogenous material. A great many other reactions have also been studied which produce monoenergetic neutrons over a wide energy range; many details of the various targets and reactions may be found in a review by Fowler and Brolley.16 Energies of accelerated particles in Van de Graaff accelerators can be extended into the 10-30 Mev region (and, perhaps, even higher) through the use of the “tandem” principle proposed by Bennett16 and Alvarez” and improved and reduced to practice by the High Voltage Engineering Corp.18~~~ Figures 5 and 6 illustrate the principles of two-stage and threestage tandem accelerators. In the two-stage tandem, positive ions from a conventional source are passed through an “electron-adding canal” containing hydrogen gas where a fraction of the positive ions are converted to negative ions by the addition of two electrons. The resultant negative ions are separated from the residual beam by an analyzing magnet and accelerated to a belt-charged, positive high-voltage terminal. In the positive terminal, the beam again passes through a gas-filled channel where the gas strips most of the negative ions of the added electrons. The resultant, high-energy, positive-ion beam is again accelerated as it passes from the terminal to ground. The accelerated beam thus has twice the energy of a beam originating a t an ion source in the terminal. However, because of the inefficiency of conversion to negative ions, intensities are only of the order of a few microamperes. Three-stage tandem accelerators are of two types, and Fig. 6 shows the “neutral-beam injection’’ version. The beam from the usual type of positive-ion source is first neutralized by gas and then drifts to a negatively charged terminal. In the negative terminal, electrons are added to make negative ions which are accelerated from the negative terminal to ground. At this high energy, they then enter a second pressure tank for a procedure like that described for the two-stage tandem machine, resulting in a positive-ion beam of three times the energy produced by the terminal voltage. I n principle, this positive-ion beam could be returned to the first machine and accelerated to its negative terminal, thus giving a fourth stage of acceleration, but termination of acceleration within a high-voltage termiJ. L. Fowler and J. E. Brolley, Jr., Revs. Modern Phys. 28, 103 (1956). W. H. Bennett and P. F. Darby, Phys. Rev. 49, 97, 422, 881 (1936). 17 L. W. Alvares, Rev. Sci. Instr. 22, 705 (1951). 1s R. J. Van de Graaff, Nuclear Instr. & Mefh. 8, 195 (1960). 19 P. H. Rose, Nuclear Znstr. & Meth. 11, 49 (1961). 16

18

Positive high-voltage terminal

ou

FIG.5. Two-stage tandem Van de Graaff generator.

Positive high-voltage terminal

Negative high-voltage terminal

I Neutraliz

source

FIG.6. Three-stage tandem Van de Graaff generator.

590

3.

SOURCES O F NUCLEAR PARTICLES A N D RADIATIONS

nal is not very suitable for experimentation. In the alternate, “negativeion injection” form of three-stage tandem, a negative-ion source is housed inside the terminal of a vertically mounted Van de Graaff machine and accelerated from the negative terminal to ground. The negative beam is then bent 90” to enter a two-stage tandem machine for two more accelerations. Two-stage accelerators have already reached over 15 Mev and three-stage machines are expected to provide 25-30 Mev in the near future.

Insulating rotor Negative high-voltage terminal

I

L charging supply

-

FIQ.7. SAMES electrostatic generator for negative polarity.

The principle of the SAMES electrostatic generator is illustrated in Fig. 7. In this generator, the belt of the Van de Graaff is replaced by a ceramic cylinder revolving around its axis in an atmosphere of hydrogen. SAMES generator or accelerator units are available a t voltages up to 2 Mv with currents up to 5 ma and are to be found in many European laboratories for nuclear physics.

3.2.1.3. Nuclear Reactors.* This is a descriptive section which should acquaint the experimentalist a t the reactor with the main properties of the machine which supplies radiation for his use. A more adequate discussion of design methods would draw heavily on the fieIds of metaIlurgy, chemical engineering, and mechanical engineering, with emphasis on

-

* Section 3.2.1.3 is by H.

Kouts.

3.2.

ARTIFICIAL SOURCES

59 1

material properties and heat transfer. It would require a treatment of the theory of neutron transport which is beyond the need of those who regard the reactors as a tool.’ 3.2.1.3.1. ‘GENERAL FEATURES. The steady operation of the chain reaction in a nuclear reactor is a consequence of a balance between competitive yields on the average 2.42 neutrons. When processes. The fission of U236 on the average this many neutrons lead to one further fission in the course of their absorption or loss in any other way from the reactor, the critical condition exists. Only the average behavior will be discussed here. The fluctuations in the fission rate do not have any great influence on the operating research reactor. The nuclear processes which compete with fission in destroying the neutrons in the reactor are: radiative capture in the fissionable nuclei, radiative capture in other nuclei, (n,2n), (n,a), (n,p) processes to a n almost negligible degree. The one other process which must be taken into account in establishing the steady chain reaction is the leakage of neutrons from the reactor. This leakage is the diffusion of the neutron gas outward from the high-intensity source region where it is produced. Leakage could only be avoided in an infinite reactor with a spacially constant source strength and therefore no variation in the neutron density. The operating level of the nuclear reactor is controlled by adjusting the competitive process to a balance, once the desired point is reached. Usually control is accomplished by adding or subtracting material which has a high-capture cross section. This material is contained in so-called poison control elements. More rarely, the amount of fissionable material in the core can be varied by means of insertion or removal of fuel control elements. Sometimes the fuel and the poison form successive sections of a single control element. These elements are said to have fuel followers. As poison is inserted, fuel is removed, and also conversely. Some reactors achieve neutron balance control through variation of the leakage probability. These are usually fast reactors, in which the mean energy a t which neutrons are absorbed is several kev or more. I n this region, poison elements are less useful, because the radiative capture cross sections are not especially large. 1 A partial bibliography of this subject is as follows: A. M. Weinberg and E. P. Wigner, “The Physical Theory of Neutron Chain Reactors.” Univ. of Chicago Press, Chicago, Illinois, 1958; S. Glasstone and M. Edlund, “The Elements of Nuclear Reactor Theory.” Van Nostrand, New York, 1952; H. Etherington, Ed., “Nuclear Engineering Handbook,” 1st Ed., McGraw-Hill, New York, 1958.

592

3.

SOURCES OF NUCLEAR PARTICLES AND RADIATIONS

Most research reactors are of the thermal variety, with the adjective referring to the energy distribution reached by the neutrons before they are absorbed. In a thermal reactor, the neutrons are slowed down (moderated) by nuclear scattering until they are nearly in thermal equilibrium with the kinetic motion of the nuclei in the reactor core. In this region, the large fission cross section of U2s6 (the common research reactor fuel) is relatively favorable for the fission process. The real reason for the dominance of thermal systems among research reactors is not this advantage, which is minor. It is that most of the research of present interest which can use the high intensities of neutrons from these machines is done with neutrons of relatively low energy. Many kinds of experiments use thermal reactors as neutron sources. Some examples are: neutron diffraction studies of solid state structure, low-energy neutron cross-section measurements using crystal spectrometers or time-of-flight energy analysis, nuclear level scheme studies based on radioactive decay or capture gamma ray spectroscopy, liquid state structure studies based on measurements of lowenergy inelastic scattering, and a host of applications to other fields which use radioactive tracer techniques. The neutron energy degradation takes place primarily through elastic scattering by light nuclei, but in some reactors inelastic scattering by heavier nuclei also plays an important part. The probability of radiative capture must be kept small during moderation. The useful moderators are therefore those which are efficient in reducing the neutron energy, but which have low-capture cross sections. The dominant materials for this purpose in research reactors are light water, heavy water, beryllium, and graphite. The change in nuclear binding energy during fission is given mostly to kinetic energy of the fission fragments. The remainder appears as kinetic energy of the neutrons, and energy of beta particles and cascade gamma rays from the radioactive decay of the fission products. Some is carried away by neutrinos. The breakdown is as follows:2 fission fragment KE neutron KE /!?-particleKE 7-rays neutrinos

168 MeV, 5, 7J

13J

11.

Additional energy, a few Mev per fission, depending on the design of the reactor, is liberated by neutron radiative capture.

* H. Etherington, ed., “Nuclear Engineering Handbook,” 1st Ed., p. 2-2. McGrawHill, New York, 1958.

3.2.

ARTIFICIAL SOURCES

593

All of this energy except that carried by neutrinos is eventually transformed to heat in the reactor. This heat must be removed if the temperatures of reactor components are to be kept reasonable. The heat removal methods are those commonly used when large amounts of energy must be dissipated. That is, a fluid is circulated past the fuel-bearing elements, and the heat transferred is carried away to a heat exchanger whose secondary circuit contains air, or which uses ground water or water circulated through a cooling tower. Other, but similar, methods are occasionally used. The excess neutrons and the gamma rays from fission and fission products complicate the design problem because they must be shielded against. The source strengths are very large. A rule of thumb is that about 10 curies of gamma rays are produced by 1 watt of steady reactor operation. These gamma rays have energies from about 12 Mev downwards, with the peak at a few MeV. The reactor itself absorbs some of this radiation, but enough escapes to lead to severe personnel hazard and interference with experiments unless thick shielding is provided. The shield thickness required for most reactors varies from about 3 to 5 meters of ordinary concrete or from 1.5 to 2.5 meters of special high-density concrete. Pool reactors use thick water layers for the same purpose. The shielding thickness required is not very sensitive to the power level or the type of reactor. In addition, the fission products themselves would be severe health hazards if they were freed. Much of the engineering design of a reactor is aimed at making such a release very improbable. Fission produces neutrons over the entire fueled region of the reactorthat is, throughout the reactor core. The only practical limitation on the source strength is the ability to remove the heat generated by the fission process. Standard methods now in use permit heat removal of up to about 1.5 kw/cm3, which provides a maximum steady source strength of about 1014neutrons/cm3-sec. Improvements in design should permit this yield to be increased by a sizable factor in the near future. Altogether, a l-Mw research reactor produces 7.2 X 10l6neutrons/sec, of which about 3 X 101s/sec are needed to produce further fissions. Typically, about 2 X 10I6/sec are lost in radiative capture in the fuel, moderator, coolant, and structural material. The rest are available for research uses. In fact, though, it happens that the intensities of available neutron beams are by no means comparable to those which would be available from point sources of the same strength. The intensity is reduced considerably by the diffuseness of the reactor core, and by the need for collimation. Furthermore, the energy distribution of the neutrons is continuous. Any experiments requiring monoenergetic neutrons suffer further losses by the energy selection process.

3.

594

SOURCES OF N U C L E A R PARTICLES A N D RADIATIONS

The experimental facilities are of two kinds: those which remove neutron beams to equipment outside the reactor shield, and those which expose material to radiation in the reactor core. Recent advances in design have led to considerable improvement in the efficiencies of both kinds of facilities. Beams have been increased in output and have had the intensity of unwanted components of the radiation reduced. Irradiation facilities are now placed in regions where the neutron spectrum is best for the experiments to be done. 3.2.1.3.2. FEATURES OF THE PROCESSES INVOLVING NEUTRONS. The only fissionable material commonly used in research reactors is U236.U23s must be synthesized in reactors by neutron capture in Th232.It is, therefore, more expensive. Furthermore, it is difficult to handle, being an unusual health hazard because of its high alpha decay rate and also because of large gamma activity accompanying decay products of U232 which is also produced through the (n,2n) process on the U233as it is made. Pu239is also relatively difficult to use because of the large alpha decay rate, and it must be synthesized by irradiation of UZ3* in a reactor. Only a few research reactors are not of the thermal type. We shall, therefore, concentrate here on only this class. The nuclear properties of U236of primary interest for reactor use are the following :s of

=

582 barns (fission cross section),

oc = 100 barns (capture cross section), a = 0.092 (capture to fission ratio), v = 2.43 (neutrons emitted per fission), = 2.074 (neutrons emitted per neutron absorbed).

These numbers are those applicable t o 2200 m/s (room temperature) neutrons; the averages over the distribution of neutron absorptions in a reactor are slightly different. The neutrons are mostly evaporation products from the highly excited fission products, emitted immediately after the fission.4 About 0.65% of the neutrons appear a considerable period of time after fission.s These delayed neutrons are emitted from fission products which are a t first cooled by evaporation until further neutron emission is no longer energetically possible. After a beta decay, emission of another neutron 3

R. Sher and J. Felberbaum, “Least Squares Analysis of the 2200 m/s Parameters

of U*ss, lJ236, and Pu*3@. BNL-722, June, 1962. 4 1. Halpern, Ann. Rev. Nuclear Sn’. 9, 245-342 (1959). 6

G . R. Keepin and T. F. Wimett, Nucleonics 16, No. 10, pp. 86-90 (1958).

3.2.

595

ARTIFICIAL SOURCES

10 9-

076-

5-

3 0)

.d

4-

h

-

.$ Y

d +

2-

1

0

1

3

2

I

4

5

I

6

Neutron energy (mev)

FIG. 1. Energy distribution of prompt fission neutrons.

is sometimes possible and so takes place. The delayed neutrons are then emitted with a half-life characteristic of the beta decay preceding them. The energy distribution of the prompt neutrons is continuous, as shown in Fig. 1. The average energy is approximately 2.5 MeV; the most probable energy is about 0.6 MeV. The distribution can be accounted for by the evaporation model. The delayed neutrons have lower energies, in the range of a few hundred kev. Six separate groups have been identified,6with characteristics shown in Table I. The existence of the delayed neutrons makes control of a nuclear reactor a much simpler matter. The neutron balance can be regulated more coarsely, because there is a partially delayed response to any deviation from the steady state. The kinetic behavior of a reactor is best discussed using the multiplicaTABLE I . Delayed Neutron from U*as Fission Half-life (sec-1) 54.51 f 0 . 9 4 21.84 rt 0 . 5 4 6.00 f 0 . 1 7

2.23 f 0.06 0.496 f 0.029 0.179 k 0.017

Relative yield 0.038 0.213 0.188 0.407 0.128 0.026

f 0.003 & 0.005

0.016 *f* 0.007 0.008 f 0.003

3.

596

SOURCES O F NUCLEAR PARTICLES AND RADIATIONS

100

5 0

J

; 10

1 10-3

10-2

Ak Reactivity (F)

FIQ.2. Asymptotic reactor period for positive reactivity.

tion factor k , which is the factor by which Y would have to be divided to cause steady operation. k is also very nearly the relative return of neutrons in a single fission cycle. The reactivity is defined as k - l / k = p." If the reactivity is positive, the fission rate in the reactor increases. If the reactivity is negative, the fission rate decreases. In either case the change in fission rate assumes an asymptotic exponential form, with a 6

S. Glasstone and M. Edlund, ibid., p. 298.

3.2.

597

ARTIFICIAL SOURCES

stable period T given by the ((inhourformula” P = 1/T

+ 2 Pi/(l + XiT)

(3.2.1.3.1)

i

where 1 is the average time between generations of prompt neutrons.

I varies in value in thermal reactors from about 50 psec for some light water moderated systems to about 1 msec for most graphite or heavy

Note: Curve has asymptote at 80.6 secondsseconds ~

5 x 10-5

lo-‘

10-3 Ak -Reactivity (-

r)

FIG.3. Asymptotic reactor period for negative reactivity.

water moderated systems. For normal reactivity changes associated with reactor control, the first term in Eq. (3.2.1.3.1)is negligible, and the relation between reactivity and stable period is seen to invoIve only the delayed neutron properties. Figure 2 is a plot of the asymptotic positive period vs. reactivity. Figure 3 is a similar plot for negative reactivities. Inspection will show that controllable increases of reactor power accompany values of p less than the total delayed neutron fraction 8. When p = 8, the reactor is

598

3.

SOURCES OF NUCLEAR PARTICLES AND RADIATIONS

critical on prompt neutrons alone, and above this point the braking effect of the delayed neutrons is lost. The previous statements on the relation between the reactivity and an asymptotic period must be modified to take into account feedback effects. The temperatures of reactor components have noticeable effects on the reactivity of the system, these effects being associated with several causes. Among the most important are the following: (a) As the moderator temperature is changed, the average energy of neutrons in near equilibrium with the moderator thermal motion also changes. Relative absorption cross sections of components are altered, and the neutron balance is upset. The effect is usually that of reactivity reduction with increasing temperature ; the absorption cross section of UZs6diminishes faster with energy increase in this region than do the capture cross sections of most other substances used.’ These generally have capture cross sections which are nearly 1/v. Thus the fraction of neutron absorptions causing fission decreases with rising moderator temperature. (b) The rate a t which neutrons leak from the reactor core is proportional to their speed. The increase of average thermal neutron speed with moderator temperature leads to an increased leakage fraction and thus a reduced fission fraction. The effect is a reduction of reactivity with increasing moderator temperature. (c) Reactors with low uranium enrichment have appreciable capture of neutrons in the UZ3* resonances. An increase in fuel temperatulje leads t o increased resonance capture and thus reduced fission probability, through doppler broadening of the resonances. (d) Most coolant materials capture neutrons. I n reactors with solid fuel, the greater expansion coefficients of coolants cause expulsion of neutron absorbers from the core region when the temperature is increased. The relative probability of fission increases as the temperature is raised; so does the reactivity. A number of other effects act on the reactivity as the temperature changes. There are influences of structural expansion, of changing spacial neutron distribution, etc. Each of these has an associated temperature coefficient of reactivity] which is the derivative of reactivity with temperature. It is considered essential for safety that the net coefficient be negative. Any unwanted increase in reactivity which causes power and temperature rises will then be opposed by a natural restoring action. There are also time changes in reactivity which are associated with burnup of the uranium. As the amount of uranium decreases, the reactivity falls for two reasons. The first is a decrease in the number of nuclei available for fission. The second is the production of new nuclei which BNL-326, Suppl. No. 1 to 2nd Ed., p. 118 (January 1, 1960).

3.2.

ARTIFICIAL SOURCES

599

compete for neutron capture. These are the fission products, a number of which have large cross sections for neutron capture. The fission product which affects reactivity the most is Xe136.It is primarily a daughter product of decay although there is evidence of a small direct yield also. and the precursor Te136are products of 6 % of fissions. Te136has a half-life of only 2 min. For this reason, the II36 can be considered to be formed directly, because it has a half-life of 6.7 hr. The daughter xenon has a half-life of 9.2 hr. The xenon thermal cross section is 2.7 X los barns, associated with a resonance strategically located a t 0.084 ev. The combination of high yield and cross section causes appreciable disturbance to the neutron balance and the reactivity a t reactor powers above a few hundred kev. At about 1 Mev a typical research reactor operating at steady state for periods greater than the half-lives concerned will have somewhat over 0.01 in reactivity tied up in the Xe136 capture. At higher powers this effect increases, to a theoretical maximum between 0.04 and 0.05 in most cases. The precise maximum differs from one reactor to another, depending on the neutron spectrum, the neutron flux shape, and other details. It is a true maximum, though, because a t the higher powers the burnup by the higher neutron intensity reduces the xenon content. The limiting case occurs when everywhere the xenon production rate is equal to its rate of destruction by neutron capture, with decay being negligible. cross section prevents this fission product from being The smaller destroyed significantly by neutron capture. Therefore, even when the Xe136 density is capture-limited, there is a large production rate. If the reactor is shut down, the xenon continues to be formed a t the same rate, while its rate of destruction falls to the level set by radioactive decay. The xenon density therefore rises, and the reactivity effect increases. A xenon transient occurs; if the reactor is left shut down, the xenon reactivity effect can rise to a very high value, 30% or more not being uncommon in very high powered systems. Since few reactors have provision for compensating such extreme changes through control rod motion, the reactor must then stay shut down until the iodine and the xenon have both decayed enough. A large xenon transient will last for from 1 to 2 days, Figure 4 shows the appearance of such a reactivity transient. Xenon transients become important at reactor powers above about 1 Mw. The second most important fission product is SmL42.This nuclide is stable. Therefore its reactivity effect increases steadily with reactor life until the burnup and production rates are equal. At this point about 0.012 of reactivity is taken by samarium capture. After reactor shutdown, there is an increase in the reactivity held by the samarium, because of the loss of burnup.

2

ln

2

0

600

0 0

-!

8

? N

? rl

9 rl

2

0

0

h In

rl 0

X 0

0

9

!!i

3.2.

ARTIFICIAL SOURCES

601

The other fission products have effects which are usually lumped into an average capture cross section of about 100 barns/fission. The burnup of uranium and the increased poisoning of the fuel combine to limit the duration of use of reactor fuel elements. Once the system cannot be held critical a t the steady power desired, refueling is needed. Fuel burnup in research reactors which use highly enriched uranium is commonly taken t o from 20% to 40% of the initial uranium charge. These reactors use alloy fuel which suffers damage from irradiation and from fission product gas evolution beyond this point. Reactors with natural uranium metal fuel are limited by radiation damage to only a few per cent burnup. Good design makes the reactivity limit and the metallurgical limit coincide. 3.2.1.3.3. ENGINEERING. 3.2.1.3.3.1. Fuel Elements. The fuel elements in most research reactors contain the uranium in an aluminum alloy. A sandwich construction is usedls the alloy being a thin layer faced with uranium-free aluminum. All edges are also protected by a picture framing which is integral with the surface cladding. The most common elements are made of plates. The clad, alloy, and picture framing are rolled as a unit, the whole being held together by edge welding and pressure bonding. Occasional elements of other shapes, such as cylinders, are coextruded sandwiches. The cladding serves to keep the fission products in the metal matrix. It is, therefore, thick compared with the path length of fission products (about 0.02 mm in aluminum), and even more important, is thick enough so that corrosion and erosion will not bare the alloy core to the coolant stream. The clad thickness is also chosen to be compatible with manufacturing tolerances, so that it is not too thin in any one place. A common choice is an alloy core about 0.5 mm thick in a cladding of the same thickness. Alloys of from 14% to 30% uranium by weight are common. At the higher value and beyond, it becomes necessary to soften the alloy with silicon, t o make the rolling properties compatible with those of the clad. Uranium metal fuel rods must be canned, because the metal is chemically active. The canning material is usually aluminum tubing, though British reactors use a magnesium alloy (Magnox). Some reactors are fueled with sintered UOz pellets, which are clad in a way similar to the metal. Some research reactors are homogeneous or semihomogeneous, with the fuel dissolved in or mixed with the moderator. These will be discussed separately. 8 J. E. Cunningham and E. J. Boyle, PTOC. 1st Intern. Cord. on the Peaceful Uses of Atomic Energy, Geneva 1965, P/953 9, pp. 203, 234 (1956).

602

3.

SOURCES OF N U C L E A R PARTICLES A N D RADIATIONS

3.2.1.3.3.2. Coolant. The common heat transfer agents in research reactors are light water, heavy water, and air. Water is the most efficient agent, having a high heat transfer coefficient and a high heat capacity. It is therefore preferred for high-performance reactors. Air is used only to cool some of the large graphite moderated reactors. Separated Li’ would be an excellent coolant, although the reactor it is used in would be expensive. The corrosion problem would be difficult to overcome. The limit on heat removal is set by the surface temperature of the fuel elements. This must be kept below the melting point. The limit is called the burnout point;Bin water-cooled systems it is expressed in terms of heat generation rate rather than temperature, which is more difficult to estimate. The burnout heat flux depends on the water temperature, the pressure, the channel flow rate, the channel dimensions, and perhaps other variables as well. The physics of the burnout process is poorly understood; thus it is standard practice to design the heat transfer aspects of the reactor conservatively. A substantial burnout margin is allowed for. High performance systems are often pressurized to raise the boiling point, since burnout will not occur until some point beyond the onset of surface boiling. The reason for avoiding fuel element melting is primarily safety. It is desirable to prevent the presence of fission products in the coolant. Furthermore, there appear to be no grounds to rule out possible exothermic metal-water reactions which could propagate. These could lead to pressures which might rupture the reactor vessel, causing severe radiation hazard. Though there has been no evidence of such a sequence in the cases of accidental burnout which have taken place, the policy has been t o remain conservative. 3.2.1.3.3.3.Moderalor. The neutron moderating material is usually the same as the coolant, though there are notable exceptions. The material must contain an abundance of light nuclei, to slow down the neutrons. It must have a lowcapture cross section in both the thermal and the epithermal regions. It must not be subject to severe radiation damage, which is caused principally by fast neutrons. It must, of course, exist as a manageable solid or liquid. The suitable nuclides are hydrogen, deuterium, carbon, beryllium, and oxygen. The materials which have been found acceptable are light water, heavy water, beryllium, beryllium oxide, and graphite. All have been used to some extent, though beryllium and its oxide are found only in a few high-performance systems. C. F. Bondla, in “Nuclear Engineering Ilandbook” (H. Etherington, ed.), 1st Ed. Chapter 9-3. McGraw-Hill, New York, 1958.

3.2.

ARTIFICIAL

SOURCES

603

3.2.1.3.3.4. Material Strength. The maintaining of structural integrity of a reactor is of paramount importance. Any breaking, bending, or melting of any part of the reactor core can cause quick reactivity changes which could lead to accidents liberating fission products. Therefore, careful attention is given to pertinent design details. The problem is complicated by the high ievels of radioactivity induced in all materials in the reactor proper. These cause repair of faulty components to be always difficult and sometimes impossible. Components must be overdesigned with respect to corrosion and stress. Allowance must be made for progressive radiation damage over the history of the system. l o Fabrication of reactor vessels and of associated pipes must conform to approved standards which have been set by the appropriate technical societies. 3.2.1.3.3.5. Control Rods and Drives.11 The materials used for control rods have large thermal neutron cross sections. In low-flux systems ( 0). EXAMPLE : Figure 2 shows the location of the focal planes and principal planes for the first magnet traversed by the beam in the two-magnet system discussed below (see Table I). The value of K = K-lI2 is taken as 1.396 m. Thus K = 0.513 m-2. Taking as a n example 3.0-Bev protons ( p = 3.825 Bev/c, e / c p = 0.784 X lo-'), one finds from Eq. (4.2.6) th a t dH,/dz = 654 gauss/cm. For a magnet of semi-aperture A/2 = 15.24 cm, this gives a field of 9,970 gauss a t the pole tips. The corresponding value of Hero for 3.0-Bev protons is: c p / e = 1.276 X lo7 gauss-cm. It will be assumed that the magnet has an effective length I = 1.19 m. One thus finds: X = 1.222 m, fl = 1.856 m. I n Fig. 2, the origin of the 5 axis is a t the object (source) which is at a distance of S1 = 4.57 m from the magnet entrance. The magnet entrance and exit are therefore located a t x = 4.57 m, and x = 5.76 m, respectively. The coordinates of the focal planes and principal planes are as follows: x F , O = 3.35, T F J = 6.98, x p , = ~ 5.21, X P J = 5.13 m. It is seen from Fig. 2 th a t the principal planes are very close to each other. Moreover, the average of .cp,o and X P . I [ ~ ( ~ P ,X~P J ) = 5.17 m] corresponds to the center of the magnet. These results have general validity, and can be easily proved as follows. If f p . 0 and Z P , denote ~ the coordinates of the principal planes Po,r and with respect, to the magnet entrance, we have

+

z p , o = K(CSC a zp,I = K ( a

- cot a) - CSC a).

+ cot

(4.2.17) (4.2.18)

698

4. I

BEAM TRANSPORT SYSTEMS

I

I

I

I

1

0

-fl-:-

I

2

4

3

7

8

9

+

b

DISTANCE x

1

-1-

D

A

0

1

21-

-1-

sl-

1

6

5

(IN M E T E R S )

FIG. 2. Schematic view of magnet showing focal planes and principal planes for direction 1 for which magnet action is focusing. The magnet shown in this figure has a length I = 1.19 m, and I[ was taken aa 1.396 m. The origin of the z coordinates cor' = 4.57 m from the entrance of the magresponds to an object placed a t a distance 51 net. The image I1 is located a t x = 8.01 m, corresponding to an image distance TI= 2.25 m.

From Eqs. (4.2.17) and (4.2.18), one obtains +(zp,o

+ z p , ~=)

iCYK

I n order to prove that the difference we note that

61

41.

= z p , ~- 5p.1

(4.2.19) is very small,

In the above example, a = 0.852, and Eq. (4.2.20) gives: a1 = 0.077 m, in good agreement with the value obtained above (5.21 - 5.13 = 0.08 m). Figure 2 shows the image 11produced by the magnet. Ilis located a t 5 = 8.01 m (image distance T I = 2.25 m). It should be noted that the Newtonian form of the lens equation used above (Eq. 4.2.16) is, of course, equivalent to the Gaussian form: (1/4

+ (1/d

=

l/fi

(4.2.21)

4.2.

BEAM BENDING AND FOCUSING SYSTEMS

699

where ul is the distance from the object to the object principal plane Po.1, and T~ is the distance from the image to FIvl.In the present example, we have: c1 = 5.206, T~ = 2.886, f , = 1.856 m, and Eq. (4.2.21) can be easily verified. 4.2.2.2. Defocusing Action of the Magnetic lens. Now we consider the direction along which the magnet action is defocusing. This direction (y or z ) will be labeled 2. The object and image distances will be denoted by S2 and T z , respectively. The convention with respect to the signs of Sz and Tz is the same as for S1 and TI. Thus if the effective object is to the left of the entrance face and the image is to the left of the exit face, S2 is positive and Tz is negative. I t may be noted that in the lens equations (4.2.16) and (4.2.24), we have in general S1 = Sz,provided that the object is anastigmatic, i.e., the effective source in the vertical direction coincides with the effective source in the horizontal direction. This condition is automatically satisfied if the particles to be focused are secondaries produced in a target which then acts as the source for both the vertical and horizontal directions. On the other hand, the beam of particles may emerge from a region of magnetic field having considerable focusing action, for which the effective vertical and horizontal objects presented to the focusing system are separated by a large distance. An example of this situation is encountered in the focusing properties of the external proton beam of the Brookhaven Cosmotron.’ The proton beam which emerges from the external magnet shims is approximately parallel in the horizontal plane (object a t infinity), whereas in the vertical direction, the particles appear to come from a point (vertical crossover7) inside the external shims. For the defocusing case, the object and image focal planes are a t a distance p : (4.2.22) p = K coth a from the field boundaries4 [see Eq. (4.2.127)]. I n contrast to the situation ~ t o the right for direction 1 (for a < 7r/2), the object focal plane F o , lies ~ to the left of the entrance boundary, and the image focal plane F I , lies of the exit boundary (see Fig. 3). The focal length f 2 is given by4 [see Eq. (4.2.115)]: fi =

- (p2

-

,2)1/2

=

- K csch

a.

(4.2.23)

In the same manner as in Eqs. (4.2.13)-(4.2.16)one obtains the following lens equation from Eqs. (4.2.22) and (4.2.23): (4.2.24) ( 8 2 -k p)(Tz -k p ) = p2 - K ~ . R. L. Cool, G. Friedlander, 0. Piccioni, S. L. Ridgway, and R. M. Sternheimer, Brookhaven National Laboratory Report BNL 498 (1-24) (1958).

4.

700

BEAM TRANSPORT SYSTEMS

The object principal plane Po,z is located at a distance / fi/to the left of the object focal plane FO,*.Similarly, the image principal plane is a t a distance lfzl t o the right of Fr,z.

EXAMPLE : Figure 3 shows the location of the focal planes and principal planes for the direction 2 for the magnet considered above in connection with Fig. 2. I

I

I

I

I

I

I

I

I

4

I 5

1

2

I 3

I

I

6

I 7

8

I

0

1

I 9

DISTANCE x (IN METERS)

FIG. 3. Schematic view of magnet showing focal planes and principal planes for direction 2 for which magnet action is defocusing. The magnet shown in this figure is the same as for Fig. 1 ( I = 1.19 m, K = 1.396 m), and the origin of the z coordinates is defined in the same manner as in Fig. 1 (object a t distance SZ= 4.57 m from magnet entrance). The image Zz is located at z = 4.065 m, corresponding to an image distance Ta= -1.695m.

The origin of the x axis is again taken a t the location of the object so th a t Sz = 4.57 m. With I = 1.19 m, K = 1.396 m, one obtains p = 2.018 m, f 2 = -1.457 m. The resulting coordinates of the focal planes and principal planes are as follows: X F , O = 6.59; X F J = 3.74; xp.0 = 5.13; xp#r= 5.20 m. Similarly as in Fig. 2, we again have x p j o = x ~ ,and ~ ,the average: +(xp.g z p , f ) = 5.17 m, corresponds to the center of the magnet. These results can be proved in the same manner as in Eqs. (4.2.17)(4.2.20). By expanding the hyperbolic functions (coth a and csch cy), one obtains

+

62 E

XP,O

- XPJ

%K

(-$+&)

(4.2.25)

4.2.

BEAM BENDING A N D FOCUSING SYSTEMS

701

which is approximately the negative of ti1 for the focusing case (direction 1). Thus 8 2 is also expected to be very small. The result given by Eq. (4.2.25) is: ti2 = -0.067 m, in good agreement with the value obtained above (5.13-5.20 = -0.07 m). The image I2produced by the magnet is located a t x = 4.065 m, corresponding t o a n image distance T2 = -1.695 m measured from the exit of the magnet. The fact that Tz is negative implies, of course, that the image is virtual, i.e., the outcoming rays effectively diverge from the image point. I n similarity to Eq. (4.2.21), we can also obtain the image distance from the following Gaussian form of the lens equation [as a n alternative to Eq. (4.2.24)]: (4.2.26) (l/UZ) (1/72) = l/fZ

+

where uz and r 2 are the distances from the object to the object principal plane and from the image t o the image principal plane PI,^, respectively. As discussed above, we have S1 = Sz in Eqs. (4.2.16) and (4.2.24), unless the effective source is astigmatic. However, the image distances T I and T 2 are, of course, quite different. Indeed T1 is generally positive because of the focusing action in direction 1, whereas T2 is negative (defocusing). The condition TI = T2 would correspond to double focusing, and this cannot be achieved with a single quadrupole lens, as was discussed in Section 4.2.1. 4.2.2.3. Focusing Properties of a Two-Magnet System. We shall now obtain the focusing properties of a system of two quadrupole magnets with field gradients of opposite sign. The two magnets will be denoted by a and b. The beam enters the system through magnet a and leaves the system through magnet b. The effective lengths of the two magnet sections will be called 1, and l b . The separation between the inner field boundaries (exit of a and entrance of b) will be denoted by d. The values of K for the two sections will be called K, and Kb. According t o an elementary theorem of geometrical opticsls a system of two lenses a and b, with focal lengths fa and f b , respectively, acts as a single thick lens with focal length f = - f a f b / q , where q is the distance between the internal focal planes of the system, i.e., between the image focal plane of a, FI,, and the object focal plane of b, Fob. The sign of q is taken as positive if Fob is to the right of Fro (assuming that the particle beam travels from left to right). The object focal plane Fo of the equivalent lens is a t a distance f u z / q from the object focal plane FOoof magnet a, 8 See, for example, G. S. Monk, “Light, Principles and Experiments,” p. 21. McGraw-Hill, New York, 1937.

702

4.

BEAM TRANSPORT SYBTEMS

and is in front (to the left) of Foaif q is positive. The image focal plane FI of the combined system is a t a distance f b 2 / qfrom the image focal plane F I b of magnet b ; if q > 0, then FI is to the right of Fib. EXAMPLE : Figure 4 shows the focal planes for the two-magnet system which is discussed below, for the direction 1 for which the first magnet (a) is focusing. The characteristics of magnet a have already been discussed in

0

I

2

3

4

5

6

i

7

8

9

DISTANCE x ( I N METERS)

FIG. 4. Schematic view of two-magnet system showing focal planes and principal planes for direction 1 for which magnet a is focusing. The two-magnet system shown in this figure is discussed in the text (la = h = 1.19 m, d = 0.280 m, I C ~= 1.396 m, Kb = 1.478 m). The origin of the x coordinates corresponds to an object placed at a distance S1. = 4.57 m from the entrance of magnet a.

connection with Fig. 2. Magnet b has the same length as a ( l b = 1.19 m), and the value of Kb is 1.478 m. The effective separation d is 0.280 m. For magnet b, one finds: p b = 2.219 m, and f l b = - 1.655 m. The coordinates of the focal planes of a and b (with respect to the object at a distance S1, = 4.57 m from the entrance of a) are as follows: ( x F , O ) ~ = 3.35; ( ~ p . 1= ) ~ 6.98; ( X F , O ) b = 8.26; ( x F , I ) ~ = 5.01 m. The distance q1 is equal to: ( X F , O ) b - ( x F , I ) ~= 8.26 - 6.98 = 1.28 m. The focal planes and principal planes of the equivalent single lens are also shown in Fig. 4. The x coordinates of these planes are as follows: X F . O = 0.652, X F , I = 7.154, x p . 0 = 3.057, X P J = 4.749 m. We note that in this case, the principal

4.2.

703

BEAM BENDING AND FOCUSINQ SYSTEMS

planes are separated by a considerable distance, unlike the situation for a single quadrupole magnet (Figs, 2 and 3). The focal length of the equivalent system is: f l = 2.405 m. It may be noted that in Fig. 4, we have omitted for simplicity the subscript 1 from the designation of the focal planes and principal planes; e.g., Foa instead of ( F o , ~ ) ~ . In view of the preceding results for a single lens (magnet) and the theorem for combining the action of two lenses, we obtain the following equation4which determines the image distance for the direction 1 (y or z ) , defined as the direction along which the first magnet (a) is focusing: (s1a

f

Ula>(Tlb

+

=

Ulb)

(4.2.27)

J1

+

where J 1is the square of the focal length f l of the system; Sla U 1 , and Tlb U l b are the distances X O . ~and of the object and image from the object focal plane and image focal plane of the system, respectively. Thus Eq. (4.2.27) is equivalent to Eq. (4.2.13) for the Newtonian form of the lens equation. S 1 , is the object distance measured from the entrance face of a, and T , b is the distance from the exit of b to the image. The constants J1,UI., and u 1 b are given by

+

J1

=flz =

u~a Ulb

(Ka2

E=

-ha -

=

Pb

-

-k

(Ka2

(h2-

- Kbz)/ql2

Xa2)(Mb2

+

XaZ)/ql

Kb2)/ql

(4.2.28) (4.2.29) (4.2.30)

where q1is the distance between the internal focal planes, and is given by: =

41

d

- X u -I- p b .

(4.2.31)

In Eqs. (4.2.28)-(4.2.31), A, 3 K . cot a! and C(b = Kb coth p, where = l a / ~ aand p = Ib/Kb. UIa is the distance from the object focal plane F o , of ~ the system to the effective entrance of a, and U l b is the distance from the exit of b to the image focal plane RS1. A similar set of equations describes the focusing action in the direction for which magnet a is defocusing. The lens equation is given by? a

(825

Uza)(Tzb

+

Uzb)

= Jz

(4.2.32)

where Sza and Ts are the object and image distances measured from the entrance of a and the exit of b, respectively. The constants J z , U2.,and U z b are obtained from the following expressions : J z = fzz = (Pa2 U 2 a = Pa

-

(Pa2

Ka2)(Kb2

- Ka2)/q2

+ hb2)/qZ

-Xb

+

-

(Kb2

d

Pa

- Xb

where pa = K . coth a, and

hb

= Kb cot p.

u2b

qz

= =

+

Xb2)/qz2

(4.2.33) (4.2.34) (4.2.35) (4.2.36)

704

4.

BEAM TRANSPORT SYSTEMS

In order to obtain double focusing, we must have: Tlb

=

(4.2.37)

T2b.

For simplicity, i t will be assumed in the following that the object is anastigmatic, so that S1, = S z a = S,. We shall now show that for a given particle momentum, given values of K~ and Kb, and for a given geometry of the magnets (la, l b , d ) , there exists a t most one set of values of Saand Tb (object and image distances) for which the system is double focusing. In order t o vary the focusing conditions for a given momentum, i t is then necessary to vary the currents in the two magnets, so as to change K~ and Kb. For simplicity, we consider the case where the two sections are equal in length and have field gradients of the same magnitude (though, of course, with opposite signs). Thus 1, = l b , and Ka = Kb. We then have: U1, = u 2 b ; Uza = Ulb; and J 1 = J 2 . Equations (4.2.27) and (4.2.32) for the focusing become:

(8, (Sa

+

ula>(Tb

Ulb)(Tb

+

Ulb)

=

u~a)=

(4.2.38) (4.2.391

J1 J1.

From Eq. (4.2.38), one finds Tb = - U l b

-I- J l / ( S a -I- UU).

(4.2.40)

Upon inserting this result into Eq. (4.2.39), one obtains after some algebraic reductions :

Sa2 which gives4

8a

+

+ U,) + + Ulb) + $[(U,a -

Sa(U1a

= -+(ula

UlaUlb

- J1

ulb)2

=

0

(4.2.41)

~ J I ] ' ' ~ . (4.2.42)

If this expression for S, is positive, it corresponds to a case of double focusing. We note that in this arrangement, we have Sa = T b , i.e., equal object and image distances, so that the focusing is symmetrical with respect to the center of the magnet system (point midway between the two magnets). This result is also obvious on general grounds, since we have assumed that ua = Kb, and Ia = la. The same procedure as in Eqs. (4.2.38)-(4.2.42) can also be used when the two magnets have unequal lengths and/or different field gradients. In this case, there is again only at most one set of values of S, and Tb for which double focusing occurs. Thus one obtains S a < T b by making the quadrupole magnet a stronger than magnet b, i.e., a! > p. Similarly double focusing with Sa > Tb is obtained by makihg LY < 0. If either of the magnets, e.g., b, is considerably weaker than the other

4.2.

BEAM BENDING

AND FOCUSING

705

SYSTEMS

I ~

I

/

sKh ‘ 8 , 3 W 8 )

I

~

sK(7/2,r/2)l

1 0.5

1.0

1.5

2.0

2.5

3.0

3.5

dK

FIG.5. Object distance S K and image distance T K for double focusing by various two-magnet systems, as a function of the separation dK. The numbers a f i e d to each curve are the values of (Y = l . / ~and @ = I ~ / Kfor the two sections (a and b). SK,T K , and dK represent the values of S, T,and d, respectively, in units K-lIp = K .

(a), then there may not be any possibility of double focusing if magnet b is unable to compensate for the strong focusing and defocusing produced by magnet a in the directions 1 and 2. Figure 5 shows the results of calculations for various double focusing systems, as previously obtained in reference 4.The curves give the object or image distance S K or TK as a function of the separation d~ between the two magnets. Here S K , T K ,and dK are the distances S,, Tt,,and d in units of K-”2 = K . It was assumed that the two sections have field gradients of the same magnitude (K,, = Kb = K ) but that they may have

706

4. BEAM

TRANSPORT SYSTEMS

different lengths. The different curves correspond to different values of L/tC = LY and & / K = p. The notation used is ~ K ( L Y p) , and TR(LY, p). For the cases with a = p (1, = &,), SK and T K are equal. Figure 5 shows that SK and T K generally decrease rapidly as d K is increased. This illustrates the well-known fact that for a two-lens system consisting of a convergent and a divergent lens, the focusing power is increased by increasing the separation of the lenses. Of course, eventually one may lose in effective aperture if the separation is made too large [see below, Eqs. (4.2.46) and (4.2.47)]. For the system with a = 7r/4, p = 37r/16, the lens b is rather weak compared to a. As a result, for d K < 0.5, there is no longer any double focusing position. Actually, for d K = 0.5, the image distance T K becomes infinite, and it remains very large as d~ is increased up to 0.7. Even for d K = 1.0, we have T K = 5.9 as compared to SIC = 2.3, which shows that the relatively small difference a - /3 = x/16 leads to widely different values of SK and T K ,even when the separation dK is appreciable. I t should be noted that the equations for the focusing in both the vertical and horizontal directions are quite complicated, so that it has not been possible to obtain expressions which give the required values of LY and p for double focusing in terms of the object and image distances S and T . I n general, a and p must be obtained by a procedure of successive approximations, i.e., by assuming a set of values a! and 8, and calculating T for both the vertical and horizontal directions. These values of T will be denoted by T , and Ti,, respectively. In general, one finds T , # Th, and both T , and TI, are different from the desired value of T . Then a and 0 must be suitably changed until a set (a$) is obtained for which T , and Th are both equal to the required T . If one has a strong focusing system a t one’s disposal, with a given geometry (la, la, and d fixed), the general focusing properties can be obtained by calculating the object and image distances S and T for double focusing for various assumed values of K,, and Kb. Then, on a graph of K. versus p , where p = K ~ / K . , one can plot a set of curves of constant S (but varying T ) and another set of curves each of which has constant T (but varying S). The intersection of the curve S = SOwith the curve T = To gives the required values of K. and p for double focusing a t S = Soand T = T o . An example of such a plot is shown in Fig. 6. These curves were calculated for the large strong focusing system in use a t the Brookhaven C o s m o t r ~ nThis . ~ system has an aperture A = 30.5 cm. Each quadrupole has a length & = ib = 1.012 m, and the separation of the iron of the two 0 R. M. Sternheimer, Brookhaven Cosmotron Internal Reports RMS-53 and 55 (1955).

4.2.

BEAM BENDING AND FOCUSING SYSTEMS

707

FIG.6. Plot of K, vs. p for double focusing a t various object distances S and image distances T for a two-magnet system with I, = b = 3.90 ft. = 1.189 rn, d = 0.92 f t = 0.280 m. The abscissa p is defined as: p = Kb/Ke. The intersection of curve of constant S with a curve of constant !/' gives the required values of K, and p for double focusing. The solid curves are curves of constant T,while the dashed curves pertain to constant S. The values of S, T,and K~ in this figure are given in feet (conversion factor: 1 f t = 0.305m).

magnets is d = 0.457 m. As was noted above, the effective length 1 of each magnet should include a correction for the fringing field a t the entrance and exit of the magnet. According to the work of Dayton et a1.,l0 the correction A1 a t each end of the magnet is: A1 = 0.29A1where A is the aperture of the quadrupole, i.e., the distance between opposite pole tips (see Fig. 1). Thus the effective length of the quadrupoles is: 1. = lb

=

1.189 m

and the effective separation is: d = 0.280 m.ll This system has been used extensively to double-focus the external proton beam of the Cosmotron, with an effective object distance S i% 4.6 m, and for various values of the image distance T. As an example, for S = 15 ft = 4.57 m, T = 30 f t = 9.14 m, Fig. 6 gives K~ = 4.58 ft = 1.396 m, p = 1.060, so that Kb = 1.480 m. This is the double focusing system which has been considered I. E. Dayton, F. C. Shoemaker, and R. F. Moeley, Rev. Sci. Instr. 26,485 (1954). However, according to recent measurements of F. Eisler (Cosmotron Internal Report FE-4, January, 1960), A1 is only 0.17A. If thia value of the fringing field correction is used, the effective length of thc magnets becomes I, = .!b = 1.116 m, and the effective separation d = 0.354 m. ID 11

708

4.

BEAM TRANSPORT SYSTEMS

above in connection with Fig. 4. In the discussion concerning Fig. 2, it was shown that for 3.0-Bev protons, the required field H a in magnet a is 9,970 gauss. Similarly, the field in section b is given by: H b = (Kb/Ka)Ho =

9,970 X (1.396/1.480)’

=

8,870 gauss.

(4.2.43)

The required field strengths H , and Ha are proportional to the momentum p . Thus for 2.0-Bev 7 mesons ( p = 2.135 Bev/c), one finds:

Ha

=

9,970 X (2.135/3.825)

=

5,565 gauss,

H b = 4,951 gauss.

The curves of Fig. 6 are applicable to any system with I, = zb and with the same value of d / l a = 0.280/1.189 = 0.235. It is necessary only to scale all distances appropriately. TABLEI. Values of the Pammeters for the Vertical and Horizontal Focusing by a Two-Magnet System’ Parameter A, P6

91 u 1 a

Ulb Jl1’2

= f,

Tib Pa Ab

92

uz. U2b 521’2

Tzb

= fz

Value 1.222 2.219 1.277 -3.920 0.075 2.405 8.778 2.018 1.423 0.875 -0.408 -6.245 3.420 9.053

1, = l b = 1.189 m, d = 0.280 m, K. = 1.396 m, K ) = 1.478 m. The object distance Sa is taken &B 4.57 m, and the system gives double focusing at an image &tance Tb S 8.9 m. All lengths are given in meters. 0

As an example of the focusing equations [Eqs. (4.2.27)-(4.2.36)], we have given in Table I a list of the values of the various focusing parameters for the two-magnet system described above (Za = zb = 1.189 m, d = 0.280 m, ua = 1.396 m, Kb = 1.480 m). It should be noted that the same type of strong focusing action which is obt,ained with quadrupole magnets can also be achieved b y means of electrostatic lenses. Since the force on a particle in a n electric field E is: F, = eE, as compared to the magnetic force e(v X H)/c [Eqs. (4.2.2)

4.2.

BEAM BENDING AND FOCUSING SYSTEMS

709

and (4.2.9)], the equivalent electric field must be of the form: E , = key, E. = kez, where k, is a constant given by. k, = ( u / c ) ( d H , / d z ) = ( u / c ) k . Since the electrostatic unit (e.s.u.) of potential, the statvolt, equals 300 volts, the field gradient k , in volts per cm per cm which is equivalent to a magnetic field gradient dH,/dz, is given by ke.volts/crn/orn=

~ O O ( V / C( d) H y / d Z ) g o u s s / c r n .

In several applications, magnetic or electrostatic strong focusing systems have been used to focus external beams from cyclotrons and Van de Graaff generators, with a resultant large increase of intensity at the image position (by a factor of the order of 10 to 30).12 In designing a magnet system for double focusing, one is generally interested in the lateral magnification x1and xz in the directions 1 and 2. The lateral magnification x, is defined as d y I / d y o or dzl/dzo, where dyo and dzo are small horizontal and vertical displacements at the object, and dyr, dz1 are the resulting displacements of the trajectory a t the image. The absolute magnitude lxal is equal to b,/a,, where a, is the size (height or width) of the object (source) and b, is the size of the image along the direction 2. xI (i = 1, 2) is given by x 1

= -"ft/Xa

=

--J:fz/(S%a-k Um)

(4.2.44)

where fi is the focal length for the direction i, and x, is the distance between the object and the object focal plane Fo., Qf the system. We note , follows that for equal lengths and equal field gradients (1, = la,K, = K ~ ) it directly from Eqs. (4.2.38) and (4.2.39) that xlxz = 1. For such a system, x1 and x 2 are considerably differentin general. As defined above, x1is the magnification for the direction along which magnet a is focusing. As an example, for a = 6 = 7r/4, dx = 0.5, for which SK = T K = 3.6 (see Fig. 5 ) , we have: XI = -2.4, xz = -0.42. Similarly, for a = p = ~ / 4 , d K = 1.5 (SK= T K = 2.2), one finds: X I = -3.7, xz = -0.27. (Here the minus sign merely means that the image is inverted.) I n the strong focusing systems which have been used a t the Cosmotron in the past few years for secondary particle beams from a target placed in the proton beam, magnet a is in general vertically focusing ( = direction 1). This has the advantage that the horizontal magnification (xz) is quite small, l* See, for example, B. Cork and E. Zajec, University of California Radiation Laboratory Report UCRL-2182 (1953); F. B. Shull, C. E. MacFarland, and M . M. Bretscher, Rev. Sci. Znstr. 26, 364 (1954); C. J. Strumski, D. I. Cooper, L). H. Frisch, and R. L. Zimmermann, d a d . 26, 514 (1954); E.L. Hubbard and E. L. Kelly, zbzd. 26, 737 (1954); D.A. Bromley and J. A. Bruner, University of Rochester Report NYO3823 (1954). See also A. Septier, Advances an Electronics and Electron Phys. 14, 85 (1961).

710

4.

BEAM TRANSPORT SYSTEMS

thus giving a small horizontal image width which does not interfere appreciably with the energy resolution obtained from the analyzing magnet traversed by the beam. Thus, for a secondary beam, one will have in general a bending magnet behind the quadrupole focusing system in order to momentum-analyze the particles. If one assumes, for simplicity, that the bending magnet (with uniform field) has negligible focusing action, then the location and the width of the image past the bending magnet will be primarily determined by the quadrupole system. In particular, the image width b h is given by X h a h , where a h is the width of the object (target) “seen” by the quadrupole system. If the bending magnet has a momentum dispersion D , then the range of momenta accepted inside the image width is: A p = X h a h / D . Here D is defined by: D 5 d y / d p , where y is the lateral position of the image as a function of p. It is seen that A p is proportional to X h . Thus the momentum resolution will be improved if X h is decreased, and for this reason, it is advantageous to make magnet a vertically focusing and b horizontally focusing, instead of the reverse. Another consideration in the design of a strong focusing system is the effective aperture B at the entrance of magnet a ; B is defined as follows. We consider the extreme trajectories which just graze the magnet iron in either section a or b. The values of y and z a t the entrance of a, to be denoted by ye and ze, are then compared to the maximum allowable values of y and z inside the magnet system, ym and zm (which are of order A/2). Then B is defined by: B 3 Yeza/(YmZm). For magnet systems in which the first section, a, is vertically focusing, one finds y,/ym 0.4, ze/Zm 0.9 for typical cases, SO that B is of the order of 0.3-0.4. The effective area a t the entrance of magnet a is - € A 2 . The intensity a t the image is proportional to the solid angle subtended a t the source which is -eA2/S2, where S is the object distance. If the detector at the image collects all of the focused particles, the intensity gain G due to the use of the magnet system is:

-

-

G LZBA~L~/(LS~C%) (4.2.45) where L is the total distance between the source and image, and Q. is the area of the detector. For the case that the first section (a) is vertically focusing, one can easily show that ZJZm and ye/ym,to be denoted by E# and eY, respectively, are given by the following equations: (4.2.46)

4.2.

711

BEAM BENDING AND FOCUSING SYSTEMS

We have: c = c,C., so that E can be calculated from Eqs. (4.2.46) and (4.2.47), if the values of K. and Ki, for double focusing have been determined. If the first section (a) is horizontally focusing, E . will be given by Eq. (4.2.47) [instead of Eq. (4.2.46)], whereas eY will be given by Eq. (4.2.46). However, the value of E = cue. is obviously not affected by this interchange of the equations for e, and 6,.

-5 -5

I

I

22.5

1

20.0 17.5

N

15.0 h

+ 12.5 z w

5

10.0

U -I

a

2

7.5

0

5.0 2.5

0

0

2

4

6

8

DISTANCE x

10

12

14

16

18

(IN M E T E R S )

FIG.7. Trajectories with maximum amplitude yn = zm = 15.2 cm for the twomagnet system discussed in the text, which gives double focusing for S = 4.57 m, T 8.9 m (see Table I). It is assumed that magnet a is vertically focusing. We note that the angles of acceptance a t the object, 8, and 8. for the y and z directions are given by: 8, = 2cYym/Sand 6, = 2e,zm/S. The solid angle a t the object is: A n = OY8.. For the two-magnet system considered above (Table I), we have: E . = 0.956, E , = 0.409, so that B = 0.391. In order to obtain the intensity gain G from Eq. (4.2.45),we assume that the detector is a 10-cm diameter counter, so that a = 78.5 cm2. With A = 30.5 cm, S = 4.57 m, and L = 16.06 m, one finds G = 57.2. Figure 7 shows the y and z trajectories with maximum amplitude ym = z, = 15.24 cm for the two-magnet system discussed above. I t has been assumed that magnet a is vertically focusing. The values of y

712

4.

BEAM TRANSPORT SYSTEMS

and z a t the entrance of magnet a are: ye = 6.25 cm, ze = 14.58 cm, giving cu = ye/ym = 0.410, e, = ze/zm = 0.957, as was obtained above. The focal planes of the equivalent thick lens are also shown in Fig. 7. The locations of the focal planes (with respect t o the object a t S, = 4.57 m) are as follows: Z O . ~= 0.652; xi," = 7.154; x 0 , h = 4.164; X i , h = 13.48 m. The magnet system gives approximate double focusing at 2 6Z 16.1 m, corresponding to an image distance T !Z 8.8 m from the exit boundary of magnet b. 4.2.2.4. Quadrupole Magnets with Rectangular Aperture. Recently, Panofsky and Piccioni have independently proposed the use of quadrupole magnets with rectangular aperture in order to increase the solid angle of acceptance AQ a t the source for the same magnet gap area as for a system of conventional quadrupoles (with hyperbolic pole faces). The reason for this improvement is as follows. Figure 7 shows that the aperture required in the y direction for magnet a is appreciably smaller than the aperture required in the z direction. Thus by using for magnet a a quadrupole with rectangular cross section, and w, < ha (w,= width, ha = height), and correspondingly for magnet b, wb > hb, one can increase the solid angle subtended a t the source, and hence the intensity a t the image, for the same magnet area as for a conventional quadrupole. Hand and Panofsky13 have designed and built a rectangular quadrupole for use in conjunction with a conventional (15.2 cm diameter) quadrupole which precedes the rectangular magnet and produces the vertical focusing of the beam. I n the rectangular quadrupole, the desired magnetic field is obtained by means of current sheets on the four sides of the magnet, with the current flowing along the same direction on the two lateral sides, and along the opposite direction on the magnet faces which are above and below the beam. It can be shown that this configuration gives the required constant field gradient inside the aperture (dH,/dz = dH,/dy = k ) . The magnet constructed by Hand and Panofsky13 has a width w = 58.4 cm, height h = 10.2 cm, and length I = 102 cm. The maximum attainable field gradient is -430 gauss/cm. The problem of the increase of intensity obtainable with rectangular quadrupoles has been investigated by Sternheimer.14 It was found th a t the increase of intensity (as compared t o conventional quadrupoles havwhere ing the same field gradient) is given by a factor tl = ymax,a/yeXit.,, ymax,b is the maximum horizontal displacement in magnet b, and yexit,, is the value of y at the exit of a, for a given trajectory. q is given by the following expressions : la 14

L. N. Hand and W. K. H. Panofsky, Rev. Sci. Instr. 30, 927 (1959). R. M. Sternheimer, Brookhaven AGS Internal Reports RMS-6 and 7 (1959).

4.2. (1) for Sza

BEAM BENDING AND FOCUSING SYSTEMS

713

> K. (4.2.48)

where E is defined by (la/Ka)

6

(2) for SZa < K ~ : 7 =

E

E

[I

+

(la/Ka)

+ COth-'(S2a/Ka)

coth 5 Ka

+

( d 2 +2Kb2)

coth2 E

Ku

(4.2.48a)

]

'I2

+ tanh-'(Sza/Ka).

(4.2.49)

(4.2.49a)

In order to realize the intensity gain b y a factor 7, it is necessary to make Y b = q Y,, where Y, and Y b are the available horizontal apertures of a and b, respectively. (Ya,b equals the width w for a rectangular quadrupole.) The z aperture of magnet b can be made correspondingly smaller than that of a. We have: zb = Z o / { , where l is given by the following expression for Tlb > Kb: (d2 + Ka2) tanhZ81'"

(4.2.50)

Kb2

with

8E

(lb/Kb)

+ COth-'(Tib/Kb).

(4.2.50a)

A similar expression applies for Tlb < Kb. I n reference 14, the values of 7, eg, and e, were calculated for various two-magnet systems as a function of the length I ( = 1, = h,) for three values of the ratio d / l : namely, 0 , 0.4, and 0.8. As a n example, for 1 = K, one finds '1 = 1.32 for d = 0 , 7 = 1.63 for d = 0.41, and = 1.97 for d = 0.81.

A disadvantage of the rectangular quadrupoles is that they require considerably more power than conventional quadrupoles for the same aperture area and the same field gradient. 4.2.2.5. High-Current Focusing Lens. I n another recent development, the use of high-current focusing lenses has been considered. The basic idea underlying this device is that a uniform cylindrically symmetrical current flowing parallel to the direction of an incident beam of particles would produce a restoring force towards the axis of the cylinder proportional t o the distance T from the axis. I n practice, such a uniform current can be approximated by an array of thin parallel wires. A given particle will then experience the required field ( a T ) , except when it comes close to one of the conductors, in which case there may be appreciable deviations. If the particle hits a conductor, it will be scattered and thus will not contribute to the intensity a t the image.

714

4.

BEAM TRANSPORT SYSTEMS

A focusing lens of this type has been built and tested by Luckey16 at M.I.T. He used a regular hexagonal array of nineteen conductors. The spacing between adjacent conductors was 3.8 cm. The lens had a length of 100 cm, and was operated at currents up to 1000 amp per conductor which gives a gradient of -46 gauss/cm. The present limitation on the lens arises from the force on the conductors. However, it is expected that with additional mechanical constraints on the conductors, considerably larger gradients can be obtained. As a test of the focusing properties of the lens, 100-Mev electrons produced by the bremsstrahlung beam of the M.I.T. Synchrotron were focused by the lens with an object distance (from the source to the front of the lens) of 200 cm. At a distance of 200 cm behind the exit of the lens, the intensity was observed with a scintillation counter. As the lens current was varied, a maximum was obtained a t the value (-500 amp) calculated for lOO-Mev/c particles. For a 5 cm diameter source, the size of the image was -7.5 cm high by 10 cm wide. The image size did not diminish appreciably when a smaller (1.3 cm diameter) source was used. It is likely that the deviations from the required field near the conductors give rise to an image size of the order of the spacing between the conductors. The main advantage of the wire lens as compared to standard quadrupoles is that one can obtain a higher solid angle of acceptance A n at the object for the same stored energy. On the other hand, the finite size of the image and imperfect focusing constitute a serious drawback to the use of the high-current lens. 4.2.2.6. Momentum Dispersion at the Image. In connection with the design of a strong focusing system, it is useful to determine the momentum dispersion at the image, i.e., the values of dT,/dp and dTh/dp, where T , and Th are the image distances for vertical and horizontal focusing, respectively. A large value of these derivatives indicates that particles whose momentum p deviates slightly from the desired momentum po will be focused a t an appreciabIe distance from the focus for p = PO,and can therefore be largely eliminated by means of appropriate collimators. We will first obtain d T / d p for a single quadrupole magnet which acts as a convergent lens. From the definition of K [Eq. (4.2.6)], one finds dK

K

=

(4.2.51)

zp'

Similarly, dX

&=

x 4-

1 csc2 a 2P

D. Luckey, Rev. Sei. Znstr. 31, 202 (1960).

(4.2.52)

4.2.

BEAM BENDING AND FOCUSING SYSTEMS

715

From Eq. (4.2.16), one obtains

T

=

(SX

+

- A)

K')>/(&

(4.2.53)

whence

dT dX ~ )+ _ - ( P + Kdp

( S - X)'dp

2~ ~

dK -.

(S - X)dp

(4.2.54)

Upon inserting Eqs. (4.2.51) and (4.2.52) into Eq. (4.2.54), one finds (4.2.55) which is the desired result. I n connection with the calculation of d T / d p for a two-magnet system (see below), it is useful to express d T / d p in terms of the image distance T , instead of S. By means of Eq. (4.2.16), one obtains (4.2.55a) which is equivalent to Eq. (4.2.55). In order to derive d T / d p for a single divergent lens, we note that Eq. (4.2.24) gives: (4.2.56) T = - ( K2 SP)/(S PI.

+

+

By proceeding in the same manner as in Eqs. (4.2.51)-(4.2.55a), one obtains the following expressions for d T / d p :

+ d2[p(S' + + 2 ~ ' s+ (8' - K')Z csch' a]. (4.2.57) dT 1 T ( T 2 - K')Z csch2 a]. (4.2.57a) dp - 2p(K2 - p 2 ) [ p ( T 2+ + ~ K ' + dT

1

K')

d p = - 2P(S

K')

For a two-magnet system, the momentum dispersion at the image dTb/dp is given by

where ( a T a / a p ) bis the value of d T b / d p for magnet b alone, for a fixed object distance Sb from the entrance face of b. Here sb refers to the distance of the object seen by magnet b, i.e., the image formed by magnet a. Thus we have: dSb = -dT,, where d T , is the change in the position of the image formed by a due to a small change of momentum d p . This leads to the relation (4.2.59)

716

4.

BEAM TRANSPORT SYSTEMS

which has been used in Eq. (4.2.58); here aT,/ap is the momentum dispersion of magnet a alone. I n Eq. (4.2.58), aTb/aSb is to be evaluated for a constant momentum p. We first consider the case where magnet a is focusing, and magnet b defocusing (direction 1). I n order to obtain dTb/aSb, we differentiate Eq. (4.2.24) with the following result dSb(Tb f Pb) so that

+ dTb(sb + Pb)

=

0

(4.2.60) (4.2.60a)

The subscript b denotes that all quantities pertain to magnet b. Upon inserting Eqs. (4.2.55), (4.2.57a), and (4.2.60a) into Eq. (4.2.58), one obtains

where 8, is the object distance from the entrance of a, and Tb is the distance from the exit face of b to the final image. For the direction 2 for which magnet a is defocusing, one obtains by means of Eqs. (4.2.55a), (4.2.57), and (4.2.58) :

It may be noted that Eqs. (4.2.61) and (4.2.62) involve only the field gradients and the distances S, and Tb, but no intermediate quantities such as T, and sb. Thus (dTb/dp)i can be calculated directly after the parameters K, and Kb of the strong focusing system have been determined. For the two-magnet system considered above (Table I), for the direction 1, one finds (dTb/dp)l = (115.6/p)m/(Bev/c), where p is the particle momentum in Bev/c. Similarly, for direction 2, one obtains (dTb/dp)z = (27.8/p)m/(Bev/c).

As was expected, both values of (dTb/dp)i are positive, meaning that the image distances increase with increasing momentum. 4.2.2.7. Width of the Image Due to a Momentum Acceptance Ap. For a point source along the axis of the strong focusing system, the expressions

4.2.

BEAM BENDING AND FOCUSING SYSTEMS

717

for ( d T b / d p ) land ( d T b / d p ) zenable one to determine the width of the image along the y and z directions due to the momentum acceptance Ap. These widths will be denoted by w, and w,,respectively. In the present discussion, we assume that Ap is given. Ap is determined by the aperture of the quadrupoles and the bending magnet which is used to momentumanalyze the particles and which generally precedes the quadrupole pair. A rough estimate of Ap for such an arrangement will be given below (see Section 4.2.4). The average momentum of the particles will be denoted by PO, and the range of accepted momenta is assumed to extend from po - ( A p / 2 ) to p a ( A p / 2 ) . We will assume th at magnet a is vertically focusing, and consider first the horizontal direction, in order t o obtain the width wy.We consider particles of momentum PO ( A p / 2 ) . These particles will be focused a t a distance ATb,

+

+

(4.2.63)

behind the central image l o pertaining to momentum P O . For the particles with p = pa, we denote by f y ~ , othe values of y a t t,he exit of magnet b for the extreme trajectories which just graze the ( A p / 2 ) , the corresponding pole faces of magnet b. For momentum po values of y a t the exit of b can be written as follows:

+

AYE

=

5 YE.O & y ~ , i ( A p / 2 )

(4.2.64)

where YE,^ is of the order of ~ E , ~ / p The o . second term of Eq. (4.2.64) is thus very small, being of the order of Ap/po times the first term ( Y E , o ) . It can be easily shown that the extreme y coordinates for

P

=

Po

+ (AP/2)

a t the central image ZO are given by:

We will henceforth neglect the second term of Eq. (4.2.65), since it is of order ( A P ) ~ . The total width w, at the central image 10 due to the particles with momentum p a ( A p / 2 ) is obtained b y taking the difference between the two y values of Eq. (4.2.65).One finds:

+

(4.2.66)

For particles of momentum po - ( A p / 2 ) , the image lies a t a distance ATb in front of T o , where ATb is given by Eq. (4.2.63). It can be easily

718

4.

BEAM TRANSPORT SYSTEMS

shown that the resulting width w, at I0 is again given by Eq. (4.2.66). For particles whose momentum lies inside the range from P O - ( A p / 2 ) to po ( A p / 2 ) , the width of the beam a t IO will be less than the value given by Eq. (4.2.66). Therefore, Eq. (4.2.66) for w,represents the total width of the beam at lo, when the momentum acceptance is Ap. The width w,in the vertical direction is similarly given by:

+

(4.2.67)

where 5-Z E , O are the values of z at the exit of quadrupole b, for the extreme trajectories for momentum P O . The values of yg.0 and z E , ~in Eqs. (4.2.66) and (4.2.67) are given by: YE,O = luYm

ZE.O

= l z ~ m

(4.2.68)

where ym and zm are the maximum possible values of y and x inside magnets a and b. Thus ym and zm are the semi-apertures in the y and z directions: ym = A , / 2 , zm = A , / 2 , where A , and A , are the corresponding apertures of the quadrupoles, which are of the order of the distance A between opposite pole faces. In Eq. (4.2.68), and l, are quantities pertaining to the exit of b, which are similar t o the ratios c, and e, referring to the entrance of a. The expression for can be simply obtained from Eq. (4.2.47) for e, by replacing S a by Tb, p a by P b , K. by Kb, and a by p:

r, r,

In a similar manner, the expression for 3; is obtained from Eq. (4.2.46) for e, with the following result:

+

lu= Tb/(Tb2

Kb2)'I2.

(4.2.69a)

Upon inserting Eqs. (4.2.68)-(4.2.69a) into (4.2.66) and (4.2.67), one finds : (4.2.70)

For the example considered above (Table I), one obtains A , = A . = 30.5 cm, this gives:

r. = 0.499,

{, = 0.987. With

w, = 46.2(Ap/p0)cm

w, = 100.l(Ap/po)cm.

Thus for Ap/po = 0.02, the widths at the image lo (at Tb C 8.9 m) are: w, = 0.92 cm, w, = 2.00 cm.

4.2.

BEAM BENDING AND FOCUSING SYSTEMS

719

The widths w, and w, obtained above must be added to the image widths produced by the finite dimensions of the source (target) a s a result of the magnification of the system ( x h and xv). I n addition, if a bending magnet for momentum-analysis is located in front of the quadrupoles a and b, there will be an additional term in w,due to the momentum dispersion of the bending magnet. This effect will be discussed below (see Section 4.2.4).

4.2.3. The M a t r i x Method of Calculation for Strong Focusing Magnet Systems

I n connection with the design of appropriate strong focusing systems for various experimental arrangements, it is often convenient to use the matrix method of calculation rather than the method of the lens equations which has been described above. These two methods are, of course, mathematically equivalent, an d it is only a question of convenience of the calculation that determines which method is best suited to a particular problem. The matrix method was first proposed by Courant and Snyder.6 As a n illustration of the matrix method, we shall first consider a single strong focusing (quadrupole) magnet, which acts as a converging lens in the horizontal (y) direction. We consider the focusing action in the y direction. For any particular trajectory through the magnet, we denote by yo and yo' the values of y and dy/dx, respectively, a t the effective entrance boundary of the magnet. The corresponding values of y and d y / d x a t the exit boundary will be called y1 and yl', respectively. Courant and Snyder6 have shown that y1 and yl' are related t o yo and yo' by a 2 X 2 matrix, to be called M , such th at (4.2.72)

where the matrix elements Mi, depend only on the properties of the magnet (effective length I and field gradient parameter K ) . The matrix equation (4.2.72) represents the following linear relations: yi

=

Milyo

y11 = Mziyo

+ Mnyo' + M22yo'.

(4.2.73) (4.2.74)

4.2.3.1. Matrix for Focusing Lens. It can be easily shown that the matrix M for the case of a converging (focusing) lens is given b y =

where a

=

I/K.

(

cos a sin a

- K-l

K

sin a cos LY

(4.2.75)

720

4.

BEAM TRANSPORT SYSTEMS

In order to derive Eq. (4.2.75), we note that the displacement y inside the magnet is given by (4.2.76) y = Y COS[(X/K) 61

+

where Y and 6 are constants, and we assume that x = 0 at the entrance of the magnet. We consider a trajectory for which the slope y' ( = dy/dx) a t the entrance (x = 0) is zero. This gives 6 = 0, Y = yo. The values of y and y' a t the exit of the magnet are given by y1 =

yl' =

Y cos = M l l Y - (Y/rc)sin LY = M z l Y .

(4.2.77) (4.2.78)

Equations (4.2.77) and (4.2.78) yield: iM11 = COB a

Mzl

= -K-~

sin a.

(4.2.79)

In order to determine the other two matrix elements, M12and M z 2 ,we consider a trajectory for which yo = 0 (but yo' # 0). I n this case, 6 = 90" in Eq. (4.2.76), and the values of, YO' yl,and yl' are related as follows to Y , K, and a, yo' = -Y/K (4.2.80) y1 = Mlzyo' = - Y sin a (4.2.81) y11 = Mzzyo' = - (Y/K)COS a. (4.2.82) Upon solving Eqs. (4.2.80)-(4.2.82) for Mlz and MI2 M22

M22,

sin a = cos a.

=K

one obtains (4.2.83) (4.2.84)

It may be noted that the determinant IM( of the matrix M is unity: IMJ=

MllM22

- M12M21= 1

(4.2.85)

as can be easily verified from Eq. (4.2.75). This is a general property of the matrix M for any type of strong focusing system, and it can be used in numerical calculations as a check on the accuracy of the matrix elements M,. I n the vertical ( z ) direction, the focusing (or defocusing) action of a magnet system can also be represented by a matrix, which will be denoted (4.2.86) where 20 and are the values of z and dz/dx a t the entrance of the field region, and zl, 211 are the corresponding values a t the exit of the field region.

4.2.

BEAM BENDING AND FOCUSING SYSTEMS

72 1

4.2.3.2. Matrix for Defocusing lens. In the example considered above of a single quadrupole, the magnet action is defocusing in the vertical direction, and one can easily show th at the matrix N is then given by =

cosh a sinh a

(~-1

sinh a cosh a

K

(4.2.87)

The expressions for the mat,rix elements Nij in Eq. (4.2.87) can be determined in the same manner as the Mi, [Eqs. (4.2.76)-(4.2.84)] by considering two particular trajectories through the magnet, which are of the general form Z = cosh[(z/K) E] (4.2.88)

+

where the constants Z and f depend on the initial conditions, i.e., on the values of zo and z{ a t the entrance of the magnet. From Eq. (4.2.87) it is easily verified that the determinant IN1 = 1, which is a n example of the general property discussed above. The matrices M and N of Eqs. (4.2.75) and (4.2.87) transform initial values at the entrance of the magnet to final values pertaining to the exit boundary of the magnet. I n addition, one may consider a field-free region (drift space) of length zo following the magnet (see Fig. 8). We wish to find the matrix M for the horizontal displacements which transforms initial values at the entrance of the magnet to final values pertaining to the exit of the field-free region past the magnet. 4.2.3.3. M a t r i x for Field-Free Region. For this purpose, we shall use the matrix Mo which describes the action of the field-free region alone. M Ois given by:

Mo

=

(A To)-

(4.2.89)

M o relates the final values y2 and yz' at the exit of the field-free region to the initial values y1 and yl' at the entrance of this region. The matrix elements ( M & of M O can be easily derived from the equations [see Eqs. (4.2.73) and (4.2.74)l: Yz Y2'

+ xoy;

= y1 = y1I.

I n order to obtain the matrix (see Fig. 8) we note th at

A?i which transforms

(4.2.90) (4.2.90~~) yo, yo' into yz, yz'

Hence M is obtained by straightforward matrix multiplication of Mo and

722

4.

BEAM TRANSPORT SYSTEMS

M [as given by Eq. (4.2.75)]. One finds i l Z = MOM

=

(cos

a

- K - ~ x O sin a

-K-l

K

sin a

+

).

sin a 20 cos a cos a

(4.2.92)

= 1. It can be easily verified that the determinant As an illustration of the preceding discussion, Fig. 8 shows the y trajectory of a particle traversing a focusing section (of length I ) followed by a field-free region (of length 2 0 ) . The value of K is such that Z / K = s/2, FOCUS1N G SECTION

-MATRIX -MATRIX

FIELD -FREE REGION

M -MATRIX MOM= -M

Mo-

FIG.8. Schematic diagram showing horizontal (y) trajectory in a focusing section followed by a field-free region. The horizontal displacements at the three boundaries arc denoted by yo, y,, and ~ 2 The . angles qo, 71,and 72, shown in the figure, are given by: ~i= tan-’ yi’ (i = 0, 1, 2), where yo’, yl’, and y ~ ’arc the values of dy/dx at the boundaries.

and the phase angle 6 [Eq. (4.2.76)] is -30°, so that the trajectory inside the magnet is given by: y = Y COS[(?~X/~Z)

- (a/6)].

(4.2.93)

The horizontal displacements yo, yl, y2 at the three boundaries are shown in Fig. 8, together with the angles V O , ql, q2, which are given by: q; = tan-’ g i (i = 0, 1 ,2 ), where yo’, y i , yz’ are the values of the derivative cEy/dx at the boundaries. In the present example, the lens which precedes the field-free region has a defocusing action in the vertical (2) direction [matrix N , Eq. (4.2.87)]. In this case, the combined matrix which transforms the initial values zo and z< into the final values of E and d z l d x , to be denoted

m

4.2.

by

z2

iV

and

=

221,

MoN

723

BEAM BENDING AND FOCUSING SYSTEMS

is given by the matrix product of N and Ma:

=

cosh a

+

K-*

K-IXO sinh a sinh a

K

+

sinh a xo cosh a cosh a

).

(4.2.94)

We note that if the horizontal action is defocusing (instead of focusing as was assumed above), the matrix M for the horizontal motion will be given by Eq. (4.2.87) (involving the hyperbolic functions) instead of Eq. (4.2.75). Conversely, the vertical action will then be focusing, so that the matrix N for the vertical motion will be given by Eq. (4.2.75) instead of (4.2.87). I n other words, the roles of M and N are just interchanged in this case. 4.2.3.4. Matrices for Two-Magnet System. We will now obtain the transformation matrix for the system of two magnets ( a and b) separated by a field-free region, which has been discussed in Section 4.2.2. The matrix M u which relates the values of y and dy/dx a t the exit of magnet b to y and dy/dx a t the entrance of magnet a, is obtained by matrix multiplication, as follows :

Mu

=

Mu,bMdMy,a

(4.2.95)

where My,aand M y , bare the matrices M u pertaining to magnets a and b, respectively; Md is the matrix for a field-free region of length d ; thus Md = M Oof Eq. (4.2.89), with xo = d . An equation similar to (4.2.95) gives the matrix M , for the vertical motion in the two-magnet system. I n the following, we will denote the matrices M , and M , by MI and Mz, where M I pertains to the direction 1 for which magnet a is focusing, and M2 refers to direction 2, for which magnet a is defocusing. I n view of Eqs. (4.2.87) and (4.2.92), one finds = =

Ml,bM&l,a

( coshsinhp K;'

sinh 0) (cos a - (d/K,)sin -1 cosh -K, sin a

Kb

ff

K,

+

sin a d cos a cos a >. (4.2.96)

The first matrix on the right-hand side of Eq. (4.2.96) represents MI,* [Eq. (4.2.87)];the second matrix is Md2Ml,,,, as obtained from Eq. (4.2.92). By straightforward matrix multiplication, one obtains the following expressions for the matrix elements of M1: (4.2.97) cash @[COS CY - (d/K&&l a] - (Kb/K,)Sinh p Sin ff (MI)IZ= cash @ ( K a Sin a d COS a) Kb sinh p cos (4.2.98) (4.2.99) ( M 1 ) n = K:' sinh ~ [ C O Sa - (d/u,)sin a] - K ; ~ cosh p sin a (M1)22 = K;' sinh @(Ka sin a d cos a) cosh /3 cos a. (4.2.100) (M1)11 =

+

+

+

+

724

4.

BEAM TRANSPORT SYSTEMS

The matrix M Zpertaining to the direction 2 is given by

M Z= M ~ ~ b M d M 2 . a =

(-KC'cossin/3 /3

@) (cosh

sin cos @

Kb

a

4- (d/K,)sinh

K , ~

sinh a

a

K~

sinh a 4- d cosh a cosh a (4.2.101)

where the first matrix on the right-hand side is M2,b, as obtained from Eq. (4.2.75),and the second matrix represents M d M 2 , a[see Eq. (4.2.94)]. The resulting matrix elements of M z are as follows:

+

+ +

(MZ)II COS @[cash (d/Ka)sinh a] (Kb/Ka)Sin @ sinh a ( M ~ ) I= z cos @ ( K ~sinh a d cosh a ) Kb sin @ cosh a (Mz)z~ = -K:' sin @[cosha (d/K,)sinh a] K,' cos @ sinh a ( M Z ) Z= Z -K;' sin @ ( K ~sinh a d cosh a) cos @ cosh a.

+

+

+

+ +

(4.2.102) (4.2.103) (4.2.104) (4.2.105)

4.2.3.5. Matrices for Three-Magnet System. In a similar manner, one can obtain the matrices for a three-magnet system (triplet). We denote the quadrupoles by a, b, and c. The distance between magnets a and b will be called d, as before, and the distance between b and c will be denoted by g. The matrix Mi for the transformation from the entrance of a to the exit of c is given by: Mi =

Mi,CM,Mi,bMdMi,,

(i = 1, 2)

(4.2.106)

where M i , a l Md, and M i . b have their previous meanings, M , is the matrix for a field-free region of length g, and Mi,, is the matrix pertaining to magnet c. For direction 1, it will be assumed that magnets a and c are focusing, while magnet b is defocusing. Equation (4.2.106) can then be written as follows : M1 = M I , c M , ( M l , b M d M l , a ) . (4.2.107)

The matrix elements of (Ml,CMdMl,a)can be obtained from Eqs. (4.2.97)(4.2.100);M , is given by Eq. (4.2.89),and M1,,is given by Eq. (4.2.75), in which K must be replaced by K , and a by l C / K C . Similarly, for direction 2, one obtains:

M Z= M Z , C M ~ ( M Z , & ~ M P , ~ )

(4.2.108)

where ( M Z , b & f d M Z , a ) is given by Eqs. (4.2.102)-(4.2.105), and M z , , can be obtained from Eq. (4.2.87). A symmetric triplet is frequently used to obtain double focusing. In this configuration, magnets a and c are identical and have the same values of the field gradient. Moreover, the distances d and g are equal. If the

4.2.

BEAM BENDING A N D FOCUSING SYSTEMS

725

gradients are so chosen that the object and image distances are equal (S = T), then the two magnifications are equal: x1 = x2 = 1, which is desirable in some experimental arrangements. In order to make a comparison of a typical three-magnet system with a two-magnet system having essentially the same field gradient and the same length of magnetic field, we consider the following two configurations which can be used to focus low-energy particles ( p 5 300 Mev/c) from a target: (1) System A is a symmetrical triplet, with magnet lengths 1,

=

1, = 20.3 cm

b#!

=

40.6 cm

separations d = g = 10.2 cm, so th at the total length of the system is: L, = 101.6 cm. It is found that for values of the field gradients such that K~ = K~ = 33.3 cm, Kb = 36.4 cm, one obtains double focusing for object and image distances S = T = 146.3 cm. The magnification x is unity for both the horizontal and vertical directions. Assuming that magnets a and c are vertically focusing and b is horizontally focusing, one finds tg = 0.524, c, = 0.975, so that t = 0.511. The intensity a t the image is determined by the quantity

e/S2 = 0.511/(146.3)2 = 2.39 X lo-'. (2) For comparison, we consider system B consisting of two quadrupoles with lengths I, = &, = 40.6 cm, separation d = 20.4 cm, and K~ = Kb = 34.8 cm. Thus system B has the same gradient as the average gradient of system A , and moreover, the combined length of the magnets is the same (81.2 cm), as well as the total length of the system (La = 101.6 em). From Eq. (4.2.42), one finds that the double focusing condition is satisfied for S = T = 49.3 em, which is considerably smaller than the value S = T = 146.3 cm for system A . The magnifications are: l x h l = 0.239 and lxvl = 4.18. One finds tha t eII = 0.195, c2 = 0.817, so that e = 0.159. Thus t is considerably smaller than for the triplet ( t = 0.511), but because of the reduced object distance,

t/S2 = 0.159/(49.3)'

=

6.54 X lopb

is appreciably larger than for the three-magnet system. Thus the gain in intensity for system B as compared to A is a factor GO= 6.54/2.39 = 2.74. The gain Go will be further increased if the particles have a short lifetime, e.g., for K mesons. The total object-to-image distance L is:

L

=

2(146.3) 4- 101.6

=

394.2 cm

726

4.

B E A M TRANSPORT SYSTEMS

+

for the triplet system, as compared to:L = 2(49.3) 101.6 = 200.2 cm for system B. For K mesons of momentum p~ = 300 Mev/c, the fraction F of particles which do not decay in traversing a distance L is: F = e-v, with v = L r n ~ / ( ~ ~where p ~ ) ,the lifetime TK = 1.2 X sec., mK = K meson mass = 494 Mev/c2. For system A , F = 0.165, whereas for system B, F = 0.400. Thus the total intensity gain at the image, GK for K mesons is: GK = 2.74 X (0.400/0.165) = 6.64.

It can be concluded from these results that for secondary beams of short-lived particles of low momenta, it is more advantageous to use a two-magnet system as compared to a three-magnet system with the same average gradient and length, since one gains both from the larger solid angle Afl subtended a t the target and from the smaller object-toimage distance L , which reduces the decay probability 1 - F. The fact that the horizontal magnification I ~ h lis appreciably less than 1 for the two-magnet system may also be useful in order to obtain good energy resolution using a bending magnet, as was discussed above. On the other hand, for focusing external beams over large distances, where the beam incident on the system is not very divergent, it may be advantageous to use a triplet system, as was done in the external proton beams at the Cosmotron and for the antiproton beams from the Berkeley Bevatron. I n this case, the fact that we can make X h = xv = 1 is convenient, since it enables us to preserve the size and shape of the source, so that the image can then serve as an object for a subsequent similar three-magnet focusing system. 4.2.3.6. Focal length and Position of Focal Planes of a Magnet System. From the matrices M and N for a general magnet system we shall now obtain the focal length jof the magnetic system and the position of the focal planes for both the horizontal and vertical directions. We consider the matrix M of Eq. (4.2.72). The focal length f is defined as follows.' Consider a trajectory which enters the strong focusing system at a unit distance from the central axis, yo = 1, and with zero slope, yo' = 0 (i.e., the effective object is at infinity). Then the focal length f is that distance along the x axis over which the displacement y of the outcoming trajectory decreases by one unit. With the initial conditions yo = 1, y4 = 0, the values at the exit of the system yl and y; are given by : yi = Mi1 y l l = Mzi. (4.2.109) The displacement y after exit from the system is given by: 1/ = y 1 +

y1)2 = Mi1

+ Mzls

(4.2.110)

4.2.

BEAM BENDING A N D FOCUSING SYSTEMS

727

where x is measured from the exit boundary of the system. The focal length f is given by Ax, where Ax is obtained from the following equation : Ay = - 1 = M ~ ~ A x

(4.2.111)

whence

f

=

AX

=

-1/Mzl.

(4.2.112)

We note that this derivation is independent of whether the action of the magnetic system is focusing or defocusing. For a defocusing system, Ax, and hencef, is negative, i.e., the displacement y increases with increasing x [the outgoing ray (trajectory) is diverging]. Equation (4.2.109) gives the horizontal focal length, which will be denoted by f h . The vertical focal length fu is obtained in a similar manner from the matrix N, and is given by fv

=

-1/Nzi.

(4.2.113)

In the derivation of Eq. (4.2.112),we have not made use of any specific property of the strong focusing system. Hence Eqs. (4.2.112)and (4.2.113) (as well as the following equations for the location of the focal planes) are applicable t o a general strong focusing system which may consist, for instance, of several quadrupole magnets separated by field-free regions. For the case of a single magnet considered above, Eqs. (4.2.112) and (4.2.113) reduce t o the usual expressions [see Eqs. (4.2.12) and (4.2.23)]: f h = K c8C a! (4.2.114) f v = --K csch a. (4.2.115) I n order to obtain the distance d I , h of the horizontal image focal plane FIVhfrom the exit boundary of the system, we use again the initial conditions y o = 1, y< = 0 for the horizontal motion. The distance d 1 . h is then determined by the position x where the y displacement of the outcoming ray is zero, Thus yi whence

+ 91'2 d1.h

=

=

Mi1

x

=

+ M 2 1 ~= 0

-Mii/Mzi.

(4.2.116) (4.2.117)

I n Eqs. (4.2.116) and (4.2.117), 2 is measured from the exit boundary of the strong focusing system. A positive d z , h means that the horizontal image focal plane F Z , h lies to the right of the exit boundary. For the distance dr,,, of the vertical image focal plane FI,,, from the exit boundary, one obtains in a similar manner: dr." = -Nii/Nzi.

(4.2.118)

728

4.

BEAM TRANSPORT SYSTEMS

For the example discussed above, where the matrices M and N are given by Eqs. (4.2.75) and (4.2.87), one finds dr,h = dr,, =

K

cot a coth a.

-K

(4.2.119) (4.2.120)

The distance dr,h for the focusing (horizontal) direction corresponds to A, (Eq. 4.2.11), whereas the distance dr,, for the defocusing (vertical) direction corresponds to - p (Eq. 4.2.22). We note that dr.o is negative, i.e., the vertical image focal plane lies to the left of the exit face (see Fig. 3). In order to obtain the location of the horizontal object focal plane FO,h, we consider the case where the exit boundary conditions are: y1 = 1, yl' = 0 (image at infinity). The resulting equations for yo and yo' as obtained from Eq. (4.2.72) are given by M11yo Mziyo

+ MlZYOI = 1 + Mzzyo' = 0.

(4.2.121) (4.2.122)

The object focal plane Fo,h is determined by the distance x (as measured from the entrance of the magnet system) for which y = 0. The corresponding distance do,h is given by do,h

=

x

= yo/yo'.

(4.2.123)

In view of Eq. (4.2.122), we obtain do,h = -Mzz/Mzi

(4.2.124)

where a positive sign of do,h means that the horizontal object focal plane

FO,his in front (to the left) of the entrance boundary. In the same manner, the distance do,%from the entrance of the system to the vertical object focal plane Fo,, is given by do.,

=

-Nzz/Nzi.

(4.2.125)

For the case of a single magnet considered in Section 4.2.2, one finds (4.2.126) (4.2.127)

In this case, we have: do,h = dr.h, and do,, = dr,,, as was already discussed above (see Figs. 2 and 3). Referring to the matrices M and N [Eqs. (4.2.75) and (4.2.87)], we see that this result arises from the fact that Mll = M z 2and N11 = N z z for these matrices. However, for more complicated strong focusing systems, these matrix elements are in general not equal, and as a result, d0.h # dr,h and do,, # dr,v. In particular, for the

4.2.

BEAM BENDING AND FOCUSING SYSTEMS

729

strong focusing system consisting of two quadrupole magnets (with gradients of opposite sign) separated by a field-free region [see Eqs. (4.2.27), (4.2.32)],the distances do,i and d1.i to the focal planes are given by : do,; = - uia (4.2.128) dr,i = - Uib. (4.2.129) In general, do,l and df,l are different since U1, # Ulbl and similarly do.2 # d1,2. Correspondingly, for the matrices M and N for the complete two-magnet system, we have M l l # Mz2 and N11 # N Z Z . We will now obtain the expression for the lateral horizontal magnification X h in terms of the matrix elements Mij. For a ray which starts with a finite slope and zero displacement a t the object, the initial conditions yo and ya' are related as follows to the object distance 8 h (distance from object to entrance boundary of system) : ?/O/yO' =

From the equations for yl and

(4.2.130)

8h.

(Eq. 4.2.72), one finds

~ 1 '

where Th is the distance from the exit boundary of the system to the horizontal image. Now we consider a ray starting from the object with a finite yo and with zero slope (go' = 0). Thus we have (4.2.132) (4.2.133)

y1 = Mnyo Yl'

=

M21yo

a t the exit of the magnet system. The horizontal displacement D h of the trajectory a t the position of the image is given by

From Eq. (4.2.134),we find for the IateraI magnification Xh

=

DdYo

where we have made use of IMI as follows:

=

=

1/(Mzi&

+ M2z)

Xh:

(4.2.135)

1. Equation (4.2.135)can be rewritten

(4.2.136)

730

4.

BEAM TRANSPORT SYSTEMS

where X 0 , h is the distance from the object to the object focal plane FoJ,. The last expression in Eq. (4.2.136) is identical with Eq. (4.2.44) given above. For some applications, it may be useful to have an expression for X h in terms of the image distance Th (rather than Sh). By means of Eqs. (4.2.131) and (4.2.135), one obtains: ~h

= Afii

(4.2.136a)

-k MziTh.

For the vertical direction, the lateral magnification xt, is given by xu =

l/(NZlSr,

+ Nz2).

(4.2.137)

By means of Eq. (4.2.131),we can obtain the expression for the horizontal longitudinal magnification A h , which is defined as aTh/dSh, where the partial derivative is evaluated for constant momentum p of the particle. Thus one finds:

(Mz1Tn

Mil)'

(4.2.138)

where we have used the property that /MI = 1. A similar expression holds for the vertical longitudinal magnification A, = a T , / a s , :

Equations (4.2.138) and (4.2.139) can be used, for example, to determine the extent of the image along the direction of the outcoming beam when the target which constitutes the object has a large longitudinal dimension (e.g., for a liquid hydrogen target whose length may be of the order of a meter). EXAMPLE : As an example of the use of the matrix method, we give the matrices for the strong focusing system considered in Table I. From Eqs. (4.2.97)(4.2.100),one obtains for the matrix Icfl for the direction 1 : -0.0315

M I = M & k f J l l . a= -0.4160

2.526 1.630

where it is assumed that all distances are expressed in meters. From the matrix elements of M 1 ,one obtains by means of Eqs. (4.2.112), (4.2.117), (4.2.125), (4.2.131), and (4.2.135) assuming that the object distance Sla = 4.57 m: f l = 2.404 m, do,l = 3.918 m, dr,l = -0.076 m,

4.2.

BEAM BENDING AND FOCUSING SYSTEMS

73 1

Tlb = 8.76 m, and x1 = -3.68. As expected, these results are in agreement with the corresponding values obtained from the lens equations (Table I), within the small rounding errors of the numerical calculations. We note that do.1 = - Ula, and dI,l = - u l b . Similarly, upon using Eqs. (4.2.102)-(4.2.105)~one obtains for the matrix M n pertaining to direction 2 :

From the matrix elements of M z , one finds: fz = 3.421 m, do,2 = 0.411 m, dr,z = 6.244 m, Tzb= 9.06 m, and xz = -0.822. These results are in good agreement with the corresponding values of Table I.

4.2.4. The Focusing Equations for Deflecting Wedge-Shaped Magnets with Finite Field Index n In the preceding discussion, we have considered only quadrupole focusing magnets. For these magnets, the field H is zero along the central axis (y = z = 0). We shall now discuss the focusing action of deflecting magnets with nonuniform fields, which can be used simultaneously for double focusing and for a determination of the momentum of the particles. For such magnets, it is customary to consider the field index n,first introduced by Kerst and Serber,lB which is defined by: n = -(R/H)(dH/dr), where R is the radius of curvature of the normal trajectory (in the central region of the particle beam), H is the field along this trajectory, and dH/dr is the radial derivative of H. A schematic view of a wedge-shaped magnet is shown in Fig. 9. The central trajectory traverses the path MM’VQ; the path NN’V’UU’ represents a second trajectory, which will be discussed below. The center of curvature of the central trajectory is P ; the distance ( P M ) = ( P V ) = (PQ) is the radius of curvature R of the particles. The entrance and exit angles at the magnet boundaries are denoted by s and t. These angles are to be taken as positive if the path of the particles at the entrance or exit of the magnet is on the same side of the normal to the boundary as the center of curvature of the central trajectory. Thus in Fig. 9, s is positive and t is negative. The focusing depends critically on the entrance and exit angles s and t. For the case of a uniform field (n = 0 ) ,the equations for the focusing action have been derived by Camac” and C r o s ~ . ~These ~ * ’ ~equations were extended by D. W. Kerst and R. Serber, Phys. Rev. 60, 53 (1941). M. Camac, Rev. Sci. Instr. 22, 197 (1951). W. G. Cross, Rev. Sci. Znstr. 22, 717 (1951). l9 References to earlier works on the focusing action of wedge-shaped magnets are given in the review article of K. T. Bainbridge, in “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 1, pp. 566-614. Wiley, New York, 1953. l6

l7

732

4.

BEAM TRANSPORT SYSTEMS

Sternheimerato the case of a wedge-shaped magnet with nonuniform field (n # 0). As will be discussed below, the resulting focusing equations can be written in the form of the lens equations (4.2.16) and (4.2.24), provided that &, Ti,K , A, and p are appropriately defined.4 I n practice, the field is made nonuniform inside the magnet by tilting the upper and lower pole faces with respect to each other, so that the gap height g increases or decreases along the width of the magnet (from A to B in

P

FIG. 9. View of a wedge-shaped magnet showing the central particle trajectory MM'VQ and a second particle trajectory NN'V'UU'.

Fig. 9). Positive n (radial falloff of the field) corresponds to the case of increasing g, as one goes radially outward with respect to the center of curvature P of the central trajectory. Magnetic spectrometers with nonuniform fields to provide double focusing have been extensively used in mass spectroscopy experiments.20-26 4.2.4.1. Vertical Focusing by the Fringing Field of the Magnet. For the case of a single magnet with n = 0, the only vertical forces acting 80 N. Svartholm and K. Siegbahn, Arkiv Mat. Astron. Fysik SSA, No. 21 (1946); N. Svartholm, ibid. SSA, No. 24 (1946). F. B. Shull and D. N. Dennison, Phys. Rev. 71, 681 (1947); 72, 256 (1947). 1) D. L. Judd, Rev. Sci. Znstr. 21, 213 (1950). 9s E. S. Rosenblum, Rev. Sei. Znstr. 21, 586 (1950). a 4 N. Svartholm, Arkiv Fysik 2,115 (1951); E. Arbman and N. Svartholm, ibid. 10, 1

(1955). 26 N. E. Alseevsky, G . P. Prudkovsky, G . I. Kusourov, and S. I. Filimnov, Doklady Akad. Nauk S.S.S.R. 100, 229 (1955). Z6 A. V. Dubroniv and G. V. Balabina, Doklady Akad. Nuuk S.S.S.R. 102, 719 (1955).

4.2.

BEAM BENDING AND FOCUSING SYSTEMS

733

on the particles occur at the entrance and the exit of the magnet. We will first consider the situation at the magnet entrance. Referring to Fig. 9, we introduce a coordinate system xyz with origin a t M , the x axis perpendicular to the entrance face ( A B ) and pointing towards the interior region of the magnet, the y axis along the magnet entrance ( M B ) , and the z axis pointing out of the paper. It will be assumed that at the magnet entrance, the vertical component of the field varies from 0 to its value H , inside the magnet over a distance Ax along the x axis. From Maxwell’s equations, we obtain : (4.2.140)

Equation (4.2.140)shows that there is a field component H , given by:

H,

=

(4.2.141)

(H,/Ax)z

which extends over the region of the fringing field (of width Ax). The resulting vertical force F:’ acting on the particle is given by:

FP) = - e v ~ ’ H , / c = -e(v/c)sin s ( H , / A x ) z

(4.2.142)

where v:) = v sin s is the y component of the velocity v. The force FF) acts during a time At = A x / ( v cos s), so that the resulting change Ap:’ of the z component of the momentum is given by: Apr’

=

F:’ At

=

- ( e H , tan s / c ) z

=

- ( p tan s / R ) z .

(4.2.143)

I n a similar manner, one finds that at the exit of the magnet, the z component of the momentum changes by the following amount: Apr’

=

- ( e H , tan t/c)z

=

- ( p tan t/R)z.

(4.2.144)

Aside from the terms (4.2.143) and (4.2.144) which describe the action of the fringing fields, there exists also a vertical force F , inside the magnet, if the field H is nonuniform (n # 0). The force F, arises from the radial component of the field H , inside the magnet, which is determined by Maxwell’s equations: (4.2.145)

One thus obtains :

H,

(4.2.146)

= -(nH,/R)z.

The force F, is then given by:

F,

=

evH,/c

=

-[(evnH,)/cR]z

=

-(pvn/R2)z.

(4.2.147)

734

4.

BEAM TRANSPORT SYSTEMS

It is seen that the terms A p r ) , Ap:", and F , will act as restoring forces provided that s, 2, and n, respectively, are positive. Thus in Fig. 9, the magnet action is vertically focusing a t the magnet entrance (s > 0) and vertically defocusing a t the exit ( t < 0). 4.2.4.2. Horizontal Focusing by a Wedge-Shaped Magnet. For the horizontal focusing, the situation is somewhat different. There is no horizontal focusing or defocusing action at the entrance or the exit of the magnet. Nevertheless, the sign and magnitude of the angles s and t determine the focusing (or defocusing) action of the magnet in the following manner. Referring to Fig. 9, we see that for a second trajectory NN'V'UU' (originating a t the object a t a finite S), which arrives above the central trajectory M M ' V Q a t the entrance of the magnet, the deflection angle Q' inside the magnet should be larger than the angle (p for the central trajectory in order that the particle shall be focused at the image. If re' is the distance between the second trajectory and the central trajectory at the magnet entrance, the second trajectory will traverse an additional path length: Al, = -re' tan s in the magnetic field. Similarly, at the exit of the magnet, if rt' is the distance between the two trajectories, the second trajectory will traverse an additional path length : Alt

=

-rt' tan t.

In Fig. 9, r,' is given by the distance ( M N ) .The distance IAl,] is given by: ( M M ' ) = ( N N ' ) = r,' tan s. Similarly, a t the exit of the magnet, we have rt' = ( U Q ) and AZt = ( U U ' ) = rt' tan 111. It may be noted that for the first-order theory here presented to be applicable, it is necessary that r'/R be (T1- A,) whence Ti

=

= K,,~

+ Xu2

(six,+ Ku')/(#I - A").

(4.2.154) (4.2.155)

From the value of TI, one can obtain T, by means of Eq. (4.2.151). One finds T, = T1/[l (TI/R)tan t ] . (4.2.156)

+

From Eqs. (4.2.150)-(4.2.156) one can also obtain the following alternative expression3 for T,: T,

=

+ + tan t]-'

R[n1/2tan(n1/2p 5 )

(4.2.157)

where the angle 4 is defined by: tan 6 = -n-'/Z[(R/S,)

- tan s].

(4.2.157a)

4.2.4.4. Horizontal Focusing for n < 1. A similar situation exists for the horizontal focusing. I n this case, the reduced distances S2and T2 are defined as follows:4 (Sh/R)tan s] (4.2.158) Sz = S h / [ l Tz Th/[l (T&/R)tant ] (4.2.159)

+

+

where s h and Th are the horizontal object and image distances, respectively. The lens equation is given by

(Sz- b)(Tz - A h ) where

Ah

and

Kh

=

Kh2

+

Ah2

(4.2.160)

are defined by: Ah 3 Kh

R(1 - n)-'" cot[(l - n)112$0] R(1 - n)-'".

(4.2.161) (4.2.162)

From Eq. (4.2.160), we obtain

+ Kh2)/(SZ -

Ah).

(4.2.163)

Th = Tz/[l - (T2/R)tan t ] .

(4.2.164)

Tz

= (SZAh

Finally, from Eq. (4.2.159),

From Eqs. (4.2.158)-(4.2.164), one can also obtain the following alternative expression for Th: Th = R ( (1

- n)l/Ztan[(l

+ 31 - tan t]-1

- n)1/2p

(4.2.165)

4.2.

BEAM BENDING AND FOCUSING SYSTEMS

737

where q is defined by: tan q

= - ( 1 - n)-1'2[(R/Sh) + tan

s].

(4.2.166)

EXAMPLE : As a n example of the preceding results, we discuss the case of a single wedge-shaped magnet for which n = 0.4, cp = 40°, s = 30°, and t = -24.8'. For an object distance S ( = S, = sh) = 3.OR, this magnet gives double focusing a t a n image distance T = 4.9612. Considering first the focusing action in the vertical direction, one finds from Eqs. (4.2.150)-(4.2.156) : S1 = -4.098R, Xu = 3.345R1

K , = 1.581R TI = 1.506R

and

T,

=

4.96R.

For the horizontal direction, Eqs. (4.2.158)-(4.2.164) yield the following results: 82 = 1.098R, Kh = 1.291R1 Ah = 2.150R1 TP = -3.829R1 and Th = 4.96R. For a magnet with given n and cp and specified values of the object distance S ( = S, = 8,) and the entrance angle s, there is (at most) one angle t which will give double focusing; the image distance T (= T,= T h ) is completely determined by the values of n, cp, S, and s. In order to obtain the values of t and T for double focusing after T, and T 2have been determined from Eqs. (4.2.155) and (4.2.163),we make use of the following relation [cf. Eqs. (4.2.156) and (4.2.164)]:

From Eq. (4.2.167), one obtains tan t T

= =

R(T1 - Tz)/(2TlT2) 2TiT2/(Ti 7'2).

+

(4.2.168) (4.2.169)

Figures 4-6 of reference 3 show plots of T and t (for double focusing) as a function of the entrance angle s for three values of the angle of deflection: cp = 20°, 40°, and 90'. The curves were calculated for S = l R , 2R, 3R, and w , and for the following n values: n = 0 and 0.4 for all three p values, and, in addition, n = -0.4 for lp = 90". Similar calculations of the double focusing for cp = 180' with n = 0.8 and 0.9 have been carried out by Karmohapatro.21 S. B. Karmohepatro, Indian J . Phys. 32, 26 (1958).

738

4.

BEAM TRANSPORT SYSTEMS

4.2.4.5. Horizontal Defocusing for n > 1. For the case n > 1, the equations for the vertical focusing (Eqs. 4.2.150-4.2.1574 are unchanged. However, in the horizontal direction, there is now defocusing, so that Eqs. (4.2.160)-(4.2.166) must be suitably modified. The reduced distances Sz and T z are still given by Eqs. (4.2.158) and (4.2.159). I n place of Eq. (4.2.160), the lens equation is given by Eq. (4.2.24), in which p and K are replaced by p h and Kh defined as follows: /lh Kh

R(n - I)-'/' coth[(n - 1)'Izp] R(n - 1)-"'.

(4.2.170) (4.2.171)

From Eq. (4.2.24) one obtains the following equation which determines the reduced horizontal image distance Tz:

Tz =

-(Kh2

+ SZPh)/(sZ +

Ph).

(4.2.172)

The actual horizontal image distance T h can now be obtained from T z by means of Eq. (4.2.164). 4.2.4.6. Vertical Defocusing for n < 0. For the case n < 0, Eqs. (4.2.158)-(4.2.166) apply for the horizontal focusing, but Eqs. (4.2.152)(4.2.157a) must be modified, since the magnet action is vertically defocusing in this case. S1 and T 1are still given by Eqs. (4.2.150) and (4.2.151). In analogy to Eq. (4.2.24), the lens equation is now given by (S1

where p,, and

K~

+

+ p.1

= pU2-'.K

(4.2.173)

= Rlnl-'/2 coth(ln11/2p)

(4.2.174) (4.2.175)

pY)(T1

are defined by: pv I(,

~%(n(-'/~.

The reduced vertical image distance

Ti =

-(G'

-k

T1

is obtained from

sipv)/(si

f

pv).

(4.2.176)

The actual image distance T , is then obtained from 2'1 by means of Eq. (4.2.156). As an alternative, T , can be obtained from the following equation^,^ which are appropriate for n < 0, provided that 1 - (S,/R) t a n s < (S ,/R)In11123 T , = R[tan t - lnI1/ztanh(ln11/2p t')]-l (4.2.177)

+

where the angle

[' is defined by: tanh 5'

E

Inl-1/2[(R/S,)- tan s].

(4.2.178)

In comparing Eqs. (4.2.150) and (4.2.151) with Eqs. (4.2.158) and (4.2.159), we note that the sign of the terms involving tan s and tan t is

4.2.

BEAM BENDING AND FOCUSING SYSTEMS

739

negative for vertical focusing, whereas it is positive for horizontal focusing. Aside from this result, there are the usual differences concerning the dependence on n:16the factor n1/2for the vertical focusing is replaced by (1 - n)II2 for the horizontal focusing. Similarly, Eqs. (4.2.165) and (4.2.166) for Ta can be obtained from Eqs. (4.2.157) and (4.2.157a) for T, by merely reversing the signs of tan s and tan t , and replacing n by (1 - n). In reference 4, Sternheimer has obtained the expressions for the momentum dispersion D and the magnifications xu and Xh for a wedgeshaped magnet with arbitrary field index n and arbitrary entrance and exit angles s and 1. For the momentum dispersion D, defined as d y l d p , where d y is the lateral displacement of the image due to a small change of momentum p , one finds: (1) F o r n < 1:

(2) For n

> 1:

+ (Tn/R)tan 21

{[l - cosh(n - 1)1'2p][l -

(Th/R)(?Z - l)%inh(n - 1)112p] (4.2.180)

where Th is the horizontal image distance (as measured from the exit face of the magnet). For the case n > 0, the vertical magnification at the image xv is given by

+

xu = sec & { c o ~ ( n l / ~ pt;ll)[l - (T,/R)tan t] where the angle

+

- n1/z(Tu/R)sin(n1/2p 5,))

(4.2.181)

I,,, is defined by: Ern tan-'(n-'"%an s). (4.2.182) < 0, the appropriate expressions for xu can be found in 3

For the case n Table 111 of reference 4. For n < 1, the horizontal magnification Xh =

+

Xh

is given by

+

sec 6,(cos[(l - n)'/zp 6,][l (Th/R)tan t] - (1 - n)1/2(Th/R)sin[(l- n)'12p

+

6m]}

(4.2.183)

where 6, is defined by: ,6

= - tan-l[(l - n)-'4an s].

(4.2.184)

740

4.

BEAM TRANSPORT SYSTEMS

The expressions for Xh for the case n > 1 are given in Table I11 of reference 4. 4.2.4.7. Focusing Properties of Wedge-Shaped Magnets with Uniform Field. The expressions given above for n # 0 cannot be directly used for the case of a uniform field (n = 0). The image distances T,,o and Th,ofor vertical and horizontal focusing by a wedge-shaped magnet with uniform field are given byl7~l8

+

R sin cp cos s cos t s h cos(p - S)COS t T h s o = (&/R)sin(p - s - t ) - cos(cp - t)cos s' For n

D

=

=

0, the expressions for D,xu, and

(R/p)((l- cos p)[l

= SeC S{COS(p- S ) [ l

+

are as follow^^^^^^:

+ (Th/R)tan t ] + (Th/R)sin

xa = (1 - p tan s)[l - (T,/R)tan Xh

Xh

(4.2.186)

p]

(4.2.187)

- (T,/R)tan s (4.2.188) (Th/R)tan t ] - (Th/R)Sin(p - 8)). (4.2.189) t]

We note that Eqs. (4.2.187) and (4.2.189) for D and Xh can be obtained directly from Eqs. (4.2.179) and (4.2.183),respectively, by setting n = 0. In order to derive Eq. (4.2.188) from Eqs. (4.2.181) and (4.2.182),one must expand lm and xu for small n, and go to the limit n -+ 0. The longitudinal magnification A h and Aw for n = 0 can be obtained by straightforward differentiation of Eqs. (4.2.185) and (4.2.186). One finds: A

-

- cos2

cos2 t/[(Sh/R)sin(cp - s - t ) - cos((p - t)cos

a T h 1 ~ s

=

ash

A , , = - a= T v o

as,

s12

(4.2.190)

-l/((S,/R)[tans

+ (1 - (otans)tant] - (1 - (otant)}2. (4.2.191)

In connection with some applications, it may be useful to evaluate the derivatives dTh,o/dpand dT,,o/dp, which give the momentum dependence of the image distances Th.Oand T,,o for a wedge-shaped magnet with uniform field. These derivatives are analogous to the quantities dT/dp (i = 1, 2) which have been evaluated in Eqs. (4.2.51)and (4.2.62) for the case of a single quadrupole magnet and a system of two quadrupoles. In order to derive the expression for dTh,o/dp,it is convenient to start from Eqs. (4.2.165) and (4.2.166), which give for n = 0: R

= tan((o

q =

R

+ q ) - tan t -= -Dh

- tan-'(tan

s

+ R/Sh).

(4.2.192) (4.2.193)

4.2.

BEAM BENDING AND FOCUSING SYSTEMS

741

Here and in the following, we denote the denominator of Th,O by Dn. Thus

Dh = tan(q

+ 7) - tan t.

(4.2.194)

As shown by Eqs. (4.2.192) and (4.2.193), Th.0 is a function of the five variables: R, s h , s, t , and p. As the momentum p is varied, R , t , and q will vary, while sh and s remain unchanged since they pertain t o the incident beam. Therefore we obtain :

From Eq. (4.2.192), we find: (4.2.196)

Equation (4.2.193) gives : (4.2.197)

whence (4.2.198)

It can be easily shown that: aR/ap = R / p (for n = 0). In connection with the second and third terms on the right hand side of Eq. (4.2.195),we note that

(4.2.200)

The derivative a q / a p can be obtained from Eqs. (86) and (91) of reference 4 by setting n = 0. One finds 1 - = - -[sin

a(a

aP

P

(a

+ (1 - cos (a)tan t].

(4.2.201.)

It is easily shown that for a small change of momentum which results in an increment dq, the corresponding increment dt of the exit angle is equal to dq, so that: &/ap = a q / a p .

742

4. BEAM

TRANSPORT SYSTEMS

Upon inserting Eqs. (4.2.198)-(4.2.201) into Eq. (4.2.195), one obtains the following result for dTA,o/dp:

+ S,R cos2q sec2(cp+ q ) + [sec*(v + 7) - sec2 t ]

DA

X [sin cp

+ (1 - cos cp)tan t ] }

(4.2.202)

where 7 and Dh are defined by Eqs. (4.2.193) and (4.2.194), respectively. In order to derive the expression for dT,,o/dp, one proceeds essentially in the same manner as for dTh,o/dp, starting from Eqs. (4.2.157) and (4.2.157a) for T,. However, in the present case, one must first obtain aT,%/dR,aT,/dcp, and aT,/dt for arbitrary n, and then go to the limit n --f 0. One thus obtains

--dp

Ra2 pDv2 sec2 1

] [sin

+

cp

+ (1 - cos cp)tan t ]

where a and D, are defined by: a

= S,/(Sv tan s - R )

D, = ( a - q)-'

+ tan t.

(4.2.204) (4.2.205)

4.2.4.8. Discussion of a Focusing System with Momentum Analysis. A commonly used focusing arrangement consists of a bending magnet B to momentum-analyze the particles, followed by a pair of focusing quadrupoles Q. and Qb. This system is shown schematically in Fig. 10. For this arrangement, one is interested in the momentum acceptance Ap and the resulting width w of the final image (formed by QaQb). In order to obtain an estimate of the momentum acceptance Ap of the system, we will assume that the limiting momenta, po - (Ap/2) and po (Ap/2) are those for which the central trajectory just grazes the pole faces of the quadrupole magnet (Q. or Qb) which is horizontally focusing ( @ , in Fig. 10). According to the definition of ey [see Eqs. (4.2.46) and (4.2.47)], the available horizontal aperture at the entrance of QG is given by e,A,, where A , is the maximum available aperture, i.e., A , is of the order of the distance between opposite pole faces. We now have: EVA, = D Ap Sal&/dpI Ap (4.2.206)

+

+

where D is the momentum dispersion at the image I B formed by magnet B , dq/dp is the derivative which enters into D [see Eq. (4.2.201)], and S,

4.2.

BEAM BENDING AND FOCUSING SYSTEMS

743

is the effective object distance for quadrupole Q., i.e., the distance from the image I g to the entrance face of Q.. Thus we have: S, = dB - T B , where d B is the distance between the exit face of B and the entrance face of Q., and T B is the distance of the image I B from the exit face of B. The first term on the right hand side of Eq. (4.2.206) gives the width of the beam at la due to the momentum dispersion D . The second term of Eq. (4.2.206) gives the additional broadening of the beam in traversing the distance S, between I g and the entrance of Q..

0

FIG.10. Schematic view of focusing system consisting of a bending magnet B and two quadrupole magnets Qa and &a. The figure shows the central trajectories for two momenta: PO and P O - ( A p / 2 ) .

Upon using Eq. (4.2.187)for D,and Eq. (4.2.201) for dcpldp, we obtain from Eq. (4.2.206): AP { ( l- coa q)(R egA, = -

Pc

+ T s tan t ) + T g sin cp + S.[sin cp + (1 - cos cp)tan t ] )

(4.2.207)

where cp is the angle of deflection, R is the radius of curvature in the bending magnet B , and t is the exit angle from B , as shown in Fig. 10. Upon T B = ds, we obtain from Eq. (4.2.207): using the relation: S,

Q

Po

=

e,A,/i[sin

(a

+ + (1 - cos cp)tan t]dB + ~

( -1 cos

cp)].

(4.2.208)

For the case that Q. is vertically focusing and Qb is horizontally focusing, is given by Eq. (4.2.47), whereas if Q, is horizontally focusing, eg is given by Eq. (4.2.46). It should be noted that the momentum acceptance A p given by Eq. (4.2.208) represents only an estimate. In Fig. 10, the upper trajectory

e,

744

4.

B E A M TRANSPORT SYSTEMS

pertains t o the average momentum pa, while the lower trajectory is tangent to the lateral boundary of Qb, and thus represents the central trajectory for momentum pa - (Ap/2). For this momentum, particles emitted by the target a t a small angle to the central trajectory will be transmitted by QoQb, if the angle is such that the particle path lies above the central trajectory in Fig. 10. On the other hand, if the trajectory lies below the central trajectory of Fig. 10, the particles will not be accepted by QoQb. Thus for momentum p a - (Ap/2), roughly half as many particles as for momentum PO will be accepted by the present focusing arrangement. (The actual fraction will, of course, depend on the other aperture stops and collimators which may be present in the system.) It can also be seen that even if the momentum is somewhat below pa - (Ap/2), a small fraction of the particles will be accepted by the system, namely those for which the (upward) angle at the target is sufficiently large so that they will not hit the lower lateral boundary of Qb in Fig. 10. We note that the preceding discussion does not include the effect of the finite dimensions of the target and detector. A complete discussion of the various factors which enter into the width w of the final image will not be given here. However, we will describe the three effects which arise directly from the momentum spread Ap. (1) As a result of the momentum dispersion D of the bending magnet, there will be a lateral displacement at the image I B formed by B , which is given by ;AYB = D(Ap/2) (4.2.209)

+

for the particles of momentum pa (Ap/2). Here D is given by Eq. (4.2.187),if the bending magnet has uniform field, i.e., n = 0. A ~ repreB sents the width of the effective object for the quadrupole pair Q.Qb. Thus as a result of the horizontal magnification XA,Qof QoQb, there will be a width a t the final image which is given by AlY/ =

IXh.QI

AyB

(4.2.210)

where X h , Q can be obtained from Eq. (4.2.135) or (4.2.136a). The matrix elements Mij which enter into these equations are given by Eqs. (4.2.102)(4.2.105) for the arrangement of Fig. 10, in which the first quadrupole, Q., is vertically focusing, so that the horizontal direction is direction 2 in the notation of Eqs. (4.2.95)-(4.2.105). I n the reverse case, in which Q. is horizontally focusing, the Mi, are given by Eqs. (4.2.97)-(4.2.100). (2) As a result of the momentum spread, the position of t h e image formed by QaQb will be spread out, even if we would assume that the position of the image formed by the bending magnet does not depend on p . This effect has been discussed in Section 4.2.2 [see Eqs. (4.2.51)-(4.2.71)].

4.2.

BEAM BENDING AND FOCUSING SYSTEMS

745

The corresponding spread in the distance from the exit of Qb to the final image, ATa, is given b y : (4.2.211)

where i = 1 if magnet Q. is horizontally focusing, i = 2 if Qa is vertically focusing, and (dTb/dp)i is given by Eq. (4.2.61) or (4.2.62). (3) As a result of the momentum spread A p , the image formed b y the bending magnet B will also be spread out longitudinally (Le., along the direction of the outcoming beam). If this spread in the image distance is denoted by A T B , we have: A S , = A T R . We note that A T B is given by ( ~ l T h , ~ / dApp), where d T h , a / d pis obtained from Eq. (4.2.202).The resulting spread in the location of the final image is given by:

A T @ = lAil AS,

=

l A i l ( d T h , ~ / dA~ p)

(4.2.212)

where i = 1 or 2 as discussed after Eq. (4.2.211),and Ai is given b y Eq. (4.2.138), in which the matrix elements M i j can be obtained from Eqs. (4.2.97)-(4.2.100) for i = 1, and from Eqs. (4.2.102)-(4.2.105) for i = 2. The total spread ATb in the location of the final image is given by:

ATb

=

ATa

+ AT@.

(4.2.213)

As discussed in Section 4.2.2 [see Eqs. (4.2.63)-(4.2.71)], the width a t the image due to ATb is given by: AZY/ = (ATb/Tb)ra,(Au/2)

(4.2.214)

where Tb is the image distance from the exit face of Qb, A , is the effective maximum available aperture of the quadrupole pair QaQb, and {, is given by Eq. (4.2.69) for i = 1 and Eq. (4.2.69a) for i = 2. We note th a t { , A , represents the width of the particle beam a t the exit of Qb. For i = 2, we have : (4.2.215)

The total width of the final image due to A p is approximately given by:

ApYf

=

A1Yi -t A z Y / .

(4.2.216)

I n connection with Eq. (4.2.213), we note that both A T a and A T @are positive, and that they must be added to obtain the total ATb. This result arises from the fact th at with an increase in momentum by a n amount A p , A T a is such as t o increase T b , as discussed after Eq. (4.2.62), and moreover, A T @ will also act in the same direction, since a n increase of momentum will result in an increase of the image distance from B , T B , and

746

4.

BEAM TRANSPORT SYSTEMS

hence a decrease of the object distance S, from the entrance of quadrupole Q.. As is shown by Eq. (4.2.138), aTh/aS,, is always negative, so that for focusing (Tb > 0 ) , Tb will be increased as a result of the decrease of S,. Equation (4.2.216) for Apyf does not include the width a t the image due to the width of the source, i.e., the target which is in front of the bending magnet B. If the width of the target is denoted by at, the resulting width of the image, Atyr, is given by I x ~ , B Q ~ u ~ , where Xh,BQ is the horizontal magnification for the entire system BQ.Qb. Atyf must be added to Apyf to obtain the total width w of the final image. 4.2.4.9. Comparison of Strong Focusing Quadrupoles with WedgeShaped Focusing Magnets. The question arises as to whether in a particular experiment it is more advantageous t o use a system of strong focusing quadrupoles combined with an analyzing magnet with uniform field, or as an alternative, one or two bending magnets with nonuniform field and appropriate entrance and exit angles s and t. In general, for experiments with high-energy particles ( p 2 1 Bev/c), it is desirable t o use the first alternative, because the large radius of curvature R of the particles in a reasonably attainable field (-15,000 gauss) would force one to use very long bending magnets ( I 2 R / 2 ) and large entrance and exit angles (e.g., s = 50°, t = -50'). The magnetic saturation effects will be important and may make it difficult to define the effective magnetic field boundaries a t the entrance and exit of the magnet, because of the irregular shape of the fringing field. This is particularly true if the pole faces are tapered, giving a large n value to increase the focusing. In this case, model measurements are necessary to determine the effective field boundaries, and hence the effective entrance and exit angles s and t . By contrast, the equivalent quadrupole magnet system will in general be considerably shorter, and the effective field boundaries are more easily determined, because of the symmetry of the quadrupole field, and its rapid falloff outside the magnet iron. However, in certain cases, it may be desirable to use the same magnet for focusing and analyzing the particles, if one is dealing with relatively low-momentum particles with a short lifetime so that the distance L from the object to the image must be minimized. As an example of this possibility, we mention the experiment of Meyer et ~ 1on .the~scattering ~ of K+ mesons by protons, a t energies up to 90 Mev ( p = 310 Mev/c). In this experiment, the Kf mesons were focused and momentum-analyzed by a single bending magnet (45.7 cm wide, 91.4 cm long), whose pole faces were modified as follows. Each pole face was divided into two sections of equal length (45.7 cm). In the first section traversed by the beam, the pole faces were tapered so as t o give increasing gap height with increasing 28

D. I. Meyer, M. L. Perl, and D. A. Glaser, Phys. Rev. 107, 279 (1957).

4.3. BEAM

SEPARATORS

747

r , so that n was positive, which leads to vertical focusing. I n the second section, the pole faces were tilted so as to make n negative, thus giving horizontal focusing. Obviously, the two sections correspond to magnets a and b of the previous discussion, the distance d being zero in this application. Double focusing of the Kf mesons was obtained for a total object-to-image distance of 3 meters. It should be emphasized that this type of system is practical mainly for relatively low-energy particles ( p 5 0.5 Bev/c). ACKNOWLEDGMENT

I wish t o thank Dr. Luke C. L. Yuan for several helpful suggestions concerning this chapter.

4.3. Beam Separators* The production of secondary beams of particles depends upon the phase space available due to high energy collisions. The result is that high momentum secondary beams from accelerators are produced predominantly in the forward direction. Various methods of selecting the momentum, focusing the secondary particles, and separating the various charged particles have made it possible to produce “clean” beams of particles. The design of secondary beams is very dependent upon the available flux of particles, the types of detectors t o be used, and the lifetime of the desired particles. For bubble chambers and emulsion experiments, spatial separation of the background particles is required ; whereas for counters and spark chamber experiments, temporal separation is required. 4.3.1. Degrader Type Separation Early experiments were done’ using a d E l d x degrader to separate the particles of a given momentum and different masses. The low velocity 1 L. T. Kerth, D. H. Stork, R. P. Haddock, R. W. Birge, J. R. Peterson, J. Sandweiss, and M. Whitehead] Positive heavy mesons produced at the Bevatron. UCRL3031 (June, 1955); 0.Chamberlain, W. W. Chupp, A. G. Ekspong, G. Goldhaber, S. Goldhaber, E. J. Lofgren, E. SegrB, C. Wiegand, E. Amaldi, G. Baroni, C. Castagnoli, C. Franzinetti, and A. Manfredini, Phys. Rev. lo%, 921 (1956);W.H.Barkas, R. W. Birge, W. W. Chupp, A. G. Ekspong, G. Goldhaber, S. Goldhaber, H. H. Heckman, D. H. Perkins, J. Sandweiss, E. SegrB, F. M. Smith, D. H. Stork, L. Van Rossum, E. Amaldi, G. Baroni, C. Castagnoli, C. Franzinetti, and A. Manfredini, ibid. 106, 1037 (1957);and L. E. Agnew, Jr., T. Elioff, W. B. Fowler, R. L. Lander, W. M. Powell, E. SegrB, H. M. Steiner, H. 6. White, C. Wiegand, and T. Ypsilantis, ibid. 118, 1371 (1960).

* Chapters 4.3 and 4.4 are by Bruce Cork.

748

4.

BEAM TRANSPORT SYSTEMS

particles have a greater energy loss in the degrader and are deflected a greater amount in the magnetic field a t the exit of the degrader. This method has many disadvantages, including loss of the desired beam due to interactions and scattering by the degrader. For low energy beams, electromagnetic separators are much better. 4.3.2. Electromagnetic Separators

A coaxial static-electromagnetic velocity spectrometer has been used2 to give a 450 Mev/c K- and p beam. Although the solid angle for accept-

copper conductor

FIG.1. The electrostatic plates p are supported by cylindrical insulators. The side walls N, 8 are the “pole tips” of an H-shaped electromagnet. The vacuum enclosure is made by welding stainless steel plates SS to the pole tips.

ance of transmitted particles was small, several experiments were done with this spectrometer. A parallel-plate velocity spectrometer has been built3 to reject negative pions from a beam of antiprotons. The electromagnetic separator, shown in Figs. 1, 2, and 3, consisted of two parallel plates 20.3 cm wide and 580 cm long separated by a mean distance of 10.2 cm and maintained 9 J. J. Murray, A coaxial static-electromagnetic velocity spectrometer for high energy particles. UCRL-3492 (May, 1957); and N. Horwitz, J. J. Murray, R. Ross, and R. Tripp, 450 Mev/c K - and $ beams in the northwest target area of the Bevatron separated by the coaxial velocity spectrometer. UCRL-8269 (June, 1958). a C. A. Coombes, B. Cork, Pi. Galbraith, and G. R. Lambertson, Phye. Rev. 112, 1303 (1958).

4.3.

BEAM SEPARATORS

749

a t potentials of f 180 kv. The crossed magnetic field was adjusted so that antiprotons would be undeflected by the combined electric and magnetic fields, while pions were deflected and did not go through the slit. The magnetic field of approximately 200 G along the full length of the separator was supplied by water-cooled conductors. The pole tips were also a part of the vacuum vessel and the return path for the magnetic field was in iron; thus the spectrometer was an H magnet.

FIG.2. Bev-1795. End view of parallel plates of t.he crossed electric and magnetic field velocity spectrometer. The sides of the vacuum tank are made of soft iron and are the pole tips of an H magnet. The top and bottom are made of stainless steel.

A particle of velocity PO is transmitted through the slit when the magnetic field is adjusted so that P o = E / B , where E is the average electric field and B is the average magnetic field. Other particles of charge e and momentum p are displaced a t the output of the spectrometer by a distance 1J =

g[

-

;]

and have an angular deflection

The effective length of the electric field is 1.

750

4.

BEAM TRANSPORT SYSTEMS

FIG.3. Bev-1806. The 6.1 in long parallel plate velocity spectrometer. Charged particles enter from the right. High potential is applied through the coaxial feedthrough insulators on the left.

4.3.3. Radiofrequency Separators

For energies greater than a few Bev, the spatial separation of pions from K mesons becomes very difficult. Since the K meson lifetime is approximately one-half that of the pion, and 4 is less for a given momentum, the system cannot be allowed to become arbitrarily long. Also, the angular separation using a dc separator,

is proportional to l / p 3 . Very good optics will be required in the multi-Bev region. A mass sensitive deflector for high energy particles has been proposed by Panofsky.4 Such a device has been built6 and used to select 350 Mev/c 4 6

W. K. H. Panofsky, HEPL 82 (May, 1956). R. P. Phillips, HEPL 171 (June, 1959).

4.3. BEAM

SEPARATORS

751

electrons. A microwave cavity was located at the center of the optical system of a triple focusing spectrometer. This cavity was coherent with the microwave linear accelerator and deflected the electrons that were scattered from a target. The amount of deflection depended upon the time of transit between the target and the deflecting cavity. The desired particles can be selected by changing the relative phase of the microwave cavity and the accelerator. With the radio frequency spectrometer, the angular separation is

eEl

A0 = -

PPC

where E is now some effective electric field. Various systems6 have been proposed so that the net force due to the electric and magnetic fields of the cavity not be zero. It is not practical to make most accelerators coherent with a microwave cavity, so that either longer wavelengths must be used as proposed by Veksler' or the secondary beam has to be made coherent by means of a series of cavities. Rejection of unwanted particles at all phase angles can be obtained by'using circular polarization of the deflecting forces in both cavities. The deflection of particles of one velocity would be zero while other velocities would be deflected by both cavities. Since the power requirements are difficult, and the solid angle of acceptance is small, Montague8 has suggested that plane polarization be used but that an absorber be so located that all particles which pass through the separator undeflected are stopped on this central beam absorber, located behind the second cavity. The desired particles are deflected around the absorber and are brought to focus in a useful beam. A spectrometer of this type is being designed a t CERN to separate 10 Bev pions from K r n e s ~ n . ~

4.3.4.

Separation by Nuclear Interactions

At energies greater than 10 Bev, the spatial separation of particles becomes very difficult. Fermi has shownL0that the relativistic increase 8 W. K. H. Panofsky and W. A. Wenzel, Rev. Sci. Znstr. 27,967 (1956); J. P. Blewett and J. D. Kiesling, BNL/ADD/JPB-JDK (August, 1959); and M. L. Good, A radiofrequency separator for high-energy particles. UCRL-8929 (October, 1959) ; E. D. Courant and L. Marshall, Rev. Sci. Znstr. 81, 193 (1960). 7 V. I. Veksler, PTOC. Intern. Conf. on High Energy Accelerators, CERN, Geneva, 1969 p. 426 (1959). 8 B. W. Montague, CERN PS/INT. AR/P SEP/60-1 (July, 1960). 9 W. Schnell, CERN 61-5 (1961). 10 E. Fermi, Phye. Rev. 66, 1242 (1939); 67, 485 (1940).

752

4.

BEAM TRANSPORT SYSTEMS

in ionization loss above the minimum, which is caused by the relativistic lateral extension of the electric field of the moving particle, is appreciable. For values of /3 = u / c > 0.97, the ionization increases as log /3r to a maximum that is dependent upon the density effect." This increase of ionization has been used12 with a n argon-filled cloud chamber to distinguish pions from protons by droplet counting. The pions have approximately 20 % greater ionization loss than protons at lOLo eV, and good separation in the energy range of 1O1O to 10l2 eV has been achieved. Other separation methods, by means of strong interaction processes, have been used. Pure p-meson and neutrino beams have been obtained by filtering out nucleons, K mesons, and 7r mesons. The long lifetime of the p meson and the large interaction cross section of the nucleons and K mesons makes this possible. At high energy the differences in the behavior of the various types of particles again allows separation schemes to be devised. Several methods that include charge exchange of a beam of high energy particles followed by magnetic separation have been suggested. l3

4.4. Some Examples of Beam Transport Systems 4.4.1. High Momentum Beams Using Counters as Detectors A beam of 10" protons of 6 Bev energy incident on a 15.2 cm long beryllium target produces approximately three antiprotons and lo6 negative pions per millisteradian in the forward direction for a f6 % momentum interval a t 2.8 Bev/c. This beam, using a system of magnetic quadrupole focusing magnets, a parallel-plate velocity spectrometer, time-of-flight scintillators, and a gas cerenkov counter, has been used14 to measure total, differential and charge-exchange antiproton-proton cross sections in the momentum interval from 1.7 to 2.8 Bev/c. The K--p total cross section was also measured in the same beam. For experiments using scintillation counters and spark chambers as detectors, the duty cycle of the beam should be as long as possible, thus reducing the accidental rate in the counters. A beam duration of 300 msec R. Sternheimer, Phys. Rev. 88, 851 (1952);81, 256 (1953);103, 511 (1956). R. G.Kepler, C. A. d'Andlau, W. B. Fretter, and L. F. Hansen, Nuovo cimento [lo] 7, 71 (1958);and L. F. Hansen, and W. B. Fretter, Phys. Rev. 118, 812 (1960). G. Goldhaber, S. Goldhaber, and B. Peters, CERN 61-3 (January, 1961). 14 R. Armanteros, C. A. Coombes, B. Cork, G. R. Lambertson, and W. A. Wenzel, Phys. Rev. 119, 2068 (1960). 11

I*

4.4. SOME

EXAMPLES O F BEAM TRANSPORT SYSTEMS

753

was achieved by steering a portion of the circulating proton beam of the Bevatron into a thin foil a t the outer radius of the orbits. Small angle collisions caused a portion of the proton beam to become phase unstable. The increasing magnetic field of the Bevatron caused the “equilibrium orbits” to move to a smaller radius, and finally the proton struck a n inner radius target. Antiprotons produced in the forward direction in the Bevatron were deflected in the magnetic field of the Bevatron. The secondary beam

-

0

10

20

30

40

50

Scale (feet)

FIG.4. MU-18068. Antiproton beam for p - p scattering experiment 1.0 to 2.0 Bev beam. Elements are described in text. [From Phgs. Rev. 119, 2068 (1960).]

channel (Fig. 4) consisted of strong focusing magnetic quadrupoles Q1to Q,, deflecting magnet M , and steering magnets C , and Cz. The momentum was selected by locating the target at the appropriate place in the magnetic field of the Bevatron and then tuning the magnets of the channel to the corresponding momentum. The momentum width, +6%, was determined by the horizontal image of the target a t the entrance of Q2. The magnetic quadrupoles Q1 and QZ were 20.3 cm diameter triplets. Antiprotons from the target were focused a t the slit in front of M . The height of the target could be made small, 0.28 cm, and thus the height of the image at the slit was small. Separation was then possible with the

754

4.

BEAM TRANSPORT SYSTEMS

electromagnetic velocity spectrometer-a crossed electric and magnetic field, with the electric field vertical. The system of 10.2 cm diameter magnetic quadrupoles focused the antiproton beam along a 30.5 meters time-of-flight channel. Particles produced in the 183 em long methane cerenkov counter were deflected by magnet Cz,while antiprotons continued on to the hydrogen target. A t a point 4.5 meters beyond Q,,the focused beam was an image 5.1 cm in diameter.

2.0 Bev

I

127

Time of flight ( m p e r e )

FIG.5. MU-19940. Antiproton time of flight delay curve. Both scintillators and a threshold gas Cerenkov counter (in anticoincidence) were used to reject pions.

The separator, described in Chapter 4.3,consisted of parallel plates 11.6 meters long separated by 10.2 cm and at potentials of 5 180 kv. At 1.7 Bev/c the separator rejection factor of pions to antiprotons was 3. This was sufficient to reduce the pion rate in the scintillation counters so that the accidental rate in the time-of-flight coincidence circuits was low. At 2.8 Bev/c the difference between /3 and PO is small, so the electromagnetic separator was not useful. Fortunately the pion rate is lower at high momentum so that accidental counts in the scintillators are less of a problem. Identification of antiprotons was made by means of eight time-of-flight scintillation counters and one gas Cerenkov counter. The delay curve is given by Fig. 5. Tuned either to antiprotons or to K - mesons the back-

4.4.

755

SOME EXAMPLES OF BEAM TRANSPORT SYSTEMS

ground was negligible. Approximately 60 antiprotons per minute were detected at the hydrogen target at 1.7 Bev/c and 15 per minute at 2.8 Bev/c.

4.4.2. Separated Beams for Bubble Chambers The parallel-plate velocity spectrometers described above have been used to obtain good spatial separation of charged particles for bubble chamber and emulsion experiments.l6 A 1.17 Bev/c K- beam, Fig. 6 , consisted of two spectrometers 6.1 meters long and a system of focusing lenses and slits, Fig. 7. Particles of negative charge were produced by 6 Bev protons striking a 3 mm high target inside the Bevatron. Care was taken to keep the lens aberrations small so that the height of the image at the first slit was less than 5 mm. "

HYDROGEN BUBBLE \ \

QUADRUPOLE BENDING

FOCUSING

MAGNET

MAGNETS

J

J

!? i

\

FIQ.6.The 1.17 Bev/c K- beam a t the Bevatron. [From Rev. Sci. Instr.31,1054 (1960).]

With the parallel plates separated a distance of 6.35 cm and operated at a potential difference of 380 kv, the attenuation of pions when tuned to K- mesons was approximately loK,a factor of 30 a t the first slit and 3000 a t the second. A considerable amount of the background through the first slit is due to the decay of pions in the first part of the system, producing muons that are transmitted by the slit. Most of these are not transmitted by the second system. The yield was approximately 9 K- mesons per 10" protons on the target, and the background was approximately 10 beam muons, probably due to the decay of pions admitted to the second stage. The pion background a t the bubble chamber was 8%. H. K. Ticho, Proc. Intern. Cmf. on High Energy Accelerators, CERN, Genera,

lb

1969 p. 387 (1959); Rev. Sci. Znslr. 31, 1054 (1960).

Feet Qbject principal plane

.nOF-I--

5

10

Image principal plane

II

HORIZONTAL PLANE"

''

Bending magnet

Puodrupole 3

Puodrupole 2

Puodrupole I

T\

\

El

\ \

Spectromete

$1,.

I I

I

-c VERTICAL PLANE

0

,,

,

,

,, ,

,

5 10 Feet

IM

1-

BEAM

FIG. 7. MUB-226. Focal propertias of the magnetic quadrupoles used in the 1.17 Bev/c K - separated beam. [From Rev. SCi. Znstr. 31, 1054 (1960).]

Ern E-

-4

01

-4

FIG.8. MU-19685. Plan view of 1 to 4 Bev/c K - beam.:Two high pressure Cerenkov counters were used to reject pions and antiprotons. [From Phys. Rev. 123, 320 (1961).]

4.

758

BEAM TRANSPORT SYSTEMS

The beam was also tuned t o antiprotons and gave a yield of 0.5 antiprotons per 101' protons, with a 20% pion background. A similar system with three 6.1 meters long separators has been used'" to obtain a 1.65 Bev/c beam of antiprotons, with 0.8 p per 10" protons and a background of 10% pions and 250% muons. Beam

0.2"A1 hemispherical window 6810A photomultiplier tube

$" Fe magnetic shielding Lucite light pipe

I$' quartz window

Scale (inchas)

FIQ.9. MU-19686. Cross section of one of the differential high pressure Cerenkov counters.

Two separators 3.05 meters long have been used" with similar optics to give a n 800 Mev/c K + meson beam. Another beam, taking 800 Mev/c K - mesons or antiprotons from an internal target of the Bevatron, a t zero degrees production angle, has been separated'* using two 3.05 meters 1 6 J . Button, P. Eberhard, G. R. Kalbflehch, J. E. Lannutti, G. R. Lynch, B. C. Maglic, M. L. Stevenson, and N. H. Xuong, Phys. Rev. 121, 1788 (1961). 17H. Bradner, W. Chinowsky, G. Goldhaber, S. Goldhaber, T. Stubbs, D. H. Stork, and H. Ticho, The K+-p interaction at 455 MeV. UCRL-9745 (June, 1961); to be published in Phys. Rev. 1*P. Bastien, 0. Dahl, J. Murray, M. Watson, R. G. Ammar, and P. Schlein, in "Proceedings of the International Conference on Instrumentation for High Energy Physics," p. 299. Interscience, New York, 1960.

4.4.

SOME EXAMPLES OF B E A M TRANSPORT SYSTEMS

759

long separators. The yield of this beam was 10 K- mesons per 10" protons, and the background was pions and muons. When tuned to antiprotons the yield was 0.5 ?? and one pion or muon per 10" protons.

4.4.3. Special Beams At high momentum, where both spatial separation and time-of-flight separation becomes difficult, it is possiblelg to use high pressure gas

1.0

lo-' N

lo-'

I -b Y

10-3

IO-~

IO-~

'o-60

200

400 600 8 0 0

1000 1200 1400 1600 1800

Methane pressure ( p s i )

FIG. 10. MU-19707. Selection of particles by varying the pressure in the high pressure methane eerenkov counter, 1.95 Bev/c. The counting rate of all particles down the channel is MI.Mode A operation is coincidence-anticoincidence operation and mode B is anticoincidence operation only. Methane pressure in psi (equivalent to 0.07 kg/cm*).

cerenkov counters to select the desired particles. A 1 to 4 Bev/c Kmeson beam is shown in Fig. 8. The two high pressure methane cerenkov counters are shown in Fig. 9. These were constructed so th a t cerenkov light from K mesons was focused onto the photomultiplier located on the axis and cerenkov light from the pions was focused on the photocathodes of the outer ring of photomultipliers. Th e signals from the pions were then l 9 V . Cook, B. Cork, T. Hoang, D. Keefe, L. Kerth, W. Wenzel, and T. Zipf, K--p and K--n cross sections in the momentum range 1-4 Bev/c. UCRL-9386 (January, 1961); Phys. Rev. l2S, 320 (1961).

760

4.

BEAM TRANSPORT SYSTEMS

placed in anticoincidence. With two counters of this type, the rejection of pions was as shown by Fig. 10. Total and differential K--p and K--n cross sections in the momentum range 1 to 4 Bev/c have been measured. This type of temporal separation of charged particles is adequate for many high energy experiments where counters or spark chamber detectors are used.

5. STATISTICAL FLUCTUATIONS IN NUCLEAR PROCESSES* It is well recognized that we can never measure any physical magnitude exactly, that is with zero error. Progressively more elaborate experimental or theoretical efforts result only in narrowing the range of the possible error. Therefore, in reporting the result of any measurements, it is obligatory to specify the probability that the result is in error by some specified amount, because a gamble on reladive correctness is always involved in all physical determinations. The theory of statistics and fluctuations, presented synoptically here, describes the mathematical procedure involved in the reduction of experimental data of the type encountered in nearly every measurement in nuclear physics.

5.1. Frequency Distributions In any series of measurements, the frequency of occurrence of particular values is expected to follow some “probability distribution law” or “frequency distribution.” There are about a half dozen distributions which are needed most often in the statistical appraisal and interpretation of nuclear data. Of these, the Poisson distribution, or distributions derived from it, is the most common. 5.1 .l. The Binomial Distribution The binomial distribution is the fundamental frequency distribution governing random events. The normal and the Poisson frequency distributions can be derived from it. If the probability that an event will occur in a single trial has the constant value p , then the probability that it will not occur in the same single trial is q = 1 - p . In a group of z independent trials the probability P, that the event will occur x times is given by that term in the binomial expansion of ( p 4)“ in which p is raised t o the x power. For example, if z = 3 identical radioactive atoms are observed over a time

+

1 This part is chiefly an epitome, with some additions and rearrangements, of the treatment given in R. D. Evans, “The Atomic Nucleus.” McGraw-Hill, New York, 1955. See especially Chapters 26, 27, and 28, and Appendix G for documentation of statements given here without proof, and for illustrative examples.

* Part 5 is by Robley D. Evans. 761

5. STATISTICAL

762

FLUCTUATIONS

+

interval during which the probability of decay is p = for each atom, then the chance that three, two, one, or zero atoms will decay during the observation is given by the individual terms of the corresponding binomial expansion :

The probability P o of zero successes is often a very useful statistic. For example, the total proability of one or more successes is simply 1 - PO. The general analytical form of the binomial expansion is (P

+ PI’

z=o

=

1 PZ

(5.1.2)

z=z

in which any individual term can be written as

P, =

Z!

z!(z

- z)! p=(l - p ) - .

(5.1.3)

The binomial distribution, Eq. (5.1.3),cont,ains two independent parameters: the constant probability of success p , which can have any vaIue from 0 t o 1, and the t,otal number of independent trials z. It applies rigorously to those phenomena in which the total number of trials z and the total number of successes 5 are both integers, and 0 5 x _< z. In Eq. (5.1.3)the coefficient z!/z!(z - z)! is the number of unordered selections or combinations of z things (trials) taken z (successes) at a time. Figure 2 contains an example of a binomial dist,ribution, for the special case of p = 0.4,z = 40. The values chosen for p and z determine the mean value (Section 5.2.1)as m = p z = 16, and the standard deviation * ) = 3.10. (Section 5.2.4)as u = d

5.1.2. The Multinomial Distribution The binomial distribution is a special case of the multinomial distribution. The multinomial distribution describes random processes in which several independent results, each having constant probabilities p l , p 2 , . . . , p a , are possible. The separate probabilities are then given by terms of the expansion of ( p l p2 .* * where

+ +

Pl+P2+

+

* . . +p,=1

and z is the number of trials. The general term in this expansion of (pl PZ . palais Z! ,p;’pq’ . . . p;‘ (5.1.4) PZ,.,,. . . . .Z. -

+ +

+

Zl!ZZ!

.

*

.

zr.

5.1.

763

FREQUENCY DISTRIBUTIONS

in which xi is the number of occurrences of the event whose probability is pi for one trial, and x1 xz * * x8 = z is the number of trials. The multinomial distribution applies to nuclear disintegrations in which several processes compete; e.g., for Cu64nuclei let pl = probability of 0- emission, p 2 = probability of p+ emission, p 3 = probability of electron capture, and p4 = probability of no transition during a fixed interval. Note that O ! = 1. This follows from the elementary definition of the factorial n! (5.1.5) _n -- (n - I ) ! hence-I!1 = (1 - l ) ! = O!

+ + - +

5.1.3. The Normal Distribution

The normal distribution (sometimes erroneously referred to as the Gaussian error curve) is an analytical approximation t o the binomial distribution when z is very large. In the normal distribution the observed m . The quantity x is a continuous variable with limits of - CQ and dx is differential probability dP, that x will lie between x and x

+

+

(5.1.6)

. .

where e = 2.7183 , , m = true mean value of x, and u = standard deviation of the distribution of x about m. The standard deviation is a fundamental statistic (see Section 5.2.4) whose value depends upon the mean value for most frequency distributions. But in the normal distribution, u is independent of m, and u and m act as two independent and freely adjustable parameters. Note that the normal distribution of x is always symmetrical about the mean value m. The smooth curve in Fig. 2 illustrates the shape of a typical normal distribution. The points of maximum slope, at which d2(dP,/dx)/d2x = 0, fall a t 5 = m k u where the slopes have the values

d(dP,/dx)/dx

=

f l/a2

G.

Tangents t o the distribution curve at these inflection points (m k u ) intersect the z axis at x = m f 2u, and this simple geometrical relationship offers a convenient method of estimating a graphically from an experimentally determined distribution curve. The full width of the normal distribution a t half-maximum height is 2a 2/2i-n2 = 2 . 3 5 4 ~ .The full width is 2a at 0.6066 of the maximum height. Figure 1 gives the integrals of the normal distribution. These play an important role in the statistical theory of errors, which is ordinarily based

764

5.

0

STATISTICAL FLUCTUATIONS

1.0

0.5

1.5

2.0

2.5

w/a

FIG.1. Integral of the normal distribution. The ordinate P, is the fraction of the total area of the symmetric normal distribution which falls farther from the mean value than a distance w ,measured in units of the standard deviation u. Particular numerical values which find frequent use are w/u

0

Pw

1.000

0.5 0.6171

0.6745 0.5000

1 0.3173

2 0.0455

3 0.00272

on normally distributed variables. In Fig. 1 the shaded areas marked &Pwcorrespond to either of the integrals

pw= J-”.-” dP,

=

/mLI1.d P ,

(5.1.7)

and Pw is the probability that a single observation of 2 will differ from the mean value m by more than w. From Eq. (5.1.7), the probability that a single observation of x will lie between x1 = m - w1and x2 = m - w 2 is (5.1.8) P(x1 to 2 2 ) = +lPw, - P w 2 1 .

5.1.4. The Poisson Distribution Poisson’s distribution describes all random processes whose probability occurrence is small and constant. It applies to a large proportion of all observations made in experimental nuclear physics. The general conditions under which the Poisson distribution is valid can be visualized by considering the illustrative case of radioactive disintegration. Then the necessary and sufficient conditions are : of

5.1.

FREQUENCY DISTRIBUTIONS

765

1. The chance for an atom to disintegrate in any particular time interval is the same for all atoms in the group (all atoms identical). 2. The fact that an atom has disintegrated in a given time interval does not affect the chance that other atoms may disintegrate in the same time interval (all atoms independent). 3. The chance for an atom to disintegrate during a given time interval is the same for all time intervals of equal size (mean life long compared with the total period of observation). 4. The total number of atoms and the total number of equal time intervals are large (hence statistical averages significant). Let a be the true average rate of appearance of disintegration particles from such a random process. In a very short time interval, di, the probability of observing one particle is P I ( & ) = a dt; and the probability of observing no particle is Po(&) = 1 - a d t , if the time dt has been chosen so short that the probability of observing two or more particles in dt is negligible. dt), of observing x particles We now write the probability, P,(t dt. This is the sum of two mutually exclusive probabilities: in the time t either z particles in t and none in dt, or (z - 1) particles in t and one in dt. Then (5.1.9) P z ( t d t ) = P,(t) Po(dt) P,-l(t) * P l ( d t ) = P,(t) (1 - a d t ) P,_l(t) a d t . (5.1.10)

+

+

+

+

+

Rewriting Eq. (5.1.10) in differential form, we have

The solutJion2of Eq. (5.1.11) is (5.1.12) as can be verified by differentiation. But at is simply the true mean value for the number of events in time t, m = at. Writing m for at in Eq. (5.1.12) we obtain the usual general form of the Poisson distribution mz

P , = - ecm X!

(5.1.13)

in which P, is the probability of observing x events when the average for a large number of tries is m events. Although m may have any positive value, z i s restricted to positive integer values only. Note that the Poisson distribution has but one parameter, m. It can be shown easily that the

* H. Bateman, PhiZ. Mag. [6] 20, 704 (1910).

766

5.

STATISTICAL FLUCTUATIONS

binomial distribution Eq. (5.1.3) with two parameters, z and p , becomes identical with the Poisson distribution in the limiting case of p > 1) of 2, i=n

(5.2.4) i=l

The variance is thus seen to be simply the second moment of the frequency distribution taken about the true mean, m. Geometrically, if a sheet of metal is cut in the shape of the distribution, the variance is the “moment of inertia” of the sheet taken about an axis at z = m and parallel to the frequency axis.

774

5.

STATISTICAL FLUCTUATIONS

The variance for each of the frequency distributions of Chapter 5.1 can be derived using Eq. (5.2.3) over appropriate limits. This gives z=z

(x - pz)*z!pz(1 - ')"-" 21(z - x)!

u2 (binomial) =

=

pz(1 - p)

(5.2.5s)

(5.2.5~)

1-(1 Lm(t

2=b

u2 (interval) = u2 (s-fold

int.)

=

-

-

k)' :)

2

ae-nLdt = 1 a

(5.2.5d)

dt = -.S a2 ( s - l)!

(5.2.5e)

a"t"-'e-"!

5.2.3. Sample Variance I n a finite series of n observations we can never determine u2 exactly, just as we can never determine m exactly. Our estimate of the true variance u2 of the distribution must therefore be based on our imperfect estimate Z of m. Accordingly, we define a "sample uariance," S2, by analogy with Eq. (5.2.4), but with m replaced by 2. Thus the experimental data give (5.2.6) i=l

With the help of the substitution (xi - 2 ) = [(xi - m ) - (Z - m ) ] , it is easy t o show quite generally that the expected value E[S2]of S2, averaged over a number of repeated samples of n observations, is

E[SZ] = u2

-1 - u2 - = n___ d. n

n

(5.2.7)

Then our best estimate of the true variance u2 of the distribution, in terms of our total finite number n of observations, xl,x2, . . . , xn, is i=n

'c

n n - s2= uz=-7 n-1

(Xi

- f)Z.

(5.2.8)

i= 1

5.2.4. Standard Deviation * The breadth of a distribution of data is best visualized in terms of some stat,istic which has the same dimensions as the mean value. The

5.2.

STATISTICAL CHARACTERIZATIONS OF DATA

775

best suited parameter is the square root of the variance, which is called the standard deviation, u, of the distribution. In the normal distribution Eq. (5.1.6) the standard deviation is just one of two arbitrary parameters, m and u. In other distributions u depends upon p, z, a, s, or m in the manner derived in Eqs. (5.2.5) and summarized in Table I. Because of its symmetry, the normal distribution contains the fraction 0.3413 of its area between x = (m - u) and x = m, and the same fraction between x = m and x = ( m cr). Thus 68% of the area of any normal distribution lies in the domain x = m _+ u, as shown in Fig. 1. None of the other distributions is symmetric. For them, the mean differs from the median, m does not divide the area into halves, the area between m - u and m is different from the area between m and m u, and both fractional areas depend upon the other parameters of the distribution. For example, for the s = 4 interval distribution of Fig. 3, Eq. (5.1.22) will show that the domain of at below the mean value, at = 4, contains not one half but 0.567 of the area, the domain between the mean value and one standard deviation below the mean contains 0.424 of the area, while the domain between the mean value and one standard deviation above the mean value contains only 0.282 of the area. With these obvious asymmetries in mind, it is clearly risky in physics to express experimental results as a sample mean plus or minus a quantity such as the standard deviation or even the standard error unless the degree of asymmetry of the distribution is known to be small. Otherwise it is best to state the sample mean, and to state the standard deviation or the standard error, but to avoid the commonly used f sign. The expected value of the standard deviation (Table I) corresponds to an infinite population of data. In a finite sample of n independent observations, the best estimate of this standard deviation is to be obtained by evaluating the individual deviations (xi - 3) from the sample mean 3 and then computing from Eq. (5.2.8) the estimated standard deviation of the parent distribution. This is

+

+

I

i=n

(5.2.9a) or I

i=n

(5.2.9b)

n-1 in the case of interval distributions. In computing the standard deviation

u

of a series of observations xl,

776

5.

STATISTICAL FLUCTUATIONS

. . . , x,, the arithmetic can often be greatly simplified, and progress can be made on the computations while the experiment is still in progress by referring the individual readings to some arbitrary value xo, usually chosen as a round number near 2. Then it is easy t o show that Eqs. (5.2.2) and (5.2.9) become

22,

i=n

z

=

[A

‘c

20

+ -n

(xi

- x , , ) ~ ]-

(Xi

(5.2.10)

- ro)

[* (z

-

i=l

I

x0)2

(5.2.11)

in which the first square bracket can be evaluated before the final value of 2 has been determined, and the second square bracket often will be only a small correction term. A form which is often arithmetically useful when xi contains only one or two digits, but which is always analytically valid, is

c

i=n

i=n

i=l

(xi -

$2

=

(2 $2) -

n(z)2.

(5.2.12)

i=l

Where only one observation x is made n = 1, and then r n z z = x, but u is indeterminate because the spread of the data has not been sampled. If the process is known to be Poisson distributed, as in many nuclear counting experiments, then the best estimate of u is the expectation value U N & (5.2.13) as given by Eq. ( 5 . 2 . 5 ~ ) .

5.2.5. Standard Error The usual objective of a series of n independent measurements xl,z2, . . . , xn of a physical magnitude is to estimate the true mean value m, and to state the chance that the estimate 2 differs from m by less than some specified amount. The usual “specified amount” is the standard error of the mean. To evaluate the standard error we imagine that repeated samples of size n are taken, and ask how the mean values from each of these samples would be distributed about the grand average of the sample means. It is found that a series of k mean values, Z1,Z2,. . . , z k , each based on n measurements of x, will tend to exhibit a normal distribution about their grand average 2. This is true, if n is sufficiently large, even if the

5.2.

777

STATISTICAL CHARACTERIZATIONS O F DATA

parent population x is not normally distributed but, for example, is an asymmetric Poisson or interval distribution. In general, the distribution of mean values tends to be much more nearly normal than the parent population. It can be shown that in a large series of k hypothetical measurements of the sample mean 2, each based on n independent measurements of x, the grand average Z approaches the true mean m, and the sample mean which is value 3T: is normally distributed about m with a variance given by i=k

(5.2.14)

where uz is called the standard error of the mean Z, and u is the usual standard deviation of the distribution of x about 2. Because the distribution of 2 about m is nearly normal, and therefore symmetric, the chance that 2 lies within (m I az) is about 68%, by Fig. 1. The mean of a single series of n independent measurements xl, x2, . . . , x, is then to be reported as ( 2 f G), where i=n

(5.2.15) and the standard error of the mean is I U

i=n

1

(5.2.16)

n(n - 1)

Experimentalists must be sure to distinguish clearly between standard error (of the mean) and standard deviation (of the distribution). An everyday illustration occurs in nuclear counting, and can serve also as a derivation of the basic Eq. (5.2.14) for any Poisson distribution. Suppose that n independent measurements of a constant nuclear process have given xl, x2, . . . , zn in a set of n 1-minute intervals. Considered as a whole, the total number of events is v = x1 x2 * * * x, = nl. Then if the distribution is Poisson, the expected value of the standard deviation of v is l / V in a series of hypothetical repetitions of the n observations. Then the fractional standard error of v is

+ +

+

(5.2.17)

778

5.

STATISTICAL FLUCTUATIONS

This is a basic result, applicable to all counting experiments on Poisson distributions and hence nearly every counting experiment in nuclear physics. For example, if u = 104 counts, then the fractional standard error of our knowledge of the average counting rate is l/di@ = 0.01 = 1%. Because v comprises all our data, it furnishes our estimate of the mean, and its standard deviation 4;is in this case also the standard error 4; of the mean v, because we have lumped all the data as though we had made just one observation over an n-minute period. But the fractional uncertainty is

4:

where u = is the standard deviation of the distribution of xi about 3. The fractional uncertainty G / v of our over-all result must be the same, whether expressed as a ratio involving v or 2. Therefore the standard error uz of the mean 2 of n observations xl,x2, . . . , x,,is u / d i , where u is the standard deviation of xi about 2, as in Eq. (5.2.14). 5.2.6. Probable Error

The earlier literature of physics made frequent use of a statistic called the probable error. In a symmetric normal distribution, the probable error r defines a domain ( m f r ) which should include exactly one half of the observations of x. For the normal distribution, Fig. 1 shows that P, = 0.500 for w/u = 0.6745. The probable error r of a single observation was therefore taken as r = 0.67451s (5.2.19) and the probable error of a mean value as

rz

=

0.6745~.

(5.2.20)

These relationships are meaningful only for a normal distribution. They fail, and Eq. (5.2.19) is generally invalid, for any of the asymmetric frequency distributions (binomial, Poisson, interval) which are met most often in nuclear physics. The use of probable error should be discouraged, and the concept abandoned, except as a means of interpreting earlier literature where Eqs. (5.2.19) and (5.2.20) may be used to convert probable error to standard deviation and standard error. 5.2.7. Dimensions

From consideration of Eq. (5.2.13), it is evident that both x and (I must be dimensionless quantities, because u has the same dimensions

5.2.

STATISTICAL CHARACTERIZATIONS OF DATA

779

as x and fi.It is generally true that all such quantities in the distribution functions and in other statistical expressions are without dimensions. For example, 3 may be physically the average number of counts per minute, but statistically the time unit chosen is only an arbitrary interval or classification by which the data have been taken. It is to be regarded statistically as dimensionless. This can be visualized by considering time intervals measured off on a chronograph tape, in which case the particular interval used for classification might equally well be one second, or an equivalent length of tape, or even an equivalent mass of tape. The interval itself does not have the dimensions of time, length, or mass but is always statistically dimensionless, as are all the other basic statistical quantities. Although the interval distributions Eqs. (5.1.18) and (5.1.21) contain the rate a in events per unit time, it always occurs in the product at or a dt, which is again dimensionless.

5.2.8. Precision vs. Accuracy In the statistical evaluation of measurements, precision and accuracy have entirely different meanings.

-

High Accuracy

Low Accuracy

High

Precision I I

X-

3

m

X-

-x m

FIG. 5. Technical definitions of precision (small statistical error) and accuracy (small systematic error).

Precision relates only to repeatability. A result of high precision is one which has a small standard error attributable to statistical fluctuations, even if it has a large bias due to some systematic error. In the earlier statistical literature the “precision” h = 1/u 4 was used as a parameter of the normal distribution before the adoption of the present nomenclature of variance and standard deviation. Thus high precision means small UE. Accuracy relates to correctness with respect to some absolute scale. High accuracy implies freedom from bias due to systematic errors in calibrations and standards. Thus, because of the incorrect calibration of a millimeter scale, the mean range obtained for a given group of LY rays

780

5.

STATISTICAL FLUCTUATIONS

could be seriously in error (i.e., low accuracy) even if the individual measurements had a high degree of reproducibility (high precision). Thus high accuracy implies small (2 - m ). The two concepts are illustrated in Fig. 5.

5.3. Composite Distributions Most measurements or calculations in physics involve more than one source of error or of statistical fluctuations. The joint effect of simultaneous but independent sources of statistical fluctuations is considered in this chapter. 5.3.1. Combined Probabilities

Two processes A and B are said to be “independent” if the probability of A occurring is unaffected by the occurrence or nonoccurrence of B, and vice versa. The combined probability of two or more observations which are independent of each other is the product of the individual probabilities. An example occurred in the derivation given earlier for Eq. (5.1.9), in which PZ(t)and Po(dt) are clearly independent probabilities. Two processes A and B are said to be “mutually exclusive” if when one of the events occurs the other cannot occur. The combined probability of two or more observations which are mutually exclusive is the sum of the individual probabilities. The derivation of Eq. (5.1.9) again provides a clear example when the mutually exclusive probabilities P,(t)Po(dt) or P,-l(t)Pl(dt) are added to obtain their combined probability. Consider a nuclear example of combined probabilities. Suppose a certain radioactive sample contains a mixture of an a-ray emitter and an unrelated, independent, @-ray emitter, the average activities being A a-rays and B 8-rays per unit time. What is the probability M ( z ) of just 5 ( x = 0, 1, 2, . . .) 8-rays in the time interval between any two successive a rays? The probability dPt of an interval of duration 1 between any two a-rays is dPt = e-AtA dt, and the probability of x 8-rays in the time t is P, = (Bt)Ze-Bt/z!The combined probability, dPz,r,of these two independent circumstances, an a-ray interval t and x D-rays during t, is the product

(5.3.1) Now, intervals of various durations t are mutually exclusive, hence the probability M ( z ) of 2 8-rays in any interval is the sum, or integral, of

5.3.

78 1

COMPOSITE DISTRIBUTIONS

dP,,t over all possible intervals. This gives (5.3.2)

If, for example, B / A

=

M(0) =

2, then the probabilities sought are:

M(3) M(4) M(5)

0.333 0.222 = 0.148

= =

M(1)

=

M(2)

= $7

$

= = =

=

0.099

A*%= 0.066 & = 0.044

.

Finally, the occurrence of x = 0, 1, 2, . . &rays in any interval are themselves mutually exclusive processes. Hence the combined probability for a n y x in a n y interval between a-rays is

c

Z=O

M(x) =

B/A

1

1

+B/A

(1

+ B / A ) 24-(1 + B / A ) 3

+

...

=

1. (5.3.3)

The sum of all possible x in all possible intervals is unity, which in probability theory is the maximum possible probability, and represents certainty, 5.3.2. Superposition of Several Independent Random Processes The complete generalization of the Poisson distribution is usually required in nuclear problems because several independent types of radiation will actuate most detection instruments. For example, in ionization chamber measurements of a-rays, there will be present a background composed of a- and @-raysfrom radioactive contamination of the walls of the instrument, of cosmic rays, and of y-rays from the earth and surrounding building materials. If each of several such processes is itself random and follows a Poisson distribut,ion, the resulting over-all fluctuations may be predicted.6 Let z, y , z, . . . be the actual number of particles from the several independent random processes, in the interval chosen. Let them respectively produce specific effects (such as ion pairs) of a, 0, y, . . . Per particle. Then the actual effect, u, on the instrument is u=ax+py+yyz+

.

*

*

(5.3.4)

.

The true mean value, mu,of u is

mu = am, 6

+ + ym, + Om,

*

R. D. Evans and H. V. Neher, Phys. Rev. 46, 144 (1934).

.

(5.3.5)

782

5.

STATISTICAL FLUCTUATIONS

From a series of n independent measurements of u, our best estimat,e of mu is the sample mean i=n

(5.3.6)

Ui.

n

In any single observation each process produces x, y, z, . . . particles, instead of the true mean value mz, mu, m,, . . . particles. Then the net deviation due to the joint action of the separate processes will be 4 z - m,) P(y - mu) Y(Z - ma> * * . , if the physical effects of the separate processes are additive in the detection instrument. The probability that z, y, z, . . . particles will be produced by the several independent physical processes is the product of the Poisson probabilities : P, = (m,)ze-m*/z!,P, = (m,,)ye-mU/y!, P , = (mJze-mz/z!,. . . . Hence the variance of the distribution of u about its true mean mu is

+

+

+

x [ C Y ( X - m,)

+ p(y - mu) + Y(Z - m,)+

*

. - 1'.

(5.3.7a)

Upon expansion, Eq. (5.3.7a) reduces to simply uU2= a2mu

+ @'mu+ y2mz+ - -

(5.3.7b)

which gives the expected value of the standard deviation, uu, of u about the sample mean fi. In accord with the principles of Sections 5.2.4 and 5.2.5, our best experimental estimate of u u will be given by I

i=n

n-1

(5.3.8)

and of the standard error, uc, of the sample mean, fi, by (5.3.9) Equation (5.3.7) is of far-reaching importance in the statistical evaluation of all types of nuclear observations. The number of independent events x, y, z, . . .. are dimensionless quantities, but their specific effectiveness a,p, y, . . . may have any dimensions, such as ion pairs or statcoulombs per particle, or scaling-circuit output pulses per input

5.3. COMPOSITE

DISTRIBUTIONS

783

event, etc. If the weighting factors a, p, y, . , . are unequal, then the events with large specific effectiveness can easily dominate the fluctuations, u,, even if they contribute only a small share of the over-all mean value ti. In single-channel counters, the weighting factors are usually equal, a! = p = 4 = . . . Then the fluctuations are determined simply by the sum of the average number of random events from all the independent processes, that is, u, II (Y(Z g z * -)]I2.

-

+ + + -

5.3.3. Propagation of Errors Where a physical magnitude is to be obtained from the summation or the differences of independent observations on two or more physical quantities, the standard error, u 1 , 2 , .. . , of the sum or difference is given by (u1,2,...)Z

+ +.

= u1*

a22

*

*

(5.3.10)

where UI, u2, . . . are the absolute values of the standard errors of the mean values of the several quantities, expressed, of course, in the same units. Thus, (100 f 3) (6 f 4) = (106 f 5) (100 f 3) - (105 f 4) = - ( 5 f 5).

+

An everyday example is the estimation of the true counting rate, a, due to any source of radiation, when the counter has a true background counting rate, b. If the background has been determined by observing Btt, counts in a time tb, then our estimate of the true background rate is

(5.3.11) Note that the statistical uncertainty in our evaluation of the background depends inversely on the square root of the duration of the observation, as would be expected for any Poisson distribution. With the source added, ( A B)t, counts are observed in a time to, and our estimate of the true counting rate (a b) for the source plus background is

+

+

(A

I:

+ B)ta k d ( A + B)ta

-=A

+ (5.3.12)

Subtracting Eq. (5.3.11) from Eq. (5.3.12) gives the observed counting rate of the source as A B B (5.3.13) ta ta tb

+-+--

784

5.

STATISTICAL FLUCTUATIONS

Note that the background uncertainty enters twice, once as B/t, for its statistical fluctuation during the measurement of ( A B ) and once as B/tb for its fluctuation during the measurement of B alone. I n the design of counting experiments, it is clear from Eq. (5.3.13) that the uncertainty in the source strength, a, measured for a fixed time t,, can always be reduced by prolonging the independent background measurement, t b . But if laboratory conditions impose a restriction on the total time available, t, tb, then it can be shown that the fractional statistical uncertainty in the determination of a has its minimum possible value when the fraction of the time spent on the background measurement is

+

+

If a physical magnitude Y is to be obtained by multiplication or division of results of several independent observations on two or more physical magnitudes yl, yz, . . . , the fractional standard error U Y / Y in the resulting value of Y depends upon the fractional standard errors ul/yl,az/y2, . , . in the measurement of yl, y2, . . . and is given by (5.3.14)

Equation (5.3.14) is a good approximation whenever the fractional standard errors are small, that is, when n(ui/yJ2 30 use the normal distribution integrals of Fig. 1, with x = m = 4 2 F - 1, u = 1, and ( i ) P w = P. In actual practice F seldom exceeds 30.

4/22,

788

5.

STATISTICAL FLUCTUATIONS

and determine P , the probability that, on repeating the series of measurements, larger deviations from the expected values would be observed. In interpreting the value of P so obtained, we may say that, if P lies between 0.1 and 0.9, the assumed distribution very probably corresponds t o the observed one, whereas if P is less than 0.02 or more than 0.98 the assumed distribution is extremely unlikely and is to be questioned seriously. The x2 test, and especially the determination of F , may be better understood through an example. I n Table I1 we apply the x2 test to the hypothesis that the observed time intervals between successive cosmic-ray bursts are in agreement with the interval distribution Eq. (5.1.19). Experimentally, Montgomery and Montgomerye observed the time of occurrence of 213 bursts in a total of 30.8 hr; thus the average interval was T = 521.6 sec = l/a. The observed intervals, t , were then classified into eight arbitrary ranges of time intervals t l 5 t 5 t z , as shown in the table. These observed frequencies oi are given in column 4 of the table, e.g., 22 intervals had a duration of less than 50 sec, etc. The expected frequencies ei are calculated from Eq. (5.1.19), in the form e . - 213(e-at1 I

-

e-ats

1

(5.4.3)

with a = 1/521.6 sec, and are given in column 5 of the table. Because oi and ei are less than 5 for the two longest intervals, these classifications are combined with the 1000 t o 2000 sec classification before applying the x2 test. There remain six classifications in which the observed distribution can differ from the assumed interval distribution. But these six classifications are not all independent because they contain two implicit restrictions in which the observed and expected distributions are required t o match. These are the two parameters from the observed series which are used to predict ei in Eq. (5.4.3), namely Zoi = Zei = 213 events, and a = Zo@ = 1/521.6 sec. The number of degrees of freedom F , or independent ways in which oi may differ from ei, is then:

F

=

6 (classifications) - 2 (parameters)

=

4 degrees of freedom.

The calculation of x2 = 1.72 is shown in column 6 and 7 of Table 11. For F = 4, Fig. 6 shows that the probability is P = 0.8 that, in a repetition of this entire set of observations, x2 would exceed 1.72. Thus in 80 out of 100 similar experiments, the deviations from the interval distribution would be expected t o be greater than here observed. There is therefore strong support for the conclusion that the observed cosmic-ray C. G . Montgomery and D. D. Montgomery, Phys. Rev. 44, 779L (1933).

5.4.

TESTS FOR GOODNESS O F F I T

789

bursts obey the interval distribution and are therefore randomly distributed in time.

5.4.2. Poisson Index of Dispersion Often, a n experimenter obtains a small sample of data which he presumes is from a Poisson population, but which contains such a small number of independent measurements that it would be pointless to attempt t o match the data with any assumed distribution. Fortunately for nuclear physicists it has been shown that, in a series of n independent measurements xl, x2, . . . , zn on a Poisson population for which Z >> 1, the quotient (n times the sample variance)/(mean value) follows the x2 distribution with F = n - 1 degrees of freedom. We therefore define the Poisson index of dispersion as i=n x 2 = c (Xi

- 2)2

(5.4.4)

i=l

with x 2 as given by Eq. (5.4.2) for F = n - 1. The numerator is nS2 of Eq. (5.2.6); the denominator is our best estimate of u2, by Eqs. (5.2.2) and (5.2.5~).Hence the Poisson index of dispersion is a n estimator of the ratio of the sample variance t o the population variance. This ratio has a x2 distribution, with F = n - 1 degrees of freedom because only one observed parameter, the sample mean, 3, has been utilized in comparing the observed values xi with the expected dispersion. As a n illuminating illustration, compare the two sets of y-ray counting data, taken with two Geiger-Muller counters called A and X , given in Table 111. The values of xi are the total number of counts observed in successive 5-minute intervals. For counter A , the population standard deviation estimated from the = 13, and this is actually a little residuals, by Eq. (5.2.9a), is u ‘v smaller than the estimate based on the mean value by assuming a Poisson distribution] u ‘v & = 15. The value of x2 = 4.37, with F = 6, gives, from Fig. 6, the probability P = 0.6 that, in repetitions of these seven measurements, the sample variance would exceed the dispersion observed here. Note that these measurements extend from 248 t o 209, and therefore have a range of 248 - 209 = 39, or about 2 . 5 ~ .Also these two observations, 248 and 209, out of the seven observations, fall outside the domain Z f u, which is about the proportion expected from Fig. 1, i.e., P, = 0.32. These data then meet th e statistical tests for small samples. Because the primary process is surely Poisson, the counter A is judged to be operating satisfactorily.

2/-q

5.

790

STATISTICAL FLUCTUATIONS

Now consider the data from counter X. Here x 2 = 0.58, F = 5 , and hence in P = 0.99 of the repetitions we would expect greater variability among the independent observations. This dispersion is grossly subnormal and would be expected with good apparatus only once in 100 tries. We conclude that either (1) a very unusual observation has been made or (2) the instrument is faulty and devoted t o periodic spurious counts. The seasoned experimenter will surely choose t o be suspicious of the TABLE 111. Analysis of Y-Ray Counting Data

I

Counter A

1

2 3 4 5 6 7

Total

- 18

209 248 217 235 224 233 223

21 - 10 8 -3 6 -4

1589 2 = 227

F=7-1=6;

0

Counter X

1 2 3 4 5 6

324 441 100 64 9 36 16

-

Total

990 x= = 4 . 3 7

.'.P=O.6

242 241 249 246 236 250

-

-

1464 Z = 244

0

IF=6-1=5;

4 9 25 4 64 36

-2 -3 5 2 -8 6

-_ x2

142 0.58

=

:.P=O.99

counter or its electronic circuits and will proceed with further examination of the instrument. Note also that the standard deviation estimated from the residuals is only u N 4- = 5, which is much smaller than u N 6= 4%= 16, and moreover that none of the readings fall outside Z f 16. A naive experimenter would usually choose the counter X as superior to counter A because of the self-consistency and reproducibility of its readings. These are false clues. Variability comparable with or even greater than that shown by counter A must be exhibited by a reliable instrument operating on a random process. When a small sample is thought to have come from a binomial population, the binomial index of dispersion is analogous t o Eq. (5.4.4) except that in accord with Eq. (5.2.5a) the population variance estimator 2 is replaced by 2(1 - p ) , so that i.= n..

xz =

2

i-i

(Xi

2(1

- 2)'

- p)

(5.4.4a)

5.4.

791

TESTS FOR GOODNESS OF FIT

has the x2 distribution for F = n - 1 if xI has a binomial distribution with mean p z and variance pz(1 - p ) N 2(1 - p ) . 5.4.3. Confidence Interval for the Standard Deviation of a Normal Distribution

In some nuclear experiments we wish t o determine and make practical use of the standard deviation of a distribution. We recognize that our inability to measure exactly the true mean m gives rise to our everyday use of the standard deviation, and the standard error, as an estimate of our uncertainty in the mean value. But if our experimental goal is the measurement of the standard deviation, then we realize that there must also be a standard deviation of the standard deviation of the distribution. The problem arises particularly when the parent distribution is normal, rather than Poisson, for then u is a free-floating parameter which is totally unrelated to m. The x2 distribution leads directly to the estimation of confidence intervals for the standard deviation u of a normal population. It can be shown, as would be expected from the form of Eqs. (5.4.4) and (5.4.4a), that if x is normally distributed the ratio nS2/u2has a x2 distribution with F = n - 1 degrees of freedom, where S2 is the sample variance among n measurements, as given by Eq. (5.2.6), and u2 is the true variance of the normal distribution. Then, with F = n - 1, i=n

(5.4.5) i= 1

It follows that if x~.gsis the P = 0.98 value, and xi.02is the P = 0.02 value of x2 for F = n - 1, then in 96% of the cases tested, nS2/u2will lie within the interval (5.4.6) which, on rearrangement gives the 96% confidence interval for u2 as (5.4.7) Obviously any other confidence intervals can be selected by suitable choice of the x2 values in Eq. (5.4.6). As an illustration, suppose that from an aqueous radioactive source a number of supposedly equal portions have been prepared by pipetting, and are to be distributed among many laboratories as interlaboratory reference standards. It is desired to test the uniformity among n of these

792

5.

STATISTICAL FLUCTUATIONS

portions and from these measurements to quantify the inhomogeneity of the standards. From Eq. (5.4.7) we can say, with 95% confidence, that the true standard deviation u is not greater than that given by

s

lin

~ > XO.96

L

>

O

.

(5.4.8)

Then if precise measurements of a set of n = 5 standards show a sample standard deviation of S = 0.6% we can say that in 95% of such cases the true standard deviation, u, should not be greater than

On the other hand, if n = 30 standards give the same S = 0.6%) then we can put a much lower limit on u and can say with 95 % confidence that u 5 0.6 -\/30/17.7= 0.8%.

5.5. Applications of Poisson Statistics to Som Instruments Used in Nuclear Physics’

Most of the primary processes which are measured in a nuclear physics laboratory follow a Poisson distribution, or its counterpart, the interval distribution. The binomial and the normal distribution are met somewhat less frequently. Some of the more common questions of experimental design and of statistical treatment of data can be put into standard forms ready for routine use on Poisson distributions.

5.5.1. Effects of Resolving l i m e in Single-Channel Counting The resolving time p of a counter channel causes the very short intervals to be missing in the output. All counting systems are really interval counters, not particle counters. Paralyzable (Type I) counters are unable to provide a second output pulse until there is a time interval of at least p between two successive true events. Then if a is the true rate, the observed counting rate A will be or

+--

A = ae-’p ‘v a(1 - ap .) a N A(l Ap) when ap 2), the system is not determined completely by giving the angle 8i of any one final particle, and a different expression applies for the Jacobian. For a two-particle reaction, we have

da/d0

=

Jl(da/dH)

(21)

where da/dfl and da/dH are the differential cross sections in the laboratory system and the c.m. system, respectively; J 1 is the appropriate Jacobian. For a reaction involving more than two particles, we have

d2a/dE d0

=

Jz(d2a/dB d o )

(22)

where d2a/dE d0 and d 2 c / d 8 dH are the differential cross sections (with respect to both energy and angle) in the laboratory system and the c.m. system respectively, Jz is the Jacobian. We shall first consider the case of only two particles. The Jacobian J 1 is then given by J1 = (d0/dH)-' (23) where dH = sin 8[d8l = Id cos 81 and dQ = sin Ode = Id cos 01 are the elements of solid angle in the c.m. system, and the laboratory system, respectively, for the particle considered. Here and in the following, we omit the subscript i [see Eqs. (15)-(20)], with the understanding that the formula for J 1 always refers to a particular final particle. From Eqs. (15)-(20), one finds cos

where p and

A

e

=

A - ~ C OfiS+ p )

(24)

are defined by

p =

B

v/z,

=I

+ 2p cos 8 +

p2

- V2 sin2 8.

(25) (26)

Here B is the velocity of the particle in the c.m. system. Upon differ-

826

APPENDIX

2

entiating both sides of Eq. (24) with respect to cos 8, one obtains

an - 11 + p cos el _ ail - y v 2 A a ~ 2 Hence from Eq. (23) ,TI

=

+

yv2Aa’211 p cos

el-’.

Equation (28) gives J 1 in terms of the center-of-mass quantities fi and However, in some applications, it is more convenient to obtain Jl from the velocity v and angle 8 of the particle in the laboratory system. In order to derive the expression for J 1 in terms of v and 8, we define the quantities p and A as follows:

e.

p = v/v A = 1 - 2p cos 8

+ p 2 - V2 sin2 8.

(29) (30)

From Eqs. (6)-(€9, one obtains cos

e = A - ~ ~ ~ 8c o-sp ) .

(31)

Upon expressing 8 in terms of v and e, one finds

p = P~--1/2(i- vvcos

el.

Substitution of Eqs. (31) and (32) in Eq. (26) for

A

(32)

A gives

=I/(~~~A).

(33)

Finally, upon inserting Eqs. (31)-(33) into Eq. (28), one obtains

J~ = y v - 2 ~ - 1 q i - cos 81-1

(34)

which gives J 1 in terms of the laboratory quantities v and e for the case that only two particles are present in the final state. We note that in Eq. (27), we have evaluateddOldil, and have then taken its reciprocal to obtain J1. This derivative is particularly simple, since the energy E does not depend on the angle 8 in the c.m. system. We could also have obtained dil/dQ = J1 directly. However, since E is a function of 0 in the laboratory system, dil/dQ is given by

where ail/an and ail/aE are the partial derivatives for fixed E and e, respectively. Equations (28) and (34) give the relation between the cross section du/dQ for a two-particle process in the laboratory system and the cor-

APPENDIX

2

827

responding cross section du/dn in the c.m. system [see Eq. (21)]. The equations for J 1 can also be used for the two-body decay of unstable strange particles ( K mesons or hyperons). JI gives the relative number of decays per unit solid angle in the laboratory system, assuming isotropic decay in the rest system of the particle. In this case, 6 is the laboratory angle of one of the decay products with respect to the direction of the primary particle. For the case that more than two particles are present in the final state, J is given by the following determinant I

ail

aQ I

As for the two-body case, it is convenient to obtain J Z by evaluating its reciprocal:

1 an

m/aQ is given by Eq.

an I

(27). For the other derivatives, one obtains from

Eqs. (19) and (24) :

Upon inserting Eqs. (27) and (38)-(40) into Eq. (37), one obtains

which gives J z in terms of the center-of-mass quantities Z, and 8. From Eqs. (18) and (20), one can also show that Eq. (41) is equivalent to:

J z = sin 8/sin e

=

p/p.

(42)

The result sin J/sin fl has been given by Marshak.’ In order to obtain J Z in terms of the laboratory quantities v and 8, we substitute Eq. (33) into 1

R. E. Marshak, “Meson Physics,” p. 300. McGraw-Hill, New York, 1952.

828

APPENDIX

2

Eq. (41), getting J2

= yy1A-1/2.

(43)

Equations (41)-(43) give the relation between the differential cross section in the laboratory system d2a/dE dQ and the corresponding cross section in the c.m. system, d 2 a / d 8 d n [see Eq. (22)J.The equations for JZcan also be used to obtain the laboratory angle and energy distribution of secondary particles arising from the three-body decay of an unstable particle (e.g., K meson). Thus if d 2 n / d 8 d n is the probability of decay per unit energy and angle in the rest system for one of the decay products, the corresponding distribution in the laboratory system d2n/dE dQ is given by 2n d2n -d- J2-e (44) dE dQ dE dQ I n particular, for a n isotropic distribution in the rest system of the unstable particle, d 2 n / d 8 d o is independent of the angle t7 and is a function of 8 only. We summarize the preceding results: the Jacobian J is given by Eqs. (28) and (34) for the case of two particles in the final state, and by Eqs. (41)-(43) for the case of more than two particles. In these equations, p , 6, p, and A are defined by Eqs. (25), (26), (29), and (30), respectively. 2.2. Maximum Angle Criterion for the Identification of Outcoming Particles in Fundamental Collisions

For nucleon-nucleon and pion-nucleon collisions in which mesons are produced, it can be shown2 that the angle e of the recoil nucleon cannot exceed a maximum value emwhich depends on the energy of the incident nucleon or pion and on the number of mesons produced. Thus if the angle 0 for a particle observed, for example, in a cloud chamber or bubble chamber, is larger than the value of Om for the recoil nucleon, i t can be concluded that the particle cannot be a nucleon and must therefore be a pion or a K meson. There is a maximum possible angle 0, for any emitted particle (e.g., for pions) at energies sufficiently near threshold. However, for pions, em becomes 180" a t an energy not far above threshold, when the c.m. velocity ij of the emitted pion exceeds the velocity V of the c.m. system with respect to the laboratory system. In order to obtain the expression for Oml we note that the c.m. velocity tl of the particle considered will have its maximum value when the other particles (2, 3, . . . n) all move with the same velocity2 and in the direction opposite to Thus particles 2, 3, . . . n are equivalent R. M. Sternheimer, Phys. Rev. 93, 642 (1954).

2

829

irni

(45 1

APPENDIX

to a single particle of mass M , given by

M =

i=2

where mi is the mass of the ith final particle. From the definition of the c.m. system, the momentum of M is -plm, where plm is the (maximum) momentum of 1. Thus plmcan be obtained from the equation:

(pi,

+ m12)1/2+ (pl, + M2)'/2

= 2 0

(46)

where EOis the total energy in the c.m. system. Upon solving Eq. (46) for plm,one finds plm = B'/2/(2E') (47) where B is defined b y

+

B = [(b, M ) 2 -

rn12][(Eo -

- m12].

(48)

Hence the maximum possible velocity D1, of particle 1 is given b y

nl,

=

+

[~/(4~3'2rn~2~ ) ] 1 / 2 .

(49)

From Eq. (20), one finds for the laboratory angle O1 of the particle:

el

where is the angle of emission of the particle in the c.m. system. The maximum value of el is obtained from the condition

d tan Ol/d&

=

0

which gives cos

el

=

-nlm/V.

Upon inserting Eq. (52) into Eq. (50), one obtains the maximum angle

elm: where nlm is given b y Eq. (49). Figures 2 and 3 show the curves of elmfor recoil nucleons from nucleonnucleon and pion-nucleon collisions, when n = 1, 2, or 3 pions are produced. The curves are plotted as a function of the kinetic energy of the incident nucleon or pion in the laboratory system. For nucleon-nucleon collisions, Oln, increases rather slowly with the incident nucleon energy

830

APPENDIX

2 I

I

-

-

-

I

0

0.5

I .o I.5 2 .o 2.5 INCIDENT NUCLEON’ENERGY T~ (IN Bevl

I

3.0

FIQ.2. Maximum possible laboratory angle el,,, of the recoil nucleon in a nucleonnucleon collision in which n ( = 1, 2, or 3) pions are produced, as a function of the laboratory kinetic energy T N of the incident nucleon.

APPENDIX

2

83 1

TN beyond 3 Bev. Thus at TN = 6.0 Bev, BIm is 77.2" for n = 1, 66.0" for n = 3, and 56.8" for production of n = 5 pions. It may be noted that Eq. (53) is completely general, and can also be used, for example, to obtain the maximum angle of hyperons or K mesons in associated production. The existence of the maximum velocity 81, implies that the laboratory momentum p l of the particle cannot exceed a certain maximum value (to be denoted by pl,) which is a function of el. In order to obtain plm,we note that by virtue of Eq. (8) we have Elm

+

= ~v[(pL,

- V p l m cos ell

m12)1/2

(54)

where Elmis the maximum energy in the c.m. system. From Eq. (49) one finds that 81, is given by 8 1 ,

=

+

( 4 8 , ~ r n ~ 2 B)1/2/(2170).

(55)

Upon solving Eq. (54) for PI,, one obtains

Thus if the measured momentum of a particle exceeds the value of plmfor recoil nucleons and K mesons at the angle el, the particle can be identified with certainty as a pion. Equation (56) can also be used to obtain the laboratory momentum of a particle in a two-body reaction as a function of the laboratory angle 81. In this case, Elmis the c.m. total energy of the particle as obtained from Eqs. (48) and (55). However, in Eq. (48), M is now the mass of the other particle resulting from the collision (i.e., A4 = m2),ml is the mass of the particle considered, and Bo is the total energy in the c.m. system.

2.3. Kinematics of Two-Body Decay For the study of the two-body decay of unstable strange particles, it is useful t o obtain the distribution of the energies El and E2 of the decay products in the laboratory system. We have

El

=

+

~ ~ ( 8T'P1 1 cos $1)

(57)

where El and pl are the total energy and momentum of the decay particle (secondary) in the rest system of the primary; is the angle of emission in the rest system, T' is the velocity of the primary, and y v = (1 - V2)-1/2. [The rest system is that in which the decaying particle (primary) is at

el

832

APPENDIX

2

rest. This coordinate system is completely analogous to the center-of-mass system considered previously. Thus the total momentum of the decay products in the rest system is zero, in similarity to the c.m. system for LL particle reaction.] The maximum and minimum values of the laboratory total energy El are given by

The number of decays dn/dEl per unit energy interval dE1 is dn __---. -

dE1

dn d cos

d cos Sl dE1

el

(59)

On the assumption that the decay of the primary particle is isotropic in its rest system, dn/dcos& is constant (= &). Furthermore, from Eq. (57), one finds

One thus obtains

Hence the energy distribution dn/dEl is constant (i.e., independent of E l ) between E1,min and The distribution of the angle a between the two decay particles in the laboratory system has been obtained by Sternheimer.s One finds for the relative number of decays per interval da:

where C is defined as:

C = +(mo2- mI2 - m2 ).

(63)

Here mo and po are the mass and momentum of the decaying particle, mi and E; are the mass and laboratory total energy of the decay products (i = 1, 2); p1 is the momentum of either particle in the rest system. It may be noted that a is related to El and E 2 by the equation: cos a

=

ElE2 - C (E l 2- m12)1/z(E22 - m22)1/2

R. M. Sternheimer, Phys. Rev. 98, 205 (1955).

(64)

APPENDIX

833

2

The application of the kinematics of two-body decay t o the interpretation of cloud chamber pictures of the V particles ( K mesons and hyperons) has been described by Bridge4 and by Thompson.6 The relation between the energy and angular distribution of primaries produced in a reaction and the energy and angular distribution of secondaries resulting from a two-body decay of the primaries has been discussed by Sternheimer.6

2.4. Threshold Energies for Associated Production of Unstable Particles In this sect,ion, we obtain the energy thresholds for associated production of unstable particles in nucleon-nucleon, pion-nucleon, and y-nucleon collisions. The following basic reactions will be mainly considered :

+ + + + + + +

N + N-l Y K + N N N - Kf K2N ?r+N+Y+K ?r N-+ K f KN y + N - + Y + K y N-+ K f KN

+ + +

where N = nucleon, Y = hyperon. We will consider both the case where the target nucleon is at rest and the case where it moves towards the incident particle with a kinetic energy of 25 MeV. As explained below, the latter assumption gives an approximate effective threshold for associated production in a collision with a nucleus. The laboratory kinetic energies of the initial particles will be called T , and Tb; their masses are denoted by ma and mb. I n order t o obtain the expression for the threshold value of T,, we write the equations for th e total energy Eo and momentum pa in the laboratory system:

+

Eo = ma -k mb T , 4-Tb P O = (Ta2 2m,T,)ll2 - (Ti,’

+

+ 2mbTb)’‘2.

(65) (66)

In Eq. (66), it has been assumed that the target particle (usually a nucleon) moves backward in the laboratory system, i.e., towards the incident particle. The required total energy Bo in the c.m. system, which is determined by the combined mass of the reaction products, is given by 3 0

= Eo(1 - V2)”’

(67)

4 H. S. Bridge, in “Progress in Cosmic Ray Physics” (J. G. Wilson, ed.), Vol. 111, Chap. 11, pp. 209-218. Interscience, New York, 1956. 5R. W. Thompson, in “Progress in Cosmic Ray Physics” (J. G. Wilson, ed.), Vol. 111, Chap. 111, pp. 266-280. Interscience, New York, 1956. 8 R. M. Sternheimer, Phys. Rev. 99, 277 (1955).

834

APPENDIX

2

where V = PO/&. From Eqs. (65)-(67), one obtains E02

-I- 2[m,Tt, -I- mbTa -I- TaTb -k ( T G 2 2maTa)"2(Tb2 2mbTb)1/2].(68)

= (ma

Upon solving Eq. (68) for T,, one finds

[GEb - 2mamb2- pb(G2- 4ma2mb2)1/2]/(2mb2) (69) where G is defined by G = Eo2 - ma2- mb2.

T,

=

In Eq. (69), p b and Eb are the momentum and total energy of the target particle. For a target particle at rest, Eq. (69) reduces to the well-known formula :' T , = [Eoz- (ma mb)21/(2mb). (71)

+

In both (70) and (71), 80is, of course, equal to the sum of the masses of the final reaction products of (1)-(VI). Thresholds for unstable particle production by both nucleons and pions have been previously obtained.8 We have recalculated these results, using somewhat more recent mass values for the various particles. The values used weregJO(in Bev/c2); r:0.1395;K:0.4935;N:0.938;A': 1.115; 2+:1.189; 2-:1.197; %-:1.321; and 2:1.615, where 2 is the hyperon observed by Eisenberg," with Q value of -6 MeV. It has been assumed that this particle decays into a A' and a K meson and that it has strangeness12 S = - 3 . The results are given in Table I. This table is divided into 3 parts. Parts A, B, and C give the thresholds for production by nucleons, pions, and y-rays, respectively, for the reactions listed. In each case, we give two results: ( 1 ) for target nucleon at rest (Tb = 0 ) ;(2) for target nucleon of 25 Mev moving towards the incident particle (Tb = 25 Mev). The reason for choosing Tb = 25 Mev is that the nucleons in the nucleus have kinetic energies of the order of 25 Mev or less. The maximum energy is ~ 2 Mev 5 and is often referred to as the Fermi energy. For a target 7 See, for example, R. E. Marshak, "Meson Physics," p. 73. McGraw-Hill, New York, 1952. 8 R. M. Sternheimer and D. H. Wilkinson, Brookhaven Cosmotron Internal Report RMS-DHW 1 (1955). 9 A. M. Shapiro, Revs. Modern Phy8. 28, 164 (1956). 'OR.Budde, M. Chretien, J. Leitner, N. P. Samios, M. Schwartz, and J. Steinberger, Phys. Rev. 108, 1827 (1956). 11Y. Eisenberg, Phys. Rev. 96, 541 (1954). See also W. F. Fry, J. Schneps, and M. S. Swami, ibid. 97, 1189 (1955). I* M. Gell-Mann, Phys. Rev. 99, 833 (1953); T. Nakano and K. Nishijima, Progr. Theoret. Phys. (Kyoto) 10, 581 (1963); A. Pais, Phy8iCa 19, 869 (1953).

APPENDIX

835

2

nucleon with the maximum energy (-25 Mev) and moving towards the incident particle, the threshold for the reaction has the lowest value. It is seen that the thresholds for Tb = 25 Mev are markedly lower than those for Tb = 0, so that the thresholds for production in a nucleus are appreciably decreased as compared to production on a free proton (liquid hydrogen target). TABLE I. Thresholds for Unstable Particle Production (inBev) Reaction (A) N + N - + A " + K + N N -+ Z+ K N N

+

N N N N

+ + + N Z- + K + N +N-+ K + + K - + 2 N + N - + E + 2K + N 4-N - + Z + 3 K + N -+

(B) H + N - + A " a N - + Zf a N 4 2H N + K+ H N -+ E: H N-+ Z

+ + + + +

+K

+K +K + K- + N

+ 2K + 3K

(0 Y + N + A " + K

+N + y +N

+K 2- + K y + N - + K+ + K - + N y

Z+

+

TS= 0

Tb

=

25 Mev

1.581 1.784 1.807 2.493 3.740 6.796

1.105 1.263 1.281 1.818 2.800 5.218

0.760 0.890 0.904 1.356 2.221 4.489

0.578 0.680 0.692 1.050 1.736 3.537

0.910 1.040 1.054 1.506

0.723 0.826 0.837 1.196

The strange particles are generally produced by means of incident nucleons or pions. However, recently both at the Cornell and Cal. Tech. electron synchrotrons, K particles have been made by means of an energetic ?-ray beam (bremsstrahlung from the electron beam). The thresholds for production by incident y-rays are listed in part C of Table I.

EXAMPLES: As an application of the equations discussed above, we consider the reaction : p p - + x+ K+ n (VW

+

+ +

for incident protons of kinetic energy T , = 3.0 Bev. For the kinetic energy of the target proton, we assume the two possibilities considered above: (1) Tb = 0 ; (2) Tb = 25 Mev (moving backward in the laboratory system). We want t o obtain the maximum possible c.m. velocity iiz,,,, of

836

APPENDIX

2

the 2+ particle, and the corresponding laboratory energy of the 2+, if it has been emitted with the maximum velocity at c.m. angles = 0' and 60'. In the latter case, we also obtain the laboratory angle e2 of the Z+. Table I1 gives the values of the various kinematic quantities. The quantities in the laboratory system pertaining to 82 = 0" and 8, = 60" are labeled by the superscripts (1) and (a), respectively. First, the total momentum p o and the total energy Eo are obtained, together with the TABLE 11. Kinematic Quantities for the Reaction p

+p

3

Z+

+ K+ + n

for 3.0-Bev Incident Protons

PO (Bev/c) E O (Bev)

v

?"

E D(Bev) B (Bev4) o2.m

Ex,.,, (Bev) P Z , ~(Bev/c)

E',L' (Bev) E g ) (Bev) P!?~ (Bev/c) p!$,'f (Bev/c)

$g)

ezSm

3.825 4.876 0.7845 1.612 3.025 20.69 0.5344 1.407 0.752 3.219 2.744 2.385 0.651 15.3" 30.0"

3.607 4.901 0.7360 1.477 3.318 45.27 0.6488 1.563 1.014 3.411 2.860 2.448 0.878 19.7" 51.7'

velocity ' I of the c.m. system and the corresponding value of yv [Eqs. (1)-(5)]. From these results, one can deduce the total energy Eo in the c.m. system. For the 2+ to have its maximum c.m. velocity B Z , ~ ,it is necessary that the K+ particle and neutron move with the same velocity and in the same direction (opposite to tz,,). The sum of the masses of K+ and n is: M = 1.433 Bev. Upon inserting this value in Eq. (48), one , Ex,,,, from Eqs. (47), (49), and obtains B , and hence f i ~ , ~P, Z . ~and (55). If the 2+ is produced at & = 0", the laboratory total energy is EL1) = 3.219 Bev for Tb = 0 and E$) = 3.411 Bev for Tb = 25 Mev (corresponding to kinetic energies T$' = 2.030 Bev and 2.222 Bev, respectively). For the case 82 = 60', we have obtained EL2) together with the components of the momentum &,: and p;:: from Eqs. (17)-(19). From the values of and p;::, one deduces that the laboratory angle OL2' is 15.3' for Tb = 0 and 19.7' for T b = 25 MeV. The last row of Table I1 gives the maximum possible laboratory angle [see Eq. (53)].

837

2

APPENDIX

As a second example, we have considered the reaction T-

+p

4

A'

+ K"

(VIII)

at an incident pion kinetic energy of 1.0 Bev. It is assumed that the K O meson is produced at a c.m. angle eR = 60°, and we wish to obtain the energy and angle of the K" in the laboratory system, and the Jacobian J K at this angle. The intermediate quantities and the final results are listed in Table 111, both for Tb = 0 and Tb = 25 MeV. We note that the TABLE 111. Kinematic Quantities for K" Mesons Produced at e ' ~= 60° by the Reaction Tp -+ A" K" for 1.0-Bev Incident Pions

+

Quantity

PO (Bev/c) Eo (Bev) V 7-v

Eo (Bev) B (Bev4)

fi3 E K (Bev) PK

(Bev/c)

E K (Bev) P K . ~(Bev/c)

PK,, (Bev/c) OK

p_K AK JK

+

Ta

=0

1.131 2.078 0.5443 1.192 1.743 1.196 0.5365 0.585 0.314 0.799 0.567 0.272 25.6" I .015 2.823 4.471

Tb = 25 Mev 0.913 2.103 0.4341 1.110 1.895 3.217 0.6922 0.684 0.473 0.873 0.592 0.410 34.7" 0.627 1.879 2.416

c.m. velocity D K can be obtained by means of Eq. (48)for B, in which m l = VZK = 0.4935 Bev and M = m~ = 1.115 Bev. The values of E K ,OR, and J E in Table I11 pertain to 8 K = 60". An extensive discussion of the kinematics of A meson production by incident nucleons and y-rays (photoproduction) on nucleons has been given by Marshak.13 The effect of the Fermi motion on the threshold for pion production in nuclei14 and the absolute threshold for pion production16 are also discussed in Marshak's book.' Recently, LeipuneP has calculated extensive tables of the kinematics of two-body reactions and two-body decays. The two-body decays conR. E. Marshak, "Meson Physics," pp. 3 and 41. McGraw-Hill, New York, 1952. W. McMillan and E. Teller, Phys. Rev. 73, 1 (1947). 16 W. H. Barkas, Phys. Rev. 76, 1109 (1949). l6 L. B. Leipuner, "Relativistic Two-Body Kinematics," Brookhaven Cosmotron Internal Report LBL-2 (1958). l3

838

APPENDIX

2

+

+

sidered in this work include the following: 0' + n+ n-; A' --t p n-; 2- + n r - ; Z0 + ha y ; and Z+ Ao n-. Among the 14 two-body reactions investigated, five are of the type of reactions (111) and (V) p 4A' K + ; T+ p 3 2+ Kf. Four additional above, e.g., y reactions involve the O2 particle (a long-lived species of the neutral K meson which can exhibit strangeness12 S = - l), e.g., O2 p A' p; 02 p + E" Kf. Leipuner has also carried out calculations for the elastic K nucleon scattering. The remaining 4 reactions do not involve the strange particles. These reactions are: y p 4p r'; y p 4n n+; r* p + ~ * p ; and p p + d +A+. For each case, for a given incident laboratory energy (for a reaction) or kinetic energy of the parent particle (for a two-body decay), Leipuner's tables" give the following quantities as a function of the c.m. angle of the heavier of the two outgoing particles: the kinetic energies TI,Tz,the momenta p l , p 2 , the velocities vl, v2, and the angles el, 02 of the two outgoing particles (all quantities in the laboratory system). Moreover, the tables list the c.m. momentum PI ( = p z ) of either secondary particle, the velocity V of the c.m. system, the corresponding yv = (1 - V2)-lI2, and the c.m.'velocities gl and g2 of the two secondary (outgoing) particles. For the exothermic reactions involving a threshold, the threshold energy and threshold momentum are also listed. Far the two-body reactions, the tables generally extend from incident energy T, = 50 Mev (or from threshold) to T, = 2.5 Bev, at intervals of 50 MeV. For the particle decays, the range of the calculations extends from p = 50 Mev/c (momentum of the parent particle) to p = 2.5 Bev/c, at intervals of 50 Mev/c. These tables were calculated on an IBM-704 Computer.

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

el

2.5. Phase Space Factors for Two- and Three-Particle Final States It may be convenient to list here the expressions for the relativistic two-body and three-body phase space factors. These phase space factors F2 and F J are related to the density of final states per unit total energy 8, in the c.m. system. F1 and F, have the significance that if the matrix element for the formation of the particles in the final state were constant (i.e., independent of energy and angle), then F 2 and Fs would give the relative probability of various final states. For more than 3 particles in the final state, it has not been possible t o obtain an analytic expression for the phase space factor." The expression for Fz is given by

17

M. M. Block, Phy.9. Reu. 101, 796 (1956); and private communication.

APPENDIX

2

839

where El and Bz are the c.m. total energies of the two particles, 30is the total energy in the c.m. system (3, = 8, BZ),and p1 is the c.m. momentum of either particle. In order t o derive Eq. (72), we note that F 2 is the density of final states (p) per unit solid angle (in the c.m. system), and is given by

+

We have

so that

Upon inserting Eq. (75) into Eq. (73), one obtains Eq. (72) for F P . The relativistic three-body phase space factor F a has been obtained by Block,” and is given by

where D is defined by

Here pl, PI, B,, and d o l are, respectively, the c.m. momentum, kinetic energy, total energy, and solid angle of the particle considered; m2 and ma are the masses of the other two particles, 8 0 is the total energy in the c.m. system. The following derivation of Eq. (76) has been obtained by Block.“ It follows essentially the same lines as that given in reference 1 for the nonrelativistic case.lS The density ‘of final states is

where pz and dQz are the c.m. momentum and solid angle of particle 2. The c.m. momentum and total energy of particle 3 will be denoted by R. E. Marshak, “Meson Physics,” p. 81. McGraw-Hill, New York, 1952.

840

APPENDIX

2

p 3 and 8,, respectively. We have

+

p2

-pl

Upon taking the direction of

pa2 =

p3

=

(79)

-p1.

as the polar axis, we can write

- 152 cos

&)z

( ~ 1

+

~

2

sin2 2 ez

(80)

where ez is the angle between pz and -pl. Differentiation of both sides of Eq. (80) gives Padp3 = - ? j l p z d ( C O S 82). . (81) For fixed p1 and p2, the differential d 8 0 in the denominator of Eq. (78) is equal to dEa, which is given by

d83

dpa/E3.

=

(82)

From Eqs. (81) and (82), one obtains d(cos

82)

-

d8o

8 , can be written as 8 3

=

--. 8

3

p1pz

w - (m22+

p22)1/2

where W is the energy available to particles 2 and 3:

w=

8 0

- 8,.

(85)

Upon inserting Eqs. (83) and (84) into Eq. (78) one obtains

dp

=

27rp1dpl d%

/

Pl,l&¶X

P9.min

[W - (m2

+

p22)112]lj2

dpz

(86)

where the factor 2a comes from the integration over the azimuthal angle of particle 2; and p 2 , m i n are the maximum and minimum values of p2, which are obtained for & = 0’ and 180°, respectively. The corresponding values of p 3 are p2 - p1 and p 2 p1, respectively. Thus Ij 2,max and p2,min are obtained by solving the equation

+

where D is given by (77). In Eq. (86), we note that p1 dpl thus obtains

=

8, d 8 1 . One

APPENDIX

where

and have

E2,mox

p2,min.We

E2,min are the values

2

of

8,corresponding

Upon inserting Eqs. (88) and (90) into Eq. (89), one obtains Eq. (76) for the three-body phase space factor. For a reaction with 3 final particles, in which the matrix element for the formation of particle 1 is Hl(pI, gl) (after integration over the variables pertaining t o particles 2 and 3), the center-of-mass differential cross section is given b y d2a 27r IH1\2Fs (91) dF1da, ft uv where ijl is the relative velocity of the initial particles in the c.m. system. The total phase space available t o the three particles, analogous to 47rFz for the two-body case, is given by

where is the maximum possible c.m. kinetic energy of particle 1 [see Eq. (55)]. Examples : As an illustration of the three-body phase space factor F S , we have considered three examples, which are presented in Figs. 4-6. Figure 4 shows the phase space factor F for the K+ mesons from the reactions: p p

+ p - - t Ao + K+ + p + p + Z+ + K+ + n

for a n incident proton kinetic energy T , = 3.0 Bev, and a target proton at rest in the laboratory system. In this case, in Eq. (76), m2 and m3 are the masses of the A' and proton for reaction (IX), and the Zf and neutron masses for reaction (X). F is plotted against the c.m. kinetic energy FK of the K meson. If the matrix element for the K meson production were constant as a function of PK,then F would give directly the energy distribution of the K mesons in the c.m. system. I n Fig. 4, and also in Figs. 5 and 6, the phase space factor has been normalized as follows:

/o''"FdP

=

1

(93)

842

APPENDIX

2

C.MS. ENERGY OF K MESON ( I N MeV) FIG.4. The relativistic three-body phase space factor F for the K+ meson from the

+

+

+

+

+

+

p -+ A" K + p and p p --t 2+ K+ n,for an incident proton reactions: p kinetic energy T, = 3.0 Bev and a target proton at rest in the laboratory system. F is plotted against the c.m. kinetic energy of the K + meson.

9

I

I

I

I

I

I

I

I

LL1 0I0

I

I

1

F (N)

50

100

150

200 250 CMS. ENERGY OF NUCLEON OR PION ( I N MeV)

300

FIG.5. The relativistic three-body phase space factor F for the pion and the nucleons p .+ p n ?r+, for an incident proton kinetic energy from the reaction: p T p = 1.0 Bev and a target proton a t rest in the laboratory system. The abscissa represents the kinetic energy of the pion or nucleon in the c.m. system.

+

+ +

APPENDIX

2

843

where f',,,is the maximum possible value of the c.m. kinetic energy T', and is expressed in Bev. Figure 5 shows the phase space factor F for the pion and the nucleons from the reaction: p p -+ p 7E n+ (XI)

+

+ +

for an incident proton kinetic energy T, = 1.0 Bev and a target proton at rest in the laboratory system. The pion phase space factor F(r+) is obtained from Eq. (76)by taking m2 = m s = mN, whereas the nucleon phase space factor F ( N ) is obtained by taking m2 = mN and m3 = m,,

ENERGY OF p MESON OR ELECTRON ( I N MeV)

FIG.6. The relativistic three-body phase space factor F for the p meson from the K$ decay, and the electron from the K,f, decay. The abscissa represents t h e kinetic energy of the p meson or electron in the rest system of the K particle.

where mN and m, are the nucIeon and the pion mass, respectively. It may be noted that according t o the Fermi statistical theory of pion production,l@the energy distribution of the pions and nucleons is given by the corresponding phase space factors F(T+) and F ( N ) , the assumption being made that the matrix element for pion production is constant aa a function of the pion and nucleon kinetic energies. Figure 6 shows the phase space factors for the p meson from the K:* decay : K:3-+p*+no+ v (XI11 E. Fermi, Progr. Theoret. Phya. (Kyoto) 6, 570 (1950); Phy8. Rev. 92, 452 (1953); SS, 1435 (1954). l9

844

APPENDIX

2

and for the electron from the K t 3 decay:

K:3

-+

e*

+ + 7r0

V.

(XIII)

The K:3 and K:3 are relatively infrequent modes of decay of the K meson, which generally decays into two particles. (The most frequent modes of decay are: Ks2 + 2 ~ and , K$ -+ ,u* F.) For K:3 and K:3J we have: BO= mKJm2 = mz, ma = 0 in Eq. (76)) where mg and mr0are the K and T O masses, respectively. The phase space factor F is shown as a function of the p meson or electron kinetic energy in the rest system of the K particle. Phase space calculations with applications to Fermi's statistical theory of meson produ~tion'~ have been carried out by several a ~ t h o r s . ' 7 , ~ ~ - ~ 6

+

ACBNOWLED~MENT

I wish to thank Dr. Luke C. L. Yuan for several helpful suggestions concerning this chapter. C. N. Yang and R. Christian, Brookhaven Cosmotron Internal Report (1953). J. V. Lepore and R. N. Stuart, Phys. Rev. 94, 1724 (1954). zz R. H. Milburn, Revs. Modern Phys. 27, 1 (1955). * 3 J. V. Lepore and M. Neuman, Phys. Rev. 98, 1484 (1955). 24 M. M. Block, E. M. Harth, and R. M. Sternheimer, Phys. Reu. 100, 324 (1955). 26 G. E. Fialho, Phys. Rev. 106, 328 (1957). 41

Appendix 3. Properties of Elementary Particles and Particle Resonance States TABLE I. Intrinsic Properties of Elementary Particles@*b ~

Class Photon Leptons

AntiSpin Isotopic Particle particle (unitsh) spin Z

I,

Strangeness S

Mass

K+

Mass (Mev) 0 0 0 0.510976 4 0.000007 105.655 0.010 135.00 f 0 . 0 5 139.59 f 0.05 493.9 -1- 0.2

0 0 0 1 206.77 264.20 273.18 966.6

KO

497.8 4 0 . 6

974.2

Nucleons

p n

Hyperons

Ao

938.213 k 0.01 939.507 f 0.01 1115.36 k 0.14

1836.12 1838.65 2182.80

Stable (1.013 f 0.029) X 103 (2.51 k 0.09) X 10-10

Zf

1189.4

0.2

2327.7

(0.81 +0.06) -0.05

zo

1191.5 k 0 . 5

2331.8