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Advances Electronics and Electron Physics. Vol. XI

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME XI This Page Intentionally Left Blank Advances in Electronics a

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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME XI

This Page Intentionally Left Blank

Advances in

Electronics and Electron Physics EDITED BY L. MARTON National Bureau of Standards, Washington, D. C.

Assistant Editor CLAIREMARTON EDITORIAL BOARD

W. B. Nottingham E. R. Piore M. Ponte

T. E. Allibone H. B. G. Casimir L. T. DeVore W. G. Dow A. 0. C. Nier

A. Rose L. P. Smith

VOLUME XI

1969

(29

ACADEMIC PRESS

New York and London

COPYRIGHT 0 1959,

BY

ACAIIEMIC PRESSINC.

ALL EIGHTS RESERVED

X O PART OF THIS BOOK MAY R E REPRODUCEI) I U AXY FORM,

BY PHOTOSTAT, MICROFILV, OR ANY OTHER MEANS,

\\'ITIIOUT WRITTEN PERMISSION FROM THE PI'BLISIIER5.

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

I -ri itad Kingdom Edition Published by ACADEMIC PRESS INC. (LOKDON) LTI). 40 PALLMALL, LONDON S.W. 1

Lthl-al-y of C'ongl-ess ('atalog (lard A'iimbcr 49-7504

PRINTED I N T H E UNITED STATES O F AMERICA

CONTRIBUTORS TO VOLUME XI G. E. BARLOW, Australian Joint Service Staff, Washington, D . C . IT. BRAUER,Institute for Solid State Research, Theoretical Department, German Academy of Sciences, Berlin

1'. GORLICH, Friedrich-Schiller- University, Jena, Germany 0. HACHENBEHG, Heinrich-Hertz-Institute, German Academy of Sciences, Berlin I 90"):

Thus, for i = +1, we have N P / N B - = [ ~ $ . For an unpolarized p--beam, the momentum spectrum N,(x) is obtained by integrating N(z,O) [Eq. (IS)] over dQe,giving N,(Z)C~Z =

dzlN(x,e)da, = 2 ~ y 3- 2 2 ) d ~

The p+ - e+ decay is completely analogous to the p+-decay proceeds by emission of i and v according to p+

---f

e+

+v +V

p-

(23)

- e- decay. The (XI)

The spectrum N(z,O) for the positrons is given by Eq. (lS), in which 0 is the angle between the positron momentum (in the p+ rest system) and the momentum of the p+-meson. Thus, the positron spectrum is also peaked backwards with respect to the p+-meson momentum. The angular distribution NB(0) irrespective of energy is again given by Eq. (20). We shall point out here an important prediction of the two-component neutrino theory concerning the @-decay of unpolarized nuclei, which was obtained independently by Landau (16), Jackson, Treiman, and Wyld (25), Wolfenstein and Page (26),and Curtis and Lewis (97).For a P-decay in which an electron is emitted, the electron has a longitudinal polarization P (spin polarization along the direction of the momentum vector p) of -v,/c, where u, is the velocity of the electron. Thus for high-energy electrons (v, = c), the spin is completely polarized opposite to the direction of motion. For a P-decay where a positron is emitted, the positron polarization P is +v,/c along the direction of the momentum vector, i.e., for a high-energy positron, the spin is completely polarized along the direction of motion. Lee and Yang (28)2have discussed an important concept in connection See also Konopinski and Mahmoud, who were the first to consider a law of conservation of leptons ( 2 8 ~ ) .

PARITY NONCONSERVATION I N WEAK INTERACTIONS

49

with the /3 and meson decays, namely t,he concept of lepton conservation. According to this hypothesis, each of the particles v, V , eft and p* is given a lepton number 1 which is either +1 or -1. For I = fl, the particle is called a lepton; for 1 = -1, the particle is an antilepton. The principle of lepton conservation states that in a particle decay (due to a weak interaction) the total lepton number is conserved, i.e., the lepton number 1 of the decaying particle is equal to the sum of the lepton numbers 1, of the decay products: 1 = I,.

2 i

I n the assignment of lepton numbers, the hyperons, nucleons, K and ?r-mesons,and y-rays are given I = 0; the p-, e-, and v are leptons ( I = +l), while the p+, e+, and fi are antileptons ( 1 = -1). It is easily verified that all the reactions which have been proposed in the preceding discussion [(V), (VI), (VII), (X), and (XI)] satisfy the principle of lepton conservation. On the other hand, the possibilities (VIII) and (IX) for the p-decay (which have been excluded on the basis of the resulting e- spectra) would not satisfy lepton conservation; e.g., for (VIII), l ( p - ) = +1 on the lefthand side, whereas Z(e-) 2I(v) = +3 on the right-hand side. Assuming lepton conservation, we can write down the reactions for various K meson decays, on the assumption that Z(K) 0:

+

K,2+ -+ p+ I(,Z 3pK,,+ -+ p+ Ke3+ e+ ---f

+v

-

+F +v + + v + rro

Of course, the important point in these decays is that lepton conservation enables us to decide whether a v or V is emitted in a particular decay. This has special importance for the two-body decay of K,z*, since the direction of the spin polarization of v or G then determines the direction of the polarization of the p*, in the same manner as for the rr-p-decay (assuming that the spin of the K meson is zero). Thus, for the K,2+ decay at rest, the p+ is polarized antiparallel to its momentum p,+, whereas for the K,2- decay, the p--spin is polarized along the direction of motion, according to (XU) and (XIII). In both cases, the muon polarization is 100%. These predictions have been recently verified by Coombes et al. (29), using Kp2+’s produced in the 6.2-Bev proton beam of the Berkeley Bevatron. OF GARWIN,LEDERMAN, AND WEINRICH VI. THE EXPERIMENT ON THE ? r y e DECAY

Following the success of the experiment of Wu et al. (Z),Gamin et al. (3) carried out a very important experiment, also suggested by Lee and Yang, ( I ) to determine the angular distribution of the positrons from the

50

R. M. STERNHEIMER

p+-decay. The experimental arrangement is schematically shown in Fig. 5. A beam of 85-Mev r+ mesons containing a -10% p+-contamination, obtained from the Columbia Nevis cyclotron, is allowed to impinge on an %in. carbon absorber, which is followed by a thin carbon target. The purpose of the carbon absorber is to stop the pions (the range of an 85-Mev r+ is -5 in. of carbon). As noted above, the beam contains -10% p+-mesons, most of which originate from r+-p+decays occurring in the vicinity of the cyclotron target. The thickness of the carbon absorber was chosen to be

-t7 85-Mev

SHIELDING

T+ BEAM t i o o i 0

pt

COUNTER #I

BON ABSORBER

MAGNETIZING COIL CARBON TARGET MAGNETIC SHIELD ABSORBER

FIG.5. Schematic view of the experimental arrangement of Garwin et al. [R. L. Garwin, L. M. Lederman, and M. Weinrich, Phys. Rev. 106, 1415 (1957)l used to measure the angular distribution of the positrons from the decay of polarized p+ mesons.

8 in., so that a maximum number of the p+, whose range is -8 in., come to rest in the carbon target following the absorber. The passage of a fast p+ through the carbon absorber is indicated by a fast coincidence count of counters No. 1 and No. 2 (see Fig. 5). Around the carbon target, there is a magnetizing coil which produces a vertical magnetic field (i.e., perpendicular to the plane of the figure). If the p+ is longitudinally polarized along its direction of motion, as predicted by the two-component theory, this field H will make the spin precess around H;i.e., the spin vector will rotate through an angle wt, where w = (p/sh)H, with p = magnetic moment of muon, s = spin of p+ = t = time

s;

PARITY NONCONSERVATION I N WEAK INTERACTIONS

51

elapsed between the arrival of the p+ at the carbon target and the p+ decay. The decay positrons are counted in the counter telescope No. 3 4 (see Fig. 5 ) . Between counters No. 3 and No. 4, a n absorber can be interposed in order to exclude positrons with energies below a certain value (e.g., 35 Mev in some of the tests in this experiment). As the magnetic field H is increased, for a given fixed time delay t, the p+ spin will precess by an increasing amount proportional to H . With the angular distribution of Eq. (20), there will be a variation of the counting rate in the No. 3-4 telescope, with maxima and minima depending on the orientation of the p+ spin with respect to the counters. Garwin et al. (3) thus obtained the sinelike dependence of the counting rate on the magnetizing current, which is shown in Fig. 6. The theoretical curve shown in Fig. 6 includes the effect

MAGNETIZING CURRENT (AMPERES)

FIQ.6. The variation of the counting rate in the counter telescope No. 3-4 of Garwin et al. [R. L. Garwin, L. M. Lederman, and M. Weinrich, Phys. Rev. 106, 1415 (1957)l as a function of the current in the magnetizing coil around the carbon target.

of the smearing due to the finite gate width of the counter telescope. (Positrons which are emitted within a time interval from T = 0.75 to 2.0 psec after the p+ stops in the carbon target are counted.) The calculated curve, which is based on Eq. (20) with 4 = +1, is in good agreement with the data, thus providing a strong confirmation of the two-component neutrino theory, which in turn implies nonconservation of both P and C invariance in the r-p-e decay. I n obtaining the amount of precession corresponding to a given magnetizing field H , Garwin et al. (3) assumed that the gyromagnetic ratio of

52

R. M. STERNHEIMER

the p+ meson is 2.00, the same as for the electron. Thus, the agreement obtained implies that the actual gyromagnetic ratio of the p + is 2.00 f 0.10, i.e., the magnetic moment is eh/(m,c) within the experimental uncertainties, where nz, is the p-meson mass. As a check on their experiment, Garwin et al. (3) decreased the thickness of carbon absorber to -5 in., so that most of the T+ mesons stopped in the carbon target inside the magnetic field, instead of the carbon absorber. In this case, no variation of counting rate with field H was observed, and the positron counting rate increased by a factor of -10. These effects would be expected, since pions decaying at rest in the target now provide most of the muons, which are emitted in all directions and therefore give no variation of the positron counting rate as H is increased. For negative muons, Garwin el al. (3) have also detected a backwardpeaked asymmetry, and have verified that the magnetic moment is negative and approximately equal to that of the pf. Following the experiments of Wu et al. ( 2 ) ,Garwin et al. (S), and Friedman and Telegdi (4),a great number of experiments on parity nonconservation have been carried out. In the following discussion, we shall restrict ourselves to experiments on the longitudinal polarization of P-particles from unpolarized nucleis Thus, we shall not discuss the numerous experiments on the asymmetries in T-p-e decay, P-decay of oriented nuclei, Kparticle and hyperon decay, which furnish important information on parity nonconservation in the weak decay interactions. VII. CIRCULARPOLARIZATION OF THE BREMSSTRAHLUNG EMITTED BY LONGITUDINALLY POLARIZED ELECTRONS

A method of demonstrating the longitudinal polarization of electrons emitted in 0-decay consists in measuring the circular polarization of the bremsstrahlung photons which are radiated by the electrons. The circular polarization of the bremsstrahlung was first discussed by McVoy (SO), in connection with the experiment of Goldhaber, Grodzins, and Sunyar (31) on the polarization of the electrons from the 0-decay of YsO. McVoy’s (SO) calculation proceeds in the same general manner as the original Bethe-Heitler calculation (32) for unpolarized electrons, except for the fact that the initial and final electron spin states are not summed over, so that one obtains the dependence of the cross section on the polarization state of the y-ray for a given direction of polarization of the incident electron. McVoy (SO) has carried out this calculation only for the case that the photon is emitted in the forward direction. This case is of greatest

* The survey of the literature pertaining to these experiments on the longitudinal polarization of 8-particles was completed in April, 1958.

PARITY NONCONSERVATION I N WEAK INTERACTIONS

53

interest for relativistic electrons, since the photons are then emitted predominantly a t small angles. The definition of right-circular polarization adopted by McVoy (SO) (and also by other workers in this field) is the opposite of the old definition used in optical work. Thus, if the z axis is along the direction of propagation of the photon (propagation vector k), the polarization vector is e = (e, iSe,)/dZ for circularly polarized light, where 8, and e, are unit polarization vectors in the x and y directions, respectively (the xyz coordinate system is taken as right-handed). According to the present definition, 6 = +1 corresponds to right-circularly polarized light, while 6 = -1 corresponds to left-circular polarization.

+

1.0

0.8

i

0.6

0.2

0 PHOTON ENERGY

(Mevl

FIQ.7. The circular polarization P of bremsstrahlung emitted in the forward direction by forward-spin electrons of kinetic energy TO= 2 MeV, as a function of the photon energy. This figure is taken from the work of K. W. McVoy [Phys.Rev. 106, 828 (1957); 110, 1484 (1958), Fig. 11 and is reprinted with the permission of the author and the Editor of the Physical Review.

For a given incoming electron kinetic energy To,the polarization P increases with increasing photon energy hv from P = 0 a t the lower end of the bremsstrahlung spectrum (hv = 0) to a maximum value P,,, a t the upper end (hv = To). Figure 7 shows the curve for P vs hv for To = 2 MeV, as obtained by McVoy (SO). The polarization P is defined as

P = (R - L)/(R

+ L)

(24)

where R and L are the cross sections for producing a right-handed and left-

54

R. M. STERNHEIMER

handed photon, respectively, in the forward direction from an electron with forward spin (spin parallel to direction of motion). The maximum polarization P,,, (for hv = To)is a relatively simple function of the incident electron energy. P,,, is given by

+

where Eo ( = To m) is the total energy of the incident e l e ~ t r o n whose ,~ momentum is denoted by p,. Figure 8 shows a plot of P,,, vs. To,as obtained 1.0 I

I

I

I

a

ELECTRON ENERGY T,(IN

MeV)

FIG.8. The maximum circular polarization P , of bremsstrahlung emitted in the forward direction by forward-spin electrons, as a function of the electron kinetic energy To.Pmaxis obtained from the equation given by McVoy [Phys. Rev. 106, 828 (1957); 110, 1484 (1958)l; see Eq. (25) of text.

from Eq. ( 2 5 ) .It is seen that P,, rapidly approaches 100% as the incident electron energy becomes relativistic. At high electron energies, the photons of maximum energy in the forward direction are essentially 100% rightcircularly polarized for incident forward-spin electrons and 100% leftcircularly polarized for incident backward-spin electrons (spin antiparallel t o direction of motion). I n a recent paper, Fronsdal and Uberall (33) have extended McVoy's results t o obtain the circular polarization of the bremsstrahlung emitted at an arbitrary angle 0 to the direction of the incident electron. Their results 4

It is assumed that the units are such that e

=

1.

PARITY NONCONSERVATION I N WEAK INTERACTIONS

55

for the polarization P as a function of photon energy hv, for finite 8, are in general similar to those obtained by McVoy (SO) for 0 = 0". In an important experiment of Goldhaber et al. ( S I ) , the circular polarization of the bremsstrahlung was used to detect the longitudinal polarization of the f-rays from the decay of YgO.The circular polarization of the bremsstrahlung y-rays was established by measuring the Compton scattering of the y-rays in an iron electromagnet which was magnetized to saturation either parallel or antiparallel to the direction of the y-rays. As was SCALE:

I

u

ANALYZING MAGNET DOUBLE 'MU METAL SHIELD

FIG.9. Schematic view of the experimental arrangement of Goldhaber et al. used t o demonstrate the circular polarization of the bremsstrahlung emitted by the electrons from the 0 decay of YgO. This figure is taken from the paper of Goldhaber et al. [M. Goldhaber, L. Grodzins, and A. W. Sunyar, Phys. Rev. 106, 826 (1957), Fig. 11 and is reprinted with the permission of the authors and the Editor of the Physical Review.

first demonstrated experimentally by Gunst and Page (34, see also 35), the Compton scattering cross section for right- or left-circularly polarized y-rays has a spin-dependent part, which reverses sign when the orientation of the spin of the target electron is reversed. For iron, there are two electrons per atom (3d electrons) whose spin aligns itself with the direction of the external magnetic field. The arrangement of the apparatus used by Goldhaber et al. (31) is shown in Fig. 9. The Sr90 Ygosource was encased in

+

56

R. M. STERNHEIMER

Monel metal (Z,ff = 28; 60% Ni, 33% Cu, 6.5% Fe) in which most of the bremsstrahlung was produced. The p--rays from the decay of Ygohave a maximum energy of 2.24 MeV, for which (v/c),,, = 0.98. According to the two-component neutrino theory, the P--rays should have a polarization -v/c, i.e., the spin vector of the electrons should be approximately opposite to their direction of motion. The bremsstrahlung y-rays are filtered in the iron of the magnet; i.e., a fraction of the y’s suffers Compton scattering and thereby disappears from the beam (which goes downward in Fig. 9). The y-rays are then counted in a 3 X 3-in. NaI(T1) scintillation counter, for which the pulse height gives directly the energy of the y-ray. (All of the energy of the y-ray is deposited in the NaI(T1) crystal and therefore contributes to the pulse height.) The counting rate was found to be different for the field direction up and field direction down; these two counting rates will be denoted by N+ and N-, respectively. At 1.8-Mev photon energy, Goldhaber et al. (31) obtained a relative difference 6 = 0.07 f 0.005, where 6 = (N- - N+)/[>.i(N- N + ) ] . Assuming a reasonable value for the effective path length in the magnet (4% in.), the value of 6 should be 6 = 0.08, if the photons were completely circularly polarized. The actual observed value of 6 therefore indicates a high degree of circular polarization (-90%) for the high-energy photons (-1.8 MeV.). From the fact that 6 is positive (N- > N+), one can conclude that the photons are left-circularly polarized. This result is expected from the theory of McVoy (SO), if the P--particles are polarized antiparallel to their direction of motion (backward spin), as is required by the two-component neutrino theory. Thus, Goldhaber et al. (31) have established the longitudinal polarization of the p--rays from YgO,which is a first-forbidden transition of unique shape AJ = 2, yes (meaning parity change of the nuclear states). It should be noted that the experiment of Goldhaber et al. (31) was one of the first to demonstrate the longitudinal polarization of 0-particles from unpolarized nuclei. The circular polarization of the internal bremsstrahlung has been discussed by Cutkosky (36), Ford (37),and Pytte (38). The internal bremsstrahlung is an effect caused by the changing dipole moment of the atom as the electronic charge of the @-particleis suddenly shifted from the nucleus to the external region of the atomic electrons. A summary of the experimental and theoretical information on the inner bremsstrahlung has been given by Wu (39). Cutkosky (36) has discussed in detail the internal bremsstrahlung accompanying K capture. This author has also pointed out that for a general 0-decay (not involving K capture) the energetic inner bremsstrahlung y-rays will be right- or left-circularly polaiized, if they accompany slow positrons or slow electrons, respectively. On the other hand, low-energy y-rays have a linear polarization correlated with the direction of the 0-particle

+

PARITY NONCONSERVATION I N WEAK INTERACTIONS

57

Ford (37) has shown that the degree of circular polarization P ( k ) of the inner bremsstrahlung increases rapidly with increasing photon energy k ( = hv). For the inner bremsstrahlen from P32(maximum electron energy Tmax = 1.70 Mev), P ( k ) is 0.72 for k/T,,, = 0.5, P ( k ) = 0.96 for k/T,,, = 0.8, and P ( k ) = 1 for k/T,,, = 1. Thus, the dependence of P ( k ) on k is very similar to the corresponding k dependence of P for the external bremsstrahlung, as discussed by McVoy (30) (see Fig. 7). The results of Pytte (38) are similar to those obtained by Ford (37). Pytte has calculated the polarization P ( k ) of the inner bremsstrahlung from P32and S35(T,,, = 167 kev). This author has also considered the effects of the nuclear Coulomb field on the y-ray spectrum and the polarization. In a recent experiment, Schopper and Galster (40) have verified the predictions of the theory of P ( k ) for the inner bremsstrahlung from a SrgO Ygosource. The calculated values of P ( k ) obtained by Ford (37) and Pytte (38)are in good agreement with the observations of Schopper and Galster (40),who measured the circular polarization of the inner bremsstrahlung by means of the Compton scattering in magnetized iron (34, 35) for photon energies from 0 to -1.8 MeV. The circular polarization is nearly complete at the upper end of the spectrum, showing that there is a maximum violation of parity conservation in the P-decay interaction. In the same experiment (40), the polarization of the ordinary (external) bremsstrahlung emitted by the P-decay electrons was also investigated and was found to be in reasonable agreement (especially a t high energies, -2 MeV) with the calculations of McVoy (SO) and the assumption that the polarization P of the electrons is - v / c .

+

VIII. DETERMINATION O F THE LONGITUDINL4LPOLARIZATION OF P-RAYSBY THE METHODOF SCATTERIKG ON POLARIZED ELECTRONS (MOLLERSCATTERING) Aside from the detection of the circular polarization of the bremsstrahlung, an independent way of establishing the longitudinal polarization of the P-decay electrons is the Mgller (41) scattering of the polarized electrons on polarized electrons in the target material (e.g., the ferromagnetic 3rl electrons in an iron sample). The dependence of the relativistic Mgller electron-electron scattering (41) on the directions of polarization of the two electrons has been investigated by Bincer (42)and by Ford and Mullin (43). Bincer (42) has investigated the electron-electron scattering for the special case that both electrons are longitudinally polarized along their relative direction of motion before the scattering, and that the initial spin directions (before the scattering) are either parallel or antiparallel. These two cross sections will be denoted by &, and &, respectively. Bincer (42)

58

R. M. STERNHEIMER

found that 4p/+ais different from 1 for all energies and scattering angles ( Z O O ) , and can be as low as 0. The ratio + p / + a is given by 4, = +a

1

22 + (3s+ + x: + ( 2 + 3s -

z2))pZ 2Z))PZ

+ (1 + 4p4 + (5 - 42 +

s”P4

(26)

where P is the velocity of either electron in the center-of-mass system (to be abbreviated as c.m. system), and s E cos2 8, where 8 is the scattering angle in the c.m. system. Equation (26) shows that = 1 for 3 = O”, independently of p. decreases with increasing S up to S = n/2, where i t reaches its minimum value, which is given by

1 .o

.8

.6

9, -

%

.4

.2

0

0

.I



.2

.3

4

.5

W

FIG. 10. The ratio &,/& for electron-electron scattering, as a function of the relative kinetic energy transfer w. This figure is taken from the work of A. M. Bincer [Phys. Rev. 107, 1434 (1957), Fig. I], and is reprinted with the permission of the author and the Editor of the Physical Review.

59

PARITY NONCONSERVATION I N WEAK INTERACTIONS

i ~ 0 as ,8 + 0 and becomes $6 for 6 + 1. Figure Thus, ( ~ # ~ / & ) ~ approaches 10 shows a plot of vs. w for several values of y. Here w is the relative kinetic energy transfer in the laboratory system; i.e., w = W / T , where W is the kinetic energy lost by the incident electron in the collision (W equals the kinetic energy of the secondary electron) and T is the initial kinetic energy of the incident electron. In Fig. 10, the values of y affixed t o the various curves represent y = E/m = ( T / m ) 1, where E is the total energy of the incident electron in the laboratory system. I n view of the symmetry of the problem with respect to the two electrons, &,/+a is the same for 8 and 180' - 8, and also for w and 1 - w.We have

+

(28)

Equation (26) can be rewritten as follows:

& &

+ +

+

- y2(1 6~ 9)- 2y(l - X) 1 - x2 89 - 2 ~ (4 5~ x2) 4 - 6~ 2x2

The expression for

+ +

+

[Eq. (27)] can also be rewritten in terms of

y:

For w = 0.5, the two outgoing electrons are symmetric with respect to the incident direction and make an angle 8, (in the laboratory system) with this direction, which is given by sin2 O8

= 2/(y

+ 3)

(31)

The strong dependence of the electron-electron scattering cross section on the initial directions of polarization of the two electrons indicates that, it should be possible to measure the longitudinal polarization of electrons from 0-decay by scattering from an iron target in which the spins of the two ferromagnetic 3d electrons have been aligned parallel or antiparallel to the direction of the incident (@-decay)electron. This method has been used successfully in a few experiments and confirms that the longitudinal polarization is - v/c within the experimental uncertainties (see below). The same spin-dependent effect is expected to occur when longitudinally polarized positrons (from /3 decay) are scattered on polarized electrons. This effect for positrons has also been considered by Bincer (42). The general expression for will not be reproduced here, since it is quite complicated. This complication arises partly from the fact that &,/4ais no longer invariant for 8 t)180' - 8 (8 = c.m. angle of scattering), since the positron and electron are distinguishable particles. Figure 11 shows the curves of &,/+a vs. w for various values of y of the incident positron. In the nonrela-

60

R. M. STERNHEIMER

tivistic case (y = l), +,/& = 1 a t all angles 8, i.e., for all values of the w I 1). As can be seen from Fig. 11, however, for energy transfer w(0 I relativistic energies ( y > I), deviates appreciably from 1, except near w = 0 (for w = 0, = 1 for all y). At very high positron energies (Y +. ~0 1, +,/& is given by "'= -1 (1 6 cos2 8 cos' 8) = 1 - 4w 6w2 - 4w3 2w4 (32) +a 8

+

+

+

+

This expression (for y -+ 00) is symmetric with respect to w = 0.5. It is 1 for w = 0 and w = 1 and attains its minimum value, $6, for w = 0.5.

W

FIG.11. The ratio +p/+O for positron-electron scattering, as a function of the relative kinetic energy transfer w. This figure is taken from the work of A. M. Bincer [Phys. Rev. 107, 1434 (1957), Fig. 21 and is reprinted with the permission of the author and the Editor of the Physical Review.

from 1 for positron-electron scattering The large deviations of indicate that the Moller scattering can also be used to detect the longitudinal polarization of positrons from 0-decay. Bincer (42) has also investigated the spin dependence of the scattering cross section for two fermions which are neither identical to each other, nor each other's antiparticle. This theory is applicable to the p - e scattering, for example, and indicates large deviations of +,/& from 1, provided that the energy of the p-meson is sufficiently high (T,,2 2 Bev). In a separate publication, Bincer (44) has studied the influence of a possible anomalous magnetic moment (e.g., for the neutron, proton) on the polarization effects in the scattering of fermions by fermions. These results are applicable, for example, to p - p , p - n, and e - ?a scattering.

PARITY NONCONSERVATION I N W E A K INTERACTIONS

61

Ford and Mullin (43)have obtained a general expression for the scattering of electrons by electrons for arbitrary spin orientations of the two electrons. I n general, they find two effects: (1) a dependence of the cross section on the relative spin orientation and (2) a n updown asymmetry in the cross section. For the special case that the incident electron is polarized either parallel or antiparallel to its direction of motion, but for arbitrary direction of the spin of the target electron, Ford and Mullin (43) obtain for the cross section da- --

dl2

+

+

To2 - { [(2y2- 1)2(4- 3 sin2 8) (y2 - 1)2 (sin4 8 4 7 ( r 2 - 1)' sin4 e 4 sin2 a)] - [(2y2- 1)(4y2- 3) sin2 8 - (7* - 1) sin4a)] cos # - 27(r2- 1) cos 3 sin3 8 sin # cos p \ (33)

I I I I

/ /

// Y'

FIG. 12. Diagram showing spins and momenta of incident and target electrons in electron-electron scattering and notation used by G. W. Ford and C. J. Mullin [Phys. Rev. 108, 477 (1957), Fig. 31.

z/m,

where ro = e2/mc2(classical electron radius), T = where is the total energy of either electron in the c.m. system, 8 is the c.m. scattering angle; Ic. and cp are the polar and azimuthal angles, respectively, of the spin l2of electron 2 (target electron) with respect, to the incident direction and the scattering plane. Figure 12 shows the notation used. I n this figure, and ll are the c.m. momentum and spin, respectively, of the incident electron;

62

R . M. STERNHEIMER

p' is the momentum of one of the outgoing electrons in the c.m. system. For the case $ = 0" or 180", Eq. (33) reduces to the results obtained by Bincer (42).The spin of the incident electron is assumed to be parallel to its direction of motion, so that the cross sections &, and 4aof Bincer (42) correspond to # = 0" and 180°, respectively. The spin dependence of du-/dQ is manifested by the terms proportional to cos $ and to sin I)cos cp. Both terms change sign when either of the spin directions is reversed. (A reversal of the spin 1 2 corresponds to changing $ to 180" - $, and cp to 180" cp). In addition, the term proportional to sin $ cos cp represents an up-down asymmetry with respect to the yz plane in Fig. 12. (A reflection with respect to the yz plane changes cp to 180" - Q, but obviously leaves $ unchanged.) As pointed out by Ford and Mullin (43)the coefficient of cos # can be obtained by measuring the cross sections for parallel and antiparallel spins, whereas the coefficient of the sin $ cos cp term can be determined by obtaining the difference in the cross sections a t cp = 0 and cp = a when the two spins are perpendicular to each other (I) = 90"). In order to discuss these effects, Ford and Mullin (43)introduce two quantities A'(Te,T) and B'(3,y) defined as follows:

+

* e-) A

B*(B,T)

3

( C J Y

=

da*(# = 0) - du*(# = a) = 0) du*(# = a)

+

- du*(*

du*(# = ~/2,cp= 0)

&*(I)

= a/2,cp = 0)

+

- du*($

+

(34)

= a/2,cp = a) a/2,cp = a)

du*($ =

(35)

where the superscripts and - refer to incident positrons and electrons, respectively. [For positrons, Ford and Mullin (43) have also obtained the explicit expression for the scattering cross section du+/dQ for arbitrary # and cp, similar to Eq. (33) for incident electrons.] A- and B- measure the strengths of the cos $ and sin # cos cp terms in (33), respectively. At 0 = a/2, both A- and A+ have their maximum values and are given by

For 7 ---f 03, both Eqs. (36) and (37) approach the same value, -Jd. In the nonrelativistic limit (T + l), A-(a/2,7) + - 1, i.e., du-(# = 0) = 0, which is due to the effect of the Pauli exclusion principle (42). The asymmetry coefficient B-(B,y) for electron-electron scattering is given by

B-(B,r)

= -

27(r2 - 1) sin2 8 cos 8 (27' - 1)2(4- 3 sin2 6) (T2 - 1)2(sin48

+

+ 4 sin2 8)

(38)

PARITY NONCONSERVATION IN WEAK INTERACTIONS

63

At 8 = 7r/3 and a laboratory energy of 1 Mev (corresponding to =?GZ 2), we have B- = -0.05. Similarly small values are obtained for B+ at 1 MeV. It may be noted that B- and B+ vanish both at nonrelativistic energies ( y -+ 1) and at very high energies (7 -+ a).It can be concluded that the coefficient of the up-down asymmetry term ( cc sin $j cos cp) is generally much smaller than the coefficient of the cos $j term. Ford and Mullin (43)have also considered depolarization effects in electron-electron and muon-electron scattering. For an electron with initial spin direction along the direction of motion, the probability that the final spin be parallel ( E = + l ) or antiparallel ( E = -1) to the initial spin is given by P(E,W)

1

+

E

= --

€(?

2

+

- 1)2(2y 1)2w (272 - 1)2

+ 0(w2>

(39)

where w is the fractional energy transfer in the laboratory system. Equation (39) shows that the probability of a spin flip is proportional to the energy transfer w, for small values of w. The term O(w2) represents a quantity of order w2 which is therefore unimportant for small w. For high-energy electrons, the great majority of the collisions correspond to a very small energy loss, and therefore the depolarization effects are expected to be negligible. Thus, as pointed out by Ford and Mullin (43),a l-Mev electron has a most probable fractional loss w 5 On the other hand, at low energies, as the electron is brought to rest and therefore suffers large fractional energy losses, the depolarization becomes important. In the nonrelativistic limit, the exact expression for P(E,w)is given by P(E,W)

1 + E

- -2E 1 - 3W2 w+3w2

=-

2

For nonrelativistic longitudinally polarized muons, the probability that a scattering will result in no spin-flip ( E = +I) or in a spin-flip ( E = -1) is given by

&(€,a)=

~

+ '2

E

[

(3

!f /34 sin2 P2

(i)+ (i)]

- sin4

sin6

(41)

where p = muon mass, ,8 = velocity of p in laboratory system, 8 = c.m. scattering angle of muon. The fractional energy loss w is given by

Thus, the second term of Eq. (41) (involving the square bracket) is of order ewP2(m/p), which is extremely small. Hence, for muons originating from pions which decay at rest, the depolarization is expected to be negligible

64

R. M. STERNHEIMER

until the muon is brought to rest. After the muon is brought to rest, there could, however, be some depolarization (by a spin-flip interaction of the 1.1 with the surrounding electrons) before the 1.1 meson decays. The first experiment using the Mgller (electron-electron) scattering to detect the electron polarization was carried out by Frauenfelder et al. (45) for the electrons from the decay of P3*and Pr144. The.experimenta1 arrangement is schematically shown in Fig. 13. The electrons from the source are first collimated and then impinge on a magnetized Deltamax foil of thickness 2.7 mg/cm2 and having an induction R = 15,000 gauss. The plane of the foil and therefore the direction of the polarization of the ferromagnetic ( 3 4 electrons is a t an angle CII = 30” to the direction of the incident electron beam. The scattered electrons are recorded in two anthracene scintilla-

COLLIMATOR

MAGNETIZING Pnll E

W

b W I L J

FIG. 13. Schematic view of the experimental arrangement of Frauenfelder et al., used to demonstrate the longitudinal polarization of electrons from the p-decay of Pa* and Pr144,by means of the M ~ l l e r(electron-electron) scattering [H. Frauenfelder, A. 0. Hanson, N. Levine, A. Rossi, and G. De Pasquali, Phys. Rev. 107, 643 (1957)].

tion counters which are in “fast-slow” coincidence. The advantage of using two counters in coincidence to detect both scattered electrons (having about equal energies) is that this procedure eliminates the undesired large background due t o Rutherford scattering as well as the complications due to plural scattering in the foil. This latter difficulty is present when the Mott (nuclear) scattering is used t o detect the electron polarization (see Sec. IX). The fraction f of the electrons of the foil which are magnetized is .f = 0.055 f 0.004 under the conditions of the experiment ( B = 15,000 gauss). The relative difference of the counting rates 6 is defined as 6

2(Cp - Ca)/(Cp

+ Ca)

(43)

where C, and Ca are the numbers of coincideiices when the incident electron momentum and the polarizing magnetic field in the scattering foil are parallel and antiparallel, respectively. (Actually, these two directions are

PARITY NOR'CONSERVATION I N WEAK INTERACTIONS

65

not exactly parallel and antiparallel to each other; the angle between them is a = 30" and 180" - a = 150" in the two cases.) In terms of the longitudinal polarization P of the incident electrons, 6 is given by 6

=

2fcos aP(1 - e ) / ( l

+ e)

(44)

where e = &,/4,, is the ratio of the scattering cross sections for longitudinally polarized electrons with parallel and antiparallel spins, as given by Bincer (4.2). For P32(I+ --+ O+ transition), two energy groups of electrons were investigated. Group (1) has energies between 0.3 and 1.0 MeV, with a n average (v/c),, = 0.85. For this group, 6 = -0.064 f 0.007, whence from Eq. (44), P = -0.85 f 0.11. Group (2) extends from 0.8 to 1.6 MeV, with (v/c),, = 0.94. The observed 6 = -0.069 f 0.010 gives P = -0.94 f 0.16. A check was obtained by substituting an aluminum foil for the Deltamax foil. In this case, the dependence on the magnetic field direction was zero within the experimental uncertainties (6 = -0.002 f 0.009). For Pr14*(0O+ transition), electron group (1) has energies ranging from 0.4 t o 1.1 Mev, with (v/c),, = 0.86, while the value of P obtained from the measured 6 is -0.66 f 0.18. Group (2) has energies between 1.2 and 3.0 MeV, with (v/c),, = 0.97. The experimental value of P for this group is -1.05 f 0.25. With the Pr144source, the aluminum scatterer again gave a negligible value of 6. It can be concluded from this experiment that both for P32and the electrons are polarized opposite to their direction of motion, and that the magnitude of the polarization is v/c within the limits of the experimental errors. By using the same method of the Mgller electron-electron scattering, Benczer-Koller et al. (&) have recently shown that the polarization P for the electrons from YgOand AuIg8is -v/c. In this experiment, the scattering foil was a piece of Supermendur, 2 x in. thick, which was mounted on a steel frame, making an angle of 30" with respect to the incident electron beam. The distance from the source to the center of the foil could be varied from 11to 29 em. A strong-focusing lens consisting of two quadrupole magnets was also used to focus the high-energy electrons. The detectors were two plastic scintillators in fast-slow coincidence, which were placed a t an angle of 35 f 11" to the incident electron beam in a symmetric arrangement [w = 0.5, see Eq. (31)]. Extensive tests were made on the collimation of the electron beam arriving a t the magnetic foil. It was found that the use of the strong focusing magnets increased the intensity of the high-energy electrons and suppressed the low-energy end of the spectrum, as is desired, since the high-energy electrons are the ones which give rise to the electron-electron scattering, ---f

66

R. M. STERNHEIMER

whereas the low-energy electrons give unwanted coulomb (nuclear) scattering, which constitutes an undesirable background. By the use of the strong focusing magnets, the intensity of the electrons from Adg8in the highenergy region (500-960 kev) was increased by a factor of 4. The observed differences 6 were as follows: For the 0- from Ygo (2-) ---f ZrgO(O+),with velocities v between 0 . 9 5 ~ and 0.98c, 6 with the use of the focusing magnets was (f6.95 f 1.60)%, which gives a polarization P = (-0.93 f 0.21)vlc. A control experiment in which an aluminum foil replaced the Supermendur gave 6 = (-0.89 f l . l l ) % for electrons from Ygoin the same velocity range (0 .9 5 ~to 0.98~).For the 0- from Aulg8 (2-) -+ Hglg8 (2+) with velocities v between 0.89~and 0 . 9 4 ~6~with the focusing magnets was (+6.65 f 1.33)%, giving P = (-1.02 f 0.19)vlc. Again a control experiment with an aluminum foil gave zero difference within the experimental errors [6 = (-0.21 f 1.30)%], as is, of course, t o be expected, since A1 has no polarized electrons, which could be aligned with the external magnetic field. The difference 6 used above is defined as 6 E 2(Np - Na)/(Np

+

Na)

(45)

where N , and N , are the numbers of coincidence counts when the spins of the incident electrons and the (ferromagnetic 3d) target electrons are parallel and antiparallel, respectively. Since the spin aligns itself in a direction opposite t o the magnetic field direction, the present N , corresponds to C, of Eq. (43), and N , corresponds to C,, so that the present 6 is equivalent t o -6 of Frauenfelder et al. (45). The polarization P is obtained from the relation

P =-

;

6 [f cos a

(; ;;$;3]-1

where f is the fraction of polarized electrons per atom, a = angle between the incident beam and the plane of the foil (a = 30"), and 4,/& is the ratio of the scabtering cross sections determined by Bincer (42) which for the conditions of this experiment (y 3, w between % and $5) is of the order of 0.1 (see Fig. 10). The fraction f was obtained by the authors (46) from the relationship:

-

B =

+ 4?r(26fpoNo)

(47)

where B is the induction, B = 13,000 gauss, which was obtained for a magnetic field H = 2.5 oersteds. (The remanence of the Supermendur was 11,000 gauss.) I n Eq. (47) po = Bohr magneton = 0.9270 X gausscm3, No = number of atoms per cm3 = 8.46 X ~ m - ~Thus, . one obtains f = 0.051, which shows that the effective f is somewhat less than

67

PARITY NONCONSERVATION IN WEAK INTERACTIONS

the maximum expected value 2/26 = 0.077, assuming two polarized 3d electrons. The result P = - u / c of the experiment of Benczer-Koller et al. (46) provides a strong confirmation of the prediction of the two-component neutrino theory. For the case of Aulgn,which is a first-forbidden transition with AJ = 0, yes (meaning parity change), the experiments of Boehm and Wapstra (47) on the @-7circular polarization correlation (see See. X) show that there is a large amount of interference between the Gamow-Teller and the Fermi interaction terms. For such a @-transition,according to the twinneutrino theory of Goeppert-Mayer and Telegdi (48),and Preston (49),the polarization should be much less than v/c. Thus, the experimental result of full (u/c) polarization for AulSnprovides a strong argument against the validity of the twin-neutrino theory.

IX. DETERMINATION OF THE POLARIZATION OF ELECTRONS FROM P-DECAYBY MOTT SCATTERING OF

THE

ELECTRONS ON NUCLEI

The Mott scattering (50) of the electrons from decay has also been used to measure their polarization. In this type of experiment, the longitudinal polarization of the electrons is first transformed into a transverse polarization, for instance, by deflecting the electrons through -90' by means of an electrostatic field. Another method consists in scattering the electrons by a large angle, as will be discussed below. After the particles have thus acquired a substantial amount of transverse polarization ( d l p ) , they are scattered by an angle 0 in the plane perpendicular to the (d,p) plane, which we assume to be horizontal, and the up-down asymmetry is observed. That is, the intensity of the electrons scattered through an angle 0 in the upward direction is different from the intensity of the electrons scattered through the same angle e in the downward direction. As was first shown by Mott (51) in 1929, the asymmetry in the scattering of transversely polarized electrons is largest. for heavy elements and for large scattering angles ( 0 90" - 150"). In connection wit,h the transformation of the longitudinal polarization of the beta particles into a transverse polarization, Case (52) has given a simple discussion of the behavior of the spin of a Dirac particle in an external electromagnetic field. The results of Case (52) have been previously derived by Tolhoek and de Groot (53). Let p be the (ordinary) kinetic momentum of the particle (electron, muon, etc.). In the presence of an electromagnetic field with vector potential A, p is given by

-

p =P where

- (e/c)A

(48)

P is the total momentum of the particle [which is represented by

68

R . M. STERNHEIMER

the quantum mechanical operator (A/i)V, where V is the gradient operator]. The Hamiltonian X of the system: particle field is given by X = cpld

*

+ p + p3mc2+ e+

(49)

where p1 and p3 are the usual matrices introduced by Dirac (54),and 4 is the scalar potent,ial for the external field. From the commutator [X,d p], Case (56) obtains

-

where E is the external electric field. Thus, for a pure magnetic field (E = 0), d p is a constant of the motion. From this property, two conclusions can be drawn: 1. A state of longitudinal polarization cannot be changed to a state of transverse polarization by using purely magnetic fields. 2. A longitudinally polarized beam will never be depolarized on passing through a purely magnetic field. For the case of a pure electric field, Case (52) considers the commutator [%,PI and finds

-

From Eqs. (50) and (51), one obtains

An important application of this equation concerns the p-meson experiment of Garwin et al. ( 3 ) .If in slowing down the p-mesons, only those moving in the initial direction (z direction) are considered, Eq. (52) becomes du,/dt = 0

(53)

Hence, if these p's are originally longitudinally polarized, they will remain so, with the same amount of polarization, after the slowing down. For the situation where both electric and magnetic fields are present, Case (52) has also derived appropriate equations for dp/dt and p * ( d d / d t ) . He has thus shown that when a longitudinally polarized beam moves perpendicular to a magnetic field, Eq. (52) is still valid. Tolhoek (35) has discussed in detail the rotation of the spin vector in transverse and longitudinal electric and magnetic fields. We shall first consider the case of a transverse electric field, such as exists between the plates of a cylindrical condenser (electrostat,ic deflector). For this case, Tolhoek (35) obtains

PARITY NONCONSERVATION IN WEAK INTERACTIONS

A a/ A y = T,/E,

69 (54)

where ACYis the angle by which the spin vector d is rotated, AT is the angle of deflection of the beam, and T , and E, are the kinetic and total energies of the electron, respectively. For nonrelativistic electrons, T J E , v2/2c2 is negligible, so that A a Z 0. Hence, in this case, the spin direction remains unchanged, and by deflecting the beam through go", an initial longitudinal polarization can be transformed into a transverse polarization. In order to accomplish the same objective for relativistic electrons, the deflection angle must be larger than go", namely, (r/2)(1 - Te/Ee)+. For a transverse magnetic field, Tolhoek (35) gives the result A a/ A y = 1

(55)

independently of the electron energy. Thus, for a pure transverse magnetic field, the spin vector follows exactly the momentum vector, and therefore the degree of polarization (longitudinal or transverse) is unchanged. This result is, of course, identical with that obtained from Eq. (50) with E = 0. For a beam with transverse polarization P (and d perpendicular to the plane of scattering) the ratio of scattered intensities in both azimuthal directions (upward and downward in the example discussed above) is given by

where a ( @is a function, first calculated by Mott (51),which depends on the atomic number of the scatterer, the incident electron energy, and the angle of scattering 8. The most complete recent calculation of n(e) has been carried out by Sherman (55),who has tabulated a(@at intervals of 15" for vaiious values of the electron velocity p, for three elements: mercury ( Z = 80), cadmium ( Z = 48), and aluminum (Z= 13). In addition to the function a(e), the rcal and imaginary parts of the Coulomb wave functiors F and G are also tabulated, together with the differential cross section & / d a (for unpolarized incident electrons). As mentioned above, la(0)l is largest for heavy elements and large values of 8. Thus, for p = 0.6 ( T , = 128 kev) and 2 = 80, a(0) = -0.062 a t e = 60", -0.271 a t 90°, -0.424 a t 120°, and -0.337 a t 150". The function a(0)is zero a t e = 0" and 180" for all energies, and a t 0 = 1 for all angles 0. In spite of the increase of [ a ( @from 0.271 at 90" t o 0.424 a t 120" and 0.418 a t 135" in the above example (0= 0.6), it has been found desirable to work a t -90' because of the rapid decrease of the cross section with increasing angle. Among the earlier determinations of a ( @ ,we may mention the calculations of Mott (52),Bartlett and Watson (56),and Mohr and Tassie (5'7).

70

R. M. STERNHEIMER

It may be remarked that the function a(e) also enters into the related problem (not directly applicable here) of the double scattering of an initially unpolarized beam of electrons. (51) I n this case, the polarization P after a single scattering through an angle el is given by a(&),and the direction of d after the scattering is perpendicular to the plane of the scattering. After a second scattering, the relative intensity of the beam as a function of the angle cp between the first and the second plane of scattering is given by I(el,ez,cp) = 1 where

02

+ a(el>a(ez)cos

CP

(57)

is the angle of the second scattering.

LEAD SHIELD

FIG.14. Schematic view of the experimental arrangement of de-Shalit et al. [A. de-Shalit, S. Kuperman, H. J. Lipkin, and T. Rothem, Phys. Rev. 107, 1459 (1957)l used to determine the longitudinal polarization of electrons from the p-decay of P32,by means of the Mott (nuclear) scattering.

Equation (57) applies if there is no magnetic field between the two scatterers. The situation where there is a magnetic field between scatterers 1 and 2 has been discussed by Mendlowitz and Case (58). This type of experiment (with magnetic field) can be used to determine the anomalous magnetic moment of the electron (59), i.e., the deviation from 2 of the gyromagnetic ratio g. As an example of the detection of the polarization of /3 particles by means of the Mott scattering, we shall discuss the experiment of de-Shalit et aZ.(60) on the polarization of the electrons from the @decay of P32.The

PARITY NONCONSERVATION I N WEAK INTERACTIONS

71

experimental arrangement is schematically shown in Fig. 14. The electrons are first scattered through 90" by a semicircular aluminum foil u1 (0.05 cm thick), which transforms their longitudinal polarization into a partial transverse polarization. In Fig. 14, the foil ul is in a plane perpendicular t o the plane of the paper. The diameter of the semicircle describing the foil is along the line joining the source S to the foil u2 (2.5 mg/cm2 Au) where the electrons undergo a second scattering. This property of u1 ensures that the first scattering angle is 90". A lead shield placed midway between S and u2 prevents direct (nonscattered) electrons emitted by the source from reaching u2. The second scattering (at uz) takes place in the plane of the paper in Fig. 14, and the electrons which are scattered to the right and to the left by 75" are recorded in the counters C R and C L .

FIG. 15. Diagram showing momentum and magnetic moment of electrons in the double scattering experiment of de-Shalit et al. [A. de-Shalit, S. Kuperman, H. J. Lipkin, and T. Rothem, Phys. Rev. 107, 1459 (1957)l. This figure applies for the nonrelativistic electrons from the 8-decay. For relativistic energies, the situation is more complicated [see references (61) and (62)l.

The situation as concerns the relative direction of the magnetic moment and the electron momentum is shown schematically in Fig. 15 for the case of nonrelativistic electrons. The electron is initially polarized with its spin s in the direction opposite to its direction of motion. Thus, since the magnetic moment is p = (eTa/mc)s, where e, the charge of the electron, is negative, p will be parallel to the electron momentum p, as shown in Fig. 15. After the scattering a t ul, p is rotated by go", but the direction of p is unchanged, so that p is then a t right angles to p (transverse polarization).

72

R. M. STERXHEIMER

Finally, with p pointing upward, the right-left asymmetry is measured. A simple qualitative argument given by de-Shalit et al. (60) shows that with the magnetic moment direction as shown in Fig. 15, there will be more particles scattered to the left (into the plane of the paper) than to the right (out of the plane). This is indeed observed, as will now be discussed. The measured right-left asymmetry was

N L - N R = (5.1 =k 0.6) X lo-' ~ ( N L NR)

+

where NR and NL are the counting rates in the counters C R and CL, respectively. The sign of the asymmetry (NL > N R ) shows that the electrons are initially polarized longitudinally, with the spin pointing backwards. The magnitude of the asymmetry is compatible with full polarization, i.e., P = -v/c. As noted above, the considerations presented in connection with Fig. 15 apply to nonrelativistic electrons. However, in the experiment of deShalit et al. (GO), only relativistic electrons with energies T, between 0.9 Mev and the maximum energy 1.7 Mev were included. Gursey (61) has given a treatment of the Coulomb scattering of polarized relativistic electrons (see also Tassie, 62). This author has calculated that for the experiment of de-Shalit et al. (GO), with a mean kinetic energy (T,) = 1.15 MeV, the asymmetry 6 is expected to be 9% for complete longitudinal polarization of the incident electron beam. The difference between the observed value, (5.1 f 0.6)%, and the theoretical result can probably be attributed to effects of plural scattering in the two scatterers u1 and U Z . In a later experiment, Lipkin et al. (63) measured the 0-ray polarization for AuIg8by the same method as used by de-Shalit (60). They found that both for Sn and Au foils used as scatterers C T ~the , asymmetry was the same for AuIg8electrons as for P3' electrons in the same energy range. Thus, for a gold foil of thickness 1.3 X cm, 6 = (8.6 f 1.0)% for Aulg8,as compared with 6 = (8.7 f 0.7)0j, for P32.This result implies that the AuIg8 electrons are fully polarized, if one accepts the result of full polarization for the P32electrons obtained by Frauenfelder et al. (45). By comparing in this manner the observed asymmetries for two nuclei, it is not necessary t o make complicated corrections for various experimental effects (e.g., plural scattering) which would enter into an absolute determination of the polarization (61). As discussed above, the result of full polarization for A d g 8 was also obtained by Benczer-Koller et al. (46) from a measurement of the MGller (electron-electron) scattering of the 0-particles in magnetized iron. One of the first experiments on the electron polarization by means of the Mott scattering was carried out by Frauenfelder et al. (64),who measured the polarization of the electrons from the Co60decay. In this experi-

PaRITY NONCONSERVATION I N W E A K INTERACTIONS

73

ment, the electrons were deflected by 108" in an electrostatic field, so that the spin was approximately perpendicular to the direction of motion after the deflection. The polarization analyzer consisted of a gold scattering foil (0.15 or 0.05 mg/cm2), and the asymmetry in the scattering was measured for scattering angles e in the region from -95" to 140". The measurements were carried out for three different groups of Co60electrons having energies T , = 50, 68, and 77 kev. The values of the polarization P obtained from the data are -0.04 for T , = 50 kev (v/c = 0.41), -0.16 for T , = 68 kev (v/c = 0.47), and P = -0.40 and -0.35 for the two runs a t T , = 77 kev (V/C = 0.49). The left-right asymmetry N L / N Rof the counting rates N L and N R was very pronounced for the two runs a t 77 kev. The two values of NL/"R were 1.35 =t 0.06 and 1.30 rt 0.09; these asymmetry ratios will be denoted by R, and Rb, respectively. In view of Eq. (56), P is given by

where R = N L / N R .From the tables of Sherman (55),the value of a ( @for Z = 80, 0 = 110", p = 0.49, is a ( 0 ) = -0.37. One obtains for R,: P(R,) = -0.40 rt 0.06, and for Rb: P(&) = -0.35 =t0.09. Thus, the observed polarization is a large fraction (7040%) of the predicted value, P = - v / c = -0.49. The relatively small discrepancy could be due to depolarization effects in the source and in the analyzer. In an experiment similar to that of Frauenfelder et al. (64),De Waard and Poppema (65) have measured the longitudinal polarization of the /3particles from Co60, P32,Tm170, and Aulg8.The electrons were deflected by 90" in an electrostatic field, and were subsequently scattered from a gold foil (thickness -200pg/cm2) a t angles from 50" to 87". The electron velocity in this experiment was v = 0 . 6 6 ~( T , = 168 kev). For Co60,the experimental value of the polarization was P = -0.49 + 0.11, as compared with the theoretical prediction, P = -0.66. It was suggested by Cavanagh et al. (66) that the observed lPl would be increased to a value close to 0.66, if the necessary correction for plural scattering in the gold foil were applied to the measured asymmetry. Cavanagh et al. (66) have pointed out that, in order to obtain more accurate results with the Mott scattering method, it is advantageous to use a system of crossed electric and magnetic fields, instead of a pure electrostatic field, to produce the transverse polarization. I n this case, the focusing condition for the particles becomes identical with the condition for turning the spin through 90". Cavanagh et al. (66) have used this method to determine the longitudinal polarization of the electrons from Co60. The crossed fields satisfy the condition E / H = 0, where E = electric field, H = mag-

74

R. M. STERNHEIMER

netic field, and 0 = v/c of the electrons. As a result, the electrons having the desired velocity travel along the central axis of the crossed-field region without any deflection. The electrons were injected from a thin-lens spectrometer. The value of 0 was taken as 0.6, to make use of the fact that the asymmetry function [a(O)i for 90" and 2 = 80 is highest for 0 = 0.6 [a(90°)= -0.2711 (55). This corresponds to a kinetic energy T, = 128 kev. The electrons are then scattered through 90" by a thin gold foil, which is placed in a transmission position at 60" to the incident beam, in order to reduce effects of plural scattering in the foil (35). Concerning the spin rotator, the two plates providing the electric field were 20 cm long, with a gap distance of 2.8 cm. The voltage could be vaned up to 70 kv applied symmetrically to both plates, and the magnetic field H was of the order of 100 oersteds. The required electric field for H = 100 oersteds and 0 = 0.6 is

E = 300 X 100 X 0.6

=

18,000 v/cm,

(60)

and the required potential difference AV across the 2.8-em gap is, therefore,

AV The angle

=

18 X 2.8 = 50.4 kv

x (in radians) by which the spin is turned is given by x = 300Hoe=stlCm(1 - P2)"/P,,/,

(61) (67) (62)

where Hoerst is the field in oersteds, Zc, = length of the plates in em, and p,,/, is the momentum in ev/c. The detector consists of a scintillation counter which counts electrons scattered by 90" in the gold foil. The reason for choosing 90' and not some larger angle where la(@/ is higher than for 90" is that the cross section du/dQ decreases rapidly with angle, as was mentioned above. Thus, for 0 = 120°, where la(0)l has its maximum value for 0 = 0.6 and 2 = 80 [a(120") = -0.4241, du/dQ is down by a factor 2.1 from its value a t 90" (du/dQ = 2.00 X lo3 barn/ster a t 120" as compared with 4.29 X lo3 barn/ster at 90") (55). As a result, the observed asymmetry would be changed appreciably from the single-scattering value by the admixture of some plural scattering, if the angle e were too large, so that the plural scattering would predominate. The thickness of the gold foil must also be held small enough so that the corrections due to secondary processes (plural scattering) will be negligible. A discussion of these effects is given in the review article of Tolhoek (35). In their double scattering experiment, Ryu, Hashimoto, and Nonaka (68) found that the thickness t of the gold foil for the scattering (analyzer) should not exceed 100 pg/cm2, to ensure that secondary corrections are not excessive. Actually, aside from several runs with t 100 pg/cm2, the experiment of Cavanagh et al. (66) was also

-

75

PARITY NONCONSERVATION I N WEAK INTERACTIONS

carried out for several thicknesses t between 100 pg/cm2 and 1 mg/cm2 to estimate the corrections due to plural scattering. Cavanagh et al. (66) obtained approximately the expected sin cp dependence of the asymmetry of the counting rate on the angle p between the spin direction and the azimuthal angle of the detector. The latter could be rotated in a plane perpendicular to the central axis of the spin rotator, so as to vary cp. Thus, for a gold foil of thickness 180 pg/cm2, the counting rate N was given by

N = No[l

+ A sin (9+ S)]

(63)

where the size of the asymmetry A was 0.11, and the const.ant phase angle 6 = 25" arose from certain instrumental misalignments. [The beam made a small angle (-3") with the axis of the spin rotator and was also displaced from the axis by a small amount.] With increasing thickness t, 6 increases. Thus, 6 = 60" for t = 770 pg/cm2 of Au. However, such large thicknesses were not weighted strongly in obtaining the asymmetry A ( t ) extrapolated to zero thickness t. The latter, A(O),is 0.159. This value must be increased by 10% to take into account the multiple scattering and the wide-angle scattering in the source. This gives A = 0.173 f 0.035. In obtaining A(O),the authors (66) used the approximate relation A ( t ) = A(O)/(l ct), where c is a constant if double scattering is the dominant process producing the secondary effects. From the experimental value of A and from the value (55) of a(90") = -0.267 for 2 = 79, /3 = 0.6, one obtains P = A / a = -0.65 f 0.13, which is consistent with the prediction of the two-component neutrino theory, P = -/3 = -0.6. Cavanagh et al. (66) have also measured P for 129-kev electrons from Aulg8,and have obtained P = (-0.97 f 0.20)v/c,in good agreement with the theoretical prediction and with the experimental results of BenczerKoller et al. (46) and Lipkin et al. (63). An experiment similar to that of Cavanagh et al. (66), using crossed electric and magnet,ic fields, and Mott scattering, has been recently carried out by Alikhanov et al. (67).These workers have measured the longitudinal polarization P of the electrons from a Sr-Y source, corresponding to the transitions: SrgO(O+) -+ YgO(2-) -+ ZrgO(O+),and Srsg((45+) -+ Ysg(>5-). The energy of the electrons involved in the experiment was T,= 300 kev (0 = 0.78). The effective length of path in the crossed fields was I = 27 cm. The gap between the condenser plates was 12 mm. The gold scattterer was placed in the transmission position a t 45" to the beam axis. Electrons scattered through an angle of 90 f 4" were counted by two Geiger counters in coincidence. The counters could be rotated about the beam axis in a plane perpendicular to the direction of the beam before the scattering. The calculated asymmetry for the scattering of 300-kev electrons a t

+

76

R. M. S T E R N H E I M E R

angles cp = 90" and 270" to the spin direction was: Bcalc = 21.8%, assuming that the polarization P = -v/c. The actual measured asymmetry was B,, = (17.4 f 4.3)70. Thus, the experimental value of the polarization IPI is

lP1

=

*

(17.421.84.3) c!! = (0.80 f 0.20)

However, the measured asymmetry should be increased by 13% to correct for multiple scattering effects. This gives jPI = (0.90 f 0.23)v/c, which is in essential agreement with the theoretical prediction. An additional experiment was carried out by Alikhanov et al. (67)using 750-kev electrons from the decay of Ygoand SrS9.I n this case, the measured = (7.8 f 2.5)%, as compared with the calculated asymmetry was, , , ,6 value Gcalc = 6.8%. This gives lP1 = (1.15 f 0.4)v/c for 750-kev electrons. The experiment was also repeated for T,= 300 kev under slightly different conditions ( E = 20 kv/cm, H = 86 oersteds, scattering angle = 105"). The azimuthal asymmetry was 35.5%. The resulting electron polarization IPI is (1.10 f 0.19)v/c. The mean value of IPl for both experiments at 300 kev is1'21 = (1.02 f 0.15)v/c, in good agreement with the prediction of the two-component neutrino theory. In all cases, the sign of the asymmetry was that to be expected for electrons whose spin direction is opposite to the direction of motion.

X. EXPERIMENTS ON THE LONGITUDINAL POLARIZATION OF POSITRONS FROM P-DECAY. /3 - y CIRCULAR POLARIZATION CORRELATION EXPERIMENTS The circular polarization of the bremsstrahlung, as well as the Moller and the Mott scattering, which were discussed above, have been used primarily for the electron P-emitters. Various other methods based on the annihilation properties of positrons have been used for the positron 0-emitters. In this section we shall describe these experiments on the longitudinal polarization of positrons and shall also give a brief discussion of the P - y circular polarization correlation experiments, which have given valuable information on the P-decay interaction for various electron emitters. The experiment of Page and Heinberg (69) on the polarization of positrons from Na22is based on the properties of the triplet (1 3Ss,,0) and singlet (1 states of positroniuni (70).In certain gases such as argon or propane, the positronium is formed with rather large kinetic energies, and is subsequently thermalized (i.e., slowed down) a t such a rate that the singlet state, with lifetime 7 1O-Io sec, retains most of its initial motion a t the time of annihilation, whereas for the triplet state, with 7 3 X sec,

-

-

PARITY NONCONSERVATION I N WEAK INTERACTIONS

77

most of the initial motion is lost by the time it undergoes two-photon annihilation (of course, in the presence of a magnetic field). Therefore, the angle O,, between the two annihilation photons mill be on the average closer to 180" for the 3X1.0states than for the 'So,ostates. If one requires strict angular correlation a t O,, = 180", with a suitable gas (e.g., argon) the relative efficiency for triplet/singlet states can be made -1.5, in the presence of the background of the other two-quantum annihilation events. When a magnetic field H is applied to the gas sample, the positrons are preferentially captured into the triplet state if the positron spin is antiparallel to H and the singlet state if the spin is parallel to H. Making use of this property, Page and Heinberg (69) applied fields H of 10-15 kilogauss to various gas samples and obtained the relative difference 6 in the annihilation yield (for O,, very close to 180") with the field H parallel and antiparallel to the direction of motion of the positrons from the Na22source. From the observed values and the sign of 6, Page and Heinberg (69) concluded that the positrons are polarized along their direction of motion, the value of the polarization P being greater than 0.4 (u/c), where (v/c) = 0.75 is the average value of v/c for the ef from Na22.The fact that less than the full expected value of P (= (v/c)) was obtained may be due to several interfering effects: (1) backscattering from the NaZ2source and its mounting; (2) partial depolarization of the positrons prior t o the formation of positronium; (3) depolarization of the ef in the positronium "atom" before the annihilation process takes place (mixing of the magnetic substates of 1s positronium). Hanna and Preston (71) have demonstrated the longitudinal polarization of the positrons from Cu6*by annihilation of the positrons in magnetized iron. A suitable part of the annihilation spectrum was selected by appropriate collimation, namely that part which corresponds to a large momentum for the electrons of the target material (Fe) against which the positrons annihilate. This was done by obscuring the central part of the angular distribution of the annihilation radiation (angle O,, between the two 7's = lSO"), so that only values of O,, which differ from 180" by more than 8.5 milliradians were included. With this arrangement, it was found that the annihilation (two-y) yield Y is consistently higher by (5 f l)% with the magnetizing field H (around the iron) parallel to the positron direction of motion than with H antiparallel. As a check on the experiment, the Fe sample was replaced by a Cu sample, and the field-dependent effect on Y was found to vanish. Hanna and Preston (71) have interpreted their results as follows. The high-momentum Fe electrons which are involved in this experiment are mostly the polarized 3d electrons, whose spin is aligned antiparallel to the direction of the applied field H.Thus the fact that the annihilation rate is larger for H parallel to the direction of motion of the positrons indicates that the positrons are polarized parallel to their direc-

78

R. M. STERNHEIMER

tion of motion and still retain a substantial part of their original polarization a t the time of the annihilation. In a recent paper, Hanna and Preston (72) have given the results of additional experiments with their arrangement, using samples of Fe, Fe-Co, Ni, Cu, and Gd, in which the annihilation takes place. The asymmetry in this experiment was defined as 6 = (N+ - N - ) / N - , where N+ and N- denote the counting rates for field parallel and antiparallel, respectively, t o the direction of motion of the positrons, and the eclipsing angle was a,,= 8 milliradians, i.e., only annihilation events with angles a = 180" - e,, > a,,were included. The values of 6 were obtained as a function of positron energy by interposing various thicknesses of A1 foil between the CuMsource and the sample. Thus, for a n Fe-Co sample, it was found that 6 increases from (5.4 f 0.8)% at T, = 0.33 Mev to (11 f 2.5)% at T,= 0.50 MeV. This increase is not primarily due to the variation of v/c (= polarization P ) which increases by only 9% (from v/c = 0.79 a t 0.33 Mev to 0.86 a t 0.50 Mev). The increase of 6 is rather due to the improved directionality of the high-energy positrons, which results in an increase of the polarization along the direction of the magnetic field in the sample. With a thin source (0.002 in.), the Fe-Co sample gives a somewhat smaller 6 a t 0.33 MeV: 6 = (4.4 f 1.2)%, as compared with (5.4 f 0.8)% for the thick source (0.005 in.). The reason for this difference is that with increasing thickness, the emerging positrons have on the average a higher energy a t creation, and a correspondingly higher polarization P. A steel sample and an Armco sample gave values of 6 of the same order as for Fe-Co. By contrast, a Ni sample gave 6 = 0 within the experimental errors [6 = (-0.3 f 0.9)% a t 0.33 Mev], even though the Curie temperature for Ni, Tc = 631" K, is considerably above room temperature. Hanna and Preston (72) attribute this unexpected result for Ni to a possible difference of the spatial and momentum distribution of the polarized electrons in the solid, as compared t o Fe and Co. The authors (72) also mention the possibility that the positron waves inside the Ni sample may not penetrate the regions where the polarized electrons are found with appreciable probability. For gadolinium a t -100" C, 6 was also zero [6 = (0.0 f 1.8)% a t 0.43 Mev], even though Gd is strongly ferromagnetic at this temperature (Curie temperature Tc = 289" K). This result is probably due to the fact that the ferromagnetic electrons of Gd, being 4 j electrons, are localized in the internal regions of the atom and are therefore very effectively shielded from the incident positrons. A Cu sample gave no effect, as would be expected from the absence of ferromagnetism. It is apparent from these results that investigations with polarized positrons can give valuable information about the spatial and momentum distribution of the polarized electrons in ferromagnetic materials, and hence ultimately about the wave functions of these electrons.

PARITY NONCONSERVATION I N WEAK INTERACTIONS

79

In an important experiment, Deutsch et al. (73) showed that the positrons from Ga'j6and C134are polarized along their direction of motion. This was accomplished by making use of the fact that high-energy photons from two-quantum annihilation are almost 100% circularly polarized in the direction of the positron spin (74). The circular polarization was detected by means of the Compton scattering in magnetized iron, i.e., from the dependence of the transmission of the y-rays on the direction of the applied magnetic field (34, 35). The annihilation took place in a Lucite converter. As a typical result, for y-rays of energy T, = 3 Mev from Ga@,the observed difference in transmission with a thick iron analyzer (12 cm) was (8.4 f 0.5)%, as compared with the theoretical value (8.8 f 1.0)%, which assumes full polarization of the positrons ( P = +v/c). These values pertain to annihilation quanta with energies above 2 MeV. The results for CP4are somewhat less certain, but they do indicate that the average polarization is again along the direction of the positron spin. The CP4data are also compatible with full polarization within the experimental uncertainties. Since CP4 is a pure Fermi transition, this experiment shows that parity nonconservation is not restricted to Gamow-Teller transitions, but is a property of the general /%decay interaction, as predicted by the two-component neutrino theory. Ga66is also probably a pure Fermi transition, although no definite conclusions can be drawn, until the parity of this nuclide is definitely established as positive. By the method of annihilation-in-flight in a magnetic material (74), Frankel et al. (75) have also found that the positrons from Ga66are highly polarized along the direction of motion. These experimental results for Ga66are important, since an earlier experiment by Frauenfelder et al. (76) indicated little or no polarization. Boehm et al. (77) have measured the positron polarization for the mirror transition N13by observing the circular polarization of the annihilation-inflight quanta. The photons were produced in a carbon sample which contains a small amount of NI3, obtained by deuteron bombardment of the sample in a 3-Mev Van de Graaff generator. In the same manner as in the experiment of Goldhaber et al. ( S I ) , the circular polarization was detected by means of the Compton scattering in an Armco iron magnet, which was magnetized by means of two coils. The difference in counting rate for the two opposite field directions was measured for the following y-energies : 620, 830, 1,040, and 1,140 kev. The calculated values of the circular polarization a t these energies are 36%, 59%, 70%, and 74%, respectively, assuming full polarization for the positrons ( P = +v/c). From these values and from the energy dependence of the Compton scattering (34), one finds that the relative counting rate difference 6 should increase from -0 a t 620 kev to 2.1% at 1,140 kev. The experimental results are in good agreement with the theoretical curve and indicate that the positron polarization is ($0.93 f 0.20)u/c for "3. From the ft value (lifetime) of this

80

R. M. STERNHEIMER

transition, Winther and Kofoed-Hansen (78) have deduced that the ratio of the squares of the matrix elements, (McT/2/IMF12, is 0.40. The result of full polarization therefore shows that the Fermi part of this transition contributes strongly to the observed polarization and, in fact, the measurements are compatible with full polarization for the Fermi part. If only the Gamow-Teller interaction would contribute to the polarization, the calculated difference 6 would be only 0.65% at 1,140kev, in definite disagreement with the observed result (2.0 & 0.6)%. It may be noted that the results of Boehm et al. (77) concerning full polarization for the Fermi interaction are in good agreement with the results of Deutsch et al. (73),which have been discussed above. A large positive polarization for the positrons from N13 has also been observed by Hanna and Preston (79),who used their method of annihilation in magnetized iron, which has been described above (71, 7 2 ) . These authors obtained comparable values for the counting rate ratios N + / N (-1.1) for N13and for Cu64(pure Gamow-Teller transition) with the same experimental arrangement. Important information on the nature of the 0-decay interaction has been obtained from the 0 - y circular polarization correlation experiments, which have been carried out by Boehm and Wapstra (47), Schopper et al. (80), and Lundby et al. (81). The basic idea underlying these experiments is the following. On account of the nonconservation of parity in the P-decay, the residual nucleus after P-decay will be polarized, even if the initial nucleus was unpolarized (as is generally the case). If the residual nucleus is in an excited state and emits a y-ray, the y-ray will be circularly polarized. The angular distribution of circularly polarized y-rays emitted a t an angle e to the preceding 0-rays is given by

w(e,

=

1 A A ( U / Ccos )

e

(65)

+

where A is a constant coefficient, v is the electron velocity, and the sign applies t o right-hand and the - sign to left-hand circular polarization. The theoretical expressions for A for various types of /3-transitions have been derived by Alder, Stech, and Winther (82). The experimental arrangement used by Boehm and Wapstra (47) is shown in Fig. 16. Above the source, there is a polarization-analyzer magnet, which serves t o determine the ciIcular polarization of the y-rays by means of the Compton scattering in magnetized iron (34) in the same manner as in the experiment of Goldhaber et al. (31).The magnet consists of a hollow Armco core which is magnetized by means of a coil. The y-rays from the source are scattered through an angle of =52" on the inside of the Armco cylinder and are then counted in a NaI crystal, which is connected by a light pipe to a photomultiplier. The direct y-rays (from the source to the

PARITY NONCONSERVATION I N WEAK INTERACTIONS

81

NaI crystal) are suppressed by a lead absorber. The @-rays(traveling downward in Fig. 16) are counted in an anthracene crystal, also connected to a light pipe and a photomultiplier. A metal shield around each photomultiplier is used to eliminate the effect of stray magnetic fields. As a result, a reversal of the field direction in the analyzing magnet changed the single pand y-counting rates by only (0.02 f 0.02)y0and (0.07 f 0.02)70, respectively.

.\

LEAD ANTHRACENE LIGHT PIPE (TO PHOTOMULTIPLIER)

FIG.16. Schematic view of the experimental arrangement of Boehm and Wapstra [Phys. Rev. 106, 1364; 107, 1202, 1462 (1957); 109, 456 (1958)], used to measure the B-r circular polarization correlation for several p emitters.

The coincidences between the P-particles and the scattered y-rays were measured with a fast-slow coincidence circuit with a resolving time of 0.02 psec. The efficiency of the analyzer in this arrangement was calculated by Alder (see Boehm and Wapstra, 4'7) and is given by E

=

2.90k(l

+ 0.13k)/(l + 0.36k + 0.09k2)

(66)

where e is defined as the percentage difference of the counting rates for opposite directions of the (saturated) magnetic field, for completely circularly polarized y-rays of energy kmc2. E must be multiplied by the cosine of the average angle between the p- and y-radiations (148'). This efficiency

82

R. M. STERNHEIMER

function E was checked by measuring the counting rate differences 6 for the bremsstrahlung emitted by the ,&particles from P32and Tml'O. The expected value of 6 was obtained from the circular polarization of the bremsstrahlung spectrum, as calculated by McVoy (SO), and from the theoretical efficiency function E [Eq. (66)l. In this manner, it was found that the longitudinal polarization P of the 0-decay electrons is (-0.97 f O.O6)v/c for P32and (-0.93 f O.O7)v/c for TmI7". These results for P are in good agreement with those obtained from other experiments, thus providing a check on the accuracy of the function used for E [Eq. (66)]. Boehm and Wapstra (47) have obtained the coefficients A for the following nuclides: Na24,S C ~S~C, ~V48, ~ , CoSslCo6O, and Adg8.In general, the value of A gives some information about the ratio x, defined as x = u ~ / M G T / M F Iwhere , MGT and MF are the Gamow-Teller and Fermi where ~, matrix elements, respectively, for the 0-transition, and a = C G T ~ / C F CGT and C F are the Gamow-Teller and Fermi coupling constants in the fundamental Odecay interaction. An experimental value of a was deduced by Kofoed-Hansen and Winther (83) from a study of theft values (0-decay lifetimes) in mirror transitions. These authors obtain a = 1.3. The ratio R = x2/(1 x2) represents the fractional amount of GamowTeller interaction for the particular 0-transition. ( R can vary between 0 and 1.) The theoretical expression (82) for A involves an interference term I , which takes on a particularly simple form if the two-component neutrino theory is valid and if, moreover, either only S and T, or only V and A , interactions occur. In this case, the absolute value of I is III = (CGT/CF)-' = a-%. For the case of S C ~the ~ , value of A is particularly large: A = +0.33 f 0.04. From this result, Boehm and Wapstra (47) deduce that 111 must be larger than 0.5 and obtain the following estimate of the ratio of the Gamow-Teller to Fermi matrix element: (MGTI/IMFI= 2.2. It may be noted that the theoretical value of A for a pure Gamow-Teller transition for Sc46would be +0.08, showing that the admixture of Fermi interaction results in a sizable change of A (from 0.08 to 0.33). For Co60, Boehm and Wapstra (47) have obtained A = -0.41 =t 0.08, which is in good agreement with the experimental values A = -0.34 rrt 0.04 of Schopper et al. (80) and A = - 0.32 f 0.07 of Lundby et al. (81).These results are also in good agreement with the theoretical value (82) A = -0.33. For Sc44and V48,the values of A are (algebraically) larger than for a pure Gamow-Teller interaction (e.g., the measured value for Sc44is -0.02 f 0.04, as compared with the theoretical prediction A = -0.17 for a pure Gamow-Teller transition). This deviation indicates an appreciable amount of interference between the Gamow-Teller and Fermi interactions. Assum-

+

PARITY NONCONSERVATION I N WEAK INTERACTIONS

83

ing the maximum value for I , the authors (47) find IMGT/MF[= 5 for both Sc44and V48. The maximum asymmetry is found for Aulg8,with a value of A = f0.52 & 0.09. The work of Boehm and Wapstra (47‘) (particularly the a pure V T and a pure SA interaction. On the experiment on S C ~excludes ~) other hand, the results are in good agreement with the pure V A interaction which has been proposed in references 18-20. Moreover, these 0 - y circular polarization correlation experiments disprove the validity of the twin-neutrino theory (48,49) and can also be used to rule out a large breakdown of time-reversal invariance. ACKNOWLEDGMENTS I wish to thank Dr. G. Feinberg for several very helpful discussions concerning parity nonconservation. I am also indebted to Dr. S. Pasternack and Dr. L. C. L. Yuan for valuable comments. REFERENCES 1. T. D. Lee and C. N. Yang, Phys. Rev. 104,254 (1956). 2. C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Rev. 106, 1413 (1957). S. R. L. Garwin, L. M. Lederman, and M. Weinrich, Phys. Rev. 106, 1415 (1957). 4. J. I. Friedman and V. L. Telegdi, Phys. Rev. 106, 1681 (1957). 5. R. H. Dalitz, Phil Mag. 44, 1068 (1953); Phys. Rev. 94, 1046 (1954); see also E. Fabri, Nuovo cimento 11, 479 (1954). 6. T. D. Lee and C. N. Yang, “Elementary Particles and Weak Interactions,” p. 10. Brookhaven Natl. Laboratory Report BNL 443 (T-Ql), 1957. 7 . J. M. Blatt and V. F. Weisskopf, “Theoretical Nuclear Physics,” pp. 24, 431, and 798. Wiley, New York, 1952. 8. See, for example, L. Wolfenstein, Ann. Rev. Nuclear Sci. 6,43 (1956). 9. E. Ambler, M. A. Grace, H. Halban, N. Kurti, H. Durand, C. E. Johnson, and H. R. Lemmer, Phil Mag. 44,216 (1953). 10. T. D. Lee, R. Oehme, and C. N. Yang, Phys. Rev. 106,340 (1957). 11. W. Pauli, in “Niels Bohr and the Development of Physics.” Pergamon Press, London, 1955; G. Luders, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 28, No. 5 (1954); J. Schwinger, Phys. Rev. 91, 720, 723 (1953); 94, 1366, 1576 (1954). 12. E. Ambler, R. W. Hayward, D. D. Hoppes, R. P. Hudson, and C. S. Wu, Phys. Rev. 106, 1361 (1957). IS. H. Postma, W. J. Huiskamp, A. R. Miedema, M. J. Steenland, H. A. Tolhoek, and C. J. Gorter, Physica 23, 259 (1957). 14. J. I. Friedman and V. L. Telegdi, Phys. Rev. 106, 1290 (1957). 15. T. D. Lee and C. N. Yang, Phys. Rev. 106, 1671 (1957). 16. L. D. Landau, Nuclear Phys. 3, 127 (1957); A. Salam, Nuovo cimento 6, 299 (1957). 17. B. M. Rustad and S. L. Ruby, Phys. Rev. 97, 991 (1955). 18. R. P. Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958). 19. E. C. G. Sudarshan and R. E. Marshak, Proc. Padua-Venice Conj. on Mesons and Recently Discovered Particles, p. V-14 (1957); Phys. Rev. 109, 1860 (1958). 20. J. J. Sakurai, Bull. Am. Phys. SOC[2] 3, 10 (1958); Nuovo cimento 7, 649 (1958); see also R. E. Behrends, Phys. Rev. 109,2217 (1958).

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81. M. Goldhaber, L. Grodzins, and A. W. Sunyar, Phys. Rev. 109, 1015 (1958); L. Grodzins, ibid. 109, 1014 (1958). 22. W. B. Herrmannsfeldt, D. R. Maxson, I-’.Stahclin, and J. S. Allen, Phys. Rev. 107, 641 (1957). 23. G. Culligan, S. G. F. Frank, J. R. IIolt, J. C. Kluyver, and T. Massam, Nature 180, 751 (1957). $4. L. Michel, Proc. Phys. SOC.A63, 514 (1950). 26. J. D. Jackson, S. B. Treiman, and H. W. Wyld, Phys. Rev. 106, 517 (1957). 26. L. Wolfenstein and L. A. Page, Bull. Am. Phys. SOC.[2] 2, 190 (1957). 27. R. B. Curtis and R. R. Lewis, Phys. Rev. 107, 543 (1957). 28. T. D. Lee and C. N. Yang, “Elementary Particles and Weak Interactions,” p. 54. Brookhaven Natl. Laboratory Report BNL 443 (T-91), 1957. 28u. E. Konopinski and H. M. Mahmoud, Phys. Rev. 92, 1045 (1953). 29. C. A. Coombes, B. Cork, W. Galbraith, G. R. Lambertson, and W. A. Wenzel, Phys. Rev. 108, 1348 (1957). 30. K. W. McVoy. Phys. Rev. 106,828 (1957); 110, 1484 (1958); see also K. W. McVoy and F. J. Dyson, ibid. 106, 1360 (1957). 31. M. Goldhaber, L. Grodains, and A. W. Sunyar, Phys. Rev. 106, 826 (1957). 32. H. A. Bethe and W. Heitler, Proc. Roy. Soc. 8146,83 (1934). 33. C. Fronsdal and H. tfberall, Phys. Rev. 111, 580 (1958). 34. S. B. Gunst and L. A. Page, Phys. Rev. 92,970 (1953). 36. H. A. Tolhoek. Revs. Modern Phys. 28, 277 (1956). 36. R. E. Cutkosky, Phys. Rev. 107, 330 (1957). 37. G. W. Ford, Phys. Reo. 107, 321 (1957). 38. A. Pytte, Phys. Rev. 107, 1681 (1957). 39. C. S. Wu, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 649. Interscience, New York, 1955. 40. H. Schopper and S. Galster, Nuclear Phys. 6, 125 (1958). 41. C. Mnller, Ann. Physik [5]14, 531 (1932). 42. A. M. Bincer, Phys. Rev. 107, 1434 (1957). 43. G. W. Ford and C. J. Mullin, Phys. Rev. 108,477 (1957); 110, 1485 (1958). See also P. Stehle, ibid. 110, 1458 (1958); A. RBczka and R. Rgcaka, ibid. 110, 1469 (1958). 44. A. M. Bincer, Phys. Rev. 107, 1467 (1957). 46. H. Frauenfelder, A. 0. Hanson, N. Levine, A. Rossi, and G. De Pasquali, Phys. Rev. 107, 643 (1957). 46. N. Benczer-Koller, A. Schwarzschild, J. B. Vise, and C. S. Wu, Phys. Rev. 109, 85 (1958). 47. F. Boehm and A. H. Wapstra, Phys. Rev. 106, 1364; 107, 1202 ,1462 (1957); 109, 456 (1958). 48. M. Goeppert-Mayer and V. L. Telegdi, Phys. Rev. 107, 1445 (1957). 49. M. A. Preston, Can. J . Phys. 36, 1017 (1957). 60. N. F. Mott, Proc. Roy. Sac. A126, 259 (1930). 61. N. F. Mott, Proc. Roy. SOC.A124, 425 (1929); A136 ,429 (1932). 62. K. M. Case, Phys. Rev. 106, 173 (1957). 65. H. A. Tolhoek and R. S. de Groot, Physica 17, 17 (1951). 64. P. A. M. Dirac, “The Principles of Quantum Mechanics.” Oxford University Press, London and New York, 1935. 65. N. Sherman, Phys. Rev. 103, 1601 (1956). 66. J. H. Bartlett and R. E. Watson, Proc. Am. Acud. Arts Sci. 74,53 (1940). 57. C. B. 0. Mohr and L. J. Tassie, Proc. Phys. SOC.A67, 711 (1954); C. B. 0. Mohr, Proc. Roy. SOC.8182, 189 (1943).

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58. H. Mendlowite and K. M. Case, Phys. Rev. 97, 33 (1955); 100, 1551 (1955). 59. W. H. Louisell, R. W. Pidd, and H. R. Crane, Phys. Rev. 94, 7 (1954). 60. A. de-Shalit, S. Kuperman, H. J. Lipkin, and T. Rothem, Phys. Rev. 107, 1459 (1957). 61. F. Gursey, Phys. Rev. 107, 1734 (1957). 61. L. J . Tassie, Phys. Rev. 107, 1452 (1957). 63. H. J. Lipkin, S. Kuperman, T. Rothem, and A. de-Shalit, Phys. Rev. 109,223’(1958). 64. H. Frauenfelder, R. Bobone, E. von Goeler, N. Levine, H. R. Lewis, R. N. Peacock, A. Rossi, and G. De Pasquali, Phys. Rev. 106,386 (1957). 66. H. De Waard and 0. J. Poppema, Physica 23, 597 (1957). 66. P. E. Cavanagh, J. F. Turner, C. F. Coleman, G. A. Gard, and B. W. Ridley, Phil. Mag. [8] 2, 1105 (1957). 67. A. I. Alikhanov, G. P. Eliseiev, V. A. Lubimov, and B. V. Ershler, Nuclear Phys. 6, 588 (1958). 68. N. Ryu, K. Hashimoto, and I. Nonaka, J . Phys. SOC.J a p a n 8, 575 (1953). 69. L. A. Page and M. Heinberg, Phys. Rev. 106, 1220 (1957). 70. M. Heinberg and L. A. Page, Phys. Rev. 107, 1589 (1957). 71. S. S. Hanna and R. S. Preston, Phys. Rev. 106, 1363 (1957). 7e. S. S. Hanna and R. S. Preston, Phys. Rev. 109,716 (1958); see also R. S. Preston and S. S.Hanna, Phys. Rev. 110, 1406 (1958). 73’. M. Deutsch, B. Gittleman, R. W. Bauer, L. Grodzins, and A. W. Sunyar, Phys. Rev. 107, 1733 (1957). 74. L. A. Page, Phys. Rev. 106,394 (1957). 76. S. Frankel, P. G. Hansen, 0. Nathan, and G. M. Temmer, Phys. Rev. 108, 1099 (1957). 76. H. Frauenfelder, A. 0. Hanson, N. Levine, A. Rossi, and G. De Pasquali, Phys. Rev. 107, 910 (1957). 77. F. Boehm, T. B. Novey, C. A. Barnes, and B. Stech, Phys. Rev. 108, 1497 (1957). 78. A. Winther and 0. Kofoed-Hansen, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 27, No. 14 (1953). 79. S.S. Hanna and R. S. Preston, Phys. Rev. 108, 160 (1957). 80. H. Schopper, Phil. Mag. [8] 2, 710 (1957); H. Appel, H. Schopper, and S. D. Bloom, Phys. Rev. 109, 2211 (1958). 81. A. Lundby, A. P. Patro, and J. P. Stroot, Nuovo cimento 6, 745 (1957); 7, 891 (1958). 81. K. Alder, B. Stech, and A. Winther, Phys. Rev. 107,728 (1957); see also M. Morita and R. S. Morita, Phys. Rev. 107, 1316 (1957). 83. 0. Kofoed-Hansen and A. Winther, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 30, No.20 (1956).

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Quantum Efficiency of Detectors for Visible and Infrared Radiation R. CLARKJONES Research Laboratory, Polaroid Corporation, Cambridge, Massachusetts Page I. Introduction and Summary.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 A. Introduction. .... B. Summary ....... . . . . . . . . . . . . . . . . . . 89 C. Elementary Detector Concepts .................... 91 92 11. Responsive Quantum Efficiency., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Detective Quantum ............................ 94 A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :. 94 B. Elementary Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 1. The Ideal Detector.. .................... 2. Definition of the Detective 3. The Quasi-Ideal Detector. . . . . . . . . . . . . . . . . . . . . . . 99 4. Alternative Expression for C. A More Rigorous Discussion.. . ........................ 101 1. The Ideal Detector.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2. Definition of the Detective Quantum Efficiency.. . . . . . . . . . . . . . . . . . . 104 3. The Quasi-Ideal Detector.. . . . . . . . . . . . . . . . . . . . . . . . IV. Detectivity and Contrast Detectivity A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Some Definitions. ........................................... 107 C. Properties of the .-Ma Curve.. ................................ 108 D. Increasing &D by a Local Source of Radiation.. . . . . . . . . . . . . . . . . . . . . . . 109 E. Increasing &D by a Neutral Filter. . ............................. 110 F. The Useful Range of a Detector; Underloading and Overloading.. ...... 111 G. Method of Comparing Television Camera Tubes with Photographic Films. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Photoemissive Tubes. ...................... A. Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 B. Responsive Quantum Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Detective Quantum Efficiency .................... 120 VI. Photoconductive Cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Introduction.. . . . . . . . . . . . . . . . . . ............................. 121 B. Responsive Quantum Efficiency. . ............................. 123 C. Detective Quantum Efficiency.. .. ............................. 125 1. Cadmium Sulfide Cells.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 2. Lead Sulfide Cells.. . . . . . . . . . . . . . . . . . . . .................... 126 .................... 128 3. Conclusions. ......................... VII. Television Camera Tubes. .......... ............................... 128 A. Introduction ............................ . . . . . . . . . . . . . . . . . . . . . . . . . 128 B. Responsive Quantum Efficiency.. .................................. 129

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Page C. Detective Quantum Efficiency... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 1. The Basic Data.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 ........................ 131 2. Derivation of a Working Formula for QD.. 3. Results for Image Orthicons.. . . . . . . . . . . . . . . . . . . . . . . . 4. Detective Quantum EAiciency of the 6326 Vidicon.. . . . . . . . . . . . . . . . 135 5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 VIII. Photographic Negatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 A. Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Responsive Quantum Efficiency.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 C. Detective Quantum Efficiency.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 1. Derivation of a Working Formula for Q D . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Description of the Films.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Sensitometric Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Granularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5. Wavelength Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6. Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 IX. Human Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 B. Responsive Quantum Efficiency., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 C. Detective Quantum Efficiency... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 ................................. 157 1. The Signal-to-Noise Ratio k 2. The Experimental Data.. .. ................................ 160 3. Derivation of a Working Formula for Q B . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4. Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 X. Other Detectors.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Heat Detectors in General.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 €3. The Golay Pneumatic Heat Detector. . . . . . . . . . . . . . . . . . . 174 C. Thermocouples and Bolometers. ......... . . . . . . . . . . . . . . . 174 D. Back-Biased p-n Junctions. ................................. 175 E. Photovoltaic Cells. . . . . . . . ......................... F. Photoelectromagnetic Detectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 G. Photosynthesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . .................................... 178

I. INTRODUCTION ASD SUMMARY A . Introduction In this review of the quantum efficiency of radiation detectors, chief emphasis is given to photoemissive detectors, photoconductive detectors, television camera tubes, photographic negatives, and t8hehuman eye. Writers have employed many different kinds of quantum efficiency. Nearly all of them relate to responsivity and are accordingly called a responsive quantum effciency. There is one kind of quantum efficiency, however, the kind here called detective quantum efficiency, that is of particular importance in connection with the detecting ability of detectors. The concept of detective quantum efficiency was first formulated by Rose,

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and his review (1) of image detectors in Vol. I of this series makes extensive use of the concept. The definition of the detective quantum efficiency involves the concept of “ambient radiation.” The ambient radiation is now defined as a certain part of the total radiation incident on the detector. There are three principal sources of such radiation: (1) the signal radiation that is to be detected, usually varying with time; (2) the blackbody radiation field produced by the detector and its environment; (3) other steady radiation, such as daylight, moonlight, or steady manmade illumination. The combination of (2) and (3) we call “ambient radiation.” Ambient radiation is the steady radiation that falls on the detector. There are two quite different kinds of situations in which one may be interested in the detectivity of a radiation detector: the photon noise of the ambient radiation may be negligible, or it may be the dominant noise. I n the first situation the noise produced in the output of the detector by the quantum fluctuations of the steady ambient radiation is small compared with the other noises in the output. Thermocouples, bolometers, and photoconductive cells are usually operated under this condition. The extreme case of the first kind of situation occurs when the only radiation incident on the detector is the blackbody radiation appropriate to the temperature of the detector and the signal radiation that is to be detected. This extreme situation was the only one considered in the writer’s review (2) of detector performance in Vol. V of this series. I n contrast with the first situation, the second situation involves an ambient radiation field whose intensity is such that the statistical fluctuations in this “steady” field produce the dominant noise in the output of detector-that is to say, the detectivity is limited by the photon noise of the ambient radiation field. Human vision and multiplier phototubes usually operate under this condition. The intensity of the incident radiation required to make the photon noise the dominant noise depends, of course, on the kind of detector. It is difficult t o make the photon noise dominant in the output of a lead sulfide cell, whereas i t is only under unusual circumstances that the photon noise is not the dominant noise in the output of a multiplier phototube. The concept of detective quantum efficiency is the appropriate means to characterize the detecting ability of a detector in the second kind of situation, and the primary purpose of this review is to describe the detecting performance of a substantial number of detectors from this point of view.

B. Summary Some of the quantitative results of this article are summarized in Figs. 1 and 2. Figure 1 shows the detective quantum efficiency (DQE) of three RCA television camera tubes, four different Kodak films, and human

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foveal vision plotted against the intensity of the ambient radiation as measured in ergs per square centimeters. In Fig. 1 all of the other parameters, such as radiation wavelength, area of the signal, etc., have been adjusted to the value that maximizes the DQE. Figure 2 shows the DQE of the same group of detectors plotted against the wavelength of the radiation, with all of the other parameters adjusted to the value that maximizes the ordinate. Data on the way that the DQE depends on the other parameters are given in the appropriate sections. 101

I

I

I

I

I

I

I

IMAGE ORTHICONS

VISION

FILM

I

,/ /

~

16~

-I

10

I

10

2 10

EXPOSURE IN ERGSICM~

FIG. 1. The detective quantum efficiency &O plotted versus the ambient exposure. Only the three image-forming detectors are summarized in this figure: television camera tubes, photographic negatives, and human vision, which are the subjects of Secs. VII, VIII and IX. The dashed lines have slopes of plus or minus 1 and indicate the way that the performance can be improved by the use of added ambient illumination or by the use of neutral filters, as discussed in Sec. IV. The results shown are for optimum choice of all of the other parameters that affect the DQE, such as wavelength, size of signal area, signal duration, etc. For human vision the method of calculating the exposure U from the luminance B is described in Sec. IX,C,4. For the camera tubes, U is calculated from the irradiation H of Sec. VI1,C on the basis of an exposure duration of $50 sec.

The concepts of responsive and detective quantum efficiency are the subjects of Sew. I1 and 111. Section I V shows in a quite general way how the detective quantum efficiency is related to detectivity and (‘contrast detectivity” (3). Two kinds of non-image-forming detectors (photoemissive tubes and photoconductive cells) are discussed in Secs. V and VI, and this is followed by the discussion of three kinds of image-forming detectors (television camera tubes, photographic negatives, and human vision) in Secs. VII, VIII, and IX. A number of other detectors are considered in Sec. X.

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C . Elementary Detector Concepts This section is now concluded with a brief summary of the more elementary concepts relating to radiation detectors. The responsivity R of a detector is the ratio of the detector output (usually in volts or amperes) to detector input (usually in watts or lumens). Thus, the responsivity may be expressed in volts per watt. The responsivity of a detector usually depends on the operating temperature, the wavelength of the radiation, the modulation frequency of the radiation, the sensitive area of the detector, and the speed of response. I

I

I

I

FIG.2. The detective quantum efficiency Q D plotted versus the wavelength of the signal radiation and ambient radiation. The detectors are the same as those involved in Fig. 1. The results are for optimum choice of all of the other parameters that affect the DQE, such as amount of ambient radiation, size of signal area, signal duration, etc.

The noise N of a detector is the rms fluctuation in the output expressed in terms of the detector output. Thus, N may be expressed in rms volts, or rms amperes. The noise equivalent input (NEI) is the rms fluctuation in the output expressed in terms of the detector input. Thus, the NEI may be expressed in watts or in lumens. When the NEI is expressed in watts, it is called the

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noise equivalent power and is denoted PN. Sometimes, and with photographic negatives, it is convenient to introduce the noise equivalent energy E N . I n terms of the responsivity R and the noise N , the noise equivalent input is defined by

NEI = N / R

(1.1)

The noise equivalent input is psychologically upside-down. A detector with a greater detecting ability has a smaller noise equivalert input. To avoid this difficulty it has been customary to use the term sensitivity to denote the right-side-up concept. But the term sensitivity has been used to denote both the reciprocal of noise equivalent input and the simple concept of responsivity. To avoid the ambiguity of the term sensitivity, the author in 1952 introduced (4) the term detectivity t o denote the reciprocal of the noise equivalent input. The detectivity D is defined by 9 =

R/N

(1.2)

The detectivity depends on the five quantities listed above on which the responsivity depends and also on the frequency bandwidth of the noise. If the radiation input is measured in terms of its power P, then the responsivity R determines the signal output S :

RP and the detectivity D determines the signal-to-noise ratio : S

S

=

(1.3)

= DP

The operation of many kinds of detectors is described in the recent book by Smith, Jones, and Chasmar ( 5 ) . A review of detectors with emphasis on their detecting ability was given in Vol. V of this series ( 2 ) . Neither of these references considers detectors from the point of view of detective quantum efficiency. 11. RESPONSIVE QUANTUMEFFICIENCY The term eficiency without the term quantum appended usually means the ratio of an output power to an input power, or perhaps the ratio of an output energy or output free energy to an input energy. But the term quantum efficiency means something else. The responsive type of quantum efficiency is always the ratio of the numbers of two kinds of countable events. For example, the quantum efficiency of a simple vacuum phototube is usually defined as the ratio of the number of electrons that reach the anode and flow in the external circuit to the number of photons that are incident on the photocathode.

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Thus, we define the responsive quantum efficiency (RQE) of a detector as the ratio of the number of countable output events to the number of photons that act on the device. It may serve a useful purpose to review briefly the various definitions of RQE that have been used for detectors. For any detector, the number of input events may be either the number of incident photons or the number of absorbed photons. Historically, the choice has been different for different types of detectors. With photoemissive tubes the incident photons are usually counted, whereas with photoconductive cells it is the absorbed photons that are usually counted. With photoconductive cells, the method used in all of the early work of Gudden and Pohl is to consider the output event as the flowing of a n electron in the external circuit. This, of course, made the RQE depend on the applied voltage. All workers today consider the creation of an electron-hole pair to be the output event. The changeover of point of view in this respect is well described by Rose (6). With respect to human vision, various workers have studied the minimum number of absorbed photons that are required to elicit a sensation of vision in the dark-adapted eye. The reciprocal of this number is the ratio of the number of perceptions of light to the number of absorbed photons and thus may be considered to be a RQE. Rose in 1942 (7) defined the performance in terms of the number of photons required to produce a just detectable signal in a single resolution element. This method of evaluation was abandoned by Rose (8) in 1946 in favor of the detective quantum efficiency that is defined below in Sec. 111. Photographic negatives have been left until last because with this detector we find the greatest variety of possible definitions of responsive quantum efficiency. With this detector the input event may be either the incident photon, the absorbed photon, or the photon that is absorbed in a photographically relevant manner. The output event may be either the absorption of a photon in a photographically relevant manner or the event of a grain’s becoming developable. Furthermore, the number of input events may be either the actual number under a given set of conditions or the minimum number required to effect the chosen output event. There are undoubtedly other possible definitions of the RQE of a photographic negative, but the list given is sufficient to show the complexity. These examples are perhaps sufficient to indicate the wide variety of the possible definitions of a RQE. Thus, when anyone says, “The quantum efficiency of detector A is 2070,’1very little information is carried by this statement until he states just what kind of (responsive) quantum efficiency he is talking about. The RQE may be greater than unity. For example, the responsive quantum efficiency of a multiplier phototube may be 100,000, if this num-

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ber of electrons reach the anode for each photon incident on the photocathode. In contrast, the detective quantum efficiency defined in the next section cannot be greater than unity.

111. DETECTIVEQUANTUM EFFICIENCY A. Introduction

As we saw in the preceding section, the term quantum efficiency has usually meant responsive quantum efficiency, with the result that a simple phototube may have a quantum efficiency of 0.1, whereas a multiplier phototube with the same kind of sensitive surface may have a quantum efficiency of 100,000. This situation is one that cries for some unifying concept, and I believe that the concept of detective quantum efficiency introduced by Rose (8) is the answer t o this need.’ The reader may have in mind the notion of a “fundamental” kind of quantum efficiency, which is probably expressed in some intuitive terms such as the following. Suppose N i photons of a certain wavelength are incident on a detector. Of these photons, a certain number N , will be absorbed. Usually, not all of the absorbed photons will be effective in stimulating an electrical output of the detector; specifically, suppose that N, of the photons are effective in producing the excitation that contributes to the electrical output. Then most of us would be willing to agree that the ratio N,,”,, the ratio of effective to incident quanta, is a good measure of a ‘(fundamental” kind of quantum efficiency. The definition of a “fundamental” quantum efficiency in the preceding paragraph is not an operational one. With many kinds of detectors, including the human eye and photographic negatives, it is difficult or impossible t o measure the number N,,or even to say exactly what we mean by “effective” photons. The concept of detective quantum efficiency, however, does have a clean-cut operational definition : The detective quantum efficiency of an actual detector is defined as the square of the ratio of the measured detectivity of the detector to the maximum possible detectivity on the given signal in the presence of the given ambient radiation. The method of calculation is suggested by the following example. Suppose we have an “ideal” detector of unit quantum efficiency, by which we mean in this paragraph that one electron flows in the external circuit for each incident photon. Since the photons are statistically independent (see l The concept was formulated by Rose. The name “detective quantum efficiency” was introduced by the author. It is called “equivalent quantum efficiency” by Feligett (9).

QUANTUM EFFICIENCY OF DETECTORS

95

below for qualifications), the electrons that flow in the external circuit are statistically independent, which is another way of saying that the noise in the output is related to the average current by the ordinary shot noise formula. Second, suppose that the detector is then changed so that it responds to only one-half of the incident quanta (accomplished, for example, by placing a filter with a transmission of 0.5 over the detector) and that an amplifier with a gain of 2 is added. Then the responsive quantum efficiency of the second detector is the same as that of the first, but observation of the noise shows that the mean square noise current of the second detector is twice that of the first. Thus, the detectivity of the second detector is less than that of the first by the factor 0.707. If we postulate that the “ideal” detector just described has the maximum possible detectivity in the presence of the given ambient radiation, then it follows from the definition of detective quantum efficiency that the second detector has a detective quantum efficiencyof 0.5. This conclusion obviously accords with our intuitive feeling that the ‘(fundamental” quantum efficiency of the second detector is only half that of the first, since only one-half of the incident photons are actually used in the second detector. In Sec. B we define carefully the concept of an ideal detector, and we then immediately define the detective quantum efficiency in terms of this ideal detector. Then finally we define a quasi-ideal detector as the equivalent of an ideal detector that is degraded by a filter of transmittance F , so that F is the fractional utilization, and then show that the detective quantum efficiency of this quasi-ideal detector is equal to F. The discussion in Sec. B is throughout elementary, in the sense that the emphasis is on the physical concepts. In Sec. C, the concept of detective quantum efficiency is redefined in a more rigorous way.

B. Elementary Discussion 1. The Ideal Detector. An ideal detector is one that makes fully effective use of every photon incident upon the sensitive area of the detector. In this section the detecting ability of an ideal detector is derived. It is supposed that the signal and the ambient radiation both consist of radiation within a narrow band of radiation frequencies and that the band is the same for the signal and for the ambient radiation. It is further supposed that the (modulation) frequency response of the detector may be characterized by an integration time T ;this assumption is made only to simplify the presentation, and the final results will be expressed in a form independent of this assumption. The strength of the ambient radiation will be specified by the average

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R. CLARK JONES

number Ma of photons that reach the sensitive area of the detector in the period of duration T. Similarly, the strength of the steady signal is defined by the average number M , of signal photons that reach the sensitive area in the integration time T. For the sake of simplicity, it is supposed that M , is small compared with Ma. Because of the statistical independence (or near independence) of the individual photons, the number of ambient photons will not be the same in successive integration periods. The average number will be M a as defined above, but the actual number 311, in any given period of length T will usually differ from Ma. The deviation from the mean value in any given period is 311, - Ma. The average value of the square of this deviation is called the mean-square deviation, and the square root of the mean-square deviation is called the root-mean-square (rms) deviation, and is here denoted by AMa: AMa = [((ma- Ma)’)Ix (3.1) where the angle brackets indicate that the quantity within is averaged over a large number of integration periods. Lewis (10) has presented results showing the magnitude of the fluctuation AMRfor radiation from a thermal source. His results are equivalent to =

OMa

(3.2)

where O is a factor that may usually be taken to be unity in the visible and infrared regions. O is defined and discussed below in Sec. C. I n the remainder of this Sec. B, the factor O will be set equal to unity. Thus, one has

AM,

=

Ma%

(3.3)

This result may be derived simply if we assume that the individual photons occur randomly and independently. The number 311, then has a Poisson distribution, and the result (3.3) follows a t once. It turns out, however, that the concept of a sequence of random events is not so simple after all. Fry (11) has examined this concept in detail and finds that there are two ways in which a sequence of events may be random. The events may be “individually a t random” or they may be “collectively a t random.” The reader is referred to Fry’s lucid account for definitions and examples of these concepts. Fry shows that if and only if a sequence of events is random in both of these senses do the events have a Poisson distribution. By a Poisson distribution, we mean that if M a is the mean number in intervals of length T, the probability that exactly 311, events occur in this interval is given by p (311,) = M,3n0e-IMn/~, !

(3.4)

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97

It is easily confirmed that this expression gives M a as the mean number, and more detailed calculation shows that the rms deviation from the mean is given by (3.5) in confirmation of (3.3). (The calculation makes use of the fact that meansquare deviation from the mean is the difference between the mean squared number and the square of the mean number.) The rms noise N , measured in photon numbers, is thus AM,

N

=

(3.6)

=

and the signal S measured in photon numbers is equal to the number M , of signal photons:

s = M,

(3.7)

All these photon numbers are, of course, those that pertain to the integration period T. The signal-to-noise ratio is thus given by

M,/M,” and the noise equivalent number of signal photons is given by SIN

SN

=

= M,N =

Ma”

(3.8)

(3.9)

Usually, because of imperfections in the detector, the actual number of photons required to produce a noise equivalent signal output is larger then the value just given. But we here define an ideal detector as a detector that does achieve the detecting ability corresponding to the last equation. 2. Definition of the Detective Q u a n t u m Eficiency. We are now prepared to define the detective quantum efficiency (DQE) of an actual detector. Suppose that we measure the signal-to-noise ratio ( S I N ) , of an actual detector on a given signal and in the presence of a given ambient radiation. Then the detective quantum efficiency Q D is the square of the ratio of the measured signal-to-noise ratio (SIN),, to the signal-to-noise ratio of the ideal detector on the given signal in the presence of the given ambient radiation: (3.10)

It is clear from this definition that the DQE cannot be greater than unity. The DQE is unity only for the (nonexistent) ideal detector described above and is less than unity for any detector that is less than ideal. We recall from Sec. I1 that the RQE is often greater than unity. It is important to note that M a is the mean number of photons incident

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R. CLARK JONES

on the sensitive area. If the detector and environment are all at the same temperature, this incident flux will be balanced by a n exactly equal flux of reflected or emitted photons, so that the net flux is zero. But Ma is not the net fiux; it is the incident flux. It may help the reader to imagine that the ideal detector is perfectly black and a t absolute zero; then the only photons are the incident photons. In fact, an ideal detector would have to be perfectly black and at absolute zero, but this fact is a consequence of the definition, not a part of the definition (of an ideal detector). The definition of DQE given above is general in that there is no restriction on the amount of the ambient radiation. It is important to realize, however, that the DQE will accord with our intuitive understanding of its significance only when the ambient radiation is sufficient in amount that its fluctuation produces the dominant noise in the output of the detector. The quantum fluctuations in the ambient radiation will always produce noise in the output of the detector, but in practical situations it may be that the noise due to these fluctuations (photon noise) may be buried under other noises of larger magnitude. Only when the dominant noise in the detector output is photon noise may we expect to obtain a value for the DQE that is independent of the amount of the ambient radiation and that accords with the intuitive notion of a “fundamental” quantum efficiency. The definition of the DQE given in the text immediately preceding Eq. (3.10) is the same as the definition in Sec. II1,A only if the ideal detector does have the maximum possible detectivity in the presence of the given ambient. The ideal detector gives equal weight to every incident photon, SO one might ask whether an even higher detectivity might be achieved by giving unequal weights to the various incident photons. For example, the detector might split the integration period T into 100 equal subperiods and give the number of photons that are incident in each subperiod a weight that depends on the number of photons that occur in that subperiod. Such a detector would be nonlinear in the sense that the output voltage would not be proportional to the incident power. Thus, the question here emphasized is whether the maximum conceivable detectivity is given by a linear detector that gives equal weight to every incident photon. I believe the answer is yes: the ideal detector defined above does have the maximum possible detectivity. But I do not have a formal proof, nor does this question appear to be discussed in the literature. My intuitive proof is based on the fact that both the signal photons and the noise photons, considered either separately or in combination, occur collectively and individually at random, with the result that there is simply no way of making an a priori judgment that any given photon is more likely to be a signal photon than any other photon. If, for example, all the signal photons arrived singly and the background photons arrived two a t a

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time, one could easily devise a detector that would reject all of the noise photons. But the essential fact here is that there is no basis for making such a distinction. So, in the absence of a basis for discrimination, the detector must give equal weight to every incident photon. I am indebted to three specialists in noise theory for discussion of this question: Dr. David Middleton, Dr. David Van Meter, and Professor Norbert Wiener. The considerations of the preceding paragraph are perhaps most nearly implicit in a recent paper by Middleton ( 2 2 ) . 3. T h e Quasi-Ideal Detector. The quasi-ideal detector is defined as the combination of an ideal detector, as defined above, covered by a filter that transmits a fraction F of the incident photons. Thus, the quasi-ideal detector is a detector that makes maximal use of the fraction F of the incident photons. We shall show that the detective quantum efficiency QD of a quasi-ideal filter is equal to F. This is the basic justification for calling QD a “quantum efficiency.” Speaking loosely, we may say that the DQE is equal to the fraction of incident photons that are actually utilized by the detector. This demonstration is also the justification for defining the DQE as the square of the ratio of signal-to-noise ratios. If any other power than the second were used in the definition, the DQE would not be equal to the fractional utilization F of a quasi-ideal detector. For the quasi-ideal detector, the effective number S of signal photons is not M , (as i t is for the ideal detector), but is rather FM.:

S = FM,

(3.11)

Similarly, the effective mean number of ambient photons is not Ma, but is rather FM,. Then the rms fluctuation N in the effective number of background photons is given by

N

=

(FMa)’

(3.12)

The square of the signal-to-noise ratio is therefore given by

(X/N)’

=

M,2/FMa

(3.13)

and if this signal-to-noise ratio be considered to be to the “measured” signal-to-noise ratio of the quasi-ideal detector, then it follows a t once from Eq. (3.10) that the detective quantum efficiency QD of the quasi-ideal detector is given by QD

=

F

(3.14)

4. Alternative Expressions for the Detective Quantum Eficiency. We now proceed to express the DQE in terms of quantities that are more directly related to experiment than the photon numbers Ma and M*N.In

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R. CLARK J O N E S

various applications, we may be concerned with the energy El the exposure U , the power P, or the irradiation H . These quantities are related to the number of photons M by E = P T = H A T = &M

(3.15)

where & is the energy of a single photon: & =

(1.9857 X 10-l2 erg)/X

(3.16)

where X is the wavelengths in microns. In some applications, the signal may be expressed in terms of its contrast C defined by

c = M,/Ma

(3.17)

In terms of the contrast C, the DQE may be written in the following four different forms: (3.18) (3.19) (3.20) (3.21) In terms of noise equivalent parameters, the DQE may be writ,ten in the following four different forms: (3.22) (3.23) (3.24) (3.25) Other forms are also possible but these eight forms are more than sufficient for the purposes of this article. For electrical detectors, it is convenient to transform the expression so tJhatit involves a frequency bandwidth Af rather than an integration time T. Since the relation between these two parameters is 2TAf = 1, Eq. (3.24) becomes QD

=

2&PaA

=

2EPaD2Af

(3.26)

where D = 1/PN is the detectivity. This is the form of the expression for the detective quantum efficiency for detectors with an electrical output. The detective quantum efficiency may be defined in still another way, as the ratio of the mean-square fluctuation in the incident power to the square of the measured noise equivalent power of the detector: QD

=

((Pa

-

paI2)/pN2

Jn this form, it is perhaps most clear that the reciprocal of

QD

is a kind

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101

of noise figure. 1 / & differs ~ from the usual noise figure, however, in that the reference noise is radiation noise = photon noise, whereas the reference noise of the ordinary noise figure is Johnson noise. Photon noise and Johnson noise differ in concept and in amount. (In the special case of the radio antenna, they become identical in concept.) The detective quantum efficiency has been defined only for nearly monochromatic radiation. Both the signal radiation and the ambient radiation must be nearly monochromatic and of the same wavelength. This is not essential to the definition of the detective quantum efficiency; the ambient radiation could be permitted to have a n arbitrary spectrum. But this degree of freedom would greatly complicate the expressions and the concepts, and all in all it does seem preferable to restrict the ambient radiation to being nearly monochromatic and of the same wavelength as the signal. A slight departure from this principle is made only in Sec. X, in connection with heat detectors. Several persons have suggested that the word “quantum” be removed from detective quantum efficiency so that it becomes “detective efficiency.” The concept behind the suggestion is that the word “quantum” implies that the concept is applicable only to “quantum” detectors, by which is meant detectors like photoemissive and photoconductive detectors, as distinct from detectors like thermocouples and radio antennas. Actually, the concept of a “quantum” detector cannot be made rigorous except by enumeration. I feel that the advantage of retaining the close relation of the concept t o the responsive quantum efficiency makes it desirable to retain the word (‘quantum,” but the reader should be prepared to find other writers using the phrase “detective efficiency.” The remainder of this section may be omitted by those who are satisfied with the definitions given so far. For those, however, who are uncomfortable about the lack of rigor, we now present a more rigorous discussion.

C . A More Rigorous Discussion I n this part we derive the expression for the DQE of a detector with more attention to some kinds of details than in the preceding part. The same order of discussion will be employed: the ideal detector, the definition of the DQE, and the quasi-ideal detector. 1. The Ideal Detector. The results of Lewis ( l o ) ,based on the quantum statistics of an unidirectional stream of photons, state that the mean-square fluctuation in the power incident on the surface of a detector is given by AP2 = 2GP,Odvdf

(3.27)

where AP2 is the mean-square fluctuation of the power in the radiation frequency band of width dv, and in the fluctuation frequency band of width

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R. CLARK JONES

df. E is the energy hv of a photon. P, is the power per unit radiation frequency. fi is called the coherence factor and is defined by =

Q

(1 - e - - h v / k y

(3.28)

where v is the radiation frequency and T is the thermodynamic temperature of the radiation. (For a detailed discussion of the significance of the radiation temperature T , see Planck's book, I S , or an article, ref. 14, by the writer.) If the radiation reaches the detector from a blackbody of absolute temperature T and if the radiation suffers neither absorption nor scattering in its path from the source to the detector, then the temperature of the radiation is equal t o the temperature of the source. The factor 8 is a measure of the degree to which the photons are clumped in the radiation from a thermal source. Photons are Bose particles, and Bose particles in thermal equilibrium have a very fundamental tendency t o clump. The average occupation number of the quantum states of the radiation field is equal to 8 - 1. A few numerical values follow. At a radiation wavelength of 10 p and a temperature T = 300" K, D has the value 1.007. Q has the same value for A = 1.0 p and T = 3000' K. For both of these cases, hv/kT has the value 5. But for A = 10 p and T = 3000" K, hv/kT is about 55, and 8 is therefore about 2.5. Thus, for high temperatures and long wavelengths, the factor D departs significantly from unity. D reaches very large values in the radio region. At the wavelength of 1 meter and T = 300°K, D has the value 500,000. The large value means that the photons in thermal radiation a t radio wavelengths are very highly clumped. The importance of the factor 8 for the detailed understanding of the ideal heat detector is discussed in Sec. I1 of ref. 2. The coherence factor D may also be expressed in terms of the spectral radiance of the radiation : D

=

1

+ c2N,/2hv3

(3.29)

where N , is the spectral radiance of the radiation (assumed to be unpolarized). For a given geometry of the radiation incident on the detector the spectral radiance is proportional to the power P,, so that one may also write 8 = 1

+ constant - P,

(3.30)

where the "constant" is independent of the radiation temperature T and of the power P,. The total ambient power is given by

Pa =

/omPu(v)dv

(3.31)

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103

and the total fluctuation of the power in the fluctuation bandwidth d j is given by

AP2 = 2df

/omhvP,(v)Q(v,T(v))dv

(3.32)

where the symbolism Q(v,T(v)) is intended to indicate that the radiation temperature T of the radiation depends only on the radiation frequency v. (T could also depend on the direction of incidence of the radiation and could differ for the two opposite states of polarization (14), but no useful purpose would be served by introducing this degree of generality in the present discussion.) We now define the ideal radiation detector as the combination of a detector proper and an ideal amplifier of gain G . The gain G is supposed to depend on the frequency f of the modulation of the incident power. The detector proper causes one electron to flow in the electrical output circuit for each photon that is incident on the detector. The combination, which we call the ideal detector, thus converts changes in incident power into changes in current in accordance with the transfer ratio I / P = e G ( f ) / a , where e is the charge of the electron. It then follows that the mean-square fluctuation in the current in the output of the ideal detector is given by A12 = 2

/om (e2/hv)P,(v)Q(v,T(v))dv/om G2(f)df

(3.33)

We now suppose that the ambient power described by P, is confined to a narrow band of radiation wavelength centered a t the frequency VO. Then the last expression may be written in the compact form

AP

=

(2ezGm2/hvo)P,SLAf

(3.34)

where Af and SL are defined by (3.35) (3.36) and where G , is the maximum value of G (f). We now consider the response to a radiation signal. Let P, be the power of a radiation signal that is modulated sinusoidally at the frequency So. More precisely Pa is the rms value of the deviation of the instantaneous power from it mean value. Like the ambient power, the signal power P, is supposed t o be confined to a narrow band of radiation frequencies centered a t the frequency yo. The rms current output due to the radiation signal is then given by

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R. CLARK JONES

The noise equivalent value PN of the radiation power P , is obtained by setting I,2 equal to AP. One thus finds PN= ~ 2&P,PAf[Gm2/G2(fo)]

(3.38)

where & now denotes hvo. Finally, if the frequency fo of the radiation signal is equal t o the frequency that maximizes G(f), the square bracket is unity and the last equation becomes

P N= ~ 2&P,PAf

(3.39)

This is the desired expression for the noise equivalent power of an ideal detector. The last three factors are defined by Eqs. (3.31), (3.36), and (3.35). 2. Definition of the Detective Q u a n t u m Eficiency. We are now ready to define the detective quantum efficiency QD. We suppose that the noise equivalent power is measured with a given amount of ambient radiation incident on the detector. The measured value is denoted PNm and its reciprocal, the measured detectivity, is denoted %. Then the DQE is defined by &D

=

(pN/pNm)2

(3.40)

This relation may be written in two other ways:

QD

=

a ) m 2 P ~=2 2&P,Dm2PAf

(3.41)

There is a hidden subtlety, however, in the result just given, that relates to the factor P. This subtlety is discussed in connection with the quasi-ideal detector, in See. 3 below. Except for the factor P, the expression just found for the detective quantum efficiency is formally the same as that given by (3.26). There are real differences, however. Perhaps the most important difference is that (3.41) has been established for a detector of arbitrary frequency response; the bandwidth Af has a precise definition (3.35) in terms of the frequency response of the detector. The derivation of (3.41) has made clear that it holds only for narrow radiation frequency bandwidths, and thus the detective quantum efficiency defined by (3.41) may be and should be considered to depend on the wavelength of the radiation used to measure it. 3. T h e Quasi-Ideal Detector. Just as in Sec. B, we here consider a quasiideal detector defined as a detector that is ideal in every respect except that it makes use of only a fraction F of the incident photons, instead of making use of all of them. Also as in See. B, the purpose of this See. 3 is primarily to justify the definition of the detective quantum efficiency as the square of the ratio of the two signal-to-noise ratios. Still another purpose is to bring out an ambiguity in the definition of the quasi-ideal detector that has an important

QUANTUM EFFICIENCY OF DETECTORS

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bearing on the suitability of the definition of detective quantum efficiency given in Sec. 2 above. In order to find the noise equivalent power PN of the quasi-ideal detector we must replace Pv, Pa, and P, in Sec. III,C,2 by FP,, FP,, and FP,. We must also consider how the introduction of the factor F influences the quantity Q. This is a subtle question. We assume that the quasi-ideal detector is the combination of an ideal detector and a filter of transmittance F. Suppose (first supposition) that the filter is a homogeneous layer of absorbing material. Then the filter reduces the spectral radiance of the ambient radiation, and the proper expression for the coherence factor of the radiation that has passed through the filter is QF =

1

+ c2FN,/2hv3

(3.42)

where N , is the spectral radiance of the incident ambient radiation. But now suppose alternatively (second supposition) that the filter is a wire screen that blocks off a fraction 1 - F of the cross section of the incident radiation. Then the spectral radiance is unchanged by the filter, and the coherence factor is as given by Eq. (3.29). On the basis of the first supposition, the noise equivalent power of the quasi-ideal detector is given by

PN2 = 2EPaQFAf/F

(3.43)

whereas on the second supposition it is given by

Plv2 = 2EPaQAf/F

(3.44)

Then on the basis of the first supposition the DQE of the quasi-ideal detector is QD =

QF/QF

(3.45)

whereas on the second supposition it is given by QD

=

F

(3.46)

Thus, the success of our aim to show that the detective quantum efficiency of a quasi-ideal detector is equal to F turns out to depend on how we imagine our quasi-ideal detector to be constructed. This difficulty is closely related to a current controversy. Both Fellgett (16) and this writer (16)have considered the ultimate performance of heat detectors as given by a statistical mechanical argument, and (probably incorrectly) we both indicated that these results were very general and should hold for all radiation detectors, including photoemissive tubes. In a recent article Hanbury-Brown and Twiss (17) disagreed with our conclusions and indicated a different conclusion for photoemissive tubes.

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R. CLARK J O N E S

Now the difference between our two different conclusions is just the difference bet,ween the two kinds of filters indicated above. Fellgett and the writer have shown that the noise equivalent power of an “ideal” heat detector with emissivity E is given by Eq. (3.44) with F set equal to E. Hanbury-Brown and Twiss find that the noise equivalent energy of a photoemissive surface with responsive quantum efficiency q is given by Eq. (3.43) with F set equal to q. Thus, the Fellgett-Jones results accords with the second supposition, whereas the Hanbury-Brown and Twiss result accords with the first supposition. With hindsight we can see that these results are reasonable. I n the case of a heat detector, the temperature of the detecting element is necessarily in equilibrium with its surroundings; the chief ambient radiation is the blackbody radiation appropriate to the temperature of the environment. If now we imagine placing a Glter over a heat detector of unit emissivity, it is clear that the filter must have the same temperature as the detector, since otherwise it would change drastically the operating temperature of the detector. But if the filter has the same temperature as the detector, then its interposition does not change the ambient radiation that falls on the detector: the filter radiates just as much radiation as it absorbs. Thus, it is clear that the filter does not change the spectral radiance of the ambient radiation, and therefore only the second supposition is valid. But the situation is different with a photoemissive tube. Here the ambient radiation is not blackbody radiation of the temperature of the photoemissive surface; the phototube does not respond significantly to such radiation. Thus, the argument just given for the second supposition does not carry through for the phototube. My conclusion is as follows. For heat detectors, the expression (3.41) with z;2 defined by (3.29) is the correct expression for the DQE. For photoemissive tubes and other detectors not in thermal equilibrium with the ambient radiation, perhaps the factor 0 should be replaced by (3.47)

We choose not to make this replacement; if we did, we would find that Q D would occur on both sides of Eq. (3.41), and we would find that Q D would be given as the positive root of a quadratic equation. We wish to avoid this complication. Furthermore, the decision to stick with Eq. (3.41) leaves us with a single formula for the DQE, albeit a somewhat arbitrary formula. (The current status of the controversy mentioned above is as follows: I prepared a report dated March 1, 1958, and circulated it to everyone I knew to be interested in the subject; I described the area of agreement, and showed just where the two groups disagreed. I emphasized that both arguments seem unassailable in their respective field of application and

QUANTUM EFFICIENCY O F DETECTORS

107

that the real problem that remained was to reconcile the two results. Dr. Hanbury-Brown replied in a letter dated April 10, 1958, agreed that there was a real problem, and indicated that he knew of two graduate students who were working on the problem, without success so far. Dr. Twiss replied in a letker dated May 13, 1958, and suggested what seems to me a convincing resolution of the problem. His solution suggests that the second supposition is correct for a detector in which the ingoing and outgoing radiation fluxes are equal, whereas the first supposition holds when these fluxes are very unequal. His result also includes the intermediate case where the fluxes have an arbitrary ratio.)”

IV. DETECTIVITY AND CONTRAST DETECTIVITY A. Introduction In this section we shall discuss the relation between the detective quantum efficiency and the other two useful approaches to detector performance: detectivity and contrast detectivity. We shall also show that some detectors have a restricted range of useful ambient radiation. For example, if the exposure of a photographic negative is less t,han a certain amount, the DQE can actually be increased by preexposing the film. On the other hand, with some detectors such as the vidicon, photographic negatives, and (probably) human vision, there is a critical value of background radiation above which the DQE can actually be increased by covering the detector with a filter that attenuates both the signal radiation and the ambient radiation. It will be shown that these two limits, the lower and the upper limit on the ambient radiation, correspond to the values of the ambient radiation that maximize, respectively, the detectivity and the contrast detectivity.

B. Some Dejinitions For reference, we repeat here the definition of the detective quantum efficiency:

The detectivity a> for the purposes of this section is now defined as the reciprocal of the noise equivalent number of signal photons:

* Note added in proof: This resolution of the controversy is described in a manuscript “Fluctuations in photon streams,” by P. B. Fellgett, R. Clark Jones, and R. Q. Twiss, which has been submitted to Nature.

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R. CLARK J O N E S

The contrast detectivity3 is the noise equivalent value of the ratio M,/M,:

Thus, if, for example, the noise equivalent contrast is 0.8%, the contrast detectivity is 125. It is immediately obvious that QD is equal to the product of 3 and D,: QD

c

SLOPE

(4.4)

= 99,

+I

log Ma FIG.3. A schematic plot showing the detective quantum efficiency &D plotted against the amount of the irradiation of the detecting surface. If the irradiation is less than the amount a t the point A (where the curve has a slope of +l),the detector is underloaded, and the DQE can be increased by adding additional ambient radiation. If the ambient irradiation is more than that of the point C (where the curve has a slope of -l), the detector is overloaded, and the DQE can be increased by placing a neutral filter over the detector. All detectors can be overloaded, but only some detectors can be underloaded (see Sec. IV for a full discussion).

C. Properties of the QD-vs-M, Curve Consider the imaginary detector whose detective quantum efficiency Q D is plotted versus the ambient photon number M a with logarithmic coordinates in Fig. 3, and consider further the two straight lines with slopes of plus and minus one that are tangent to the curve (at the points A and C). It is not meant to imply that all detectors have a curve with slopes of both and -1. A multiplier phototube has no point A , and furthermore, it

+

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probably has no point C until the radiation is so intense that it heats up the photocathode. Photographic negatives always have points A and C. The point B represents the maximum value of the DQE with respect to variation of the ambient radiation Ma. It will now be shown that the points A and C correspond to the ambients that maximize D and D, respectively. I n general the slope of the curve in Fig. 3 is given by (4.5)

At the point A , this slope is + l . If one sets the above expression equal t o +1, and performs a little reduction, one finds d9 -dMa =o

Similarly, a t the point C, the slope given by (4.5) is -1. One finds

The last two equations are, of course, the formal conditions that D and D,, respectively, be stationary with respect to variation of Ma.

D. Increasing QD by a Local Source of Radiation We shall show that if the ambient radiation is less than the amount that corresponds to the point A in Fig. 3, the DQE of the detector can be increased by deliberately increasing the steady radiation that falls on the detector. This can be done, for example, by letting a local source of steady radiation act on the detector. Consider a point (on the curve in Fig. 3) that is to the left of point A , such as the point D. We shall show that the effective value of the DQE can be raised vertically from D to the point E if, without changing the number of signal photons, the number of ambient photons is increased (by a local source) so that the total number or ambient photons is the same as at the point A. In this Sec. D and also in Sec. E, we shall simplify the calculations as much as possible by supposing that the number of signal photons in the original signal to be a given fixed number. Subscripts will be added to M a to indicate the point to which the value of Ma refers. I n the argument to be given, we shall throughout use the definition (4.1) of the DQE: the ratio of the signal-to-noise ratio squared ( S / N ) 2in the detector output to the square of the signal-to-noise ratio LW,~/M,in the radiation input to the detector. At the point A , the DQE is the ratio of the output signal-to-noise squared ( S / N ) A *to the input signal-to-noise squared, M?/Ma* :

110

R. CLARK JONES

To compute the value of the DQE at the point D or El we note that the input signal-to-noise squared is M , 2 / M a ~Furthermore, . if the ambient number of photons is increased by the local source so that the number of photons is the same as a t A , both the number of signal photons and the number of ambient photons is the same as a t A , and therefore the output signal-to-noise squared is the same as a t A . Thus, the value of the DQE with the local source is

By eliminating M , between the last two equations, one finds QLS =

(MaD/MaA)&A

(4.10)

In words, the value of the DQE with the local source is the value at A multiplied by the ratio M a D / M a ~whence , by simple geometrical reasoning i t follows that QLS is the value a t the intersection of the line with slope +1 and the vertical line through D; that is to say, QLs is the value of the DQE a t the point E. This is what we set out to prove.

E. Increasing Q D by a Neutral Filter We shall show that if the ambient radiation is greater than the amount corresponding to the point C, the DQE of the detector can be increased by placing a neutral filter over the detector. The filter attentuates the signal radiation and the ambient radiation by the same amount. Consider a point (on the curve in Fig. 3) that is to the right of point C, such as the point F. We shall show that the effective value of the DQE can be raised vertically from the point F to G if a neutral filter is placed over the detector that reduces the number of ambient photons that reach the detector t o the amount a t C. The value of the DQE at the point C is the ratio of the output signal-tonoise squared ( S / N ) c 2to the input signal-to-noise squared Ms2/Mac: (4.11)

The transmittance T of the filter that reduces the ambient M,F to Mac is

T

=

Mac/MaF

(4.12)

To compute the value of the DQE with the ambient MaF and with the filter in place, we note that the effect of the filter is to reduce the number of signal photons by the factor T. Thus the output signal-to-noise squared is T2

QUANTUM EFFICIENCY OF DETECTORS

111

times ( S / N ) c 2 .The input signal-to-noise ratio is Ma2/M,F. The value of the DQE with the filter in place is therefore (4.13) By elimiiiating T and M , among the last three equations, one finds &fil

= (MaC/MaF)&c

(4.14)

I n words, the value of the DQE with the filter in place is the value a t C multiplied by the ratio M,c/M,F. From simple geometrical reasoning it follows that & f l l is the value of the DQE a t the intersection of the line with slope - 1 and the vertical line through F ; that is to say, Q f l l is the value of the DQE a t the point G. This is what we desired to prove.

F. The Useful Range of a Detector; Underloading and Overloading It is a simple consequence of the results established in the preceding two parts that the useful range of the detector itself is limited to the range of ambients lying between A and C . If the ambient radiation is greater than the amount a t C, the detector is ((overloaded”: the performance of the detector can be improved by placing over the detector a filter that reduces the ambient radiation to the value that maximizes the contrast detectivity. If, on the other hand, the ambient radiation is less than that a t A , the detector is “uiiderloaded”: the performance of the detector can be improved by deliberately increasing the ambient radiation by a local source to the value that maximizes the energy detectivity. All detectors without exception can be overloaded : all detectors can be destroyed by sufficiently intense radiation. But not all detectors can be underloaded. Human vision and photoemissive tubes cannot be underloaded. But photographic negatives can always be underloaded. G. Method of Comparing Television Camera Tubes with Photographic Films An uiiderloaded detector may show strikingly poor performance. A good example is presented by the series of pictures shown in Fig. 4, reproduced with permission of Rose, Weimer, and Law (18). These pictures compare the performance of the image orthicon and Super-XX film. (We shall show that there is a sense in which the comparison favored the image orthicon in that no effort was made to correct for the underloading of the film.) I n each of the four pictures shown in Fig. 4, the model is on the right, and an image, picked up by an image orthicon and shown on the screen of a television receiver, is shown on the left. The four pictures cover a hundredfold change in the illumination of the model. The photographs in Fig. 4 were all taken with a 35-mm still camera, with Super-XX film and

112

R. CLARK J O N E S

with an exposure duration of 1/30 sec. Both the television camera and the 35-mm camera used an j / 2 lens. Under these conditions both the image orthicon and the film receive approximately equal numbers of signal photons and equal numbers of ambient photons in the 1/30-sec frame time; and, therefore, the signal-tonoise ratios in the images were proportional to the square roots of the detective quantum efficiencies. The four pictures show that the image orthicon continues to provide a clear signal even when the photographic system fails to provide any image a t all. Now this is precisely what we would expect from the results shown in Fig. 1, unless measures are taken to correct the underloading of the film.

0.02

0.2

0.07

2

FIG.4. Four pictures showing the comparison made by Rose, Weimer, and Law of the performance of image orthicons and Super-XX film. A 35-mm photographic camera and a television camera both viewed the same subject, and the pictures show the view obtained with the photographic camera. The model is on the right in each picture, and a television receiver showing the image picked up by the television camera is on the left. The two cameras used lenses of the same aperture and focal length. The luminance of the model varied over a hundredfold range and is indicated under each picture, in footlamberts. The text explains how this comparison is partial to the image orthicon, since no effort was made to correct the underloading of the film in these tests.

QUANTUM EFFICIENCY OF DETECTORS

113

(Films and image orthicons have been much improved since 1946; thus, in invoking data from Fig. 1 we must rather arbitrarily select the items to be compared. We choose the Royal-X film and the 5820 image orthicon.) Figure 1 shows that although the Royal-X film has its highest DQE a t about erg/cm2, the DQE has a very low value indeed a t one-third of this ambient exposure. Thus, for values of the ambient exposure less than about 3 X lo-* erg/cm2, the film gave no image a t all. But a t the same ambient exposure, the 5820 image orthicon has a DQE greater than 1%, and the DQE is falling with a slope that is less than fl. Consider, however, the effect of suitable additional ambient exposure, provided, for example, by either preexposing or postexposing the film. The Royal-X curve is thereby converted to the curve shown by the dashed line. At the ambient exposure of 3 X 10-4 erg/cm2, the Royal-X film then has a DQE of about 0.3%, compared with about 1.6% for the 5820. Under these conditions, both the film and the image orthicon would provide a substantial signal-to-noise ratio, the SIN for the image orthicon being 2.3 times the SIN of the film. For lower ambient exposures, the two curves both decrease, with the separation between them increasing slightly and with an asymptotic separation by the factor ten, so that the image orthicon provides about three times the signal-to-noise ratio of the film. If we are to obtain a good image a t low ambient exposures, the film must be suitably preexposed, and the resulting negative must be printed with higher than normal contrast. It may a t first seem that these measures are more complex than those used with an image orthicon a t low light levels, but it should be recalled that a reduction of ambient light requires adjustments of the image orthicon also; the beam current must be reduced, the video gain must be increased, and it may be necessary to retrim the shading adjustments. But none of these considerations is intended to obscure the fact that the curve for the 5820 image orthicon shown in Fig. 1 is for all ambients well above the curve for Royal-X film. I n summary, this section has called attention to the serious consequences of underloading photographic films. When the underloading is left uncorrected, dramatic demonstrations of the poor performance of photographic films are possible: Fig. 4, for example.

v. PHOTOEMISSIVE

TUBES

A . Introduction Of all radiation detectors, photoemissive tubes are the most simple to discuss in terms of quantum efficiency. The specification of the responsive quantum efficiency of the photocathode is an almost complete specification of the performance in the presence of ambient radiation.

114

R. CLARK JONES

The discovery of photoemissive surfaces that have substantial quantum efficiency in the visible region of the spectrum is a fascinating story. The history is described by Sommer (19) and by Zworykin and Ramberg (20). The investigation of the photoelectric properties of bulk metals was followed by studies of thin films of alkali metals by Ives @ I ) , Campbell (22),and K. T. Bainbridge. These investigations led to the development of three photosurfaces that have been of outstanding importance during the last twenty to thirty years: the silver-cesium-oxygen surface (used for the S-1 response) developed by Holler (23) in 1928, the cesium-antimony surface (used for the S-4 response) developed by Gorlich (24) in 1935, and the bismuth-silver-cesium-oxygen surface (widely used for television camera tubes) developed by Sommer (25, 26) in 1939. The Ag-Cs-0 surface is outstanding in its unique response in the infrared, the Cs-Sb surface for its high quantum efficiency in the blue, and the Ag-Bi-O-Cs surface for its moderately high quantum efficiency throughout the visible spectrum. Several recent developments will now be described. One of the difficulties in making surfaces of high quantum efficiency is that the mean free path for the photoelectrons is shorter than the mean free path for the exciting photons. Thus, if the layer is made thick enough to absorb most of the light, many of the photoelectrons are stopped inside the layer. An ingenious way of mitigating this limitation is employed in the 7029 multiplier phototube: the cesium-antimony is deposited on top of an opaque aluminum mirror. This arrangement effectively doubles the optical thickness of the layer without increasing the path length for the photoelectrons. A markedly superior photosurface has recently been developed by Sommer (27, 28). At all wavelengths this surface has a higher responsive quantum efficiency than the antimony-cesium surface. This new surface is variously called a multialkali surface or a trialkali surface. It employs the elements antimony, potassium, sodium, and cesium. The responsive quantum efficiency of this surface, based on data given in the RCA HB-3 Tube Handbook, is shown by curve A in Fig. 5. The last decade has seen an impressive development in the use of scintillation counters in nuclear research. These counters involve a scintillating material, and one or more multiplier phototubes. Tubes have been developed for this application that have photocathodes of very large area, reduced dispersion of transit time, and very high gain and may deliver peak currents of many amperes. A group of articles on this subject is collected in the November, 1956, issue of Nuclear Science Transactions (29). A special problem surrounds the use of multiplier phototubes that have Ag-Cs-0 photocathodes (S-1 response). These tubes are very difficult to make, and although quite a number of such tubes have been manufactured and tested (30-32) they have shown a persistent tendency to lose infrared response during operational life.

QUANTUM EFFICIENCY OF DETECTORS

115

Mr. Bennett Sherman (Farrand Optical Company) examined in 1955 a number of such tubes manufactured up to 12 months or more previously by DuMont, Farnsworth, RCA, and Cinetel and found that none of them showed the classical S-1 response; they had no response at 8,000 A (33). Dr. Gerald E. Kron (Lick Observatory) has done precision photometry and colorimetry since 1938 with Ag-Cs-0 surfaces. Until 1955, he used gas-filled phototubes with an electrometer amplifier (34). Beginning in 1955, he has been using a 12-stage Lallemand multiplier phototube with a Ag-Cs-0 photocathode and with solid silver-magnesium-alloy dynodes. In a letter dated October 30, 1957, Dr. Kron wrote: Our techniques in using these photosurfaces (Ag-Cs-0) differ quite a bit from the usual, I think. First of all, we do not even turn on voltage unless the tube is refrigerated with dry ice. The room temperature dark current is so large that I believe operation even in the unilluminated condition with full cathode voltage would probably spoil a tube. Secondly, our illumination level is very low. We almost never operate with a cathode current of more than 10-l’ amp. Thirdly, we keep the cathode voltage no higher than 90 volts; this means running the first stage of a multiplier a t a lower voltage than the others but it does preserve the cathode. I think it possible that the chief cause for the deterioration of the cathodes in commercial multipliers may be simply operating them without refrigeration. It may be that the normal thermal dark current, which we know to be large, is so large that it alone is enough t o spoil the cathode during service periods.

Both Dr. Sommer and Dr. Engstrom (RCA) have written (35)that with Ag-Cs-0 surfaces the loss of infrared response with use is usually found only in multiplier tubes and that the loss may be associated with release of oxygen from the dynodes during electron bombardment. Dr. Engstrom says further, While we have not entirely solved the problem of loss of infrared sensitivity in the new RCA 7102, we have very much reduced it. I n fact, in many applications, particularly those involving low current levels, the tube is quite satisfactory and will give reasonably long life. The improvement in this characteristic has been obtained by a very strenuous processing procedure.

B. Responsive Quantum Eficiency If R is the responsivity (in amperes per watt) of the photocathode for radiation of the wavelength X (in microns), the responsive quantum efficiency QR is given by QR

= 1.2396R/X

(5.1)

where the number 1.2396 is hc/e in appropriate units. This relation permits the immediate calculation of the RQE from data on the radiant responsivity of the photocathode, in amperes per watt.

116

R. CLARK JONES

Through the courtesy of Dr. Ralph W. Engstrom and Mr. R. G. Stoudenheimer (RCA at Lancaster), the writer has received a substantial amount of unpublished information about the characteristics of RCA phototubes. This section is confined to data on RCA phototubes. The response-vs-wavelength curves of RCA phototubes are given as one of a number of S-responses. A given S-response denotes a specific relative response-vs-wavelength curve (plotted in the RCA HB-3 Tube Handbook) and does not denote any particular kind of surface. The writer here follows this convention. TABLE I. PROPERTIES A N D NATUREOF THE PHOTOCATHODES USED IN RCA PHOTOTUBES Designation of response

s-1 s-3 s-4 s-5

s-8 s-9 s-10 s-11 9-13 S-17 s-20 (1

Cathode surface used in 1958 Ag-Cs-0 Ag-Rb-0 Sb-Cs (opaque layer) Sb-Cs (opaque layer in 9741 glass bulb) Bi-Cs (opaque layer) Sb-Cs (semitransparent) Ag-Bi-O-Cs (semitransparcnt) Same as for S-9 but a thinner layer Sb-Cs (semitransparent on fused silica window) Sb-Cs (thin layer on aluminum mirror) Sb-K-Na-Cs

Wavelength of Median responsive maximum response, quantum efficiency, angstroms per cent" 8,000 4,200 4,000 3,400

0.42 0.57 14.0 18.2

4,200 4,800 5,400 4,400

0.68 6.45 3.6 15.8

4,400

13.2

4,900

21.5

4,200

18.9

The value tabulated is the maximum value listed in Table 11.

At any one date, of course, a given kind of surface is used to obtain each of the S-responses. Table I lists for each of the current S-responses (used for photoemissive tubes) the surface that was employed in 1958, the wavelength of maximum response, and the highest value of the median RQE that is obtained with that surface. Inspection of the second column of Table I reveals the importance of Gorlich's cesium-antimony surface: six of the eleven responses were obtained with various forms of that surface in 1958. The properties of most RCA phototubes, including gas and multiplier types, are listed in Table 11. Only cathode responsivities are shown. The values of the RQE listed in the last column are calculated by Eq. (5.1)

117

QUANTUM EFFICIENCY OF DETECTORS

from the radiant responsivities listed in the second column. The data in Table I1 were assembled from a variety of sources listed in the statement a t the end of the table. One should note particularly in Table I1 the very high responsivity in microamperes per lumen (of 2870" K radiation) of the trialkali surface; it is two or three times the responsivity obtainable with prior surfaces. All of the RQE values given in Tables I and I1 are given for the (designcenter) wavelength a t which the responsivity has its maximum value. Since the RQE is proportional to the responsivity divided by the wavelength, the RQE will be slightly higher than the values given for a wavelength slightly shorter than the wavelength given in Table I. The only phototube for which

00

I

I

I

I

4000

5000

6000

7000

8000

WAVELENGTH IN ANGSTROMS

FIG.5. The responsive quantum efficiency &R of four important photosurfaces plotted versus the wavelength in angstroms. Curve A : The RQE of the 7265 multiplier tube with an 5-20 response and a trialkali photocathode; Curve A also represents the RQE of the 7037 image orthicon. Curve B : The RQE of the 6810-A multiplier tube with an S-11 response and a Sb-Cs photocathode. Curve C: The RQE of the 7612 multiplier tube with an S-10 response and a Ag-Bi-0-Cs- photocathode. Curve D: The RQE of the 5280 and 6849 image orthicons with Ag-Bi-0-Cs photocathodes. The figure shows clearly the markedly superior performance of the new trialkali surface in the important red region from 6,000 t o 7,000 A (see also Figs. 2 and 8). The circles indicate the point where the responsivity has its maximum value.

118

R. CLARK JONES

TABLE 11. RESPONSIVITY A N D QUANTUM EFFICIENCY OF RCA PHOTOTUBES Cathode responsivity

amperes per watt

Luminous, microamperes per lumen (28700K)

V M

0.0019 0.0016 0.0018 0.0024 0.00135 0.0016 0,0018 0.0018 0.0016 0.0020 0.0027 0.0027

23* 18* 20 27* 15* 18* 20 20 18* 23* 30 30

G V

0.0019 0.0018

M G V V M V G G G G V M M

0.04 0.04 0.045 0.045 0.03 0.03 0.035 0.03 0.035 0.03 0.045 0.02 0.02

M V

Tube designationu

Ratio, lumens per watt

Responsive quantum efficiency, per cent

-

S-1 phototubes: 1P40 868 917,919 918 920 921 922 925 927 930 6570 7102 S-3 phototubes: 1P29 926 S-4phototubes: 1P21 1P37 1P39 929 931-A 934 5581 5582 5583 5584 5652 6323,6328 6472 S-5 phototubes: 1P28 935 S-8 phototubes lP22 S-9 phototubes: 1P42 S-10 phototubes: 6217 S-11 phototubes: 2020 5819

G G

90 90 90 90 90 90 90 90 90 90 90 90

0.30 0.25 0.28 0.38 0.21 0.25 0.28 0.28 0.25 0.32 0.42 0.42

270 280

0.57 0.53

1000 1000 1000 1000

20*

1000 1000 1000 1000

12.4 12.5 14.0 14.0 9.3 8.7 11.9 9.3 11.9 9.3 14.0 6.2 6.2

0.05 0.043

40 35

1250 1230

18.2 15.6

M

0.0023

3

768

0.68

V

0.025

37

675

6.45

M

0.0156

40

390

3.6

M M

0.04 0.04

50 50

800 800

11.3 11.3

V G G G V V

G G

7* 6.5 40 40* 45 45 30 30 35* 30* 35* 30* 45 20*

1000

1000 1000 1000 1000

119

QUANTUM EFFICIENCY OF DETECTORS

6199 6342 6372 6655 6810-A S-13 phototubes: 6903 S-17 phototubes: 7029 S-20 phototubes: 7265 The following three (S-11) 6810-A (5-17) 7029 (S-20) 7265

M M M M M

0.036 0.048 0.027 0.04 0.056

45 60 33 50 70

800 800 800 800 800

10.2 13.5 7.5 11.3 15.8

M

0.047

60

780

13.2

M

0.085

125

680

21.5

0.064 150 426 give maximum observed responsivities: 0.8 100 800 0.122 180 680 0.10 225 444

18.9

M entries M M M

22.5 30.8 29.5

a The letter following the tube designation indicates whether the tube is a vacuum phototube (V), a gas phototube (G), or a multiplier phototube (M). The values given for the cathode responsivity are median values, except for the last three rows, where maximum observed values are given. The values of the radiant cathode responsivity and the values of the responsive quantum efficiency are for the wavelength of maximum response given in Table I. The wavelength of maximum response is the design-center value. Most of the data are taken from the 1955 RCA Publication No. CRPD-105, “Photosensitive Devices and Cathode-Ray Tubes.” Data for the 6810A, 6903, 7029, and 7235 are from the RCA HB-3 Tube Handbook. The maximum observed values in the last three rows are given in a letter dated Sept. 2, 1958 from Dr. Ralph W. Engstrom. The cathode responsivities of the gas phototubes are not given in the publications cited; these were supplied in a letter dated May 21, 1957, from Mr. R. G. Stoudenheimer, along with the values for a few of the multiplier phototubes; values so obtained are indicated by an asterisk.

this method of calculation gives a slightly misleading result is the 6217 multiplier phototube with the S-10 response; this tube has the RQE of 3.6% a t the wavelength 5,400 A, where the responsivity has its maximum value, but has the RQE of 4.7% at 3,800 A. Figure 5 shows the RQE of four photosurfaces plotted against the wavelength of the radiation. From the top down, the three solid curves are for the trialkali surface (Sb-K-Na-Cs), the Sb-Cs surface, and the Ag-Bi-0-Cs surface, as represented respectively by the 7265, 6810-A, and 6217 multiplier phototubes, and representing, respectively, the 5-20, S-11, and S-10 responses; all these tubes have semitransparent photocathodes. The dashed curve represents the 5820 and 6849 image orthicons. The strikingly superior performance of the trialkali surface in the important red region from 6,000 to 7,000 A is well shown by curve A in Fig. 5. All of the results shown in the figure are based on the median responsivity as given in the RCA-HB-3 Tube Handbook. One notes in Fig. 5 that all of the curves drop sharply between 4,000

120

R . CLARK JONES

and 3,000 A. A recent article by Spicer (36) indicates that this drop for the two upper curves (and probably for the other two) is due entirely to the absorption of the glass envelope. Spicer's paper contains a number of plots of the responsive quantum efficiency versus the energy of the photon for various alkali-metal-antimony photocathodes; all the curves rise toward a horizontal asymptote as the photon energy increases to the maximum measured energy of 4.5 volts (2,750 A).

C . Detective Quantum Eficiency It is often possible in practice to achieve a detective quantum efficiency that is as high as 0.8 or 0.9 times the responsive quantum efficiency. The first necessary condition is that the ambient radiation be sufficient to make the photocurrent large compared with the thermionic dark current. If I, is the photocurrent and Id the dark current and if everything else is ideal, the detective quantum efficiency Q D is related to the responsive quantum efficiency Q R by QD-- I , QR I p -k Id

(5.2)

If one is concerned with a simple vacuum phototube (not a multiplier phototube), another necessary condition is that the shot noise of the photocurrent must be large compared with the Johnson noise of the load resistance R. I n a unit frequency bandwidth, the former mean-square noise voltage is 2eI,R2 and the latter is 4kTR. If everything else is ideal, then the relation between QD and QR is QD = QR

eI,R eI,R -k 2kT

(5.3)

When T is 300" K, the last equation becomes

!&QR

EP

E,

+ 0.0518 volt

(5.4)

where E, is the I R drop across the load resistor. Thus, the photocurrent must produce a drop across the load resistor that is large compared with 0.0518 volt. If the amplifier has additional noise above the noise of the load resistor, the voltage drop must be correspondingly larger. When the photocurrents are very small, as in astronomical work, the tubes must be refrigerated to reduce the thermionic dark current and very large load resistances must be used. Kron (34) in his work with a simple S-1 phototube used load resistances as large as 2.5 X 1013ohms. A very good answer to the problem of uncomfortably large load resistances is the multiplier phototube and, to a lesser extent, the gas phototube.

QUANTUM EFFICIENCY OF DETECTORS

121

The dynode chain in a multiplier tube and the gas in a gas phototube provide amplification and introduce relatively little additional noise. In most cases the additional noise serves to reduce the detective quantum efficiency by a factor of not less than about 0.7. The theory of the noise produced by the amplification process in a mukiplier phototube is developed in a fundamental paper by Shockley and Pierce (37'). They find that if (1) the noise in the cathode current is shot noise, if (2) a t each dynode the number of secondary electrons for each primary electron has a Poisson distribution, and if (3) the gain of each dynode is the same, then the amplification process increases the mean square noise more than the signal squared by the factor: Mm-1 M ( m - 1)

(5.5)

where M is the total gain of the dynode chain and m is the gain of each dynode. In practical multiplier tubes where M is very large compared with m, the factor reduces to m/(m

- 1)

(5 * 6)

If everything else is ideal, the relation between the responsive quantum efficiency of the cathode QR and the detective quantum efficiency QD is QD/QR

= 1

- m-l

(5.7)

For a typical dynode with gain m = 4 , Q d is 0.75 times Qr. The corresponding theory for the noise produced by the amplification process in a gas phototube was developed by Rajchman and Synder (38). They suppose that the number of secondary electrons produced a t each collision has a Poisson distribution. Their result, stated without proof, is that the amplification increases the mean square noise more than the square of the signal by the factor 1 G-l, where G is the total current gain produced by the gas. If G = 5, the factor is 1.2 and Q D is 0.83 times QR. In summary, the use of a multiplier phototube or a gas phototube reduces the DQE that may be attained to a value that is not less than 0.7 or 0.8 times the RQE of the photocathode. The values of RQE shown in Tables I and I1 are numbers that the DQE can approach but not quite achieve.

+

VI. PHOTOCONDUCTIVE CELLS

A . Introduction Since World War I1 a number of different kinds of photoconductive cells have become important in industrial and military technology. These

122

R. CLARK JONES

include cadmium sulfide and selenide cells for the visible spectrum; lead sulfide, telluride, and selenide cells for”the region out to not more than 10 p ; and doped germanium cells with response extending out to 50 I.( or more. Cells that have significant response beyond 3 or 4 I.( must usually be cooled to attain their best performance. In any discussion of the properties of photoconductive cells, the concept of the absorption edge plays a major role: the absorption coefficient for radiation is small for wavelengths longer than that of the edge and rises rapidly for shorter wavelengths. The rising absorption for wavelengths shorter than the edge is due to the production of free electron-hole pairs, and the photoconduction is due to the movement of these charge carriers. One of the two carriers (electron or hole) usually has a very short lifetime in photoconductors and is trapped a t once. The other carrier has a longer lifetime and is responsible for the photoconduction. We now sketch briefly some of the other concepts, including transit time and photoconductive gain. At equilibrium in a given volume for the photoconductor, the number of free carriers (produced by the radiation) is

M

=

Fr

(6.1)

where F is the number of pairs produced per second and r is the lifetime of the carrier. The transit time T , required for the carrier to travel from one electrode t o the other under the influence of the bias voltage V is

T , = L2/pV (6.2) is the mobility of the carrier and L is the interelectrode distance.

where p The photocurrent produced by the radiation is then

I

=

eFr/T,

(6.3)

where e is the magnitude of the charge of the electron. If just one electronic charge were transferred from one electrode to the other for each pair that was produced by the radiation, the photocurrent would be I = eF. But Eq. (6.3) indicates that the actual photocurrent is greater than this by the factor r/Tr. This ratio may thus be considered as the ‘(gain” G of the photoconductor:

G

=

r/T,

=

rpV/L2

(6.4)

The gain G may also be written in a form that is analogous to Eq. (5.1) :

G

=

1.24R/X&~

(6.5)

where R is the responsivity of the call in amperes per watt, X is the wavelength in microns, and QR is the RQE defined by Eq. (6.6) below.

QUANTUM EFFICIENCY O F DETECTORS

123

The photoconductive gain can be quite large. A developmental RCA cadmium selenide cell C7218 has a median responsivity of 13,500 amp per incident watt at 0.72 p with a polarizing voltage of 75 volts. If one supposes that all the incident photons produce pairs (QR = l), one calculates with Eq. (6.5) that the gain G is 23,300. If the more plausible assumption is made that roughly one-half of the incident photons produce pairs (QR = 0.5), the computed photoconductive gain G is 46,500. The theory given above is due to Rose. (8). When one compares the RQE and the DQE of a photoconductive cell, it is important to note that, following established practice, the RQE is defined in terms of the absorbed photons, whereas the DQE is defined in terms of the incident photons. Thus, the DQE cannot be greater than the absorptance.2 Even if a cell with a RQE of unity is ideal in every other way, the DQE cannot be greater than the absorptance. This Sec. VI is confined t o true photoconductive cells, cells that a t a given irradiation obey Ohm’s law; p-n junctions are considered separately in Sec. X.

B. Responsive Quantum Eficiency Since the event produced by a photon is the creation of an electron-hole pair, it is clear that responsive quantum efficiency should be defined as the ratio of the number of pairs produced per second t o the number n of photons absorbed per second: QR

=

F/n

(6.6)

(As stated in Sec. 11, it is customary with photoconductive detectors to consider the absorption of a photon as the input event, instead of the incidence of a photon.) This, however, was not the definition of the RQE that was used prior to 1937 by Gudden and Pohl (see Nix, 39) in their extensive pioneer work on photoconductivity. They defined the RQE as the ratio of the photocurrent (measured in electronic charges per second) to the number n of photons absorbed per second : QR

=

I/en

(6.7)

By comparing the last two equations with Eqs. (6.3) and (6.4), one sees that the RQE used by Gudden and Pohl is equal to the product of Q R as properly defined and the photoconductive gain G. Since G may be large compared with unity, we can see why the early workers were puzzled by “anomalous” photocurrents, currents for which QR as defined by Gudden and Pohl was greater than unity.

* The absorptance is defined as the fraction of the incident light that is absorbed. The absorptance plus the transmittance plus the reflectance is equal to unity.

124

R. CLARK JONES

The first careful measurement of the RQE of a photoconductor was carried out by Goucher (40) on a sample of nearly intrinsic germanium. This was a sample in which the lifetime and mobilities of both of the carriers had been measured. Goucher found that the RQE was unity over the range from 1.0 to 1.7 p , with a probable error of 10 or 15%. Goucher’s careful experiment was carried out to test a hypothesis that has come to be widely accepted by solid-state physicists-the hypothesis that all photoconductors wit,h a sharp absorption edge have a RQE of unity for wavelengths just shorter than the edge and for some distance toward shorter wavelengths. The basis for this hypothesis is easy to understand: as the wavelength moves into the absorption edge from longer wavelengths, the absorption coefficient increases by several orders of magnitude-by six or seven orders of magnitude for germanium (41,42). All this extra absorption is due to the much increased cross section for pair production. Thus, if the absorption coefficient has increased by three orders of magnitude, all but one part in lo3 is due to pair production. This is merely another way of saying that 99.9% of the absorbed photons produce pairs and that the RQE is therefore 99.9%. This hypothesis must be used with care, of course. As the wavelength is decreased through the visible and into the ultraviolet, other electronic absorption mechanisms will set in and will compete with pair production. The net result is a drop in the RQE. And if one is dealing with a complex photoconductor of unknown structure, such as lead sulfide evaporated films, one cannot be sure that other absorption mechanisms may not be setting in just inside the absorption edge. This hypothesis may be in error in the opposite direction, also. Smith and Dutton (43) present evidence that the RQE of lead sulfide films rises above unity and is inversely proportional to the wavelength for wavelengths between 0.6 and 0.2 p. For wavelengths shorter than 0.6 p , this corresponds, on the average, to the production of one electron-hole pair per 2.1 electron volts of energy in the absorbed radiation. In summary, the hypothesis is probably true for most photoconductors under most conditions, but is far from being a law of nature. We now describe a few specific results about the RQE of photoconductors. In 1957 Lummis and Petritz (44)reported a RQE of about 60% for lead sulfide films, and a year later Spencer (45) reported a RQE of nearly 100% for the same kind of filmsQ3 By combining the results of photoconductive and photoelectromagnetic Earlier, in 1956, Wolfe (46) reported a value of only 0.25%, but this report has since been found (46) to be based on an incorrect premise as t o the origin of the noise in these films.

QUANTUM EFFICIENCY OF DETECTORS

125

measurements, Moss (47’) has found that the RQE of single crystals of lead sulfide is roughly unity at 2 p. Moss has also reported (48) a similar finding about the RQE of single crystals of indium antimonide. Mollwo (49) has found rough confirmation of unity RQE in single crystals of zinc oxide. Significant measurements have also been made over a wide range of energy in germanium. To be sure the measurements now to be reported were made on p-n junctions, but such measurements relate t o the RQE of germanium as a material even though the germanium was in the form of a junction rather than in the form of photoconductive cell. While studying the photovoltaic effect in germanium p-n junction excited with X-rays, Backovsky, Malkovska, and Tauc (50) found the short-circuit current to be linearly proportional to the absorbed power of the X-rays and not proportional to the number of X-ray photons. The conclusion is that the RQE is proportional to the energy of the X-ray photons. According to Drahokoupil, Malkovska, and Tauc (51))the absorbed energy required t o form one electron-hole pair is about 2.5 electron volts, which energy corresponds to a 0.5-p photon. Similar results were obtained by McKay (52) using excitation by alpha particles; the result was a n energy per pair of 3.0 0.4 electron volts per pair. Finally, Koc (53) studied the RQE of germanium p-n junctions over the wavelength range 0.3 t o 2.0 p ; he found an RQE of unity for wavelengths longer than 0.6 p and an RQE greater than unity and equal to 0.6/X for wavelengths between 0.6 and 0.3 p. Goucher’s pioneer measurement of the RQE of photoconductivity in germanium over the range from 1.0 to 1.7 p has already been reported. I n summary, there is good reason to suppose that the RQE of all photoconductors that show a marked absorption edge is substantially loo%, particularly for the octave between the edge wavelength and one-half of the edge wavelength.

C. Detective Quantum, Eficiency Rose (6) has pointed out that in a photoconductive cell, there is a statistical fluctuation in the number of the free carriers and also a fluctuation in their lifetime. These two fluctuations contribute equally to the meansquare noise voltage in the output. The result is that the mean-square noise in the output, when referred to the input, is never less than twice the noise in the steady ambient radiation. The consequence is that the maximum possible detective quantum efficiency of a photoconductive cell is one-half. I n this respect photoconductive cells must be distinguished from back-biased p-n junctions (Sec. X,D), in which there is no correspond-

126

R. CLARK JONES

ing fluctuation in the lifetime; the lifetime of a carrier is equal to the transit time between electrodes. Thus, in back-biased p-n junctions, the maximum possible DQE is unity. Measurements of the detective quantum efficiency (DQE) of photoconductive cells are few in number. 1. C a d m i u m SulJide Cells. Shulman (54)has reported values of the DQE close to 100% for a cadmium sulfide crystal cell. His values are valid at not just one light level, but over a range of 5 to 1 in cell illumination. He assumed a unit responsive quantum efficiency, measured the photoconductive gain G, and calculated the power spectrum of the noise in the cell output that would be produced by photon noise alone. This power spectrum is compared with the measured power spectrum in Shulman’s Fig. 1. The measured power spectrum is about twice the calculated spectrum, except in the vicinity of 100 cps, where the ratio is 1.5. Shulman states that if the correction due to surface reflection is made, the comput’ed curve must be multiplied by the factor 1.25. One concludes, therefore, that the DQE is about 60% and would be higher if a nonreflecting coating were used. The fact that the observed noise was in fact photon noise was confirmed by Shulman in an interesting way. If one has an ideal photoconductor of constant DQE and varies the photocurrent a t constant bias voltage by varying the intensity of the light, the mean-square noise should be proportional t o the current. But if one varies the current at constant light by varying the bias voltage, one is effectively varying the photoconductive gain G, and one would expect the mean-square noise to vary as the square of the photocurrent. These behaviors were in fact observed, over a 10-to-1 range of current for both kinds of variation. The absolute values of the light intensity and of the currents involved are not given by Shulman. Van Vliet et al. (55) have reported similar and quite extensive measurements on the noise characteristics of cadmium sulfide cells. With respect to order of magnitude, they estimate a DQE of about 10% for modulation frequencies below 1,000 cps and for light within the absorption edge. The exact value of the DQE is found to depend slightly on the amount of ambient light. For frequencies larger than 1,000 cps, the DQE begins to decrease because of spontaneous trapping fluctuations. 2. Lead Xuljide Cells. Some information is available also about the DQE of lead sulfide cells. Wolfe’s conclusion (46),which he presented as a determination of the RQE, was actually close to a determination of the DQE. He found that the detectivity of the cells he measured was 5y0 of that of an ideal detector operating a t room temperature and limited by fluctuations in both the incident and emitted photons. This corresponds to a detectivity 3.5Yo of that of a n ideal detector limited only by the fluctuations in the incident

QUANTUM EFFICIENCY O F DETECTORS

127

photons. We conclude that the detective quantum efficiency of his cells was 0.12% [0.0012 = (0.035)2]. A lead sulfide cell with a typical responsivity versus wavelength curve (such as that shown in Fig. 13 of ref. 2) responds effectively to only 1/20,000 of the total power in room temperature blackbody radiation. Since the latter is 0.05 watt/cm2, the effective fraction of the blackbody radiation is 2.5 X watt/cm2. From this we conclude that if a lead sulfide cell does have a DQE close to its absorptance, it will do so only for ambient irradiations greater than 800 X 2.5 X =2 X watt/cm2. Free (56) has measured the DQE of a group of lead sulfide cells for blue light; he found values lying in the range from 25 to 100%. The source of the ambient blue light was an overvoltaged ribbon filament lamp, monochromatized by a (quartz prism) Perkin Elmer monochromator with 2-mm wide slits. A separate chopped source was used to measure the detectivity. The irradiation of the ambient light was not independently measured, but from the fact that it reduced the resistance of the lead sulfide cells t o 85% of the dark resistance, the irradiation is computed to be 0.5 X watt/cm2. The tests of Wolfe and Free were carried out on Eastman Kodak chemically deposited lead sulfide cells operated a t room temperature. In summary, we conclude that lead sulfide cells have a DQE close to their absorptance, but only for irradiations greater than about watt/ cm2. For smaller irradiations, the DQE is proportional t o the irradiation. I n marked contrast to the low detective quantum efficiency of roomtemperature lead sulfide cells, Watts (5'7) has described a cooled lead sulfide cell that as interpreted by Moss (58), has a DQE of (1.3)-2 = 59%. This result, of course, breaches the theoretical limit of 50%, but the accuracy of the result is not such that the discrepancy is significant. This cell was a t a temperature of 110" K, but it was in an enclosure whose temperature was a t 200" K. Thus, the ambient radiation was 200" K blackbody radiation. Fellgett (59) has described a lead telluride cell a t 90" K, in a room temperature enclosure, that has a DQE of (1.9)-2 = 28%. Further details about these cells, and a derivation of the noise figures of 1.3 and 1.9, will be found in reference 2, pp. 71 and 75. This writer was rather disturbed by the wide difference between the low DQE of the room temperature cells and the high DQE of the cooled cells as described above. It was therefore gratifying to receive a letter dated February 20, 1959 from Dr. Harry E. Spencer of the Eastman Kodak Company in which he reported the calculation of the DQE for two lead sulfide cells over a wide range of temperature, as shown in the following table :

128 ~~~

R. CLARK JONES ~

Cell

T (deg. K)

QD

(%I

PbS-6-4

302 275 207 100

0.14 0.46 36 78

N179-8-4

302 275 207 100

0.028 0.18 14 100

Dr. Spencer reported that the values of Q D in this table are approximate only; the extreme error is probably less than a factor of two. The data in this table indicate that the DQE of the same lead sulfide cell can vary from 10-4 to unity as the temperature drops from room temperature to 100°K. 3. Conclusions. From the limited number of results presented above, perhaps one is permitted to speculate that many photoconductive cells have values of the DQE close to their absorptance for sufficiently high values of the ambient radiation. As improvements are made in these cells, it is to be expected that the amount of ambient radiation required for a DQE close to the absorptance will be reduced. This reduction will also increase the detectivity in the presence of a negligible amount of ambient radiation.

VII. TELEVISION CAMERATUBES A . Introduction I n this section we consider the two kinds of television camera tubes that are of commercial importance: the image orthicon and the vidicon. The responsive quantum efficiency (RQE) of several image orthicons is described. The detective quantum efficiency (DQE) of two RCA image orthicons (the 5820 and 6849) is computed from unpublished signal, noise, and resolution data generously supplied by the Radio Corporation of America. We present also (Sec. C,5) the DQE of the RCA 6326 vidicon. To be sure, DQE is not a fully appropriate criterion for the vidicon, since the noise of a vidicon and its amplifier is quite independent of the amount of ambient radiation. But in order to be able to compare the performance of vidicons with that of image orthicons, we must discuss the vidicon from the point of view of the DQE. The DQE is discussed as a function of the wavelength of the photocathode irradiation, of the line number of the target, and of the amount of the ambient irradiation of the cathode. The chief results are presented in Table 111and in Figs. 8 through 14. The maximum value of the DQE with

QUANTUM EFFICIENCY OF DETECTORS

129

TABLE111. THESIGNAL-TO-NOISE RATIOR, THE PHOTOCATHODE IRRADIATION H, AND THE DETECTIVE QUANTUM EFFICIENCY ALL FOR THE IRRADIATION THATMAXIMIZES THE DQE OF THE Two IMAGE ORTHICONS. 5820

R H QIMX

16.2 5 . 6 5 X 102.65%

6849 8.1 8 . 6 3 X 10-lo watts/cm2 4.35%

respect to wavelength, line number, and ambient radiation is found to be about 2.5% for the 5820, about 4.5% for the 6949, and only about 0.1% for the 6326 vidicon.

B. Responsive QuantumEficiency

So far as this writer knows, the responsive quantum efficiency of image orthicons has been defined in only one way: as the responsive quantum efficiency of the photocathode-that is, the ratio of the number of photoelectrons to the number of incident photons. Accordingly, the responsive quantum efficiency of several RCA image orthicons has already been covered in Sec. V,B. Curve D of Fig. 5 shows the RQE of the 5820 and the 6849; curve A shows the RQE of the new 7037 image orthicon with the trialkali photocathode: the photocathode of the 7037 is identical with that of the 7265 multiplier tube. C . Detective QuantumEficiency In this section the detective quantum efficiency (DQE) is calculated for two RCA image orthicons and one RCA vidicon as a function of photocathode illumination, size of signal area, and the radiation wavelength. The two image orthicons are the RCA 5820 and the RCA 6849, the latter being the wide-spaced version of the former. The vidicon is the RCA 6326. The vidicon is treated separately in Sec. 4. 1. The Basic Data. The writer is much indebted to Mr. F. David Marschka, Dr. George A. Morton, and Dr. Benjamin H. Vine, of the Radio Corporation of America for supplying unpublished information on the performance of these image orthicons. Most of this information is shown in Figs. 6 and 7. Figure 6 shows the electrical signal-to-noise ratio R of the two camera tubes as a function of the photocathode irradiation in watts per square centimeter of 395-mp monochromatic radiation. The signal is the peak-topeak output when the orthicon sees a pattern that has high contrast, between areas of large angular subtense. (Such a pattern will be called briefly a large-area black-to-white transition.) The noise is the rms electrical noise

130

R. CLARK JONES

voltage in a bandwidth that is slightly greater than 4.5 Mc. The beam current was separately adjusted for each of the experimental points on which the curves in Fig. 6 are based. The original data supplied by RCA involved an abscissa equal to the photocathode illumination in lumens per square foot of radiation from a bank of Sylvania “white” fluorescent lamps. Dr. Keith Butler of Sylvania Electric Products has kindly supplied the relative radiant output of these lamps per unit wavelength interval. By combining this information with the relative dpectral sensitivity of the image orthicon as given by the RCA HB-3 Tube Handbook, I calculate that 480 lumens are equivalent to 1 watt of 395-mp radiation. (This ratio is slightly greater than the ratio found below of 450 lumens of 2870” K tungsten radiation per watt of 395-mp radiation.) ILLU MI NATION I N LUMENS /FT

10-l0

I O - ~

IRRADIATION

I o-8

H IN WATTS/CM*

FIG.6. The signal-to-noise ratio of the two image orthicons plotted versus the cathode illumination in lumens (of “white” fluorescent light) per square foot. An alternative scale indicates the cathode irradiation in watts (of 395-mp radiation) per square centimeter. The signal-to-noise ratio is the ratio of the peak signal voltage (for a black-towhite transition) t o the rms noise voltage in a bandwidth slightly greater than 4.5 Me.

Figure 7 shows the way tha$ the resolution of the two camera tubes depends on the photocathode irradiation. The four curves, two for each tube, show the line numbers a t which the response has decreased to 0.5 and 0.25 of the large-area response. These data also were originally supplied with the abscissa given in lumens per square foot, of fluorescent radiation. Curve C of Fig. 5 in Sec. V, which is based on data in the RCA HB-3 Tube Handbook, indicates that the RQE of the two image orthicons has its maximum value a t 395 mp, where the RQE is 5.21% and the responsivity is 0.0166 amp/watt. RCA has supplied the additional information that the response-versus-

131

QUANTUM EFFICIENCY O F DETECTORS

wavelength curve in the HB-3 Tube Handbook corresponds to a responsivity of 36.6 pa/lumen of 2870" K radiation, from which we conclude that 453 lumens of 2870" K radiation is equivalent to 1watt of 395-mpradiation. RCA has supplied the additional information that the image orthicons now being manufactured have a higher responsivity than is indicated above, and that the particular tubes used to obtain the data in Fig. 6 had a responsivity about 60 pa/lumen and about 0.027 amp/watt. Thus, for these newer tubes the RQE is about 8.1% and the ratio of responsivities is 450 lumens/ watt. Thus, the abscissa in Figs. 6 and 7 may be converted t o photocathode current in amperes per square centimeter by multiplying the abscissa by 0.027 amp/watt. Alternatively, the abscissas may be converted to lumens per square centimeter by multiplying the abscissa by 480 or 450 lumens/ watt for the two kinds of light mentioned above. ILLUMINATION I N

lo-*

1000

>

+

LUMEN SIFT.^

I

6849 5 0 % AMPLITUDE,

35' C.

395 m p

.10-10

10-9 10-8 IRRADIATION H IN WATTSICM'

10-7

FIG.7. The television line number of the two image orthicons for two different amplitude responses plotted versus the cathode illumination and the cathode irradiation. The curves labeled "25 percent amplitude" indicates the line number at which the electrical response is $4 of the response for any very small line number and similarly for the other two curves.

2. Derivation of a Working Formula jor Q D . The photocathode of the image orthicons has the dimensions 2.44 by 3.25 cm and thus has the area

A, The integration time orthicon is

=

7.93 cm2

(7.1)

T of the normal mode of operation of the image

T = 130 sec

(7.2)

At 395 mp, the energy of a single photon as given by Eq. (3.16) is

132

R. CLARK JONES

E

=

5.025 X

joule

(7.3)

With the help of the last two equations and Eq. (3.21), one has the following preliminary expression for the detective quantum efficiency: QD

= 1.508 X 10-1’(S/N),2/HAC2

(7.4)

where A is the photocathode area (in square centimeters) that is illuminated by the signal, C is the contrast, and H is the ambient irradiation of the cathode in watts per square centimeter. If the image orthicon had no resolution limitations, then we could say that the electrical signal-to-noise ratio CR in a 4.5-Mc bandwidth would be equal t o the signal-to-noise ratio ( S I N ) , for a signal spot of size equal to the smallest area that can be resolved by such a bandwidth. If A , is the photocathode area, this smallest area Aminis given by A,/(2 X 4.5 X 106/30) Amin = AC/300,000

Amin =

(7.5) (7.6)

To derive the signal-to-noise ratio ( S I N ) , for larger signal areas, we note that the frequency bandwidth required to transmit a signal spot of area A is inversely proportional to A. The required bandwidth is 4.5 Mc for the area Aminand is only 15 cps for a signal that uniformly covers the photocathode. Since, furthermore, the noise has a flat spectrum, the signalto-noise ratio for a signal spot of area A is given by

( S I N )m

=

(A/Amin) %CR

(7.7)

or by

( S / N ) , = (300,000A/AC)”CR

(7.8)

where A , is the area of the photocathode. If the area of the photocathode A , = 7.93 cm2 is inserted in the last expression and the result substituted in Eq. (7.4), one finds

Q~

=

5.71

x

10-13~2/~

(7.9)

This is the final “working” expression for the DQE in terms of the measured quantities R and H . In this expression, QD is a fraction (not in per cent), and H is in watts per square centimeter. It is particularly to be noted that the area A of the signal does not appear in this expression. Actually, of course, the area A fails to appear because of our assumption that the performance is not limited by the resolution capability; for sufficiently small areas the DQE will decrease, and this decrease is examined in Sec. 3. 3. Results for Image Orthicons. The fact indicated by Eq. (7.9) that the DQE varies as R 2 / H means that the curves of constant DQE in Fig. 6 are straight lines with a positive slope of one-half. The point where each of the two curves has a slope of one-half is indicated by the open circles in Fig. 6.

133

QUANTUM EFFICIENCY O F DETECTORS

The values of R and H at these two points are indicated in Table 111. The last row of Table I11 shows the value of the DQE computed for these two points by Eq. (7.9), in per cent. These values of the DQE are of course the maximum values with respect to radiation wavelength, image size, and photocathode irradiation. The value of the DQE under other conditions is related to its maximum value by QD =

Q,&'xF$H

(7.10)

where the three F's are factors with a maximum value of unity, which depend respectively on the radiation wavelength, line number, and illumination. I

I

I

I

I

I

I

I -

-

5820 AND 6849

-

-

WAVELENGTH IN MILLIMICRONS 300

4 00

500

600

700

FIG.8. The relative detective quantum efficiency plotted versus wavelength. The curve is normalized so that its maximum value is unity. The curve applies to both kinds of image orthicon. The nominal responsive quantum efficiency is shown by curve D in Fig. 5 .

The factor FA that takes into account the variation of QD with the wavelength A is easily shown to vary with wavelength in proportion t o the responsive quantum efficiency of the photocathode, which in turn is proportional t o the responsivity (in amperes per watt) divided by the wavelength, as indicated by Eq. (5.1). Suppose, for example, that we consider a wavelength where the responsive quantum efficiency is just half its value a t 395 mp. This means that both the signal and ambient photon fluxes M , and M a must be doubled in order to produce t,he same photocurrents as before. It then follows directly from Eq. (3.10) that the DQE is reduced to half its previous value. (The experimental data on the responsivity used to construct Fig. 8 were taken from the HB-3 Tube Handbook.) To assure that FArepresents the variation of the DQE with wavelength,

134

R. CLARK J O N E S

the ambient photon flux must vary inversely as the function FA in order that the photocathode current be held a t its optimum value. With reference t o Fig. 8, one sees that this is equivalent to the statement that the photocathode current be held a t 0.00150 pa for the 5820 and a t 0.000237 pa microampere for the 6849.

1.c LLF

>

0

z

w

0 LL LL W

5

I-

z a 0. W

2

I-

0 W

IW 0 W

I

l-

a _I

w

cr 0.0

I

1

40

I

I 100

I

200

T V LINE NUMBER

I 400

\

I

v

FIG. 9. The relative detective quantum efficiency plotted versus the television line number. The ordinate is equal t o the square of the amplitude line-number response. The two image orthicon curves apply only when the cathode illumination is such as to maximize the detective quantum efficiency (see Fig. 10).

The detective quantum efficiency Qmaxapplies to a target so large as to be completely resolved. The DQE is smaller for targets that are not fully resolved. Schade (60) has shown how the response of an image system for a target of any size and shape may be calculated from the line-number response of the system by Fourier methods. For the sake of brevity, we omit this transformation and simply show how the DQE depends on the line number of a simple pattern that may be described adequately by a narrow range of line numbers. The factor F , shown in Fig. 9 is proportional to the square of the line-number response. The shape of this curve depends on the irradiation level, as shown by Fig. 7. The two curves in Fig. 9 are both for the irradiation level shown in Table 111, for which Q D is a maximum for large-area targets.

135

QUANTUM EFFICIENCY O F DETECTORS

The development of the full significance of the function F, would require the introduction of two-dimensional Fourier analysis (61, 62), and would be a sufficiently extensive discussion that its length would not be in proper proportion to its importance for this section. Figure 10 shows the function FH plotted as a function of the irradiation H for a wavelength of 395 mp. This function is equal to the ratio of R2/H as given by Fig. 6 to the maximum value of R 2 / H . The function plott,ed in Fig. 10 is for radiation of the wavelength 395 mp. For other wavelengths, the curve should be shifted to the right by a factor equal to the reciprocal of FA. ILLUMINATION IN

LUMEN SIFT.^

C

0 W

FH

2

k

-I W

.0.1 "O

~

[L

395

mp

IRRADIATION H IN WATTS/CM2 -10 10

-9 10

-8 10

-7 10

FIG.10. The relative detective quantum efficiency of the two image orthicons plotted versus the cathode illumination and the cathode irradiation. The curves are normalized. With the data in this figure and the data in Figs. 7,8, and 9 and Table 111, one may compute the detective quantum efficiency of either image orthicon for any combination of radiation wavelength, line number, and cathode illumination.

4. Detective Q u a n t u m Eltfciency of the 6326 Vidicon. The detective quantum efficiency is best adapted to describing the performance of radiation detectors whose noise is due to the fluctuation in the arrival of the ambient photons a t the sensitive surface. The vidicon is not a member of this class of detectors. The noise at the output of the amplifier associated with the vidicon is quite independent of the level of ambient illumination. Accordingly, if the vidicon were being compared with detectors in general, it would be more suitable to evaluate it by the methods of reference 2. But in this section we are not interested in comparing the vidicon with detectors in general. Rather, we wish to compare its performance with other camera tubes, the image orthicon in particular. Since the image orthicon is suitably evaluated by means of the detective quantum efficiency, we shall use this method also for the vidicon.

136

R. CLARK J O N E S

The signal-to-noise ratio of the vidicon camera tube is substantially degraded by the noise of the best available video amplifier. The noise level depends to some extent on the degree of frequency compensation used in the amplifier, which compensation corrects the horizontal response for the aperturing effect of the scanning beam and for the shunt capacity of the tube. The electrical signal-to-noise ratio R shown in Fig. 11 and the F,

200 -

100 -

-

40 -

20 10 -

-

I

I

I 1

-6

10

I

I

I I -5 10

I

I

I 1 -4

10

FIG.11. The signal-to-noise ratio R of the 6326 vidicon for a black-to-white transition and also the quantity r R to be used for calculating the signal-to-noise ratio for a small signal, plotted versus the cathode illumination in lumens per square foot of 2870' K radiation and the cathode irradiation in watts per square centimeter of 435-mp radiation.

function shown in Fig. 9 are for no compensation and represent the situation in which the spectrum of the iioise a t the output of the amplifier is approximately flat. The curves in Figs. 9 and 11 are based on Figs. 8 and 10 of the RCA Bulletin describing the 6326 vidicon and on the information (kindly supplied by Mr. A. D. Cope and Dr. Benjamin H. Vine of RCA) to the effect that the noise current referred to the output of the tube is about 1.5 X amp in a 4.5-Mc bandwidth. The curves in Fig. 11 are for a large-area black-to-white transition and are for monochromatic light of wavelength 435 mp. From the data in Figs. 8 and 10 of the 6326 Bulletin one finds that the tube's responsivity/wavelength ratio has a maximum a t this wavelength and that 1 watt of 435-mp radiation produces the same response as 520 lumens.

QUANTUM EFFICIENCY O F DETECTORS

137

Unlike the response of the image orthicon, which is linear for illuminations below the “knee” of the characteristic curve, the response of the vidicon is nonlinear, with a “gamma” less than unity. Thus, the signal-tonoise ratio from a large-area target of small contrast C will not be CR, but will rather be yCR, where y is the gradient of the log-current-output-vslog-irradiation curve. A plot of y R vs H is also shown in Fig. 11. At 435 mp, the energy of a photon is & = 4.56 X 10-19 joule

(7.11)

We shall base our calculation on the assumption that the integrat’ion time of the vidicon is 1/30 sec. This assumption, which is very sound for image orthicons, is not exact for vidicons. These camera tubes have appreciable carryover from one frame to the next, which is small a t the recommended illumination of 30 1umens/ft2, but which becomes quite marked a t much lower illuminations; this integration increases significantly the signal-tonoise ratio at lower illuminations, a t the cost of blurring rapid motion. In the absence of detailed information about the amount of temporal integration as a function of illumination, we make the simple assumption that the integration time is

T = 1/30 sec

(7.12)

As will be apparent shortly, the detective quantum efficiency has a maximum value for H = 4 lumens/ft2, and thus this assumption is sound for the illuminations of greatest interest. By exactly the same type of argument as that used in Sec. VI1,2, we find

(X/N),

=

(300,000 A/A,)’yCR

(7.13)

where A , is the sensitive area of the vidicon, which has the dimensions 0.5 by 0.375 in. :

A,

=

1.342 cm2

(7.14)

With the help of Eq. (3.21), the last four equations yield QD

=

3.06 X 10-l2Y2R2/H

(7.15)

This is the “working” equation for the DQE of the vidicon. The DQE varies as y2R2/H.The point on the curve in Fig. 11 where this ratio has its maximum value is indicated by a n open circle. For this watt/cm2, whence one has point, y R = 47.5, and H = 8.28 X

Q,,

=

0.084%

(7.16)

This is the maximum value of the DQE with respect to the radiation wavelength, size of the signal image, and amount of ambient radiation. The

138

R. CLARK J O N E S

factors FA,F,,, FH, defined as in the preceding section, are plotted in Figs. 9, 12, and 13. The function FA is proportional to the responsivity (plotted in Fig. 10 of the Tube Bulletin) divided by the wavelength. The function F, is proportional to the square of the response versus line number. The function FH is equal to the ratio of -y2R2/Hto its maximum value. Many of the comI

I

I

I

I

I

I

I

I

I

FIG.12. The relative detective quantum efficiency of the 6326 vidicon plotted versus the radiation wavelength. The curve is normalized.

> ILLUMINATION IN LUMENSIFT*

I

I I

I

-6

10

I

I

I I

-5

10

IRRADIATION H IN WATTS/CM*

I

I -4

10

FIG.13. The relative detective quantum efficiency of the 6326 vidicon plotted versus the cathode illumination and the cathode irradiation. The curve is normalized.

139

QUANTUM EFFICIENCY O F DETECTOIZS

ments made in the preceding section about the F functions apply also to the F functions of this section. The fact that the limiting noise of the vidicon is amplifier noise has a n important consequence. If an image orthicon had a DQE of only O.O840j, (the value for the vidicon) under the best conditions, the RQE of the photocathode would have to be increased by the factor 52 to equal the 4.35% DQE of the 6849. But since the noise of the vidicon is amplifier noise, the responsivity of the vidicon’s photocathode would have to be increased only by the factor 7.2 = (52)% in order that the DQE rise to 4.35%, provided that the noise of the vidicon proper continues to remain below the amplifier noise. This is one of the consequences of the fact (mentioned a t the beginning of this Sec. 4) that the vidicon is not a member of the class of detectors that are best described in terms of their detective quantum

l,06849-



0.1

162

I

I

I

10

0 I-

a I-

K

-

CT 0

w n

z -01 0 0

,.--E too-

~

--,,------

5822, /

6849///;’ / / /,,/

/

/

/

4-

/

YR,

0



6326

+T-\

R

?

/

/

W

10-

//

/

10-8

Io

-~

IO-~

IO-~

1 6 ~

FIG.14. A summary plot showing both the detective quantum efficiency Q D (solid curves) and the signal-to-noise ratio (dashed curves) of all three camera tubes plotted versus the cathode illumination and the cathode irradiation. The values of Q D assume optimum choice of radiation wavelength and television line number.

5. Discussion. Figure 14 shows the electrical signal-to-noise ratio and the DQE of image orthicons and vidicons when the camera sees a pattern with contrast between large areas, which pattern is illuminated by light of the optimum wavelength. This figure summarizes the most important results of this paper. The accuracy of the results presented here is not high. The data supplied by RCA were laboratory dat,a not obtained for the purposes of this review; the noise levels are based on peak-to-peak noise amplitudes as observed on a n oscilloscope. Since the DQE is inversely proportional to the mean-

140

R. CLARK JONES

square noise voltage, it is clear that there is room for appreciably error in the results. I mould guess that the numbers found are probably between % and 35 of the correct values for the image orthicons, and the result for the vidicon may have a somewhat larger range of probable error. I feel confident that the values of the DQE found for the image orthicons are approximat,ely correct : I had expected the detective quantum efficiency to be about of the responsive quantum efficiency, and the result found accords with this expectation. It is also in agreement with expectations based on the internal parameters of the image orthicons. The values of DQE found in Table 111, about 2.5 and 4.5%, are from 45 to $6 of the responsive quantum efficiency of about 8%. I share with Dr. Rose the feeling that these high efficiencies are a technical accomplishment of the first rank. The writer knows of no other image system that can quite come up to this performance. As indicated in Secs. VIII and IX, the human eye and photographic negatives both have a DQE under the best conditions of about 1%. The new image orthicon with the improved cathode (RCA 7037) has a n RQE of 19.2% a t 400 mp, as indicated by curve A of Fig. 5, and there is every reason t o expect that such tubes will have a DQE approaching 10%. To summarize, Fig. 14 shows the signal-to-noise ratio and the DQE of the three image tubes discussed in this report. The maximum signal-tonoise ratio of the vidicon is slightly higher than that of either of the image orthicons, but the DQE of the vidicon is much lower, because the illumination required to achieve the high signal-to-noise ratio is so much larger. VIII. PHOTOGRAPHIC NEGATIVES

A. Introduction Nearly all the fundamental investigations on the behavior of photographic materials and of the individual grains have in one way or another contributed t o our understanding of the responsive quantum efficiency. Prominent in this field are the names of Silberstein, Trivelli, and Webb in this country, and Berg, Burton, Gurney, Mitchell, and Mott in England. Important in these investigations have been the shape of the density-vs-logexposure curve, the intermittency effect, and reciprocity law failure. An important tool has been the counting of developed grains in single-grainlayer films. The broad outline of the Mott-Gurney theory (63, 64) of the photographic process, announced in 1938, is still the accepted theory. This theory views the silver halide grain as a photoconductor. The absorbed photon produces an electron-hole pair. The electron is mobile. Interstitial silver ions are also mobile. At special sites within or on the surface of the grain,

QUANTUM EFFICIENCY O F DETECTORS

141

the electrons and silver ions combine to form a collection of silver atoms. This bit of metallic silver is the latent image speck. Sixteen years before the pioneer publication by Gurney and Mott, Silberstein (65)in 1922 made the first effort,to understand the photographic process in terms of the fact that light arrives a t the film in discrete bundles (the photons). He assumed that the effective absorption of a single photon was sufficient to make the grain developable. I n the same year, Svedberg (66) established that incidence of a single alpha particle is sufficient to make a grain developable; the same fact was established for X-rays by Silberstein and Trivelli (67) in 1930. In 1928 Silberstein (68) generalized this 1922 concept by the assumption that a small but finite number of photons must be effectively absorbed to make a grain developable. Silberstein and Webb (69) reported in 1934 that the intermittency effect could be understood only in terms of the quantum nature of light. I n 1927 Wightman and Quirk (70) suggested that the formation of a developable grain proceeds in two stages. In the first stage, a nondevelopable “subspeck” is formed. The subspeck is later converted into a developable “full speck” by further action of light. A large literature has developed about this concept, and the concept is now fully accepted and has a great deal of evidence t o support it. In 1938 Webb and Evans (71) and Berg and Mendelssohn (72) showed that the reciprocity law failure was due t o the instability of the subspeck in its initial stage of formation. I n 1946 and 1948 Burton and his co-workers reported (73-77) an impressive series of experiments using the “double exposure” technique. From these experiments it was concluded that the formation of the subspeck required the effective absorption of a t least two photons and that the full speck required a t least two more photons. (In the more recent literature, the subspeck is called the latent subimage speck and the full speck is called the latent image speck.) I n 1950, Webb (78) and Katz (79) showed conclusively that the effective absorption of two photons within a critical period of length r is required to form a latent subimage speck on the basis of measurements on reciprocity law failure. Webb indicated that after the latent subimage is formed, the grain must absorb about six more photons to make the grain developable. In 1954 Maerker (80) determined that the length of the critical period was 3 or 4 sec a t room temperature. I n 1957 Mitchell and Mott (81)and Mitchell (82) published a refinement of the Mott-Gurney theory, as part of which the latent subimage speck requires the absorption of two photons, and the latent image speck requires the absorption of one more; the minimum latent image speck consists of four silver atoms with a unit positive charge. The effort to use the shape of the density-vs-log-exposures curve to estimate the number of photons required to make a grain developable was

142

R. CLARK J O N E S

carried further by Webb (83, 84)in 1939 and 1941, and by Burton (85) in 1951, but the conclusion of all of this work was that the shape of this curve is determined primarily by the fact that the required number is widely different for different grains within a given emulsion. Several workers have measured the number of incident photons required to make a grain developable. As summarized by Webb (86) in 1948, the published measurements range from 200 to 1,350 photons/grain for wavelengths near 4,000 A. Taking the number to be 400, Webb indicates that XOof these incident photons are absorbed by the grain and that of the 40 absorbed, only 10 are effective photographically. For further information on the theory of the photographic process, the reader is referred to the excellent reviews by Berg (87) and by Mitchell (88),and to the incomparable book (89) edited by Mees.

B. Responsive Quantum Eficiency One definition of the responsive quantum efficiency is the ratio of the number of developed grains to the number of incident photons. Under optimum conditions of wavelength, exposure, and development, this ratio was found above to be about $&o = 0.2y0. Another definition is the ratio of the number of grains made developable to the number of effectively absorbed photons. The ratio predicted by the Mott-Mitchell theory is 55 = 33%. Actual measurements carried out under optimum conditions indicate the ratio HO= 10%. These results, which vary from 0.2 to 33%, may be compared with the results of 0.3 to 0.9% for the detective quantum efficiency (see Sec. C ) .

C . Detective Quantum Eficiency In this section the detective quantum efficiency (DQE) of four Eastman Kodak films (abbreviated names: Royal-X, Tri-X, Plus-X, and Pan-X) is computed from sensitometric and granularity data generously supplied by the manufacturer. The DQE of a film depends on the wavelength of the radiation and on the amount of the preexposure. For each of the films, a curve of the DQE versus the preexposure (for a radiation wavelength of 430 mp) is presented in Fig. 18. The DQE passes through a maximum as the pre-exposure is increased. The maximum values of the DQE for 430-mp radiation are found t o be 0.90,0.59, 0.62 and 0.30y0 for the four films, and these maxima occur for preexposures of 0.0011, 0.0040, 0.010, and 0.018 erg/cm2. 1. Derivation of a Working Formula for Q D . The DQE of a given photographic negative may be expected to depend on the following: 1. The amount of ambient exposure 2. The spectral distribution of the radiation signal 3. The method of development

QUANTUM EFFICIENCY OF DETECTORS

143

In this Sec. C, the dependence of the DQE on the first two items will be discussed. Standard developing conditions are assumed. I n application to photography, the concepts formulated in Sec. I11 may be interpreted as follows. The “noise” in the photographic negative is the density fluctuation from place to place on the surface: if one measures the density with an aperture of area A a t a large number of different places on the developed negative, the measured densities will not all be the same. The set of measured densities may be characterized by a mean density that will be denoted simply by D and by a rms deviation from D that is denoted by u. Suppose, then, that the entire surface of a negative is uniformly preexposed by the ‘(ambient” radiation and that on one small region of area A, a small additional radiation “signal” is incident. The noise equivalent value of this signal is the value that produces a density increment equal to the rms density fluctuation u measured with apertures of the same area A . The value of u will depend, of course on the area A of the aperture, and indeed where there is now good evidence (90) that u varies as A% for apertures substantially larger than the size of the grains in the emulsion. The numerical values given in this Sec. C are for a n arbitrarily chosen aperture 10 p in diameter. The symbol U is used to indicate the exposure of the film in ergs per square centimeter. The preexposure is denoted by U,. The amount of additional exposure that produces a density increment equal to the rms fluctuation may be called the noise equivalent exposure UN and is given by

U,

=

udU/dD

(8.1)

where d U / d D is the slope of the U-vs-D curve a t the point where the exposure U is equal to U,. In practice, the slope dD/dU is determined by taking the ratio of small finite increments AD and AU. Thus, UN may be written

UN = uAU/AD

(8.2)

The detective quantum efficiency as given by (3.24) now may be written

where U, is the exposure a t the middle of the range AU. The chief results derived in this section are for 430-mp radiation. At this wavelength, the energy of a photon is & = 4.6180 X 10-l2 erg

based on h = 6.6238 X erg-sec and c area of the 10-p circular aperture is

A

=

78.54

x

=

(8.4) 2.9979 X 1Olo cm/sec. The

lo-* cm2

(8.5)

144

R. CLARK JONES

The last three relations then yield QD

= 0.5880 X 10-5Uv,(AD/~loAU)2

(8.6)

This is the “working” expression for the DQE of photographic negatives. U, and AU must be expressed in ergs per square centimeter. 2. Description of the Films. The Eastman Kodak Company has very generously provided sufficient data for the calculation of the DQE of four of its current films. The four films are: 1. Kodak Royal-X Pan Film, Code 6128 2. Eastman Tri-X Panchromatic Negative Film, Type 5233 3. Eastman Plus-X Panchromatic Negative Film, Type 4231 4. Kodak Panatomic-X Film, Code 5240 The data supplied by Eastman Kodak for these films are representative of the films at the time of manufacture, but it should be recognized that their characteristics can be expected to vary with manufacturing tolerances and may change as improvements are made. The first is sheet film, and the last three are 35-mm roll films. The films were manfactured in February, 1957. Table I V shows the abbreviations used in this review for these films, the developing conditions, and the gamma to which the films were developed. TABLE IV. DATACONCERNING THE FOUR PHOTOGRAPHIC FILMS Material

Abbreviation

Time

Developer

Gamma

1 2

Royal-X Tri-X Plus-x Pan-X

5 min 6.5 min 6 . 5 min 6 rnin

DK-50 SD-28 SD-28 D-76

0.65 0.54 0.79 0.67

3 4

5. Sensitometric Data. Eastman Kodak has supplied density-vs-logexposure curves for each of the materials. The exposures were made to radiation that has passed through a narrow-band Wratten filter whose effective wavelength was 430 mp. The exposures through the filter were calibrated against exposures made with a prism monochromator. The duration of the sensitometric exposures was 15 sec. There was a significant reciprocity failure at this duration, and the exposures were corrected at a density of 0.38 above base to the exposure that would be required for a 0.1-sec exposure. The reciprocity correction amounted to 0.45, 0.34, 0.29, and 0.29 log exposure units for material