Reference Data on Atomic Physics and Atomic Processes (Springer Series on Atomic, Optical, and Plasma Physics)

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Springer Series on

ATOMIC , OPTICAL , AND PLASMA PHYSICS

51

Springer Series on

ATOMIC , OPTICAL , AND PLASMA PHYSICS The Springer Series on Atomic, Optical, and Plasma Physics covers in a comprehensive manner theory and experiment in the entire field of atoms and molecules and their interaction with electromagnetic radiation. Books in the series provide a rich source of new ideas and techniques with wide applications in fields such as chemistry, materials science, astrophysics, surface science, plasma technology, advanced optics, aeronomy, and engineering. Laser physics is a particular connecting theme that has provided much of the continuing impetus for new developments in the field. The purpose of the series is to cover the gap between standard undergraduate textbooks and the research literature with emphasis on the fundamental ideas, methods, techniques, and results in the field.

47 Semiclassical Dynamics and Relaxation By D.S.F. Crothers 48 Theoretical Femtosecond Physics Atoms and Molecules in Strong Laser Fields By F. Großmann 49 Relativistic Collisions of Structured Atomic Particles By A. Voitkiv and J. Ullrich 50 Cathodic Arcs From Fractal Spots to Energetic Condensation By A. Anders 51 Reference Data on Atomic Physics and Atomic Processes By B.M. Smirnov

Vols. 20–46 of the former Springer Series on Atoms and Plasmas are listed at the end of the book

Boris M. Smirnov

Reference Data on Atomic Physics and Atomic Processes With 95 Figures

Professor Boris M. Smirnov Russian Academy of Sciences Joint Institute for High Temperatures Izhorskaya ul. 13/19, 127412 Moscow, Russia E-mail: [email protected]

Springer Series on Atomic, Optical, and Plasma Physics ISBN 978-3-540-79362-5

ISSN 1615-5653

e-ISBN 978-3-540-79363-2

Library of Congress Control Number: 2008929556 © Springer-Verlag Berlin Heidelberg 2008 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting and production: VTEX, Vilnius Cover concept: eStudio Calmar Steinen Cover design: WMX Design GmbH, Heidelberg SPIN 12254521 57/3180/VTEX Printed on acid-free paper 987654321 springer.com

Preface

Each scientist works with certain information and collects it in the course of professional activity. In the same manner, the author collected data for atomic physics and atomic processes. This information was checked in the course of the author’s professional activity and was published in the form of appendices to the corresponding books on atomic and plasma physics. Now it has been decided to publish these data separately. This book contains atomic data and useful information about atomic particles and atomic systems including molecules, nanoclusters, metals and condensed systems of elements. It also gives information about atomic processes and transport processes in gases and plasmas. In addition, the book deals with general concepts and simple models for these objects and processes. We give units and conversion factors for them as well as conversion factors for spread formulas of general physics and the physics of atoms, clusters and ionized gases since such formulas are used in professional practice by each scientist of this area. This book includes numerical information from some reference books for physical units and constants [1–4] and for numerical parameters of atomic particles and processes [2, 5–10] (in the most degree [2]). We also use data of some reviews and original papers. The methodical peculiarity of this book consists in representation of some physical parameters in the form of periodic tables. This form simplifies the information retrieval because it only uses uniform information. If the data relate to a restricted number of elements, they are given in the form of specified tables. Some part of the book is devoted to atomic spectra, which are given in the form of the Grotrian diagrams for atoms with the electron valence shell s, s 2 , and also the valence shell p k for light atoms. This information has not changed during the last decades. We use the Grotrian diagrams from [11]; diagrams for the lowest atom states are taken from [12]. Along with the Grotrian diagrams, some concepts and formulas of atomic physics are represented. This book also contains basic concepts of the physics of atomic systems and the simplest models for their description. These models may be a basis for simple estimations of the parameters of some atomic objects and processes. For example, the model of a hard sphere describes atom–cluster collisions, the liquid drop model is convenient for the analysis of cluster evaporation and other cluster processes and the model of a degenerate electron gas may be used for the metal plasma. In these

vi

Preface

cases numerical parameters of models are given for certain objects, as they follow from measured object parameters. As a scientist who has used the data about atomic and plasma physics contained herein to fulfill some estimations for certain problems of this area, the author intends this book to be used by scientists and advanced students. In the first stage of information collection, the author was a user of these data, and the basis of this book is Appendices to books [12–16] for certain aspects of atomic and plasma physics. Therefore the author hopes that this book will be useful both for specialists and for advanced students of this physical area. Moscow, May 2008

Boris M. Smirnov

Contents

1

2

3

Fundamental Constants, Elements, Units . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fundamental Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Elements and Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Physical Units and Conversion Factors . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Systems of Physical Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Conversion Factors for Units . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Conversion Factors in Formulas of General Physics with Atomic Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 2 2 6

Elements of Atomic and Molecular Physics . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Properties of Atoms and Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Properties of the Hydrogen Atom and Hydrogen-Like Ions . 2.1.2 Properties of the Helium Atom and Helium-Like Ions . . . . . . 2.1.3 Quantum Numbers of a Light Atom . . . . . . . . . . . . . . . . . . . . . 2.1.4 Shell Atom Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Schemes of Coupling of Electron Momenta in Atoms . . . . . . 2.1.6 Parameters of Atoms and Ions in the Form of Periodic Tables 2.2 Atomic Radiative Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 General Formulas and Conversion Factors for Atomic Radiative Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Radiative Transitions between Atom Discrete States . . . . . . . 2.2.3 Absorption Parameters and Broadening of Spectral Lines . . . 2.3 Interaction Potential of Atomic Particles at Large Separations . . . . . 2.4 Properties of Diatomic Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Bound States of Diatomic Molecule . . . . . . . . . . . . . . . . . . . . . 2.4.2 Correlation between Atomic and Molecular States . . . . . . . . . 2.4.3 Excimer Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 11 15 17 18 24 29 33

Elementary Processes Involving Atomic Particles . . . . . . . . . . . . . . . . . . 3.1 Parameters of Elementary Processes in Gases and Plasmas . . . . . . . . 3.2 Inelastic Collisions of Electrons with Atoms . . . . . . . . . . . . . . . . . . . . 3.3 Collision Processes Involving Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Atom Ionization by Electron Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Atom Ionization in Gas Discharge Plasma . . . . . . . . . . . . . . . . . . . . . .

71 71 76 78 85 87

9

33 38 39 43 47 47 53 66

viii

Contents

3.6 3.7 3.8 3.9

Ionization Processes Involving Excited Atoms . . . . . . . . . . . . . . . . . . Electron–Ion Recombination in Plasma . . . . . . . . . . . . . . . . . . . . . . . . Attachment of Electrons to Molecules . . . . . . . . . . . . . . . . . . . . . . . . . Processes in Air Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 93 95 96

4

Transport Phenomena in Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.1 Transport Coefficients of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2 Ion Drift in Gas in External Electric Field . . . . . . . . . . . . . . . . . . . . . . 103 4.3 Conversion Parameters for Transport Coefficients . . . . . . . . . . . . . . . . 111 4.4 Electron Drift in Gas in Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.5 Diffusion of Excited Atoms in Gases . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5

Properties of Macroscopic Atomic Systems . . . . . . . . . . . . . . . . . . . . . . . . 115 5.1 Equation of State for Gases and Vapors . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2 Basic Properties of Ionized Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3 Parameters and Rates of Processes Involving Nanoparticles . . . . . . . 120 5.4 Parameters of Condensed Atomic Systems . . . . . . . . . . . . . . . . . . . . . 126

A

Atomic Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

1 Fundamental Constants, Elements, Units

1.1 Fundamental Physical Constants Table 1.1. Fundamental Physical Constants Electron mass

me = 9.10939 × 10−28 g

Proton mass

mp = 1.67262 × 10−24 g

Ratio of proton and electron masses

1 m(12 C) = 1.66054 × 10−24 g ma = 12 mp /me = 1836.15

Ratio of atomic and electron masses

ma /me = 1822.89

Electron charge

e = 1.602177 × 10−19 C = 4.8032 × 10−10 e.s.u.

Atomic unit of mass

e2 = 2.3071 × 10−19 erg cm Planck constant

h = 6.62619 × 10−27 erg s

Light velocity

h¯ = 1.05457 × 10−27 erg s c = 2.99792 × 1010 cm/s

Fine-structure constant

α = e2 /(hc) ¯ = 0.07295

Inverse fine-structure constant

2 = 137.03599 1/α = hc/e ¯

Bohr radius

a0 = h¯ 2 /(me e2 ) = 0.529177 Å

Rydberg constant

R = me e4 /(2h¯ 2 ) = 13.6057 eV = 2.17987 × 10−18 J

Bohr magneton

−24 J/T μB = eh/(2m ¯ e c) = 9.27402 × 10

= 9.27402 × 10−21 erg/Gs

Avogadro number

NA = 6.02214 × 1023 mol−1

Stephan–Boltzmann constant

σ = π 2 /(60h¯ 3 c2 ) = 5.669 × 10−12 W/(cm2 K4 )

Molar volume

R = 22.414 l/mol

Loschmidt number

L = NA /R = 2.6867 × 1019 cm−3

Faraday constant

F = NA e = 96485.3 C/mol

2

1 Fundamental Constants, Elements, Units

1.2 Elements and Isotopes There are about 2700 stable and 2000 radioactive isotopes in nature [17]. Stable isotopes relate to elements with a nuclear charge below 83, excluding technetium 43 Tc and promethium 61 Pm, and also to elements with a nuclear charge in the range 90–93 (thorium, protactinium, uranium, neptunium). The diagram Pt1 contains standard atomic masses for elements, taking into account their occurrence in the Earth’s crust [2, 18]. If stable isotopes of a given element are absent, the masses are given in square brackets. These masses are given in atomic mass units (amu) where the unit is taken 1/12 mass of the carbon isotope 12 C from 1961 (see Table 1.1). There is in diagram Pt2 the occurrence of stable isotopes in the Earth’s crust, and diagram Pt3 contains the lifetimes of stable isotopes [2, 18–20]. These lifetimes are expressed in days (d) and years (y) and are given for isotopes whose lifetime exceeds 2 hours (0.08 d).

1.3 Physical Units and Conversion Factors 1.3.1 Systems of Physical Units The unit system is a set of units through which various physical parameters are expressed. The basis of any mechanical system of units is the fact that the value of any dimensionality may be expressed through three dimensional units—length, mass and time. The CGS system, which is based on the centimeter, gram and second [3], is the oldest system of units and was introduced by British association for the Advancement of Science in 1874. The system of International Units (SI) [4] was adopted in 1960 (the conference des Poids et Mesure, Paris) and is based on the meter (m), kilogram (kg) and second (s). Other bases may be used for specific units. In transferring from mechanics to other physics branches, additional units or assumptions are required. In particular, along with mechanical units, we deal with electric and magnetic units below. Then the base of SI units along with the above mechanical units contain the unit of an electric current ampere (A). Table 1.2 contains the SI units and their connection with the base units. Note that we express the thermodynamic temperature through energy units below. In spreading the CGS system of units to electrostatics, one can use the Coulomb law for the force F between two charges e1 and e2 that are located at a distance r in a vacuum. This force is given by F = 0

e1 e2 r2

where 0 is the vacuum permittivity. Defining a charge unit from this formula under the assumption 0 = 1, we construct in this manner the CGSE system of units that describes physical parameters of mechanics and electrostatics. In the same way, in constructing of the CGSM system of units on the basis of the mechanical CGS system of units, the assumption is used that the vacuum magnetic conductivity is

Pt1. Standard atomic weights of elements and their natural occurrence in Earth’s crust

1.3 Physical Units and Conversion Factors 3

Pt2. Natural occurrence of stable isotopes

4 1 Fundamental Constants, Elements, Units

5

Pt3. Long-lived radioactive isotopes

1.3 Physical Units and Conversion Factors

6

1 Fundamental Constants, Elements, Units Table 1.2. Basic SI units [1, 2, 4]

Quantity

Name

Symbol

Connection with base units

Frequency

hertz

Hz

1/s

Force

newton

N

m kg/s2

Pressure

pascal

Pa

kg/(m s2 )

Energy

joule

J

m2 kg/s2

Power

watt

W

m2 kg/s3

Charge

coulomb

C

As

Electric potential

volt

V

W/A

Electric capacitance

farad

F

A2 s4 /(m2 kg)

Electric resistance

ohm



m2 kg/(A2 s3 )

Conductance

siemens

S

A2 s3 /(m2 kg)

Inductance

henry

H

m2 kg/(A2 s2 )

Magnetic flux

weber

Wb

m2 kg/(A s2 )

Magnetic flux density

tesla

T

kg/(A s2 )

one μ0 = 1. Because of units of electrostatic CGS (CGSE system of units) and electromagnetic CGS (CGSM system of units) are used, we give some conversions between these systems and SI units below. For atomic systems, the system of atomic units (or Hartree atomic units) is of importance because parameters of atomic systems are expressed through atomic parameters. In construction the system of atomic units, the fact is used that the parameter of any dimensionality may be built on the basis of three parameters of different dimensionality. As a basis for the system of atomic units, the following three parameters are taken: the Planck constant h¯ = 1.05457×10−27 erg s, the electron charge e = 1.60218 × 10−19 C and the electron mass me = 9.10939 × 10−28 g (we take the vacuum permittivity and the magnetic conductivity of a vacuum to be one). The system of atomic units constructed on these parameters is given below in Table 1.3. 1.3.2 Conversion Factors for Units Tables 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, and 1.10 contain conversion factors between units used. The specific electric field strength E/p or E/Na (E is the electric field strength, p is the pressure, Na is the number density of atoms) is a spread unit in physics of ionized gases. The unit of E/Na is 1 Townsend (Td) [21]. The connection between units of the above quantities is as follows: 1

V = 0.3535 Td; cm · Torr

1 Td = 2.829

V ; cm · Torr

1 Td = 1 × 10−17 V cm2 .

1.3 Physical Units and Conversion Factors

7

Table 1.3. System of atomic units Parameter

Symbol, formula

Value

Length

a0 = h¯ 2 /(me2 ) v0 = e2 /h¯ τ0 = h¯ 3 /(me4 ) ν0 = me4 /h¯ 3 ε0 = me4 /h¯ 2

5.2918 × 10−9 cm

Velocity Time Frequency Energy Power Electric voltage Electric field strength Momentum Number density Volume

ε0 /τ = m2 e8 /h¯ 5 ϕ0 = me3 /h¯ 2 E0 = me5 /h¯ 4 p0 = me2 /h¯ N0 = a0−3 V0 = a03

2.1877 × 108 cm/s 2.4189 × 10−17 s 4.1341 × 1016 s−1 27.2114 eV = 4.3598 × 10−18 J 0.18024 W 27.2114 V 5.1422 × 109 V/cm 1.9929 × 10−19 g cm/s 6.7483 × 1024 cm−3 1.4818 × 10−25 cm3 = 0.089240 cm3 /mol 2.8003 × 10−17 cm2

Square, cross section

σ0 = a02

Rate constant Dipole moment

k0 = v0 a02 = h¯ 3 /(m2 e2 ) K0 = v0 a05 = h¯ 9 /(m5 e8 ) ea0 = a0 = h¯ 2 /(me)

Magnetic moment

h¯ 2 /(me) = 2μB /α

2.5418 × 10−18 erg/Gs

Electric current

I = e/τ = me5 /h¯ 3 j0 = N0 v0 = m3 e8 /h¯ 7 i0 = eN0 v0 = m3 e9 /h¯ 7 J = ε0 N0 v0 = m4 e12 /h¯ 9

6.6236 × 10−3 A

Three body rate constant

6.126 × 10−9 cm3 /s 9.078 × 10−34 cm6 /s 2.5418 × 10−18 esu = 2.5418 D

Flux Electric current density Energy flux

= 2.5418 × 10−21 J/T 1.476 × 1033 cm−2 s−1 2.3653 × 1014 A/cm2 6.436 × 1015 W/cm2

6.2415 × 1018 6.2415 × 1011 1 8.6174 × 10−5 1.2398 × 10−4 4.1357 × 10−9 4.3364 × 10−2 1.0364 × 10−2

107 1 1.6022 × 10−12 1.3807 × 10−16 1.9864 × 10−16 6.6261 × 10−21 6.9477 × 10−28 1.6605 × 10−28

1 0.1 1 Torr 133.332 1 atma 1.01325 × 105 b 9.80665 × 104 1 at 1 bar 105 a atm—physical atmosphere b at = kg/cm2 —technical atmosphere

7.2429 × 1022 7.2429 × 1015 11604 1 1.4388 4.7992 × 10−5 503.22 120.27

1K 5.0341 × 1022 5.0341 × 1015 8065.5 0.69504 1 3.3356 × 10−5 349.76 83.594

1 cm−1

10 1 1333.32 1.01325 × 106 9.80665 × 105 106

1 dyn/cm2

1 kcal/mol

1.3595 × 10−3 1.01332 1 1.0197

1.0197 × 10−6 1.3158 × 10−3 1 0.96785 0.98693

9.8693 × 10−7

1 760 735.56 750.01

7.5001 × 10−4

1.0197 × 10−5

9.8693 × 10−6

7.5001 × 10−3

1.4393 × 1020 1.4393 × 1013 23.045 1.9872 × 10−3 2.8591 × 10−3 9.5371 × 10−9 1 0.23901

1 atma

1 atb

1.5092 × 1027 1.5092 × 1020 2.4180 × 108 2.0837 × 104 2.9979 × 104 1 1.0485 × 107 2.5061 × 106

1 MHz

1 Torr

Table 1.5. Conversion factors for units of pressure

1 eV

1 erg

1 Pa = 1 N/m2

1 10−7 1.6022 × 10−19 1.3807 × 10−23 1.9864 × 10−23 6.6261 × 10−28 6.9477 × 10−21 1.6605 × 10−21

1 Pa = 1 N/m2 1 dyn/cm2

1J 1 erg 1 eV 1K 1 cm−1 1 MHz 1 kcal/mol 1 kJ/mol

1J

Table 1.4. Conversion factors for units of energy

10−5 10−6 1.33332 × 10−3 1.01325 0.980665 1

1 bar

6.0221 × 1020 6.0221 × 1013 96.485 8.3145 × 10−3 1.1963 × 10−2 3.9903 × 10−7 4.184 1

1 kJ/mol

8 1 Fundamental Constants, Elements, Units

1.3 Physical Units and Conversion Factors

9

Table 1.6. Conversion factors for units of electric voltage

1V 1 CGSE 1 CGSM

1V

1 CGSE

1 CGSM

1 299.792 10−8

3.33564 × 10−3 1 3.33564 × 10−11

108 2.99792 × 1010 1

Table 1.7. Conversion factors for units of electric field strength

1 V/cm 1 CGSE 1 CGSM

1 V/cm

1 CGSE

1 CGSM

1 299.792 10−8

3.33564 × 10−3 1 3.33564 × 10−11

108 2.99792 × 1010 1

Table 1.8. Conversion factors for units of electric resistance 1 1 CGSE 1 CGSM

1

1 CGSE

1 CGSM

1 8.98755 × 1011 10−9

1.11265 × 10−12 1 1.11265 × 10−21

109 8.98755 × 1020 1

Table 1.9. Conversion factors for units of magnetic field strength

1 Oe 1 CGSE 1 A/m

1 Oe

1 CGSE

1 A/m

1 3.33564 × 10−11 0.012566

2.99792 × 1010 1 1.11265 × 10−21

79.5775 2.65442 × 10−9 1

Table 1.10. Conversion factors for units of magnetic induction

1 CGSE 1 T = 1 Wb/m2 1 Gs

1 CGSE

1 T = 1 Wb/m2

1 Gs

1 3.33564 × 10−7 3.33564 × 10−11

2.99792 × 106

2.99792 × 1010 104 1

1 10−4

1.3.3 Conversion Factors in Formulas of General Physics with Atomic Particles Explanations to Table 1.11: 1. The particle velocity is v = mass

√ 2ε/m, where ε is the energy, m is the particle

10

1 Fundamental Constants, Elements, Units Table 1.11. Conversion factors for formulas involving atomic particles

1

Formulaa √ v = C ε/m

2

√ v = C T /m

3

ε = Cv 2

4

ω = Cε

5 6 7

ω = C/λ ε = C/λ ωH = CH /m

8

√ rH = C εm/H

9

p = CH 2

Number

Factor C

Units used

5.931 × 107 cm/s 1.389 × 106 cm/s 5.506 × 105 cm/s 1.289 × 104 cm/s 1.567 × 106 cm/s 1.455 × 104 cm/s 3.299 × 10−12 K 6.014 × 10−9 K 2.843 × 10−16 eV 5.182 × 10−13 eV 1.519 × 1015 s−1 1.309 × 1011 s−1 1.884 × 1015 s−1 1.2398 eV 1.759 × 107 s−1 9655 s−1 3.372 cm 143.9 cm 3.128 × 10−2 cm 1.336 cm 4.000 × 10−3 Pa = 0.04 erg/cm3

ε in eV, m in e.m.u.a ε in eV, m in a.m.u.a ε in K, m in e.m.u. ε in K, m in a.m.u. T in eV, m in a.m.u. ε in K, m in a.m.u. v in cm/s, m in e.m.u. v in cm/s, m in a.m.u. v in cm/s, m in e.m.u. v in cm/s, m in a.m.u. ε in eV ε in K λ in μm λ in μm H in Gs, m in e.m.u. H in Gs, m in a.m.u. ε in eV, m in e.m.u., H in Gs ε in eV, m in a.m.u., H in Gs ε in K, m in e.m.u., H in Gs ε in K, m in a.m.u., H in Gs H in Gs

a e.m.u. is the electron mass unit (m = 9.108 × 10−28 g), a.m.u. is the atomic mass unit e (ma = 1.6605 × 10−24 g)

√ 2. The average particle velocity is v = 8T /(πm) with the Maxwell distribution function of particles on velocities; T is the temperature expressed in energetic units, m is the particle mass 3. The particle energy is ε = mv 2 /2, where m is the particle mass, v is the particle velocity 4. The photon frequency is ω = ε/h¯ , where ε is the photon energy 5. The photon frequency is ω = 2πc/λ, where λ is the wavelength 6. The photon energy is ε = 2π h¯ c/λ 7. The Larmor frequency is ωH = eH /(mc) for a charged particle of a mass m in a magnetic field of strength H √ 8. The Larmor radius of a charged particle is rH = 2ε/m/ωH , where ε is the energy of a charged particle, m is its mass, ωH is the Larmor frequency 9. The magnetic pressure is pm = H 2 /(8π)

2 Elements of Atomic and Molecular Physics

2.1 Properties of Atoms and Ions 2.1.1 Properties of the Hydrogen Atom and Hydrogen-Like Ions The hydrogen atom includes a bound electron located in the nuclear Coulomb field. The electron position in the field of the Coulomb center is described by three space coordinates: r is the distance of the electron from the Coulomb center, θ is the polar electron angle, and ϕ is the azimuthal angle. In addition, the spin electron state is characterized by the spin electron projection σ on a given direction. In the non-relativistic limit, space and spin coordinates are separated, and the space wave function has the form (2.1) ψ(r, θ, ϕ) = Rnl (r)Ylm (θ, ϕ). Here, separation of variables is used for an electron located in the Coulomb center field, so that Rnl (r) is the radial wave function, and Ylm is the angle wave function. In addition, n is the principle quantum number, l is the angular momentum and m is the angular momentum projection onto a given direction. These quantum numbers are whole values, and n ≥ 1, n ≥ l + 1, l ≥ 0, |m| ≤ l. The binding state energy εn that counts off from the continuous spectrum boundary is given by εn = −

me e4 2h¯ 2 n2

.

(2.2)

Here, n is the principle electron quantum number, e is the electron charge and me is the electron mass. As is seen, the states of an electron that is located in the Coulomb field are degenerated with respect to the electron momentum l and its projection m onto a given direction (and with respect to the spin projection σ , since the electron Hamiltonian is independent of its spin). For notations of electron states (electron terms) are used its quantum numbers n, l, where the principal quantum number n is given as a value, whereas the quantum number l is denoted by letters so that notations s, p, d, f, g, h relate to states with the angular momenta l = 0, 1, 2, 3, 4, 5, correspondingly. For example, the notation 4f refers to a state with quantum numbers n = 4, l = 3. The angle wave function of an electron located in the Coulomb center field is satisfied to the Schrödinger equation

12

2 Elements of Atomic and Molecular Physics

Table 2.1. The angle electron wave function in the hydrogen atom for small electron momenta l [22–24] l

m

0

0

1

0

1

±1

2

0

2

±1

2

±2

3

0

3

±1

3

±2

3

±3

4

0

4

±1

4

±2

4

±3

4

±4

Ylm (θ, ϕ) √1

4π

3 4π · cos θ



3 · sin θ · exp(±iϕ) ± 8π    5 · 3 cos2 θ − 1 4π 2 2  15 ± 8π · sin θ · cos θ · exp(±iϕ)  1 15 · sin2 θ · exp(±2iϕ) 2 8π  7 3 4π · (5 cos θ − 3 cos θ)  2 ± 18 21 π · sin θ · (5 cos θ − 1) exp(±iϕ)  1 105 · sin2 θ cos θ · exp(±2iϕ) 4 2π  3 ± 18 35 π · sin θ · exp(±3iϕ)  9 35 15 3 4 2 4π · ( 8 cos θ − 4 cos θ + 8 )  ± 38 π5 · sin θ · (7 cos3 θ − 3 cos θ) · exp(±iϕ)  5 · sin2 θ · (7 cos2 θ − 1) · exp(±2iϕ) ± 38 2π  3 ± 38 35 π · sin θ · cos θ · exp(±3iϕ)  3 35 4 16 2π · sin θ · exp(±4iϕ)

  ∂ ∂Ylm 1 ∂ 2 Ylm + l(l + 1)Ylm = 0 sin2 θ + ∂ cos θ ∂ cos θ sin2 θ ∂ϕ 2

(2.3)

and is given by the formula [22–24]  Ylm (θ, ϕ) =

2l + 1 (l − m)! 4π (l + m)!

1/2 Plm (cos θ ) exp(imϕ).

(2.4)

These angle wave functions of an electron are represented in Table 2.1. The radial wave function of an electron in the hydrogen atom is the solution of the Schrödinger equation   1 d2 2 l(l + 1) Rnl = 0, (rR) + 2ε + (2.5) − r dr 2 r r2

2.1 Properties of Atoms and Ions

13

Table 2.2. Electron radial wave functions Rnl for the hydrogen atom [22–24] State

Rnl

1s

2 exp(−r)

√1 1 − r exp(−r/2) 2

2s

2 1 √ r exp(−r/2) 24   2 2r 2 exp(−r/3) √ 1 − 2r + 3 27 3 3

2p 3s



2 · r 1 − r exp(−r/3) 3 6

3p

2 27

3d 4s

4 √ · r 2 exp(−r/3) 81 30   1 1 − 3r + r 2 − r 3 exp(−r/4) 4 4 8 192

4p

1 16





5 · r · 1 − r + r2 3 4 80



exp(−r/4)

1√ · r 2 · 1 − r exp(−r/4) 12

4d

64 5 1√ · r 3 · exp(−r/4) 768 35

4f

that, with accounting for boundary conditions, has the form    2 (n + l)! 1 r Rnl (r) = l+2 (2r)l exp − (2l + 1)! (n − l − 1)! n n   2r , × F −n + l + 1, 2l + 2, n

(2.6)

where F is degenerate hypergeometric function. The following normalization condition takes place for the radial wave function ∞ 2 Rnl (r)r 2 dr = 1. (2.7) 0

Table 2.2 lists the expressions of the radial electron wave functions for lowest hydrogen atom states. Table 2.3 contains expressions for average quantities r n  of the hydrogen atom, where r is the electron distance from the center, and Table 2.4 lists the values of these quantities for lowest states of the hydrogen atom. ∞ 2 r n  = Rnl (r)r n+2 dr. (2.8) 0

Fine structure splitting in the hydrogen atom is determined by spin–orbit interaction and corresponds according to its nature to the interaction between spin and

14

2 Elements of Atomic and Molecular Physics

Table 2.3. Formulas for average values r n  given in atomic units, where r is the electron distance from the Coulomb center [23] Value

Formula

r

r −1 

1 · [3n2 − l(l + 1)] 2 n2 · [5n2 + 1 − 3l(l + 1)] 2 n2 · [35n2 (n2 − 1) − 30n2 (l + 2)(l − 1) + 3(l + 2)(l + 1)l(l − 1)] 8 n4 · [63n4 − 35n2 (2l 2 + 2l − 3) + 5l(l + 1)(3l 2 + 3l − 10) + 12] 8 n−2

r −2 

[n3 (l + 1/2)]−1

r −3 

[n3 (l + 1) · (l + 1/2) · l]−1

r −4 

[3n2 − l(l + 1)][2n5 · (l + 3/2) · (l + 1) · (l + 1/2) · l · (l − 1/2)]−1

r 2  r 3  r 4 

Table 2.4. Average values r n  for lowest states of the hydrogen atom expressed in atomic units [23] State 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f

r 1.5 6 5 13.5 12.5 10.5 24 23 21 18

r 2  3 42 30 207 180 126 648 600 504 360

r 3  7.5 330 210 3442 2835 1701 18720 16800 13100 7920

r 4  22.5 2880 1680 6.136 × 104 4.420 × 104 2.552 × 104 5.702 × 105 4.973 × 105 3.629 × 105 1.901 × 105

r −1  1 0.25 0.25 0.111 0.111 0.111 0.0625 0.0625 0.0625 0.0625

r −2  2 0.25 0.0833 0.0741 0.0247 0.0148 0.0312 0.0104 0.00625 0.00446

r −3  – – 0.0417 – 0.0123 0.0247 – 5.21 × 10− 3 1.04 × 10−3 3.72 × 10−4

r −4  – – 0.0417 – 0.0137 5.49 × 10−3 – 5.49 × 10−4 2.60 × 10−4 3.7 × 10−5

magnetic field due to angular electron rotation. The fine structure splitting is given by       1 Z 2 eh¯ 2 1 ˆ  2l + 1 Z 2 eh¯ 2 1 · lˆs = δf = 2 mc 4 mc r3 r3  2 2 e Z4 = . 3 h¯ c 2n l(l + 1)

(2.9)

Here, angle brackets mean an average over the electron space distribution and this formula relates to the hydrogen-like ion where Z is the Coulomb center charge. Since the parameter [e2 /(h¯ c)]2 /4 = 1.33×10−5 is small, the fine structure splitting is small for the hydrogen atom. Accounting for the fine structure splitting gives the quantum numbers of the hydrogen atom as lsj mj instead of numbers lmsσ for degenerate levels, where s = 1/2 is the electron spin, j = l ± 1/2 is the total

2.1 Properties of Atoms and Ions

15

Table 2.5. Quantum defect for atoms of alkali metals [26, 27] Atom δs δp δd δf

Li 0.399 0.053 0.002 –

Na 1.347 0.854 0.0145 0.0016

K 2.178 1.712 0.267 0.010

Rb 3.135 2.65 1.34 0.0164

Cs 4.057 3.58 2.47 0.033

electron momentum and m, σ, mj are projections of the orbital momentum, spin and total momentum onto a given direction. Note that the lower level corresponds to the total electron momentum j = l − 1/2. The behavior of highly excited (Rydberg) states of atoms is close to that of the hydrogen atom because the Coulomb electron interaction with the center is the main interaction. But a short-range electron interaction with an atomic core leads to a displacement of atom energetic levels, and the electron energy, instead of that by formula (2.2), is given by [25] εn = −

me e4 2h¯ 2 n2ef

=−

me e4 2h¯ 2 (n − δl )2

,

(2.10)

where nef is the effective principal quantum number and δl is the so-called quantum defect. Since it is determined by a short-range electron–core interaction, its value decreases with an increase of the electron angular momentum l. Table 2.5 contains the values of the quantum defect for alkali metal atoms [26, 27]. In addition, Fig. A.1 in Appendix A gives spectrum of the hydrogen atom. 2.1.2 Properties of the Helium Atom and Helium-Like Ions In order to explain the atom’s nature, structure and spectrum, we use a one-electron approximation when the atom wave function is the combination of products of the one-electron wave functions. In construction the atom wave functions we are based on the Pauli exclusion principle [28], according to which location of two electrons is prohibited in an identical electron state, and the total electron wave function is zero if the coordinates of the two electrons with an identical spin direction coincide. We include in the Hamiltonian of electrons, alongside the spin–orbit interaction, the exchange interaction whose potential may be represented in the form Vex = A(|r1 − r2 |)ˆs1 sˆ2 .

(2.11)

Here r1 , r2 are electron coordinates and sˆ1 , sˆ2 are their spins. Note that the nature of exchange interaction is non-relativistic and is connected with the symmetry of the total wave function. For the system of two electrons (helium atom or helium-like ion), the total wave function according to the Pauli principle changes sign as a result of permutation of electrons. But the total wave function is the product of the space and spin electron wave functions. For the total spin S = 1 the spin wave function is symmetric with respect to electron permutation, and for the total spin S = 0

16

2 Elements of Atomic and Molecular Physics

Fig. 2.1. The lowest excited states of the helium atom. The excitation energy for a given state is expressed in eV, the wavelengths λ and the radiative lifetimes τ are indicated for a corresponding radiative transition

the spin wave function changes sign at permutation of electrons. Hence, the space wave function of two electrons is symmetric with respect to electron coordinates if the electron total spin is zero and is antisymmetric if the total electron spin is one. Thus, the total electron spin determines the symmetry of the space electron wave function, and this fact may be taken into account formally by the introduction of an exchange interaction term (2.11) in the electron Hamiltonian. Thus the spectrum of the helium atom (Fig. A.2 in Appendix A) is divided into two independent parts, with the total electron spin S = 0 and S = 1 with a symmetric space wave function of electrons in the first case and with antisymmetric space wave function of electrons in the second case. The radiative transitions between states that belong to different parts are practically absent because of conservation of the total electron spin in radiative transitions due to first approximations in the expansion over a small relativistic parameter. In addition, atom levels related to the total spin S = 1 are located below the corresponding levels of states S = 0 with identical other quantum numbers. One can see it in the Grotrian diagram Fig. A.2 for the helium atom and in Fig. 2.1, where the lowest excited states of the helium atom are given. As a result, we represent the electron Hamiltonian in the form h¯ 2 Ze2 Ze2 e2 h¯ 2 1 − 2 − − + Hˆ = − 2me 2me r1 r2 |r1 − r2 | + Aˆs1 sˆ2 + B ˆl1 sˆ1 + B ˆl2 sˆ2 .

(2.12)

An electrostatic interaction of electrons with a core and also with each other, the exchange interaction of electrons and the interaction of an electron spin with its orbit are included in this Hamiltonian. Since spin–orbit interaction is of importance for excited electron states, interaction of a spin with a foreign orbit is not important. Let us use the electron Hamiltonian (2.12) for the analysis of the spectrum of the helium atom that is given in Fig. A.2 and the spectra of helium-like ions. In the ground atom state, both electrons are located in the state 1s, and the ground atom

2.1 Properties of Atoms and Ions

17

Table 2.6. Dependence of interactions of helium-like ions on the nuclear charge Z Parameter

Z-dependence

Ionization potential Potential of exchange interaction Spin–orbit interaction Rate of one-photon radiative transition Rate of two-photon radiative transition

Z2 Z Z4 Z4 Z8

state is 1s 2 1 S, i.e. the total atom spin is zero, S = 0. The atom state with S = 1 and the states 1s of both electrons are not realized because the space wave function is antisymmetric with respect to electron permutation and is zero for identical electron states. The quantum numbers of an excited helium atom are its orbital momentum and spin, which can be equal to zero or one. In the latter case, the atom quantum number is the total atom momentum, which is a sum of the orbital momentum and spin (see Fig. A.2). In the case of helium-like ions, which consist of the Coulomb center of a charge Z and two electrons, the contribution of different interactions to the total energy varies with variation of the center charge, which follows from Table 2.6. Correspondingly, the spectrum of helium-like ions changes with variation of Z. 2.1.3 Quantum Numbers of a Light Atom One can ignore relativistic interactions for light atoms in the first approximation, and the electron Hamiltonian for such an atom in accordance with the expression (2.12) has the form   Ze2  e2 h¯ 2  +A sˆi sˆk . i − + Hˆ = − 2me ri |ri − rk | i

i

i,k

(2.13)

i,k

Here i, k are electron numbers, ri , rk are electron coordinates, and sˆi , sˆk are the operators of electron spins. ˆ and the total electron spin Sˆ for The operators of the total electron momentum L an atom are given by   ˆli , sˆi , Sˆ = (2.14) Lˆ = i

i

where ˆli and sˆi are the operators of the angular momentum and spin for i-th electron. Because these operators commute with the Hamiltonian (2.13) [29], the atom quantum numbers are LSML MS , where L is the angular atom momentum, S is the total atom spin, and ML , MS are the projections of these momenta onto a given direction. If we add to the Hamiltonian (2.13) the operator of spin–orbit interaction in the ˆ the energy levels for given atom quantum numbers L and S are split, and form B Lˆ S, the atom eigen states are characterized by the quantum numbers LSJ MJ , where J is the total atom momentum, and MJ is its projection onto a given direction.

18

2 Elements of Atomic and Molecular Physics

Fig. 2.2. The lowest excited states of the carbon atom. Energies of excitation for corresponding states are indicated inside rectangular boxes, accounting for their fine structure, and are expressed in cm−1 . Wavelengths of radiative transitions are given inside arrows and are expressed in Å. The radiative lifetimes of states are placed in triangular boxes and are expressed in s

The following notations are used to describe an atom state. The number 2S + 1 of spin projections and the multiplicity of spin states are given above. The angular atom momentum is given by letters S, P , D, F, G, etc. for values of the angular momentum L = 0, 1, 2, 3, 4, etc. The total atom momentum J is given as a subscript. For example, 3 D3 corresponds to atom quantum numbers S = 1, L = 2, J = 3. This method of description of atom states is suitable for light atoms. But this scheme may be used also for heavy atoms to classify atom states, though relativistic effects are of importance for these atoms. The Grotrian diagrams for the lowest states of some atoms are given in Figs. 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8 and 2.9 [12]. In particular, fine splitting of the ground state of the tellurium atom (Fig. 2.9) is approximately 20 times more than that for the oxygen atom with the same electron structure (Fig. 2.4), and the character of fine splitting in these cases is different. Nevertheless, using the same classification method for both cases is convenient. 2.1.4 Shell Atom Scheme The atom shell model distributes bound electrons over electron shells, so that each shell contains electrons with identical quantum numbers nl. This corresponds to one-electron approximation if the field that acts on a test electron consists of the Coulomb field of the nucleus and a self-consistent field of other electrons. The sequence of shell filling for an atom in the ground state corresponds to that for a center with a screened Coulomb field that acts on a test electron. Diagrams Pt4, Pt5 give the valence electron shells for atoms and negative ions in the ground state. In addition, the ionization potentials of atoms and their first ions in the ground states and the electron affinities for atoms are given in these diagrams [2, 30, 31].

2.1 Properties of Atoms and Ions

19

Fig. 2.3. The lowest excited states of the nitrogen atom. Energies of excitation for corresponding states are indicated inside rectangular boxes, accounting for their fine structure, and are expressed in cm−1 . Wavelengths of radiative transitions are given inside arrows and are expressed in Å. The radiative lifetimes of states are placed in triangular boxes and are expressed in s, 15 means 1 × 105

Fig. 2.4. The lowest excited states of the oxygen atom. Notations are similar to those of Fig. 2.3. The oscillator strength of a radiative transition is given in parentheses near the wavelength

Valence electrons determine properties of atoms and atom interaction. One can construct the valence electron shell to consist of one electron and atomic core. This parentage scheme allows one to analyze atom properties due to one-electron transitions. Then the electron wave function of an atom ΨLSML MS (1, 2, . . . , n) is expressed through the wave function of an atomic core ΨL S  ML MS (2, . . . , n) and

20

2 Elements of Atomic and Molecular Physics

Fig. 2.5. The lowest excited states of the silicon atom. Notations are similar to those of Fig. 2.4

Fig. 2.6. The lowest excited states of the phosphorus atom. Notations are similar to those of Fig. 2.4

through the wave function of a valence electron ψl 1 mσ (1) in the following manner 2 [32–35]: ΨLSML MS (1, 2, . . . , n)   1 l = √ Pˆ GLS (l, n) L S  m n     L ML S MS mσ

L ML

× ψl 1 mσ (1) · ΨL S  ML MS (2, . . . , n). 2

L ML

 1 2

σ

S MS

S MS



2.1 Properties of Atoms and Ions

21

Fig. 2.7. The lowest excited states of the sulfur atom. Notations are similar to those of Fig. 2.4

Fig. 2.8. The lowest excited states of the selenium atom. Notations are similar to those of Fig. 2.4

Here n is a number of valence electrons and the operator Pˆ permutes coordinates and spin of a test electron that is denoted by an argument 1 with those of valence electrons of the atomic core; LSML MS are the atom quantum numbers, L S  ML MS are the quantum numbers of the atomic core, l 12 mσ are the quantum numbers of a test electron, GLS L S  (l, n) is the parentage or Racah coefficient [32, 33] and the Clebsh–Gordan coefficients result from summation of momenta (orbital and spin momenta) of a test electron and atomic core into an atom momentum. Note that the one-electron wave function, by analogy with formula (2.1), has the form (2.15) ψ(r, θ, ϕ) = Rnl (r)Ylm (θ, ϕ)χ1/2,σ where the angular wave function Ylm is given by formula (2.4), χ1/2,σ is the spin electron wave function, and the radial wave function Rnl (r) is normalized by the condition (2.6) and satisfies to the Schrödinger equation (2.5) far from the core. The Coulomb interaction potential between the electron and the core takes place at large

22

2 Elements of Atomic and Molecular Physics

Fig. 2.9. The lowest excited states of the tellurium atom. Notations are similar to those of Fig. 2.4

electron distances r from the core, where the solution of this equation has the form 1

Rnl = Ar γ

−1

exp(−γ r),

rγ 1; rγ 2 1.

(2.16) √ Here γ = (−2ε) in atomic units, and the electron behavior far from the core is determined by two asymptotic parameters, an exponent γ and an amplitude A. The parentage scheme relates to a non-relativistic approximation when the orbital momentum and spin are quantum numbers for an atom and its atomic core, and this scheme accounts for the spherical atom symmetry if non-relativistic interactions are small. Then LSML MS , i.e. the atom angular momentum, spin and their projections on a given direction, are the quantum numbers. In this approximation the atom states are degenerated over the angular momentum and spin projections, i.e. identical energies correspond to different projections on a given direction for these momenta. The simplest construction of electron shells relates to s- and pvalence electrons, and the values of parentage coefficients for these cases are given in Table 2.7. Fractional parentage coefficients satisfy some relations that follow from the definition of these quantities. The normalization of the atomic wave function takes the form  2 GLS (2.17) L S  (l, n, v) = 1. L S  v

The total number of valence electrons equals 4l+2 if this electron shell is completed. The analogy between electrons and vacancies connects the parameters of electron shells in the following manner [32–35]:   (4l + 3 − n)(2S  + 1)(2L + 1) 1/2 LS L+L +S+S  −l−1/2 · GL S  (l, n, v) = (−1) n(2S + 1)(2L + 1)  

S × GL LS (l, 4l + 3 − n, v).

(2.18)

2.1 Properties of Atoms and Ions

23

Table 2.7. Fractional parentage coefficients for electrons of s and p shells [11, 35] Atom

Atomic core

GLS L S 

Atom

Atomic core

s(2 S) s 2 (1 S) p(2 P ) p 2 (3 P ) p 2 (1 D) p 2 (1 S) p 3 (4 S)

(1 S) s(2 S) (1 S) p(2 P ) p(2 P ) p(2 P ) p 2 (3 P ) p 2 (1 D) p 2 (1 S) p 2 (3 P ) p 2 (1 D) p 2 (1 S) p 2 (3 P ) p 2 (1 D)

1 1 1 1 1 1 1 0 0 √ 1/ √ 2 −1/ 2 0 √ −1/ 2 √ − 5/18

p 3 (2 P ) p 4 (3 P )

p 2 (1 S) p 3 (4 S) p 3 (2 D) p 3 (2 P ) p 3 (4 S) p 3 (2 D) p 3 (2 P ) p 3 (4 S) p 3 (2 D) p 3 (2 P ) p 4 (3 P ) p 4 (1 D) p 4 (1 S) p 5 (2 P )

p 3 (2 D)

p 3 (2 P )

p 4 (1 D)

p 4 (1 S)

p 5 (2 P )

p 6 (1 S)

GLS L S  √ 2/3 √ −1/ 3 √ 5/12 −1/2 0 √ 3/4 −1/2 0 0 1 √ 3/5 √ 1/√3 1/ 15 1

The parentage scheme relates to light atoms where relativistic effects are small. Then, within the framework of the LS scheme of momentum summation, the angular atom momentum is the sum of angular momenta of an atomic core and test electron, and the atom spin is the sum of spins of these particles. Then, as a result of the interaction of an atom angular momentum L and spin S, these momenta are summed into the total atom momentum J , so that atom quantum numbers are LSJ MJ , where MJ is the projection of the total atom momentum on a given direction. As is seen, splitting of the electron terms with given values of L and S due to spin–orbit interaction leads to atom quantum numbers LSJ in contrast to quantum numbers LS in neglecting spin–orbit interaction. Table 2.8 contains the numbers of electron terms and electron levels for atoms with non-completed electron shells. The parentage scheme is simple for valence s and p-electrons, and the values of fractional parentage coefficients are represented in Table 2.7 for these cases. In the case of d and f valence electrons, removal of one electron can lead to different states of an atomic core at identical values of the atom angular momentum and spin. To distinguish these states, one more quantum number v, the seniority, is introduced. Table 2.9 contains the number of electron states and electron terms for atoms with valence d-electrons. The ground atom state for a given electron shell follows from the Hund empiric law [36], according to which the maximum atom spin corresponds to the ground electron state, and the total atom momentum is minimal if the shell is filled below one half and is maximal from possible ones if the electron shell is filled by more than half. For example, the nitrogen atom with the electron shell p 3 has three electron terms, 4 S, 2 D, 2 P , and the electron term 4 S corresponds to its ground state. The ground state of the aluminum atom with the electron shell p is characterized by the total momentum 1/2 (the state 2 P1/2 ), and the ground state of the chlorine atom

24

2 Elements of Atomic and Molecular Physics Table 2.8. States of atoms with non-filled electron shells

Shell configuration s s2 p, p 5 p2 , p4 p3 d, d 9 d 2, d 8 d 3, d 7 d 4, d 6 d5 f, f 13 f 2 , f 12 f 3 , f 11 f 4 , f 10 f 5, f 9 f 6, f 8 f7

Number of terms 1 1 1 3 3 1 5 8 18 16 1 7 17 47 73 119 119

Number of levels 1 1 2 5 5 2 9 19 40 37 2 13 41 107 197 289 327

Statistical weight 2 1 6 15 20 10 45 120 210 252 14 91 364 1001 2002 3003 3432

Table 2.9. Electron terms of atoms with filling electron shells d n [12] n 0, 10 1, 9 2, 8 3, 7 4, 6 5

Electron terms 1S 2D 1 S, 3 P , 1 D, 3 F, 1 G 2 P , 4 P , 2 D(2), 2 F, 4 F, 2 G, 2 H 1 S(2), 3 P (4), 1 D(2), 3 D, 5 D, 1 F, 3 F (2), 1 G(2), 3 G, 3 H, 1 J 2 S, 6 S, 2 P , 4 P , 2 D(3), 2 F (2), 4 F, 2 G(2), 4 G, 2 H, 2 J

Number of states 1 10 45 120

Number of electron terms 1 2 9 19

210

40

252

37

with the electron shell p 5 is 2 P3/2 (the total momentum is 3/2). The ground states of atoms with d and f filling electron shells are given in Tables 2.10 and 2.11. Note that the parentage scheme holds true for light atoms both for the ground and lower excited atom states, and it is violated for heavy atoms both due to an increase of the role of relativistic effects and because of overlapping electron shells with different nl. 2.1.5 Schemes of Coupling of Electron Momenta in Atoms In construction of electron shells we were based above on the LS scheme of electron momentum summation. Along with this, jj scheme of electron momentum summation is possible. In particular, taking an atom to be composed from an atomic core with the angular momentum l and spin s and from a valence or excited electron

2.1 Properties of Atoms and Ions

25

Table 2.10. The ground states for atoms with filling d-shell [12] Electron shell d d2 d3 d4 d5 d6 d7 d8 d9

Term of the ground state 2D 3F 4F

3/2

2

3/2 5D 0 6S 5/2 5D 4 4F 9/2 3F 4 2D 5/2

Atoms with this electron shell Sc(3d), Y(4d), La(5d), Lu(5d), Ac(6d), Lr(6d) Ti(3d 2 ), Zr(4d 2 ), Hf(5d 2 ), Th(6d 2 ) V(3d 3 ), Ta(5d 3 ) W(5d 4 ) Mn(3d 5 ), Tc(4d 5 ), Re(5d 5 ) Fe(3d 6 ), Os(5d 6 ) Co(3d 7 ), Ir(5d 7 ) Ni(3d 8 ) –

Table 2.11. The ground state of atoms with a filling f -shell [12] Electron shell f f2 f3 f4 f5 f6 f7 f8 f9 f 10 f 11 f 12 f 13

Term of the ground state 2F 5/2 3H 4 4I 9/2 5I 4 6H 5/2 7F 0 8S 7/2 7F 6 6H 15/2 5I 8 4I 15/2 3H 6 2F 5/2

Atoms with this electron shell – – Pr(4f 3 ) Nd(4f 4 ) Pm(4f 5 ) Sm(4f 6 ), Pu(5f 6 ) Eu(4f 7 ), Am(5f 7 ) – Tb(4f 9 ), Bk(5f 9 ) Dy(4f 10 ), Cf(5f 10 ) Ho(4f 11 ), Es(5f 11 ) Er(4f 12 ), Fm(5f 12 ) Tm(4f 13 ), Md(5f 13 )

with the angular momentum le and spin se , one can realize the summation of these momenta into the total atom momentum in the schemes l + le = L, l + s = j,

s + se = S, S + L = J (LS coupling), le + se = je , j + je = J (jj coupling).

A certain hierarchy of interaction corresponds to each scheme of momentum summation, and we will follow this connection. Let us represent the Hamiltonian of interacting electron and atomic core in the form Hˆ = Hˆ core + Hˆ e + Aˆssˆe + B ˆlˆs + bˆle sˆe .

(2.19)

The terms Hˆ core and Hˆ e in this sum account for the electron kinetic energy and electrostatic interaction for an atomic core and valence electron, respectively; the third term describes an exchange interaction of the valence electron and atomic core; and the last two terms are responsible for spin–orbit interaction for an atomic

26

2 Elements of Atomic and Molecular Physics Table 2.12. Electron terms for filling of electron shells with valence p-electrons

LS scheme p p2

p3

p4

p5 p6

LS term 2P 2P 3P 3P 3P 1D 1S 4S 2D 2D 2P 2P 3P 3P 3P 1D 1S 2P 2P 1S

J 1/2 3/2 0 1 2 2 0 3/2 3/2 5/2 1/2 3/2 2 0 1 2 0 3/2 1/2 0

jj scheme [1/2]1 [3/2]1 [1/2]2 [1/2]1 [3/2]1 [1/2]1 [3/2]1 [3/2]2 [3/2]2 [1/2]2 [3/2]1 [1/2]1 [3/2]2 [1/2]1 [3/2]2 [1/2]1 [3/2]2 [3/2]3 [1/2]1 [3/2]3 [1/2]2 [3/2]2 [1/2]21 [3/2]2 [1/2]1 [3/2]3 [3/2]4 [1/2]2 [3/2]3 [1/2]1 [3/2]4 [1/2]2 [3/2]4

J 1/2 3/2 0 1 2 2 0 3/2 3/2 5/2 1/2 3/2 2 0 1 2 0 3/2 1/2 0

core and valence electron. It is of importance that this Hamiltonian commutes with the total atom momentum [37] Jˆ = ˆl + sˆ + ˆle + sˆe .

(2.20)

This means that the total atom momentum is the quantum atom number for both schemes of momentum summation at any relation between exchange and spin–orbit interactions. Next, the LS-coupling scheme is valid in the limit of a weak spin– orbit interaction, whereas the jj -coupling scheme corresponds to a weak exchange interaction. In reality, the LS-coupling scheme is valid for light atoms, where the relativistic interactions are small, whereas the jj -coupling scheme may be used as a model for heavy atoms. Nevertheless, the LS-coupling scheme is used often for notations of electron terms of heavy atoms, and Table 2.12 contains electron terms of atoms with filling p shells for both coupling schemes. Notations for jj -coupling schemes are such that the total momentum of an individual electron is indicated in square parentheses and the total number of such electrons is given as a right superscript. Thus, roughly in accordance with the hierarchy of interaction inside the atom, we have two basic types of momentum coupling. If the electrostatic interaction exceeds a typical relativistic interaction, we have the LS-type of momentum summation; in another case we obtain the jj -type of momentum coupling. Really, the

2.1 Properties of Atoms and Ions

27

Fig. 2.10. The lowest excited states of the argon atom. Excitation energies count off from the lowest excited state and are given in cm−1 . The wavelengths λ and radiative lifetimes τ are indicated for a corresponding radiative transition

Fig. 2.11. The lowest excited states of the neon atom. Energies of excitation for corresponding states are indicated inside right rectangular boxes and are expressed in cm−1 . Radiative lifetimes of corresponding states are given inside left rectangular boxes in ns. Wavelengths of radiative transitions are indicated inside arrows and are expressed in Å, the rates of radiative transitions are represented in parentheses near the wavelength and are expressed in 106 s−1

28

2 Elements of Atomic and Molecular Physics

Fig. 2.12. The lowest excited states of the argon atom. Notations are similar to those of Fig. 2.11

electrostatic interaction Vel inside the atom corresponds to splitting of atom levels of the same electron shell, while the relativistic interaction may be characterized by the fine splitting δ of atom levels. The specific case relates to atoms of inert gases, and Fig. 2.10 gives the lowest group of excited states of the argon atom. On this example one can demonstrate the ways of classifications of these states. The structure of the electron shell for these states is 3p 5 4s, and, along with notations of the LS-coupling scheme, the notations of the jj -coupling scheme are used. In the last case the total momentum of the atomic core is given inside square brackets, and the total atom momentum is given as a right subscript. In addition, the Pashen notations are used for lowest excited states of inert gas atoms. The electron terms of the lowest group of excited states are denoted as 1s5 , 1s4 , 1s3 , 1s2 , and the subscript decreases with an increase of excitation. In addition, the lowest excited states of inert gases are given in Figs. 2.11, 2.12, 2.13, and 2.14. Pashen notations for the next group of excited states with the electron shell np 5 (n + 1)p (n is the principal quantum number of electrons of the valence shell) are from 2p10 up 2p1 as excitation increases. In notations of the jj -scheme of momentum coupling, the state notation includes the state of an ex-

2.1 Properties of Atoms and Ions

29

Fig. 2.13. The lowest excited states of the krypton atom. Notations are similar to those of Fig. 2.11

cited electron, the core state in square brackets, and the total atom momentum as a subscript.

2.1.6 Parameters of Atoms and Ions in the Form of Periodic Tables Diagram Pt4 gives the ionization potentials for atoms and their first ions in the ground states on the basis of the periodic table of elements together with construction of the electron shell and the electron term for the ground atom state within the framework of the LS scheme of momentum coupling. Because of the periodical character of the electron shell structure, we obtain the periodical dependence for atom parameters [36, 38]. The same is given in the diagram Pt5 for negative atomic ions. Because of a short-range interaction for a valence electron and an atomic core, in contrast to atoms, the number of bound states for a negative ion is finite (it is often one or zero). The binding energies of excited states of negative ions are also represented in diagram Pt5 in the cases when they exist (in particular, for elements of the fourth group). Diagram Pt6 contains parameters of lower excited states of atoms on the basis of the periodic table. Parameters of lower resonant states from which a dipole radiative

2 Elements of Atomic and Molecular Physics

Pt4. Ionization potentials of atoms and ions

30

31

Pt5. Electron affinities of atoms

2.1 Properties of Atoms and Ions

2 Elements of Atomic and Molecular Physics

Pt6. Lowest excited states of atoms

32

2.2 Atomic Radiative Transitions

33

Fig. 2.14. The lowest excited states of the xenon atom. Notations are similar to those of Fig. 2.11

transition is possible in the ground state are given in diagram Pt7. Diagram Pt8 lists the splitting energies for lower atom states, and the polarizabilities of atoms in the ground state are given in diagram Pt9.

2.2 Atomic Radiative Transitions 2.2.1 General Formulas and Conversion Factors for Atomic Radiative Transitions Table 2.13 contains the conversion factors in formulas for photon parameters and the radiative lifetime of resonantly excited states. Explanation of Table 2.13. 1. The photon energy ε = h¯ ω, where ω is the photon frequency. 2. The photon frequency is ω = ε/h¯ . 3. The photon frequency is ω = 2πc/λ, where λ is the wavelength, c is the light speed. 4. The photon energy is ε = 2π hc/λ. ¯

2 Elements of Atomic and Molecular Physics

Pt7. Resonantly excited atom states

34

Pt8. Splitting of lowest atom levels

2.2 Atomic Radiative Transitions 35

2 Elements of Atomic and Molecular Physics

Pt9. Atomic polarizability

36

2.2 Atomic Radiative Transitions

37

Table 2.13. The conversion factors for radiative transition between atom states Number

Formula

Conversion factor C

Units used

1

ε = Cω

C = 4.1347 × 10−15 eV C = 6.6261 × 10−34 J

2

ω = Cε

3 4 5

ω = C/λ ε = C/λ fo∗ = Cωd 2 g∗

6 7

fo∗ = Cd 2 g∗ /λ 1/τ∗o = Cω3 d 2 go

8 9

1/τ∗o = Cd 2 go /λ3 1/τ∗o = Cω2 go fo∗ /g∗

10

1/τ∗o = Cfo∗ go /(g∗ λ2 )

ω in s−1 ω in s−1 ε in eV ε in K ε in eV λ in µm ω in s−1 , d in Da ε = hω ¯ in eV , d in Da λ in µm, d in Da ω in s−1 , d in Da ε = hω ¯ in eV, d in Da λ in µm, d in Da ω in s−1 , d in Da ε = hω ¯ in eV; d in Da λ in µm, d in Da

C = 1.519 × 1015 s−1 C = 1.309 × 1011 s−1 C = 1.884 × 1015 s−1 1.2398 eV 1.6126 × 10−17 0.02450 0.03038 3.0316 × 10−40 s−1 1.06312 × 106 s−1 2.0261 × 106 s−1 1.8799 × 10−23 s−1 4.3393 × 107 s−1 6.6703 × 107 s−1

a D is Debye, 1D = ea = 2.5418 × 10−18 CGSE o

5. The oscillator strength for a radiative transition from a lower o to an upper ∗ state of an atomic particle that is averaged over lower states o and is summed over upper states ∗ is equal to fo∗ =

2me ω 2me ω 2 |o|D|∗|2 g∗ = d g∗ , 2 3h¯ e 3h¯ e2

where d = o|D|∗ is the matrix element for the operator of the dipole moment of an atomic particle taken between transition states. Here me , h¯ are atomic parameters, g∗ is the statistical weight of the upper state, and ω = (ε∗ − εo )/h¯ is the transition frequency, where εo , ε∗ are the energies of transition states. 6. The oscillator strength for radiative transition is [23, 29] fo∗ =

4πcme 2 d g∗ . 3h¯ e2 λ

Here λ is the transition wavelength; other notations are the same as above. 7. The rate of the radiative transition is [34, 35, 39] 1 4ω3 2 = B∗o = d go . τ∗o 3h¯ c3 Here B is the Einstein coefficient; other notations are as above. 8. The rate of radiative transition is given by 1 32π 3 2 = B∗o = d go . τ∗o 3h¯ λ3

38

2 Elements of Atomic and Molecular Physics

Table 2.14. Parameters of radiative transitions involving lowest states of the hydrogen atom, so that i is the lower state and j is the upper transition state [22, 23] Transition 1s–2p 1s–3p 1s–4p 1s–5p 2s–3p 2s–4p 2s–5p 2p–3s 2p–3d 2p–4s 2p–4d 2p–5s 2p–5d 3s–4p 3s–5p

fij 0.4162 0.0791 0.0290 0.0139 0.4349 0.1028 0.0419 0.014 0.696 0.0031 0.122 0.0012 0.044 0.484 0.121

τj i , ns 1.6 5.4 12.4 24 5.4 12.4 24 160 15.6 230 26.5 360 70 12.4 24

Transition 3p–4s 3p–4d 3p–5s 3p–5d 3d–4p 3d–4f 3d–5p 3d–5f 4s–5p 4p–5s 4p–5d 4d–5p 4d–5f 4f –5d 4f –5g

fij 0.032 0.619 0.007 0.139 0.011 1.016 0.0022 0.156 0.545 0.053 0.610 0.028 0.890 0.009 1.345

τj i , ns 230 36.5 360 70 12.4 73 24 140 24 360 70 24 140 70 240

Here λ is the wavelength of this transition; other notations are the same as above. 9. The rate of radiative transition is 1 2ω2 e2 go = fo∗. τ∗o me c3 g∗ 10. The rate of radiative transition is 1 8π 2 go fo∗. = τ∗o h¯ g∗ λ2 c 2.2.2 Radiative Transitions between Atom Discrete States The strongest radiative transitions between discrete atom states in a non-relativistic case connect the atom states with nonzero matrix elements of the dipole moment operator. This follows from the above formulas for the oscillator strength of radiative transitions and leads to certain selection rules for the radiative transitions. For transitions between hydrogen atom states when one electron is located in the Coulomb field, these selection laws have the form [23, 29] l = ±1,

m = 0, ±1.

(2.21)

Table 2.14 contains parameters of radiative transitions for the hydrogen atom. Radiative transitions between states with zero matrix elements of the dipole moment operator are weaker than those for dipole radiation transitions. Such transitions are named as forbidden radiative transitions. As the transition energy increases, the

2.2 Atomic Radiative Transitions

39

Table 2.15. Times of radiative transitions between lowest states for helium-like ions Ion

Z

τ (21 P → 11 S), s τ (23 P → 11 S), s τ (21 S → 11 S), s τ (23 S → 11 S), s

Li+

3 10 20 30 40 50 –

3.9×10−11 1.1×10−13 6.0×10−15 1.3×10−15 5 × 10−16 2 × 10−16 3.6

Ne+8

Ca+18 Zn+28 Zr+38 Sn+48 ln τ − dd ln Z

5.6 × 10−5 1.8 × 10−10 2.1 × 10−13 8.1 × 10−15 1.4 × 10−15 5 × 10−16 9.5

5.1 × 10−4 1.0 × 10−7 1.2 × 10−9 1.0 × 10−10 2 × 10−11 4 × 10−12 5.8

49 9.2 × 10−5 7.0 × 10−8 1.1 × 10−9 6 × 10−11 6 × 10−12 7.3

difference in radiative transitions for resonant and forbidden transitions decreases. This is demonstrated by the data of Table 2.15, where the times of identical radiative transitions are compared for helium-like ions. Spectra of atoms and ions involving lowest excited states can be represented in the form of the Grotrian diagrams that indicate the positions of excited states of atoms and ions, and parameters of radiative transitions with their participation. These diagrams are given in some books [11, 40–48]. These diagrams are most obvious for atoms and ions with simple electron shells. We give below the Grotrian diagrams for atoms with the electron shells s, s 2 , and with the electron shell p k for light atoms. These diagrams are taken from [11] and are given in Figs. A.1–A.28 in Appendix A. In addition, Figs. 2.1–2.14 contain lowest levels of some atoms and parameters of radiative transitions between them [12]. The most information for the rates of radiative transitions between atom states is collected by NIST information center [49–53]. In addition, along with notations for excited states of gas atoms with valence p-electrons given in Table 2.12 for LS- and jj -coupling schemes, the Pashen notations for excited inert gas atoms are represented in Fig. 2.10, as well as they are given in Figs. 2.11, 2.12, 2.13, and 2.14. 2.2.3 Absorption Parameters and Broadening of Spectral Lines The character of absorption and emission of photons in a gas or plasma depends on properties of this system. We consider these processes when they are determined by transition between atom discrete states. Then absorption and emission of resonant photons, the energy of which is close to the difference of atom state energies, is determined by broadening of spectral lines [35, 54], i.e. broadening of energies of atom states that partake in the radiative process. Let us introduce the distribution function aω over frequencies of emitting photons, so that aω dω is the probability that the photon frequency ranges from ω up to ω + dω. As the probability, the frequency distribution function of photons is normalized as (2.22) aω dω = 1.

40

2 Elements of Atomic and Molecular Physics

Because spectral lines are narrow, in scales of frequencies of emitting photons, the distribution function of photons is aω = δ(ω − ω0 ),

(2.23)

where ω0 is the central frequency of an emitting photon. Hence, the rate and times of spontaneous radiative emission are independent of the width and shape of a spectral line, i.e. of the photon distribution function aω . But it is not so when photons are generated or absorbed as a result of interaction between the radiation field and atoms. Such transitions are of two types, absorption of photons and induced emission of photons. The first process is described by the scheme h¯ ω + A → A∗ ,

(2.24)

and the process of induced emission is nh¯ ω + A∗ → (n + 1)h¯ ω + A,

(2.25)

where A, A∗ denote an atom in the lower and upper states of transition, h¯ ω is the photon, and n is a number of identical incident photons. The cross section of absorption σabs as a result of transition between two atom states is given by [35, 39] σabs =

π 2 c2 π 2 c 2 gf a ω . Aa = ω ω2 ω 2 g0 τ

(2.26)

In the same manner, we have the following formula for the cross section of stimulated photon emission σem [35, 39]: σem =

g0 π 2 c2 π 2 c 2 aω = Ba = σabs . ω 2 2 gf ω ω τ

(2.27)

Here A, B are the Einstein coefficients, indices o and f correspond to the lower and upper atom states, g0 , gf are the statistical weights of these atom states, and τ is the radiative lifetime of the upper atom state with respect to the radiative transition to the lower atom state. The absorption coefficient kω is   N f g0 (2.28) kω = N0 σabs − Nf σem = N0 σabs 1 − N 0 gf where N0 , Nf are the number density of atoms in the lower and upper states of transition. The absorption coefficient kω for a weak radiation intensity Iω is defined as dIω = −kω Iω , (2.29) dx where x is the direction of radiation propagation and, according to definition, the radiation flux does not perturb a gas where it propagates. In the case where population

2.2 Atomic Radiative Transitions

41

of atom states is determined by the Boltzmann formula, the absorption coefficient is equal to    hω ¯ . (2.30) kω = N0 σabs 1 − exp − T Thus, the character of photon absorption depends on broadening of spectral lines, and we give below three types of broadening that are of importance for excited gases. The basis of Doppler broadening is the Doppler effect, according to which an emitting frequency ω0 at a relative velocity vx of a radiating particle with respect to a receiver is conceived as a frequency   vx , (2.31) ω = ω0 1 + c where c is the light velocity. Hence, for radiating atomic particles with the Maxwell distribution function on velocities, the distribution function of photons aω has the form     1 mc2 1/2 mc2 (ω − ω0 )2 , (2.32) · exp − aω = ω0 2πT 2T ω02 where ω0 is the frequency for a motionless particle, m is the particle mass and T is the gas temperature expressed in energetic units. The Lorentz (or impact) mechanism of broadening of spectral lines results from single collisions of a radiating atom with surrounding particles. As a result of these collisions, the spectral line is shifted and broadens, and the distribution function aω of radiating photons has the Lorenz form [35] aω =

ν .  2π (ω − ω0 + ν)2 + ( ν2 )2

(2.33)

Here ν = Nvσt , where N is the number density of perturbed particles, and σt is the total cross section of collision of radiating and surrounding particles for an upper state of a radiating particle under assumption that collision in the lower particle state is not important. Next, ν = N vσ ∗ , and if the cross section is determined by a large number of collision momenta, we have σt σ ∗ , and one can ignore the spectral line shift. In the classical case where the main contribution to the total cross section results from many collision momenta, the total cross section σt is given by σt = πRt2 ,

Rt U (Rt ) ∼ 1, h¯ v

(2.34)

where v is the collision velocity, U (R) is the difference of interaction potentials for the upper and lower states of transition, and Rt is the Weiskopf radius. The criterion of validity of the Lorenz broadening (2.33) is based on the assumption that the probability to locate for two or more surrounding particles in a region of a strong interaction with a radiating atom is small, which gives N Rt3 1.

(2.35)

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2 Elements of Atomic and Molecular Physics

Table 2.16. Broadening parameters for spectral lines of alkali metal atoms. λ is the photon wavelength for resonant transition, τ is the radiative lifetime of the resonantly excited atoms, ωD , ωL are given by formula (2.36), NDL is expressed in 1016 cm−3 and Nt is given in 1018 cm−3 . The temperature of alkali metal atoms is 500 K [16] Element Li Na Na K K Rb Rb Cs Cs

Transition

λ, nm

τ , ns

22 S → 32 P 2 1/2 → 3 P1/2 2 3 S1/2 → 32 P3/2 42 S1/2 → 42 P1/2 42 S1/2 → 42 P3/2 52 S1/2 → 52 P1/2 52 S1/2 → 52 P3/2 62 S1/2 → 62 P1/2 62 S1/2 → 62 P3/2

670.8 589.59 589.0 769.0 766.49 794.76 780.03 894.35 852.11

27 16 16 27 27 28 26 31 27

32 S

ωD , 109 s−1 8.2 4.5 4.5 2.7 2.7 1.7 1.8 1.2 1.3

ωL /N , 10−7 cm3 /s 2.6 1.6 2.4 2.0 3.2 2.0 3.1 2.6 4.0

NDL

Nt

16 15 9.4 6.5 4.2 4.5 2.9 2.3 1.6

3.2 2.5 1.4 1.2 0.6 0.7 0.4 0.3 0.2

Table 2.16 compares the Doppler ωD and Lorenz ωL widths for transition between the ground and resonantly excited states of an alkali metal atom when these widths are given by the formulas  T 1 ωD = ω0 ; ωL = N vσt , (2.36) 2 2 mc and also the number density NDL of atoms when the line widths for these mechanisms are coincided. One more mechanism of broadening of spectral lines due to interaction with surrounding atoms takes place at large number densities of surrounding atoms. As a matter of fact, in this case, a system of interacting atoms emits radiation instead of individual atoms, though this interaction is small [39, 55]. One can assume surrounding atoms to be motionless in the course of radiation, and the shift of the spectral line of a radiating atom corresponds to a certain configuration of surrounding atoms 1 U (Rk ), (2.37) ω ≡ ω − ω0 = h¯ k where Rk is the coordinate of k-th atom in the frame of reference where the radiating atom is the origin. The photon distribution function is equal for the wing of a spectral line in the case of a uniform distribution of surrounding atoms aω dω = N · 4πR 2 dR;

aω =

4NπR 2 h¯ , dU/dR

(2.38)

where U (R) is the shift of a spectral line due to a surrounding atom located at a distance R from the radiating atom. Table 2.16 contains the number density of atoms Nt = Rt−3 for transition between the Lorenz and quasi-static mechanisms of broadening.

2.3 Interaction Potential of Atomic Particles

at Large Separations

43

Table 2.17. Radiative parameters and the absorption coefficient for the center line of resonant radiative transitions in atoms of the first and second groups of the periodical system of elements [14, 16] Element Transition H 12S → 22P He 11S → 21P Li 22S → 32P Be 21S → 21P 2 Na 3 S1/2 → 32P1/2 Na 32S1/2 → 32P3/2 Mg 31S → 31P 2 K 4 S1/2 → 42P1/2 K 42S1/2 → 42P3/2 Ca 41S → 41P 2 Cu 4 S1/2 → 42P1/2 Cu 42S1/2 → 42P3/2 Zn 41S → 41P Rb 52S1/2 → 52P1/2 Rb 52S1/2 → 52P3/2 Sr 51S → 51P 2 Ag 5 S1/2 → 52P1/2 Ag 52S1/2 → 52P3/2 Cd 51S → 51P 2 Cs 6 S1/2 → 62P1/2 Cs 62S1/2 → 62P3/2 Ba 61S → 61P 2 Au 6 S1/2 → 62P1/2 Au 62S1/2 → 62P3/2 Hg 61S → 61P

λ, nm 121.57 58.433 670.8 234.86 589.59 589.0 285.21 769.0 766.49 422.67 327.40 324.75 213.86 794.76 780.03 460.73 338.29 328.07 228.80 894.35 852.11 553.55 267.60 242.80 184.95

τ , ns 1.60 0.56 27 1.9 16 16 2.1 27 27 4.6 7.0 7.2 1.4 28 26 6.2 7.9 6.7 1.7 31 27 8.5 6.0 4.6 1.3

g∗ /g0 3 3 3 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

vσt , 10−7 cm3 /s k0 , 105 cm−1 0.516 8.6 0.164 18 5.1 1.6 9.6 1.4 3.1 1.1 4.8 1.4 5.5 3.4 4.1 0.85 6.3 1.1 7.3 2.5 1.1 2.1 1.7 2.8 3.3 4.8 3.9 0.91 6.2 1.2 9.4 1.7 1.1 2.0 1.7 2.9 3.3 4.5 5.3 0.77 8.1 1.0 9.0 1.9 0.49 1.9 0.75 4.2 2.2 5.7

In the case of radiative transition between the resonantly excited and ground atom state in a parent gas or vapor, the interaction potential depends as R −3 on the distance between atoms, and the total cross section σt  1/v, so that vσt is independent of the collision velocity. Hence, the absorption coefficient in the line center k0 is independent of the number density of atoms and their temperature. Table 2.17 contains their values for transitions between the ground and lowest resonantly excited states of alkali metal and alkali earth metal atoms.

2.3 Interaction Potential of Atomic Particles at Large Separations We give below a long-range interaction involving atoms or atoms and ions at large distances between them that will allow us to separate different interaction types

44

2 Elements of Atomic and Molecular Physics

Fig. 2.15. Geometry of interaction for bound electrons of different atoms

and find the condition of formation of chemical bonds. A long-range interaction of atomic particles is determined by distribution of valence electrons, and the operator of atom interaction V results from Coulomb interaction of valence electrons with electrons of another atom and its core =− V

 i

  Z1 Z2 e2 e2 e2 e2 − + + . |ri − R| |rk + R| |rk + R − ri | R k

(2.39)

i,k

Here R is the vector joined nuclei, and the coordinates of valence electrons ri , rk are counted off their nuclei (see Fig. 2.15). Expanding the operator (2.39) over a small parameter r/R, after an average over the electron distribution we obtain different types of a long-range interaction between atomic particles. The strongest term is proportional to R −2 and is the ion interaction with the atom dipole moment if it is not zero, as it takes place for ion interaction with an excited hydrogen atom, which state is characterized by parabolic quantum numbers n, n1 , n2 , m. Then the interaction potential has the form [23, 24] U (R) =

3Z h¯ 2 (n2 − n1 ) 2me R 2

(2.40)

where Z is the ion charge expressed in electron charges. The next expansion term is proportional to R −3 and is an interaction of ion charge with the atom quadrupole moment U (R) =

Ze2 Q , R3

(2.41)

where Q is the component of the quadrupole moment tensor on the axis that joins nuclei. In the one-electron approximation this quantity is [56, 57] Q=2

 i

ri2 P2 (cos θi ) = 2

 li (li + 1) − 3m2 i 2 r , (2li − 1)(2li + 3) i

(2.42)

i

where ri , θi are the spherical coordinates of the valence electron. The quadrupole moment is zero for a completed electron shell and for an average over the momentum projections. The values of the quadrupole moment for atoms with a filling p-shell are given in Table 2.18 [12, 58]. The next expansion term for the long-range interaction potential is interaction of a charge with an induced atom moment and, for a unit ion charge, is given by

2.3 Interaction Potential of Atomic Particles

at Large Separations

45

Table 2.18. The quadrupole moment Q of atoms with filling p-shell for the LS-coupling scheme; M is the atom momentum projection onto the molecular axis Atom states M=0 |M| = 1

(p)2 P 4/5 −2/5

(p 2 )2 P −4/5 2/5

(p 3 )4 S 0 –

(p 4 )3 P 4/5 −2/5

(p 5 )2 P −4/5 2/5

Table 2.19. Given in atomic units, the constant of van der Waals interaction C6 between two identical atoms of inert gas Interacting atoms C6

He–He 1.5

Ne–Ne 6.6

Ar–Ar 68

Kr–Kr 130

Xe–Xe 270

Fig. 2.16. Regions of coordinates of valence electrons that are responsible for atom interaction at large distances. 1—region of location of atomic electrons; 2—region that is responsible for long-range atom interaction at large separations; 3—region of electron coordinates that is responsible for exchange interaction of atoms.

U (R) = −

αe2 2R 4

(2.43)

where α is the atom polarizability, the values of which are given in diagram Pt9. The interaction potential of two atoms due to induced dipole moments (van der Waals interaction) when one of these takes a completed shell has the form U (R) = −

C6 , R6

(2.44)

and the values of C6 for two inert gas atoms are given in Table 2.19. Exchange interaction of atomic particles results from overlapping of wave functions of valence electrons and is determined by coordinate regions as shown in Fig. 2.16. Because long-range and exchange interactions between atomic particles are given by different coordinate regions at large distances between them, the total interaction potential is the sum of these interaction potentials. According to the nature of the exchange interaction potential (R), we have the following estimation in the case of exchange by one electron located in the field of identical atomic cores: (R) ∼ |ψ(R/2)|2 ∼ exp(−γ R), where ψ(r) is the wave function of the valence electron, and γ 2 = 2me J /h¯ 2 with the ionization potential J . This interaction potential with exchange by one electron

46

2 Elements of Atomic and Molecular Physics

corresponds to ion–atom interaction. Interaction of two atoms is accompanied by exchange by two valence electrons, and the exchange interaction potential of two atoms at large separations is (R) ∼ exp(−2γ R). In particular, the exchange interaction potential of two hydrogen atoms is (R) ∼ exp(−2R/a0 ) at large distances between them. The exchange interaction potential may have a different sign that determines the possibility of formation of a chemical bond for a given electron term at the approach of atomic particles. If this sign is negative, a covalence chemical bond is formed at moderate distances between atomic particles and the number of attractive and repulsed electron terms is approximately equal. In particular, we consider interaction of a structureless ion and a parent atom with a valence s-electron. At large distances between nuclei R, the eigen functions (ψg and ψu ) of a system of interacting ion and atom correspond to the even (gerade) and odd (ungerade) states, and are given by ψ(1) + ψ(2) ψ(1) − ψ(2) ; ψu = (2.45) ψg = √ √ 2 2 at large distances between nuclei. Here ψ(1), ψ(2) corresponds to electron location in the field of the first and second atomic cores. Correspondingly, if an electron is located in the field of one core, its state is a combination of the even and odd states. In particular, the wave function for electron location in the field of the first atomic core is ψg + ψ u . (2.46) ψ(1) = √ 2 As is seen, the atomic electron term is split in the even εg (R) and odd εu (R) molecular electron terms at finite distances R between nuclei, and the even term corresponds to attraction while the odd term corresponds to repulsion. The exchange interaction potential (R) is the difference of the energies of these states and, for the s-valence electron in the field of structureless cores, this quantity at large R is equal to [59] (R) ≡ |εg (R) − εu (R)| = A2 R 2/γ −1 exp(−Rγ − 1/γ ),

(2.47)

where A, γ are the parameters of the asymptotic expression (2.16) for the electron wave function. Another type of exchange interaction of two atoms A–B is realized if an electron term A–B correlates with an electron term A− –B + at large separations, where a valence electron of one atom transfers to another atom. The Coulomb interaction of positive and negative ions leads to attraction, and atoms form an ion–ion bond in this case. This bond is typical for atoms with a large electron affinity, in particular, for halogen atoms, and is realized into molecules consisting of halogen and alkali metal atoms. Excimer molecules have such a chemical bond. Another type of exchange interaction takes place when one atom is excited. The nature of this interaction consists in penetration of an excited electron to a nonexited atom when it is located in the field of the parent core. In the limiting case

2.4 Properties of Diatomic Molecules

47

of a highly excited atom this interaction potential is given by the Fermi formula [60–62] 2π h¯ 2 L |Ψ (R)|2 , (2.48) U (R) = me where R is the non-excited atom coordinate, Ψ (R) is the wave function of an excited electron, and L is the electron scattering length for a non-excited atom. The Fermi formula (2.48) is valid if the size of electron orbit as well as the distance R between atoms significantly exceeds the size of a non-excited atom. This interaction may be used as a model for systems and processes involving interaction of excited atoms [63]. As is seen, there are many forms of interaction between atomic particles. At large distances between particles, these interactions may be separated, which allows one to ascertain the role of certain interactions.

2.4 Properties of Diatomic Molecules 2.4.1 Bound States of Diatomic Molecule A diatomic molecule is a bound state of two atoms. The energy ε of a given electron state as a function of a distance R between motionless nuclei is the electron term. A small parameter me /μ, where me is the electron mass and μ is the reduced mass of nuclei, allows us to separate the electron, vibration and rotation degrees of freedom. Indeed, a typical difference of energies for neighboring electron terms is ε0 ∼ 1 eV, a typical energy difference between neighboring vibration states is ∼(me /μ)1/2 ε0 and a typical energy difference for neighboring rotation states is ∼(me /μ)ε0 . We now give standard notations for the molecule energy [64]. The total excitation energy of the molecule T , accounting for the separation of degrees of freedom, may be represented in the form [64] T = Te + G(v) + Fv (J ),

(2.49)

where Te is the excitation energy of an electron state that corresponds to excitation of this electron term from the ground state of the molecule, G(v) is the excitation energy of a vibrational state and Fv (J ) is the excitation energy of a rotational state. Formula (2.49) includes the main part of the vibrational and rotational energy for a weakly excited molecule. Near the minimum of the electron term that corresponds to the stable molecule state, the vibrational and rotational energy of a molecule, accounting for the first expansion terms, has the form 2 G(v) = hω ¯ e x(v + 1/2) , ¯ e (v + 1/2) − hω

Fv (J ) = Bv J (J + 1),

Bv = Be − αe (v + 1/2).

(2.50)

Here v and J are the vibration and rotation quantum numbers (J is the rotation momentum), which are whole numbers starting from zero. The spectroscopic parameters h¯ ωe , hω ¯ e x are the vibration energy and the anharmonic parameter, Be = h¯ 2 /(2I )

48

2 Elements of Atomic and Molecular Physics Table 2.20. The cases of Hund coupling [66, 67] Hund case a b c d e

Relation Ve Vm Vr Ve Vr Vm Vm Ve Vr Vr Ve Vm Vr Vm Ve

Quantum numbers Λ, S, Sn Λ, S, SN Ω L, S, LN , SN J, JN

is the rotation constant, where the inertia momentum I = μRe2 , Re is the distance between atoms for the electron term minimum, the equilibrium distance. Bv is the rotation constant for a vibrational excited state. These parameters for diatomic molecules and molecular ions with identical nuclei are contained in periodic tables Pt10, Pt11 and Pt12 [2, 65, 64]. We give below the classification of interactions inside the diatomic molecule, keeping the standard Hund scheme of momentum coupling [24, 66, 67]. Then we are based on three interaction types inside the molecule: the electrostatic interaction Ve (interaction between the orbital angular momentum of electrons and the molecular axis), spin–orbit interaction Vm and interaction between the orbital and spin electron momenta with rotation of the molecular axis Vr . A certain scheme of coupling is determined by the hierarchy of these interactions, and possible coupling schemes are given in Table 2.20. Each type of momentum coupling gives certain quantum numbers that describe the electron molecule state, correspond to a given hierarchy of interactions and are represented in Table 2.20. Indeed, denote by L the electron angular momentum; S the total electron spin; j the total electron momentum, that is, the sum of the angular and spin momenta (j = L + S); n the unit vector along the molecular axis; and K the rotation momentum of nuclei. Depending on the hierarchy of interactions, we can obtain the following quantum numbers: Λ, the projection of the angular momentum of electrons onto the molecular axis; Ω, the projection of the total electron momentum J onto the molecular axis; and Sn , the projection of the electron spin onto the molecular axis. LN , SN , JN are projections of these momenta onto the direction of nuclei rotation momentum K. Note that the operator of the total molecule momentum J = L + S + K = j + K commutes with the molecule Hamiltonian, and the eigenvalues of the operator J are the molecule quantum numbers for various cases of Hund coupling. The character of momentum coupling under consideration describes not only the molecule structure and quantum numbers of a diatomic molecule, but is of importance for dynamics of atomic collisions [68–71]. Then in the course of collision of two atomic particles, when a distance between two colliding atomic particles varies, the character of momentum coupling may also be changed. Transition from one coupling scheme to another leads simultaneously to a change in the quantum numbers of colliding particles. This may be responsible for some transitions in atomic collisions [12, 70, 71].

Pt10. Parameters of homonuclear molecules

2.4 Properties of Diatomic Molecules 49

Pt11. Parameters of homonuclear positive diatomic ions

50 2 Elements of Atomic and Molecular Physics

Pt12. Parameters of homonuclear negative diatomic ions

2.4 Properties of Diatomic Molecules 51

52

2 Elements of Atomic and Molecular Physics

The classical Hund scheme of momentum coupling inside a molecule gives a qualitative description of this problem. In reality, the number of interactions is greater, and the hierarchy of interactions is more complex than that in the Hund cases of momentum coupling inside a molecule. We demonstrate this on a simple example of interaction between a halogen ion and its atom at large distances between them that are responsible for the charge exchange process in the collision of these atomic particles. For definiteness, we take interaction between these particles at a distance R0 , so that the cross section of resonant charge exchange is πR02 /2 at the collision energy of 1 eV. Because this distance is large compared to an atom or ion size, this simplifies the problem and allows us separate various types of interaction. Let us enumerate interactions in the quasi-molecule X2+ (X is the halogen atom in the ground electron state), which have the form Vex ,

UM =

QMM , R3

Um =

QMM qmm , R5

(R),

δi ,

δa ,

Vrot .

(2.51)

Here we divide the electrostatic interaction Ve of Table 2.20 into four parts: the exchange interaction, Vex , inside the atom and ion that is responsible for electrostatic splitting of levels inside an isolated atom and ion; the long-range interaction, UM , of the ion charge with the quadrupole moment of the atom; the long-range interaction, Um , which is responsible for the splitting of ion levels; and the ion–atom exchange interaction potential, , which is determined by electron transition between atomic cores. Instead of the relativistic interaction Vm of Table 2.20, we introduce separately the fine splitting of levels δi for the ion (the splitting of 3 P0 and 3 P2 ion levels) and δa (the splitting of 3 P1/2 and 3 P3/2 atom levels) for the atom. Here M, m are the projections of the atom and ion angular momenta on the molecular axis, R is an ion–atom distance, Qik is the tensor of the atom quadrupole moment and qik is the quadrupole moment tensor for the ion. Taking for simplicity the impact parameter of ion–atom collision to be R0 , we have for the rotation energy at the closest approach of colliding particles h¯ v , Vrot = R0 where v is the relative ion–atom velocity, and we use the expression below for estimation of the Coriolis interaction. As is seen, the number of possible coupling cases is larger than that in the classical case. Of course, only a small fraction of these cases can be realized. Table 2.21 lists the values of the above interactions for the halogen molecular ion at a distance R0 between nuclei that is responsible for the cross section of the resonant charge exchange process. Comparing different types of interaction, we obtain the following hierarchy of interactions in this case Vex δi ,δa UM Um ,Vrot .

(2.52)

Comparing this with the data of Table 2.20, we obtain an intermediate case between cases “a” and “c” of the Hund coupling, but the hierarchy sequence (2.52) does not correspond exactly to any one of the Hund cases.

2.4 Properties of Diatomic Molecules

53

Table 2.21. Parameters of interaction of a positive halogen ion with a parent atom in the ground electron states at a distance R0 that is responsible for resonant charge exchange at the collision energy of 1 eV [72] R0 , a0 δa , cm−1 δi , cm−1 Vex , cm−1 δi /Vex UM , cm−1 UM /δa Vrot , cm−1 (R0 ), cm−1

F

Cl

Br

I

10.6 404 490 20873 0.023 341 0.84 30 23

13.8 882 996 11654 0.085 407 0.46 17 14

15.1 3685 3840 11410 0.34 448 0.12 10 8.4

17.2 7603 7087 13727 0.52 372 0.049 7.1 6.1

Thus, we conclude that the classical Hund cases of electron momentum coupling in diatomic molecules may be used for qualitative description of the molecule structure and its quantum numbers, but in reality the interaction inside molecules is more complex. 2.4.2 Correlation between Atomic and Molecular States When two atoms form a molecule, electron states of these atoms as well as the character of coupling of electron momenta in atoms and molecules determine possible electron states of the forming molecules. In other words, there is a correlation between electron states of atoms and molecules [73–76]. We consider this below for a molecule consisting of two identical atoms. If a molecule consists of identical atoms, a new symmetry occurs that corresponds to the electron reflection with respect to the plane that is perpendicular to the molecular axis and divides it into two equal parts. The electron wave function of an even (gerade) state conserves its sign as a result of this operation and the wave function of the odd (ungerade) state changes a sign when the electrons are reflected with respect to the symmetry plane. The corresponding states are denoted by g and u, respectively. The operator of electron reflection with respect to any plane passed through the molecular axis commutes with the electron Hamiltonian also, and the parity of a molecular state is denoted by + or − depending on conservation or change of the sign of the electron wave function in this operation. The parity of the molecular state is the sum of parities of atomic states when atoms are removed on infinite distance and this value does not change at the variation of a distance between nuclei. Thus, in the “a” case of the Hund coupling, molecular quantum numbers are the projection Λ of the angular momentum on the molecular axis, the total molecule spin S and its projection onto a given direction, the evenness and parity of this state which is degenerated with respect to the spin projection. Table 2.22 represents the correlation between states of two identical atoms and a forming molecule [77] in the case “a” of Hund coupling when electrostatic interaction dominates, which

54

2 Elements of Atomic and Molecular Physics

Table 2.22. Correlation between states of two identical atoms in the same electron states and the molecule consisting of these atoms in the case “a” of Hund coupling [77]. The number of molecule states of a given symmetry is indicated in parentheses if it is not 1 Atomic states

Dimer states

1S

1Σ + g 1Σ +, 3Σ + g u 1Σ +, 3Σ +, 5Σ + g u g 1Σ +, 3Σ +, 5Σ +, 7Σ + g u g u 1 Σ + (2), 1 Σ − , 1 Π , 1 Π , 1 Δ g u g g u 1 Σ + (2), 1 Σ − , 1 Π , 1 Π , 1 Δ , 3 Σ + (2), 3 Σ − , 3 Π , 3 Π , 3 Δ g u g g u u g g u g 1 Σ + (2), 1 Σ − , 1 Π , 1 Π , 1 Δ , 3 Σ + (2), 3 Σ − , 3 Π , 3 Π , 3 Δ , g u g g u u g u u g 5 Σ + (2), 5 Σ − , 5 Π , 5 Π , 5 Δ g u g g u 1 Σ + (2), 1 Σ − , 1 Π , 1 Π , 1 Δ , 3 Σ + (2), 3 Σ − , 3 Π , 3 Π , 3 Δ g u g g u u g g u g 5 Σ + (2), 5 Σ − , 5 Π , 5 Π , 5 Δ , 7 Σ + (2), 7 Σ − , 7 Π , 7 Π , 7 Δ g u g g u g g u u g 1 Σ + (3), 1 Σ − (2), 1 Π (2), 1 Π (2), 1 Δ (2), 1 Δ , 1 Φ , 1 Φ , 1 Γ g u g u g u g g u 1 Σ + (3), 1 Σ − (2), 1 Π (2), 1 Π (2), 1 Δ (2), 1 Δ , 1 Φ , 1 Φ , 1 Γ , g u g u g u g g u 3 Σ + (3), 3 Σ − (2), 3 Π (2), 3 Π (2), 3 Δ , 3 Δ (2), 3 Φ , 3 Φ , 3 Γ g u g u g u u u g 1 Σ + (3), 1 Σ − (2), 1 Π (2), 1 Π (2), 1 Δ (2), 1 Δ , 1 Φ , 1 Φ , 1 Γ , g u g u g u g g u 3 Σ + (3), 3 Σ − (2), 3 Π (2), 3 Π (2), 3 Δ , 3 Δ (2), 3 Φ , 3 Φ , 3 Γ , g u g u g u u u g 5 Σ + (3), 5 Σ − (2), 5 Π (2), 5 Π (2), 5 Δ (2), 5 Δ , 5 Φ , 5 Φ , 5 Γ g u g u g u g g u

2S 3S 4S 1P 2P 3P 4P 1D 2D 3D

includes the exchange interaction in atoms. This coupling character relates to molecules consisting of light atoms with the LS scheme of momentum coupling in atoms. The “a” Hund case corresponds to light atoms for which the LS scheme of momentum coupling holds true. Figures 2.17, 2.18, 2.19, 2.20 and 2.21 [11] give the potential curves or electron terms for lowest electron states of hydrogen, helium, carbon, nitrogen and oxygen diatomic molecules, where the “a” Hund case is realized, and the structure of valence electron shells is s, s 2 , 2p 2 , 2p 3 and 2p 4 , respectively. These figures also contain electron terms of the corresponding molecular ions. In these cases an electron state of a molecule is denoted by letters X, A, B, C, etc. for the states with zero total spin and by letters a, b, c, etc. for the states with one total spin. The projection of the molecule angular momentum onto the molecular axis as a quantum number is denoted by letters Σ, Π, Δ, etc. when the electron momentum projection is one, two, three, etc. The total molecule spin is the quantum number in the “a” Hund case and the multiplicity of the spin state 2S + 1 (S is the molecule spin) is given as a left superscript for the state notation. Next, the state parity (+ or −) is given as a right superscript, and the state evenness for a molecule consisting of identical atoms (g or u) is represented as a right subscript. These notations are used in Figs. 2.17, 2.18, 2.19, 2.20 and 2.21. The correlation between atomic and molecular states in the “c” case of Hund coupling, when the coupling in atoms corresponds to the jj -coupling scheme, may be fulfilled in the same manner. This correlation is given in Table 2.23, that is taken from [77].

2.4 Properties of Diatomic Molecules

55

Fig. 2.17. Potential curves of hydrogen molecule (H2 ) involving excited hydrogen atoms Table 2.23. Correlation between states of two identical atoms in the same electron states and the molecule consisting of these atoms in the case “c” of Hund coupling [77]. The number of molecule states of a given symmetry is indicated in parentheses if it is not 1 Atom states J J J J J

=0 = 1/2 =1 = 3/2 =2

Dimer states 0+ g − 1u , 0+ g , 0u − 2g , 1u , 1g , 0+ g (2), 0u − 3u , 2g , 2u , 1g , 1u (2), 0+ g (2), 0u (2) − 4g , 3g , 3u , 2g (2), 2u , 1g (2), 1u (2), 0+ g (3), 0u (2)

56

2 Elements of Atomic and Molecular Physics

Fig. 2.18. Potential curves of helium molecule (He2 ) involving excited helium atoms. The energy of electron terms expressed in cm−1 starts from the ground vibration and the lowest electron excited state. The energy of electron terms in eV begins from the ground vibration and electron states

In the Hund “c” case [66, 67], when the spin–orbit interaction potential exceeds the level splitting for a different projection of the angular electron momentum, the total electron momentum, that is, the molecule quantum number, is denoted by

2.4 Properties of Diatomic Molecules

57

Fig. 2.19. Potential curves of diatomic carbon molecule (C2 )

a large value. In addition, depending on the behavior of the electron wave function as a result of the reflection with respect to the plane passed through molecule axis, the electron state is denoted by superscripts + or − after the electron momentum. Transition from the Hund case “a” to “c” leads to the following change in term notations for a diatomic molecule: X 1 Σg+ → 0+ g;

a 3 Σu+ → 1u , 0− u;

A1 Σu+ → 0+ u;

b3 Σg+ → 1g , 0− g . (2.53)

As an example, Fig. 2.22 contains electron terms for the argon molecule that consists of atoms in the ground Ar(3p 6 ) and excited Ar(3p 5 4s) states. States of an excited atom at large separations are given for the LS scheme of momentum coupling. The same behavior of electron terms takes place for molecules of other inert gases. We note that the quantum numbers of the molecule electron state are defined accurately only to a restricted number of interactions as they take place in the Hund

58

2 Elements of Atomic and Molecular Physics

Fig. 2.20. Potential curves of diatomic nitrogen molecule (N2 )

cases of momentum coupling. In particular, a simple description relates to light atoms when non-relativistic interactions dominate. The description of molecules consisting of heavy atoms, when different types of interaction partake in molecule construction, is in reality outside of the Hund scheme. In particular, let us return to the case of Table 2.21 for the interaction of a halogen positive ion with a parent atom at large distances that are responsible for the resonant charge exchange process. Then, according to the hierarchy of interactions in the formula (2.52), the

2.4 Properties of Diatomic Molecules

59

Fig. 2.21. Potential curves of diatomic oxygen molecule (O2 )

molecule quantum numbers are J MJ j and the evenness (g or u), where J is the total atom momentum, MJ is its projection on the molecular axis and j is the total ion momentum. Fig. 2.23 contains electron terms of the molecular diatomic ion Cl+ 2 in a range of large distances between nuclei [72]. Table 2.24 contains parameters of the lowest excited states of inert gas diatomic molecules, so that Re is the equilibrium distance between nuclei, De is the minimum interaction potential that corresponds to this distance and τ is the molecule radiative lifetime. One more peculiarity of interaction between atoms follows from the character of exchange interaction. At large separation, the exchange interaction potential is given by the Fermi formula (2.48), and the electron scattering length L in this formula is positive for helium and neon atoms and is negative for argon, krypton and

60

2 Elements of Atomic and Molecular Physics

Fig. 2.22. Potential curves of diatomic argon molecule consisting of atoms in the ground Ar(2p 6 ) and lowest excited Ar(2p 5 3s) states. They start from the lowest excited states of the atom located at large distances from the argon atom in the ground state and correspond to only odd molecule states and correspond to only odd molecule states

Table 2.24. Parameters of lowest excited states of inert gas molecules Molecule, state He2 (a 3 Σu+ ) He2 (A1 Σu+ ) Ne2 (a 3 Σu+ ) Ar2 (1u , 0− u) Ar2 (0+ u) Kr2 (0+ u) Xe2 (1u , 0− u) Xe2 (0+ u)

Re , Å 1.05 1.06 1.79 2.4 2.4

De , eV 2.0 2.5 0.47 0.72 0.69

3.03 3.02

0.79 0.77

τ , 10−7 s 800 0.28 360 0.5 30 0.6 11 0.6

2.4 Properties of Diatomic Molecules

61

Fig. 2.23. Potential curves of the chlorine molecular ion Cl+ 2 resulting from interaction of the atom Cl(2 P ) and ion Cl+ (3 P ) in the lowest electron states at large separations, which are responsible for the resonant charge exchange process [72]. The excitation energies are counted off from the ground ion and atom states with infinite distance between them and are expressed in cm−1 . The indicated quantum numbers of electron terms are J MJ j and the evenness, where J , j are the total momenta of the atom and ion, and MJ is the projection of the total atom momentum onto the molecular axis

62

2 Elements of Atomic and Molecular Physics

Fig. 2.24. Potential curves of molecule CH Table 2.25. The distance Rh between nuclei and the hump height ε for the interaction potential of excited and non-excited atoms of helium and neon Molecule, state

Rh , Å

ε, eV

He2 (a 3 Σu+ ) He2 (A1 Σu+ ) Ne2 (a 3 Σu+ ) Ne2 (A1 Σu+ )

3.1 2.8 2.6 2.5

0.06 0.05 0.11 0.20

xenon atoms. This means that repulsion at large distances in helium and neon excited molecules is changed by attraction at moderate distances between interacting atoms. Therefore, the interaction potential contains a hump at large separations, and the parameters of this hump are given in Table 2.25.

2.4 Properties of Diatomic Molecules

63

Fig. 2.25. Potential curves of molecule NH Table 2.26. The polarizabilities α of diatomic homonuclear molecules expressed in a03 Molecule H2 Li2 N2 O2 F2 Na2

α 5.42 230 11.8 10.8 93 260

Molecule Al2 Cl2 K2 Br2 Rb2 Cs2

α 130 31 500 474 530 700

In addition, Table 2.26 contains the polarizabilities of homonuclear diatomic molecules, and the diagram Pt13 gives the electron affinities of homonuclear diatomic molecules. The properties of diatomic molecules including different atoms with nearby ionization potentials are similar to those of homonuclear diatomic molecules. This is confirmed by Figs. 2.24, 2.25, and 2.26, which contain potential curves for the lowest states of molecules CH, NH, OH, and diagram Pt14, which gives the affinities

2 Elements of Atomic and Molecular Physics

Pt13. Electron affinity of molecules

64

Pt14. Affinity of atoms to hydrogen and oxygen atoms

2.4 Properties of Diatomic Molecules 65

66

2 Elements of Atomic and Molecular Physics

Fig. 2.26. Potential curves of molecule OH

of various atoms to the hydrogen and oxygen atom, i.e. the dissociation energies of corresponding diatomic molecules. Note that alongside with quantum numbers of electron states, the letters X, A, B, C etc. are used for notations of excited states in the course of an excitation increase. For light atoms when electron terms with different spins are independent the notations a, b, c etc. are used if the total molecule spin differs from that of the ground molecule state. In addition, a prime may be used for the states which became known when a certain sequence of states was created. 2.4.3 Excimer Molecules Excimer molecules consist of an atom in the ground state and an atom in excited state, and these atoms form a strong chemical bond. When these atoms are in the ground state, the bond is weak. Spread excimer molecules include a halogen atom in the ground state and an inert gas atom in the lowest excited states with an ex-

2.4 Properties of Diatomic Molecules

67

Fig. 2.27. Potential curves of excimer molecule XeF

cited s-electron. These excimer molecules are analogous to molecules consisting of halogen and alkali metal atoms in the ground state, and the chemical bond in these excimer molecules results from a partial transition of an excited s-electron to the halogen atom. Interaction of positive and negative ions in these molecules determines a strong chemical bond in them. Electron terms of the excimer molecule XeF are given in Fig. 2.27 as an example of an excimer molecule. If atoms are found in the ground states, the chemical bond between them is not realized since an exchange interaction involving an atom with the completed electron shell corresponds to repulsion. Then valence electrons remain in the field of parent atoms, so that the quantum molecular numbers are the total molecule spin, the projection of the angular electron momentum onto the molecular axis and also the molecule symmetry for its reflection with respect to the plane passed through the molecular axis. Excitation of the molecule leads to the transition of an excited electron to the halogen atom, and the quantum number of a formed molecule is the total momentum of an inert gas atomic core. Figure 2.28 contains parameters of some states of excimer molecules under consideration [78], so that Re is the equilibrium distance between nuclei that corresponds to the interaction potential minimum, and De is the potential well depth, i.e. the difference of the interaction potential at infinite distance between nuclei and the equilibrium one Re . Radiative transitions in excimer molecules are the basis of excimer lasers [79–81], and Fig. 2.29 shows radiative transitions between electron states of excimer molecules, whereas the radiative lifetime for some states of excimer molecules and the wavelengths of the band middle for corresponding radiative transitions are represented in Fig. 2.28.

2 Elements of Atomic and Molecular Physics

Fig. 2.28. Parameters of excimer molecules

68

2.4 Properties of Diatomic Molecules

Fig. 2.29. Character of radiative transitions in excimer molecule

69

3 Elementary Processes Involving Atomic Particles

3.1 Parameters of Elementary Processes in Gases and Plasmas Properties of rare atomic systems, gases and plasmas, are determined by various processes of collision of atomic particles [82]. In a general consideration of atomic collisions [83, 84], we use as a characteristic of collision of atomic particles the collision cross section. By definition, the differential cross section of collision of two atomic particles dσ that is introduced in the center-of-mass frame of reference, is the ratio of a number of scattered particles per unit time per unit solid angle dΩ to the flux of incident particles. The parameters of elastic scattering of particles when the particle internal state is not changed are given in Fig. 3.1 in the classical limit. The classical parameters of particle collision are the impact collision parameter ρ, a distance of closest approach r0 , and the scattering angle ϑ. If the interaction potential U (R) of colliding particles is spherically symmetric (R is the distance between particles), the conservation of the angular momentum of particles in the course of collision leads to the following relation between the impact collision parameter and the distance of closest approach [85]:   ρ2 U (r0 ) , (3.1) = 1 − ε r02 where ε = μv 2 /2 is the kinetic energy of colliding particles in the center-of-mass frame of reference, μ is the reduced mass of colliding particles, and v is their relative

Fig. 3.1. Parameters of classical scattering in the center-of-mass frame of reference for colliding particles. The arrow indicates the direction of the particle trajectory in the center-of-mass frame of reference. ρ is the impact parameter of collision, r0 is the distance of closest approach, and ϑ is the scattering angle. 1 is the particle trajectory, 2 are its asymptotic lines

72

3 Elementary Processes Involving Atomic Particles

Fig. 3.2. 1—the interaction potential of atomic particles for the hard sphere model; 2—real interaction potential of atomic particles

velocity. In the case of a monotonic dependence ϑ(ρ) of the scattering angle on the impact collision parameter, the differential cross section of scattering in the classical limit is dσ = 2πρ dρ. (3.2) The model of hard spheres is a convenient model for the scattering of classical particles in a strongly varied potential, and the interaction potential for this model corresponds to a solid wall of a radius R0 , as it is shown in Fig. 3.2. Within the framework of this model, when the cross section is independent of the collision  velocity, the differential cross section dσ and the transport cross section σ ∗ = (1− cos ϑ) dσ are correspondingly equal dσ = πR02 d cos ϑ,

σ ∗ = πR02

(3.3)

since according to Fig. 3.3 ρ/R0 = sin α. The transport cross section of scattering σ ∗ relates to the transport of particles in a gas, whereas the total cross section σt = dσ characterizes a shift of the phase of an electromagnetic wave that interacts with an atomic particle when this phase shift follows from collisions of atomic particles. Therefore, the total cross section of particle scattering may be responsible for the broadening of spectral lines resulting from the transition between discrete states of atomic particles. This cross section is infinite (h¯ → 0) in the classical limit because weak scattering takes place at any large impact parameter of particle collisions. The classical description holds true in the case when the main contribution to the scattering cross section is given by large collision momenta l = μρv/h¯  1. In the case of a spherically symmetric interaction potential between colliding particles, scattering for different collision momenta l proceeds independently, and any total parameter of scattering is the sum of these parameters for each collision momentum l. The characteristic of particle scattering is the scattering phase δl , that is,

3.1 Parameters of Elementary Processes in Gases and Plasmas

73

Fig. 3.3. The character of scattering within the framework of the hard sphere model

the parameter of the wave function of colliding atomic particles at large distances between them. The connection between the differential cross section of scattering dσ = 2π|f (ϑ)|2 d cos ϑ, the scattering amplitude f (ϑ), the diffusion or transport cross section σ ∗ and the total scattering cross section σt are [24, 86, 87] f (ϑ) =

∞   1  (2l + 1) e2iδl − 1 Pl (cos ϑ), 2iq l=0

∞ 4π  σ∗ = 2 (l + 1) sin2 (δl − δl+1 ), q

(3.4)

l=0

∞ 4π  σt = 2 (2l + 1) sin2 δl . q l=0

Here q is the phase vector of colliding particles in the center-of-mass frame of reference, so that the energy of particles in a center-of-mass system is ε = h¯ 2 q 2 /2μ, where μ is the reduced mass of colliding particles. Electron–atom collisions in gases and plasmas are of importance both for plasma processes involving excited atom states and for transport processes [84, 88, 89]. Because of a finite number of scattering momenta in electron–atom collisions at not-high energies, these collisions correspond to the quantum case. In the limit of small electron wave numbers q, the electron scattering phase is δ0 = −Lq, and this is the definition of the scattering length L. Correspondingly, the cross sections of electron–atom scattering at small electron energies are equal to σ ∗ (0) = σt (0) = 4πL2 .

(3.5)

Other scattering phases have a stronger dependence on the wave vector in the limit of its low value; in particular, for short-range electron–atom interaction, this de-

74

3 Elementary Processes Involving Atomic Particles

Fig. 3.4. The diffusion cross section of electron scattering on the xenon atom [94–98] Table 3.1. Parameters of the total cross section σt for electron scattering on inert gas atoms [26, 99] Atom

He

Ne

Ar

Kr

Xe

L/a0 σt (ε = 0), Å2 εmin , eV σt (εmin ), Å2

1.2 5.1 – –

0.2 0.14 – –

−1.6 9.0 0.18 0.88

−3.5 43 0.32 3.8

−6.5 150 0.44 13

pendence has the form δl ∼ q 2l+1 . Expansion of the scattering phases at small electron energies [90] with accounting for the polarization electron–atom interaction U (r) = −αe2 /2r 4 at large distances r between them along with a short-range interaction (2.48) gives for the scattering amplitude at low electron energies [91] f (ϑ) = −L −

ϑ παq sin . 2a0 2

(3.6)

If an electron is scattered on a non-structured atom and the scattering electron–atom length is negative, the zeroth phase becomes equal to zero at low electron energies where other scattering phases are small. This leads to a sharp minimum in the cross section of electron–atom scattering that takes in the case of electron scattering on argon, krypton and xenon atoms and is called the Ramsauer effect [92, 93]. Table 3.1 gives parameters of the total cross section σt = dσ of electron scattering on atoms of inert gases at small electron energies and in a range of the Ramsauer minimum. In addition, Fig. 3.4  gives the dependence on the electron energy for the diffusion cross section σ ∗ = (1 − cos θ )dσ on the xenon atom [94–98], where θ is the scattering angle. In addition to Fig. 3.4, Table 3.2 contains the diffusion cross section σ ∗ depending on the electron energy according to evaluations [99] that are made on the basis of experimental data for kinetic coefficients. Note that in the case of argon, krypton

3.1 Parameters of Elementary Processes in Gases and Plasmas

75

Table 3.2. The diffusion cross section σ ∗ of electron scattering on atoms of inert gases [99] expressed in Å2 as a function of the electron energy ε ε, eV 0 0.001 0.002 0.005 0.01 0.02 0.04 0.06 0.08 0.10 0.15 0.20 0.25 0.30 0.35 0.4 0.5 0.6 0.7 0.8 1.0 1.5 2.0 2.5 3 4 5 6 7 8 10

He

Ar

Kr

Xe

4.95 4.96 4.99 5.14 5.27 5.38 5.58 5.63 5.71 5.80 6.00 6.20 6.26 6.32 6.38 6.44 6.55 6.66 6.74 6.79 6.87 6.98 6.97 6.95 6.93 6.8 6.6 6.3 5.9 5.5 5.0

10.0 8.35 7.80 6.61 5.60 4.15 2.50 1.60 1.00 0.59 0.23 0.10 0.091 0.15 0.24 0.33 0.51 0.68 0.86 1.05 1.38 2.07 2.70 3.37 4.10 6.00 7.60 9.3 11.0 14.0 14.6

39.7 39.7 37.7 30.0 26.2 21.4 15.3 11.7 9.13 7.23 4.56 2.37 1.30 0.86 0.55 0.26 0.10 0.15 0.27 0.42 0.80 1.84 3.00 4.40 6.00 10.0 14.0 16.0 17.0 16.5 15.5

176 170 160 139 116 89.0 50.2 34.8 25.6 20.4 13.6 8.4 5.9 3.3 2.5 1.6 0.53 0.38 0.38 0.55 1.15 3.50 7.50 11.5 16.5 24.5 28 26 27 26 20

and xenon, the Ramsauer minimum in electron–atom scattering is sharper for the diffusion cross section σ ∗ than that for the total cross section σt . In contrast to the above case of electron scattering on atoms with completed electron shells, there are different channels of electron scattering on atoms with noncompleted electron shells. In particular, for electron scattering on atoms of alkali metals, these channels are characterized by different total spins of an incident electron and atom that corresponds to ignoring relativistic effects. In particular, the cross section of scattering for electrons of zero energy is given by the following formula instead of (3.7):   (3.7) σ ∗ (0) = σt (0) = π L20 + 3L21 ,

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3 Elementary Processes Involving Atomic Particles

Table 3.3. The parameters of scattering of a slow electron on atoms of alkali metals and parameters of the autodetaching state 3 P of the autodetaching state of the alkali metal negative ion [26, 27] L0 /a0 L1 /a0 Er , eV Γr , eV

Li 3.6 −5.7 0.06 0.07

Na 4.2 −5.9 0.08 0.08

K

Rb

0.56 −15 0.02 0.02

2.0 −17 0.03 0.03

Cs −2.2 −24 0.011 0.008

where L0 and L1 are the electron scattering lengths on atoms for zero and one spin, respectively. The parameters of this formula are given in Table 3.1. Note that the autodetachment resonance 3 P that corresponds to an autodetaching state of the alkali metal negative ion gives a remarkable contribution to the cross section of electron–atom scattering. Rough values of the parameters of this state, the excitation energy Er and the width Γr of the autodetaching state level are also represented in Table 3.3. Along with the cross section σ , the rate constant vσ is the characteristic of particle collision, where v is the relative velocity of colliding particles. In particular, the rate constant of inelastic scattering of particles is given by kij (v) = vσij (v),

(3.8)

where σij is the cross section of inelastic collision with transition from a state i to a state j . When colliding particles are distributed over velocities, the mean rate constant is averaged over velocities of collisions according to the formula  kij (v) = f (v)kij (v) dv, (3.9) where the normalization of the velocity distribution function f (v) is given by f (v) dv = 1. The rate constants of particle collisions determine evolution of excited states in gases and plasmas. If excited states relate to atoms the states of which are denoted by indices i and j and transition between these states result from collisions with particles of other types (for example, electrons) the number density of which is NB , the balance equation for the number density of particles Ni in a state i has the form   dNi = −Ni NB kij + NB Nj kij . dt j

(3.10)

j

3.2 Inelastic Collisions of Electrons with Atoms Processes of excitation and quenching of excited atoms by electron impact proceed according to the scheme

3.2 Inelastic Collisions of Electrons with Atoms

e + A ↔ e + A∗ .

77

(3.11)

The excitation cross section by electron impact near the threshold has the form [24, 54] √ (3.12) σex  ε − ε, ε − ε ε, where ε is the energy of an incident electron and ε is the excitation energy. This gives for the quenching cross section according to the principle of detailed balance 1 , (3.13) v where v is the velocity of an incident electron. Correspondingly, the rate constant of atom quenching in collisions with a slow electron kq is independent of the electron energy. Table 3.4 gives parameters of the lowest resonantly excited states of some atoms with valence s-electrons. These parameters include the excitation energy ε, the wavelength λ for the radiative transition into the ground state, and the radiative lifetime τ∗0 for this transition; f is the oscillator strength for this transition, and the quenching rate constant is kq for a slow electron. In addition, Table 3.5 contains the rate constants of quenching by a slow electron impact for some metastable states. The problem of inelastic electron–atom collisions may be solved under special conditions [24, 54, 86]. One can suggest the following formula [12] for the rate constant of the quenching of the resonantly excited state by a slow electron when dipole interaction of atoms dominates σq 

kq =

k0 7/2 ε

. (3.14) ·τ If the excitation energy ε is expressed in eV, and the radiative lifetime of the excitation state τ is given in ns, the rate constant k0 in formula (3.14), as it follows from experimental data, equals [100] k0 = (4.4 ± 0.7) × 10−5 cm3 /s. Table 3.4. Parameters of resonantly excited states of atoms with s-valence electrons and the quenching rate in collisions of such atoms with a slow electron [100] Atom, transition

ε, eV

λ, nm

f

H(21 P → 11 S) He(21 P → 11 S) He(21 P → 21 S) He(23 P → 23 S) Li(22 P → 22 S) Na(32 P → 32 S) K(42 P1/2 → 42 S1/2 ) K(42 P3/2 → 42 S1/2 ) Rb(52 P1/2 → 52 S1/2 ) Rb(52 P3/2 → 52 S1/2 ) Cs(62 P1/2 → 62 S1/2 ) Cs(62 P3/2 → 62 S1/2 )

10.20 21.22 0.602 1.144 1.848 2.104 1.610 1.616 1.560 1.589 1.386 1.455

121.6 58.43 2058 1083 670.8 589 766.9 766.5 794.8 780.0 894.4 852.1

0.416 0.276 0.376 0.539 0.74 0.955 0.35 0.70 0.32 0.67 0.39 0.81

τ∗0 , ns 1.60 0.555 500 98 27 16.3 26 25 28 26 30 27

kq , 10−8 cm3 /s 0.79 0.18 51 27 19 20 31 32 32 33 46 43

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3 Elementary Processes Involving Atomic Particles

Table 3.5. The rate constants for quenching of metastable atom states in collisions with a slow electron [100] Atom, transition

ε, eV

kq , 10−10 cm3 /s

He(23 S → 11 S) Ne(23 P2 → 21 S) Ar(33 P2 → 32 S) Kr(43 P2 → 42 S0 ) Xe(52 P2 → 52 S0 )

19.82 16.62 11.55 9.915 8.315

31 2.0 4.0 3.4 19

The principle of detailed balance establishes the relation between the rates of a direct and reverse process that correspond to time reversal [101]. The principle of detailed balance gives the following connection between the cross section of atom excitation by electron impact σex (ε) and the cross section of atom quenching by electron impact σq (ε − ε) [102] σex (ε) =

g∗ (ε − ε) σq (ε − ε), g0 ε

(3.15)

where ε is the energy of a fast electron and g0 , g∗ are the statistical weights of atoms in the ground and excited states. From this it follows the connection between the excitation rate constant kex of an atom by electron impact, and the rate constant kq of atom quenching by electron impact kex = kq

g∗ g0



ε − ε . ε

(3.16)

Using formula (3.14) for the quenching rate constant, one can obtain the following expression for the excitation rate constant near the threshold: √ k0 g∗ ε − ε . (3.17) kex = g0 ε 4 · τ

3.3 Collision Processes Involving Ions Ions as a charged component of a plasma are of importance for plasma properties that result from collisions in a plasma involving ions [103–106]. Ion–atom collisions with variations in the ion momentum have a classical character, i.e. nuclei are moving in these collisions according to the classical law. These processes include elastic ion–atom collisions and the charge exchange process. Elastic scattering of ions on atoms at low energies is determined by their interaction at large distances with the polarization interaction potential (2.43) between them. The peculiarity of the interaction potential (2.43) is the possibility of particle capture that means the

3.3 Collision Processes Involving Ions

79

approach of colliding particles up to R = 0. The cross section of capture σcap for the polarization interaction potential is [85]

αe2 , (3.18) σcap = 2π μg 2 where μ is the reduced mass of colliding ion and atom, and g is their relative velocity. In reality, the interaction potential differs from the polarization potential (2.43) at distances compared with atomic size and includes ion–atom repulsion at small separations. Therefore, the capture cross section (3.18) means only a strong approach of colliding ion and atom. Nevertheless the diffusion cross section of ion– atom collision for the polarization interaction potential (2.43) between them, σia∗ is close to the capture cross section (3.18) and is equal to [107] σia∗ = 1.10σcap . Resonant charge exchange in ion collision with a parent atom proceeds according to the scheme + + A, → A (3.19) A+ + A where the tilde marks one of the colliding particles. Usually, at room temperature and higher collision energies, the cross section of resonant charge exchange remarkably exceeds the cross section of ion–atom elastic collision, and therefore transport of ions in the parent gas is determined by resonant charge exchange. Considering the resonant charge exchange process as a result of state interference, we have for the electron wave function Ψ (t) instead of formula (2.46)  

 t  t ψg ψu i i (3.20) Ψ (t) = √ exp − εg dt + √ exp − εu dt , 2h¯ −∞ 2h¯ −∞ 2 2 where εg , εu are the energies of the even and odd states at a given distance between nuclei. From this one can find the probability of resonant charge exchange as a result of ion–atom collision [108]  ∞ 2  i Δ dt, Δ = |εg − εu |, (3.21) P = Ψ (∞)|ψ(1)  = sin2 ζ, ζ = 2h¯ −∞ and ζ is the phase shift between even and odd states as a result of collision. This exhibits the interference nature of charge exchange, so that the electron state is the mixture of the even and odd states, and a phase shift between these states leads to the electron transition from the field of one core to another one. In turn, the phase shift is determined by the exchange interaction potential (2.46) in the course of collision. The expression (3.21) for the probability of the resonant charge process corresponds to a two-state approximation when the atom and ion electron states are not degenerated, which takes place if an atom A of the process (3.19) relates to atoms of the first and second groups of the periodic tables, i.e. for atoms with valent selectrons. Nevertheless, this expression may be used within the framework of some

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3 Elementary Processes Involving Atomic Particles

models [109–111] in other cases. In particular, the hydrogen-like model by Rapp and Francis [109] using the hydrogen wave functions in determination of the ion– atom exchange interaction potential is popular. But one can construct in this case a strict asymptotic theory by expansion of the cross section over a small parameter that exists in reality. In contrast to model evaluations, the asymptotic theory allows us to estimate the accuracy of the results within the theory because this accuracy is expressed through a small parameter of the theory. We show this below. Note the classical character of nuclear motion at not-small collision energies [112] (say, above those at room temperature). If an atom is found in a highly excited state, the electron transition in the resonant charge exchange process has a classical character also [112] and proceeds when a barrier between fields of two cores disappears. We consider another case related to atoms in the ground and lowest excited states, when the transition has a tunnel character. Then we have a weak logarithm dependence of the cross section of resonant charge exchange σres on the collision velocity v [113, 114]

 v∗ , (3.22) σres (v) = C ln2 v where C and v∗ are constants. Indeed, taking the basic dependence of the exchange interaction potential (R) on an ion–atom distance R as (R)  exp(−γ R) according to formula (2.47), one can represent the cross section of resonant charge exchange in the form σres (v) =



2 v0 π 1 R0 + ln , 2 γ v

(3.23)

where πR02 /2 is the cross section at the collision velocity v0 . Diagram Pt15 gives the cross sections of resonant charge exchange involving ions and atoms of various elements in the ground states. The collision energy at which the cross section is given, corresponds to the laboratory frame of reference, where an atom is motionless. We will be guided by collision energies of eV, at which the parameter γ R0 in formula (3.23) is large. Diagram Pt16 gives values of the parameter γ R0 for ions and atoms of various elements in the ground states. From this it follows that this parameter for various elements for the collision energy of 1 eV ranges approximately from 10 to 15. On the basis of this, one can construct a strict asymptotic theory of the resonant charge exchange by expansion of the cross section of this process over a small parameter 1/γ R0 . The first term of this expansion leads to the cross section of resonant charge exchange in the form [115] σres (v) =

π 2 R , 2 0

ζ (R0 ) =

e−C = 0.28, 2

(3.24)

where C = 0.577 is the Euler constant. Let us ignore elastic ion–atom scattering in the charge exchange process; i.e. particles move along straightforward trajectories and the ion–atom distance R varies in time as R 2 = ρ 2 = v 2 t 2 , where ρ is the

Pt15. Cross sections of resonant charge exchange

3.3 Collision Processes Involving Ions 81

Pt16. Parameters of cross section of resonant charge exchange

82 3 Elementary Processes Involving Atomic Particles

3.3 Collision Processes Involving Ions

83

impact parameter of collision, and v is the relative velocity of colliding particles. Then the cross section of resonant charge exchange is given by [59] σres (v) =

π 2 R , 2 0

1 v

πR0 (R0 ) = 0.28, 2γ

(3.25)

where the exchange interaction potential (R) in this formula is determined by formula (2.46). Then the cross section (3.25) of resonant charge exchange is expressed through the parameters of the asymptotic electron wave function, when a valence electron is located in the atom far from the core. Because this theory represents the cross section as an expansion over a small parameter 1/γ R0 , one can estimate the accuracy of the expression (3.25) within the framework of the theory by comparison with the results where the next terms of expansion over a small parameter are taken into account. In the hydrogen case (that is, for the process H+ + H), the cross section of resonant charge exchange at the ion energy of 1 eV is [116] (173 ± 2)a02 as an averaging of the results and accounting for the first and the second terms of expansion in different versions of this expansion. Evidently, the accuracy of this ∼1% is the best that one can expect from formula (3.25). We estimate the data of diagram Pt15 for the cross section of the resonant charge exchange of elements of the first and second groups of the periodic table to be better than 2% when ions and their atoms are found in the ground states. Resonant charge exchange for ions and atoms of other element groups of the periodic table becomes more complicated because the process of resonant charge exchange is entangled with the processes of turning of atom angular momenta and transition between states of fine structure. In particular, Fig. 2.23 gives the lowest electron terms in the case of collision Cl+ (3 P )+Cl(2 P ) [72], which may be responsible for resonant charge exchange. We will be guided by the average cross section of resonant charge exchange that is averaged over initial values of atom and ion momenta and their projections. In particular, there are 18 electron even and odd terms in the case of interaction Cl+ (3 P ) + Cl(2 P ) (Fig. 2.18) where the quantum numbers are the total momenta J and j of an atom and ion and MJ is the atom momentum projection onto the impact parameter of collision. Let us consider the case when a valence p-electron is located in the fields of structureless cores. It is convenient to compare the parameters in this case with those for an s-electron if the asymptotic radial wave functions (2.16) of a transferred electron are identical in both cases. Then the exchange interaction potential of ion and atom with a valence p-electron is expressed through that Δ0 for an s-electron given by formula (2.47) as [69] (2l + 1)(l + |m|)! , (3.26) Δlm (R) = Δ0 (R) · (l − |m|!)|m|! where l, m are the electron momentum and its projection onto the molecular axis. Table 3.6 gives the partial cross sections of resonant charge exchange with transition p-electron between two structureless cores, so that σ0 is the cross section with participation of an s-electron, σ10 is the cross section involving a p-electron if the momentum projection on the impact parameter of collision is zero, σ11 is this cross section when the momentum projection equals to ±1, and

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3 Elementary Processes Involving Atomic Particles

Table 3.6. The partial cross sections of resonant charge exchange with transition p-electron between two structureless cores [117] R0 γ

6

8

10

12

14

16

σ10 /σ0 σ11 /σ0 σ /σ0 σ1/2 /σ0 σ3/2 /σ0

1.40 1.08 1.19 1.18 1.18

1.29 0.98 1.08 1.10 1.10

1.23 0.94 1.04 1.07 1.06

1.19 0.95 1.01 1.05 1.04

1.16 0.912 0.99 1.04 1.03

1.14 0.91 0.99 1.03 1.02

Table 3.7. The reduced average cross sections of resonant charge exchange for atoms of groups 3 (σ3 ), 4 (σ4 ) and 5 (σ5 ) of the periodical system of elements in the case “a” of Hund coupling [58, 117] R0 γ σ3 /σ0 σ4 /σ0 σ5 /σ0

6 1.17 1.50 1.44

8 1.09 1.32 1.29

10 1.05 1.23 1.22

12 1.03 1.18 1.17

14 1.02 1.14 1.14

16 1.03 1.12 1.11

the average cross section is σ = σ10 /3 + 2σ11 /3. These cross sections relate to the “a” case of Hund coupling when spin–orbit interaction is ignored. The cross sections σ1/2 , σ3/2 correspond to case “c” of Hund coupling when the spin–orbit splitting of levels is relatively large. As is seen, in the case “c” of Hund coupling, the cross section is independent practically on the total electron momentum. In addition, the average cross sections are close for both cases of Hund coupling. Note that the resonant charge exchange process for elements of 3 and 8 groups of the periodical system of elements [58] relates just to this one-electron case. The accuracy of the cross sections of diagram Pt15 is determined by the accuracy of asymptotic coefficients A mostly and is estimated for these cases to be better than 5%. The average cross section of resonant charge exchange for ions and atoms with noncompleted shells depends on the initial distributions over electron states of ions and atoms as well as on the character of momentum coupling in them, as follows from the analysis for halogens, oxygen and nitrogen [72, 118, 119]. Table 3.7 gives the average cross sections for resonant charge exchange with respect to the cross sections for s-electrons in the case “a” of Hund coupling when spin–orbit interaction may be ignored. Note that transition to elements of groups 6, 7 and 8 from elements of groups 3, 4 and 5 results from change of electrons by holes. The accuracy of the cross sections of diagram Pt15 in these cases is estimated to be better than 10%. As is seen, though the partial cross sections differ remarkably, the average cross sections are close for different types of momentum coupling. Representing the velocity dependence for the cross section of resonant charge exchange in the form

α v0 , (3.27) σres (v) = σres (v0 ) v

3.4 Atom Ionization by Electron Impact

85

we obtain from formula (3.23) α=

2 1. γ R0

In reality, the exponent α in formula (3.27) differs from this value because of the approximated character of formula (3.23). The values of this exponent for the resonant charge exchange process involving various elements are given in diagram Pt16.

3.4 Atom Ionization by Electron Impact Among various processes of electron collisions with atomic particles, ionization processes are of importance for plasma properties and evolution. Atom ionization by electron impact proceeds according to the scheme e + A → 2e + A+ .

(3.28)

The simplest model of atom ionization is the Thomson model [120], which ignores the interaction between impact and bound electrons during the collision act. If the energy exchange in collisions of these electrons owing to the Coulomb interaction between them ε exceeds the ionization potential, a release of the bound electron proceeds. The cross section of energy exchange between an incident and bound electron is given by the Rutherford formula [85] dσ = 2πρ dρ =

πe4 d ε , ε( ε)2

if the energy exchange ranges from ε to ε + d ε. Here electrons are assumed to be classical and the energy exchange is assumed to be small compared to the collision energy ε, i.e. ε  ε. Because atom ionization corresponds to bond breaking for a bound electron, the ionization process proceeds when the exchange energy exceeds the atom ionization potential J , i.e. ε > J . This gives the ionization cross section σion within the framework of the Thomson model [120]

  ε 1 πe4 1 − . (3.29) dσ = σion = ε J ε J In the case of several valence electrons, summation in the Thomson formula takes place over valence electrons. In derivation of the Thomson formula (3.29), a bound electron assumes to be motionless. One can refuse from this assumption and then, in the limit of high energies of an incident electron (ε  J ), we obtain instead of the Thomson formula (3.29) [121, 122]

 me v 2 πe4 1+ , (3.30) σion = Jε 3J where me is the electron mass, v is the velocity of a bound electron, and an overline means an average over velocities of a bound electron. In the limit v = 0, this

86

3 Elementary Processes Involving Atomic Particles

formula is transformed into the Thomson formula (3.29). Since the average kinetic energy of a bound electron is compared to the atom ionization potential, taking into account the velocity distribution for a bound electron leads to a remarkable change of the ionization cross section compared with the Thomson formula (3.29). In particular, if a bound electron is located mostly in the Coulomb field of an atomic core, we have me v 2 /2 = J , and the formula (3.30) differs from the Thomson formula (3.29) by a factor 5/3. Note also that the threshold dependence of the cross section of atom ionization on the electron energy differs slightly on the linear one [123], though practically this difference is not of importance. One more peculiarity of the ionization cross section relates to a logarithmic dependence of the cross section on the collision energy in the quantum case. After discovering this fact in 1930 [124], the interest to classical models of the ionization process drops. Indeed, the classical treatment leads to the dependence 1/ε for the ionization cross section σion at large energies ε of an incident electron. The logarithm dependence for the quantum approach is determined [125] by large impact parameters of collision when the ionization probability is small or is zero under classical consideration. But, as it is shown by Kingston [126–128] as a result of the numerical analysis, values of the ionization cross sections from the quantum (Born approximation) and classical approach are close for excited atoms at large collision energies. This means that the different energy dependence for the classical and quantum ionization cross sections is not of principle. Note that the basis of the Thomson formula is the Rutherford formula for elastic scattering of two free electrons on small angles. But the Rutherford formula conserves its form if an exchange energy is compared to the energy of colliding electrons. In addition, the Coulomb cross section is identical for the classical and quantum cases. Therefore, the Thomson formula may be recommended for rough calculations, especially if experimental data are taken into account in such evaluations. Indeed, assuming that atom ionization by electron impact has the classical character, we obtain a general formula for the cross section of atom ionization

 πe4 ε . (3.31) σion = 2 f J J This dependence follows from the dimensionality considerations, and f (x) is some universal function that is equal to f (x) = 1/x − 1/x 2 for the Thomson model. Figure 3.5 gives this function, that follows from experimental data [129–135] for the ionization cross sections of atoms with valence one or two s-electrons. This figure shows that the scaling law (3.31) is better, the higher the electron energy is. We are based on the Thomson model of atom ionization by electron impact because of the simplicity and transparency of this model, so that this model gives the main peculiarities of this process. Note that the threshold law for the cross section of atom ionization differs slightly from the linear dependence on the energy excess for an incident electron [123] that is determined by the character of electron release [16]. This has a fundamental importance, but not a practical one. Next, semiempirical models for evaluation of the cross section of the ionization process

3.5 Atom Ionization in Gas Discharge Plasma

87

Fig. 3.5. The reduced ionization cross section of atoms with valence s-electrons, based on experimental data: [129] for e + H, [130] for e + He, [131, 132] for e + He(23 S), [133] for e + H2 , [134] for e + He+ , [135] for e + Li

start from the limit of large energies of an incident electron where they transfer to the Born approximation and give an accuracy of approximately 20%. This follows, for example, from comparison of the results of the Deutsch–Märk [136–139] and McQuire [140–142] models that in turn correspond to the sum of experiments [143–145].

3.5 Atom Ionization in Gas Discharge Plasma A gas discharge plasma is a self-maintaining ionized gas where the ionization balance results under the action of an external electric field [146–148]. The ionization processes determine properties of a gas discharge plasma [82, 149]. Formation of free electrons proceeds in electron–atom ionization collisions and is characterized by the first Townsend coefficient α that accounts for generation of electrons and is defined by the balance equation for the number density Ne of electrons dNe = αNe , dx

(3.32)

88

3 Elementary Processes Involving Atomic Particles

Fig. 3.6. The reduced first Townsend coefficient for helium according to experimental data [150–153]

where x directs along the electric field, and 1/α is the mean free path with respect to electron multiplication. The average rate constant of ionization kion in a gas discharge plasma is then α kion = w , (3.33) Na where w is the drift electron velocity, i.e. the average velocity of electron motion in a gas under the action of an electric field. Figures 3.6, 3.7, 3.8, 3.9 and 3.10 contain the dependence of the reduced first Townsend coefficient of inert gases on the reduced electric field strength according to experimental data. One can add to these data the book [162] and the review [163] where various parameters of processes in a gas discharge plasma are collected and analyzed. The values of the first Townsend coefficient may be used in the balance equation for a gas discharge plasma with some reservations. There are several channels of atom ionization in a plasma and we restrict by direct atom ionization in electron–atom collisions, and hence these data are suitable for such a regime of ionization in a gas discharge plasma. At low values of the first Townsend coefficients that correspond to low electric field strengths, the ionization channels involving excited atoms will dominate, i.e. this type of ionization corresponds to moderate and high electric field strengths. Next, atom ionization proceeds owing to a tail of the energy distribution function of electrons when the electron energy exceeds the atom ionization potential. We assume that the tail of the electron distribution function is established as a result of elastic and inelastic electron collisions with atoms. At not low electron number densities, electron–electron collisions influence the tail of the

3.5 Atom Ionization in Gas Discharge Plasma

89

Fig. 3.7. The reduced first Townsend coefficient for neon according to experimental data [154–157, 152]

Fig. 3.8. The reduced first Townsend coefficient for argon according to experimental data [154, 155, 158–160]

energy distribution function, and this possibility is excluded from our consideration. Hence, these data relate to the case of a low electron number density. Stepwise transitions in atom ionization are also not suitable if we use the first Townsend coefficient as a parameter of atom ionization. Thus, focusing on electron–

90

3 Elementary Processes Involving Atomic Particles

Fig. 3.9. The reduced first Townsend coefficient for krypton according to experimental data [154, 159–161]

Fig. 3.10. The reduced first Townsend coefficient for xenon according to experimental data [154, 159]

atom ionization in a gas discharge plasma and using the first Townsend coefficient, we deal with a certain regime of a gas discharge plasma that corresponds to low electron number densities and electric field strengths that are not low.

3.6 Ionization Processes Involving Excited Atoms

91

3.6 Ionization Processes Involving Excited Atoms Excited atoms are present in a gas discharge plasma and may be of importance for its properties. One of the ionization channels in a gas discharge plasma is determined by the Penning process that proceeds in a pair collision process according to the scheme [164, 165] (3.34) A + B ∗ → e + A+ + B, where the ionization potential of an atom A is lower than the excitation energy of an atom B. Therefore the level A + B ∗ of the system of colliding atoms is above the boundary of continuous spectrum, i.e. it is an autoionizing state, and this state decays at moderate distances between colliding atoms when the width of this autoionizing level becomes enough for its decay during collision. Table 3.8 contains the cross sections of the Penning process involving metastable atoms of inert gases. As a result of another process of atom ionization involving excited atoms, associative ionization, a molecular ion is formed. Table 3.9 gives the parameters of this process with participation of two excited alkali metal atoms when this process proceeds according to the scheme 2  (3.35) 2M P → M2+ + e − εi , where εi is the energy released in this process, and kas is the rate constant of this process. This process of associative ionization is of importance for a photoresonant plasma. Figure 3.11 shows the behavior of electron terms and some of these are responsible for this process. As it follows from Table 2.22, there are 12 electron terms Table 3.8. The rate constant of the Penning process (3.34) at room temperature [166, 167]. The rate constants are given in cm3 /s Colliding atoms He(23 S) 4 ×10−12 8 ×10−11 1 ×10−10 1.5 ×10−10 3 ×10−11 9 ×10−11 3 ×10−10

Ne Ar Kr Xe H2 N2 O2

He(21 S)

Ne(3 P2 )

Ar(3 P2 )

Kr(3 P2 )

Xe(3 P2 )

4 ×10−11 3 ×10−10 5 ×10−10 6 ×10−10 4 ×10−11 2 ×10−10 5 ×10−10

– 1 ×10−10 7 ×10−12 1 ×10−10 5 ×10−11 8 ×10−11 2 ×10−10

– – 5 ×10−12 2 ×10−10 9 ×10−11 2 ×10−10 4 ×10−11

– – – 2 ×10−10 3 ×10−11 4 ×10−12 6 ×10−11

– – – – 2 ×10−11 2 ×10−11 2 ×10−10

Table 3.9. Parameters of the process (3.35) at temperature 500 K [16] A∗

εi , eV

kas , cm3 /s

Na(32 P ) K(42 P ) Rb(52 P ) Cs(62 P )