springer series on atomic optical and plasma physics...

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 compre-hensive manner theory and experiment in the entire field of atoms and moleculesand their interaction with electromagnetic radiation. Books in the series providea rich source of new ideas and techniques with wide applications in fields suchas chemistry, materials science, astrophysics, surface science, plasma technology,advanced optics, aeronomy, and engineering. Laser physics is a particular con-necting theme that has provided much of the continuing impetus for new develop-ments in the field. The purpose of the series is to cover the gap between standardundergraduate textbooks and the research literature with emphasis on the funda-mental ideas, methods, techniques, and results in the field.

47 Semiclassical Dynamics and RelaxationBy D.S.F. Crothers

48 Theoretical Femtosecond PhysicsAtoms and Molecules in Strong Laser FieldsBy F. Großmann

49 Relativistic Collisions of Structured Atomic ParticlesBy A. Voitkiv and J. Ullrich

50 Cathodic ArcsFrom Fractal Spots to Energetic CondensationBy A. Anders

51 Reference Data on Atomic Physics and Atomic ProcessesBy 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 Dataon Atomic Physicsand Atomic Processes

With 95 Figures

Professor Boris M. SmirnovRussian Academy of SciencesJoint Institute for High TemperaturesIzhorskaya ul. 13/19, 127412 Moscow, RussiaE-mail: [email protected]

Springer Series on Atomic, Optical, and Plasma Physics ISSN 1615-5653

ISBN 978-3-540-79362-5 e-ISBN 978-3-540-79363-2

Library of Congress Control Number: 2008929556

© Springer-Verlag Berlin Heidelberg 2008

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Preface

Each scientist works with certain information and collects it in the course of profes-sional activity. In the same manner, the author collected data for atomic physics andatomic processes. This information was checked in the course of the author’s pro-fessional activity and was published in the form of appendices to the correspondingbooks on atomic and plasma physics. Now it has been decided to publish these dataseparately.

This book contains atomic data and useful information about atomic particlesand atomic systems including molecules, nanoclusters, metals and condensed sys-tems of elements. It also gives information about atomic processes and transportprocesses in gases and plasmas. In addition, the book deals with general conceptsand simple models for these objects and processes. We give units and conversionfactors for them as well as conversion factors for spread formulas of general physicsand the physics of atoms, clusters and ionized gases since such formulas are used inprofessional practice by each scientist of this area.

This book includes numerical information from some reference books for phys-ical units and constants [1–4] and for numerical parameters of atomic particles andprocesses [2, 5–10] (in the most degree [2]). We also use data of some reviews andoriginal papers. The methodical peculiarity of this book consists in representationof some physical parameters in the form of periodic tables. This form simplifiesthe information retrieval because it only uses uniform information. If the data re-late to a restricted number of elements, they are given in the form of specified ta-bles.

Some part of the book is devoted to atomic spectra, which are given in the formof the Grotrian diagrams for atoms with the electron valence shell s, s2, and alsothe valence shell pk for light atoms. This information has not changed during thelast decades. We use the Grotrian diagrams from [11]; diagrams for the lowest atomstates are taken from [12]. Along with the Grotrian diagrams, some concepts andformulas of atomic physics are represented.

This book also contains basic concepts of the physics of atomic systems andthe simplest models for their description. These models may be a basis for simpleestimations of the parameters of some atomic objects and processes. For example,the model of a hard sphere describes atom–cluster collisions, the liquid drop modelis convenient for the analysis of cluster evaporation and other cluster processes andthe 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 followfrom measured object parameters.

As a scientist who has used the data about atomic and plasma physics containedherein to fulfill some estimations for certain problems of this area, the author in-tends this book to be used by scientists and advanced students. In the first stage ofinformation collection, the author was a user of these data, and the basis of this bookis 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 foradvanced students of this physical area.

Moscow, Boris M. SmirnovMay 2008

Contents

1 Fundamental Constants, Elements, Units . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Fundamental Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Elements and Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Physical Units and Conversion Factors . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Systems of Physical Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Conversion Factors for Units . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.3 Conversion Factors in Formulas of General Physics with

Atomic Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Elements of Atomic and Molecular Physics . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Properties of Atoms and Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Properties of the Hydrogen Atom and Hydrogen-Like Ions . 112.1.2 Properties of the Helium Atom and Helium-Like Ions . . . . . . 152.1.3 Quantum Numbers of a Light Atom . . . . . . . . . . . . . . . . . . . . . 172.1.4 Shell Atom Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.5 Schemes of Coupling of Electron Momenta in Atoms . . . . . . 242.1.6 Parameters of Atoms and Ions in the Form of Periodic Tables 29

2.2 Atomic Radiative Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.1 General Formulas and Conversion Factors for Atomic

Radiative Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.2 Radiative Transitions between Atom Discrete States . . . . . . . 382.2.3 Absorption Parameters and Broadening of Spectral Lines . . . 39

2.3 Interaction Potential of Atomic Particles at Large Separations . . . . . 432.4 Properties of Diatomic Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.4.1 Bound States of Diatomic Molecule . . . . . . . . . . . . . . . . . . . . . 472.4.2 Correlation between Atomic and Molecular States . . . . . . . . . 532.4.3 Excimer Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3 Elementary Processes Involving Atomic Particles . . . . . . . . . . . . . . . . . . 713.1 Parameters of Elementary Processes in Gases and Plasmas . . . . . . . . 713.2 Inelastic Collisions of Electrons with Atoms . . . . . . . . . . . . . . . . . . . . 763.3 Collision Processes Involving Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.4 Atom Ionization by Electron Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.5 Atom Ionization in Gas Discharge Plasma . . . . . . . . . . . . . . . . . . . . . . 87

viii Contents

3.6 Ionization Processes Involving Excited Atoms . . . . . . . . . . . . . . . . . . 913.7 Electron–Ion Recombination in Plasma . . . . . . . . . . . . . . . . . . . . . . . . 933.8 Attachment of Electrons to Molecules . . . . . . . . . . . . . . . . . . . . . . . . . 953.9 Processes in Air Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4 Transport Phenomena in Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.1 Transport Coefficients of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.2 Ion Drift in Gas in External Electric Field . . . . . . . . . . . . . . . . . . . . . . 1034.3 Conversion Parameters for Transport Coefficients . . . . . . . . . . . . . . . . 1114.4 Electron Drift in Gas in Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.5 Diffusion of Excited Atoms in Gases . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5 Properties of Macroscopic Atomic Systems . . . . . . . . . . . . . . . . . . . . . . . . 1155.1 Equation of State for Gases and Vapors . . . . . . . . . . . . . . . . . . . . . . . . 1155.2 Basic Properties of Ionized Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.3 Parameters and Rates of Processes Involving Nanoparticles . . . . . . . 1205.4 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

Atomic unit of mass ma = 112m(12C) = 1.66054 × 10−24 g

Ratio of proton and electron masses mp/me = 1836.15

Ratio of atomic and electron masses ma/me = 1822.89

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

e2 = 2.3071 × 10−19 erg cm

Planck constant h = 6.62619 × 10−27 erg s

h = 1.05457 × 10−27 erg s

Light velocity c = 2.99792 × 1010 cm/s

Fine-structure constant α = e2/(hc) = 0.07295

Inverse fine-structure constant 1/α = hc/e2 = 137.03599

Bohr radius a0 = h2/(mee2) = 0.529177 Å

Rydberg constant R = mee4/(2h2) = 13.6057 eV

= 2.17987 × 10−18 J

Bohr magneton μB = eh/(2mec) = 9.27402 × 10−24 J/T

= 9.27402 × 10−21 erg/Gs

Avogadro number NA = 6.02214 × 1023 mol−1

Stephan–Boltzmann constant σ = π2/(60h3c2) = 5.669 × 10−12 W/(cm2K4)

Molar volume R = 22.414 l/mol

Loschmidt number L = NA/R = 2.6867 × 1019 cm−3

Faraday constant F = NAe = 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]. Stableisotopes relate to elements with a nuclear charge below 83, excluding technetium43Tc and promethium 61Pm, and also to elements with a nuclear charge in the range90–93 (thorium, protactinium, uranium, neptunium). The diagram Pt1 contains stan-dard atomic masses for elements, taking into account their occurrence in the Earth’scrust [2, 18]. If stable isotopes of a given element are absent, the masses are givenin square brackets. These masses are given in atomic mass units (amu) where theunit is taken 1/12 mass of the carbon isotope 12C 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 lifetimesare expressed in days (d) and years (y) and are given for isotopes whose lifetimeexceeds 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 areexpressed. The basis of any mechanical system of units is the fact that the valueof 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 theAdvancement of Science in 1874. The system of International Units (SI) [4] wasadopted in 1960 (the conference des Poids et Mesure, Paris) and is based on themeter (m), kilogram (kg) and second (s). Other bases may be used for specific units.

In transferring from mechanics to other physics branches, additional units orassumptions are required. In particular, along with mechanical units, we deal withelectric and magnetic units below. Then the base of SI units along with the abovemechanical units contain the unit of an electric current ampere (A). Table 1.2 con-tains the SI units and their connection with the base units. Note that we express thethermodynamic temperature through energy units below.

In spreading the CGS system of units to electrostatics, one can use the Coulomblaw for the force F between two charges e1 and e2 that are located at a distance r ina vacuum. This force is given by

F = ε0e1e2

r2

where ε0 is the vacuum permittivity. Defining a charge unit from this formula underthe assumption ε0 = 1, we construct in this manner the CGSE system of unitsthat 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 CGSsystem of units, the assumption is used that the vacuum magnetic conductivity is

1.3 Physical Units and Conversion Factors 3

Pt1

.Sta

ndar

dat

omic

wei

ghts

ofel

emen

tsan

dth

eir

natu

ralo

ccur

renc

ein

Ear

th’s

crus

t


Pt2

.Nat

ural

occu

rren

ceof

stab

leis

otop

es


Pt3

.Lon

g-liv

edra

dioa

ctiv

eis

otop

es


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 A s

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) andelectromagnetic CGS (CGSM system of units) are used, we give some conversionsbetween these systems and SI units below.

For atomic systems, the system of atomic units (or Hartree atomic units) is ofimportance because parameters of atomic systems are expressed through atomic pa-rameters. In construction the system of atomic units, the fact is used that the parame-ter of any dimensionality may be built on the basis of three parameters of differentdimensionality. As a basis for the system of atomic units, the following three para-meters are taken: the Planck constant h = 1.05457×10−27 erg s, the electron chargee = 1.60218 × 10−19 C and the electron mass me = 9.10939 × 10−28 g (we takethe vacuum permittivity and the magnetic conductivity of a vacuum to be one). Thesystem 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 betweenunits used. The specific electric field strength E/p or E/Na (E is the electric fieldstrength, p is the pressure, Na is the number density of atoms) is a spread unit inphysics of ionized gases. The unit of E/Na is 1 Townsend (Td) [21]. The connectionbetween units of the above quantities is as follows:

1V

cm · Torr= 0.3535 Td; 1 Td = 2.829

V

cm · Torr; 1 Td = 1 × 10−17 V cm2.


Table 1.3. System of atomic units

Parameter Symbol, formula Value

Length a0 = h2/(me2) 5.2918 × 10−9 cm

Velocity v0 = e2/h 2.1877 × 108 cm/s

Time τ0 = h3/(me4) 2.4189 × 10−17 s

Frequency ν0 = me4/h3 4.1341 × 1016 s−1

Energy ε0 = me4/h2 27.2114 eV

= 4.3598 × 10−18 J

Power ε0/τ = m2e8/h5 0.18024 W

Electric voltage ϕ0 = me3/h2 27.2114 V

Electric field strength E0 = me5/h4 5.1422 × 109 V/cm

Momentum p0 = me2/h 1.9929 × 10−19 g cm/s

Number density N0 = a−30 6.7483 × 1024 cm−3

Volume V0 = a30 1.4818 × 10−25 cm3

= 0.089240 cm3/mol

Square, cross section σ0 = a20 2.8003 × 10−17 cm2

Rate constant k0 = v0a20 = h3/(m2e2) 6.126 × 10−9 cm3/s

Three body rate constant K0 = v0a50 = h9/(m5e8) 9.078 × 10−34 cm6/s

Dipole moment ea0 = a0 = h2/(me) 2.5418 × 10−18 esu

= 2.5418 D

Magnetic moment h2/(me) = 2μB/α 2.5418 × 10−18 erg/Gs

= 2.5418 × 10−21 J/T

Electric current I = e/τ = me5/h3 6.6236 × 10−3 A

Flux j0 = N0v0 = m3e8/h7 1.476 × 1033 cm−2 s−1

Electric current density i0 = eN0v0 = m3e9/h7 2.3653 × 1014 A/cm2

Energy flux J = ε0N0v0 = m4e12/h9 6.436 × 1015 W/cm2


Tabl

e1.

4.C

onve

rsio

nfa

ctor

sfo

run

itsof

ener

gy

1J

1er

g1

eV1

K1

cm−1

1M

Hz

1kc

al/m

ol1

kJ/m

ol

1J

110

76.

2415

×10

187.

2429

×10

225.

0341

×10

221.

5092

×10

271.

4393

×10

206.

0221

×10

20

1er

g10

−71

6.24

15×

1011

7.24

29×

1015

5.03

41×

1015

1.50

92×

1020

1.43

93×

1013

6.02

21×

1013

1eV

1.60

22×

10−1

91.

6022

×10

−12

111

604

8065

.52.

4180

×10

823

.045

96.4

85

1K

1.38

07×

10−2

31.

3807

×10

−16

8.61

74×

10−5

10.

6950

42.

0837

×10

41.

9872

×10

−38.

3145

×10

−31

cm−1

1.98

64×

10−2

31.

9864

×10

−16

1.23

98×

10−4

1.43

881

2.99

79×

104

2.85

91×

10−3

1.19

63×

10−2

1M

Hz

6.62

61×

10−2

86.

6261

×10

−21

4.13

57×

10−9

4.79

92×

10−5

3.33

56×

10−5

19.

5371

×10

−93.

9903

×10

−71

kcal

/m

ol6.

9477

×10

−21

6.94

77×

10−2

84.

3364

×10

−250

3.22

349.

761.

0485

×10

71

4.18

4

1kJ

/m

ol1.

6605

×10

−21

1.66

05×

10−2

81.

0364

×10

−212

0.27

83.5

942.

5061

×10

60.

2390

11

Tabl

e1.

5.C

onve

rsio

nfa

ctor

sfo

run

itsof

pres

sure

1Pa

=1

N/m

21

dyn/

cm2

1To

rr1

atm

a1

atb

1ba

r

1Pa

=1

N/m

21

107.

5001

×10

−39.

8693

×10

−61.

0197

×10

−510

−51

dyn/

cm2

0.1

17.

5001

×10

−49.

8693

×10

−71.

0197

×10

−610

−61

Torr

133.

332

1333

.32

11.

3158

×10

−31.

3595

×10

−31.

3333

2×

10−3

1at

ma

1.01

325

×10

51.

0132

5×

106

760

11.

0133

21.

0132

5

1at

b9.

8066

5×

104

9.80

665

×10

573

5.56

0.96

785

10.

9806

65

1ba

r10

510

675

0.01

0.98

693

1.01

971

aat

m—

phys

ical

atm

osph

ere

bat

=kg

/cm

2—

tech

nica

latm

osph

ere


Table 1.6. Conversion factors for units of electric voltage

1 V 1 CGSE 1 CGSM

1 V 1 3.33564 × 10−3 108

1 CGSE 299.792 1 2.99792 × 1010

1 CGSM 10−8 3.33564 × 10−11 1

Table 1.7. Conversion factors for units of electric field strength

1 V/cm 1 CGSE 1 CGSM

1 V/cm 1 3.33564 × 10−3 108

1 CGSE 299.792 1 2.99792 × 1010

1 CGSM 10−8 3.33564 × 10−11 1

Table 1.8. Conversion factors for units of electric resistance

1 � 1 CGSE 1 CGSM

1 � 1 1.11265 × 10−12 109

1 CGSE 8.98755 × 1011 1 8.98755 × 1020

1 CGSM 10−9 1.11265 × 10−21 1

Table 1.9. Conversion factors for units of magnetic field strength

1 Oe 1 CGSE 1 A/m

1 Oe 1 2.99792 × 1010 79.5775

1 CGSE 3.33564 × 10−11 1 2.65442 × 10−9

1 A/m 0.012566 1.11265 × 10−21 1

Table 1.10. Conversion factors for units of magnetic induction

1 CGSE 1 T = 1 Wb/m2 1 Gs

1 CGSE 1 2.99792 × 106 2.99792 × 1010

1 T = 1 Wb/m2 3.33564 × 10−7 1 104

1 Gs 3.33564 × 10−11 10−4 1

1.3.3 Conversion Factors in Formulas of General Physicswith Atomic Particles

Explanations to Table 1.11:

1. The particle velocity is v = √2ε/m, where ε is the energy, m is the particle

mass


Table 1.11. Conversion factors for formulas involving atomic particles

Number Formulaa Factor C Units used

1 v = C√

ε/m 5.931 × 107 cm/s ε in eV, m in e.m.u.a

1.389 × 106 cm/s ε in eV, m in a.m.u.a

5.506 × 105 cm/s ε in K, m in e.m.u.

1.289 × 104 cm/s ε in K, m in a.m.u.

2 v = C√

T/m 1.567 × 106 cm/s T in eV, m in a.m.u.

1.455 × 104 cm/s ε in K, m in a.m.u.

3 ε = Cv2 3.299 × 10−12 K v in cm/s, m in e.m.u.

6.014 × 10−9 K v in cm/s, m in a.m.u.

2.843 × 10−16 eV v in cm/s, m in e.m.u.

5.182 × 10−13 eV v in cm/s, m in a.m.u.

4 ω = Cε 1.519 × 1015 s−1 ε in eV

1.309 × 1011 s−1 ε in K

5 ω = C/λ 1.884 × 1015 s−1 λ in μm

6 ε = C/λ 1.2398 eV λ in μm

7 ωH = CH/m 1.759 × 107 s−1 H in Gs, m in e.m.u.

9655 s−1 H in Gs, m in a.m.u.

8 rH = C√

εm/H 3.372 cm ε in eV, m in e.m.u., H in Gs

143.9 cm ε in eV, m in a.m.u., H in Gs

3.128 × 10−2 cm ε in K, m in e.m.u., H in Gs

1.336 cm ε in K, m in a.m.u., H in Gs

9 p = CH 2 4.000 × 10−3 Pa H in Gs

= 0.04 erg/cm3

a e.m.u. is the electron mass unit (me = 9.108 × 10−28 g), a.m.u. is the atomic mass unit(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 energeticunits, m is the particle mass

3. The particle energy is ε = mv2/2, where m is the particle mass, v is the particlevelocity

4. The photon frequency is ω = ε/h, where ε is the photon energy5. The photon frequency is ω = 2πc/λ, where λ is the wavelength6. The photon energy is ε = 2πhc/λ

7. The Larmor frequency is ωH = eH/(mc) for a charged particle of a mass m ina 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 frequency9. 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 spacecoordinates: r is the distance of the electron from the Coulomb center, θ is thepolar electron angle, and ϕ is the azimuthal angle. In addition, the spin electronstate is characterized by the spin electron projection σ on a given direction. In thenon-relativistic limit, space and spin coordinates are separated, and the space wavefunction has the form

ψ(r, θ, ϕ) = Rnl(r)Ylm(θ, ϕ). (2.1)

Here, separation of variables is used for an electron located in the Coulomb centerfield, 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 isthe angular momentum projection onto a given direction. These quantum numbersare 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 = − mee4

2h2n2. (2.2)

Here, n is the principle electron quantum number, e is the electron charge and me isthe electron mass. As is seen, the states of an electron that is located in the Coulombfield 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 electronHamiltonian is independent of its spin). For notations of electron states (electronterms) 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 no-tations 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 num-bers n = 4, l = 3.

The angle wave function of an electron located in the Coulomb center field issatisfied 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 mo-menta l [22–24]

l m Ylm(θ, ϕ)

0 0 1√4π

1 0√

34π

· cos θ

1 ±1 ±√

38π

· sin θ · exp(±iϕ)

2 0√

54π

·(

32 cos2 θ − 1

2

)

2 ±1 ±√

158π

· sin θ · cos θ · exp(±iϕ)

2 ±2 12

√158π

· sin2 θ · exp(±2iϕ)

3 0√

74π

· (5 cos3 θ − 3 cos θ)

3 ±1 ± 18

√21π · sin θ · (5 cos2 θ − 1) exp(±iϕ)

3 ±2 14

√1052π

· sin2 θ cos θ · exp(±2iϕ)

3 ±3 ± 18

√35π · sin3 θ · exp(±3iϕ)

4 0√

94π

· ( 358 cos4 θ − 15

4 cos2 θ + 38 )

4 ±1 ± 38

√5π · sin θ · (7 cos3 θ − 3 cos θ) · exp(±iϕ)

4 ±2 ± 38

√5

2π· sin2 θ · (7 cos2 θ − 1) · exp(±2iϕ)

4 ±3 ± 38

√35π · sin3 θ · cos θ · exp(±3iϕ)

4 ±4 316

√352π

· sin4 θ · exp(±4iϕ)

∂

∂ cos θ

(sin2 θ

∂Ylm

∂ cos θ

)+ 1

sin2 θ

∂2Ylm

∂ϕ2+ l(l + 1)Ylm = 0 (2.3)

and is given by the formula [22–24]

Ylm(θ, ϕ) =[

2l + 1

4π

(l − m)!(l + m)!

]1/2

P ml (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

r

d2

dr2(rR) +

[2ε + 2

r− l(l + 1)

r2

]Rnl = 0, (2.5)

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)

2s 1√2

(1 − r

2

)exp(−r/2)

2p 1√24

r exp(−r/2)

3s 23√

3

(1 − 2r

3 + 2r2

27

)exp(−r/3)

3p 227

√23 · r

(1 − r

6

)exp(−r/3)

3d 481

√30

· r2 exp(−r/3)

4s 14

(1 − 3r

4 + r2

8 − r3

192

)exp(−r/4)

4p 116

√53 · r ·

(1 − r

4 + r2

80

)exp(−r/4)

4d 164

√5

· r2 · (1 − r

12

)exp(−r/4)

4f 1768

√35

· r3 · exp(−r/4)

that, with accounting for boundary conditions, has the form

Rnl(r) = 1

nl+2

2

(2l + 1)!

√(n + l)!

(n − l − 1)! (2r)l exp

(− r

n

)

× F

(−n + l + 1, 2l + 2,

2r

n

), (2.6)

where F is degenerate hypergeometric function. The following normalization con-dition takes place for the radial wave function

∫ ∞

0R2

nl(r)r2 dr = 1. (2.7)

Table 2.2 lists the expressions of the radial electron wave functions for lowest hy-drogen atom states.

Table 2.3 contains expressions for average quantities 〈rn〉 of the hydrogen atom,where r is the electron distance from the center, and Table 2.4 lists the values ofthese quantities for lowest states of the hydrogen atom.

〈rn〉 =∫ ∞

0R2

nl(r)rn+2 dr. (2.8)

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


Table 2.3. Formulas for average values 〈rn〉 given in atomic units, where r is the electrondistance from the Coulomb center [23]

Value Formula

〈r〉 12 · [3n2 − l(l + 1)]

〈r2〉 n2

2 · [5n2 + 1 − 3l(l + 1)]〈r3〉 n2

8 · [35n2(n2 − 1) − 30n2(l + 2)(l − 1) + 3(l + 2)(l + 1)l(l − 1)]〈r4〉 n4

8 · [63n4 − 35n2(2l2 + 2l − 3) + 5l(l + 1)(3l2 + 3l − 10) + 12]〈r−1〉 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

Table 2.4. Average values 〈rn〉 for lowest states of the hydrogen atom expressed in atomicunits [23]

State 〈r〉〈r2〉〈r3〉〈r4〉〈r−1〉〈r−2〉〈r−3〉〈r−4〉1s 1.5 3 7.5 22.5 1 2 – –2s 6 42 330 2880 0.25 0.25 – –2p 5 30 210 1680 0.25 0.0833 0.0417 0.04173s 13.5 207 3442 6.136 × 104 0.111 0.0741 – –3p 12.5 180 2835 4.420 × 104 0.111 0.0247 0.0123 0.01373d 10.5 126 1701 2.552 × 104 0.111 0.0148 0.0247 5.49 × 10−3

4s 24 648 18720 5.702 × 105 0.0625 0.0312 – –4p 23 600 16800 4.973 × 105 0.0625 0.0104 5.21 × 10−3 5.49 × 10−4

4d 21 504 13100 3.629 × 105 0.0625 0.00625 1.04 × 10−3 2.60 × 10−4

4f 18 360 7920 1.901 × 105 0.0625 0.00446 3.72 × 10−4 3.7 × 10−5

magnetic field due to angular electron rotation. The fine structure splitting is givenby

δf = 1

2

(Z2eh

mc

)2⟨ 1

r3

⟩· ⟨ls⟩ = 2l + 1

4

(Z2eh

mc

)2⟨ 1

r3

⟩

=(

e2

hc

)2Z4

2n3l(l + 1). (2.9)

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


Table 2.5. Quantum defect for atoms of alkali metals [26, 27]

Atom Li Na K Rb Csδs 0.399 1.347 2.178 3.135 4.057δp 0.053 0.854 1.712 2.65 3.58δd 0.002 0.0145 0.267 1.34 2.47δf – 0.0016 0.010 0.0164 0.033

electron momentum and m, σ,mj are projections of the orbital momentum, spinand total momentum onto a given direction. Note that the lower level correspondsto the total electron momentum j = l − 1/2.

The behavior of highly excited (Rydberg) states of atoms is close to that ofthe hydrogen atom because the Coulomb electron interaction with the center is themain interaction. But a short-range electron interaction with an atomic core leads toa displacement of atom energetic levels, and the electron energy, instead of that byformula (2.2), is given by [25]

εn = − mee4

2h2n2ef

= − mee4

2h2(n − δl)2, (2.10)

where nef is the effective principal quantum number and δl is the so-called quantumdefect. Since it is determined by a short-range electron–core interaction, its valuedecreases with an increase of the electron angular momentum l. Table 2.5 containsthe values of the quantum defect for alkali metal atoms [26, 27]. In addition, Fig. A.1in 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-electronapproximation when the atom wave function is the combination of products of theone-electron wave functions. In construction the atom wave functions we are basedon the Pauli exclusion principle [28], according to which location of two electronsis prohibited in an identical electron state, and the total electron wave function iszero 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, theexchange interaction whose potential may be represented in the form

Vex = A(|r1 − r2|)s1s2. (2.11)

Here r1, r2 are electron coordinates and s1, s2 are their spins. Note that the nature ofexchange interaction is non-relativistic and is connected with the symmetry of thetotal wave function. For the system of two electrons (helium atom or helium-likeion), the total wave function according to the Pauli principle changes sign as a resultof permutation of electrons. But the total wave function is the product of the spaceand spin electron wave functions. For the total spin S = 1 the spin wave functionis symmetric with respect to electron permutation, and for the total spin S = 0


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

the spin wave function changes sign at permutation of electrons. Hence, the spacewave function of two electrons is symmetric with respect to electron coordinates ifthe 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 wavefunction, and this fact may be taken into account formally by the introduction of anexchange interaction term (2.11) in the electron Hamiltonian.

Thus the spectrum of the helium atom (Fig. A.2 in Appendix A) is divided intotwo independent parts, with the total electron spin S = 0 and S = 1 with a symmet-ric space wave function of electrons in the first case and with antisymmetric spacewave function of electrons in the second case. The radiative transitions betweenstates that belong to different parts are practically absent because of conservationof the total electron spin in radiative transitions due to first approximations in theexpansion over a small relativistic parameter. In addition, atom levels related to thetotal spin S = 1 are located below the corresponding levels of states S = 0 withidentical other quantum numbers. One can see it in the Grotrian diagram Fig. A.2for the helium atom and in Fig. 2.1, where the lowest excited states of the heliumatom are given.

As a result, we represent the electron Hamiltonian in the form

H = − h2

2me�1 − h2

2me�2 − Ze2

r1− Ze2

r2+ e2

|r1 − r2|+ As1s2 + B l1s1 + B l2s2. (2.12)

An electrostatic interaction of electrons with a core and also with each other, theexchange interaction of electrons and the interaction of an electron spin with itsorbit are included in this Hamiltonian. Since spin–orbit interaction is of importancefor 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 ofthe helium atom that is given in Fig. A.2 and the spectra of helium-like ions. In theground atom state, both electrons are located in the state 1s, and the ground atom


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

Parameter Z-dependence

Ionization potential Z2

Potential of exchange interaction Z

Spin–orbit interaction Z4

Rate of one-photon radiative transition Z4

Rate of two-photon radiative transition Z8

state is 1s2 1S, i.e. the total atom spin is zero, S = 0. The atom state with S = 1 andthe states 1s of both electrons are not realized because the space wave function isantisymmetric with respect to electron permutation and is zero for identical electronstates. The quantum numbers of an excited helium atom are its orbital momentumand spin, which can be equal to zero or one. In the latter case, the atom quantumnumber is the total atom momentum, which is a sum of the orbital momentum andspin (see Fig. A.2). In the case of helium-like ions, which consist of the Coulombcenter of a charge Z and two electrons, the contribution of different interactionsto the total energy varies with variation of the center charge, which follows fromTable 2.6. Correspondingly, the spectrum of helium-like ions changes with variationof 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

H = − h2

2me

∑i

�i −∑

i

Ze2

ri+

∑i,k

e2

|ri − rk| + A∑i,k

si sk. (2.13)

Here i, k are electron numbers, ri , rk are electron coordinates, and si , sk are theoperators of electron spins.

The operators of the total electron momentum L and the total electron spin S foran atom are given by

L =∑

i

li , S =∑

i

si , (2.14)

where li and si 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 quan-tum numbers are LSMLMS , where L is the angular atom momentum, S is the totalatom 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 theform BLS, the energy levels for given atom quantum numbers L and S are split, andthe atom eigen states are characterized by the quantum numbers LSJMJ , where J isthe total atom momentum, and MJ is its projection onto a given direction.


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

The following notations are used to describe an atom state. The number 2S + 1of spin projections and the multiplicity of spin states are given above. The angularatom momentum is given by letters S, P,D, F,G, etc. for values of the angular mo-mentum L = 0, 1, 2, 3, 4, etc. The total atom momentum J is given as a subscript.For example, 3D3 corresponds to atom quantum numbers S = 1, L = 2, J = 3.

This method of description of atom states is suitable for light atoms. But thisscheme may be used also for heavy atoms to classify atom states, though relativisticeffects are of importance for these atoms. The Grotrian diagrams for the loweststates 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) isapproximately 20 times more than that for the oxygen atom with the same electronstructure (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 eachshell contains electrons with identical quantum numbers nl. This corresponds toone-electron approximation if the field that acts on a test electron consists of theCoulomb field of the nucleus and a self-consistent field of other electrons. The se-quence of shell filling for an atom in the ground state corresponds to that for a centerwith a screened Coulomb field that acts on a test electron. Diagrams Pt4, Pt5 givethe valence electron shells for atoms and negative ions in the ground state. In addi-tion, the ionization potentials of atoms and their first ions in the ground states andthe electron affinities for atoms are given in these diagrams [2, 30, 31].


Fig. 2.3. The lowest excited states of the nitrogen atom. Energies of excitation for corre-sponding 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 andare expressed in Å. The radiative lifetimes of states are placed in triangular boxes and areexpressed in s, 15 means 1 × 105

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

Valence electrons determine properties of atoms and atom interaction. One canconstruct the valence electron shell to consist of one electron and atomic core.This parentage scheme allows one to analyze atom properties due to one-electrontransitions. Then the electron wave function of an atom ΨLSMLMS

(1, 2, . . . , n) isexpressed through the wave function of an atomic core ΨL′S′M ′

LM ′S(2, . . . , n) and


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

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

through the wave function of a valence electron ψl 1

2 mσ(1) in the following manner

[32–35]:

ΨLSMLMS(1, 2, . . . , n)

= 1√nP

∑

L′M ′LS′M ′

Smσ

GLSL′S′(l, n)

[l L′ L

m M ′L ML

] [12 S′ S

σ M ′S MS

]

× ψl 1

2 mσ(1) · ΨL′S′M ′

LM ′S(2, . . . , n).


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 thoseof Fig. 2.4

Here n is a number of valence electrons and the operator P permutes coordinatesand spin of a test electron that is denoted by an argument 1 with those of valenceelectrons of the atomic core; LSMLMS are the atom quantum numbers, L′S′M ′

LM ′S

are the quantum numbers of the atomic core, l 12mσ are the quantum numbers of

a test electron, GLSL′S′(l, n) is the parentage or Racah coefficient [32, 33] and the

Clebsh–Gordan coefficients result from summation of momenta (orbital and spinmomenta) 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 theform

ψ(r, θ, ϕ) = Rnl(r)Ylm(θ, ϕ)χ1/2,σ (2.15)

where the angular wave function Ylm is given by formula (2.4), χ1/2,σ is the spinelectron wave function, and the radial wave function Rnl(r) is normalized by thecondition (2.6) and satisfies to the Schrödinger equation (2.5) far from the core. TheCoulomb interaction potential between the electron and the core takes place at large


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

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

Rnl = Ar1γ

−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 or-

bital 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 inter-actions are small. Then LSMLMS , i.e. the atom angular momentum, spin and theirprojections on a given direction, are the quantum numbers. In this approximationthe atom states are degenerated over the angular momentum and spin projections,i.e. identical energies correspond to different projections on a given direction forthese momenta. The simplest construction of electron shells relates to s- and p-valence electrons, and the values of parentage coefficients for these cases are givenin Table 2.7.

Fractional parentage coefficients satisfy some relations that follow from the de-finition of these quantities. The normalization of the atomic wave function takes theform ∑

L′S′v

[GLS

L′S′(l, n, v)]2 = 1. (2.17)

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 electronshells in the following manner [32–35]:

GLSL′S′(l, n, v) = (−1)L+L′+S+S′−l−1/2 ·

[(4l + 3 − n)(2S′ + 1)(2L′ + 1)

n(2S + 1)(2L + 1)

]1/2

× GL′S′LS (l, 4l + 3 − n, v). (2.18)


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

Atom Atomic core GLSL′S′ Atom Atomic core GLS

L′S′

s(2S) (1S) 1 p3(2P) p2(1S)√

2/3s2(1S) s(2S) 1 p4(3P) p3(4S) −1/

√3

p(2P) (1S) 1 p3(2D)√

5/12p2(3P) p(2P) 1 p3(2P) −1/2p2(1D) p(2P) 1 p4(1D) p3(4S) 0p2(1S) p(2P) 1 p3(2D)

√3/4

p3(4S) p2(3P) 1 p3(2P) −1/2p2(1D) 0 p4(1S) p3(4S) 0p2(1S) 0 p3(2D) 0

p3(2D) p2(3P) 1/√

2 p3(2P) 1p2(1D) −1/

√2 p5(2P) p4(3P)

√3/5

p2(1S) 0 p4(1D) 1/√

3p3(2P) p2(3P) −1/

√2 p4(1S) 1/

√15

p2(1D) −√5/18 p6(1S) p5(2P) 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 an-gular atom momentum is the sum of angular momenta of an atomic core and testelectron, and the atom spin is the sum of spins of these particles. Then, as a re-sult of the interaction of an atom angular momentum L and spin S, these momentaare summed into the total atom momentum J , so that atom quantum numbers areLSJMJ , where MJ is the projection of the total atom momentum on a given direc-tion. As is seen, splitting of the electron terms with given values of L and S due tospin–orbit interaction leads to atom quantum numbers LSJ in contrast to quantumnumbers LS in neglecting spin–orbit interaction. Table 2.8 contains the numbers ofelectron terms and electron levels for atoms with non-completed electron shells.

The parentage scheme is simple for valence s and p-electrons, and the valuesof fractional parentage coefficients are represented in Table 2.7 for these cases. Inthe case of d and f valence electrons, removal of one electron can lead to differentstates 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 withvalence d-electrons.

The ground atom state for a given electron shell follows from the Hund empiriclaw [36], according to which the maximum atom spin corresponds to the groundelectron state, and the total atom momentum is minimal if the shell is filled belowone half and is maximal from possible ones if the electron shell is filled by more thanhalf. For example, the nitrogen atom with the electron shell p3 has three electronterms, 4S, 2D, 2P , and the electron term 4S corresponds to its ground state. Theground state of the aluminum atom with the electron shell p is characterized by thetotal momentum 1/2 (the state 2P1/2), and the ground state of the chlorine atom


Table 2.8. States of atoms with non-filled electron shells

Shell configuration Number of terms Number of levels Statistical weights 1 1 2s2 1 1 1p, p5 1 2 6p2, p4 3 5 15p3 3 5 20d, d9 1 2 10d2, d8 5 9 45d3, d7 8 19 120d4, d6 18 40 210d5 16 37 252f, f 13 1 2 14f 2, f 12 7 13 91f 3, f 11 17 41 364f 4, f 10 47 107 1001f 5, f 9 73 197 2002f 6, f 8 119 289 3003f 7 119 327 3432

Table 2.9. Electron terms of atoms with filling electron shells dn [12]

n Electron terms Number of states Number of electron terms

0, 10 1S 1 11, 9 2D 10 22, 8 1S, 3P, 1D, 3F, 1G 45 93, 7 2P, 4P, 2D(2), 2F, 4F, 2G, 2H 120 194, 6 1S(2), 3P(4), 1D(2), 3D, 5D,

1F, 3F(2), 1G(2), 3G, 3H, 1J 210 405 2S, 6S, 2P, 4P, 2D(3), 2F(2),

4F, 2G(2), 4G, 2H, 2J 252 37

with the electron shell p5 is 2P3/2 (the total momentum is 3/2). The ground statesof 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 groundand lower excited atom states, and it is violated for heavy atoms both due to anincrease of the role of relativistic effects and because of overlapping electron shellswith 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 electronmomentum summation. Along with this, jj scheme of electron momentum summa-tion is possible. In particular, taking an atom to be composed from an atomic corewith the angular momentum l and spin s and from a valence or excited electron


Table 2.10. The ground states for atoms with filling d-shell [12]

Electron shell Term of the ground state Atoms with this electron shell

d 2D3/2 Sc(3d), Y(4d), La(5d), Lu(5d), Ac(6d), Lr(6d)

d2 3F2 Ti(3d2), Zr(4d2), Hf(5d2), Th(6d2)

d3 4F3/2 V(3d3), Ta(5d3)

d4 5D0 W(5d4)

d5 6S5/2 Mn(3d5), Tc(4d5), Re(5d5)

d6 5D4 Fe(3d6), Os(5d6)

d7 4F9/2 Co(3d7), Ir(5d7)

d8 3F4 Ni(3d8)

d9 2D5/2 –

Table 2.11. The ground state of atoms with a filling f -shell [12]

Electron shell Term of the ground state Atoms with this electron shell

f 2F5/2 –f 2 3H4 –f 3 4I9/2 Pr(4f 3)

f 4 5I4 Nd(4f 4)

f 5 6H5/2 Pm(4f 5)

f 6 7F0 Sm(4f 6), Pu(5f 6)

f 7 8S7/2 Eu(4f 7), Am(5f 7)

f 8 7F6 –f 9 6H15/2 Tb(4f 9), Bk(5f 9)

f 10 5I8 Dy(4f 10), Cf(5f 10)

f 11 4I15/2 Ho(4f 11), Es(5f 11)

f 12 3H6 Er(4f 12), Fm(5f 12)

f 13 2F5/2 Tm(4f 13), Md(5f 13)

with the angular momentum le and spin se, one can realize the summation of thesemomenta into the total atom momentum in the schemes

l + le = L, s + se = S, S + L = J (LS coupling),

l + s = j, le + se = je, j + je = J (jj coupling).

A certain hierarchy of interaction corresponds to each scheme of momentumsummation, and we will follow this connection. Let us represent the Hamiltonian ofinteracting electron and atomic core in the form

H = Hcore + He + Asse + B ls + blese. (2.19)

The terms Hcore and He in this sum account for the electron kinetic energy andelectrostatic interaction for an atomic core and valence electron, respectively; thethird term describes an exchange interaction of the valence electron and atomiccore; and the last two terms are responsible for spin–orbit interaction for an atomic


Table 2.12. Electron terms for filling of electron shells with valence p-electrons

LS scheme LS term J jj scheme J

p 2P 1/2 [1/2]1 1/22P 3/2 [3/2]1 3/2

p2 3P 0 [1/2]2 03P 1 [1/2]1[3/2]1 13P 2 [1/2]1[3/2]1 21D 2 [3/2]2 21S 0 [3/2]2 0

p3 4S 3/2 [1/2]2[3/2]1 3/22D 3/2 [1/2]1[3/2]2 3/22D 5/2 [1/2]1[3/2]2 5/22P 1/2 [1/2]1[3/2]2 1/22P 3/2 [3/2]3 3/2

p4 3P 2 [1/2]1[3/2]3 23P 0 [1/2]2[3/2]2 03P 1 [1/2]21[3/2]2 11D 2 [1/2]1[3/2]3 21S 0 [3/2]4 0

p5 2P 3/2 [1/2]2[3/2]3 3/22P 1/2 [1/2]1[3/2]4 1/2

p6 1S 0 [1/2]2[3/2]4 0

core and valence electron. It is of importance that this Hamiltonian commutes withthe total atom momentum [37]

J = l + s + le + se. (2.20)

This means that the total atom momentum is the quantum atom number for bothschemes of momentum summation at any relation between exchange and spin–orbitinteractions. 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 exchangeinteraction. In reality, the LS-coupling scheme is valid for light atoms, where therelativistic interactions are small, whereas the jj -coupling scheme may be used asa model for heavy atoms. Nevertheless, the LS-coupling scheme is used often fornotations of electron terms of heavy atoms, and Table 2.12 contains electron termsof atoms with filling p shells for both coupling schemes. Notations for jj -couplingschemes are such that the total momentum of an individual electron is indicated insquare parentheses and the total number of such electrons is given as a right super-script.

Thus, roughly in accordance with the hierarchy of interaction inside the atom,we have two basic types of momentum coupling. If the electrostatic interaction ex-ceeds a typical relativistic interaction, we have the LS-type of momentum sum-mation; in another case we obtain the jj -type of momentum coupling. Really, the


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

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


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

electrostatic interaction Vel inside the atom corresponds to splitting of atom levelsof the same electron shell, while the relativistic interaction may be characterized bythe fine splitting δ of atom levels.

The specific case relates to atoms of inert gases, and Fig. 2.10 gives the lowestgroup of excited states of the argon atom. On this example one can demonstrate theways of classifications of these states. The structure of the electron shell for thesestates is 3p54s, and, along with notations of the LS-coupling scheme, the notationsof the jj -coupling scheme are used. In the last case the total momentum of theatomic core is given inside square brackets, and the total atom momentum is givenas a right subscript. In addition, the Pashen notations are used for lowest excitedstates of inert gas atoms. The electron terms of the lowest group of excited statesare denoted as 1s5, 1s4, 1s3, 1s2, and the subscript decreases with an increase ofexcitation.

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 elec-tron shell np5(n + 1)p (n is the principal quantum number of electrons of thevalence shell) are from 2p10 up 2p1 as excitation increases. In notations of thejj -scheme of momentum coupling, the state notation includes the state of an ex-


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

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

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 theground states on the basis of the periodic table of elements together with construc-tion of the electron shell and the electron term for the ground atom state within theframework of the LS scheme of momentum coupling. Because of the periodicalcharacter of the electron shell structure, we obtain the periodical dependence foratom parameters [36, 38]. The same is given in the diagram Pt5 for negative atomicions. 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 isoften one or zero). The binding energies of excited states of negative ions are alsorepresented in diagram Pt5 in the cases when they exist (in particular, for elementsof the fourth group).

Diagram Pt6 contains parameters of lower excited states of atoms on the basis ofthe periodic table. Parameters of lower resonant states from which a dipole radiative


Pt4

.Ion

izat

ion

pote

ntia

lsof

atom

san

dio

ns


Pt5

.Ele

ctro

naf

finiti

esof

atom

s


Pt6

.Low

este

xcite

dst

ates

ofat

oms

2.2 Atomic Radiative Transitions 33

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

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

2.2 Atomic Radiative Transitions

2.2.1 General Formulas and Conversion Factors for Atomic RadiativeTransitions

Table 2.13 contains the conversion factors in formulas for photon parameters andthe radiative lifetime of resonantly excited states.

Explanation of Table 2.13.

1. The photon energy ε = hω, 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 = 2πhc/λ.


Pt7

.Res

onan

tlyex

cite

dat

omst

ates


Pt8

.Spl

ittin

gof

low

esta

tom

leve

ls


Pt9

.Ato

mic

pola

riza

bilit

y


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 ω in s−1

C = 6.6261 × 10−34 J ω in s−1

2 ω = Cε C = 1.519 × 1015 s−1 ε in eVC = 1.309 × 1011 s−1 ε in K

3 ω = C/λ C = 1.884 × 1015 s−1 ε in eV4 ε = C/λ 1.2398 eV λ in µm5 fo∗ = Cωd2g∗ 1.6126 × 10−17 ω in s−1, d in Da

0.02450 �ε = hω in eV , d in Da

6 fo∗ = Cd2g∗/λ 0.03038 λ in µm, d in Da

7 1/τ∗o = Cω3d2go 3.0316 × 10−40 s−1 ω in s−1, d in Da

1.06312 × 106 s−1 �ε = hω in eV, d in Da

8 1/τ∗o = Cd2go/λ3 2.0261 × 106 s−1 λ in µm, d in Da

9 1/τ∗o = Cω2gofo∗/g∗ 1.8799 × 10−23 s−1 ω in s−1, d in Da

4.3393 × 107 s−1 �ε = hω in eV; d in Da

10 1/τ∗o = Cfo∗go/(g∗λ2) 6.6703 × 107 s−1 λ in µm, d in Da

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

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 summedover upper states ∗ is equal to

fo∗ = 2meω

3he2|〈o|D|∗〉|2g∗ = 2meω

3he2d2g∗,

where d = 〈o|D|∗〉 is the matrix element for the operator of the dipole momentof an atomic particle taken between transition states. Here me, h are atomicparameters, 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

3he2λd2g∗.

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

τ∗o

= B∗o = 4ω3

3hc3d2go.

Here B is the Einstein coefficient; other notations are as above.8. The rate of radiative transition is given by

1

τ∗o

= B∗o = 32π3

3hλ3d2go.


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 fij τji , ns Transition fij τji , ns1s–2p 0.4162 1.6 3p–4s 0.032 2301s–3p 0.0791 5.4 3p–4d 0.619 36.51s–4p 0.0290 12.4 3p–5s 0.007 3601s–5p 0.0139 24 3p–5d 0.139 702s–3p 0.4349 5.4 3d–4p 0.011 12.42s–4p 0.1028 12.4 3d–4f 1.016 732s–5p 0.0419 24 3d–5p 0.0022 242p–3s 0.014 160 3d–5f 0.156 1402p–3d 0.696 15.6 4s–5p 0.545 242p–4s 0.0031 230 4p–5s 0.053 3602p–4d 0.122 26.5 4p–5d 0.610 702p–5s 0.0012 360 4d–5p 0.028 242p–5d 0.044 70 4d–5f 0.890 1403s–4p 0.484 12.4 4f –5d 0.009 703s–5p 0.121 24 4f –5g 1.345 240

Here λ is the wavelength of this transition; other notations are the same asabove.

9. The rate of radiative transition is

1

τ∗o

= 2ω2e2go

mec3g∗fo∗.

10. The rate of radiative transition is

1

τ∗o

= 8π2go

hg∗λ2cfo∗.

2.2.2 Radiative Transitions between Atom Discrete States

The strongest radiative transitions between discrete atom states in a non-relativisticcase connect the atom states with nonzero matrix elements of the dipole moment op-erator. This follows from the above formulas for the oscillator strength of radiativetransitions and leads to certain selection rules for the radiative transitions. For tran-sitions between hydrogen atom states when one electron is located in the Coulombfield, 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 mo-

ment operator are weaker than those for dipole radiation transitions. Such transitionsare named as forbidden radiative transitions. As the transition energy increases, the


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

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

Li+ 3 3.9×10−11 5.6 × 10−5 5.1 × 10−4 49Ne+8 10 1.1×10−13 1.8 × 10−10 1.0 × 10−7 9.2 × 10−5

Ca+18 20 6.0×10−15 2.1 × 10−13 1.2 × 10−9 7.0 × 10−8

Zn+28 30 1.3×10−15 8.1 × 10−15 1.0 × 10−10 1.1 × 10−9

Zr+38 40 5 × 10−16 1.4 × 10−15 2 × 10−11 6 × 10−11

Sn+48 50 2 × 10−16 5 × 10−16 4 × 10−12 6 × 10−12

− d ln τd ln Z

– 3.6 9.5 5.8 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 radiativetransitions are compared for helium-like ions.

Spectra of atoms and ions involving lowest excited states can be representedin the form of the Grotrian diagrams that indicate the positions of excited statesof atoms and ions, and parameters of radiative transitions with their participation.These diagrams are given in some books [11, 40–48]. These diagrams are mostobvious for atoms and ions with simple electron shells. We give below the Grotriandiagrams for atoms with the electron shells s, s2, and with the electron shell pk forlight atoms. These diagrams are taken from [11] and are given in Figs. A.1–A.28in Appendix A. In addition, Figs. 2.1–2.14 contain lowest levels of some atoms andparameters of radiative transitions between them [12]. The most information for therates of radiative transitions between atom states is collected by NIST informationcenter [49–53]. In addition, along with notations for excited states of gas atomswith valence p-electrons given in Table 2.12 for LS- and jj -coupling schemes, thePashen notations for excited inert gas atoms are represented in Fig. 2.10, as well asthey 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 onproperties of this system. We consider these processes when they are determined bytransition between atom discrete states. Then absorption and emission of resonantphotons, the energy of which is close to the difference of atom state energies, isdetermined by broadening of spectral lines [35, 54], i.e. broadening of energies ofatom states that partake in the radiative process. Let us introduce the distributionfunction aω over frequencies of emitting photons, so that aω dω is the probabilitythat the photon frequency ranges from ω up to ω + dω. As the probability, thefrequency distribution function of photons is normalized as

∫aω dω = 1. (2.22)


Because spectral lines are narrow, in scales of frequencies of emitting photons, thedistribution function of photons is

aω = δ(ω − ω0), (2.23)

where ω0 is the central frequency of an emitting photon. Hence, the rate and timesof spontaneous radiative emission are independent of the width and shape of a spec-tral line, i.e. of the photon distribution function aω. But it is not so when photonsare generated or absorbed as a result of interaction between the radiation field andatoms. Such transitions are of two types, absorption of photons and induced emis-sion of photons. The first process is described by the scheme

hω + A → A∗, (2.24)

and the process of induced emission is

nhω + A∗ → (n + 1)hω + A, (2.25)

where A,A∗ denote an atom in the lower and upper states of transition, hω is thephoton, and n is a number of identical incident photons. The cross section of ab-sorption σabs as a result of transition between two atom states is given by [35, 39]

σabs = π2c2

ω2Aaω = π2c2

ω2

gf

g0

aω

τ. (2.26)

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

σem = π2c2

ω2Baω = π2c2

ω2

aω

τ= g0

gf

σabs. (2.27)

Here A,B are the Einstein coefficients, indices o and f correspond to the lower andupper atom states, g0, gf are the statistical weights of these atom states, and τ is theradiative lifetime of the upper atom state with respect to the radiative transition tothe lower atom state.

The absorption coefficient kω is

kω = N0σabs − Nf σem = N0σabs

(1 − Nf

N0

g0

gf

)(2.28)

where N0, Nf are the number density of atoms in the lower and upper states oftransition. The absorption coefficient kω for a weak radiation intensity Iω is definedas

dIω

dx= −kωIω, (2.29)

where x is the direction of radiation propagation and, according to definition, the ra-diation flux does not perturb a gas where it propagates. In the case where population


of atom states is determined by the Boltzmann formula, the absorption coefficient isequal to

kω = N0σabs

[1 − exp

(− hω

T

)]. (2.30)

Thus, the character of photon absorption depends on broadening of spectrallines, and we give below three types of broadening that are of importance for excitedgases. The basis of Doppler broadening is the Doppler effect, according to which anemitting frequency ω0 at a relative velocity vx of a radiating particle with respect toa receiver is conceived as a frequency

ω = ω0

(1 + vx

c

), (2.31)

where c is the light velocity. Hence, for radiating atomic particles with the Maxwelldistribution function on velocities, the distribution function of photons aω has theform

aω = 1

ω0

(mc2

2πT

)1/2

· exp

[−mc2(ω − ω0)

2

2T ω20

], (2.32)

where ω0 is the frequency for a motionless particle, m is the particle mass and T isthe gas temperature expressed in energetic units.

The Lorentz (or impact) mechanism of broadening of spectral lines results fromsingle collisions of a radiating atom with surrounding particles. As a result of thesecollisions, 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 + ( ν

2 )2] . (2.33)

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

σt = πR2t ,

RtU(Rt)

hv∼ 1, (2.34)

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

NR3t 1. (2.35)


Table 2.16. Broadening parameters for spectral lines of alkali metal atoms. λ is the photonwavelength 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 in1018 cm−3. The temperature of alkali metal atoms is 500 K [16]

Element Transition λ, nm τ , ns �ωD,109 s−1

�ωL/N ,10−7 cm3/s

NDL Nt

Li 22S → 32P 670.8 27 8.2 2.6 16 3.2Na 32S1/2 → 32P1/2 589.59 16 4.5 1.6 15 2.5Na 32S1/2 → 32P3/2 589.0 16 4.5 2.4 9.4 1.4K 42S1/2 → 42P1/2 769.0 27 2.7 2.0 6.5 1.2K 42S1/2 → 42P3/2 766.49 27 2.7 3.2 4.2 0.6Rb 52S1/2 → 52P1/2 794.76 28 1.7 2.0 4.5 0.7Rb 52S1/2 → 52P3/2 780.03 26 1.8 3.1 2.9 0.4Cs 62S1/2 → 62P1/2 894.35 31 1.2 2.6 2.3 0.3Cs 62S1/2 → 62P3/2 852.11 27 1.3 4.0 1.6 0.2

Table 2.16 compares the Doppler �ωD and Lorenz �ωL widths for transition be-tween the ground and resonantly excited states of an alkali metal atom when thesewidths are given by the formulas

�ωD = ω0

√T

mc2; �ωL = 1

2Nvσt, (2.36)

and also the number density NDL of atoms when the line widths for these mecha-nisms are coincided.

One more mechanism of broadening of spectral lines due to interaction withsurrounding atoms takes place at large number densities of surrounding atoms. Asa matter of fact, in this case, a system of interacting atoms emits radiation insteadof individual atoms, though this interaction is small [39, 55]. One can assume sur-rounding atoms to be motionless in the course of radiation, and the shift of the spec-tral line of a radiating atom corresponds to a certain configuration of surroundingatoms

�ω ≡ ω − ω0 = 1

h

∑k

U(Rk), (2.37)

where Rk is the coordinate of k-th atom in the frame of reference where the radiatingatom is the origin. The photon distribution function is equal for the wing of a spectralline in the case of a uniform distribution of surrounding atoms

aωdω = N · 4πR2dR; aω = 4NπR2h

dU/dR, (2.38)

where U(R) is the shift of a spectral line due to a surrounding atom located ata distance R from the radiating atom. Table 2.16 contains the number density ofatoms Nt = R−3

t for transition between the Lorenz and quasi-static mechanisms ofbroadening.

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 reso-nant radiative transitions in atoms of the first and second groups of the periodical system ofelements [14, 16]

Element Transition λ, nm τ , ns g∗/g0 vσt, 10−7 cm3/s k0, 105 cm−1

H 12S → 22P 121.57 1.60 3 0.516 8.6He 11S → 21P 58.433 0.56 3 0.164 18Li 22S → 32P 670.8 27 3 5.1 1.6Be 21S → 21P 234.86 1.9 3 9.6 1.4Na 32S1/2 → 32P1/2 589.59 16 1 3.1 1.1Na 32S1/2 → 32P3/2 589.0 16 2 4.8 1.4Mg 31S → 31P 285.21 2.1 3 5.5 3.4K 42S1/2 → 42P1/2 769.0 27 1 4.1 0.85K 42S1/2 → 42P3/2 766.49 27 2 6.3 1.1Ca 41S → 41P 422.67 4.6 3 7.3 2.5Cu 42S1/2 → 42P1/2 327.40 7.0 1 1.1 2.1Cu 42S1/2 → 42P3/2 324.75 7.2 2 1.7 2.8Zn 41S → 41P 213.86 1.4 3 3.3 4.8Rb 52S1/2 → 52P1/2 794.76 28 1 3.9 0.91Rb 52S1/2 → 52P3/2 780.03 26 2 6.2 1.2Sr 51S → 51P 460.73 6.2 3 9.4 1.7Ag 52S1/2 → 52P1/2 338.29 7.9 1 1.1 2.0Ag 52S1/2 → 52P3/2 328.07 6.7 2 1.7 2.9Cd 51S → 51P 228.80 1.7 3 3.3 4.5Cs 62S1/2 → 62P1/2 894.35 31 1 5.3 0.77Cs 62S1/2 → 62P3/2 852.11 27 2 8.1 1.0Ba 61S → 61P 553.55 8.5 3 9.0 1.9Au 62S1/2 → 62P1/2 267.60 6.0 1 0.49 1.9Au 62S1/2 → 62P3/2 242.80 4.6 2 0.75 4.2Hg 61S → 61P 184.95 1.3 3 2.2 5.7

In the case of radiative transition between the resonantly excited and groundatom state in a parent gas or vapor, the interaction potential depends as R−3 on thedistance between atoms, and the total cross section σt � 1/v, so that vσt is inde-pendent of the collision velocity. Hence, the absorption coefficient in the line centerk0 is independent of the number density of atoms and their temperature. Table 2.17contains their values for transitions between the ground and lowest resonantly ex-cited states of alkali metal and alkali earth metal atoms.

2.3 Interaction Potential of Atomic Particlesat Large Separations

We give below a long-range interaction involving atoms or atoms and ions at largedistances between them that will allow us to separate different interaction types


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 ofatomic particles is determined by distribution of valence electrons, and the operatorof atom interaction V results from Coulomb interaction of valence electrons withelectrons of another atom and its core

V = −∑

i

e2

|ri − R| −∑

k

e2

|rk + R| +∑i,k

e2

|rk + R − ri | + Z1Z2e2

R. (2.39)

Here R is the vector joined nuclei, and the coordinates of valence electrons ri , rk arecounted off their nuclei (see Fig. 2.15). Expanding the operator (2.39) over a smallparameter r/R, after an average over the electron distribution we obtain differenttypes of a long-range interaction between atomic particles. The strongest term isproportional to R−2 and is the ion interaction with the atom dipole moment if itis 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 theinteraction potential has the form [23, 24]

U(R) = 3Zh2(n2 − n1)

2meR2(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) = Ze2Q

R3, (2.41)

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

Q = 2∑

i

r2i P2(cos θi) = 2

∑i

li (li + 1) − 3m2i

(2li − 1)(2li + 3)r2i , (2.42)

where ri, θi are the spherical coordinates of the valence electron. The quadrupolemoment is zero for a completed electron shell and for an average over the mo-mentum projections. The values of the quadrupole moment for atoms with a fillingp-shell are given in Table 2.18 [12, 58].

The next expansion term for the long-range interaction potential is interactionof 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-couplingscheme; M is the atom momentum projection onto the molecular axis

Atom states (p)2P (p2)2P (p3)4S (p4)3P (p5)2P

M = 0 4/5 −4/5 0 4/5 −4/5|M| = 1 −2/5 2/5 – −2/5 2/5

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

Interacting atoms He–He Ne–Ne Ar–Ar Kr–Kr Xe–XeC6 1.5 6.6 68 130 270

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

U(R) = − αe2

2R4(2.43)

where α is the atom polarizability, the values of which are given in diagram Pt9. Theinteraction potential of two atoms due to induced dipole moments (van der Waalsinteraction) 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 func-

tions of valence electrons and is determined by coordinate regions as shown inFig. 2.16. Because long-range and exchange interactions between atomic particlesare given by different coordinate regions at large distances between them, the totalinteraction potential is the sum of these interaction potentials. According to the na-ture of the exchange interaction potential �(R), we have the following estimationin the case of exchange by one electron located in the field of identical atomic cores:

�(R) ∼ |ψ(R/2)|2 ∼ exp(−γR),

where ψ(r) is the wave function of the valence electron, and γ 2 = 2meJ/h2 withthe ionization potential J . This interaction potential with exchange by one electron


corresponds to ion–atom interaction. Interaction of two atoms is accompanied byexchange by two valence electrons, and the exchange interaction potential of twoatoms at large separations is �(R) ∼ exp(−2γR). In particular, the exchange inter-action potential of two hydrogen atoms is �(R) ∼ exp(−2R/a0) at large distancesbetween them.

The exchange interaction potential may have a different sign that determines thepossibility of formation of a chemical bond for a given electron term at the approachof atomic particles. If this sign is negative, a covalence chemical bond is formed atmoderate distances between atomic particles and the number of attractive and re-pulsed electron terms is approximately equal. In particular, we consider interactionof a structureless ion and a parent atom with a valence s-electron. At large distancesbetween nuclei R, the eigen functions (ψg and ψu) of a system of interacting ionand atom correspond to the even (gerade) and odd (ungerade) states, and are givenby

ψg = ψ(1) + ψ(2)√2

; ψu = ψ(1) − ψ(2)√2

(2.45)

at large distances between nuclei. Here ψ(1), ψ(2) corresponds to electron locationin the field of the first and second atomic cores. Correspondingly, if an electron islocated 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 atomiccore is

ψ(1) = ψg + ψu√2

. (2.46)

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 termcorresponds to attraction while the odd term corresponds to repulsion. The exchangeinteraction potential �(R) is the difference of the energies of these states and, forthe s-valence electron in the field of structureless cores, this quantity at large R isequal to [59]

�(R) ≡ |εg(R) − εu(R)| = A2R2/γ−1 exp(−Rγ − 1/γ ), (2.47)

where A, γ are the parameters of the asymptotic expression (2.16) for the electronwave function.

Another type of exchange interaction of two atoms A–B is realized if an elec-tron term A–B correlates with an electron term A−–B+ at large separations, wherea valence electron of one atom transfers to another atom. The Coulomb interactionof positive and negative ions leads to attraction, and atoms form an ion–ion bond inthis 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 alkalimetal atoms. Excimer molecules have such a chemical bond.

Another type of exchange interaction takes place when one atom is excited. Thenature of this interaction consists in penetration of an excited electron to a non-exited 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]

U(R) = 2πh2L

me|Ψ (R)|2, (2.48)

where R is the non-excited atom coordinate, Ψ (R) is the wave function of an ex-cited electron, and L is the electron scattering length for a non-excited atom. TheFermi formula (2.48) is valid if the size of electron orbit as well as the distanceR between atoms significantly exceeds the size of a non-excited atom. This inter-action may be used as a model for systems and processes involving interaction ofexcited atoms [63]. As is seen, there are many forms of interaction between atomicparticles. 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 electronstate 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 ofnuclei, 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)1/2ε0and a typical energy difference for neighboring rotation states is ∼(me/μ)ε0.

We now give standard notations for the molecule energy [64]. The total excita-tion 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 excitationof this electron term from the ground state of the molecule, G(v) is the excitationenergy 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 aweakly excited molecule. Near the minimum of the electron term that correspondsto the stable molecule state, the vibrational and rotational energy of a molecule,accounting for the first expansion terms, has the form

G(v) = hωe(v + 1/2) − hωex(v + 1/2)2,(2.50)

Fv(J ) = BvJ (J + 1), Bv = Be − αe(v + 1/2).

Here v and J are the vibration and rotation quantum numbers (J is the rotation mo-mentum), which are whole numbers starting from zero. The spectroscopic parame-ters hωe, hωex are the vibration energy and the anharmonic parameter, Be = h2/(2I )


Table 2.20. The cases of Hund coupling [66, 67]

Hund case Relation Quantum numbersa Ve Vm Vr Λ,S, Sn

b Ve Vr Vm Λ,S, SN

c Vm Ve Vr Ω

d Vr Ve Vm L, S, LN, SN

e Vr Vm Ve J, JN

is the rotation constant, where the inertia momentum I = μR2e , Re is the distance

between atoms for the electron term minimum, the equilibrium distance. Bv is therotation constant for a vibrational excited state. These parameters for diatomic mole-cules 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 arebased on three interaction types inside the molecule: the electrostatic interaction Ve(interaction between the orbital angular momentum of electrons and the molecularaxis), spin–orbit interaction Vm and interaction between the orbital and spin electronmomenta with rotation of the molecular axis Vr. A certain scheme of coupling isdetermined by the hierarchy of these interactions, and possible coupling schemesare given in Table 2.20.

Each type of momentum coupling gives certain quantum numbers that describethe electron molecule state, correspond to a given hierarchy of interactions and arerepresented in Table 2.20. Indeed, denote by L the electron angular momentum; Sthe total electron spin; j the total electron momentum, that is, the sum of the angularand spin momenta (j = L+S); n the unit vector along the molecular axis; and K therotation momentum of nuclei. Depending on the hierarchy of interactions, we canobtain the following quantum numbers: Λ, the projection of the angular momentumof electrons onto the molecular axis; Ω , the projection of the total electron momen-tum J onto the molecular axis; and Sn, the projection of the electron spin onto themolecular axis. LN , SN , JN are projections of these momenta onto the direction ofnuclei rotation momentum K. Note that the operator of the total molecule momen-tum J = L + S + K = j + K commutes with the molecule Hamiltonian, and theeigenvalues of the operator J are the molecule quantum numbers for various casesof Hund coupling.

The character of momentum coupling under consideration describes not only themolecule structure and quantum numbers of a diatomic molecule, but is of impor-tance for dynamics of atomic collisions [68–71]. Then in the course of collision oftwo atomic particles, when a distance between two colliding atomic particles varies,the character of momentum coupling may also be changed. Transition from onecoupling scheme to another leads simultaneously to a change in the quantum num-bers of colliding particles. This may be responsible for some transitions in atomiccollisions [12, 70, 71].


Pt1

0.Pa

ram

eter

sof

hom

onuc

lear

mol

ecul

es


Pt1

1.Pa

ram

eter

sof

hom

onuc

lear

posi

tive

diat

omic

ions


Pt1

2.Pa

ram

eter

sof

hom

onuc

lear

nega

tive

diat

omic

ions


The classical Hund scheme of momentum coupling inside a molecule givesa qualitative description of this problem. In reality, the number of interactions isgreater, and the hierarchy of interactions is more complex than that in the Hund casesof momentum coupling inside a molecule. We demonstrate this on a simple exampleof interaction between a halogen ion and its atom at large distances between themthat are responsible for the charge exchange process in the collision of these atomicparticles. For definiteness, we take interaction between these particles at a distanceR0, so that the cross section of resonant charge exchange is πR2

0/2 at the collisionenergy of 1 eV. Because this distance is large compared to an atom or ion size, thissimplifies the problem and allows us separate various types of interaction.

Let us enumerate interactions in the quasi-molecule X+2 (X is the halogen atom

in the ground electron state), which have the form

Vex, UM = QMM

R3, Um = QMMqmm

R5, �(R), δi, δa, Vrot. (2.51)

Here we divide the electrostatic interaction Ve of Table 2.20 into four parts: the ex-change interaction, Vex, inside the atom and ion that is responsible for electrostaticsplitting of levels inside an isolated atom and ion; the long-range interaction, UM , ofthe 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 exchangeinteraction potential, �, which is determined by electron transition between atomiccores. Instead of the relativistic interaction Vm of Table 2.20, we introduce sepa-rately the fine splitting of levels δi for the ion (the splitting of 3P0 and 3P2 ionlevels) and δa (the splitting of 3P1/2 and 3P3/2 atom levels) for the atom. Here M,m

are the projections of the atom and ion angular momenta on the molecular axis, R isan ion–atom distance, Qik is the tensor of the atom quadrupole moment and qik isthe quadrupole moment tensor for the ion. Taking for simplicity the impact para-meter of ion–atom collision to be R0, we have for the rotation energy at the closestapproach of colliding particles

Vrot = hv

R0,

where v is the relative ion–atom velocity, and we use the expression below for esti-mation of the Coriolis interaction. As is seen, the number of possible coupling casesis larger than that in the classical case. Of course, only a small fraction of these casescan be realized.

Table 2.21 lists the values of the above interactions for the halogen molecularion at a distance R0 between nuclei that is responsible for the cross section of theresonant charge exchange process. Comparing different types of interaction, we ob-tain 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 betweencases “a” and “c” of the Hund coupling, but the hierarchy sequence (2.52) does notcorrespond exactly to any one of the Hund cases.


Table 2.21. Parameters of interaction of a positive halogen ion with a parent atom in theground electron states at a distance R0 that is responsible for resonant charge exchange at thecollision energy of 1 eV [72]

F Cl Br IR0, a0 10.6 13.8 15.1 17.2δa, cm−1 404 882 3685 7603δi, cm−1 490 996 3840 7087Vex, cm−1 20873 11654 11410 13727δi/Vex 0.023 0.085 0.34 0.52UM , cm−1 341 407 448 372UM/δa 0.84 0.46 0.12 0.049Vrot, cm−1 30 17 10 7.1�(R0), cm−1 23 14 8.4 6.1

Thus, we conclude that the classical Hund cases of electron momentum cou-pling in diatomic molecules may be used for qualitative description of the moleculestructure and its quantum numbers, but in reality the interaction inside molecules ismore 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 char-acter of coupling of electron momenta in atoms and molecules determine possibleelectron states of the forming molecules. In other words, there is a correlation be-tween electron states of atoms and molecules [73–76]. We consider this below for amolecule consisting of two identical atoms.

If a molecule consists of identical atoms, a new symmetry occurs that corre-sponds to the electron reflection with respect to the plane that is perpendicular tothe molecular axis and divides it into two equal parts. The electron wave function ofan even (gerade) state conserves its sign as a result of this operation and the wavefunction of the odd (ungerade) state changes a sign when the electrons are reflectedwith respect to the symmetry plane. The corresponding states are denoted by g andu, respectively. The operator of electron reflection with respect to any plane passedthrough the molecular axis commutes with the electron Hamiltonian also, and theparity of a molecular state is denoted by + or − depending on conservation orchange of the sign of the electron wave function in this operation. The parity of themolecular state is the sum of parities of atomic states when atoms are removed on in-finite distance and this value does not change at the variation of a distance betweennuclei. Thus, in the “a” case of the Hund coupling, molecular quantum numbersare the projection Λ of the angular momentum on the molecular axis, the total mole-cule spin S and its projection onto a given direction, the evenness and parity of thisstate which is degenerated with respect to the spin projection. Table 2.22 representsthe correlation between states of two identical atoms and a forming molecule [77]in the case “a” of Hund coupling when electrostatic interaction dominates, which


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

Atomic states Dimer states1S 1Σ+

g2S 1Σ+

g , 3Σ+u

3S 1Σ+g , 3Σ+

u , 5Σ+g

4S 1Σ+g , 3Σ+

u , 5Σ+g , 7Σ+

u1P 1Σ+

g (2), 1Σ−u , 1Πg, 1Πu, 1Δg

2P 1Σ+g (2), 1Σ−

g , 1Πg, 1Πu, 1Δg, 3Σ+u (2), 3Σ−

g , 3Πg, 3Πu, 3Δu3P 1Σ+

g (2), 1Σ−u , 1Πg, 1Πu, 1Δg, 3Σ+

u (2), 3Σ−g , 3Πg, 3Πu, 3Δu,

5Σ+g (2), 5Σ−

u , 5Πg, 5Πu, 5Δg4P 1Σ+

g (2), 1Σ−g , 1Πg, 1Πu, 1Δg, 3Σ+

u (2), 3Σ−g , 3Πg, 3Πu, 3Δu

5Σ+g (2), 5Σ−

u , 5Πg, 5Πu, 5Δg, 7Σ+u (2), 7Σ−

g , 7Πg, 7Πu, 7Δg1D 1Σ+

g (3), 1Σ−u (2), 1Πg(2), 1Πu(2), 1Δg(2), 1Δu, 1Φg, 1Φu, 1Γg

2D 1Σ+g (3), 1Σ−

u (2), 1Πg(2), 1Πu(2), 1Δg(2), 1Δu, 1Φg, 1Φu, 1Γg,3Σ+

u (3), 3Σ−g (2), 3Πg(2), 3Πu(2), 3Δg, 3Δu(2), 3Φg, 3Φu, 3Γu

3D 1Σ+g (3), 1Σ−

u (2), 1Πg(2), 1Πu(2), 1Δg(2), 1Δu, 1Φg, 1Φu, 1Γg,3Σ+

u (3), 3Σ−g (2), 3Πg(2), 3Πu(2), 3Δg, 3Δu(2), 3Φg, 3Φu, 3Γu,

5Σ+g (3), 5Σ−

u (2), 5Πg(2), 5Πu(2), 5Δg(2), 5Δu, 5Φg, 5Φu, 5Γg

includes the exchange interaction in atoms. This coupling character relates to mole-cules 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 mo-mentum coupling holds true. Figures 2.17, 2.18, 2.19, 2.20 and 2.21 [11] give thepotential curves or electron terms for lowest electron states of hydrogen, helium, car-bon, nitrogen and oxygen diatomic molecules, where the “a” Hund case is realized,and the structure of valence electron shells is s, s2, 2p2, 2p3 and 2p4, respectively.These figures also contain electron terms of the corresponding molecular ions. Inthese cases an electron state of a molecule is denoted by letters X,A,B,C, etc. forthe states with zero total spin and by letters a, b, c, etc. for the states with one totalspin. The projection of the molecule angular momentum onto the molecular axisas a quantum number is denoted by letters Σ , Π , Δ, etc. when the electron mo-mentum projection is one, two, three, etc. The total molecule spin is the quantumnumber in the “a” Hund case and the multiplicity of the spin state 2S + 1 (S is themolecule spin) is given as a left superscript for the state notation. Next, the stateparity (+ or −) is given as a right superscript, and the state evenness for a mole-cule consisting of identical atoms (g or u) is represented as a right subscript. Thesenotations 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 Hundcoupling, when the coupling in atoms corresponds to the jj -coupling scheme, maybe fulfilled in the same manner. This correlation is given in Table 2.23, that is takenfrom [77].


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 andthe molecule consisting of these atoms in the case “c” of Hund coupling [77]. The number ofmolecule states of a given symmetry is indicated in parentheses if it is not 1

Atom states Dimer states

J = 0 0+g

J = 1/2 1u, 0+g , 0−

uJ = 1 2g, 1u, 1g, 0+

g (2), 0−u

J = 3/2 3u, 2g, 2u, 1g, 1u(2), 0+g (2), 0−

u (2)

J = 2 4g, 3g, 3u, 2g(2), 2u, 1g(2), 1u(2), 0+g (3), 0−

u (2)


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

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


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

a large value. In addition, depending on the behavior of the electron wave func-tion as a result of the reflection with respect to the plane passed through moleculeaxis, the electron state is denoted by superscripts + or − after the electron momen-tum. Transition from the Hund case “a” to “c” leads to the following change in termnotations for a diatomic molecule:

X1Σ+g → 0+

g ; a3Σ+u → 1u, 0−

u ; A1Σ+u → 0+

u ; b3Σ+g → 1g, 0−

g . (2.53)

As an example, Fig. 2.22 contains electron terms for the argon molecule that con-sists of atoms in the ground Ar(3p6) and excited Ar(3p54s) states. States of anexcited atom at large separations are given for the LS scheme of momentum cou-pling. The same behavior of electron terms takes place for molecules of other inertgases.

We note that the quantum numbers of the molecule electron state are definedaccurately only to a restricted number of interactions as they take place in the Hund


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

cases of momentum coupling. In particular, a simple description relates to lightatoms when non-relativistic interactions dominate. The description of moleculesconsisting of heavy atoms, when different types of interaction partake in moleculeconstruction, is in reality outside of the Hund scheme. In particular, let us returnto the case of Table 2.21 for the interaction of a halogen positive ion with a par-ent atom at large distances that are responsible for the resonant charge exchangeprocess. Then, according to the hierarchy of interactions in the formula (2.52), the


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

molecule quantum numbers are JMJ j and the evenness (g or u), where J is thetotal atom momentum, MJ is its projection on the molecular axis and j is the totalion momentum. Fig. 2.23 contains electron terms of the molecular diatomic ion Cl+2in a range of large distances between nuclei [72].

Table 2.24 contains parameters of the lowest excited states of inert gas diatomicmolecules, so that Re is the equilibrium distance between nuclei, De is the minimuminteraction potential that corresponds to this distance and τ is the molecule radiativelifetime. One more peculiarity of interaction between atoms follows from the char-acter of exchange interaction. At large separation, the exchange interaction potentialis given by the Fermi formula (2.48), and the electron scattering length L in this for-mula is positive for helium and neon atoms and is negative for argon, krypton and


Fig. 2.22. Potential curves of diatomic argon molecule consisting of atoms in the groundAr(2p6) and lowest excited Ar(2p53s) states. They start from the lowest excited states of theatom located at large distances from the argon atom in the ground state and correspond toonly odd molecule states and correspond to only odd molecule states

Table 2.24. Parameters of lowest excited states of inert gas molecules

Molecule, state Re, Å De, eV τ , 10−7 s

He2(a3Σ+u ) 1.05 2.0 800

He2(A1Σ+u ) 1.06 2.5 0.28

Ne2(a3Σ+u ) 1.79 0.47 360

Ar2(1u, 0−u ) 2.4 0.72 0.5

Ar2(0+u ) 2.4 0.69 30

Kr2(0+u ) 0.6

Xe2(1u, 0−u ) 3.03 0.79 11

Xe2(0+u ) 3.02 0.77 0.6


Fig. 2.23. Potential curves of the chlorine molecular ion Cl+2 resulting from interaction of

the atom Cl(2P) and ion Cl+(3P) in the lowest electron states at large separations, whichare responsible for the resonant charge exchange process [72]. The excitation energies arecounted off from the ground ion and atom states with infinite distance between them and areexpressed in cm−1. The indicated quantum numbers of electron terms are JMJ j and theevenness, where J , j are the total momenta of the atom and ion, and MJ is the projection ofthe total atom momentum onto the molecular axis


Fig. 2.24. Potential curves of molecule CH

Table 2.25. The distance Rh between nuclei and the hump height �ε for the interactionpotential of excited and non-excited atoms of helium and neon

Molecule, state Rh, Å �ε, eV

He2(a3Σ+u ) 3.1 0.06

He2(A1Σ+u ) 2.8 0.05

Ne2(a3Σ+u ) 2.6 0.11

Ne2(A1Σ+u ) 2.5 0.20

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


Fig. 2.25. Potential curves of molecule NH

Table 2.26. The polarizabilities α of diatomic homonuclear molecules expressed in a30

Molecule α Molecule α

H2 5.42 Al2 130Li2 230 Cl2 31N2 11.8 K2 500O2 10.8 Br2 474F2 93 Rb2 530Na2 260 Cs2 700

In addition, Table 2.26 contains the polarizabilities of homonuclear diatomicmolecules, and the diagram Pt13 gives the electron affinities of homonuclear di-atomic molecules.

The properties of diatomic molecules including different atoms with nearby ion-ization potentials are similar to those of homonuclear diatomic molecules. This isconfirmed by Figs. 2.24, 2.25, and 2.26, which contain potential curves for the low-est states of molecules CH, NH, OH, and diagram Pt14, which gives the affinities


Pt1

3.E

lect

ron

affin

ityof

mol

ecul

es


Pt1

4.A

ffini

tyof

atom

sto

hydr

ogen

and

oxyg

enat

oms


Fig. 2.26. Potential curves of molecule OH

of various atoms to the hydrogen and oxygen atom, i.e. the dissociation energies ofcorresponding diatomic molecules. Note that alongside with quantum numbers ofelectron states, the letters X, A, B, C etc. are used for notations of excited states inthe course of an excitation increase. For light atoms when electron terms with dif-ferent spins are independent the notations a, b, c etc. are used if the total moleculespin differs from that of the ground molecule state. In addition, a prime may be usedfor 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 excitedstate, and these atoms form a strong chemical bond. When these atoms are in theground state, the bond is weak. Spread excimer molecules include a halogen atomin the ground state and an inert gas atom in the lowest excited states with an ex-


Fig. 2.27. Potential curves of excimer molecule XeF

cited s-electron. These excimer molecules are analogous to molecules consistingof halogen and alkali metal atoms in the ground state, and the chemical bond inthese excimer molecules results from a partial transition of an excited s-electronto the halogen atom. Interaction of positive and negative ions in these moleculesdetermines a strong chemical bond in them.

Electron terms of the excimer molecule XeF are given in Fig. 2.27 as an exam-ple of an excimer molecule. If atoms are found in the ground states, the chemicalbond between them is not realized since an exchange interaction involving an atomwith the completed electron shell corresponds to repulsion. Then valence electronsremain in the field of parent atoms, so that the quantum molecular numbers are thetotal molecule spin, the projection of the angular electron momentum onto the mole-cular axis and also the molecule symmetry for its reflection with respect to the planepassed through the molecular axis. Excitation of the molecule leads to the transitionof an excited electron to the halogen atom, and the quantum number of a formedmolecule is the total momentum of an inert gas atomic core. Figure 2.28 containsparameters of some states of excimer molecules under consideration [78], so thatRe is the equilibrium distance between nuclei that corresponds to the interactionpotential minimum, and De is the potential well depth, i.e. the difference of the in-teraction 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 mole-cules, whereas the radiative lifetime for some states of excimer molecules and thewavelengths of the band middle for corresponding radiative transitions are repre-sented in Fig. 2.28.


Fig

.2.2

8.Pa

ram

eter

sof

exci

mer

mol

ecul

es


Fig. 2.29. Character of radiative transitions in excimer molecule

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 variousprocesses of collision of atomic particles [82]. In a general consideration of atomiccollisions [83, 84], we use as a characteristic of collision of atomic particles the col-lision cross section. By definition, the differential cross section of collision of twoatomic particles dσ that is introduced in the center-of-mass frame of reference, isthe ratio of a number of scattered particles per unit time per unit solid angle dΩ tothe flux of incident particles. The parameters of elastic scattering of particles whenthe 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, adistance of closest approach r0, and the scattering angle ϑ . If the interaction poten-tial U(R) of colliding particles is spherically symmetric (R is the distance betweenparticles), the conservation of the angular momentum of particles in the course ofcollision leads to the following relation between the impact collision parameter andthe distance of closest approach [85]:

ρ2

r20

=[

1 − U(r0)

ε

], (3.1)

where ε = μv2/2 is the kinetic energy of colliding particles in the center-of-massframe 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 collid-ing particles. The arrow indicates the direction of the particle trajectory in the center-of-massframe of reference. ρ is the impact parameter of collision, r0 is the distance of closest ap-proach, 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—realinteraction potential of atomic particles

velocity. In the case of a monotonic dependence ϑ(ρ) of the scattering angle on theimpact collision parameter, the differential cross section of scattering in the classicallimit is

dσ = 2πρ dρ. (3.2)

The model of hard spheres is a convenient model for the scattering of classicalparticles in a strongly varied potential, and the interaction potential for this modelcorresponds to a solid wall of a radius R0, as it is shown in Fig. 3.2. Within theframework of this model, when the cross section is independent of the collisionvelocity, the differential cross section dσ and the transport cross section σ ∗ = ∫

(1−cos ϑ) dσ are correspondingly equal

dσ = πR20 d cos ϑ, σ ∗ = πR2

0 (3.3)

since according to Fig. 3.3 ρ/R0 = sin α. The transport cross section of scatter-ing σ ∗ relates to the transport of particles in a gas, whereas the total cross sectionσt = ∫

dσ characterizes a shift of the phase of an electromagnetic wave that inter-acts with an atomic particle when this phase shift follows from collisions of atomicparticles. Therefore, the total cross section of particle scattering may be responsiblefor the broadening of spectral lines resulting from the transition between discretestates of atomic particles. This cross section is infinite (h → 0) in the classicallimit because weak scattering takes place at any large impact parameter of particlecollisions.

The classical description holds true in the case when the main contribution tothe scattering cross section is given by large collision momenta l = μρv/h � 1.In the case of a spherically symmetric interaction potential between colliding par-ticles, scattering for different collision momenta l proceeds independently, and anytotal parameter of scattering is the sum of these parameters for each collision mo-mentum 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 distancesbetween them. The connection between the differential cross section of scatteringdσ = 2π |f (ϑ)|2 d cos ϑ , the scattering amplitude f (ϑ), the diffusion or transportcross section σ ∗ and the total scattering cross section σt are [24, 86, 87]

f (ϑ) = 1

2iq

∞∑l=0

(2l + 1)(e2iδl − 1

)Pl(cos ϑ),

σ ∗ = 4π

q2

∞∑l=0

(l + 1) sin2(δl − δl+1), (3.4)

σt = 4π

q2

∞∑l=0

(2l + 1) sin2 δl.

Here q is the phase vector of colliding particles in the center-of-mass frame of ref-erence, so that the energy of particles in a center-of-mass system is ε = h2q2/2μ,where μ is the reduced mass of colliding particles.

Electron–atom collisions in gases and plasmas are of importance both for plasmaprocesses involving excited atom states and for transport processes [84, 88, 89].Because of a finite number of scattering momenta in electron–atom collisions atnot-high energies, these collisions correspond to the quantum case. In the limit ofsmall electron wave numbers q, the electron scattering phase is δ0 = −Lq, and thisis the definition of the scattering length L. Correspondingly, the cross sections ofelectron–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 limitof its low value; in particular, for short-range electron–atom interaction, this de-


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 1.2 0.2 −1.6 −3.5 −6.5σt(ε = 0), Å2 5.1 0.14 9.0 43 150εmin, eV – – 0.18 0.32 0.44σt(εmin), Å2 – – 0.88 3.8 13

pendence has the form δl ∼ q2l+1. Expansion of the scattering phases at smallelectron energies [90] with accounting for the polarization electron–atom interac-tion U(r) = −αe2/2r4 at large distances r between them along with a short-rangeinteraction (2.48) gives for the scattering amplitude at low electron energies [91]

f (ϑ) = −L − παq

2a0sin

ϑ

2. (3.6)

If an electron is scattered on a non-structured atom and the scattering electron–atomlength is negative, the zeroth phase becomes equal to zero at low electron energieswhere other scattering phases are small. This leads to a sharp minimum in the crosssection of electron–atom scattering that takes in the case of electron scattering onargon, krypton and xenon atoms and is called the Ramsauer effect [92, 93]. Table 3.1gives parameters of the total cross section σt = ∫

dσ of electron scattering on atomsof inert gases at small electron energies and in a range of the Ramsauer minimum. Inaddition, Fig. 3.4 gives the dependence on the electron energy for the diffusion crosssection σ ∗ = ∫

(1 − cos θ)dσ on the xenon atom [94–98], where θ is the scatteringangle.

In addition to Fig. 3.4, Table 3.2 contains the diffusion cross section σ ∗ depend-ing on the electron energy according to evaluations [99] that are made on the basisof 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 He Ar Kr Xe

0 4.95 10.0 39.7 1760.001 4.96 8.35 39.7 1700.002 4.99 7.80 37.7 1600.005 5.14 6.61 30.0 1390.01 5.27 5.60 26.2 1160.02 5.38 4.15 21.4 89.00.04 5.58 2.50 15.3 50.20.06 5.63 1.60 11.7 34.80.08 5.71 1.00 9.13 25.60.10 5.80 0.59 7.23 20.40.15 6.00 0.23 4.56 13.60.20 6.20 0.10 2.37 8.40.25 6.26 0.091 1.30 5.90.30 6.32 0.15 0.86 3.30.35 6.38 0.24 0.55 2.50.4 6.44 0.33 0.26 1.60.5 6.55 0.51 0.10 0.530.6 6.66 0.68 0.15 0.380.7 6.74 0.86 0.27 0.380.8 6.79 1.05 0.42 0.551.0 6.87 1.38 0.80 1.151.5 6.98 2.07 1.84 3.502.0 6.97 2.70 3.00 7.502.5 6.95 3.37 4.40 11.53 6.93 4.10 6.00 16.54 6.8 6.00 10.0 24.55 6.6 7.60 14.0 286 6.3 9.3 16.0 267 5.9 11.0 17.0 278 5.5 14.0 16.5 26

10 5.0 14.6 15.5 20

and xenon, the Ramsauer minimum in electron–atom scattering is sharper for thediffusion cross section σ ∗ than that for the total cross section σt.

In contrast to the above case of electron scattering on atoms with completedelectron shells, there are different channels of electron scattering on atoms withnoncompleted electron shells. In particular, for electron scattering on atoms of al-kali metals, these channels are characterized by different total spins of an incidentelectron and atom that corresponds to ignoring relativistic effects. In particular, thecross section of scattering for electrons of zero energy is given by the followingformula instead of (3.7):

σ ∗(0) = σt(0) = π(L2

0 + 3L21

), (3.7)


Table 3.3. The parameters of scattering of a slow electron on atoms of alkali metals and pa-rameters of the autodetaching state 3P of the autodetaching state of the alkali metal negativeion [26, 27]

Li Na K Rb CsL0/a0 3.6 4.2 0.56 2.0 −2.2L1/a0 −5.7 −5.9 −15 −17 −24Er, eV 0.06 0.08 0.02 0.03 0.011Γr, eV 0.07 0.08 0.02 0.03 0.008

where L0 and L1 are the electron scattering lengths on atoms for zero and onespin, respectively. The parameters of this formula are given in Table 3.1. Note thatthe autodetachment resonance 3P that corresponds to an autodetaching state of thealkali metal negative ion gives a remarkable contribution to the cross section ofelectron–atom scattering. Rough values of the parameters of this state, the excitationenergy Er and the width Γr of the autodetaching state level are also represented inTable 3.3.

Along with the cross section σ , the rate constant vσ is the characteristic ofparticle 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 rateconstant 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 ingases and plasmas. If excited states relate to atoms the states of which are denotedby indices i and j and transition between these states result from collisions withparticles 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

dt= −NiNB

∑j

kij + NB

∑j

Njkij . (3.10)

3.2 Inelastic Collisions of Electrons with Atoms

Processes of excitation and quenching of excited atoms by electron impact proceedaccording to the scheme

3.2 Inelastic Collisions of Electrons with Atoms 77

e + A ↔ e + A∗. (3.11)

The excitation cross section by electron impact near the threshold has the form[24, 54]

σex �√

ε − �ε, ε − �ε �ε, (3.12)

where ε is the energy of an incident electron and �ε is the excitation energy. Thisgives for the quenching cross section according to the principle of detailed balance

σq � 1

v, (3.13)

where v is the velocity of an incident electron. Correspondingly, the rate constant ofatom quenching in collisions with a slow electron kq is independent of the electronenergy. Table 3.4 gives parameters of the lowest resonantly excited states of someatoms with valence s-electrons. These parameters include the excitation energy �ε,the wavelength λ for the radiative transition into the ground state, and the radiativelifetime τ∗0 for this transition; f is the oscillator strength for this transition, andthe quenching rate constant is kq for a slow electron. In addition, Table 3.5 con-tains the rate constants of quenching by a slow electron impact for some metastablestates.

The problem of inelastic electron–atom collisions may be solved under specialconditions [24, 54, 86]. One can suggest the following formula [12] for the rateconstant of the quenching of the resonantly excited state by a slow electron whendipole interaction of atoms dominates

kq = k0

�ε7/2 · τ. (3.14)

If the excitation energy �ε is expressed in eV, and the radiative lifetime of theexcitation state τ is given in ns, the rate constant k0 in formula (3.14), as it followsfrom 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 thequenching rate in collisions of such atoms with a slow electron [100]

Atom, transition �ε, eV λ, nm f τ∗0, ns kq, 10−8 cm3/s

H(21P → 11S) 10.20 121.6 0.416 1.60 0.79He(21P → 11S) 21.22 58.43 0.276 0.555 0.18He(21P → 21S) 0.602 2058 0.376 500 51He(23P → 23S) 1.144 1083 0.539 98 27Li(22P → 22S) 1.848 670.8 0.74 27 19Na(32P → 32S) 2.104 589 0.955 16.3 20K(42P1/2 → 42S1/2) 1.610 766.9 0.35 26 31K(42P3/2 → 42S1/2) 1.616 766.5 0.70 25 32Rb(52P1/2 → 52S1/2) 1.560 794.8 0.32 28 32Rb(52P3/2 → 52S1/2) 1.589 780.0 0.67 26 33Cs(62P1/2 → 62S1/2) 1.386 894.4 0.39 30 46Cs(62P3/2 → 62S1/2) 1.455 852.1 0.81 27 43


Table 3.5. The rate constants for quenching of metastable atom states in collisions with aslow electron [100]

Atom, transition �ε, eV kq, 10−10 cm3/s

He(23S → 11S) 19.82 31Ne(23P2 → 21S) 16.62 2.0Ar(33P2 → 32S) 11.55 4.0Kr(43P2 → 42S0) 9.915 3.4Xe(52P2 → 52S0) 8.315 19

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

σex(ε) = g∗g0

(ε − �ε)

εσq(ε − �ε), (3.15)

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

kex = kqg∗g0

√ε − �ε

�ε. (3.16)

Using formula (3.14) for the quenching rate constant, one can obtain the followingexpression for the excitation rate constant near the threshold:

kex = k0g∗√

ε − �ε

g0�ε4 · τ. (3.17)

3.3 Collision Processes Involving Ions

Ions as a charged component of a plasma are of importance for plasma propertiesthat result from collisions in a plasma involving ions [103–106]. Ion–atom colli-sions with variations in the ion momentum have a classical character, i.e. nuclei aremoving in these collisions according to the classical law. These processes includeelastic ion–atom collisions and the charge exchange process. Elastic scattering ofions on atoms at low energies is determined by their interaction at large distanceswith the polarization interaction potential (2.43) between them. The peculiarity ofthe 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 forthe polarization interaction potential is [85]

σcap = 2π

√αe2

μg2, (3.18)

where μ is the reduced mass of colliding ion and atom, and g is their relative veloc-ity.

In reality, the interaction potential differs from the polarization potential (2.43)at distances compared with atomic size and includes ion–atom repulsion at smallseparations. Therefore, the capture cross section (3.18) means only a strong ap-proach of colliding ion and atom. Nevertheless the diffusion cross section of ion–atom collision for the polarization interaction potential (2.43) between them, σ ∗

ia isclose to the capture cross section (3.18) and is equal to [107] σ ∗

ia = 1.10σcap.Resonant charge exchange in ion collision with a parent atom proceeds accord-

ing to the schemeA+ + A → A+ + A, (3.19)

where the tilde marks one of the colliding particles. Usually, at room temperatureand higher collision energies, the cross section of resonant charge exchange remark-ably exceeds the cross section of ion–atom elastic collision, and therefore transportof ions in the parent gas is determined by resonant charge exchange.

Considering the resonant charge exchange process as a result of state interfer-ence, we have for the electron wave function Ψ (t) instead of formula (2.46)

Ψ (t) = ψg√2

exp

(− i

2h

∫ t

−∞εg dt ′

)+ ψu√

2exp

(− i

2h

∫ t

−∞εu dt ′

), (3.20)

where εg, εu are the energies of the even and odd states at a given distance betweennuclei. From this one can find the probability of resonant charge exchange as a resultof ion–atom collision [108]

P = ∣∣〈Ψ (∞)|ψ(1)〉∣∣2 = sin2 ζ, ζ = i

2h

∫ ∞

−∞Δ dt, Δ = |εg − εu|, (3.21)

and ζ is the phase shift between even and odd states as a result of collision. Thisexhibits the interference nature of charge exchange, so that the electron state is themixture of the even and odd states, and a phase shift between these states leads tothe electron transition from the field of one core to another one. In turn, the phaseshift is determined by the exchange interaction potential (2.46) in the course ofcollision.

The expression (3.21) for the probability of the resonant charge process corre-sponds to a two-state approximation when the atom and ion electron states are notdegenerated, which takes place if an atom A of the process (3.19) relates to atomsof the first and second groups of the periodic tables, i.e. for atoms with valent s-electrons. Nevertheless, this expression may be used within the framework of some


models [109–111] in other cases. In particular, the hydrogen-like model by Rappand Francis [109] using the hydrogen wave functions in determination of the ion–atom exchange interaction potential is popular. But one can construct in this casea strict asymptotic theory by expansion of the cross section over a small parameterthat exists in reality. In contrast to model evaluations, the asymptotic theory allowsus to estimate the accuracy of the results within the theory because this accuracy isexpressed 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 excitedstate, the electron transition in the resonant charge exchange process has a classicalcharacter also [112] and proceeds when a barrier between fields of two cores disap-pears. We consider another case related to atoms in the ground and lowest excitedstates, when the transition has a tunnel character. Then we have a weak logarithmdependence of the cross section of resonant charge exchange σres on the collisionvelocity v [113, 114]

σres(v) = C ln2(

v∗v

), (3.22)

where C and v∗ are constants. Indeed, taking the basic dependence of the exchangeinteraction potential �(R) on an ion–atom distance R as �(R) � exp(−γR) ac-cording to formula (2.47), one can represent the cross section of resonant chargeexchange in the form

σres(v) = π

2

[R0 + 1

γln

(v0

v

)]2

, (3.23)

where πR20/2 is the cross section at the collision velocity v0. Diagram Pt15 gives

the cross sections of resonant charge exchange involving ions and atoms of variouselements in the ground states. The collision energy at which the cross section isgiven, corresponds to the laboratory frame of reference, where an atom is motion-less.

We will be guided by collision energies of eV, at which the parameter γR0 informula (3.23) is large. Diagram Pt16 gives values of the parameter γR0 for ionsand atoms of various elements in the ground states. From this it follows that this pa-rameter for various elements for the collision energy of 1 eV ranges approximatelyfrom 10 to 15. On the basis of this, one can construct a strict asymptotic theory ofthe resonant charge exchange by expansion of the cross section of this process overa small parameter 1/γR0. The first term of this expansion leads to the cross sectionof resonant charge exchange in the form [115]

σres(v) = π

2R2

0, ζ(R0) = e−C

2= 0.28, (3.24)

where C = 0.577 is the Euler constant. Let us ignore elastic ion–atom scattering inthe charge exchange process; i.e. particles move along straightforward trajectoriesand the ion–atom distance R varies in time as R2 = ρ2 = v2t2, where ρ is the


Pt1

5.C

ross

sect

ions

ofre

sona

ntch

arge

exch

ange


Pt1

6.Pa

ram

eter

sof

cros

sse

ctio

nof

reso

nant

char

geex

chan

ge


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

2R2

0,1

v

√πR0

2γ�(R0) = 0.28, (3.25)

where the exchange interaction potential �(R) in this formula is determined by for-mula (2.46). Then the cross section (3.25) of resonant charge exchange is expressedthrough the parameters of the asymptotic electron wave function, when a valenceelectron is located in the atom far from the core.

Because this theory represents the cross section as an expansion over a smallparameter 1/γR0, one can estimate the accuracy of the expression (3.25) withinthe framework of the theory by comparison with the results where the next termsof 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 exchangeat the ion energy of 1 eV is [116] (173 ± 2)a2

0 as an averaging of the results andaccounting for the first and the second terms of expansion in different versions ofthis expansion. Evidently, the accuracy of this ∼1% is the best that one can expectfrom formula (3.25). We estimate the data of diagram Pt15 for the cross sectionof the resonant charge exchange of elements of the first and second groups of theperiodic table to be better than 2% when ions and their atoms are found in the groundstates.

Resonant charge exchange for ions and atoms of other element groups of theperiodic table becomes more complicated because the process of resonant chargeexchange is entangled with the processes of turning of atom angular momenta andtransition between states of fine structure. In particular, Fig. 2.23 gives the lowestelectron terms in the case of collision Cl+(3P)+Cl(2P) [72], which may be respon-sible for resonant charge exchange. We will be guided by the average cross sectionof resonant charge exchange that is averaged over initial values of atom and ion mo-menta and their projections. In particular, there are 18 electron even and odd termsin the case of interaction Cl+(3P)+Cl(2P) (Fig. 2.18) where the quantum numbersare the total momenta J and j of an atom and ion and MJ is the atom momentumprojection onto the impact parameter of collision. Let us consider the case when avalence p-electron is located in the fields of structureless cores. It is convenient tocompare the parameters in this case with those for an s-electron if the asymptoticradial 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-electronis expressed through that Δ0 for an s-electron given by formula (2.47) as [69]

Δlm(R) = Δ0(R) · (2l + 1)(l + |m|)!(l − |m|!)|m|! , (3.26)

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 sec-tion with participation of an s-electron, σ10 is the cross section involving ap-electron if the momentum projection on the impact parameter of collision iszero, σ11 is this cross section when the momentum projection equals to ±1, and


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

R0γ 6 8 10 12 14 16

σ10/σ0 1.40 1.29 1.23 1.19 1.16 1.14σ11/σ0 1.08 0.98 0.94 0.95 0.912 0.91σ/σ0 1.19 1.08 1.04 1.01 0.99 0.99σ1/2/σ0 1.18 1.10 1.07 1.05 1.04 1.03σ3/2/σ0 1.18 1.10 1.06 1.04 1.03 1.02

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

R0γ 6 8 10 12 14 16σ3/σ0 1.17 1.09 1.05 1.03 1.02 1.03σ4/σ0 1.50 1.32 1.23 1.18 1.14 1.12σ5/σ0 1.44 1.29 1.22 1.17 1.14 1.11

the average cross section is σ = σ10/3 + 2σ11/3. These cross sections relate tothe “a” case of Hund coupling when spin–orbit interaction is ignored. The crosssections σ1/2, σ3/2 correspond to case “c” of Hund coupling when the spin–orbitsplitting 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. Inaddition, 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 groupsof 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 accuracyof asymptotic coefficients A mostly and is estimated for these cases to be betterthan 5%.

The average cross section of resonant charge exchange for ions and atoms withnoncompleted shells depends on the initial distributions over electron states of ionsand atoms as well as on the character of momentum coupling in them, as followsfrom the analysis for halogens, oxygen and nitrogen [72, 118, 119]. Table 3.7 givesthe average cross sections for resonant charge exchange with respect to the crosssections for s-electrons in the case “a” of Hund coupling when spin–orbit interactionmay be ignored. Note that transition to elements of groups 6, 7 and 8 from elementsof groups 3, 4 and 5 results from change of electrons by holes. The accuracy of thecross 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 crosssections are close for different types of momentum coupling.

Representing the velocity dependence for the cross section of resonant chargeexchange in the form

σres(v) = σres(v0)

(v0

v

)α

, (3.27)

3.4 Atom Ionization by Electron Impact 85

we obtain from formula (3.23)

α = 2

γR0 1.

In reality, the exponent α in formula (3.27) differs from this value because of the ap-proximated character of formula (3.23). The values of this exponent for the resonantcharge 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, ionizationprocesses are of importance for plasma properties and evolution. Atom ionizationby 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 ignoresthe interaction between impact and bound electrons during the collision act. If theenergy exchange in collisions of these electrons owing to the Coulomb interactionbetween them �ε exceeds the ionization potential, a release of the bound electronproceeds. The cross section of energy exchange between an incident and boundelectron is given by the Rutherford formula [85]

dσ = 2πρ dρ = πe4d�ε

ε(�ε)2,

if the energy exchange ranges from �ε to �ε + d�ε. Here electrons are assumed tobe classical and the energy exchange is assumed to be small compared to the colli-sion energy ε, i.e. ε � �ε. Because atom ionization corresponds to bond breakingfor a bound electron, the ionization process proceeds when the exchange energy ex-ceeds the atom ionization potential J , i.e. �ε > J . This gives the ionization crosssection σion within the framework of the Thomson model [120]

σion =∫ ε

J

dσ = πe4

ε

(1

J− 1

ε

). (3.29)

In the case of several valence electrons, summation in the Thomson formula takesplace over valence electrons.

In derivation of the Thomson formula (3.29), a bound electron assumes to bemotionless. One can refuse from this assumption and then, in the limit of highenergies of an incident electron (ε � J ), we obtain instead of the Thomson for-mula (3.29) [121, 122]

σion = πe4

Jε

(1 + mev2

3J

), (3.30)

where me is the electron mass, v is the velocity of a bound electron, and an over-line means an average over velocities of a bound electron. In the limit v = 0, this


formula is transformed into the Thomson formula (3.29). Since the average kineticenergy of a bound electron is compared to the atom ionization potential, taking intoaccount the velocity distribution for a bound electron leads to a remarkable changeof the ionization cross section compared with the Thomson formula (3.29). In par-ticular, if a bound electron is located mostly in the Coulomb field of an atomiccore, we have mev2/2 = J , and the formula (3.30) differs from the Thomson for-mula (3.29) by a factor 5/3. Note also that the threshold dependence of the crosssection 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 de-pendence of the cross section on the collision energy in the quantum case. Afterdiscovering this fact in 1930 [124], the interest to classical models of the ionizationprocess drops. Indeed, the classical treatment leads to the dependence 1/ε for theionization cross section σion at large energies ε of an incident electron. The loga-rithm dependence for the quantum approach is determined [125] by large impactparameters of collision when the ionization probability is small or is zero underclassical consideration. But, as it is shown by Kingston [126–128] as a result of thenumerical analysis, values of the ionization cross sections from the quantum (Bornapproximation) and classical approach are close for excited atoms at large colli-sion energies. This means that the different energy dependence for the classical andquantum ionization cross sections is not of principle.

Note that the basis of the Thomson formula is the Rutherford formula for elas-tic scattering of two free electrons on small angles. But the Rutherford formulaconserves its form if an exchange energy is compared to the energy of collidingelectrons. In addition, the Coulomb cross section is identical for the classical andquantum cases. Therefore, the Thomson formula may be recommended for roughcalculations, especially if experimental data are taken into account in such evalu-ations. Indeed, assuming that atom ionization by electron impact has the classicalcharacter, we obtain a general formula for the cross section of atom ionization

σion = πe4

J 2f

(ε

J

). (3.31)

This dependence follows from the dimensionality considerations, and f (x) is someuniversal function that is equal to f (x) = 1/x − 1/x2 for the Thomson model.Figure 3.5 gives this function, that follows from experimental data [129–135] forthe ionization cross sections of atoms with valence one or two s-electrons. Thisfigure 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 be-cause of the simplicity and transparency of this model, so that this model gives themain peculiarities of this process. Note that the threshold law for the cross sec-tion of atom ionization differs slightly from the linear dependence on the energyexcess for an incident electron [123] that is determined by the character of elec-tron 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 onexperimental data: [129] for e + H, [130] for e + He, [131, 132] for e + He(23S), [133] fore + H2, [134] for e + He+, [135] for e + Li

start from the limit of large energies of an incident electron where they transferto the Born approximation and give an accuracy of approximately 20%. This fol-lows, 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 bal-ance results under the action of an external electric field [146–148]. The ionizationprocesses determine properties of a gas discharge plasma [82, 149]. Formation offree electrons proceeds in electron–atom ionization collisions and is characterizedby the first Townsend coefficient α that accounts for generation of electrons and isdefined by the balance equation for the number density Ne of electrons

dNe

dx= αNe, (3.32)


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 respectto electron multiplication. The average rate constant of ionization kion in a gas dis-charge plasma is then

kion = wα

Na, (3.33)

where w is the drift electron velocity, i.e. the average velocity of electron motion ina 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 firstTownsend coefficient of inert gases on the reduced electric field strength accord-ing 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 collectedand analyzed. The values of the first Townsend coefficient may be used in the bal-ance equation for a gas discharge plasma with some reservations. There are severalchannels of atom ionization in a plasma and we restrict by direct atom ionization inelectron–atom collisions, and hence these data are suitable for such a regime of ion-ization in a gas discharge plasma. At low values of the first Townsend coefficientsthat correspond to low electric field strengths, the ionization channels involving ex-cited atoms will dominate, i.e. this type of ionization corresponds to moderate andhigh electric field strengths. Next, atom ionization proceeds owing to a tail of theenergy distribution function of electrons when the electron energy exceeds the atomionization potential. We assume that the tail of the electron distribution function isestablished as a result of elastic and inelastic electron collisions with atoms. At notlow 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 firstTownsend coefficient as a parameter of atom ionization. Thus, focusing on electron–


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 lowelectron 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 itsproperties. One of the ionization channels in a gas discharge plasma is determinedby the Penning process that proceeds in a pair collision process according to thescheme [164, 165]

A + B∗ → e + A+ + B, (3.34)

where the ionization potential of an atom A is lower than the excitation energy ofan atom B. Therefore the level A + B∗ of the system of colliding atoms is abovethe boundary of continuous spectrum, i.e. it is an autoionizing state, and this statedecays at moderate distances between colliding atoms when the width of this au-toionizing level becomes enough for its decay during collision. Table 3.8 containsthe cross sections of the Penning process involving metastable atoms of inert gases.

As a result of another process of atom ionization involving excited atoms, as-sociative ionization, a molecular ion is formed. Table 3.9 gives the parameters ofthis process with participation of two excited alkali metal atoms when this processproceeds according to the scheme

2M(2

P) → M+

2 + e − �εi, (3.35)

where �εi is the energy released in this process, and kas is the rate constant of thisprocess. This process of associative ionization is of importance for a photoresonantplasma. Figure 3.11 shows the behavior of electron terms and some of these are re-sponsible 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(23S) He(21S) Ne(3P2) Ar(3P2) Kr(3P2) Xe(3P2)

Ne 4 ×10−12 4 ×10−11 – – – –Ar 8 ×10−11 3 ×10−10 1 ×10−10 – – –Kr 1 ×10−10 5 ×10−10 7 ×10−12 5 ×10−12 – –Xe 1.5 ×10−10 6 ×10−10 1 ×10−10 2 ×10−10 2 ×10−10 –H2 3 ×10−11 4 ×10−11 5 ×10−11 9 ×10−11 3 ×10−11 2 ×10−11

N2 9 ×10−11 2 ×10−10 8 ×10−11 2 ×10−10 4 ×10−12 2 ×10−11

O2 3 ×10−10 5 ×10−10 2 ×10−10 4 ×10−11 6 ×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(32P) <0 4 × 10−11

K(42P) 0.1 9 × 10−13

Rb(52P) 0.2 4 × 10−13

Cs(62P) 0.33 7 × 10−13


Fig. 3.11. Positions of electron terms that are responsible for the process of associative ion-ization 2M(2P) → e + M+

2 accounting for two resonantly excited atoms of alkali metals

Table 3.10. The rate constants of the process (3.36) given in 10−33 cm6/s at room tem-perature. The accuracy classes: A—the accuracy is better than 10%, B—better than 20%,C—better than 30%, D—worse than 30%

A∗ He(3P2) Ne(3P2) Ne(3P1) Ar(3P2) Ar(3P0) Ar(3P1) Ar(1P1) Kr(3P2)

K 0.23(B) 0.5(C) 5.8(C) 10(A) 11(D) 12(D) 14(D) 36(C)

A∗ Kr(3P0) Kr(3P1) Kr(1P1) Xe(3P2) Xe(3P0) Xe(3P1) Hg(3P0) Hg(3P1)

K 54(D) 30(C) 1.6(A) 55(D) 40(D) 70(C) 250(A) 160(C)

corresponding to two interacting atoms of alkali metals in the resonantly excitedstates 2P . Some repulsive terms intersect with the electron term of the molecularion M+

2 . If the intersection cross section is attained in the collision of two reso-nantly excited atoms M(2P), the state of the colliding atoms becomes autoioniza-tion at lower distances between them and then it can be decomposed according tothe scheme (3.35).

Excited atoms create radiation of a gas discharge plasma. This radiation mayhave a form of separate spectral lines when photons result from radiative transitionsin atoms and can form radiation bands when radiation is created by molecular par-ticles. In a dense gas excited atoms may be transformed into excited molecules as aresult of three body processes according to the scheme

A∗ + 2A → A∗2 + A. (3.36)

Table 3.10 contains the rate constants of the process (3.36) at room temperature.

3.7 Electron–Ion Recombination in Plasma 93

3.7 Electron–Ion Recombination in Plasma

In a dense low-temperature plasma electron–ion recombination proceeds through athree body recombination process in accordance with the scheme

2e + A+ → e + A∗. (3.37)

Because the electron temperature Te expressed in energetic units is small comparedto the atom ionization potential

Te J, (3.38)

the initial bound state of an electron after its capture in the process (3.37) proceedsat a high level, the ionization potential of which is of the order of a thermal elec-tron energy. The capture of an electron in a highly excited state in the case of asmall electron temperature corresponds to a general Thomson scheme of three bodycollisions [168]. A subsequent transition to the ground electron state proceeds as aresult of many pair collisions and radiative transitions according to a general kinet-ics scheme [169–171]. The dependence on the electron temperature for this threebody recombination coefficient Kei, as it follows from the scaling law, has the form[172]

Kei = α

Ne= C

e10

m1/2e T

9/2e

. (3.39)

The constant in this expression, according to the sum of the data, is equal to [16]

C = 1.5 × 10±0.2. (3.40)

The inverse process with respect to the process (3.37) is stepwise atom ioniza-tion. In this ionization process at low electron temperature (3.38) and high elec-tron densities, many transitions between excited states proceed as a result of colli-sions with electrons unless atom ionization proceeds. According to the principleof detailed balance and formula (3.39), the rate constant of stepwise ionizationis [173]

kion = Agi

g0

mee10

h3T 3e

exp

(− J

Te

), (3.41)

where A ≈ 0.2, gi, g0 are the statistical weights of the ion and atom in the groundstate, me is the electron mass, and J is the ionization potential. Note that this for-mula is valid at low electron temperatures and, in reality, the rate constant (3.41)is very small at such temperatures, and other mechanisms of atom ionization domi-nate. Table 3.11 gives the values of the rate constant of stepwise ionization and thesedata may be considered as an upper limit for the ionization rate constant.

Electron–ion recombination that involves molecular ions, dissociative recombi-nation, proceeds according to the scheme


Table 3.11. The asymptotic values of the rate constant of stepwise ionization (in cm3/s)according to formula (3.41)

Te, eV He Ne Ar Kr Xe

0.8 2.0 ×10−18 2.7 ×10−16 3.8 ×10−13 3.3 ×10−12 3.6 ×10−11

1.0 4.9 ×10−16 3.0 ×10−14 1.0 ×10−11 5.9 ×10−11 3.8 ×10−10

2.0 2.4 ×10−10 1.8 ×10−10 3.3 ×10−9 8.0 ×10−9 2.0 ×10−8

3.0 7.8 ×10−10 2.0 ×10−9 1.4 ×10−8 2.5 ×10−8 4.6 ×10−8

Table 3.12. The rate constant of dissociative recombination α given at room temperature isexpressed in 10−7 cm3/s [177, 178]

Ion α Ion α Ion α

Ne+2 1.8 H+

3 2.3 NH+4 16

Ar+2 6.9 CO+2 3.6 CH+

5 11

Kr+2 10 H+ · CO 1.1 NH+4 · NH3 28

Xe+2 20 N+

4 20 NH+4 · (NH3)2 27

N+2 2.2 O+

4 14 H3O+ · H2O 24

O+2 2.0 H3O+ 17 H3O+ · (H2O)2 34

NO+ 3.5 CO+ · CO 14 H3O+ · (H2O)3 44

CO+ 6.8 CO+ · (CO)2 19 H3O+ · (H2O)4 50

e + AB+ → A + B∗ (3.42)

and the specifics of this process are such that it proceeds through formation of au-toionization states of forming atoms [174–176]. Table 3.12 contains the values of thedissociative recombination coefficients at room temperature. A rough dependenceof this recombination coefficient αdis on the electron temperature Te at constant gastemperature has the form

αdis ∼ T −0.5e . (3.43)

If a complex ion partakes in the recombination process, it is convenient to usethe model [166] with the introduction of a region of strong electron–ion interaction.According to this model, penetration of an electron inside a sphere of a radius R0leads to the recombination process (3.43). Then the rate of this process is given bythe formula [179]

αdis = 2e2√

2π√meTe

R0, (3.44)

3.8 Attachment of Electrons to Molecules 95

where e is the electron charge, me is the electron mass, and the radius ofthe sphere of strong electron–ion interaction R0 is equal usually to several Bohrradii.

In a plasma with negative ions the charge recombination results in mutual neu-tralization of ions according to the scheme

A+ + B− → A + B. (3.45)

The rate constant of this process may be estimated on the basis of the formula [178]

krec ≈ 5 × 10−8(

300

Te

)1/2

, (3.46)

where the recombination rate constant krec is expressed in cm3/s, and the electrontemperature Te is given in K .

3.8 Attachment of Electrons to Molecules

This process results in the capture of an electron by a molecule in an autodetach-ment level, and subsequent decay of a forming negative ion in an autodetachmentstate proceeds usually by dissociation of this negative ion [30, 180]. Since this au-todetachment state is characterized by a large lifetime, the transfer of an excesselectron energy to atoms or molecules of a gas proceeds in the course of removal ofparticles when their electron state becomes stable. Therefore, the process of electroncapture has a resonance character and results in the transition between two electronterms, as shown in Fig. 3.12. The probability of electron capture as a function of theelectron energy is shown on the right in Fig. 3.12, which testifies to the resonancecharacter of this process.

Fig. 3.12. Electron terms of a molecule and its molecular ion, which are responsible for elec-tron attachment to a molecule, and the variation of these terms proceeds along a coordinatethat determines this process (the distance between nuclei for a diatomic molecule). The prob-ability of electron capture depending on the electron energy is represented at right


Table 3.13. The rate constants of electron attachment to halomethane molecules at roomtemperature [181, 182]

Molecule Rate constant, cm3/s Molecule Rate constant, cm3/s

CH3Cl <2 ×10−15 CF2Cl2 1.6 × 10−9

CH2Cl2 5 ×10−12 CFCl3 1.9 × 10−7

CHCl3 4 ×10−9 CF3Br 1.4 × 10−8

CHFCl2 4 ×10−12 CF2Br2 2.6 × 10−7

CH3Br 7 ×10−12 CFBr3 5 × 10−9

CH2Br2 6 ×10−8 CF3I 1.9 × 10−7

CH3I 1 ×10−7 CCl4 3.5 × 10−7

CF4 <1 ×10−16 CCl3Br 6 × 10−8

CF3Cl 1 ×10−13

In reality, the resonance dependence of the cross section of electron attach-ment on the electron energy is typical for diatomic molecules with a small num-ber of autodetachment states of a forming negative ion. For complex moleculeswith a large number of autodetachment states, the dependence of the cross sec-tion on the electron energy becomes smooth, and the electron attachment processbecomes possible at zero electron energy. We give in Table 3.13 the rate constantsof electron attachment to halomethane molecules at room temperature. As shown,the rate constant of this process is nonzero at low electron energies, though theposition of an autodetachment level for electron capture is dependent of a cer-tain molecule. Therefore, the rate constants of electron attachment differ by an or-der of magnitude for different halomethane molecules that have an identical struc-ture.

The electron attachment process is of importance for electric breakdown of gasesor for gas discharge. A small concentration of electronegative molecules may give aneffective loss of electrons in a gas discharge plasma as a result of electron attachmentto molecules. This can prevent a gas from electric breakdown or change conditionsof stable gas discharge.

3.9 Processes in Air Plasma

Properties of an ionized gas that is not rare are determined by collision processesinvolving electrons, ions, excited atoms, etc. and radiative processes. In particular,this takes place in a gas discharge plasma where the dominant processes depend onthe composition and regime of a gas discharge plasma [162, 183]. In Table 3.14, wegive the rates of processes that proceed in a plasma of the Earth’s atmosphere at var-ious altitudes. The rate constants of processes in an air plasma at room temperatureinvolving negative ions are represented in Table 3.15.

3.9 Processes in Air Plasma 97

Table 3.14. Processes in an air plasma involving electrons, ions and excited atoms at roomtemperature [14]. λ is the wavelength of emitting photon, τ is the radiative lifetime, k is therate constant of a pair process and K is the rate constant of three body process

Type of process Scheme of process Rate of process

Radiation of metastable O(1S) → O(1D) + hω τ = 0.8 s, λ = 558 nmatoms O(1D) → O(3P) + hω τ = 140 s λ = 630 nm

N(2D5/2) → N(4S) + hω τ = 1.4 · 105 s, λ = 520 nmN(2D3/2) → N(4S) + hω τ = 6 · 104 s, λ = 520 nm

Recombination e + N+2 → N + N k = 2 × 10−7 cm3/s

e + O+2 → O + O k = 2 × 10−7 cm3/s

e + NO+ → N + O k = 4 × 10−7 cm3/se + N+

4 → N2 + N2 k = 2 × 10−6 cm3/sO− + O+

2 + N2 → O + O2 + N2 K = 7 × 10−26 cm6/sO−

2 + O+2 + O2 → 3O2 K = 1.6 × 10−25 cm6/s

NO−2 + NO+ + N2

→ NO + NO2 + N2 K = 1 × 10−25 cm6/sElectron e + 2O2 → O−

2 + O2 K = 3 × 10−30 cm6/sattachment e + O2 + N2 → O−

2 + N2 K = 1 × 10−31 cm6/se + O2 → O− + O k = 2 × 10−10 cm3/s,

ε = 6.5 eVDecomposition O− + O3 → e + 2O2 + 2.6 eV k = 3 × 10−10 cm3/sof negative ion O−

2 + O2 → e + O3 + 0.62 eV k = 3 × 10−10 cm3/sO− + O2 → e + O3 − 0.42 eV k < 1 × 10−12 cm3/s

Dissociation e + O2 → 2O + e k = 2 × 10−10 cm3/s,ε > 6 eV

Ion-molecular O+ + N2 → NO+ + N + 1.1 eV k = 6 × 10−13 cm3/sreaction O+ + O2 → O+

2 + O + 1.5 eV k = 2 × 10−11 cm3/sN+ + O2 → O+ + NO + 2.3 eV k = 2 × 10−11 cm3/sN+

2 + O → NO+ + N + 3.2 eV k = 1 × 10−10 cm3/sQuenching 2O2(1Δg) → O2 + O2(1Σ+

g ) k = 2 × 10−17 cm3/sO2(1Δg) + O2 → 2O2 k = 2 × 10−18 cm3/sO2(1Σ+

g ) + N2 → O2 + N2 k = 2 × 10−17 cm3/sN2(A3Σ+) + O2 → N2 + O2 k = 4 × 10−12 cm3/sO(1D) + O2 → O + O2 k = 5 × 10−11 cm3/sO(1D) + N2 → O + N2 k = 6 × 10−11 cm3/sO(1S) + O2 → O + O2 k = 3 × 10−13 cm3/sO(1S) + O → O + O k = 7 × 10−12 cm3/s

Chemical reaction O + O3 → 2O2 k = 7 × 10−14 cm3/sThree body O + 2O2 → O3 + O2 K = 7 × 10−34 cm6/sassociation O + O2 + N2 → O3 + N2 K = 6 × 10−34 cm6/s


Table 3.15. The rate constants of ion-molecular reactions involving negative ions at roomtemperature [30, 184] given in 10−11 cm3/s

Type of process Scheme of process k

Charge exchange O− + O2 → O−2 + O 10

Charge exchange O− + O3 → O−3 + O 80

Charge exchange O−2 + O → O− + O2 33

Charge exchange O−2 + O3 → O−

3 + O2 35Charge exchange NO− + O2 → O−

2 + NO 74Associative O− + O → O2 + e 20decay O− + N → NO + e 30Associative O− + NO → NO2 + e 14decay O− + N2 → N2O + e 10−3

Associative O− + O2 → O3 + e 5 × 10−4

decay O− + O3 → 2O2 + e 30Associative O− + O2(1Δg) → O3 + e 30decay O−

2 + O → O3 + e 30Associative O−

2 + N → NO2 + e 40decay O−

2 + O2 → 2O2 + e 2 × 10−7

Associative O−2 + N2 → O2 + N2 + e 2 × 10−5

decay O−2 + O2(1Δg) → 2O2 + e 20

NO− + NO → 2NO + e 0.6

4 Transport Phenomena in Gases

4.1 Transport Coefficients of Gases

Transport coefficients characterize the connection between fluxes and weak gradi-ents of some quantities. The diffusion coefficient D for atoms or molecules of a gasis the proportionality coefficient between the flux j of these atoms and the gradientof their concentration c, that is,

j = −DN∇c, (4.1)

where N is the total number density of gas atoms or molecules. If the concentrationof atoms of a given type is small (ci � 1), i.e. this component is a small admixtureto a gas, this formula may be represented in the form

j = −Di∇Ni, (4.2)

where Ni is the number density of atoms of a given component. The thermal con-ductivity of a gas κ is defined as the proportionality coefficient between the thermalflux q and the temperature gradient ∇T as

q = −κ∇T . (4.3)

The viscosity coefficient η is the proportionality coefficient between the frictionforce F per area unit of a moving gas and the gradient of a gas velocity. In the frameof reference where the direction of the average gas velocity w is x and the averagevelocity varies in the direction z, the friction force is proportional to the quantity∂wx/∂z and acts on the surface xy. Correspondingly, the viscosity coefficient isdefined as

F = −η∂wx

∂z, (4.4)

and this definition holds true both for a gas and for a liquid.Transport coefficients in a gas are determined by collision processes. Because

the elastic cross section in an atomic gas or in a molecular gas at not high tempera-tures is significantly larger than that of inelastic processes, the transport coefficientsin a gas are expressed through average cross sections of elastic collisions. The sim-ple connection between these quantities takes place in the Chapman–Enskog ap-

100 4 Transport Phenomena in Gases

proximation [185, 186], which corresponds to an expansion over a small numericalparameter. The accuracy of the first Chapman–Enskog approximation is several per-cent. The diffusion coefficient D of a test atom or molecule in a gas is given in thefirst Chapman–Enskog approximation [185, 186]

D = 3√

πT

8N√

2μσ, σ ≡ Ω(1,1)(T ) = 1

2

∫ ∞

0e−t t2σ ∗(t) dt, t = μg2

2T. (4.5)

Here T is the gas temperature expressed in energetic units, N is the number densityof gas atoms, μ is the reduced mass of a test atom and a gas atom, σ ∗(g) is thediffusion cross section of the collision of two atomic particles at a relative velocity g

of collision, and the brackets mean an average over atom velocities with the Maxwelldistribution function.

The thermal conductivity coefficient κ in the first Chapman–Enskog approxima-tion is given by [185, 186]

κ = 25√

πT

32σ2√

m, (4.6)

where m is the mass of an atom or molecule, an average of the cross section ofelastic collision is determined by the following formula:

σ2 ≡ Ω(2,2)(T ) =∫ ∞

0t2 exp(−t)σ (2)(t) dt,

t = μg2

2T, σ (2)(t) =

∫ (1 − cos2 ϑ

)dσ. (4.7)

The viscosity coefficient η of the first Chapman–Enskog approximation is de-termined by the formula [185, 186]

η = 5√

πT m

24σ2, (4.8)

where the average cross section σ2 is given by formula (4.7). In this approximationthe coefficients of thermal conductivity and viscosity are connected by the relation

κ = 15

4mη. (4.9)

Formulas (4.5), (4.6) and (4.8) for the first Chapman–Enskog approximation aresimplified if particle collision is described by the hard sphere model (3.3). Then theaverage cross sections are given by

σ = σ0; σ2 = 2

3σ0, (4.10)

where σ0 = πR20, and R0 is the hard sphere radius. This leads to the following

expressions for the transport coefficients, instead of formulas (4.5), (4.6), (4.8):

D = 3√

πT

8√

2μNσ0; κ = 75

√πT

64σ0√

m; η = 15

√πT m

48σ0. (4.11)

4.1 Transport Coefficients of Gases 101

Table 4.1. The diffusion coefficients D of atoms and molecules in a parent gas given in cm2/sand reduced to the normal number density

Gas D, cm2/s Gas D, cm2/s Gas D, cm2/sHe 1.6 H2 1.3 H2O 0.28Ne 0.45 N2 0.18 CO2 0.096Ar 0.16 O2 0.18 NH3 0.25Kr 0.084 CO 0.18 CH4 0.20Xe 0.048

Table 4.2. The parameters of formula (4.12) with the diffusion coefficient D0 at temperatureT = 300 K that is reduced to the normal number density and is expressed in cm2/s and theparameter γ of formula (4.12) is given in parentheses

Atom, gas He Ne Ar Kr XeHe 1.66(1.73) 1.08(1.72) 0.76(1.71) 0.68(1.67) 0.56(1.66)Ne 0.52(1.69) 0.32(1.72) 0.25(1.74) 0.23(1.67)Ar 0.195(1.70) 0.15(1.70) 0.115(1.74)Kr 0.099(1.78) 0.080(1.79)Xe 0.059(1.76)

Table 4.3. The diffusion coefficient of metal atoms in inert gases at temperature 300 K, ex-pressed in cm2/s and reduced to the normal number density [187]

Atom, gas He Ne Ar Kr XeLi 0.54 0.29 0.28 0.24 0.20Na 0.50 0.31 0.18 0.14 0.12K 0.54 0.24 0.22 0.11 0.087Cu 0.68 0.25 0.11 0.077 0.059Rb 0.39 0.21 0.14 0.11 0.088Cs 0.34 0.22 0.13 0.080 0.057Hg 0.46 0.21 0.11 0.072 0.056

The coefficients of self-diffusion given in Table 4.1 are reduced as usual to thenormal density of atoms or molecules N = 2.687 × 1019 cm−3 (T = 273 K,p = 1 atm) because the diffusion coefficient is inversely proportional to the num-ber density of gas atoms. Let us approximate the temperature dependence of thediffusion coefficient by

D = D0

(T

300

)γ

(4.12)

where the temperature is expressed in K. The parameters of this formula are basedon the data [188, 189] for the temperature range T = 300–1500 K and are givenin Table 4.2. Note that, within the framework of the hard sphere model accordingto formula (4.11), we have γ = 1.5 since the number density of atoms is inverselyproportional to the temperature. In addition to these data, the diffusion coefficientsof metal atoms in inert gases at temperature T = 500 K are given in Table 4.3.


Table 4.4. Gas-kinetic cross sections for collisions of atoms or molecules σg expressed in10−15 cm2

Colliding particles He Ne Ar Kr Xe H2 N2 O2 CO CO2He 1.3 1.5 2.1 2.4 2.7 1.7 2.3 2.2 2.2 2.7Ne 1.8 2.6 3.0 3.3 2.0 2.4 2.5 2.7 3.1Ar 3.7 4.2 5.0 2.7 3.8 3.9 4.0 4.4Kr 4.8 5.7 3.2 4.4 4.3 4.4 4.9Xe 6.8 3.7 5.0 5.2 5.0 6.1H2 2.0 2.8 2.7 2.9 3.4N2 3.9 3.8 3.7 4.9O2 3.8 3.7 4.4CO 4.0 4.9CO2 7.5

Table 4.5. Thermal conductivity coefficients in gases κ expressed in 10−4 W/(cm K) [190]

T , K 100 200 300 400 600 800 1000H2 6.7 13.1 18.3 22.6 30.5 37.8 44.8He 7.2 11.5 15.1 18.4 25.0 30.4 35.4CH4 – 2.17 3.41 4.88 8.22 – –NH3 – 1.53 2.47 6.70 6.70 – –H2O – – – 2.63 4.59 7.03 9.74Ne 2.23 3.67 4.89 6.01 7.97 9.71 11.3CO 0.84 1.72 2.49 3.16 4.40 5.54 6.61N2 0.96 1.83 2.59 3.27 4.46 5.48 6.47Air 0.95 1.83 2.62 3.28 4.69 5.73 6.67O2 0.92 1.83 2.66 3.30 4.73 5.89 7.10Ar 0.66 1.26 1.77 2.22 3.07 3.74 4.36CO2 – 0.94 1.66 2.43 4.07 5.51 6.82Kr – 0.65 1.00 1.26 1.75 2.21 2.62Xe – 0.39 0.58 0.74 1.05 1.35 1.64

The gas-kinetic cross section is introduced within the framework of the hardsphere model on the basis of the expression for the diffusion coefficient (4.11). Thevalue of the diffusion coefficient is used at room temperature. We have the followingconnection between the gas-kinetic cross section σg and the diffusion coefficient D

σg = 0.47

ND

√T

μ(4.13)

where N is the number density of atoms or molecules, D is the diffusion coefficient,T is room temperature expressed in energetic units, and μ is the reduced mass ofcolliding particles. The values of gas-kinetic cross sections σg obtained on the basisof formula (4.13) are given in Table 4.4.

The values of thermal conductivity coefficients of inert gases are given in Ta-ble 4.5 at a pressure of 1 atm, and Table 4.6 contains the viscosity coefficients for

4.2 Ion Drift in Gas in External Electric Field 103

Table 4.6. The viscosity coefficient of gases at atmospheric pressure given in 10−5 g/(cm s)[191]

T , K 100 200 300 400 600 800 1000H2 4.21 6.81 8.96 10.8 14.2 17.3 20.1He 9.77 15.4 19.6 23.8 31.4 38.2 44.5CH4 – 7.75 11.1 14.1 19.3 – –H2O – – – 13.2 21.4 29.5 37.6Ne 14.8 24.1 31.8 38.8 50.6 60.8 70.2CO – 12.7 17.7 21.8 28.6 34.3 39.2N2 6.88 12.9 17.8 22.0 29.1 34.9 40.0Air 7.11 13.2 18.5 23.0 30.6 37.0 42.4O2 7.64 14.8 20.7 25.8 34.4 41.5 47.7Ar 8.30 16.0 22.7 28.9 38.9 47.4 55.1CO2 – 9.4 14.9 19.4 27.3 33.8 39.5Kr – – 25.6 33.1 45.7 54.7 64.6Xe – – 23.3 30.8 43.6 54.7 64.6

these gases at atmospheric pressure when the dependence on the pressure is weak.In addition, these transport coefficients in the first Chapman–Enskog approximation[185, 186] are connected by the following relation by analogy with formula (4.9)

κ = 5cV η

2m. (4.14)

Here κ is the thermal conductivity coefficient, η is the viscosity coefficient, cV isthe heat capacity per atom or molecule that is cV = 3/2 for ideal gas, and m is themass of an atom or molecule.

4.2 Ion Drift in Gas in External Electric Field

In consideration of ion motion in a gas under the action of an external electric fieldof strength E, we restrict by limiting cases of low and high electric fields. The limitof low electric fields corresponds to the criterion

eEλ � T , (4.15)

where λ = 1/(Nσ) is the mean free path of ions in a gas, σ is the cross sectionof ion–atom scattering on large angles and N is the number density of atoms. Weassume that the ion and atom masses have the same order of magnitude. In thislimiting case the ion drift velocity, i.e. its average velocity, is proportional to theelectric field strength

w = EK, (4.16)

and this is the definition of the ion mobility K in low electric fields when it isindependent of the electric field strength and the ion drift velocity is small comparedto its thermal velocity.


The diffusion coefficient of a charged particle in a gas D is connected with theion mobility K by the Einstein relation [192, 193]

D = KT

e. (4.17)

Here T is the gas temperature, and the Einstein relation follows from the equilibriumcondition that leads to equality of the diffusion and drift fluxes. Note that, thoughthis relation is called the Einstein relation, it was derived by Townsend several yearsbefore [194, 195] and Einstein used these results in the analysis of Brownian motionof particles [21]. On the basis of formulas (4.5), (4.11), (4.17) we have for the ionmobility in the first Chapman–Enskog approximation and within the framework ofthe hard sphere model

K = 3√

πe

8√

2μT Nσo

. (4.18)

The diffusion cross section of ion–atom scattering under the polarization inter-action potential (2.43) between them is close to the capture cross section and isequal to [166, 196]

σ ∗(v) =∫

(1 − cos ϑ) dσ = 2.21π

√αe2

μg2,

i.e. it exceeds the capture cross section by about 10%. For this cross section, the ionmobility reduced to the normal number density of gas atoms N = 2.69×1019 cm−3

is given by the Dalgarno formula [107, 196]

K = 36√αμ

cm2

V · s, (4.19)

where the ion–atom reduced mass is expressed in atomic mass units (1.66×10−24 g)and the polarizability is given in atomic units (a3

0). It is important that this mobilitydoes not depend on the temperature, and ion parameters are taken into account onlyby the ion–atom reduced mass.

Table 4.7 contains the mobilities of alkali metal atoms in inert gases and nitrogenat room temperature in a zero electric field [197–202]. These data allow us to check

Table 4.7. The zero field mobility (in cm2/(V s)) of positive ions of alkali metals in inertgases and nitrogen at room temperature [197, 198, 202]

Ion, gas He Ne Ar Kr Xe H2 N2Li+ 22.9 10.6 4.6 3.7 2.8 12.4 4.15Na+ 22.8 8.2 3.1 2.2 1.7 12.2 2.85K+ 21.5 7.4 2.7 1.83 1.35 13.1 2.53Rb+ 20 6.5 2.3 1.45 1.0 13.0 2.28Cs+ 18.3 6.0 2.1 1.3 0.91 12.9 2.2


Table 4.8. The ratio between experimental and theoretical mobilities of ions in rare gases[166]

Ion, gas He Ne Ar Kr Xe H2 N2Li+ 1.27 1.17 1.06 1.07 1.06 1.02 0.89Na+ 1.38 1.24 1.07 1.06 1.09 1.13 0.97K+ 1.35 1.28 1.08 1.10 1.07 1.14 0.98Rb+ 1.29 1.26 1.08 1.07 1.0 1.14 0.99Cs+ 1.19 1.18 1.07 1.08 1.07 1.15 0.99

the accuracy of the Dalgarno formula (4.19) for the mobility of ions in a foreigngas. Indeed, Table 4.8 contains the ratio of experimental mobilities of alkali ionsin inert gases and nitrogen at room temperature [197–202] to those calculated byformula (4.19). One can see that experimental data exceeds theoretical data and thestatistical treatment gives the average ratio 1.15 ± 0.10. From this one can obtainthat the Dalgarno formula (4.19) may be used for a rough evaluation of the ionmobility of ions. One can correct the Dalgarno formula by taking into account theexperimental data, and then we have

K = 41 ± 4√αμ

cm2

V · s, (4.20)

and the indicated accuracy (∼10%) is the statistical one. The real divergence fromformula (4.20) may be more. In particular, the ratio of the mobility of alkali metalions in helium to that given by formula (4.20) is equal to 1.13 ± 0.06. Thus, onecan suggest formula (4.20) for a rough evaluation of the ion mobility in gases. Onthe basis of the Einstein relation and formula (4.20), we have for the ion diffusioncoefficient in a gas

D = 1.0 ± 0.1√αμ

cm2

s. (4.21)

When atomic ions are moving in a parent atomic gas in an electric field, thescattering of ions has a specific character according to the Sena effect [113, 114],as shown in Fig.4.1. Then the colliding ion and atom move along straightforwardtrajectories, but charge exchange changes the trajectory of a charged particle so thatelectron transfer to another atomic core leads to effective ion scattering. This takesplace, in particular, at room gas temperature, when the elastic cross section of ion–atom scattering is less than the cross section of resonant charge exchange. Thenthe cross section of resonant charge exchange determines the mobility of ions in aparent gas.

Indeed, in the absence of elastic ion–atom scattering in the center-of-mass frameof reference, the ion scattering angle is ϑ = π and the diffusion cross section of ionscattering in the absence of elastic scattering is [203]

σ ∗ =∫

(1 − cos ϑ) dσ = 2σres, (4.22)


Fig. 4.1. The Sena effect. Atom and ion are moving along straightforward trajectories andresonant charge exchange takes place, so that a valence electron transfers to another atomiccore. This leads to effective ion scattering

where σres is the cross section of the resonant charge exchange process. Then, as-suming the resonant transfer cross section σres to be independent of the collisionvelocity, we obtain from formula (4.18) for the ion mobility in a parent gas in thefirst Chapman–Enskog approach [204]

KI = 0.331e

N√

T maσres(2.2vT )(4.23)

where the typical velocity is vT = √2T/ma, and the argument of the cross section

of resonant charge exchange σres indicates a velocity at which this cross sectionis taken. The second Chapman–Enskog approach [205] exceeds the ion mobility by1/40 and gives for the ion mobility in a parent gas in the absence of elastic scattering[166, 206]

K = 0.341e

N√

T maσres(2.1vT ). (4.24)

Reducing the mobility to the normal number density of atoms N = 2.69 ×1019 cm−3, we rewrite formula (4.24) to the form [166, 206]

K = 1340√T maσres(2.1vT )

cm2

V · s, (4.25)

where the temperature T is given in Kelvin, the atom and ion mass ma = M areexpressed in units of atomic mass and the resonant charge exchange cross section isgiven in 10−15 cm2; the argument indicates the collision velocity that correspondsto the collision energy 4.5 T in the laboratory frame of reference. Diagram Pt17contains the values of the mobilities of atomic ions in parent gases at temperaturesT = 300 K, 800 K. The accuracy of this formula is approximately 10%.

Accounting for elastic ion–atom scattering due to the polarization ion–atom in-teraction, we obtain for the ion mobility in a parent gas at a zero electric field [166]


Pt17. Mobilities of atomic ions in parent gas


Table 4.9. The contribution of elastic ion–atom scattering to the mobility of atomic ions in aparent gas. K is the mobility without elastic scattering, �K is a decrease of the mobility dueto elastic ion–atom scattering [166]

Ion, gas �K/K , % Ion, gas �K/K , %He 5.8 Li 12Ne 7.8 Na 8.2Ar 11 K 8.9Kr 6.0 Rb 7.7Xe 9.2 Cs 7.5

K = K0

1 + x + x2, x =

√αe2/T

σres(√

7.5T/M)(4.26)

where K0 is the mobility in ignoring elastic scattering according to formula (4.24),and α is the atom polarizability. This formula accounts for the transition to the limitsof low and high temperatures. The contribution to the ion mobility in a parent gas ofelastic ion–atom scattering due to the polarization interaction is given in Table 4.9for inert gases and alkali metal vapors [166]. Note that taking elastic scattering intoaccount leads to a decrease the ion mobility.

The diffusion coefficient of ions in a parent gas in zero field follows from theEinstein relation (4.17) and formula (4.23) is given by

D = 0.12

√T

M

1

Nσres(√

9T/M), (4.27)

where the diffusion coefficient is expressed in cm2/s and others are explained informula (4.23).

In the case of large electric fields

eEλ T , (4.28)

the velocity distribution function of ions in the electric field direction x has thefollowing form if we ignore elastic ion–atom scattering and assume the cross sectionof resonant charge exchange to be independent of the collision velocity

f (vx) = C exp

(− Mv2

x

2eEλ

), vx > 0. (4.29)

Here C is the normalization constant, and this formula gives for the ion drift velocitywi and the average ion energy [113]

wi = vx =√

2eEλ

πM, ε = Mv2

x

2= eEλ

2. (4.30)

Accounting for the velocity dependence for the cross section of resonant chargeexchange, it is taken in formula (4.30) at the ion velocity 1.4

√eEλ/M in the labo-

ratory frame of reference (an atom is motionless). The diffusion ion coefficients in


the field direction D‖ and in the perpendicular direction to it D⊥ are given by [16]

D‖ = 0.137λw, D⊥ = T λ

Mw,

D‖D⊥

= 0.137Mw2

T= 0.087

eEλ

T. (4.31)

The ion drift velocity is given by formulas (4.15) and (4.28) in the limit oflow electric field strengths and by formula (4.30) in the limit of large electric fieldstrengths. One can construct the ion drift velocity in an intermediate range of electricfields. Indeed, let us introduce the parameter

β = eE

T Nσres, (4.32)

and in the first approximation assume the cross section of resonant charge exchangeto be independent of the collision velocity. Let us represent the ion drift velocity inthe form

w =√

2T

mΦ(β), (4.33)

and, according to formulas (4.23) and (4.30), the limiting dependencies for the func-tion Φ(β) have the form

Φ(β) = 0.48β, β � 1; Φ(β) =√

β

π, β 1.

As follows from the solution of the kinetic equation [207, 208] for the ion distribu-tion function on velocities, the reduced ion drift velocity Φ(β) may be approximatedby the dependence [100, 166]

Φ(β) = 0.48β

(1 + 0.22β3/2)1/3. (4.34)

The dependence Φ(β) for intermediate values β is given in Fig. 4.2.

Fig. 4.2. The dependence of the reduced drift velocity of atomic ions in a parent gas Φ onthe reduced electric field strength β (solid curve) in accordance with formula (4.34). Dottedcurves correspond to limiting cases


Tabl

e4.

10.C

onve

rsio

npa

ram

eter

sfo

rtr

ansp

ortc

oeffi

cien

ts

Num

ber

Form

ula

Coe

ffici

entC

Uni

ts

1.D

=C

KT

8.61

7×

10−5

cm2/s

Kin

cm2/(V

s),T

inK

1cm

2/s

Kin

cm2/(V

s),T

ineV

2.K

=C

D/T

1160

4cm

2/(V

s)D

incm

2/s,

Tin

K1

cm2/(V

s),

Din

cm2/s,

Tin

eV3.

D=

C√ T

/μ

/(N

σ1)

4.61

7·1

021cm

2/s

σ1

inÅ

2,N

incm

−3,T

inK

,μin

a.u.

m.

1.59

5cm

2/s

σ1

inÅ

2,N

=2.

687

×10

19cm

−3,T

inK

,μin

a.u.

m.

171.

8cm

2/s

σ1

inÅ

2,N

=2.

687

×10

19cm

−3,T

ineV

μin

a.u.

m.

68.1

115

cm2/s

σ1

inÅ

2,N

=2.

687

×10

19cm

−3,T

inK

,μin

e.u.

m.

7338

cm2/s

σ1

inÅ

2,N

=2.

687

×10

19cm

−3,T

ineV

,μin

e.u.

m.

4.K

=C

(√ Tμ

Nσ

1)−

11.

851

×10

4cm

2/(V

s)σ

1in

Å2,N

=2.

687

×10

19cm

−3,T

inK

,μin

a.u.

m.

171.

8cm

2/(V

s)σ

1in

Å2,N

=2.

687

×10

19cm

−3,T

ineV

,μin

a.u.

m.

7.90

4×

105

cm2/(V

s)σ

1in

Å2,N

=2.

687

×10

19cm

−3,T

inK

,μin

e.u.

m.

7338

cm2/(V

s)σ

1in

Å2,N

=2.

687

×10

19cm

−3,T

ineV

,μin

e.u.

m.

5.κ

=C

√ T/m

/σ

21.

743

×10

4W

/(c

mK

)T

inK

,min

a.u.

m.,

σ2

inÅ

2

7.44

3×

105

W/(c

mK

)T

inK

,min

e.u.

m.,

σ2

inÅ

2

6.η

=C

√ Tm

/σ

25.

591

×10

−5g/

(cm

s)T

inK

,min

a.u.

m.,

σ2

inÅ

2

7.ξ

=C

E/(T

Nσ)

1.16

0×

1020

Ein

V/cm

,Tin

K,σ

inÅ

2,N

incm

−31

×10

16E

inV

/cm

,Tin

eV,σ

inÅ

2,N

incm

−3

4.4 Electron Drift in Gas in Electric Field 111

4.3 Conversion Parameters for Transport Coefficients

Explanation of Table 4.10.

1. The Einstein relation (4.17) for the diffusion coefficient of a charged particle ina gas D = KT/e, where D, K are the diffusion coefficient and mobility of acharged particle, and T is the gas temperature.

2. The Einstein relation (4.17) for the mobility of a charged particle in a gasK = eD/T .

3. The diffusion coefficient of an atomic particle in a gas in the first Chapman–Enskog approximation D = 3

√2πT/μ/(16Nσ1), where T is the gas tempera-

ture, N is the number density of gas atoms or molecules, μ is the reduced massof a colliding particle and gas atom or molecule, and σ1 is the average crosssection of collision.

4. The mobility of a charged particle in a gas in the first Chapman–Enskog ap-proximation K = 3e

√2π/(T μ)/(16Nσ1); notations are the same as above.

5. The gas thermal conductivity in the first Chapman–Enskog approximationκ = 25

√πT /(32

√mσ2), where m is the atom or molecule mass, and σ2 is

the average cross section of collision between gas atoms or molecules; othernotations are the same as above.

6. The gas viscosity in the first Chapman–Enskog approximation η = 5√

πT m/

(24σ2); notations are the same as above.7. The parameter of ion drift in a gas in a constant electric field ξ = eE/(T Nσ),

where E is the electric field strength, T is the gas temperature, N is the numberdensity of atoms or molecules, and σ is the cross section of collision.

4.4 Electron Drift in Gas in Electric Field

At low and moderate electric field strengths, electron drift in a gas under the actionof an external electric field is determined by electron–atom elastic scattering. Thecross section of electron–atom scattering at energies up to several eV is determinedby a finite number of collision momenta, and hence this process has a quantumcharacter. A sequent of this is the Ramsauer effect for electron scattering on atomswith a negative electron scattering length. The parameters of this effect for inertgas atoms Ar, Kr, Xe are given in Tables 3.1 and 3.2. Table 4.11 contains reducedzero-field mobilities of electrons in inert gases at room temperature KeN (Ke is

Table 4.11. Parameters of electron drift in inert gas atoms at zero field and room temperature[163]

Gas He Ne Ar Kr Xe

DeNa,1021 cm−1 s−1 7.2 72 30 1.6 0.43KeNa, 1023 (cm s V)−1 2.9 29 12 0.63 0.17


the electron mobility, N is the number density of gas atoms) and the values of thereduced diffusion coefficients of electrons DeN (De is the diffusion coefficient ofelectrons), which are connected with the electron mobilities by the Einstein relation(4.17).

Being moved in an atomic gas in an external electric field, an electron obtainsenergy from the field and transfers it to atoms in elastic collisions until the electronenergy does not allow it to excite atoms. In each strong collision, an electron isscattered on a large angle, so that its momentum changes significantly, whereas itsenergy changes weakly. Therefore, the electron distribution function of velocities isclose to a spherically symmetric one. Hence, the electron drift velocity w and thediffusion coefficient for an electron in a gas D⊥ in the direction perpendicular to thefield are expressed through the spherical distribution function of electrons, and thisconnection has the form [209]

w = eE

3me

⟨1

v2

d

dv

(v3

ν

)⟩, D⊥ = 1

3

⟨(v2

ν

)⟩. (4.35)

Here brackets mean an average with the spherical distribution function of electronvelocities and the rate of electron elastic scattering in a gas is ν = Nvσ ∗(v), whereN is the number density of atoms, and v is the electron velocity. In the simplest case,when the rate of electron elastic scattering ν is independent of the electron velocityv, these formulas take the form

w = eE

meν, D⊥ = 〈v2〉

ν. (4.36)

The data of Table 4.12 for the electron drift velocity in some atomic and mole-cular gases in an external electric field show that this case is not realistic. Notethat the drift velocity depends on the reduced electric field strength E/N , whereE is the electric field strength and the unit for this quantity is Townsend (Td)(1 Td = 10−17 V cm2).

Table 4.12. The electron drift velocity w in gases, expressed in 105 cm/s [163]

E/N , Td He Ne Ar Kr Xe H2 N20.001 0.028 0.35 0.15 0.015 0.004 0.020 –0.003 0.080 0.55 0.53 0.038 0.013 0.052 –0.01 0.25 0.89 1.0 0.13 0.040 0.17 0.390.03 0.64 1.3 1.2 0.90 0.16 0.46 1.00.1 1.4 2.1 1.7 1.4 0.86 1.4 2.30.3 2.7 3.5 2.4 – 1.1 3.2 3.11 4.8 – 3.1 – 1.4 6.2 4.53 8.6 – 4.0 – 1.8 10 7.7

10 22 – 11 – 6.0 19 1830 72 – 26 – 18 37 42

100 – – 74 – 53 130 100

4.5 Diffusion of Excited Atoms in Gases 113

This character of electron processes when an electron takes energy from an ex-ternal electric field and then transfers it to atoms as a result of elastic and inelasticcollisions, takes place in a gas discharge plasma. A gas discharge plasma resultsfrom passage of electric current through a gas under the action of an external elec-tric field. Collision of electrons with atoms in a gas discharge plasma establishesa self-maintaining equilibrium inside it.

4.5 Diffusion of Excited Atoms in Gases

Table 4.13 contains the values of the diffusion coefficients for metastable atoms andmolecules in a parent gas at room temperature.

Table 4.13. Diffusion coefficients of metastable atoms or molecules in parent gases at roomtemperature reduced to the normal number density [30]

Metastable atom, molecule D, cm2/s

He(23S) 0.59He(21S) 0.52He(23Σu) 0.45Ne(3P2) 0.20Ne(3P0) 0.20Ar(3P2) 0.067Ar(3P0) 0.071Kr(3P2) 0.039Kr(3P0) 0.043Xe(3P2) 0.024Xe(3P0) 0.020N(2D) 0.24N(2D) 0.19N2(A3Σu) 0.20O2(1Δg) 0.19

5 Properties of Macroscopic Atomic Systems

5.1 Equation of State for Gases and Vapors

The equation of state for an ideal gas has the form

p = NT, (5.1)

where p is the gas pressure, T is the temperature, and N is the number density ofatoms or molecules. Conversion factors for this equation are given in Table 5.1. Thisequation may be represented in the form [210]

pV = nRT, (5.2)

where n is a gas amount expressed in moles, V is the gas volume per mole, and thegaseous constant R is equal to

R = 82.06cm3

mol

atm

K= 8.314

cm3

mol

MPa

K.

Equation (5.2) describes a rare gas when an average distance between neighbor-ing atoms or molecules N−1/3 significantly exceeds a distance R0 of a strong inter-action between gas atoms or molecules U(R0) ∼ T , where U(R) is the interactionpotential of two atoms at a distance R, and T is a thermal energy. A subsequentexpansion over a small parameter N1/3R0 may lead to the van der Waals equation

(p + a

V 2

)(V − b) = T (5.3)

using two van der Waals constants, a and b. This equation may be spread to the crit-ical point and the liquid state of the system of atoms. The lower the number density

Table 5.1. Conversion factors for the equation of gas state

Number Formula The proportionality factor C Units

1. N = Cp/T 7.339 × 1021 cm−3 p in atm, T in K9.657 × 1018 cm−3 p in Torr, T in K

2. p = CNT 1.036 × 10−19 Torr N in cm−3, T in K

116 5 Properties of Macroscopic Atomic Systems

of atoms is, the better the van der Waals equation. If we use this equation at the criti-cal point, the critical parameters [211] of this system of atoms are expressed throughthe van der Waals parameters. The critical point of an atom system is determined bythe equations (

∂p

∂V

)

T

= 0,

(∂2p

∂V 2

)

T

= 0, (5.4)

which allows us to express the critical parameters Vcr, pcr and Tcr through the para-meters of the van der Waals equation (5.3) in the following manner

Vcr = 3b, pcr = a

27b2, Tcr = 8a

27b. (5.5)

In particular, from this we obtain the relation between critical parameters within theframework of the van der Waals equation

ζ = Tcr

Vcrpcr= 8

3. (5.6)

The accuracy of the van der Waals equation decreases with an increase of the gasdensity. Its accuracy is about 30% at the critical point, as follows from comparisonof the data of Table 5.2 with formulas (5.5) and (5.6), where parameters of the vander Waals equation are used. In particular, the combination of critical parameters ζ

defined according to formula (5.6) is equal for inert gases 3.38 ± 0.13 and for mole-cular gases 3.49 ± 0.15 according to the data of Table 5.2. These values coincidewithin the limits of their accuracy, but differ from that of formula (5.6) based on thevan der Waals equation.

Table 5.3 contains critical parameters for molecular systems consisting of mole-cules AF6. These molecules are almost round, and therefore interaction between

Table 5.2. Parameters of van der Waals equation and critical parameters of inert and molec-ular gases [2, 9, 10]

a

(105 MPa cm6/mol2)b

(cm3/mol)Tcr(K)

pcr(MPa)

Vcr(cm3/mol)

ζ = TcrpcrVcr

H2 0.245 26.5 32.97 1.293 65 3.26He 0.0346 23.8 5.19 0.227 57 3.33N2 1.370 38.7 126.2 3.39 90 3.44O2 1.382 31.9 154.6 5.043 73 3.49F2 1.171 29 144.1 5.172 66 3.51Ne 0.208 16.7 44.4 2.76 42 3.18Cl2 6.343 54.2 416.9 7.99 123 3.52Ar 1.355 32 150.9 4.90 75 3.41Br2 9.75 59.1 588 10.3 127 3.74Kr 5.19 10.6 209.4 5.50 91 3.48J2 – – 819 (13) 155 –Xe 4.19 51.6 289.8 5.84 118 3.50Rn – – 377 6.28 (150) –

5.2 Basic Properties of Ionized Gas 117

Table 5.3. The critical parameters for “round” molecules AF6 and the boiling point for con-densed systems of these molecules at atmospheric pressure [212]

A Tb, K Tcr, K pcr, MPa Vcr, cm3/mol ζ = Tcr/(pcrVcr)

S 209.4 318.7 3.77 199 3.53Mo 307.2 473 4.75 226 3.66W 290.3 444 4.34 233 3.65U 329.6 505.8 4.66 250 3.61

Table 5.4. Critical parameters for some vapors and their boiling points [2, 9, 10]

Tb, K Tcr, K pcr, MPa Vcr, cm3/molLi 1615 3223 67 66Na 1156 2573 35 116P 554 994 – –S 717.8 1314 20.7 –K 119.9 209.4 5.50 91As 876 1673 – 35Se 958 1766 27.2 –Rb 961 2093 16 247Cs 944 1938 9.4 341Hg 629.9 1750 172 43

them is similar to interaction between atoms of inert gases. Correspondingly, thecombination of critical parameters ζ defined by formula (5.6) is 3.61 ± 0.06. Thisvalue is close to that of the inert gases and molecular gases of Table 5.2, but differsfrom that followed from van der Waals equation according to formula (5.6).

Table 5.4 contains critical parameters for some vapors. In this case, atoms canform bound atomic systems with strong interaction between atoms in which atomshave lost their individuality. Interaction between atoms of these vapors differs fromthat in the case of inert gases and similar systems. Therefore, the behavior of thesesystems at the critical point is also different. In particular, according to the data ofTable 5.4, the parameter ζ = Tcr

Vcrpcrfor alkali metal vapors is in average ζ = 5 ± 1,

which differs from that of inert gases.

5.2 Basic Properties of Ionized Gas

An ionized gas under consideration is an ideal plasma where Coulomb interactionof electrons and ions satisfies the gaseousness condition, that has the form

α ≡ Nee6

T 3e

� 1, (5.7)

where Ne is the number density of electrons, and Te is the electron temperatureexpressed in energetic units. According to this criterion, the interaction potential


Table 5.5. Conversion factors for plasma physics

Number Formula Proportionalityfactor C

Units

1. α = CNe/T 3 2.998 × 10−21 Ne in cm−3, T in eV4.685 × 10−9 Ne in cm−3, T in K

2. f = Cm3/2T 3/2 2.415 × 1015 cm−3 T in K, m in e.m.u.3.019 × 1021 cm−3 T in K, m in e.m.u.1.879 × 1020 cm−3 T in K, m in a.m.u.2.349 × 1026 cm−3 T in eV, m in a.m.u.

3. K = C/T9/2e 2.406 × 10−31 cm6/s Te in 1000 K

4. ωp = C√

Ne/m 5.642 × 104 s−1 Ne in cm−3, m in e.m.u.1322 s−1 Ne in cm−3, m in a.m.u.

5. rD = C√

T/Ne 525.3 cm Ne in cm−3, T in eV4.876 cm Ne in cm−3, T in K

6. i = CU3/2m−1/2l−2 5.447 × 10−8 A/cm2 m in a.m.u., l in cm, U in V2.326 × 10−6 A/cm2 m in e.m.u., l in cm, U in V

7. U = Cm1/3l4/3i−2/3 69594 B m in a.m.u., l in cm, i in A/cm2

5697 V m in e.m.u., l in cm, i in A/cm2

8. l = Cm−1/4U3/4i−1/2 2.3338 × 10−4 cm m in a.m.u., U in V, i in A/cm2

1.525 × 10−3 cm m in e.m.u., U in V, i in A/cm2

Explanations:

1. The ideality plasma parameter (5.7) α = Nee6/T 3

e2. Preexponent of the Saha formula (5.9) ξ = [mT/(2πh2)]3/2

3. The rate constant of three-body electron–ion recombination according to formulas (3.39)

and (3.40) K = 1.5 e10/(m1/2e T

9/2e )

4. The plasma frequency (5.14) ωp =√

4πNee2/me

5. The Debye–Hückel radius (5.13) rD =√

T/(8πNee2)

6. The three-halves law (5.16) i = 29π

√e

2mU3/2

l2, where l is the distance between elec-

trodes, U is the difference of electrode voltages, i is the maximum current density, e, m

is the particle charge and mass7. The three-halves law in the form U = ( 9πi

2 )2/3( 2me )1/3l4/3

8. The three-halves law in the form l = ( 29π

)1/2( e2m

)1/4 U3/4

i1/2

of two charged particles (electrons and ions) at an average distance between theme2N

1/3e is small compared to the electron thermal energy Te. Table 5.5 contains the

conversion factor for the ideality parameter.Let us consider an ionization equilibrium in a weakly ionized gas that is estab-

lished as a result of atom ionization by electron impact and three-body electron–ionrecombination according to the scheme

e + A ↔ 2e + A+. (5.8)

Then the relation between the number density of electrons Ne, ions Ni and atomsNa is given by the Saha formula [14, 213]

5.2 Basic Properties of Ionized Gas 119

NeNi

Na= gegi

ga

(meTe

2πh2

)3/2

exp

(− J

Te

). (5.9)

Here ge, gi, ga are statistical weights of electron, ion and atom, respectively; me isthe electron mass, Te is the electron temperature, and J is the atom ionization po-tential.

We define below two basic parameters of an ionized gas that are determinedby a long-range Coulomb interaction between charged plasma particles (electronsand ions). Since this interaction remains in the presence of neutral particles, theseproperties become apparent also in a weakly ionized gas. If a weak electric field isintroduced in a plasma, it causes displacement of electrons and ions, which leadsto the screening of this field. Correspondingly, the electric field strength E on adistance x from the plasma boundary is given by

E = E0 exp

(− x

rD

), (5.10)

where E0 is the electric field strength on the plasma boundary, and rD is the Debye–Hückel radius. In the same manner, the interaction potential of two particles of acharge e at a distance r between them equals

U(r) = e2

rexp

(− r

rD

), (5.11)

and the Debye–Hückel radius is given by formula [214]

rD =[

4πNee2(

1

Te+ 1

Ti

)]−1/2

. (5.12)

Here Te is the electron temperature, Ti is the ion temperature, Ne is the numberdensity of electrons or ions (a plasma is quasi-neutral). In the case of an equilibriumplasma Te = Ti = T , this formula takes the form

rD =√

T

8πNee2, (5.13)

where Te = Ti = T . As is seen, the Debye–Hückel radius rD is an important plasmaparameter.

Take a plasma that occupies an infinite space and displace the plasma electronsat some distance. Then an electric field arises that tends to return the electrons in theinitial positions. Because of inertia of electrons, their return to the initial positionshas the vibration character and the oscillation frequency ωp, the frequency of plasmaoscillations is equal to [215]

ωp =√

4πNee2

me, (5.14)


where me is the electron mass. The reciprocal value 1/ωp is a typical time of plasmareaction on the action of external fields.

The above parameters, the Debye–Hückel radius rD and plasma frequency ωp,are basic parameters that characterize collective plasma properties. The presenceof atoms and molecules in a plasma does not change collective plasma propertiesthat are determined by Coulomb interaction of charged particles because neutralparticles do not influence this interaction. Note that the ratio of the above parametersrD/ωp equals the thermal velocity of electrons, i.e.

rD

ωp=

√2T

me. (5.15)

If a current of charged particles of a density i flows between two plane electrodeswith a distance l between them, and the voltage difference is U , the relation betweenthese parameters is given by the three-halves law [216, 217]

i = 2

9π

√e

2m

U3/2

l2, (5.16)

where e is a particle charge and m is the mass of a charged particle. Conversionfactors for this formula are given in Table 5.5.

5.3 Parameters and Rates of Processes Involving Nanoparticles

A cluster is a system of identical bound atoms. We model below clusters or nanopar-ticles consisting of many atoms in a liquid drop. This liquid drop has a sphericalshape and its density coincides with that of a macroscopic system. The radius r of aliquid drop containing n atoms is given by

r = rWn1/3, (5.17)

where rW is the Wigner–Seits radius defined by the formula

rW =(

3m

4πρ

)1/3

. (5.18)

Here m is the atom mass, ρ is the density of bulk liquid.Within the framework of the liquid drop model, the cross section of atom attach-

ment to a cluster equals πr2ξ , where r is the cluster radius that significantly exceedsthe radius of action of atomic forces, and ξ is the probability of atom attachment tothe cluster after their contact. In this case the rate of cluster growth in its vapor isequal to [13]

dn

dt= πr2

√T

2πmNξ = k0ξn2/3N, k0 =

√T

2πmπr2

W. (5.19)

5.3 Parameters and Rates of Processes Involving Nanoparticles 121

Table 5.6. Conversion factors for parameters and processes involving clusters or nanoparti-cles

Number Formula Proportionalityfactor C

Units

1. rW = Cm/ρ1/3 0.7346 Å m in a.u.m., ρ in g/cm3

n = C(r/rW)3 4.189 r and rW in Å2. k0 = Cr2

W√

T/m 4.5714 × 10−12 cm3/s rW in Å, T in K,m in a.u.m.

3. v = Cρr2/η 0.2179 cm/s r in µm, ρ in g/cm3,η in 10−5g/(cm s)

0.01178 cm/s r in µm, ρ in g/cm3

v = Cr2/η η for air at p = 1 atm,T = 300 K

4. D0 = C√

T/m/(Nr2W) 1.469 × 1021 cm2/s rW in Å, N in cm−3,

T in K, m in a.u.m.0.508 cm2/s rW in Å,

N = 2.687 × 1019 cm−3,T in K, m in a.u.m.

54.69 cm2/s rW in Å,N = 2.687 × 1019 cm−3,T in eV, m in a.u.m.

5. K0 = C(√

T mNr2W)−1 1.364 × 1019 cm2/(V s) rW in Å, N in cm−3,

T in K, m in a.u.m.0.508 cm2/(V s) rW in Å,

N = 2.687 × 1019 cm−3,T in K, m in a.u.m.

54.69 cm2/(V s) rW in Å,N = 2.687 × 1019 cm−3,T in K, m in a.u.m.

Explanation:

1. The Wigner–Seits radius according to formula (5.18)2. The reduced rate constant for atom attachment to a cluster according to formula (5.19)3. The free fall velocity for a cluster of radius r that is given by formula v = 2ρgr2/(9η),

where g is the free fall acceleration, ρ is the cluster density, η is the viscosity of a matterthrough which a cluster moves

4. The diffusion coefficient for a spherical cluster consisting of n atomsDn = D0/n2/3, where in the first Chapman–Enskog approximation (4.5)D0 = 3

√2T/πm/(16Nr2

W), so that T is the gas temperature, N is the numberdensity of gas atoms, m is the mass of a cluster atom, rW is the Wigner–Seits radius

5. The zero-field mobility for a spherical cluster consisting of n atoms Kn =K0/n2/3, where in the first Chapman–Enskog approximation (4.18) K0 =3e/(8Nr2

W

√2πmT ), and notations are given above


Here N is the number density of atoms, T is the gas temperature, m is the atommass, and n is the number of cluster atoms. Note that the vapor pressure significantlyexceeds the saturated vapor pressure at a given temperature and the vapor may belocated in a buffer gas with a higher density. The values of the parameters rW andk0, the reduced rate constant of cluster growth, according to formulas (5.18) and(5.19) are given in diagram Pt18 for some metals and diagram Pt19 for lanthanidestaken from [13]. The melting and boiling points are given for macroscopic metals.

Table 5.7 gives the parameters of molecular liquids consisting of diatomic mole-cules. Then the Wigner–Seits radius rW and the reduced rate constant k0 relate toa molecule instead of an atom at room temperature T = 300 K. Here Tm, Tb arethe bulk melting and boiling points, ρliq is the density of a macroscopic liquid,the Wigner–Seits radius rW and the reduced rate constant k0 are given by formulas(5.18) and (5.19)

The total binding energy En of cluster atoms is given by [218]

En = ε0n − An2/3. (5.20)

The first term is the volume cluster energy and the second one is the surface clusterenergy, so that ε0 is the binding energy per atom for a bulk atom system, and A isthe specific surface cluster energy. The values of these energy parameters are givenin diagram Pt20. From this we have for the energy εn that is spent on one atomliberation

εn = −dEn

dn= ε0 − 2A

3n1/3. (5.21)

Values of the specific surface energy A in diagram Pt20 are expressed through thesurface tension of liquids which are taken from [2, 219, 220].

Let us include in the balance equation (5.18) cluster evaporation if the clustertemperature is T . Within the framework of the liquid drop model for a cluster on thebasis of the principle of detailed balance, we have the following equation of clustergrowth instead of (5.18)

dn

dt= k0ξ

[N − Nsat(T ) exp

(2A

3T n1/3

)], (5.22)

Table 5.7. Parameters of liquids and liquid drops consisting of diatomic molecules [2, 9, 10]

Element Tm, K Tb, K ρliq, g/cm3 rW, Å k0, 10−11 cm3/sH2 14.01 20.28 0.071 2.29 28N2 63.3 77.4 0.88 2.33 8.1O2 54.8 90.2 1.14 2.23 7.0F2 53.5 85.0 1.52 2.15 5.9Cl2 172.2 239.2 1.51 2.65 6.6Br2 265.9 331.9 3.13 (293 K) 2.72 4.6I2 386.7 457.5 (4.9) 2.73 3.7


Pt1

8.Pa

ram

eter

sof

the

liqui

ddr

opm

odel

for

liqui

dm

etal

clus

ters


Pt1

9.L

anth

anid

es


Pt2

0.E

vapo

ratio

npa

ram

eter

sof

met

alliq

uid

clus

ters


Table 5.8. Parameters of the pair interaction of inert gas atoms and liquid clusters of inertgases

Ne Ar Kr XeRe, Å 3.09 3.76 4.01 4.36D, meV 3.64 12.3 17.3 24.4Tm, K 24.6 83.7 115.8 161.2Tb, K 27.1 87.3 119.8 165.0a, Å 3.156 3.755 3.892 4.335εsub, meV 20 80 116 164A, meV 15.3 53 73 97p0, 104 atm 0.34 1.15 1.14 1.28ε0, meV 18.6 68 95 132�Hfus, meV 3.42 12.3 17.0 23.7

where Nsat is the saturated number density of atoms over a plane surface. The satu-ration vapor pressure psat over a plane surface is given by

psat = p0 · exp

(−ε0

T

). (5.23)

Diagram Pt20 contains parameters of this formula for metal clusters.If the interaction of bound atoms in clusters is small compared to the ionization

potential, atoms conserve their individuality in clusters. Table 5.8 contains the pa-rameters of pair interaction of inert gas atoms: Re, the equilibrium distance for adiatomic molecule, and D, the depth of the pair interaction potential, which werefound in [221–224] on the basis of various measurements. In addition, in Table 5.8,ε0, A are the parameters of formula (5.20), and p0 is the parameter of formula (5.23)for liquid at the triple point. Next, Tm is the melting point for the triple point, Tb isthe boiling point, a is the distance between nearest neighbors of the crystal at zerotemperature, εsub is the binding energy per atom for the crystal, and �Hfus is thefusion energy per atom. The parameters of condensed inert gases are taken from[2, 6, 8, 9].

5.4 Parameters of Condensed Atomic Systems

Among various properties of macroscopic condensed systems of atoms [225, 226],we consider the simplest systems with a weak interaction between bound atomswhen atoms conserve their individuality in bound systems of atoms, i.e. parametersof macroscopic solids and liquids consisting of these atoms are determined by thepair interaction potential of atoms. A simpler description of a condensed system ofatoms takes place when interaction between nearest neighbors dominates in thesesystems. This allows us to construct any dimensional parameter of this condensedsystem of bound atoms through three dimensional parameters: the atom mass m andtwo parameters of the pair interaction potential of atoms, the equilibrium distance Re

5.4 Parameters of Condensed Atomic Systems 127

Table 5.9. Reduced parameters of condensed inert gases [212]

Ne Ar Kr Xe Averagea/Re 1.028 1.000 0.996 0.993 1.004 ± 0.014Tm/D 0.583 0.585 0.576 0.570 0.578 ± 0.006εsub/D 5.50 6.49 6.66 6.71 6.35 ± 0.50A/D 4.20 4.31 4.22 3.98 4.2 ± 0.1ε0/D 5.1 5.5 5.5 5.4 5.4 ± 0.2�Hfus/D 0.955 0.990 0.980 0.977 0.98 ± 0.01rW/Re 0.601 0.594 0.595 0.589 0.595 ± 0.003p0R3

e /D 17 31 27 28 26 ± 6

between atoms that corresponds to the interaction potential minimum, and the min-imum potential energy −D. If these conditions are fulfilled, the scaling law takesplace and, because of a short-range character of atom interaction in a condensedsystem of bound atoms, the distance between nearest neighbors a in this condensedsystem is a = Re and the binding energy per atom is εsub = 6D [15, 212, 227].

These conditions are fulfilled more or less for condensed inert gases that leadto identical reduced values of the same quantity. This fact is demonstrated in Ta-ble 5.9 for condensed inert gases and allows us to find parameters of one inert gasthrough the parameters of another inert gas with an accuracy better than 10%. Say,for instance, to find parameters of condensed radon. The same is valid for condensedidentical molecular systems with a weak interaction (for example, AF4, AF6).

We now consider condensed systems consisting of identical atoms. Diagram Pt21gives the thermodynamic parameters of condensed systems of elements. In this di-agram, the evaporation enthalpy �Hev corresponds to a release of atoms (or mole-cules) from the surface of a macroscopic system, while the enthalpy of conversionof a condensed system in an atomic gas or vapor �Hgas respects to separationof a condensed system of bound atoms in free atoms. Therefore, one can expect�Hev < �Hgas. The aggregate state of a substance for a given element at the tem-perature 298 K is represented also in diagram Pt21, as well as the entropy of theatomic vapor state of this element at the indicated temperature.

Diagram Pt22 contains parameters of the crystal structure of a given element atzero temperature. Carbon is an exclusion and has practically two stable structures atlow temperatures. Though, formally at normal conditions T = 298 K, p = 0.1 MPa,graphite is a more stable structure, the enthalpy of transition graphite–diamond(1.9 kJ/mol) is small compared to the graphite evaporation enthalpy (711 kJ/mol)and the enthalpy of graphite conversion into a gas of carbon atoms (717 kJ/mol).Therefore, energetically, one can consider both structures to be equivalent. Valuesof the Debye temperature in diagrams Pt19, Pt22, that is, the parameter of the Debyemodel for vibration excitation of crystal states, are taken from [228].

Metal parameters at room temperature, including the density, conductivity, ther-mal conductivity, and work function, are represented in diagram Pt23. According tothe definition, a metal is a conducting material inside which electrons are presentand are characterized by a large mean free path. According to the diagram Pt23, the


Pt2

1.T

herm

odyn

amic

para

met

ers

ofel

emen

ts


Pt2

2.Pa

ram

eter

sof

crys

tals

truc

ture

sof

elem

ents

atlo

wte

mpe

ratu

res


Pt2

3.M

etal

para

met

ers

atro

omte

mpe

ratu

re


ratio of the mean free path to the Wigner–Seits radius for the crystal lattice of metalsis tens and hundreds. In addition to this, diagram Pt19 contains the parameters ofliquid and solid rare-earth metals–lanthanides.

A deeper understanding of the metals’ nature follows from the analysis of theirproperties on the basis of simple models. Within the framework of a simple model,we represent the system of metal electrons as a degenerate electron gas [225, 226],ignoring interaction of electrons with the crystal lattice and interaction betweenelectrons. This model starts from the Drude model of independent classic electrons[229, 230] and, though the momentum electron distribution in metals is more com-plex in reality [231], the model of a degenerate electron gas describes roughly thebehavior of electrons in metals. Then the parameter of this electron system is thedensity and, at zero temperature, the electron distribution function on momenta cor-responds to the electron location inside a sphere of a radius pF (Fermi momentum).The energy of electrons located on the Fermi-sphere surface εF is connected withthe electron number density Ne by the relation

εF = (3π2Ne)2/3h2

2me. (5.24)

This electron distribution accounts for the Pauli principle for particles with the spin1/2, so that two electrons cannot be located in the same state. At non-zero temper-ature, the electron distribution is close to that of a degenerate electron gas if theparameter

η = T

εF(5.25)

is small. Here T is the electron temperature expressed in energetic units. Table 5.10gives the values of the Fermi energy εF, the electron velocity on the Fermi surfacevF, the Wigner–Seits radius rW, and the parameter η at room temperature for single-valence metals, the atoms of which have one valence s-electron. Table 5.10 containsalso the values of the parameter

ξ = e2

rWεF= 25/3

3πa0N1/3e

= 0.337

a0N1/3e

, (5.26)

that is, the ratio of the Coulomb interaction potential at an average distance betweenthe nearest electrons to the Fermi energy of electrons.

In Table 5.10, there are the reduced mean free paths of electrons λ/a for thesemetals (λ is the mean free path of electrons at room temperature, a is the distancebetween the nearest atoms of this metal), the electron effective mass m∗ and theDebye temperature ΘD for these metals. The electron mobility Ke, the conductiv-ity Σ and the thermal conductivity coefficient κ for these metals are given also inTable 5.10. From these data it follows that metal electrons may be considered as adegenerated electron gas. Indeed, the parameter (5.25) is small, the mean free path λ

is large compared to the lattice constant, and the electron effective mass correspondsto the mass of a free electron.


Pt2

4.V

alen

ceel

ectr

ons

ofm

etal

s


Table 5.10. Parameters of metals for elements of the first group of the periodical systemswith s-valence electrons at room temperature [5]. The crystal lattice has the body-centeredcubic (bcc) structure or the face-centered cubic (fcc) structure (a is the lattice constant, ρ isthe metal density)

Metal Li Na K Cu Rb Ag Cs AuType of lattice bcc bcc bcc fcc bcc fcc bcc fcca, Å 3.51 4.29 5.34 3.61 5.71 4.09 6.09 4.08ρ, g/cm3 0.534 0.97 0.89 8.96 1.53 10.5 1.93 19.3Ne, 1022 cm−3 4.6 2.5 1.4 8.5 11 5.9 8.7 5.9εF, eV 4.7 3.2 2.1 7.1 1.8 5.5 1.6 5.5vF, 108 cm/s 1.3 1.0 0.86 1.6 0.79 1.4 0.74 1.4η, 10−3 5.5 8.2 13 3.7 15 4.6 17 4.6ξ 1.8 2.2 2.7 1.4 2.9 1.6 3.1 1.6λ/a 38 73 54 82 36 100 26 62ΘD, K 370 158 90 310 52 220 54 185m∗/me 1.40 0.98 0.94 1.01 0.87 0.99 0.83 0.99Ke, cm2/(V s) 18 52 58 34 44 53 38 32Σ , 1016 s−1 9.7 19 12 54 7.0 57 4.4 40κ , W/(cm K) 0.85 1.41 1.02 4.01 0.58 4.29 0.36 3.17

In addition to the data in Table 5.10, we give similar data in diagram Pt24 forsome metals according to [2]. The Sommerfeld radius rS is determined from therelation

4πr3SNe

3= 1, (5.27)

where Ne is the number density of electrons. Hence the Sommerfeld radius charac-terizes a size of a region that relates to one electron. If rS is small compared withthe lattice constant, the structure of the metal crystal is not of importance for metalelectron properties. If rS is small compared to the Bohr radius, the metallic plasmais classical. As follows from the data of Table 5.10 and diagram Pt24, the indicatedvalues of these relations are comparable. This means that, though a metallic plasmadepends on the metal crystal lattice and is not classical, these assumptions may beused for the metallic plasma as models.

A Atomic Spectra

Fig

.A.1

.Spe

ctru

mof

hydr

ogen

atom

136 A Atomic Spectra

Fig

.A.2

.Spe

ctru

mof

heliu

mat

om

A Atomic Spectra 137

Fig. A.3. Spectrum of lithium atom


Fig. A.4. Spectrum of beryllium atom


Fig. A.5. Spectrum of boron atom


Fig

.A.6

.Spe

ctru

mof

carb

onat

om


Fig

.A.7

.Spe

ctru

mof

nitr

ogen

atom


Fig

.A.8

.Spe

ctru

mof

oxyg

enat

om


Fig

.A.9

.Spe

ctru

mof

fluor

ine

atom


Fig

.A.1

0.Sp

ectr

umof

neon

atom


Fig. A.11. Spectrum of sodium atom


Fig

.A.1

2.Sp

ectr

umof

mag

nesi

umat

om


Fig. A.13. Spectrum of aluminium atom


Fig

.A.1

4.Sp

ectr

umof

silic

onat

om


Fig

.A.1

5.Sp

ectr

umof

phos

phor

usat

om


Fig

.A.1

6.Sp

ectr

umof

sulf

urat

om


Fig

.A.1

7.Sp

ectr

umof

chlo

rine

atom


Fig. A.18. Spectrum of potassium atom


Fig. A.19. Spectrum of copper atom


Fig

.A.2

0.Sp

ectr

umof

zinc

atom


Fig. A.21. Spectrum of rubidium atom


Fig

.A.2

2.Sp

ectr

umof

stro

ntiu

mat

om


Fig. A.23. Spectrum of silver atom


Fig

.A.2

4.Sp

ectr

umof

cadm

ium

atom


Fig. A.25. Spectrum of caesium atom


Fig

.A.2

6.Sp

ectr

umof

bari

umat

om


Fig

.A.2

7.Sp

ectr

umof

gold

atom


Fig

.A.2

8.Sp

ectr

umof

mer

cury

atom

References

1. L.A. Sena, Units of Physical Units and Their Dimensionalities (Mir, Moscow, 1988)2. D.R. Lide (ed.), Handbook of Chemistry and Physics, 86th edn. (CRC Press, London,

2003–2004)3. http://en.wikipedia.org/wiki/Centimetre-gram-second-system-of-units4. http://physics.nist.gov/cuu/units5. D.E. Gray (ed.), American Institute of Physics Handbook (McGraw-Hill, New York,

1971)6. A.J. Moses, The Practicing Scientists Book (Van Nostrand, New York, 1978)7. R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids, 4th edn.

(McGraw-Hill, New York, 1987)8. V.A. Rabinovich, Thermophysical Properties of Neon, Argon, Krypton and Xenon

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Index

absorption coefficient 40absorption cross section 40abundance of elements 3abundance of isotopes 4affinity to hydrogen atom 65affinity to oxygen atom 65aggregate state of elements 128air plasma, processes 97, 98angle wave function 13amplitude of scattering 73angular atom momentum 11, 17anharmonic constant of diatomic 47,

49–51associative ionization 91asymptotic atom parameters 22atomization enthalpy 128attachment of electron to molecule 95, 96autodetaching state 76, 97autoionizing state 91

balance equation for number density 76binding energy of cluster atoms 124boiling point of metals 122, 123Boltzmann formula 41broadening of spectral lines 39

capture cross section 79CGS system of units 2Chapman–Enskog approximation 100closest approach distance 71conductivity of metals 130conversion factors for units 6correlation between atomic and molecular

states 53covalence chemical bond 46critical parameters 116, 117

cross section of resonant charge exchange40, 81

cross section of collision 71

Dalgarno formula 104Debye–Hückel radius 119Debye temperature 129, 133degenerate electron gas 131density of elements at room temperature

129density of liquid clusters 122density of metals 130differential cross section 71, 72diffusion coefficient 99diffusion coefficient for atomic ion in parent

gas 107–109diffusion coefficient of clusters 122diffusion cross section 72, 73diffusion of electrons in gases 111diffusion of metastable atoms in gases 113dissociation energy for diatomic 49dissociation energy for negative diatomic

ion 51dissociation energy for positive diatomic ion

50dissociative recombination 93, 94Doppler broadening of spectral lines 41

effective principle quantum number 15Einstein coefficient 37Einstein relation 104electromagnetic (CGSM) system of units

2electron affinity for atoms 31electron affinity of diatomic molecules 64electron drift in gases 111electron–ion recombination 93electron mobility in metals 133

172 Index

electron scattering length 47, 59, 73electron term of diatomic molecule 47, 49electron term of excited atom states 32electron term of ground atom state 30, 32electron term for negative diatomic ion 51electron term of negative atomic ion 31electron term for positive diatomic ion 50electron wave function for hydrogen atom

11electrostatic (CGSE) system of units 2enthalpy of bulk atomization 128entropy of atomic vapor state 128equation of gas state 114equilibrium distance for diatomic 48, 49evaporation enthalpy 128eveness of molecular state 53, 54exchange interaction 15, 45excimer molecules 46, 66excitation cross section 77excitation energy of atoms 32

Fermi energy of metal electrons 132Fermi formula 47fine structure splitting 14first Townsend coefficient 87free fall velocity of clusters 121frequency distribution function of photons

39fundamental physical constants 1

gaseous constant 115gas discharge plasma 87, 113gas-kinetic cross section 102Grotrian diagrams 16, 18, 39, 135–162

hard sphere model 72Hartree atomic units 9helium atom 16, 136helium-like ions 17Hund empiric rule 23Hund scheme of momentum coupling in

diatomic molecule 48hydrogen atom 11, 135hydrogen-like ions 11

ideality plasma parameter 117, 118impact parameter of collision 71ion drift velocity in parent gas 103, 108,

109

ion–ion chemical bond 46ionization equilibrium 118ionization potential of atomic ions 30ionization potential of atoms 30ionization potential of diatomic 49isotopes 2

jj -scheme of momentum coupling 24, 25

lanthanides 124Larmor frequency 10lattice constant 129lifetime of resonantly excited atom state

35, 42, 43lifetime of radioactive isotopes 5liquid drop model 120long-range interaction 43Lorenz broadening of spectral lines 41LS-scheme of momentum coupling 23

mean free path 103mean free path of electrons in metalsmelting points of elements 130metal thermal conductivity 130mobility of atomic ions in parent gases

106, 107mobility of charged clusters 121mobility of ions in gases 105mutual neutralization of ions 95

nanoparticles 120

one-electron approximation 15oscillator strength for atom transition 34,

37

parameters of charge exchange cross section82

parentage atom scheme 19parity of molecule state 53, 54Pashen notations 28Pauli exclusion principle 15Penning process 91plasma frequency 119polarizability of atom 36, 45polarizability of molecule 36, 63polarization capture 79polarization interaction 45potential curves of molecules 55–63principle quantum number 11

Index 173

principle of detailed balance 78

quadrupole moment 44quantum defect 15quasi-statistic broadening of spectral lines

42quenching rate constant 77

Racah coefficient 21radial wave function 11, 13radiative transitions in hydrogen atom 38radiative transitions for heliumlike ions 39Ramsauer effect 74rate constant 76reduced binding energy for bulk atoms

124, 125reduced rate constant of cluster growth

120–122resonant charge exchange 79resonantly excited state 29, 34, 43rotational constant 48rotation energy for diatomic 47, 49rotation constant for negative diatomic ion

51rotation constant for positive diatomic ion

50Rutherford formula 85Rydberg states 15

Saha formula 118scaling law 127saturation vapor pressure 126scattering phase 72selection laws for radiation 39self-diffusion coefficient 101Sena effect 105shell atom structure 18shell of valence electrons for atom ground

state 31

shell of valence electrons for excited atomstates 32

shell of valence electrons for negative ion31

SI units 2, 6Sommerfeld radius 132, 133specific surface energy of clusters 124,

125spin–orbit interaction 13, 25splitting of atom levels 35standard atomic weights 2stepwise atom ionization 93stimulated emission cross section 40system of atomic units 7

thermal conductivity coefficient 99Thomson model for atom ionization by

electron impact 85three body recombination process 93three halves law 120total cross section 41, 72Townsend 6, 112transport cross section 72, 73types of crystal lattices 124, 129

van der Waals equation 115van der Waals interaction constant 45vibration energy for diatomic 47, 49vibration energy for negative diatomic ion

51vibration energy for positive diatomic ion

50viscosity coefficient 99

wavelength of radiative transition 33, 37Weiskopf radius 47Wigner–Seits radius 120work function for metals 130zero-field mobility of electrons in gases

111

Springer Series on


Editors-in-Chief:

Professor G.W.F. DrakeDepartment of Physics, University of Windsor401 Sunset, Windsor, Ontario N9B 3P4, Canada

Professor Dr. G. EckerRuhr-Universität Bochum, Fakultät für Physik und AstronomieLehrstuhl Theoretische Physik IUniversitätsstrasse 150, 44801 Bochum, Germany

Professor Dr. H. Kleinpoppen, EmeritusStirling University, Stirling, UK, andFritz-Haber-InstitutMax-Planck-GesellschaftFaradayweg 4-6, 14195 Berlin, Germany

Editorial Board:

Professor W.E. BaylisDepartment of Physics, University of Windsor401 Sunset, Windsor, Ontario N9B 3P4, Canada

Professor Uwe BeckerFritz-Haber-InstitutMax-Planck-GesellschaftFaradayweg 4–6, 14195 Berlin, Germany

Professor Philip G. BurkeBrook House, Norley LaneCrowton, Northwich CW8 2RR, UK

Professor R.N. ComptonOak Ridge National LaboratoryBuilding 4500S MS6125Oak Ridge, TN 37831, USA

Professor M.R. FlannerySchool of PhysicsGeorgia Institute of TechnologyAtlanta, GA 30332-0430, USA

Professor C.J. JoachainFaculté des SciencesUniversité Libre BruxellesBvd du Triomphe, 1050 Bruxelles, Belgium

Professor B.R. JuddDepartment of PhysicsThe Johns Hopkins UniversityBaltimore, MD 21218, USA

Professor K.P. KirbyHarvard-Smithsonian Center for Astrophysics60 Garden Street, Cambridge, MA 02138, USA

Professor P. Lambropoulos, Ph.D.Max-Planck-Institut für Quantenoptik85748 Garching, Germany, andFoundation for Researchand Technology – Hellas (F.O.R.T.H.),Institute of Electronic Structureand Laser (IESL),University of Crete, PO Box 1527Heraklion, Crete 71110, Greece

Professor G. LeuchsFriedrich-Alexander-UniversitätErlangen-NürnbergLehrstuhl für Optik, Physikalisches InstitutStaudtstrasse 7/B2, 91058 Erlangen, Germany

Professor P. MeystreOptical Sciences CenterThe University of ArizonaTucson, AZ 85721, USA

Springer Series on


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32 Atom, Molecule, and Cluster Beams IICluster Beams, Fast and Slow Beams,Accessory Equipment and ApplicationsBy H. Pauly

33 Atom OpticsBy P. Meystre

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35 Many-Particle Quantum Dynamicsin Atomic and Molecular FragmentationEditors: J. Ullrich and V.P. Shevelko

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38 Plasma Physicsand Controlled Nuclear FusionBy K. Miyamoto

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