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Springer Series on
ATOMIC , OPTICAL , AND PLASMA PHYSICS
51
Springer Series on
ATOMIC , OPTICAL , AND PLASMA PHYSICS
The Springer Series on Atomic, Optical, and Plasma Physics covers in a comprehensive manner theory and experiment in the entire field of atoms and molecules
and their interaction with electromagnetic radiation. Books in the series provide
a rich source of new ideas and techniques with wide applications in fields such
as chemistry, materials science, astrophysics, surface science, plasma technology,
advanced optics, aeronomy, and engineering. Laser physics is a particular connecting theme that has provided much of the continuing impetus for new developments in the field. The purpose of the series is to cover the gap between standard
undergraduate textbooks and the research literature with emphasis on the fundamental ideas, methods, techniques, and results in the field.
47 Semiclassical Dynamics and Relaxation
By D.S.F. Crothers
48 Theoretical Femtosecond Physics
Atoms and Molecules in Strong Laser Fields
By F. Großmann
49 Relativistic Collisions of Structured Atomic Particles
By A. Voitkiv and J. Ullrich
50 Cathodic Arcs
From Fractal Spots to Energetic Condensation
By A. Anders
51 Reference Data on Atomic Physics and Atomic Processes
By B.M. Smirnov
Vols. 20–46 of the former Springer Series on Atoms and Plasmas are listed at the end of the book
Boris M. Smirnov
Reference Data
on Atomic Physics
and Atomic Processes
With 95 Figures
Professor Boris M. Smirnov
Russian Academy of Sciences
Joint Institute for High Temperatures
Izhorskaya ul. 13/19, 127412 Moscow, Russia
E-mail: [email protected]
Springer Series on Atomic, Optical, and Plasma Physics
ISBN 978-3-540-79362-5
ISSN 1615-5653
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Preface
Each scientist works with certain information and collects it in the course of professional activity. In the same manner, the author collected data for atomic physics and
atomic processes. This information was checked in the course of the author’s professional activity and was published in the form of appendices to the corresponding
books on atomic and plasma physics. Now it has been decided to publish these data
separately.
This book contains atomic data and useful information about atomic particles
and atomic systems including molecules, nanoclusters, metals and condensed systems of elements. It also gives information about atomic processes and transport
processes in gases and plasmas. In addition, the book deals with general concepts
and simple models for these objects and processes. We give units and conversion
factors for them as well as conversion factors for spread formulas of general physics
and the physics of atoms, clusters and ionized gases since such formulas are used in
professional practice by each scientist of this area.
This book includes numerical information from some reference books for physical units and constants [1–4] and for numerical parameters of atomic particles and
processes [2, 5–10] (in the most degree [2]). We also use data of some reviews and
original papers. The methodical peculiarity of this book consists in representation
of some physical parameters in the form of periodic tables. This form simplifies
the information retrieval because it only uses uniform information. If the data relate to a restricted number of elements, they are given in the form of specified tables.
Some part of the book is devoted to atomic spectra, which are given in the form
of the Grotrian diagrams for atoms with the electron valence shell s, s 2 , and also
the valence shell p k for light atoms. This information has not changed during the
last decades. We use the Grotrian diagrams from [11]; diagrams for the lowest atom
states are taken from [12]. Along with the Grotrian diagrams, some concepts and
formulas of atomic physics are represented.
This book also contains basic concepts of the physics of atomic systems and
the simplest models for their description. These models may be a basis for simple
estimations of the parameters of some atomic objects and processes. For example,
the model of a hard sphere describes atom–cluster collisions, the liquid drop model
is convenient for the analysis of cluster evaporation and other cluster processes and
the model of a degenerate electron gas may be used for the metal plasma. In these
vi
Preface
cases numerical parameters of models are given for certain objects, as they follow
from measured object parameters.
As a scientist who has used the data about atomic and plasma physics contained
herein to fulfill some estimations for certain problems of this area, the author intends this book to be used by scientists and advanced students. In the first stage of
information collection, the author was a user of these data, and the basis of this book
is Appendices to books [12–16] for certain aspects of atomic and plasma physics.
Therefore the author hopes that this book will be useful both for specialists and for
advanced students of this physical area.
Moscow,
May 2008
Boris M. Smirnov
Contents
1
2
3
Fundamental Constants, Elements, Units . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Fundamental Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Elements and Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Physical Units and Conversion Factors . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Systems of Physical Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Conversion Factors for Units . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 Conversion Factors in Formulas of General Physics with
Atomic Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
2
2
2
6
Elements of Atomic and Molecular Physics . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Properties of Atoms and Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Properties of the Hydrogen Atom and Hydrogen-Like Ions .
2.1.2 Properties of the Helium Atom and Helium-Like Ions . . . . . .
2.1.3 Quantum Numbers of a Light Atom . . . . . . . . . . . . . . . . . . . . .
2.1.4 Shell Atom Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.5 Schemes of Coupling of Electron Momenta in Atoms . . . . . .
2.1.6 Parameters of Atoms and Ions in the Form of Periodic Tables
2.2 Atomic Radiative Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 General Formulas and Conversion Factors for Atomic
Radiative Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Radiative Transitions between Atom Discrete States . . . . . . .
2.2.3 Absorption Parameters and Broadening of Spectral Lines . . .
2.3 Interaction Potential of Atomic Particles at Large Separations . . . . .
2.4 Properties of Diatomic Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Bound States of Diatomic Molecule . . . . . . . . . . . . . . . . . . . . .
2.4.2 Correlation between Atomic and Molecular States . . . . . . . . .
2.4.3 Excimer Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
11
11
15
17
18
24
29
33
Elementary Processes Involving Atomic Particles . . . . . . . . . . . . . . . . . .
3.1 Parameters of Elementary Processes in Gases and Plasmas . . . . . . . .
3.2 Inelastic Collisions of Electrons with Atoms . . . . . . . . . . . . . . . . . . . .
3.3 Collision Processes Involving Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Atom Ionization by Electron Impact . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Atom Ionization in Gas Discharge Plasma . . . . . . . . . . . . . . . . . . . . . .
71
71
76
78
85
87
9
33
38
39
43
47
47
53
66
viii
Contents
3.6
3.7
3.8
3.9
Ionization Processes Involving Excited Atoms . . . . . . . . . . . . . . . . . .
Electron–Ion Recombination in Plasma . . . . . . . . . . . . . . . . . . . . . . . .
Attachment of Electrons to Molecules . . . . . . . . . . . . . . . . . . . . . . . . .
Processes in Air Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
93
95
96
4
Transport Phenomena in Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.1 Transport Coefficients of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2 Ion Drift in Gas in External Electric Field . . . . . . . . . . . . . . . . . . . . . . 103
4.3 Conversion Parameters for Transport Coefficients . . . . . . . . . . . . . . . . 111
4.4 Electron Drift in Gas in Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.5 Diffusion of Excited Atoms in Gases . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5
Properties of Macroscopic Atomic Systems . . . . . . . . . . . . . . . . . . . . . . . . 115
5.1 Equation of State for Gases and Vapors . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 Basic Properties of Ionized Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3 Parameters and Rates of Processes Involving Nanoparticles . . . . . . . 120
5.4 Parameters of Condensed Atomic Systems . . . . . . . . . . . . . . . . . . . . . 126
A
Atomic Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
1 Fundamental Constants, Elements, Units
1.1 Fundamental Physical Constants
Table 1.1. Fundamental Physical Constants
Electron mass
me = 9.10939 × 10−28 g
Proton mass
mp = 1.67262 × 10−24 g
Ratio of proton and electron masses
1 m(12 C) = 1.66054 × 10−24 g
ma = 12
mp /me = 1836.15
Ratio of atomic and electron masses
ma /me = 1822.89
Electron charge
e = 1.602177 × 10−19 C = 4.8032 × 10−10 e.s.u.
Atomic unit of mass
e2 = 2.3071 × 10−19 erg cm
Planck constant
h = 6.62619 × 10−27 erg s
h̄ = 1.05457 × 10−27 erg s
Light velocity
c = 2.99792 × 1010 cm/s
Fine-structure constant
α = e2 /(h̄c) = 0.07295
Inverse fine-structure constant
1/α = h̄c/e2 = 137.03599
Bohr radius
a0 = h̄2 /(me e2 ) = 0.529177 Å
Rydberg constant
R = me e4 /(2h̄2 ) = 13.6057 eV
= 2.17987 × 10−18 J
Bohr magneton
μB = eh̄/(2me c) = 9.27402 × 10−24 J/T
= 9.27402 × 10−21 erg/Gs
Avogadro number
NA = 6.02214 × 1023 mol−1
Stephan–Boltzmann constant
σ = π 2 /(60h̄3 c2 ) = 5.669 × 10−12 W/(cm2 K4 )
Molar volume
R = 22.414 l/mol
Loschmidt number
L = NA /R = 2.6867 × 1019 cm−3
Faraday constant
F = NA e = 96485.3 C/mol
2
1 Fundamental Constants, Elements, Units
1.2 Elements and Isotopes
There are about 2700 stable and 2000 radioactive isotopes in nature [17]. Stable
isotopes relate to elements with a nuclear charge below 83, excluding technetium
43 Tc and promethium 61 Pm, and also to elements with a nuclear charge in the range
90–93 (thorium, protactinium, uranium, neptunium). The diagram Pt1 contains standard atomic masses for elements, taking into account their occurrence in the Earth’s
crust [2, 18]. If stable isotopes of a given element are absent, the masses are given
in square brackets. These masses are given in atomic mass units (amu) where the
unit is taken 1/12 mass of the carbon isotope 12 C from 1961 (see Table 1.1).
There is in diagram Pt2 the occurrence of stable isotopes in the Earth’s crust,
and diagram Pt3 contains the lifetimes of stable isotopes [2, 18–20]. These lifetimes
are expressed in days (d) and years (y) and are given for isotopes whose lifetime
exceeds 2 hours (0.08 d).
1.3 Physical Units and Conversion Factors
1.3.1 Systems of Physical Units
The unit system is a set of units through which various physical parameters are
expressed. The basis of any mechanical system of units is the fact that the value
of any dimensionality may be expressed through three dimensional units—length,
mass and time. The CGS system, which is based on the centimeter, gram and second
[3], is the oldest system of units and was introduced by British association for the
Advancement of Science in 1874. The system of International Units (SI) [4] was
adopted in 1960 (the conference des Poids et Mesure, Paris) and is based on the
meter (m), kilogram (kg) and second (s). Other bases may be used for specific units.
In transferring from mechanics to other physics branches, additional units or
assumptions are required. In particular, along with mechanical units, we deal with
electric and magnetic units below. Then the base of SI units along with the above
mechanical units contain the unit of an electric current ampere (A). Table 1.2 contains the SI units and their connection with the base units. Note that we express the
thermodynamic temperature through energy units below.
In spreading the CGS system of units to electrostatics, one can use the Coulomb
law for the force F between two charges e1 and e2 that are located at a distance r in
a vacuum. This force is given by
F = 0
e1 e2
r2
where 0 is the vacuum permittivity. Defining a charge unit from this formula under
the assumption 0 = 1, we construct in this manner the CGSE system of units
that describes physical parameters of mechanics and electrostatics. In the same way,
in constructing of the CGSM system of units on the basis of the mechanical CGS
system of units, the assumption is used that the vacuum magnetic conductivity is
Pt1. Standard atomic weights of elements and their natural occurrence in Earth’s crust
1.3 Physical Units and Conversion Factors
3
Pt2. Natural occurrence of stable isotopes
4
1 Fundamental Constants, Elements, Units
5
Pt3. Long-lived radioactive isotopes
1.3 Physical Units and Conversion Factors
6
1 Fundamental Constants, Elements, Units
Table 1.2. Basic SI units [1, 2, 4]
Quantity
Name
Symbol
Connection with base units
Frequency
hertz
Hz
1/s
Force
newton
N
m kg/s2
Pressure
pascal
Pa
kg/(m s2 )
Energy
joule
J
m2 kg/s2
Power
watt
W
m2 kg/s3
Charge
coulomb
C
As
Electric potential
volt
V
W/A
Electric capacitance
farad
F
A2 s4 /(m2 kg)
Electric resistance
ohm
m2 kg/(A2 s3 )
Conductance
siemens
S
A2 s3 /(m2 kg)
Inductance
henry
H
m2 kg/(A2 s2 )
Magnetic flux
weber
Wb
m2 kg/(A s2 )
Magnetic flux density
tesla
T
kg/(A s2 )
one μ0 = 1. Because of units of electrostatic CGS (CGSE system of units) and
electromagnetic CGS (CGSM system of units) are used, we give some conversions
between these systems and SI units below.
For atomic systems, the system of atomic units (or Hartree atomic units) is of
importance because parameters of atomic systems are expressed through atomic parameters. In construction the system of atomic units, the fact is used that the parameter of any dimensionality may be built on the basis of three parameters of different
dimensionality. As a basis for the system of atomic units, the following three parameters are taken: the Planck constant h̄ = 1.05457×10−27 erg s, the electron charge
e = 1.60218 × 10−19 C and the electron mass me = 9.10939 × 10−28 g (we take
the vacuum permittivity and the magnetic conductivity of a vacuum to be one). The
system of atomic units constructed on these parameters is given below in Table 1.3.
1.3.2 Conversion Factors for Units
Tables 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, and 1.10 contain conversion factors between
units used. The specific electric field strength E/p or E/Na (E is the electric field
strength, p is the pressure, Na is the number density of atoms) is a spread unit in
physics of ionized gases. The unit of E/Na is 1 Townsend (Td) [21]. The connection
between units of the above quantities is as follows:
1
V
= 0.3535 Td;
cm · Torr
1 Td = 2.829
V
;
cm · Torr
1 Td = 1 × 10−17 V cm2 .
1.3 Physical Units and Conversion Factors
7
Table 1.3. System of atomic units
Parameter
Symbol, formula
Value
Length
a0 = h̄2 /(me2 )
v0 = e2 /h̄
τ0 = h̄3 /(me4 )
ν0 = me4 /h̄3
ε0 = me4 /h̄2
5.2918 × 10−9 cm
Velocity
Time
Frequency
Energy
2.1877 × 108 cm/s
2.4189 × 10−17 s
4.1341 × 1016 s−1
27.2114 eV
= 4.3598 × 10−18 J
Power
Electric voltage
Electric field strength
Momentum
Number density
Volume
ε0 /τ = m2 e8 /h̄5
ϕ0 = me3 /h̄2
E0 = me5 /h̄4
p0 = me2 /h̄
N0 = a0−3
V0 = a03
0.18024 W
27.2114 V
5.1422 × 109 V/cm
1.9929 × 10−19 g cm/s
6.7483 × 1024 cm−3
1.4818 × 10−25 cm3
= 0.089240 cm3 /mol
Square, cross section
σ0 = a02
Rate constant
Dipole moment
k0 = v0 a02 = h̄3 /(m2 e2 )
K0 = v0 a05 = h̄9 /(m5 e8 )
ea0 = a0 = h̄2 /(me)
Magnetic moment
h̄2 /(me) = 2μB /α
Electric current
I = e/τ = me5 /h̄3
j0 = N0 v0 = m3 e8 /h̄7
i0 = eN0 v0 = m3 e9 /h̄7
J = ε0 N0 v0 = m4 e12 /h̄9
Three body rate constant
2.8003 × 10−17 cm2
6.126 × 10−9 cm3 /s
9.078 × 10−34 cm6 /s
2.5418 × 10−18 esu
= 2.5418 D
2.5418 × 10−18 erg/Gs
= 2.5418 × 10−21 J/T
Flux
Electric current density
Energy flux
6.6236 × 10−3 A
1.476 × 1033 cm−2 s−1
2.3653 × 1014 A/cm2
6.436 × 1015 W/cm2
6.2415 × 1018
6.2415 × 1011
1
8.6174 × 10−5
1.2398 × 10−4
4.1357 × 10−9
4.3364 × 10−2
1.0364 × 10−2
107
1
1.6022 × 10−12
1.3807 × 10−16
1.9864 × 10−16
6.6261 × 10−21
6.9477 × 10−28
1.6605 × 10−28
1
0.1
1 Torr
133.332
1 atma
1.01325 × 105
b
9.80665 × 104
1 at
1 bar
105
a atm—physical atmosphere
b at = kg/cm2 —technical atmosphere
7.2429 × 1022
7.2429 × 1015
11604
1
1.4388
4.7992 × 10−5
503.22
120.27
1K
5.0341 × 1022
5.0341 × 1015
8065.5
0.69504
1
3.3356 × 10−5
349.76
83.594
1 cm−1
10
1
1333.32
1.01325 × 106
9.80665 × 105
106
1 dyn/cm2
1 kcal/mol
1.3595 × 10−3
1.01332
1
1.0197
1.0197 × 10−6
1.3158 × 10−3
1
0.96785
0.98693
9.8693 × 10−7
1
760
735.56
750.01
7.5001 × 10−4
1.0197 × 10−5
9.8693 × 10−6
7.5001 × 10−3
1.4393 × 1020
1.4393 × 1013
23.045
1.9872 × 10−3
2.8591 × 10−3
9.5371 × 10−9
1
0.23901
1 atma
1 atb
1.5092 × 1027
1.5092 × 1020
2.4180 × 108
2.0837 × 104
2.9979 × 104
1
1.0485 × 107
2.5061 × 106
1 MHz
1 Torr
Table 1.5. Conversion factors for units of pressure
1 eV
1 erg
1 Pa = 1 N/m2
1
10−7
1.6022 × 10−19
1.3807 × 10−23
1.9864 × 10−23
6.6261 × 10−28
6.9477 × 10−21
1.6605 × 10−21
1 Pa = 1 N/m2
1 dyn/cm2
1J
1 erg
1 eV
1K
1 cm−1
1 MHz
1 kcal/mol
1 kJ/mol
1J
Table 1.4. Conversion factors for units of energy
10−5
10−6
1.33332 × 10−3
1.01325
0.980665
1
1 bar
6.0221 × 1020
6.0221 × 1013
96.485
8.3145 × 10−3
1.1963 × 10−2
3.9903 × 10−7
4.184
1
1 kJ/mol
8
1 Fundamental Constants, Elements, Units
1.3 Physical Units and Conversion Factors
9
Table 1.6. Conversion factors for units of electric voltage
1V
1 CGSE
1 CGSM
1V
1 CGSE
1 CGSM
1
299.792
10−8
3.33564 × 10−3
1
3.33564 × 10−11
108
2.99792 × 1010
1
Table 1.7. Conversion factors for units of electric field strength
1 V/cm
1 CGSE
1 CGSM
1 V/cm
1 CGSE
1 CGSM
1
299.792
10−8
3.33564 × 10−3
1
3.33564 × 10−11
108
2.99792 × 1010
1
Table 1.8. Conversion factors for units of electric resistance
1
1 CGSE
1 CGSM
1
1 CGSE
1 CGSM
1
8.98755 × 1011
10−9
1.11265 × 10−12
1
1.11265 × 10−21
109
8.98755 × 1020
1
Table 1.9. Conversion factors for units of magnetic field strength
1 Oe
1 CGSE
1 A/m
1 Oe
1 CGSE
1 A/m
1
3.33564 × 10−11
0.012566
2.99792 × 1010
1
1.11265 × 10−21
79.5775
2.65442 × 10−9
1
Table 1.10. Conversion factors for units of magnetic induction
1 CGSE
1 T = 1 Wb/m2
1 Gs
1 CGSE
1 T = 1 Wb/m2
1 Gs
1
3.33564 × 10−7
3.33564 × 10−11
2.99792 × 106
2.99792 × 1010
104
1
1
10−4
1.3.3 Conversion Factors in Formulas of General Physics
with Atomic Particles
Explanations to Table 1.11:
1. The particle velocity is v =
mass
√
2ε/m, where ε is the energy, m is the particle
10
1 Fundamental Constants, Elements, Units
Table 1.11. Conversion factors for formulas involving atomic particles
1
Formulaa
√
v = C ε/m
2
√
v = C T /m
3
ε = Cv 2
4
ω = Cε
5
6
7
ω = C/λ
ε = C/λ
ωH = CH /m
8
√
rH = C εm/H
9
p = CH 2
Number
Factor C
Units used
5.931 × 107 cm/s
1.389 × 106 cm/s
5.506 × 105 cm/s
1.289 × 104 cm/s
1.567 × 106 cm/s
1.455 × 104 cm/s
3.299 × 10−12 K
6.014 × 10−9 K
2.843 × 10−16 eV
5.182 × 10−13 eV
1.519 × 1015 s−1
1.309 × 1011 s−1
1.884 × 1015 s−1
1.2398 eV
1.759 × 107 s−1
9655 s−1
3.372 cm
143.9 cm
3.128 × 10−2 cm
1.336 cm
4.000 × 10−3 Pa
= 0.04 erg/cm3
ε in eV, m in e.m.u.a
ε in eV, m in a.m.u.a
ε in K, m in e.m.u.
ε in K, m in a.m.u.
T in eV, m in a.m.u.
ε in K, m in a.m.u.
v in cm/s, m in e.m.u.
v in cm/s, m in a.m.u.
v in cm/s, m in e.m.u.
v in cm/s, m in a.m.u.
ε in eV
ε in K
λ in μm
λ in μm
H in Gs, m in e.m.u.
H in Gs, m in a.m.u.
ε in eV, m in e.m.u., H in Gs
ε in eV, m in a.m.u., H in Gs
ε in K, m in e.m.u., H in Gs
ε in K, m in a.m.u., H in Gs
H in Gs
a e.m.u. is the electron mass unit (m = 9.108 × 10−28 g), a.m.u. is the atomic mass unit
e
(ma = 1.6605 × 10−24 g)
√
2. The average particle velocity is v = 8T /(πm) with the Maxwell distribution
function of particles on velocities; T is the temperature expressed in energetic
units, m is the particle mass
3. The particle energy is ε = mv 2 /2, where m is the particle mass, v is the particle
velocity
4. The photon frequency is ω = ε/h̄, where ε is the photon energy
5. The photon frequency is ω = 2πc/λ, where λ is the wavelength
6. The photon energy is ε = 2π h̄c/λ
7. The Larmor frequency is ωH = eH /(mc) for a charged particle of a mass m in
a magnetic field of strength H
√
8. The Larmor radius of a charged particle is rH = 2ε/m/ωH , where ε is the
energy of a charged particle, m is its mass, ωH is the Larmor frequency
9. The magnetic pressure is pm = H 2 /(8π)
2 Elements of Atomic and Molecular Physics
2.1 Properties of Atoms and Ions
2.1.1 Properties of the Hydrogen Atom and Hydrogen-Like Ions
The hydrogen atom includes a bound electron located in the nuclear Coulomb field.
The electron position in the field of the Coulomb center is described by three space
coordinates: r is the distance of the electron from the Coulomb center, θ is the
polar electron angle, and ϕ is the azimuthal angle. In addition, the spin electron
state is characterized by the spin electron projection σ on a given direction. In the
non-relativistic limit, space and spin coordinates are separated, and the space wave
function has the form
(2.1)
ψ(r, θ, ϕ) = Rnl (r)Ylm (θ, ϕ).
Here, separation of variables is used for an electron located in the Coulomb center
field, so that Rnl (r) is the radial wave function, and Ylm is the angle wave function.
In addition, n is the principle quantum number, l is the angular momentum and m is
the angular momentum projection onto a given direction. These quantum numbers
are whole values, and n ≥ 1, n ≥ l + 1, l ≥ 0, |m| ≤ l. The binding state energy εn
that counts off from the continuous spectrum boundary is given by
εn = −
me e4
2h̄2 n2
.
(2.2)
Here, n is the principle electron quantum number, e is the electron charge and me is
the electron mass. As is seen, the states of an electron that is located in the Coulomb
field are degenerated with respect to the electron momentum l and its projection m
onto a given direction (and with respect to the spin projection σ , since the electron
Hamiltonian is independent of its spin). For notations of electron states (electron
terms) are used its quantum numbers n, l, where the principal quantum number n
is given as a value, whereas the quantum number l is denoted by letters so that notations s, p, d, f, g, h relate to states with the angular momenta l = 0, 1, 2, 3, 4, 5,
correspondingly. For example, the notation 4f refers to a state with quantum numbers n = 4, l = 3.
The angle wave function of an electron located in the Coulomb center field is
satisfied to the Schrödinger equation
12
2 Elements of Atomic and Molecular Physics
Table 2.1. The angle electron wave function in the hydrogen atom for small electron momenta l [22–24]
l
m
0
0
1
0
1
±1
2
0
2
±1
2
±2
3
0
3
±1
3
±2
3
±3
4
0
4
±1
4
±2
4
±3
4
±4
Ylm (θ, ϕ)
√1
4π
3
4π · cos θ
3 · sin θ · exp(±iϕ)
± 8π
5 · 3 cos2 θ − 1
4π
2
2
15
± 8π · sin θ · cos θ · exp(±iϕ)
1 15 · sin2 θ · exp(±2iϕ)
2 8π
7
3
4π · (5 cos θ − 3 cos θ)
2
± 18 21
π · sin θ · (5 cos θ − 1) exp(±iϕ)
1 105 · sin2 θ cos θ · exp(±2iϕ)
4 2π
3
± 18 35
π · sin θ · exp(±3iϕ)
9
35
15
3
4
2
4π · ( 8 cos θ − 4 cos θ + 8 )
± 38 π5 · sin θ · (7 cos3 θ − 3 cos θ) · exp(±iϕ)
5 · sin2 θ · (7 cos2 θ − 1) · exp(±2iϕ)
± 38 2π
3
± 38 35
π · sin θ · cos θ · exp(±3iϕ)
3
35
4
16 2π · sin θ · exp(±4iϕ)
∂
∂Ylm
1 ∂ 2 Ylm
+ l(l + 1)Ylm = 0
sin2 θ
+
∂ cos θ
∂ cos θ
sin2 θ ∂ϕ 2
(2.3)
and is given by the formula [22–24]
Ylm (θ, ϕ) =
2l + 1 (l − m)!
4π (l + m)!
1/2
Plm (cos θ ) exp(imϕ).
(2.4)
These angle wave functions of an electron are represented in Table 2.1.
The radial wave function of an electron in the hydrogen atom is the solution of
the Schrödinger equation
1 d2
2 l(l + 1)
Rnl = 0,
(rR)
+
2ε
+
(2.5)
−
r dr 2
r
r2
2.1 Properties of Atoms and Ions
13
Table 2.2. Electron radial wave functions Rnl for the hydrogen atom [22–24]
State
Rnl
1s
2 exp(−r)
√1 1 − r exp(−r/2)
2
2s
2
1
√ r exp(−r/2)
24
2
2r 2 exp(−r/3)
√
1 − 2r
+
3
27
3 3
2p
3s
2 · r 1 − r exp(−r/3)
3
6
3p
2
27
3d
4s
4
√
· r 2 exp(−r/3)
81 30
1 1 − 3r + r 2 − r 3 exp(−r/4)
4
4
8
192
4p
1
16
5 · r · 1 − r + r2
3
4
80
exp(−r/4)
1√ · r 2 · 1 − r exp(−r/4)
12
4d
64 5
1√ · r 3 · exp(−r/4)
768 35
4f
that, with accounting for boundary conditions, has the form
2
(n + l)!
1
r
Rnl (r) = l+2
(2r)l exp −
(2l + 1)! (n − l − 1)!
n
n
2r
,
× F −n + l + 1, 2l + 2,
n
(2.6)
where F is degenerate hypergeometric function. The following normalization condition takes place for the radial wave function
∞
2
Rnl
(r)r 2 dr = 1.
(2.7)
0
Table 2.2 lists the expressions of the radial electron wave functions for lowest hydrogen atom states.
Table 2.3 contains expressions for average quantities r n of the hydrogen atom,
where r is the electron distance from the center, and Table 2.4 lists the values of
these quantities for lowest states of the hydrogen atom.
∞
2
r n =
Rnl
(r)r n+2 dr.
(2.8)
0
Fine structure splitting in the hydrogen atom is determined by spin–orbit interaction and corresponds according to its nature to the interaction between spin and
14
2 Elements of Atomic and Molecular Physics
Table 2.3. Formulas for average values r n given in atomic units, where r is the electron
distance from the Coulomb center [23]
Value
Formula
r
r −1 1 · [3n2 − l(l + 1)]
2
n2 · [5n2 + 1 − 3l(l + 1)]
2
n2 · [35n2 (n2 − 1) − 30n2 (l + 2)(l − 1) + 3(l + 2)(l + 1)l(l − 1)]
8
n4 · [63n4 − 35n2 (2l 2 + 2l − 3) + 5l(l + 1)(3l 2 + 3l − 10) + 12]
8
n−2
r −2 [n3 (l + 1/2)]−1
r −3 [n3 (l + 1) · (l + 1/2) · l]−1
r −4 [3n2 − l(l + 1)][2n5 · (l + 3/2) · (l + 1) · (l + 1/2) · l · (l − 1/2)]−1
r 2 r 3 r 4 Table 2.4. Average values r n for lowest states of the hydrogen atom expressed in atomic
units [23]
State
1s
2s
2p
3s
3p
3d
4s
4p
4d
4f
r
1.5
6
5
13.5
12.5
10.5
24
23
21
18
r 2 3
42
30
207
180
126
648
600
504
360
r 3 7.5
330
210
3442
2835
1701
18720
16800
13100
7920
r 4 22.5
2880
1680
6.136 × 104
4.420 × 104
2.552 × 104
5.702 × 105
4.973 × 105
3.629 × 105
1.901 × 105
r −1 1
0.25
0.25
0.111
0.111
0.111
0.0625
0.0625
0.0625
0.0625
r −2 2
0.25
0.0833
0.0741
0.0247
0.0148
0.0312
0.0104
0.00625
0.00446
r −3 –
–
0.0417
–
0.0123
0.0247
–
5.21 × 10− 3
1.04 × 10−3
3.72 × 10−4
r −4 –
–
0.0417
–
0.0137
5.49 × 10−3
–
5.49 × 10−4
2.60 × 10−4
3.7 × 10−5
magnetic field due to angular electron rotation. The fine structure splitting is given
by
1 Z 2 eh̄ 2 1 2l + 1 Z 2 eh̄ 2 1
· l̂ŝ =
δf =
2 mc
4
mc
r3
r3
2 2
e
Z4
=
.
3
h̄c 2n l(l + 1)
(2.9)
Here, angle brackets mean an average over the electron space distribution and this
formula relates to the hydrogen-like ion where Z is the Coulomb center charge.
Since the parameter [e2 /(h̄c)]2 /4 = 1.33×10−5 is small, the fine structure splitting
is small for the hydrogen atom. Accounting for the fine structure splitting gives the
quantum numbers of the hydrogen atom as lsj mj instead of numbers lmsσ for
degenerate levels, where s = 1/2 is the electron spin, j = l ± 1/2 is the total
2.1 Properties of Atoms and Ions
15
Table 2.5. Quantum defect for atoms of alkali metals [26, 27]
Atom
δs
δp
δd
δf
Li
0.399
0.053
0.002
–
Na
1.347
0.854
0.0145
0.0016
K
2.178
1.712
0.267
0.010
Rb
3.135
2.65
1.34
0.0164
Cs
4.057
3.58
2.47
0.033
electron momentum and m, σ, mj are projections of the orbital momentum, spin
and total momentum onto a given direction. Note that the lower level corresponds
to the total electron momentum j = l − 1/2.
The behavior of highly excited (Rydberg) states of atoms is close to that of
the hydrogen atom because the Coulomb electron interaction with the center is the
main interaction. But a short-range electron interaction with an atomic core leads to
a displacement of atom energetic levels, and the electron energy, instead of that by
formula (2.2), is given by [25]
εn = −
me e4
2h̄2 n2ef
=−
me e4
2h̄2 (n − δl )2
,
(2.10)
where nef is the effective principal quantum number and δl is the so-called quantum
defect. Since it is determined by a short-range electron–core interaction, its value
decreases with an increase of the electron angular momentum l. Table 2.5 contains
the values of the quantum defect for alkali metal atoms [26, 27]. In addition, Fig. A.1
in Appendix A gives spectrum of the hydrogen atom.
2.1.2 Properties of the Helium Atom and Helium-Like Ions
In order to explain the atom’s nature, structure and spectrum, we use a one-electron
approximation when the atom wave function is the combination of products of the
one-electron wave functions. In construction the atom wave functions we are based
on the Pauli exclusion principle [28], according to which location of two electrons
is prohibited in an identical electron state, and the total electron wave function is
zero if the coordinates of the two electrons with an identical spin direction coincide.
We include in the Hamiltonian of electrons, alongside the spin–orbit interaction, the
exchange interaction whose potential may be represented in the form
Vex = A(|r1 − r2 |)ŝ1 ŝ2 .
(2.11)
Here r1 , r2 are electron coordinates and ŝ1 , ŝ2 are their spins. Note that the nature of
exchange interaction is non-relativistic and is connected with the symmetry of the
total wave function. For the system of two electrons (helium atom or helium-like
ion), the total wave function according to the Pauli principle changes sign as a result
of permutation of electrons. But the total wave function is the product of the space
and spin electron wave functions. For the total spin S = 1 the spin wave function
is symmetric with respect to electron permutation, and for the total spin S = 0
16
2 Elements of Atomic and Molecular Physics
Fig. 2.1. The lowest excited states of the helium atom. The excitation energy for a given
state is expressed in eV, the wavelengths λ and the radiative lifetimes τ are indicated for
a corresponding radiative transition
the spin wave function changes sign at permutation of electrons. Hence, the space
wave function of two electrons is symmetric with respect to electron coordinates if
the electron total spin is zero and is antisymmetric if the total electron spin is one.
Thus, the total electron spin determines the symmetry of the space electron wave
function, and this fact may be taken into account formally by the introduction of an
exchange interaction term (2.11) in the electron Hamiltonian.
Thus the spectrum of the helium atom (Fig. A.2 in Appendix A) is divided into
two independent parts, with the total electron spin S = 0 and S = 1 with a symmetric space wave function of electrons in the first case and with antisymmetric space
wave function of electrons in the second case. The radiative transitions between
states that belong to different parts are practically absent because of conservation
of the total electron spin in radiative transitions due to first approximations in the
expansion over a small relativistic parameter. In addition, atom levels related to the
total spin S = 1 are located below the corresponding levels of states S = 0 with
identical other quantum numbers. One can see it in the Grotrian diagram Fig. A.2
for the helium atom and in Fig. 2.1, where the lowest excited states of the helium
atom are given.
As a result, we represent the electron Hamiltonian in the form
Ĥ = −
h̄2
Ze2
Ze2
e2
h̄2
1 −
2 −
−
+
2me
2me
r1
r2
|r1 − r2 |
+ Aŝ1 ŝ2 + B l̂1 ŝ1 + B l̂2 ŝ2 .
(2.12)
An electrostatic interaction of electrons with a core and also with each other, the
exchange interaction of electrons and the interaction of an electron spin with its
orbit are included in this Hamiltonian. Since spin–orbit interaction is of importance
for excited electron states, interaction of a spin with a foreign orbit is not important.
Let us use the electron Hamiltonian (2.12) for the analysis of the spectrum of
the helium atom that is given in Fig. A.2 and the spectra of helium-like ions. In the
ground atom state, both electrons are located in the state 1s, and the ground atom
2.1 Properties of Atoms and Ions
17
Table 2.6. Dependence of interactions of helium-like ions on the nuclear charge Z
Parameter
Z-dependence
Ionization potential
Potential of exchange interaction
Spin–orbit interaction
Rate of one-photon radiative transition
Rate of two-photon radiative transition
Z2
Z
Z4
Z4
Z8
state is 1s 2 1 S, i.e. the total atom spin is zero, S = 0. The atom state with S = 1 and
the states 1s of both electrons are not realized because the space wave function is
antisymmetric with respect to electron permutation and is zero for identical electron
states. The quantum numbers of an excited helium atom are its orbital momentum
and spin, which can be equal to zero or one. In the latter case, the atom quantum
number is the total atom momentum, which is a sum of the orbital momentum and
spin (see Fig. A.2). In the case of helium-like ions, which consist of the Coulomb
center of a charge Z and two electrons, the contribution of different interactions
to the total energy varies with variation of the center charge, which follows from
Table 2.6. Correspondingly, the spectrum of helium-like ions changes with variation
of Z.
2.1.3 Quantum Numbers of a Light Atom
One can ignore relativistic interactions for light atoms in the first approximation,
and the electron Hamiltonian for such an atom in accordance with the expression
(2.12) has the form
Ĥ = −
Ze2 e2
h̄2 +A
ŝi ŝk .
i −
+
2me
ri
|ri − rk |
i
i
i,k
(2.13)
i,k
Here i, k are electron numbers, ri , rk are electron coordinates, and ŝi , ŝk are the
operators of electron spins.
The operators of the total electron momentum L̂ and the total electron spin Ŝ for
an atom are given by
ŝi ,
Ŝ =
(2.14)
l̂i ,
L̂ =
i
i
where l̂i and ŝi are the operators of the angular momentum and spin for i-th electron.
Because these operators commute with the Hamiltonian (2.13) [29], the atom quantum numbers are LSML MS , where L is the angular atom momentum, S is the total
atom spin, and ML , MS are the projections of these momenta onto a given direction.
If we add to the Hamiltonian (2.13) the operator of spin–orbit interaction in the
form B L̂Ŝ, the energy levels for given atom quantum numbers L and S are split, and
the atom eigen states are characterized by the quantum numbers LSJ MJ , where J is
the total atom momentum, and MJ is its projection onto a given direction.
18
2 Elements of Atomic and Molecular Physics
Fig. 2.2. The lowest excited states of the carbon atom. Energies of excitation for corresponding states are indicated inside rectangular boxes, accounting for their fine structure, and are
expressed in cm−1 . Wavelengths of radiative transitions are given inside arrows and are expressed in Å. The radiative lifetimes of states are placed in triangular boxes and are expressed
in s
The following notations are used to describe an atom state. The number 2S + 1
of spin projections and the multiplicity of spin states are given above. The angular
atom momentum is given by letters S, P , D, F, G, etc. for values of the angular momentum L = 0, 1, 2, 3, 4, etc. The total atom momentum J is given as a subscript.
For example, 3 D3 corresponds to atom quantum numbers S = 1, L = 2, J = 3.
This method of description of atom states is suitable for light atoms. But this
scheme may be used also for heavy atoms to classify atom states, though relativistic
effects are of importance for these atoms. The Grotrian diagrams for the lowest
states of some atoms are given in Figs. 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8 and 2.9 [12].
In particular, fine splitting of the ground state of the tellurium atom (Fig. 2.9) is
approximately 20 times more than that for the oxygen atom with the same electron
structure (Fig. 2.4), and the character of fine splitting in these cases is different.
Nevertheless, using the same classification method for both cases is convenient.
2.1.4 Shell Atom Scheme
The atom shell model distributes bound electrons over electron shells, so that each
shell contains electrons with identical quantum numbers nl. This corresponds to
one-electron approximation if the field that acts on a test electron consists of the
Coulomb field of the nucleus and a self-consistent field of other electrons. The sequence of shell filling for an atom in the ground state corresponds to that for a center
with a screened Coulomb field that acts on a test electron. Diagrams Pt4, Pt5 give
the valence electron shells for atoms and negative ions in the ground state. In addition, the ionization potentials of atoms and their first ions in the ground states and
the electron affinities for atoms are given in these diagrams [2, 30, 31].
2.1 Properties of Atoms and Ions
19
Fig. 2.3. The lowest excited states of the nitrogen atom. Energies of excitation for corresponding states are indicated inside rectangular boxes, accounting for their fine structure,
and are expressed in cm−1 . Wavelengths of radiative transitions are given inside arrows and
are expressed in Å. The radiative lifetimes of states are placed in triangular boxes and are
expressed in s, 15 means 1 × 105
Fig. 2.4. The lowest excited states of the oxygen atom. Notations are similar to those
of Fig. 2.3. The oscillator strength of a radiative transition is given in parentheses near the
wavelength
Valence electrons determine properties of atoms and atom interaction. One can
construct the valence electron shell to consist of one electron and atomic core.
This parentage scheme allows one to analyze atom properties due to one-electron
transitions. Then the electron wave function of an atom ΨLSML MS (1, 2, . . . , n) is
expressed through the wave function of an atomic core ΨL S ML MS (2, . . . , n) and
20
2 Elements of Atomic and Molecular Physics
Fig. 2.5. The lowest excited states of the silicon atom. Notations are similar to those
of Fig. 2.4
Fig. 2.6. The lowest excited states of the phosphorus atom. Notations are similar to those
of Fig. 2.4
through the wave function of a valence electron ψl 1 mσ (1) in the following manner
2
[32–35]:
ΨLSML MS (1, 2, . . . , n)
1
l
= √ P̂
GLS
(l,
n)
L S m
n
L ML S MS mσ
L
ML
× ψl 1 mσ (1) · ΨL S ML MS (2, . . . , n).
2
L
ML
1
2
σ
S
MS
S
MS
2.1 Properties of Atoms and Ions
21
Fig. 2.7. The lowest excited states of the sulfur atom. Notations are similar to those of Fig. 2.4
Fig. 2.8. The lowest excited states of the selenium atom. Notations are similar to those
of Fig. 2.4
Here n is a number of valence electrons and the operator P̂ permutes coordinates
and spin of a test electron that is denoted by an argument 1 with those of valence
electrons of the atomic core; LSML MS are the atom quantum numbers, L S ML MS
are the quantum numbers of the atomic core, l 12 mσ are the quantum numbers of
a test electron, GLS
L S (l, n) is the parentage or Racah coefficient [32, 33] and the
Clebsh–Gordan coefficients result from summation of momenta (orbital and spin
momenta) of a test electron and atomic core into an atom momentum.
Note that the one-electron wave function, by analogy with formula (2.1), has the
form
(2.15)
ψ(r, θ, ϕ) = Rnl (r)Ylm (θ, ϕ)χ1/2,σ
where the angular wave function Ylm is given by formula (2.4), χ1/2,σ is the spin
electron wave function, and the radial wave function Rnl (r) is normalized by the
condition (2.6) and satisfies to the Schrödinger equation (2.5) far from the core. The
Coulomb interaction potential between the electron and the core takes place at large
22
2 Elements of Atomic and Molecular Physics
Fig. 2.9. The lowest excited states of the tellurium atom. Notations are similar to those
of Fig. 2.4
electron distances r from the core, where the solution of this equation has the form
1
Rnl = Ar γ
−1
exp(−γ r),
rγ 1; rγ 2 1.
(2.16)
√
Here γ = (−2ε) in atomic units, and the electron behavior far from the core is
determined by two asymptotic parameters, an exponent γ and an amplitude A.
The parentage scheme relates to a non-relativistic approximation when the orbital momentum and spin are quantum numbers for an atom and its atomic core,
and this scheme accounts for the spherical atom symmetry if non-relativistic interactions are small. Then LSML MS , i.e. the atom angular momentum, spin and their
projections on a given direction, are the quantum numbers. In this approximation
the atom states are degenerated over the angular momentum and spin projections,
i.e. identical energies correspond to different projections on a given direction for
these momenta. The simplest construction of electron shells relates to s- and pvalence electrons, and the values of parentage coefficients for these cases are given
in Table 2.7.
Fractional parentage coefficients satisfy some relations that follow from the definition of these quantities. The normalization of the atomic wave function takes the
form
2
GLS
(2.17)
L S (l, n, v) = 1.
L S v
The total number of valence electrons equals 4l+2 if this electron shell is completed.
The analogy between electrons and vacancies connects the parameters of electron
shells in the following manner [32–35]:
(4l + 3 − n)(2S + 1)(2L + 1) 1/2
LS
L+L +S+S −l−1/2
·
GL S (l, n, v) = (−1)
n(2S + 1)(2L + 1)
S
× GL
LS (l, 4l + 3 − n, v).
(2.18)
2.1 Properties of Atoms and Ions
23
Table 2.7. Fractional parentage coefficients for electrons of s and p shells [11, 35]
Atom
Atomic core
GLS
L S Atom
Atomic core
s(2 S)
s 2 (1 S)
p(2 P )
p 2 (3 P )
p 2 (1 D)
p 2 (1 S)
p 3 (4 S)
(1 S)
s(2 S)
(1 S)
p(2 P )
p(2 P )
p(2 P )
p 2 (3 P )
p 2 (1 D)
p 2 (1 S)
p 2 (3 P )
p 2 (1 D)
p 2 (1 S)
p 2 (3 P )
p 2 (1 D)
1
1
1
1
1
1
1
0
0 √
1/ √
2
−1/ 2
0 √
−1/ 2
√
− 5/18
p 3 (2 P )
p 4 (3 P )
p 2 (1 S)
p 3 (4 S)
p 3 (2 D)
p 3 (2 P )
p 3 (4 S)
p 3 (2 D)
p 3 (2 P )
p 3 (4 S)
p 3 (2 D)
p 3 (2 P )
p 4 (3 P )
p 4 (1 D)
p 4 (1 S)
p 5 (2 P )
p 3 (2 D)
p 3 (2 P )
p 4 (1 D)
p 4 (1 S)
p 5 (2 P )
p 6 (1 S)
GLS
L S √
2/3
√
−1/ 3
√
5/12
−1/2
0
√
3/4
−1/2
0
0
1
√
3/5
√
1/√3
1/ 15
1
The parentage scheme relates to light atoms where relativistic effects are small.
Then, within the framework of the LS scheme of momentum summation, the angular atom momentum is the sum of angular momenta of an atomic core and test
electron, and the atom spin is the sum of spins of these particles. Then, as a result of the interaction of an atom angular momentum L and spin S, these momenta
are summed into the total atom momentum J , so that atom quantum numbers are
LSJ MJ , where MJ is the projection of the total atom momentum on a given direction. As is seen, splitting of the electron terms with given values of L and S due to
spin–orbit interaction leads to atom quantum numbers LSJ in contrast to quantum
numbers LS in neglecting spin–orbit interaction. Table 2.8 contains the numbers of
electron terms and electron levels for atoms with non-completed electron shells.
The parentage scheme is simple for valence s and p-electrons, and the values
of fractional parentage coefficients are represented in Table 2.7 for these cases. In
the case of d and f valence electrons, removal of one electron can lead to different
states of an atomic core at identical values of the atom angular momentum and spin.
To distinguish these states, one more quantum number v, the seniority, is introduced.
Table 2.9 contains the number of electron states and electron terms for atoms with
valence d-electrons.
The ground atom state for a given electron shell follows from the Hund empiric
law [36], according to which the maximum atom spin corresponds to the ground
electron state, and the total atom momentum is minimal if the shell is filled below
one half and is maximal from possible ones if the electron shell is filled by more than
half. For example, the nitrogen atom with the electron shell p 3 has three electron
terms, 4 S, 2 D, 2 P , and the electron term 4 S corresponds to its ground state. The
ground state of the aluminum atom with the electron shell p is characterized by the
total momentum 1/2 (the state 2 P1/2 ), and the ground state of the chlorine atom
24
2 Elements of Atomic and Molecular Physics
Table 2.8. States of atoms with non-filled electron shells
Shell configuration
s
s2
p, p 5
p2 , p4
p3
d, d 9
d 2, d 8
d 3, d 7
d 4, d 6
d5
f, f 13
f 2 , f 12
f 3 , f 11
f 4 , f 10
f 5, f 9
f 6, f 8
f7
Number of terms
1
1
1
3
3
1
5
8
18
16
1
7
17
47
73
119
119
Number of levels
1
1
2
5
5
2
9
19
40
37
2
13
41
107
197
289
327
Statistical weight
2
1
6
15
20
10
45
120
210
252
14
91
364
1001
2002
3003
3432
Table 2.9. Electron terms of atoms with filling electron shells d n [12]
n
0, 10
1, 9
2, 8
3, 7
4, 6
5
Electron terms
1S
2D
1 S, 3 P , 1 D, 3 F, 1 G
2 P , 4 P , 2 D(2), 2 F, 4 F, 2 G, 2 H
1 S(2), 3 P (4), 1 D(2), 3 D, 5 D,
1 F, 3 F (2), 1 G(2), 3 G, 3 H, 1 J
2 S, 6 S, 2 P , 4 P , 2 D(3), 2 F (2),
4 F, 2 G(2), 4 G, 2 H, 2 J
Number of states
1
10
45
120
Number of electron terms
1
2
9
19
210
40
252
37
with the electron shell p 5 is 2 P3/2 (the total momentum is 3/2). The ground states
of atoms with d and f filling electron shells are given in Tables 2.10 and 2.11.
Note that the parentage scheme holds true for light atoms both for the ground
and lower excited atom states, and it is violated for heavy atoms both due to an
increase of the role of relativistic effects and because of overlapping electron shells
with different nl.
2.1.5 Schemes of Coupling of Electron Momenta in Atoms
In construction of electron shells we were based above on the LS scheme of electron
momentum summation. Along with this, jj scheme of electron momentum summation is possible. In particular, taking an atom to be composed from an atomic core
with the angular momentum l and spin s and from a valence or excited electron
2.1 Properties of Atoms and Ions
25
Table 2.10. The ground states for atoms with filling d-shell [12]
Electron shell
d
d2
d3
d4
d5
d6
d7
d8
d9
Term of the ground state
2D
3F
4F
3/2
2
3/2
5D
0
6S
5/2
5D
4
4F
9/2
3F
4
2D
5/2
Atoms with this electron shell
Sc(3d), Y(4d), La(5d), Lu(5d), Ac(6d), Lr(6d)
Ti(3d 2 ), Zr(4d 2 ), Hf(5d 2 ), Th(6d 2 )
V(3d 3 ), Ta(5d 3 )
W(5d 4 )
Mn(3d 5 ), Tc(4d 5 ), Re(5d 5 )
Fe(3d 6 ), Os(5d 6 )
Co(3d 7 ), Ir(5d 7 )
Ni(3d 8 )
–
Table 2.11. The ground state of atoms with a filling f -shell [12]
Electron shell
f
f2
f3
f4
f5
f6
f7
f8
f9
f 10
f 11
f 12
f 13
Term of the ground state
2F
5/2
3H
4
4I
9/2
5I
4
6H
5/2
7F
0
8S
7/2
7F
6
6H
15/2
5I
8
4I
15/2
3H
6
2F
5/2
Atoms with this electron shell
–
–
Pr(4f 3 )
Nd(4f 4 )
Pm(4f 5 )
Sm(4f 6 ), Pu(5f 6 )
Eu(4f 7 ), Am(5f 7 )
–
Tb(4f 9 ), Bk(5f 9 )
Dy(4f 10 ), Cf(5f 10 )
Ho(4f 11 ), Es(5f 11 )
Er(4f 12 ), Fm(5f 12 )
Tm(4f 13 ), Md(5f 13 )
with the angular momentum le and spin se , one can realize the summation of these
momenta into the total atom momentum in the schemes
l + le = L,
l + s = j,
s + se = S, S + L = J (LS coupling),
le + se = je , j + je = J (jj coupling).
A certain hierarchy of interaction corresponds to each scheme of momentum
summation, and we will follow this connection. Let us represent the Hamiltonian of
interacting electron and atomic core in the form
Ĥ = Ĥcore + Ĥe + Aŝsˆe + B l̂ŝ + bl̂e ŝe .
(2.19)
The terms Ĥcore and Ĥe in this sum account for the electron kinetic energy and
electrostatic interaction for an atomic core and valence electron, respectively; the
third term describes an exchange interaction of the valence electron and atomic
core; and the last two terms are responsible for spin–orbit interaction for an atomic
26
2 Elements of Atomic and Molecular Physics
Table 2.12. Electron terms for filling of electron shells with valence p-electrons
LS scheme
p
p2
p3
p4
p5
p6
LS term
2P
2P
3P
3P
3P
1D
1S
4S
2D
2D
2P
2P
3P
3P
3P
1D
1S
2P
2P
1S
J
1/2
3/2
0
1
2
2
0
3/2
3/2
5/2
1/2
3/2
2
0
1
2
0
3/2
1/2
0
jj scheme
[1/2]1
[3/2]1
[1/2]2
[1/2]1 [3/2]1
[1/2]1 [3/2]1
[3/2]2
[3/2]2
[1/2]2 [3/2]1
[1/2]1 [3/2]2
[1/2]1 [3/2]2
[1/2]1 [3/2]2
[3/2]3
[1/2]1 [3/2]3
[1/2]2 [3/2]2
[1/2]21 [3/2]2
[1/2]1 [3/2]3
[3/2]4
[1/2]2 [3/2]3
[1/2]1 [3/2]4
[1/2]2 [3/2]4
J
1/2
3/2
0
1
2
2
0
3/2
3/2
5/2
1/2
3/2
2
0
1
2
0
3/2
1/2
0
core and valence electron. It is of importance that this Hamiltonian commutes with
the total atom momentum [37]
Ĵ = l̂ + ŝ + l̂e + ŝe .
(2.20)
This means that the total atom momentum is the quantum atom number for both
schemes of momentum summation at any relation between exchange and spin–orbit
interactions. Next, the LS-coupling scheme is valid in the limit of a weak spin–
orbit interaction, whereas the jj -coupling scheme corresponds to a weak exchange
interaction. In reality, the LS-coupling scheme is valid for light atoms, where the
relativistic interactions are small, whereas the jj -coupling scheme may be used as
a model for heavy atoms. Nevertheless, the LS-coupling scheme is used often for
notations of electron terms of heavy atoms, and Table 2.12 contains electron terms
of atoms with filling p shells for both coupling schemes. Notations for jj -coupling
schemes are such that the total momentum of an individual electron is indicated in
square parentheses and the total number of such electrons is given as a right superscript.
Thus, roughly in accordance with the hierarchy of interaction inside the atom,
we have two basic types of momentum coupling. If the electrostatic interaction exceeds a typical relativistic interaction, we have the LS-type of momentum summation; in another case we obtain the jj -type of momentum coupling. Really, the
2.1 Properties of Atoms and Ions
27
Fig. 2.10. The lowest excited states of the argon atom. Excitation energies count off from the
lowest excited state and are given in cm−1 . The wavelengths λ and radiative lifetimes τ are
indicated for a corresponding radiative transition
Fig. 2.11. The lowest excited states of the neon atom. Energies of excitation for corresponding states are indicated inside right rectangular boxes and are expressed in cm−1 . Radiative
lifetimes of corresponding states are given inside left rectangular boxes in ns. Wavelengths of
radiative transitions are indicated inside arrows and are expressed in Å, the rates of radiative
transitions are represented in parentheses near the wavelength and are expressed in 106 s−1
28
2 Elements of Atomic and Molecular Physics
Fig. 2.12. The lowest excited states of the argon atom. Notations are similar to those of
Fig. 2.11
electrostatic interaction Vel inside the atom corresponds to splitting of atom levels
of the same electron shell, while the relativistic interaction may be characterized by
the fine splitting δ of atom levels.
The specific case relates to atoms of inert gases, and Fig. 2.10 gives the lowest
group of excited states of the argon atom. On this example one can demonstrate the
ways of classifications of these states. The structure of the electron shell for these
states is 3p 5 4s, and, along with notations of the LS-coupling scheme, the notations
of the jj -coupling scheme are used. In the last case the total momentum of the
atomic core is given inside square brackets, and the total atom momentum is given
as a right subscript. In addition, the Pashen notations are used for lowest excited
states of inert gas atoms. The electron terms of the lowest group of excited states
are denoted as 1s5 , 1s4 , 1s3 , 1s2 , and the subscript decreases with an increase of
excitation.
In addition, the lowest excited states of inert gases are given in Figs. 2.11, 2.12,
2.13, and 2.14. Pashen notations for the next group of excited states with the electron shell np 5 (n + 1)p (n is the principal quantum number of electrons of the
valence shell) are from 2p10 up 2p1 as excitation increases. In notations of the
jj -scheme of momentum coupling, the state notation includes the state of an ex-
2.1 Properties of Atoms and Ions
29
Fig. 2.13. The lowest excited states of the krypton atom. Notations are similar to those of
Fig. 2.11
cited electron, the core state in square brackets, and the total atom momentum as a
subscript.
2.1.6 Parameters of Atoms and Ions in the Form of Periodic Tables
Diagram Pt4 gives the ionization potentials for atoms and their first ions in the
ground states on the basis of the periodic table of elements together with construction of the electron shell and the electron term for the ground atom state within the
framework of the LS scheme of momentum coupling. Because of the periodical
character of the electron shell structure, we obtain the periodical dependence for
atom parameters [36, 38]. The same is given in the diagram Pt5 for negative atomic
ions. Because of a short-range interaction for a valence electron and an atomic core,
in contrast to atoms, the number of bound states for a negative ion is finite (it is
often one or zero). The binding energies of excited states of negative ions are also
represented in diagram Pt5 in the cases when they exist (in particular, for elements
of the fourth group).
Diagram Pt6 contains parameters of lower excited states of atoms on the basis of
the periodic table. Parameters of lower resonant states from which a dipole radiative
2 Elements of Atomic and Molecular Physics
Pt4. Ionization potentials of atoms and ions
30
31
Pt5. Electron affinities of atoms
2.1 Properties of Atoms and Ions
2 Elements of Atomic and Molecular Physics
Pt6. Lowest excited states of atoms
32
2.2 Atomic Radiative Transitions
33
Fig. 2.14. The lowest excited states of the xenon atom. Notations are similar to those of
Fig. 2.11
transition is possible in the ground state are given in diagram Pt7. Diagram Pt8 lists
the splitting energies for lower atom states, and the polarizabilities of atoms in the
ground state are given in diagram Pt9.
2.2 Atomic Radiative Transitions
2.2.1 General Formulas and Conversion Factors for Atomic Radiative
Transitions
Table 2.13 contains the conversion factors in formulas for photon parameters and
the radiative lifetime of resonantly excited states.
Explanation of Table 2.13.
1. The photon energy ε = h̄ω, where ω is the photon frequency.
2. The photon frequency is ω = ε/h̄.
3. The photon frequency is ω = 2πc/λ, where λ is the wavelength, c is the light
speed.
4. The photon energy is ε = 2π h̄c/λ.
2 Elements of Atomic and Molecular Physics
Pt7. Resonantly excited atom states
34
Pt8. Splitting of lowest atom levels
2.2 Atomic Radiative Transitions
35
2 Elements of Atomic and Molecular Physics
Pt9. Atomic polarizability
36
2.2 Atomic Radiative Transitions
37
Table 2.13. The conversion factors for radiative transition between atom states
Number
Formula
Conversion factor C
Units used
1
ε = Cω
C = 4.1347 × 10−15 eV
C = 6.6261 × 10−34 J
2
ω = Cε
3
4
5
ω = C/λ
ε = C/λ
fo∗ = Cωd 2 g∗
6
7
fo∗ = Cd 2 g∗ /λ
1/τ∗o = Cω3 d 2 go
8
9
1/τ∗o = Cd 2 go /λ3
1/τ∗o = Cω2 go fo∗ /g∗
10
1/τ∗o = Cfo∗ go /(g∗ λ2 )
ω in s−1
ω in s−1
ε in eV
ε in K
ε in eV
λ in µm
ω in s−1 , d in Da
ε = h̄ω in eV , d in Da
λ in µm, d in Da
ω in s−1 , d in Da
ε = h̄ω in eV, d in Da
λ in µm, d in Da
ω in s−1 , d in Da
ε = h̄ω in eV; d in Da
λ in µm, d in Da
C = 1.519 × 1015 s−1
C = 1.309 × 1011 s−1
C = 1.884 × 1015 s−1
1.2398 eV
1.6126 × 10−17
0.02450
0.03038
3.0316 × 10−40 s−1
1.06312 × 106 s−1
2.0261 × 106 s−1
1.8799 × 10−23 s−1
4.3393 × 107 s−1
6.6703 × 107 s−1
a D is Debye, 1D = ea = 2.5418 × 10−18 CGSE
o
5. The oscillator strength for a radiative transition from a lower o to an upper ∗
state of an atomic particle that is averaged over lower states o and is summed
over upper states ∗ is equal to
fo∗ =
2me ω
2me ω 2
|o|D|∗|2 g∗ =
d g∗ ,
2
3h̄e
3h̄e2
where d = o|D|∗ is the matrix element for the operator of the dipole moment
of an atomic particle taken between transition states. Here me , h̄ are atomic
parameters, g∗ is the statistical weight of the upper state, and ω = (ε∗ − εo )/h̄
is the transition frequency, where εo , ε∗ are the energies of transition states.
6. The oscillator strength for radiative transition is [23, 29]
fo∗ =
4πcme 2
d g∗ .
3h̄e2 λ
Here λ is the transition wavelength; other notations are the same as above.
7. The rate of the radiative transition is [34, 35, 39]
1
4ω3 2
= B∗o =
d go .
τ∗o
3h̄c3
Here B is the Einstein coefficient; other notations are as above.
8. The rate of radiative transition is given by
1
32π 3 2
= B∗o =
d go .
τ∗o
3h̄λ3
38
2 Elements of Atomic and Molecular Physics
Table 2.14. Parameters of radiative transitions involving lowest states of the hydrogen atom,
so that i is the lower state and j is the upper transition state [22, 23]
Transition
1s–2p
1s–3p
1s–4p
1s–5p
2s–3p
2s–4p
2s–5p
2p–3s
2p–3d
2p–4s
2p–4d
2p–5s
2p–5d
3s–4p
3s–5p
fij
0.4162
0.0791
0.0290
0.0139
0.4349
0.1028
0.0419
0.014
0.696
0.0031
0.122
0.0012
0.044
0.484
0.121
τj i , ns
1.6
5.4
12.4
24
5.4
12.4
24
160
15.6
230
26.5
360
70
12.4
24
Transition
3p–4s
3p–4d
3p–5s
3p–5d
3d–4p
3d–4f
3d–5p
3d–5f
4s–5p
4p–5s
4p–5d
4d–5p
4d–5f
4f –5d
4f –5g
fij
0.032
0.619
0.007
0.139
0.011
1.016
0.0022
0.156
0.545
0.053
0.610
0.028
0.890
0.009
1.345
τj i , ns
230
36.5
360
70
12.4
73
24
140
24
360
70
24
140
70
240
Here λ is the wavelength of this transition; other notations are the same as
above.
9. The rate of radiative transition is
1
2ω2 e2 go
=
fo∗.
τ∗o
me c3 g∗
10. The rate of radiative transition is
1
8π 2 go
fo∗.
=
τ∗o
h̄g∗ λ2 c
2.2.2 Radiative Transitions between Atom Discrete States
The strongest radiative transitions between discrete atom states in a non-relativistic
case connect the atom states with nonzero matrix elements of the dipole moment operator. This follows from the above formulas for the oscillator strength of radiative
transitions and leads to certain selection rules for the radiative transitions. For transitions between hydrogen atom states when one electron is located in the Coulomb
field, these selection laws have the form [23, 29]
l = ±1,
m = 0, ±1.
(2.21)
Table 2.14 contains parameters of radiative transitions for the hydrogen atom.
Radiative transitions between states with zero matrix elements of the dipole moment operator are weaker than those for dipole radiation transitions. Such transitions
are named as forbidden radiative transitions. As the transition energy increases, the
2.2 Atomic Radiative Transitions
39
Table 2.15. Times of radiative transitions between lowest states for helium-like ions
Ion
Z
τ (21 P → 11 S), s τ (23 P → 11 S), s τ (21 S → 11 S), s τ (23 S → 11 S), s
Li+
3
10
20
30
40
50
–
3.9×10−11
1.1×10−13
6.0×10−15
1.3×10−15
5 × 10−16
2 × 10−16
3.6
Ne+8
Ca+18
Zn+28
Zr+38
Sn+48
ln τ
− dd ln
Z
5.6 × 10−5
1.8 × 10−10
2.1 × 10−13
8.1 × 10−15
1.4 × 10−15
5 × 10−16
9.5
5.1 × 10−4
1.0 × 10−7
1.2 × 10−9
1.0 × 10−10
2 × 10−11
4 × 10−12
5.8
49
9.2 × 10−5
7.0 × 10−8
1.1 × 10−9
6 × 10−11
6 × 10−12
7.3
difference in radiative transitions for resonant and forbidden transitions decreases.
This is demonstrated by the data of Table 2.15, where the times of identical radiative
transitions are compared for helium-like ions.
Spectra of atoms and ions involving lowest excited states can be represented
in the form of the Grotrian diagrams that indicate the positions of excited states
of atoms and ions, and parameters of radiative transitions with their participation.
These diagrams are given in some books [11, 40–48]. These diagrams are most
obvious for atoms and ions with simple electron shells. We give below the Grotrian
diagrams for atoms with the electron shells s, s 2 , and with the electron shell p k for
light atoms. These diagrams are taken from [11] and are given in Figs. A.1–A.28
in Appendix A. In addition, Figs. 2.1–2.14 contain lowest levels of some atoms and
parameters of radiative transitions between them [12]. The most information for the
rates of radiative transitions between atom states is collected by NIST information
center [49–53]. In addition, along with notations for excited states of gas atoms
with valence p-electrons given in Table 2.12 for LS- and jj -coupling schemes, the
Pashen notations for excited inert gas atoms are represented in Fig. 2.10, as well as
they are given in Figs. 2.11, 2.12, 2.13, and 2.14.
2.2.3 Absorption Parameters and Broadening of Spectral Lines
The character of absorption and emission of photons in a gas or plasma depends on
properties of this system. We consider these processes when they are determined by
transition between atom discrete states. Then absorption and emission of resonant
photons, the energy of which is close to the difference of atom state energies, is
determined by broadening of spectral lines [35, 54], i.e. broadening of energies of
atom states that partake in the radiative process. Let us introduce the distribution
function aω over frequencies of emitting photons, so that aω dω is the probability
that the photon frequency ranges from ω up to ω + dω. As the probability, the
frequency distribution function of photons is normalized as
(2.22)
aω dω = 1.
40
2 Elements of Atomic and Molecular Physics
Because spectral lines are narrow, in scales of frequencies of emitting photons, the
distribution function of photons is
aω = δ(ω − ω0 ),
(2.23)
where ω0 is the central frequency of an emitting photon. Hence, the rate and times
of spontaneous radiative emission are independent of the width and shape of a spectral line, i.e. of the photon distribution function aω . But it is not so when photons
are generated or absorbed as a result of interaction between the radiation field and
atoms. Such transitions are of two types, absorption of photons and induced emission of photons. The first process is described by the scheme
h̄ω + A → A∗ ,
(2.24)
and the process of induced emission is
nh̄ω + A∗ → (n + 1)h̄ω + A,
(2.25)
where A, A∗ denote an atom in the lower and upper states of transition, h̄ω is the
photon, and n is a number of identical incident photons. The cross section of absorption σabs as a result of transition between two atom states is given by [35, 39]
σabs =
π 2 c2
π 2 c 2 gf a ω
.
Aa
=
ω
ω2
ω 2 g0 τ
(2.26)
In the same manner, we have the following formula for the cross section of stimulated photon emission σem [35, 39]:
σem =
g0
π 2 c2
π 2 c 2 aω
=
Ba
=
σabs .
ω
2
2
gf
ω
ω τ
(2.27)
Here A, B are the Einstein coefficients, indices o and f correspond to the lower and
upper atom states, g0 , gf are the statistical weights of these atom states, and τ is the
radiative lifetime of the upper atom state with respect to the radiative transition to
the lower atom state.
The absorption coefficient kω is
N f g0
(2.28)
kω = N0 σabs − Nf σem = N0 σabs 1 −
N 0 gf
where N0 , Nf are the number density of atoms in the lower and upper states of
transition. The absorption coefficient kω for a weak radiation intensity Iω is defined
as
dIω
= −kω Iω ,
(2.29)
dx
where x is the direction of radiation propagation and, according to definition, the radiation flux does not perturb a gas where it propagates. In the case where population
2.2 Atomic Radiative Transitions
41
of atom states is determined by the Boltzmann formula, the absorption coefficient is
equal to
h̄ω
.
(2.30)
kω = N0 σabs 1 − exp −
T
Thus, the character of photon absorption depends on broadening of spectral
lines, and we give below three types of broadening that are of importance for excited
gases. The basis of Doppler broadening is the Doppler effect, according to which an
emitting frequency ω0 at a relative velocity vx of a radiating particle with respect to
a receiver is conceived as a frequency
vx
,
(2.31)
ω = ω0 1 +
c
where c is the light velocity. Hence, for radiating atomic particles with the Maxwell
distribution function on velocities, the distribution function of photons aω has the
form
1 mc2 1/2
mc2 (ω − ω0 )2
,
(2.32)
· exp −
aω =
ω0 2πT
2T ω02
where ω0 is the frequency for a motionless particle, m is the particle mass and T is
the gas temperature expressed in energetic units.
The Lorentz (or impact) mechanism of broadening of spectral lines results from
single collisions of a radiating atom with surrounding particles. As a result of these
collisions, the spectral line is shifted and broadens, and the distribution function aω
of radiating photons has the Lorenz form [35]
aω =
ν
.
2π (ω − ω0 + ν)2 + ( ν2 )2
(2.33)
Here ν = Nvσt , where N is the number density of perturbed particles, and σt is the
total cross section of collision of radiating and surrounding particles for an upper
state of a radiating particle under assumption that collision in the lower particle
state is not important. Next, ν = N vσ ∗ , and if the cross section is determined
by a large number of collision momenta, we have σt σ ∗ , and one can ignore the
spectral line shift. In the classical case where the main contribution to the total cross
section results from many collision momenta, the total cross section σt is given by
σt = πRt2 ,
Rt U (Rt )
∼ 1,
h̄v
(2.34)
where v is the collision velocity, U (R) is the difference of interaction potentials for
the upper and lower states of transition, and Rt is the Weiskopf radius. The criterion
of validity of the Lorenz broadening (2.33) is based on the assumption that the
probability to locate for two or more surrounding particles in a region of a strong
interaction with a radiating atom is small, which gives
N Rt3 1.
(2.35)
42
2 Elements of Atomic and Molecular Physics
Table 2.16. Broadening parameters for spectral lines of alkali metal atoms. λ is the photon
wavelength for resonant transition, τ is the radiative lifetime of the resonantly excited atoms,
ωD , ωL are given by formula (2.36), NDL is expressed in 1016 cm−3 and Nt is given in
1018 cm−3 . The temperature of alkali metal atoms is 500 K [16]
Element
Li
Na
Na
K
K
Rb
Rb
Cs
Cs
Transition
λ, nm
τ , ns
22 S → 32 P
2
1/2 → 3 P1/2
2
3 S1/2 → 32 P3/2
42 S1/2 → 42 P1/2
42 S1/2 → 42 P3/2
52 S1/2 → 52 P1/2
52 S1/2 → 52 P3/2
62 S1/2 → 62 P1/2
62 S1/2 → 62 P3/2
670.8
589.59
589.0
769.0
766.49
794.76
780.03
894.35
852.11
27
16
16
27
27
28
26
31
27
32 S
ωD ,
109 s−1
8.2
4.5
4.5
2.7
2.7
1.7
1.8
1.2
1.3
ωL /N ,
10−7 cm3 /s
2.6
1.6
2.4
2.0
3.2
2.0
3.1
2.6
4.0
NDL
Nt
16
15
9.4
6.5
4.2
4.5
2.9
2.3
1.6
3.2
2.5
1.4
1.2
0.6
0.7
0.4
0.3
0.2
Table 2.16 compares the Doppler ωD and Lorenz ωL widths for transition between the ground and resonantly excited states of an alkali metal atom when these
widths are given by the formulas
T
1
ωD = ω0
;
ωL = N vσt ,
(2.36)
2
2
mc
and also the number density NDL of atoms when the line widths for these mechanisms are coincided.
One more mechanism of broadening of spectral lines due to interaction with
surrounding atoms takes place at large number densities of surrounding atoms. As
a matter of fact, in this case, a system of interacting atoms emits radiation instead
of individual atoms, though this interaction is small [39, 55]. One can assume surrounding atoms to be motionless in the course of radiation, and the shift of the spectral line of a radiating atom corresponds to a certain configuration of surrounding
atoms
1
U (Rk ),
(2.37)
ω ≡ ω − ω0 =
h̄
k
where Rk is the coordinate of k-th atom in the frame of reference where the radiating
atom is the origin. The photon distribution function is equal for the wing of a spectral
line in the case of a uniform distribution of surrounding atoms
aω dω = N · 4πR 2 dR;
aω =
4NπR 2 h̄
,
dU/dR
(2.38)
where U (R) is the shift of a spectral line due to a surrounding atom located at
a distance R from the radiating atom. Table 2.16 contains the number density of
atoms Nt = Rt−3 for transition between the Lorenz and quasi-static mechanisms of
broadening.
2.3 Interaction Potential of Atomic Particles
at Large Separations
43
Table 2.17. Radiative parameters and the absorption coefficient for the center line of resonant radiative transitions in atoms of the first and second groups of the periodical system of
elements [14, 16]
Element Transition
H
12S → 22P
He
11S → 21P
Li
22S → 32P
Be
21S → 21P
2
Na
3 S1/2 → 32P1/2
Na
32S1/2 → 32P3/2
Mg
31S → 31P
2
K
4 S1/2 → 42P1/2
K
42S1/2 → 42P3/2
Ca
41S → 41P
2
Cu
4 S1/2 → 42P1/2
Cu
42S1/2 → 42P3/2
Zn
41S → 41P
Rb
52S1/2 → 52P1/2
Rb
52S1/2 → 52P3/2
Sr
51S → 51P
2
Ag
5 S1/2 → 52P1/2
Ag
52S1/2 → 52P3/2
Cd
51S → 51P
2
Cs
6 S1/2 → 62P1/2
Cs
62S1/2 → 62P3/2
Ba
61S → 61P
2
Au
6 S1/2 → 62P1/2
Au
62S1/2 → 62P3/2
Hg
61S → 61P
λ, nm
121.57
58.433
670.8
234.86
589.59
589.0
285.21
769.0
766.49
422.67
327.40
324.75
213.86
794.76
780.03
460.73
338.29
328.07
228.80
894.35
852.11
553.55
267.60
242.80
184.95
τ , ns
1.60
0.56
27
1.9
16
16
2.1
27
27
4.6
7.0
7.2
1.4
28
26
6.2
7.9
6.7
1.7
31
27
8.5
6.0
4.6
1.3
g∗ /g0
3
3
3
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
vσt , 10−7 cm3 /s k0 , 105 cm−1
0.516
8.6
0.164
18
5.1
1.6
9.6
1.4
3.1
1.1
4.8
1.4
5.5
3.4
4.1
0.85
6.3
1.1
7.3
2.5
1.1
2.1
1.7
2.8
3.3
4.8
3.9
0.91
6.2
1.2
9.4
1.7
1.1
2.0
1.7
2.9
3.3
4.5
5.3
0.77
8.1
1.0
9.0
1.9
0.49
1.9
0.75
4.2
2.2
5.7
In the case of radiative transition between the resonantly excited and ground
atom state in a parent gas or vapor, the interaction potential depends as R −3 on the
distance between atoms, and the total cross section σt 1/v, so that vσt is independent of the collision velocity. Hence, the absorption coefficient in the line center
k0 is independent of the number density of atoms and their temperature. Table 2.17
contains their values for transitions between the ground and lowest resonantly excited states of alkali metal and alkali earth metal atoms.
2.3 Interaction Potential of Atomic Particles
at Large Separations
We give below a long-range interaction involving atoms or atoms and ions at large
distances between them that will allow us to separate different interaction types
44
2 Elements of Atomic and Molecular Physics
Fig. 2.15. Geometry of interaction for bound electrons of different atoms
and find the condition of formation of chemical bonds. A long-range interaction of
atomic particles is determined by distribution of valence electrons, and the operator
of atom interaction V results from Coulomb interaction of valence electrons with
electrons of another atom and its core
=−
V
i
Z1 Z2 e2
e2
e2
e2
−
+
+
.
|ri − R|
|rk + R|
|rk + R − ri |
R
k
(2.39)
i,k
Here R is the vector joined nuclei, and the coordinates of valence electrons ri , rk are
counted off their nuclei (see Fig. 2.15). Expanding the operator (2.39) over a small
parameter r/R, after an average over the electron distribution we obtain different
types of a long-range interaction between atomic particles. The strongest term is
proportional to R −2 and is the ion interaction with the atom dipole moment if it
is not zero, as it takes place for ion interaction with an excited hydrogen atom,
which state is characterized by parabolic quantum numbers n, n1 , n2 , m. Then the
interaction potential has the form [23, 24]
U (R) =
3Z h̄2 (n2 − n1 )
2me R 2
(2.40)
where Z is the ion charge expressed in electron charges.
The next expansion term is proportional to R −3 and is an interaction of ion
charge with the atom quadrupole moment
U (R) =
Ze2 Q
,
R3
(2.41)
where Q is the component of the quadrupole moment tensor on the axis that joins
nuclei. In the one-electron approximation this quantity is [56, 57]
Q=2
i
ri2 P2 (cos θi ) = 2
li (li + 1) − 3m2
i 2
r ,
(2li − 1)(2li + 3) i
(2.42)
i
where ri , θi are the spherical coordinates of the valence electron. The quadrupole
moment is zero for a completed electron shell and for an average over the momentum projections. The values of the quadrupole moment for atoms with a filling
p-shell are given in Table 2.18 [12, 58].
The next expansion term for the long-range interaction potential is interaction
of a charge with an induced atom moment and, for a unit ion charge, is given by
2.3 Interaction Potential of Atomic Particles
at Large Separations
45
Table 2.18. The quadrupole moment Q of atoms with filling p-shell for the LS-coupling
scheme; M is the atom momentum projection onto the molecular axis
Atom states
M=0
|M| = 1
(p)2 P
4/5
−2/5
(p 2 )2 P
−4/5
2/5
(p 3 )4 S
0
–
(p 4 )3 P
4/5
−2/5
(p 5 )2 P
−4/5
2/5
Table 2.19. Given in atomic units, the constant of van der Waals interaction C6 between two
identical atoms of inert gas
Interacting atoms
C6
He–He
1.5
Ne–Ne
6.6
Ar–Ar
68
Kr–Kr
130
Xe–Xe
270
Fig. 2.16. Regions of coordinates of valence electrons that are responsible for atom interaction at large distances. 1—region of location of atomic electrons; 2—region that is responsible
for long-range atom interaction at large separations; 3—region of electron coordinates that is
responsible for exchange interaction of atoms.
U (R) = −
αe2
2R 4
(2.43)
where α is the atom polarizability, the values of which are given in diagram Pt9. The
interaction potential of two atoms due to induced dipole moments (van der Waals
interaction) when one of these takes a completed shell has the form
U (R) = −
C6
,
R6
(2.44)
and the values of C6 for two inert gas atoms are given in Table 2.19.
Exchange interaction of atomic particles results from overlapping of wave functions of valence electrons and is determined by coordinate regions as shown in
Fig. 2.16. Because long-range and exchange interactions between atomic particles
are given by different coordinate regions at large distances between them, the total
interaction potential is the sum of these interaction potentials. According to the nature of the exchange interaction potential (R), we have the following estimation
in the case of exchange by one electron located in the field of identical atomic cores:
(R) ∼ |ψ(R/2)|2 ∼ exp(−γ R),
where ψ(r) is the wave function of the valence electron, and γ 2 = 2me J /h̄2 with
the ionization potential J . This interaction potential with exchange by one electron
46
2 Elements of Atomic and Molecular Physics
corresponds to ion–atom interaction. Interaction of two atoms is accompanied by
exchange by two valence electrons, and the exchange interaction potential of two
atoms at large separations is (R) ∼ exp(−2γ R). In particular, the exchange interaction potential of two hydrogen atoms is (R) ∼ exp(−2R/a0 ) at large distances
between them.
The exchange interaction potential may have a different sign that determines the
possibility of formation of a chemical bond for a given electron term at the approach
of atomic particles. If this sign is negative, a covalence chemical bond is formed at
moderate distances between atomic particles and the number of attractive and repulsed electron terms is approximately equal. In particular, we consider interaction
of a structureless ion and a parent atom with a valence s-electron. At large distances
between nuclei R, the eigen functions (ψg and ψu ) of a system of interacting ion
and atom correspond to the even (gerade) and odd (ungerade) states, and are given
by
ψ(1) + ψ(2)
ψ(1) − ψ(2)
;
ψu =
(2.45)
ψg =
√
√
2
2
at large distances between nuclei. Here ψ(1), ψ(2) corresponds to electron location
in the field of the first and second atomic cores. Correspondingly, if an electron is
located in the field of one core, its state is a combination of the even and odd states.
In particular, the wave function for electron location in the field of the first atomic
core is
ψg + ψ u
.
(2.46)
ψ(1) = √
2
As is seen, the atomic electron term is split in the even εg (R) and odd εu (R)
molecular electron terms at finite distances R between nuclei, and the even term
corresponds to attraction while the odd term corresponds to repulsion. The exchange
interaction potential (R) is the difference of the energies of these states and, for
the s-valence electron in the field of structureless cores, this quantity at large R is
equal to [59]
(R) ≡ |εg (R) − εu (R)| = A2 R 2/γ −1 exp(−Rγ − 1/γ ),
(2.47)
where A, γ are the parameters of the asymptotic expression (2.16) for the electron
wave function.
Another type of exchange interaction of two atoms A–B is realized if an electron term A–B correlates with an electron term A− –B + at large separations, where
a valence electron of one atom transfers to another atom. The Coulomb interaction
of positive and negative ions leads to attraction, and atoms form an ion–ion bond in
this case. This bond is typical for atoms with a large electron affinity, in particular,
for halogen atoms, and is realized into molecules consisting of halogen and alkali
metal atoms. Excimer molecules have such a chemical bond.
Another type of exchange interaction takes place when one atom is excited. The
nature of this interaction consists in penetration of an excited electron to a nonexited atom when it is located in the field of the parent core. In the limiting case
2.4 Properties of Diatomic Molecules
47
of a highly excited atom this interaction potential is given by the Fermi formula
[60–62]
2π h̄2 L
|Ψ (R)|2 ,
(2.48)
U (R) =
me
where R is the non-excited atom coordinate, Ψ (R) is the wave function of an excited electron, and L is the electron scattering length for a non-excited atom. The
Fermi formula (2.48) is valid if the size of electron orbit as well as the distance
R between atoms significantly exceeds the size of a non-excited atom. This interaction may be used as a model for systems and processes involving interaction of
excited atoms [63]. As is seen, there are many forms of interaction between atomic
particles. At large distances between particles, these interactions may be separated,
which allows one to ascertain the role of certain interactions.
2.4 Properties of Diatomic Molecules
2.4.1 Bound States of Diatomic Molecule
A diatomic molecule is a bound state of two atoms. The energy ε of a given electron
state as a function of a distance R between motionless nuclei is the electron term.
A small parameter me /μ, where me is the electron mass and μ is the reduced mass of
nuclei, allows us to separate the electron, vibration and rotation degrees of freedom.
Indeed, a typical difference of energies for neighboring electron terms is ε0 ∼ 1 eV,
a typical energy difference between neighboring vibration states is ∼(me /μ)1/2 ε0
and a typical energy difference for neighboring rotation states is ∼(me /μ)ε0 .
We now give standard notations for the molecule energy [64]. The total excitation energy of the molecule T , accounting for the separation of degrees of freedom,
may be represented in the form [64]
T = Te + G(v) + Fv (J ),
(2.49)
where Te is the excitation energy of an electron state that corresponds to excitation
of this electron term from the ground state of the molecule, G(v) is the excitation
energy of a vibrational state and Fv (J ) is the excitation energy of a rotational state.
Formula (2.49) includes the main part of the vibrational and rotational energy for a
weakly excited molecule. Near the minimum of the electron term that corresponds
to the stable molecule state, the vibrational and rotational energy of a molecule,
accounting for the first expansion terms, has the form
G(v) = h̄ωe (v + 1/2) − h̄ωe x(v + 1/2)2 ,
Fv (J ) = Bv J (J + 1),
Bv = Be − αe (v + 1/2).
(2.50)
Here v and J are the vibration and rotation quantum numbers (J is the rotation momentum), which are whole numbers starting from zero. The spectroscopic parameters h̄ωe , h̄ωe x are the vibration energy and the anharmonic parameter, Be = h̄2 /(2I )
48
2 Elements of Atomic and Molecular Physics
Table 2.20. The cases of Hund coupling [66, 67]
Hund case
a
b
c
d
e
Relation
Ve Vm Vr
Ve Vr Vm
Vm Ve Vr
Vr Ve Vm
Vr Vm Ve
Quantum numbers
Λ, S, Sn
Λ, S, SN
Ω
L, S, LN , SN
J, JN
is the rotation constant, where the inertia momentum I = μRe2 , Re is the distance
between atoms for the electron term minimum, the equilibrium distance. Bv is the
rotation constant for a vibrational excited state. These parameters for diatomic molecules and molecular ions with identical nuclei are contained in periodic tables Pt10,
Pt11 and Pt12 [2, 65, 64].
We give below the classification of interactions inside the diatomic molecule,
keeping the standard Hund scheme of momentum coupling [24, 66, 67]. Then we are
based on three interaction types inside the molecule: the electrostatic interaction Ve
(interaction between the orbital angular momentum of electrons and the molecular
axis), spin–orbit interaction Vm and interaction between the orbital and spin electron
momenta with rotation of the molecular axis Vr . A certain scheme of coupling is
determined by the hierarchy of these interactions, and possible coupling schemes
are given in Table 2.20.
Each type of momentum coupling gives certain quantum numbers that describe
the electron molecule state, correspond to a given hierarchy of interactions and are
represented in Table 2.20. Indeed, denote by L the electron angular momentum; S
the total electron spin; j the total electron momentum, that is, the sum of the angular
and spin momenta (j = L + S); n the unit vector along the molecular axis; and K the
rotation momentum of nuclei. Depending on the hierarchy of interactions, we can
obtain the following quantum numbers: Λ, the projection of the angular momentum
of electrons onto the molecular axis; Ω, the projection of the total electron momentum J onto the molecular axis; and Sn , the projection of the electron spin onto the
molecular axis. LN , SN , JN are projections of these momenta onto the direction of
nuclei rotation momentum K. Note that the operator of the total molecule momentum J = L + S + K = j + K commutes with the molecule Hamiltonian, and the
eigenvalues of the operator J are the molecule quantum numbers for various cases
of Hund coupling.
The character of momentum coupling under consideration describes not only the
molecule structure and quantum numbers of a diatomic molecule, but is of importance for dynamics of atomic collisions [68–71]. Then in the course of collision of
two atomic particles, when a distance between two colliding atomic particles varies,
the character of momentum coupling may also be changed. Transition from one
coupling scheme to another leads simultaneously to a change in the quantum numbers of colliding particles. This may be responsible for some transitions in atomic
collisions [12, 70, 71].
Pt10. Parameters of homonuclear molecules
2.4 Properties of Diatomic Molecules
49
Pt11. Parameters of homonuclear positive diatomic ions
50
2 Elements of Atomic and Molecular Physics
Pt12. Parameters of homonuclear negative diatomic ions
2.4 Properties of Diatomic Molecules
51
52
2 Elements of Atomic and Molecular Physics
The classical Hund scheme of momentum coupling inside a molecule gives
a qualitative description of this problem. In reality, the number of interactions is
greater, and the hierarchy of interactions is more complex than that in the Hund cases
of momentum coupling inside a molecule. We demonstrate this on a simple example
of interaction between a halogen ion and its atom at large distances between them
that are responsible for the charge exchange process in the collision of these atomic
particles. For definiteness, we take interaction between these particles at a distance
R0 , so that the cross section of resonant charge exchange is πR02 /2 at the collision
energy of 1 eV. Because this distance is large compared to an atom or ion size, this
simplifies the problem and allows us separate various types of interaction.
Let us enumerate interactions in the quasi-molecule X2+ (X is the halogen atom
in the ground electron state), which have the form
Vex ,
UM =
QMM
,
R3
Um =
QMM qmm
,
R5
(R),
δi ,
δa ,
Vrot .
(2.51)
Here we divide the electrostatic interaction Ve of Table 2.20 into four parts: the exchange interaction, Vex , inside the atom and ion that is responsible for electrostatic
splitting of levels inside an isolated atom and ion; the long-range interaction, UM , of
the ion charge with the quadrupole moment of the atom; the long-range interaction,
Um , which is responsible for the splitting of ion levels; and the ion–atom exchange
interaction potential, , which is determined by electron transition between atomic
cores. Instead of the relativistic interaction Vm of Table 2.20, we introduce separately the fine splitting of levels δi for the ion (the splitting of 3 P0 and 3 P2 ion
levels) and δa (the splitting of 3 P1/2 and 3 P3/2 atom levels) for the atom. Here M, m
are the projections of the atom and ion angular momenta on the molecular axis, R is
an ion–atom distance, Qik is the tensor of the atom quadrupole moment and qik is
the quadrupole moment tensor for the ion. Taking for simplicity the impact parameter of ion–atom collision to be R0 , we have for the rotation energy at the closest
approach of colliding particles
h̄v
,
Vrot =
R0
where v is the relative ion–atom velocity, and we use the expression below for estimation of the Coriolis interaction. As is seen, the number of possible coupling cases
is larger than that in the classical case. Of course, only a small fraction of these cases
can be realized.
Table 2.21 lists the values of the above interactions for the halogen molecular
ion at a distance R0 between nuclei that is responsible for the cross section of the
resonant charge exchange process. Comparing different types of interaction, we obtain the following hierarchy of interactions in this case
Vex δi ,δa UM Um ,Vrot .
(2.52)
Comparing this with the data of Table 2.20, we obtain an intermediate case between
cases “a” and “c” of the Hund coupling, but the hierarchy sequence (2.52) does not
correspond exactly to any one of the Hund cases.
2.4 Properties of Diatomic Molecules
53
Table 2.21. Parameters of interaction of a positive halogen ion with a parent atom in the
ground electron states at a distance R0 that is responsible for resonant charge exchange at the
collision energy of 1 eV [72]
R0 , a0
δa , cm−1
δi , cm−1
Vex , cm−1
δi /Vex
UM , cm−1
UM /δa
Vrot , cm−1
(R0 ), cm−1
F
Cl
Br
I
10.6
404
490
20873
0.023
341
0.84
30
23
13.8
882
996
11654
0.085
407
0.46
17
14
15.1
3685
3840
11410
0.34
448
0.12
10
8.4
17.2
7603
7087
13727
0.52
372
0.049
7.1
6.1
Thus, we conclude that the classical Hund cases of electron momentum coupling in diatomic molecules may be used for qualitative description of the molecule
structure and its quantum numbers, but in reality the interaction inside molecules is
more complex.
2.4.2 Correlation between Atomic and Molecular States
When two atoms form a molecule, electron states of these atoms as well as the character of coupling of electron momenta in atoms and molecules determine possible
electron states of the forming molecules. In other words, there is a correlation between electron states of atoms and molecules [73–76]. We consider this below for a
molecule consisting of two identical atoms.
If a molecule consists of identical atoms, a new symmetry occurs that corresponds to the electron reflection with respect to the plane that is perpendicular to
the molecular axis and divides it into two equal parts. The electron wave function of
an even (gerade) state conserves its sign as a result of this operation and the wave
function of the odd (ungerade) state changes a sign when the electrons are reflected
with respect to the symmetry plane. The corresponding states are denoted by g and
u, respectively. The operator of electron reflection with respect to any plane passed
through the molecular axis commutes with the electron Hamiltonian also, and the
parity of a molecular state is denoted by + or − depending on conservation or
change of the sign of the electron wave function in this operation. The parity of the
molecular state is the sum of parities of atomic states when atoms are removed on infinite distance and this value does not change at the variation of a distance between
nuclei. Thus, in the “a” case of the Hund coupling, molecular quantum numbers
are the projection Λ of the angular momentum on the molecular axis, the total molecule spin S and its projection onto a given direction, the evenness and parity of this
state which is degenerated with respect to the spin projection. Table 2.22 represents
the correlation between states of two identical atoms and a forming molecule [77]
in the case “a” of Hund coupling when electrostatic interaction dominates, which
54
2 Elements of Atomic and Molecular Physics
Table 2.22. Correlation between states of two identical atoms in the same electron states and
the molecule consisting of these atoms in the case “a” of Hund coupling [77]. The number of
molecule states of a given symmetry is indicated in parentheses if it is not 1
Atomic states
Dimer states
1S
1Σ +
g
1Σ +, 3Σ +
g
u
1Σ +, 3Σ +, 5Σ +
g
u
g
1Σ +, 3Σ +, 5Σ +, 7Σ +
g
u
g
u
1 Σ + (2), 1 Σ − , 1 Π , 1 Π , 1 Δ
g
u
g
g
u
1 Σ + (2), 1 Σ − , 1 Π , 1 Π , 1 Δ , 3 Σ + (2), 3 Σ − , 3 Π , 3 Π , 3 Δ
g
u
g
g
u
u
g
g
u
g
1 Σ + (2), 1 Σ − , 1 Π , 1 Π , 1 Δ , 3 Σ + (2), 3 Σ − , 3 Π , 3 Π , 3 Δ ,
g
u
g
g
u
u
g
u
u
g
5 Σ + (2), 5 Σ − , 5 Π , 5 Π , 5 Δ
g
u
g
g
u
1 Σ + (2), 1 Σ − , 1 Π , 1 Π , 1 Δ , 3 Σ + (2), 3 Σ − , 3 Π , 3 Π , 3 Δ
g
u
g
g
u
u
g
g
u
g
5 Σ + (2), 5 Σ − , 5 Π , 5 Π , 5 Δ , 7 Σ + (2), 7 Σ − , 7 Π , 7 Π , 7 Δ
g
u
g
g
u
g
g
u
u
g
1 Σ + (3), 1 Σ − (2), 1 Π (2), 1 Π (2), 1 Δ (2), 1 Δ , 1 Φ , 1 Φ , 1 Γ
g
u
g
u
g
u
g
g
u
1 Σ + (3), 1 Σ − (2), 1 Π (2), 1 Π (2), 1 Δ (2), 1 Δ , 1 Φ , 1 Φ , 1 Γ ,
g
u
g
u
g
u
g
g
u
3 Σ + (3), 3 Σ − (2), 3 Π (2), 3 Π (2), 3 Δ , 3 Δ (2), 3 Φ , 3 Φ , 3 Γ
g
u
g
u
g
u
u
u
g
1 Σ + (3), 1 Σ − (2), 1 Π (2), 1 Π (2), 1 Δ (2), 1 Δ , 1 Φ , 1 Φ , 1 Γ ,
g
u
g
u
g
u
g
g
u
3 Σ + (3), 3 Σ − (2), 3 Π (2), 3 Π (2), 3 Δ , 3 Δ (2), 3 Φ , 3 Φ , 3 Γ ,
g
u
g
u
g
u
u
u
g
5 Σ + (3), 5 Σ − (2), 5 Π (2), 5 Π (2), 5 Δ (2), 5 Δ , 5 Φ , 5 Φ , 5 Γ
g
u
g
u
g
u
g
g
u
2S
3S
4S
1P
2P
3P
4P
1D
2D
3D
includes the exchange interaction in atoms. This coupling character relates to molecules consisting of light atoms with the LS scheme of momentum coupling in atoms.
The “a” Hund case corresponds to light atoms for which the LS scheme of momentum coupling holds true. Figures 2.17, 2.18, 2.19, 2.20 and 2.21 [11] give the
potential curves or electron terms for lowest electron states of hydrogen, helium, carbon, nitrogen and oxygen diatomic molecules, where the “a” Hund case is realized,
and the structure of valence electron shells is s, s 2 , 2p 2 , 2p 3 and 2p 4 , respectively.
These figures also contain electron terms of the corresponding molecular ions. In
these cases an electron state of a molecule is denoted by letters X, A, B, C, etc. for
the states with zero total spin and by letters a, b, c, etc. for the states with one total
spin. The projection of the molecule angular momentum onto the molecular axis
as a quantum number is denoted by letters Σ, Π, Δ, etc. when the electron momentum projection is one, two, three, etc. The total molecule spin is the quantum
number in the “a” Hund case and the multiplicity of the spin state 2S + 1 (S is the
molecule spin) is given as a left superscript for the state notation. Next, the state
parity (+ or −) is given as a right superscript, and the state evenness for a molecule consisting of identical atoms (g or u) is represented as a right subscript. These
notations are used in Figs. 2.17, 2.18, 2.19, 2.20 and 2.21.
The correlation between atomic and molecular states in the “c” case of Hund
coupling, when the coupling in atoms corresponds to the jj -coupling scheme, may
be fulfilled in the same manner. This correlation is given in Table 2.23, that is taken
from [77].
2.4 Properties of Diatomic Molecules
55
Fig. 2.17. Potential curves of hydrogen molecule (H2 ) involving excited hydrogen atoms
Table 2.23. Correlation between states of two identical atoms in the same electron states and
the molecule consisting of these atoms in the case “c” of Hund coupling [77]. The number of
molecule states of a given symmetry is indicated in parentheses if it is not 1
Atom states
J
J
J
J
J
=0
= 1/2
=1
= 3/2
=2
Dimer states
0+
g
−
1u , 0+
g , 0u
−
2g , 1u , 1g , 0+
g (2), 0u
−
3u , 2g , 2u , 1g , 1u (2), 0+
g (2), 0u (2)
−
4g , 3g , 3u , 2g (2), 2u , 1g (2), 1u (2), 0+
g (3), 0u (2)
56
2 Elements of Atomic and Molecular Physics
Fig. 2.18. Potential curves of helium molecule (He2 ) involving excited helium atoms. The
energy of electron terms expressed in cm−1 starts from the ground vibration and the lowest
electron excited state. The energy of electron terms in eV begins from the ground vibration
and electron states
In the Hund “c” case [66, 67], when the spin–orbit interaction potential exceeds
the level splitting for a different projection of the angular electron momentum, the
total electron momentum, that is, the molecule quantum number, is denoted by
2.4 Properties of Diatomic Molecules
57
Fig. 2.19. Potential curves of diatomic carbon molecule (C2 )
a large value. In addition, depending on the behavior of the electron wave function as a result of the reflection with respect to the plane passed through molecule
axis, the electron state is denoted by superscripts + or − after the electron momentum. Transition from the Hund case “a” to “c” leads to the following change in term
notations for a diatomic molecule:
X 1 Σg+ → 0+
g;
a 3 Σu+ → 1u , 0−
u;
A1 Σu+ → 0+
u;
b3 Σg+ → 1g , 0−
g . (2.53)
As an example, Fig. 2.22 contains electron terms for the argon molecule that consists of atoms in the ground Ar(3p 6 ) and excited Ar(3p 5 4s) states. States of an
excited atom at large separations are given for the LS scheme of momentum coupling. The same behavior of electron terms takes place for molecules of other inert
gases.
We note that the quantum numbers of the molecule electron state are defined
accurately only to a restricted number of interactions as they take place in the Hund
58
2 Elements of Atomic and Molecular Physics
Fig. 2.20. Potential curves of diatomic nitrogen molecule (N2 )
cases of momentum coupling. In particular, a simple description relates to light
atoms when non-relativistic interactions dominate. The description of molecules
consisting of heavy atoms, when different types of interaction partake in molecule
construction, is in reality outside of the Hund scheme. In particular, let us return
to the case of Table 2.21 for the interaction of a halogen positive ion with a parent atom at large distances that are responsible for the resonant charge exchange
process. Then, according to the hierarchy of interactions in the formula (2.52), the
2.4 Properties of Diatomic Molecules
59
Fig. 2.21. Potential curves of diatomic oxygen molecule (O2 )
molecule quantum numbers are J MJ j and the evenness (g or u), where J is the
total atom momentum, MJ is its projection on the molecular axis and j is the total
ion momentum. Fig. 2.23 contains electron terms of the molecular diatomic ion Cl+
2
in a range of large distances between nuclei [72].
Table 2.24 contains parameters of the lowest excited states of inert gas diatomic
molecules, so that Re is the equilibrium distance between nuclei, De is the minimum
interaction potential that corresponds to this distance and τ is the molecule radiative
lifetime. One more peculiarity of interaction between atoms follows from the character of exchange interaction. At large separation, the exchange interaction potential
is given by the Fermi formula (2.48), and the electron scattering length L in this formula is positive for helium and neon atoms and is negative for argon, krypton and
60
2 Elements of Atomic and Molecular Physics
Fig. 2.22. Potential curves of diatomic argon molecule consisting of atoms in the ground
Ar(2p 6 ) and lowest excited Ar(2p 5 3s) states. They start from the lowest excited states of the
atom located at large distances from the argon atom in the ground state and correspond to
only odd molecule states and correspond to only odd molecule states
Table 2.24. Parameters of lowest excited states of inert gas molecules
Molecule, state
He2 (a 3 Σu+ )
He2 (A1 Σu+ )
Ne2 (a 3 Σu+ )
Ar2 (1u , 0−
u)
Ar2 (0+
u)
Kr2 (0+
u)
Xe2 (1u , 0−
u)
Xe2 (0+
u)
Re , Å
1.05
1.06
1.79
2.4
2.4
De , eV
2.0
2.5
0.47
0.72
0.69
3.03
3.02
0.79
0.77
τ , 10−7 s
800
0.28
360
0.5
30
0.6
11
0.6
2.4 Properties of Diatomic Molecules
61
Fig. 2.23. Potential curves of the chlorine molecular ion Cl+
2 resulting from interaction of
the atom Cl(2 P ) and ion Cl+ (3 P ) in the lowest electron states at large separations, which
are responsible for the resonant charge exchange process [72]. The excitation energies are
counted off from the ground ion and atom states with infinite distance between them and are
expressed in cm−1 . The indicated quantum numbers of electron terms are J MJ j and the
evenness, where J , j are the total momenta of the atom and ion, and MJ is the projection of
the total atom momentum onto the molecular axis
62
2 Elements of Atomic and Molecular Physics
Fig. 2.24. Potential curves of molecule CH
Table 2.25. The distance Rh between nuclei and the hump height ε for the interaction
potential of excited and non-excited atoms of helium and neon
Molecule, state
Rh , Å
ε, eV
He2 (a 3 Σu+ )
He2 (A1 Σu+ )
Ne2 (a 3 Σu+ )
Ne2 (A1 Σu+ )
3.1
2.8
2.6
2.5
0.06
0.05
0.11
0.20
xenon atoms. This means that repulsion at large distances in helium and neon excited molecules is changed by attraction at moderate distances between interacting
atoms. Therefore, the interaction potential contains a hump at large separations, and
the parameters of this hump are given in Table 2.25.
2.4 Properties of Diatomic Molecules
63
Fig. 2.25. Potential curves of molecule NH
Table 2.26. The polarizabilities α of diatomic homonuclear molecules expressed in a03
Molecule
H2
Li2
N2
O2
F2
Na2
α
5.42
230
11.8
10.8
93
260
Molecule
Al2
Cl2
K2
Br2
Rb2
Cs2
α
130
31
500
474
530
700
In addition, Table 2.26 contains the polarizabilities of homonuclear diatomic
molecules, and the diagram Pt13 gives the electron affinities of homonuclear diatomic molecules.
The properties of diatomic molecules including different atoms with nearby ionization potentials are similar to those of homonuclear diatomic molecules. This is
confirmed by Figs. 2.24, 2.25, and 2.26, which contain potential curves for the lowest states of molecules CH, NH, OH, and diagram Pt14, which gives the affinities
2 Elements of Atomic and Molecular Physics
Pt13. Electron affinity of molecules
64
Pt14. Affinity of atoms to hydrogen and oxygen atoms
2.4 Properties of Diatomic Molecules
65
66
2 Elements of Atomic and Molecular Physics
Fig. 2.26. Potential curves of molecule OH
of various atoms to the hydrogen and oxygen atom, i.e. the dissociation energies of
corresponding diatomic molecules. Note that alongside with quantum numbers of
electron states, the letters X, A, B, C etc. are used for notations of excited states in
the course of an excitation increase. For light atoms when electron terms with different spins are independent the notations a, b, c etc. are used if the total molecule
spin differs from that of the ground molecule state. In addition, a prime may be used
for the states which became known when a certain sequence of states was created.
2.4.3 Excimer Molecules
Excimer molecules consist of an atom in the ground state and an atom in excited
state, and these atoms form a strong chemical bond. When these atoms are in the
ground state, the bond is weak. Spread excimer molecules include a halogen atom
in the ground state and an inert gas atom in the lowest excited states with an ex-
2.4 Properties of Diatomic Molecules
67
Fig. 2.27. Potential curves of excimer molecule XeF
cited s-electron. These excimer molecules are analogous to molecules consisting
of halogen and alkali metal atoms in the ground state, and the chemical bond in
these excimer molecules results from a partial transition of an excited s-electron
to the halogen atom. Interaction of positive and negative ions in these molecules
determines a strong chemical bond in them.
Electron terms of the excimer molecule XeF are given in Fig. 2.27 as an example of an excimer molecule. If atoms are found in the ground states, the chemical
bond between them is not realized since an exchange interaction involving an atom
with the completed electron shell corresponds to repulsion. Then valence electrons
remain in the field of parent atoms, so that the quantum molecular numbers are the
total molecule spin, the projection of the angular electron momentum onto the molecular axis and also the molecule symmetry for its reflection with respect to the plane
passed through the molecular axis. Excitation of the molecule leads to the transition
of an excited electron to the halogen atom, and the quantum number of a formed
molecule is the total momentum of an inert gas atomic core. Figure 2.28 contains
parameters of some states of excimer molecules under consideration [78], so that
Re is the equilibrium distance between nuclei that corresponds to the interaction
potential minimum, and De is the potential well depth, i.e. the difference of the interaction potential at infinite distance between nuclei and the equilibrium one Re .
Radiative transitions in excimer molecules are the basis of excimer lasers [79–81],
and Fig. 2.29 shows radiative transitions between electron states of excimer molecules, whereas the radiative lifetime for some states of excimer molecules and the
wavelengths of the band middle for corresponding radiative transitions are represented in Fig. 2.28.
2 Elements of Atomic and Molecular Physics
Fig. 2.28. Parameters of excimer molecules
68
2.4 Properties of Diatomic Molecules
Fig. 2.29. Character of radiative transitions in excimer molecule
69
3 Elementary Processes Involving Atomic Particles
3.1 Parameters of Elementary Processes in Gases and Plasmas
Properties of rare atomic systems, gases and plasmas, are determined by various
processes of collision of atomic particles [82]. In a general consideration of atomic
collisions [83, 84], we use as a characteristic of collision of atomic particles the collision cross section. By definition, the differential cross section of collision of two
atomic particles dσ that is introduced in the center-of-mass frame of reference, is
the ratio of a number of scattered particles per unit time per unit solid angle dΩ to
the flux of incident particles. The parameters of elastic scattering of particles when
the particle internal state is not changed are given in Fig. 3.1 in the classical limit.
The classical parameters of particle collision are the impact collision parameter ρ, a
distance of closest approach r0 , and the scattering angle ϑ. If the interaction potential U (R) of colliding particles is spherically symmetric (R is the distance between
particles), the conservation of the angular momentum of particles in the course of
collision leads to the following relation between the impact collision parameter and
the distance of closest approach [85]:
ρ2
U (r0 )
,
(3.1)
=
1
−
ε
r02
where ε = μv 2 /2 is the kinetic energy of colliding particles in the center-of-mass
frame of reference, μ is the reduced mass of colliding particles, and v is their relative
Fig. 3.1. Parameters of classical scattering in the center-of-mass frame of reference for colliding particles. The arrow indicates the direction of the particle trajectory in the center-of-mass
frame of reference. ρ is the impact parameter of collision, r0 is the distance of closest approach, and ϑ is the scattering angle. 1 is the particle trajectory, 2 are its asymptotic lines
72
3 Elementary Processes Involving Atomic Particles
Fig. 3.2. 1—the interaction potential of atomic particles for the hard sphere model; 2—real
interaction potential of atomic particles
velocity. In the case of a monotonic dependence ϑ(ρ) of the scattering angle on the
impact collision parameter, the differential cross section of scattering in the classical
limit is
dσ = 2πρ dρ.
(3.2)
The model of hard spheres is a convenient model for the scattering of classical
particles in a strongly varied potential, and the interaction potential for this model
corresponds to a solid wall of a radius R0 , as it is shown in Fig. 3.2. Within the
framework of this model, when the cross section is independent of the collision
velocity, the differential cross section dσ and the transport cross section σ ∗ = (1−
cos ϑ) dσ are correspondingly equal
dσ = πR02 d cos ϑ,
σ ∗ = πR02
(3.3)
since according to Fig. 3.3 ρ/R0 = sin α. The transport cross section of scattering σ ∗ relates to the transport of particles in a gas, whereas the total cross section
σt = dσ characterizes a shift of the phase of an electromagnetic wave that interacts with an atomic particle when this phase shift follows from collisions of atomic
particles. Therefore, the total cross section of particle scattering may be responsible
for the broadening of spectral lines resulting from the transition between discrete
states of atomic particles. This cross section is infinite (h̄ → 0) in the classical
limit because weak scattering takes place at any large impact parameter of particle
collisions.
The classical description holds true in the case when the main contribution to
the scattering cross section is given by large collision momenta l = μρv/h̄ 1.
In the case of a spherically symmetric interaction potential between colliding particles, scattering for different collision momenta l proceeds independently, and any
total parameter of scattering is the sum of these parameters for each collision momentum l. The characteristic of particle scattering is the scattering phase δl , that is,
3.1 Parameters of Elementary Processes in Gases and Plasmas
73
Fig. 3.3. The character of scattering within the framework of the hard sphere model
the parameter of the wave function of colliding atomic particles at large distances
between them. The connection between the differential cross section of scattering
dσ = 2π|f (ϑ)|2 d cos ϑ, the scattering amplitude f (ϑ), the diffusion or transport
cross section σ ∗ and the total scattering cross section σt are [24, 86, 87]
f (ϑ) =
∞
1 (2l + 1) e2iδl − 1 Pl (cos ϑ),
2iq
l=0
∞
4π σ∗ = 2
(l + 1) sin2 (δl − δl+1 ),
q
(3.4)
l=0
∞
4π σt = 2
(2l + 1) sin2 δl .
q
l=0
Here q is the phase vector of colliding particles in the center-of-mass frame of reference, so that the energy of particles in a center-of-mass system is ε = h̄2 q 2 /2μ,
where μ is the reduced mass of colliding particles.
Electron–atom collisions in gases and plasmas are of importance both for plasma
processes involving excited atom states and for transport processes [84, 88, 89].
Because of a finite number of scattering momenta in electron–atom collisions at
not-high energies, these collisions correspond to the quantum case. In the limit of
small electron wave numbers q, the electron scattering phase is δ0 = −Lq, and this
is the definition of the scattering length L. Correspondingly, the cross sections of
electron–atom scattering at small electron energies are equal to
σ ∗ (0) = σt (0) = 4πL2 .
(3.5)
Other scattering phases have a stronger dependence on the wave vector in the limit
of its low value; in particular, for short-range electron–atom interaction, this de-
74
3 Elementary Processes Involving Atomic Particles
Fig. 3.4. The diffusion cross section of electron scattering on the xenon atom [94–98]
Table 3.1. Parameters of the total cross section σt for electron scattering on inert gas atoms
[26, 99]
Atom
He
Ne
Ar
Kr
Xe
L/a0
σt (ε = 0), Å2
εmin , eV
σt (εmin ), Å2
1.2
5.1
–
–
0.2
0.14
–
–
−1.6
9.0
0.18
0.88
−3.5
43
0.32
3.8
−6.5
150
0.44
13
pendence has the form δl ∼ q 2l+1 . Expansion of the scattering phases at small
electron energies [90] with accounting for the polarization electron–atom interaction U (r) = −αe2 /2r 4 at large distances r between them along with a short-range
interaction (2.48) gives for the scattering amplitude at low electron energies [91]
f (ϑ) = −L −
ϑ
παq
sin .
2a0
2
(3.6)
If an electron is scattered on a non-structured atom and the scattering electron–atom
length is negative, the zeroth phase becomes equal to zero at low electron energies
where other scattering phases are small. This leads to a sharp minimum in the cross
section of electron–atom scattering that takes in the case of electron scattering on
argon, krypton and xenon atoms and is called the Ramsauer effect [92, 93]. Table 3.1
gives parameters of the total cross section σt = dσ of electron scattering on atoms
of inert gases at small electron energies and in a range of the Ramsauer minimum. In
addition, Fig. 3.4
gives the dependence on the electron energy for the diffusion cross
section σ ∗ = (1 − cos θ )dσ on the xenon atom [94–98], where θ is the scattering
angle.
In addition to Fig. 3.4, Table 3.2 contains the diffusion cross section σ ∗ depending on the electron energy according to evaluations [99] that are made on the basis
of experimental data for kinetic coefficients. Note that in the case of argon, krypton
3.1 Parameters of Elementary Processes in Gases and Plasmas
75
Table 3.2. The diffusion cross section σ ∗ of electron scattering on atoms of inert gases [99]
expressed in Å2 as a function of the electron energy ε
ε, eV
0
0.001
0.002
0.005
0.01
0.02
0.04
0.06
0.08
0.10
0.15
0.20
0.25
0.30
0.35
0.4
0.5
0.6
0.7
0.8
1.0
1.5
2.0
2.5
3
4
5
6
7
8
10
He
Ar
Kr
Xe
4.95
4.96
4.99
5.14
5.27
5.38
5.58
5.63
5.71
5.80
6.00
6.20
6.26
6.32
6.38
6.44
6.55
6.66
6.74
6.79
6.87
6.98
6.97
6.95
6.93
6.8
6.6
6.3
5.9
5.5
5.0
10.0
8.35
7.80
6.61
5.60
4.15
2.50
1.60
1.00
0.59
0.23
0.10
0.091
0.15
0.24
0.33
0.51
0.68
0.86
1.05
1.38
2.07
2.70
3.37
4.10
6.00
7.60
9.3
11.0
14.0
14.6
39.7
39.7
37.7
30.0
26.2
21.4
15.3
11.7
9.13
7.23
4.56
2.37
1.30
0.86
0.55
0.26
0.10
0.15
0.27
0.42
0.80
1.84
3.00
4.40
6.00
10.0
14.0
16.0
17.0
16.5
15.5
176
170
160
139
116
89.0
50.2
34.8
25.6
20.4
13.6
8.4
5.9
3.3
2.5
1.6
0.53
0.38
0.38
0.55
1.15
3.50
7.50
11.5
16.5
24.5
28
26
27
26
20
and xenon, the Ramsauer minimum in electron–atom scattering is sharper for the
diffusion cross section σ ∗ than that for the total cross section σt .
In contrast to the above case of electron scattering on atoms with completed
electron shells, there are different channels of electron scattering on atoms with
noncompleted electron shells. In particular, for electron scattering on atoms of alkali metals, these channels are characterized by different total spins of an incident
electron and atom that corresponds to ignoring relativistic effects. In particular, the
cross section of scattering for electrons of zero energy is given by the following
formula instead of (3.7):
(3.7)
σ ∗ (0) = σt (0) = π L20 + 3L21 ,
76
3 Elementary Processes Involving Atomic Particles
Table 3.3. The parameters of scattering of a slow electron on atoms of alkali metals and parameters of the autodetaching state 3 P of the autodetaching state of the alkali metal negative
ion [26, 27]
L0 /a0
L1 /a0
Er , eV
Γr , eV
Li
3.6
−5.7
0.06
0.07
Na
4.2
−5.9
0.08
0.08
K
Rb
0.56
−15
0.02
0.02
2.0
−17
0.03
0.03
Cs
−2.2
−24
0.011
0.008
where L0 and L1 are the electron scattering lengths on atoms for zero and one
spin, respectively. The parameters of this formula are given in Table 3.1. Note that
the autodetachment resonance 3 P that corresponds to an autodetaching state of the
alkali metal negative ion gives a remarkable contribution to the cross section of
electron–atom scattering. Rough values of the parameters of this state, the excitation
energy Er and the width Γr of the autodetaching state level are also represented in
Table 3.3.
Along with the cross section σ , the rate constant vσ is the characteristic of
particle collision, where v is the relative velocity of colliding particles. In particular,
the rate constant of inelastic scattering of particles is given by
kij (v) = vσij (v),
(3.8)
where σij is the cross section of inelastic collision with transition from a state i
to a state j . When colliding particles are distributed over velocities, the mean rate
constant is averaged over velocities of collisions according to the formula
kij (v) = f (v)kij (v) dv,
(3.9)
where the normalization of the velocity distribution function f (v) is given by
f (v) dv = 1.
The rate constants of particle collisions determine evolution of excited states in
gases and plasmas. If excited states relate to atoms the states of which are denoted
by indices i and j and transition between these states result from collisions with
particles of other types (for example, electrons) the number density of which is NB ,
the balance equation for the number density of particles Ni in a state i has the form
dNi
= −Ni NB
kij + NB
Nj kij .
dt
j
(3.10)
j
3.2 Inelastic Collisions of Electrons with Atoms
Processes of excitation and quenching of excited atoms by electron impact proceed
according to the scheme
3.2 Inelastic Collisions of Electrons with Atoms
e + A ↔ e + A∗ .
77
(3.11)
The excitation cross section by electron impact near the threshold has the form
[24, 54]
√
(3.12)
σex ε − ε, ε − ε ε,
where ε is the energy of an incident electron and ε is the excitation energy. This
gives for the quenching cross section according to the principle of detailed balance
1
,
(3.13)
v
where v is the velocity of an incident electron. Correspondingly, the rate constant of
atom quenching in collisions with a slow electron kq is independent of the electron
energy. Table 3.4 gives parameters of the lowest resonantly excited states of some
atoms with valence s-electrons. These parameters include the excitation energy ε,
the wavelength λ for the radiative transition into the ground state, and the radiative
lifetime τ∗0 for this transition; f is the oscillator strength for this transition, and
the quenching rate constant is kq for a slow electron. In addition, Table 3.5 contains the rate constants of quenching by a slow electron impact for some metastable
states.
The problem of inelastic electron–atom collisions may be solved under special
conditions [24, 54, 86]. One can suggest the following formula [12] for the rate
constant of the quenching of the resonantly excited state by a slow electron when
dipole interaction of atoms dominates
σq kq =
k0
7/2
ε
.
(3.14)
·τ
If the excitation energy ε is expressed in eV, and the radiative lifetime of the
excitation state τ is given in ns, the rate constant k0 in formula (3.14), as it follows
from experimental data, equals [100] k0 = (4.4 ± 0.7) × 10−5 cm3 /s.
Table 3.4. Parameters of resonantly excited states of atoms with s-valence electrons and the
quenching rate in collisions of such atoms with a slow electron [100]
Atom, transition
ε, eV
λ, nm
f
H(21 P → 11 S)
He(21 P → 11 S)
He(21 P → 21 S)
He(23 P → 23 S)
Li(22 P → 22 S)
Na(32 P → 32 S)
K(42 P1/2 → 42 S1/2 )
K(42 P3/2 → 42 S1/2 )
Rb(52 P1/2 → 52 S1/2 )
Rb(52 P3/2 → 52 S1/2 )
Cs(62 P1/2 → 62 S1/2 )
Cs(62 P3/2 → 62 S1/2 )
10.20
21.22
0.602
1.144
1.848
2.104
1.610
1.616
1.560
1.589
1.386
1.455
121.6
58.43
2058
1083
670.8
589
766.9
766.5
794.8
780.0
894.4
852.1
0.416
0.276
0.376
0.539
0.74
0.955
0.35
0.70
0.32
0.67
0.39
0.81
τ∗0 , ns
1.60
0.555
500
98
27
16.3
26
25
28
26
30
27
kq , 10−8 cm3 /s
0.79
0.18
51
27
19
20
31
32
32
33
46
43
78
3 Elementary Processes Involving Atomic Particles
Table 3.5. The rate constants for quenching of metastable atom states in collisions with a
slow electron [100]
Atom, transition
ε, eV
kq , 10−10 cm3 /s
He(23 S → 11 S)
Ne(23 P2 → 21 S)
Ar(33 P2 → 32 S)
Kr(43 P2 → 42 S0 )
Xe(52 P2 → 52 S0 )
19.82
16.62
11.55
9.915
8.315
31
2.0
4.0
3.4
19
The principle of detailed balance establishes the relation between the rates of a
direct and reverse process that correspond to time reversal [101]. The principle of
detailed balance gives the following connection between the cross section of atom
excitation by electron impact σex (ε) and the cross section of atom quenching by
electron impact σq (ε − ε) [102]
σex (ε) =
g∗ (ε − ε)
σq (ε − ε),
g0
ε
(3.15)
where ε is the energy of a fast electron and g0 , g∗ are the statistical weights of
atoms in the ground and excited states. From this it follows the connection between
the excitation rate constant kex of an atom by electron impact, and the rate constant
kq of atom quenching by electron impact
kex = kq
g∗
g0
ε − ε
.
ε
(3.16)
Using formula (3.14) for the quenching rate constant, one can obtain the following
expression for the excitation rate constant near the threshold:
√
k0 g∗ ε − ε
.
(3.17)
kex =
g0 ε 4 · τ
3.3 Collision Processes Involving Ions
Ions as a charged component of a plasma are of importance for plasma properties
that result from collisions in a plasma involving ions [103–106]. Ion–atom collisions with variations in the ion momentum have a classical character, i.e. nuclei are
moving in these collisions according to the classical law. These processes include
elastic ion–atom collisions and the charge exchange process. Elastic scattering of
ions on atoms at low energies is determined by their interaction at large distances
with the polarization interaction potential (2.43) between them. The peculiarity of
the interaction potential (2.43) is the possibility of particle capture that means the
3.3 Collision Processes Involving Ions
79
approach of colliding particles up to R = 0. The cross section of capture σcap for
the polarization interaction potential is [85]
αe2
,
(3.18)
σcap = 2π
μg 2
where μ is the reduced mass of colliding ion and atom, and g is their relative velocity.
In reality, the interaction potential differs from the polarization potential (2.43)
at distances compared with atomic size and includes ion–atom repulsion at small
separations. Therefore, the capture cross section (3.18) means only a strong approach of colliding ion and atom. Nevertheless the diffusion cross section of ion–
atom collision for the polarization interaction potential (2.43) between them, σia∗ is
close to the capture cross section (3.18) and is equal to [107] σia∗ = 1.10σcap .
Resonant charge exchange in ion collision with a parent atom proceeds according to the scheme
+ + A,
→ A
(3.19)
A+ + A
where the tilde marks one of the colliding particles. Usually, at room temperature
and higher collision energies, the cross section of resonant charge exchange remarkably exceeds the cross section of ion–atom elastic collision, and therefore transport
of ions in the parent gas is determined by resonant charge exchange.
Considering the resonant charge exchange process as a result of state interference, we have for the electron wave function Ψ (t) instead of formula (2.46)
t
t
ψg
ψu
i
i
(3.20)
Ψ (t) = √ exp −
εg dt + √ exp −
εu dt ,
2h̄ −∞
2h̄ −∞
2
2
where εg , εu are the energies of the even and odd states at a given distance between
nuclei. From this one can find the probability of resonant charge exchange as a result
of ion–atom collision [108]
∞
2
i
Δ dt, Δ = |εg − εu |, (3.21)
P = Ψ (∞)|ψ(1) = sin2 ζ, ζ =
2h̄ −∞
and ζ is the phase shift between even and odd states as a result of collision. This
exhibits the interference nature of charge exchange, so that the electron state is the
mixture of the even and odd states, and a phase shift between these states leads to
the electron transition from the field of one core to another one. In turn, the phase
shift is determined by the exchange interaction potential (2.46) in the course of
collision.
The expression (3.21) for the probability of the resonant charge process corresponds to a two-state approximation when the atom and ion electron states are not
degenerated, which takes place if an atom A of the process (3.19) relates to atoms
of the first and second groups of the periodic tables, i.e. for atoms with valent selectrons. Nevertheless, this expression may be used within the framework of some
80
3 Elementary Processes Involving Atomic Particles
models [109–111] in other cases. In particular, the hydrogen-like model by Rapp
and Francis [109] using the hydrogen wave functions in determination of the ion–
atom exchange interaction potential is popular. But one can construct in this case
a strict asymptotic theory by expansion of the cross section over a small parameter
that exists in reality. In contrast to model evaluations, the asymptotic theory allows
us to estimate the accuracy of the results within the theory because this accuracy is
expressed through a small parameter of the theory. We show this below.
Note the classical character of nuclear motion at not-small collision energies
[112] (say, above those at room temperature). If an atom is found in a highly excited
state, the electron transition in the resonant charge exchange process has a classical
character also [112] and proceeds when a barrier between fields of two cores disappears. We consider another case related to atoms in the ground and lowest excited
states, when the transition has a tunnel character. Then we have a weak logarithm
dependence of the cross section of resonant charge exchange σres on the collision
velocity v [113, 114]
v∗
,
(3.22)
σres (v) = C ln2
v
where C and v∗ are constants. Indeed, taking the basic dependence of the exchange
interaction potential (R) on an ion–atom distance R as (R) exp(−γ R) according to formula (2.47), one can represent the cross section of resonant charge
exchange in the form
σres (v) =
2
v0
π
1
R0 + ln
,
2
γ
v
(3.23)
where πR02 /2 is the cross section at the collision velocity v0 . Diagram Pt15 gives
the cross sections of resonant charge exchange involving ions and atoms of various
elements in the ground states. The collision energy at which the cross section is
given, corresponds to the laboratory frame of reference, where an atom is motionless.
We will be guided by collision energies of eV, at which the parameter γ R0 in
formula (3.23) is large. Diagram Pt16 gives values of the parameter γ R0 for ions
and atoms of various elements in the ground states. From this it follows that this parameter for various elements for the collision energy of 1 eV ranges approximately
from 10 to 15. On the basis of this, one can construct a strict asymptotic theory of
the resonant charge exchange by expansion of the cross section of this process over
a small parameter 1/γ R0 . The first term of this expansion leads to the cross section
of resonant charge exchange in the form [115]
σres (v) =
π 2
R ,
2 0
ζ (R0 ) =
e−C
= 0.28,
2
(3.24)
where C = 0.577 is the Euler constant. Let us ignore elastic ion–atom scattering in
the charge exchange process; i.e. particles move along straightforward trajectories
and the ion–atom distance R varies in time as R 2 = ρ 2 = v 2 t 2 , where ρ is the
Pt15. Cross sections of resonant charge exchange
3.3 Collision Processes Involving Ions
81
Pt16. Parameters of cross section of resonant charge exchange
82
3 Elementary Processes Involving Atomic Particles
3.3 Collision Processes Involving Ions
83
impact parameter of collision, and v is the relative velocity of colliding particles.
Then the cross section of resonant charge
exchange is given by [59]
σres (v) =
π 2
R ,
2 0
1
v
πR0
(R0 ) = 0.28,
2γ
(3.25)
where the exchange interaction potential (R) in this formula is determined by formula (2.46). Then the cross section (3.25) of resonant charge exchange is expressed
through the parameters of the asymptotic electron wave function, when a valence
electron is located in the atom far from the core.
Because this theory represents the cross section as an expansion over a small
parameter 1/γ R0 , one can estimate the accuracy of the expression (3.25) within
the framework of the theory by comparison with the results where the next terms
of expansion over a small parameter are taken into account. In the hydrogen case
(that is, for the process H+ + H), the cross section of resonant charge exchange
at the ion energy of 1 eV is [116] (173 ± 2)a02 as an averaging of the results and
accounting for the first and the second terms of expansion in different versions of
this expansion. Evidently, the accuracy of this ∼1% is the best that one can expect
from formula (3.25). We estimate the data of diagram Pt15 for the cross section
of the resonant charge exchange of elements of the first and second groups of the
periodic table to be better than 2% when ions and their atoms are found in the ground
states.
Resonant charge exchange for ions and atoms of other element groups of the
periodic table becomes more complicated because the process of resonant charge
exchange is entangled with the processes of turning of atom angular momenta and
transition between states of fine structure. In particular, Fig. 2.23 gives the lowest
electron terms in the case of collision Cl+ (3 P )+Cl(2 P ) [72], which may be responsible for resonant charge exchange. We will be guided by the average cross section
of resonant charge exchange that is averaged over initial values of atom and ion momenta and their projections. In particular, there are 18 electron even and odd terms
in the case of interaction Cl+ (3 P ) + Cl(2 P ) (Fig. 2.18) where the quantum numbers
are the total momenta J and j of an atom and ion and MJ is the atom momentum
projection onto the impact parameter of collision. Let us consider the case when a
valence p-electron is located in the fields of structureless cores. It is convenient to
compare the parameters in this case with those for an s-electron if the asymptotic
radial wave functions (2.16) of a transferred electron are identical in both cases.
Then the exchange interaction potential of ion and atom with a valence p-electron
is expressed through that Δ0 for an s-electron given by formula (2.47) as [69]
(2l + 1)(l + |m|)!
,
(3.26)
Δlm (R) = Δ0 (R) ·
(l − |m|!)|m|!
where l, m are the electron momentum and its projection onto the molecular axis.
Table 3.6 gives the partial cross sections of resonant charge exchange with
transition p-electron between two structureless cores, so that σ0 is the cross section with participation of an s-electron, σ10 is the cross section involving a
p-electron if the momentum projection on the impact parameter of collision is
zero, σ11 is this cross section when the momentum projection equals to ±1, and
84
3 Elementary Processes Involving Atomic Particles
Table 3.6. The partial cross sections of resonant charge exchange with transition p-electron
between two structureless cores [117]
R0 γ
6
8
10
12
14
16
σ10 /σ0
σ11 /σ0
σ /σ0
σ1/2 /σ0
σ3/2 /σ0
1.40
1.08
1.19
1.18
1.18
1.29
0.98
1.08
1.10
1.10
1.23
0.94
1.04
1.07
1.06
1.19
0.95
1.01
1.05
1.04
1.16
0.912
0.99
1.04
1.03
1.14
0.91
0.99
1.03
1.02
Table 3.7. The reduced average cross sections of resonant charge exchange for atoms of
groups 3 (σ3 ), 4 (σ4 ) and 5 (σ5 ) of the periodical system of elements in the case “a” of Hund
coupling [58, 117]
R0 γ
σ3 /σ0
σ4 /σ0
σ5 /σ0
6
1.17
1.50
1.44
8
1.09
1.32
1.29
10
1.05
1.23
1.22
12
1.03
1.18
1.17
14
1.02
1.14
1.14
16
1.03
1.12
1.11
the average cross section is σ = σ10 /3 + 2σ11 /3. These cross sections relate to
the “a” case of Hund coupling when spin–orbit interaction is ignored. The cross
sections σ1/2 , σ3/2 correspond to case “c” of Hund coupling when the spin–orbit
splitting of levels is relatively large. As is seen, in the case “c” of Hund coupling,
the cross section is independent practically on the total electron momentum. In
addition, the average cross sections are close for both cases of Hund coupling.
Note that the resonant charge exchange process for elements of 3 and 8 groups
of the periodical system of elements [58] relates just to this one-electron case.
The accuracy of the cross sections of diagram Pt15 is determined by the accuracy
of asymptotic coefficients A mostly and is estimated for these cases to be better
than 5%.
The average cross section of resonant charge exchange for ions and atoms with
noncompleted shells depends on the initial distributions over electron states of ions
and atoms as well as on the character of momentum coupling in them, as follows
from the analysis for halogens, oxygen and nitrogen [72, 118, 119]. Table 3.7 gives
the average cross sections for resonant charge exchange with respect to the cross
sections for s-electrons in the case “a” of Hund coupling when spin–orbit interaction
may be ignored. Note that transition to elements of groups 6, 7 and 8 from elements
of groups 3, 4 and 5 results from change of electrons by holes. The accuracy of the
cross sections of diagram Pt15 in these cases is estimated to be better than 10%.
As is seen, though the partial cross sections differ remarkably, the average cross
sections are close for different types of momentum coupling.
Representing the velocity dependence for the cross section of resonant charge
exchange in the form
α
v0
,
(3.27)
σres (v) = σres (v0 )
v
3.4 Atom Ionization by Electron Impact
85
we obtain from formula (3.23)
α=
2
1.
γ R0
In reality, the exponent α in formula (3.27) differs from this value because of the approximated character of formula (3.23). The values of this exponent for the resonant
charge exchange process involving various elements are given in diagram Pt16.
3.4 Atom Ionization by Electron Impact
Among various processes of electron collisions with atomic particles, ionization
processes are of importance for plasma properties and evolution. Atom ionization
by electron impact proceeds according to the scheme
e + A → 2e + A+ .
(3.28)
The simplest model of atom ionization is the Thomson model [120], which ignores
the interaction between impact and bound electrons during the collision act. If the
energy exchange in collisions of these electrons owing to the Coulomb interaction
between them ε exceeds the ionization potential, a release of the bound electron
proceeds. The cross section of energy exchange between an incident and bound
electron is given by the Rutherford formula [85]
dσ = 2πρ dρ =
πe4 dε
,
ε(ε)2
if the energy exchange ranges from ε to ε + dε. Here electrons are assumed to
be classical and the energy exchange is assumed to be small compared to the collision energy ε, i.e. ε ε. Because atom ionization corresponds to bond breaking
for a bound electron, the ionization process proceeds when the exchange energy exceeds the atom ionization potential J , i.e. ε > J . This gives the ionization cross
section σion within the framework of the Thomson model [120]
ε
1
πe4 1
−
.
(3.29)
dσ =
σion =
ε
J
ε
J
In the case of several valence electrons, summation in the Thomson formula takes
place over valence electrons.
In derivation of the Thomson formula (3.29), a bound electron assumes to be
motionless. One can refuse from this assumption and then, in the limit of high
energies of an incident electron (ε J ), we obtain instead of the Thomson formula (3.29) [121, 122]
me v 2
πe4
1+
,
(3.30)
σion =
Jε
3J
where me is the electron mass, v is the velocity of a bound electron, and an overline means an average over velocities of a bound electron. In the limit v = 0, this
86
3 Elementary Processes Involving Atomic Particles
formula is transformed into the Thomson formula (3.29). Since the average kinetic
energy of a bound electron is compared to the atom ionization potential, taking into
account the velocity distribution for a bound electron leads to a remarkable change
of the ionization cross section compared with the Thomson formula (3.29). In particular, if a bound electron is located mostly in the Coulomb field of an atomic
core, we have me v 2 /2 = J , and the formula (3.30) differs from the Thomson formula (3.29) by a factor 5/3. Note also that the threshold dependence of the cross
section of atom ionization on the electron energy differs slightly on the linear one
[123], though practically this difference is not of importance.
One more peculiarity of the ionization cross section relates to a logarithmic dependence of the cross section on the collision energy in the quantum case. After
discovering this fact in 1930 [124], the interest to classical models of the ionization
process drops. Indeed, the classical treatment leads to the dependence 1/ε for the
ionization cross section σion at large energies ε of an incident electron. The logarithm dependence for the quantum approach is determined [125] by large impact
parameters of collision when the ionization probability is small or is zero under
classical consideration. But, as it is shown by Kingston [126–128] as a result of the
numerical analysis, values of the ionization cross sections from the quantum (Born
approximation) and classical approach are close for excited atoms at large collision energies. This means that the different energy dependence for the classical and
quantum ionization cross sections is not of principle.
Note that the basis of the Thomson formula is the Rutherford formula for elastic scattering of two free electrons on small angles. But the Rutherford formula
conserves its form if an exchange energy is compared to the energy of colliding
electrons. In addition, the Coulomb cross section is identical for the classical and
quantum cases. Therefore, the Thomson formula may be recommended for rough
calculations, especially if experimental data are taken into account in such evaluations. Indeed, assuming that atom ionization by electron impact has the classical
character, we obtain a general formula for the cross section of atom ionization
πe4
ε
.
(3.31)
σion = 2 f
J
J
This dependence follows from the dimensionality considerations, and f (x) is some
universal function that is equal to f (x) = 1/x − 1/x 2 for the Thomson model.
Figure 3.5 gives this function, that follows from experimental data [129–135] for
the ionization cross sections of atoms with valence one or two s-electrons. This
figure shows that the scaling law (3.31) is better, the higher the electron energy is.
We are based on the Thomson model of atom ionization by electron impact because of the simplicity and transparency of this model, so that this model gives the
main peculiarities of this process. Note that the threshold law for the cross section of atom ionization differs slightly from the linear dependence on the energy
excess for an incident electron [123] that is determined by the character of electron release [16]. This has a fundamental importance, but not a practical one. Next,
semiempirical models for evaluation of the cross section of the ionization process
3.5 Atom Ionization in Gas Discharge Plasma
87
Fig. 3.5. The reduced ionization cross section of atoms with valence s-electrons, based on
experimental data: [129] for e + H, [130] for e + He, [131, 132] for e + He(23 S), [133] for
e + H2 , [134] for e + He+ , [135] for e + Li
start from the limit of large energies of an incident electron where they transfer
to the Born approximation and give an accuracy of approximately 20%. This follows, for example, from comparison of the results of the Deutsch–Märk [136–139]
and McQuire [140–142] models that in turn correspond to the sum of experiments
[143–145].
3.5 Atom Ionization in Gas Discharge Plasma
A gas discharge plasma is a self-maintaining ionized gas where the ionization balance results under the action of an external electric field [146–148]. The ionization
processes determine properties of a gas discharge plasma [82, 149]. Formation of
free electrons proceeds in electron–atom ionization collisions and is characterized
by the first Townsend coefficient α that accounts for generation of electrons and is
defined by the balance equation for the number density Ne of electrons
dNe
= αNe ,
dx
(3.32)
88
3 Elementary Processes Involving Atomic Particles
Fig. 3.6. The reduced first Townsend coefficient for helium according to experimental data
[150–153]
where x directs along the electric field, and 1/α is the mean free path with respect
to electron multiplication. The average rate constant of ionization kion in a gas discharge plasma is then
α
kion = w ,
(3.33)
Na
where w is the drift electron velocity, i.e. the average velocity of electron motion in
a gas under the action of an electric field.
Figures 3.6, 3.7, 3.8, 3.9 and 3.10 contain the dependence of the reduced first
Townsend coefficient of inert gases on the reduced electric field strength according to experimental data. One can add to these data the book [162] and the review
[163] where various parameters of processes in a gas discharge plasma are collected
and analyzed. The values of the first Townsend coefficient may be used in the balance equation for a gas discharge plasma with some reservations. There are several
channels of atom ionization in a plasma and we restrict by direct atom ionization in
electron–atom collisions, and hence these data are suitable for such a regime of ionization in a gas discharge plasma. At low values of the first Townsend coefficients
that correspond to low electric field strengths, the ionization channels involving excited atoms will dominate, i.e. this type of ionization corresponds to moderate and
high electric field strengths. Next, atom ionization proceeds owing to a tail of the
energy distribution function of electrons when the electron energy exceeds the atom
ionization potential. We assume that the tail of the electron distribution function is
established as a result of elastic and inelastic electron collisions with atoms. At not
low electron number densities, electron–electron collisions influence the tail of the
3.5 Atom Ionization in Gas Discharge Plasma
89
Fig. 3.7. The reduced first Townsend coefficient for neon according to experimental data
[154–157, 152]
Fig. 3.8. The reduced first Townsend coefficient for argon according to experimental data
[154, 155, 158–160]
energy distribution function, and this possibility is excluded from our consideration.
Hence, these data relate to the case of a low electron number density.
Stepwise transitions in atom ionization are also not suitable if we use the first
Townsend coefficient as a parameter of atom ionization. Thus, focusing on electron–
90
3 Elementary Processes Involving Atomic Particles
Fig. 3.9. The reduced first Townsend coefficient for krypton according to experimental data
[154, 159–161]
Fig. 3.10. The reduced first Townsend coefficient for xenon according to experimental data
[154, 159]
atom ionization in a gas discharge plasma and using the first Townsend coefficient,
we deal with a certain regime of a gas discharge plasma that corresponds to low
electron number densities and electric field strengths that are not low.
3.6 Ionization Processes Involving Excited Atoms
91
3.6 Ionization Processes Involving Excited Atoms
Excited atoms are present in a gas discharge plasma and may be of importance for its
properties. One of the ionization channels in a gas discharge plasma is determined
by the Penning process that proceeds in a pair collision process according to the
scheme [164, 165]
(3.34)
A + B ∗ → e + A+ + B,
where the ionization potential of an atom A is lower than the excitation energy of
an atom B. Therefore the level A + B ∗ of the system of colliding atoms is above
the boundary of continuous spectrum, i.e. it is an autoionizing state, and this state
decays at moderate distances between colliding atoms when the width of this autoionizing level becomes enough for its decay during collision. Table 3.8 contains
the cross sections of the Penning process involving metastable atoms of inert gases.
As a result of another process of atom ionization involving excited atoms, associative ionization, a molecular ion is formed. Table 3.9 gives the parameters of
this process with participation of two excited alkali metal atoms when this process
proceeds according to the scheme
2 (3.35)
2M P → M2+ + e − εi ,
where εi is the energy released in this process, and kas is the rate constant of this
process. This process of associative ionization is of importance for a photoresonant
plasma. Figure 3.11 shows the behavior of electron terms and some of these are responsible for this process. As it follows from Table 2.22, there are 12 electron terms
Table 3.8. The rate constant of the Penning process (3.34) at room temperature [166, 167].
The rate constants are given in cm3 /s
Colliding atoms He(23 S)
4 ×10−12
8 ×10−11
1 ×10−10
1.5 ×10−10
3 ×10−11
9 ×10−11
3 ×10−10
Ne
Ar
Kr
Xe
H2
N2
O2
He(21 S)
Ne(3 P2 )
Ar(3 P2 )
Kr(3 P2 )
Xe(3 P2 )
4 ×10−11
3 ×10−10
5 ×10−10
6 ×10−10
4 ×10−11
2 ×10−10
5 ×10−10
–
1 ×10−10
7 ×10−12
1 ×10−10
5 ×10−11
8 ×10−11
2 ×10−10
–
–
5 ×10−12
2 ×10−10
9 ×10−11
2 ×10−10
4 ×10−11
–
–
–
2 ×10−10
3 ×10−11
4 ×10−12
6 ×10−11
–
–
–
–
2 ×10−11
2 ×10−11
2 ×10−10
Table 3.9. Parameters of the process (3.35) at temperature 500 K [16]
A∗
εi , eV
kas , cm3 /s
Na(32 P )
K(42 P )
Rb(52 P )
Cs(62 P )
<0
0.1
0.2
0.33
4 × 10−11
9 × 10−13
4 × 10−13
7 × 10−13
92
3 Elementary Processes Involving Atomic Particles
Fig. 3.11. Positions of electron terms that are responsible for the process of associative ionization 2M(2 P ) → 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 temperature. The accuracy classes: A—the accuracy is better than 10%, B—better than 20%,
C—better than 30%, D—worse than 30%
A∗
K
He(3 P2 ) Ne(3 P2 ) Ne(3 P1 ) Ar(3 P2 )
0.23(B) 0.5(C)
5.8(C)
10(A)
A∗
K
Kr(3 P0 )
54(D)
Kr(3 P1 )
30(C)
Kr(1 P1 )
1.6(A)
Ar(3 P0 )
11(D)
Ar(3 P1 )
12(D)
Ar(1 P1 )
14(D)
Kr(3 P2 )
36(C)
Xe(3 P2 ) Xe(3 P0 ) Xe(3 P1 ) Hg(3 P0 ) Hg(3 P1 )
55(D)
40(D)
70(C)
250(A)
160(C)
corresponding to two interacting atoms of alkali metals in the resonantly excited
states 2 P . Some repulsive terms intersect with the electron term of the molecular
ion M2+ . If the intersection cross section is attained in the collision of two resonantly excited atoms M(2 P ), the state of the colliding atoms becomes autoionization at lower distances between them and then it can be decomposed according to
the scheme (3.35).
Excited atoms create radiation of a gas discharge plasma. This radiation may
have a form of separate spectral lines when photons result from radiative transitions
in atoms and can form radiation bands when radiation is created by molecular particles. In a dense gas excited atoms may be transformed into excited molecules as a
result 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 a
three 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 compared
to the atom ionization potential
(3.38)
Te J,
the initial bound state of an electron after its capture in the process (3.37) proceeds
at a high level, the ionization potential of which is of the order of a thermal electron energy. The capture of an electron in a highly excited state in the case of a
small electron temperature corresponds to a general Thomson scheme of three body
collisions [168]. A subsequent transition to the ground electron state proceeds as a
result of many pair collisions and radiative transitions according to a general kinetics scheme [169–171]. The dependence on the electron temperature for this three
body recombination coefficient Kei , as it follows from the scaling law, has the form
[172]
α
e10
= C 1/2 9/2 .
(3.39)
Kei =
Ne
me Te
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 ionization. In this ionization process at low electron temperature (3.38) and high electron densities, many transitions between excited states proceed as a result of collisions with electrons unless atom ionization proceeds. According to the principle
of detailed balance and formula (3.39), the rate constant of stepwise ionization
is [173]
gi me e10
J
,
(3.41)
kion = A
exp
−
g0 h̄3 Te3
Te
where A ≈ 0.2, gi , g0 are the statistical weights of the ion and atom in the ground
state, me is the electron mass, and J is the ionization potential. Note that this formula 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 dominate. Table 3.11 gives the values of the rate constant of stepwise ionization and these
data may be considered as an upper limit for the ionization rate constant.
Electron–ion recombination that involves molecular ions, dissociative recombination, proceeds according to the scheme
94
3 Elementary Processes Involving Atomic Particles
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
1.0
2.0
3.0
2.0 ×10−18
2.7 ×10−16
3.8 ×10−13
3.3 ×10−12
3.6 ×10−11
3.8 ×10−10
2.0 ×10−8
4.6 ×10−8
4.9 ×10−16
2.4 ×10−10
7.8 ×10−10
3.0 ×10−14
1.8 ×10−10
2.0 ×10−9
1.0 ×10−11
3.3 ×10−9
1.4 ×10−8
5.9 ×10−11
8.0 ×10−9
2.5 ×10−8
Table 3.12. The rate constant of dissociative recombination α given at room temperature is
expressed 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
Kr+
2
Xe+
2
N+
2
O+
2
NO+
CO+
6.9
10
20
2.2
CO+
2
H+ · CO
3.6
1.1
N+
4
O+
4
20
14
17
3.5
H3 O+
CO+ · CO
6.8
CO+ · (CO)2
19
2.0
14
e + AB + → A + B ∗
CH+
5
NH+
4 · NH3
NH+
4 · (NH3 )2
H3 O+ · H2 O
H3 O+ · (H2 O)2
H3 O+ · (H2 O)3
H3 O + · (H2 O)4
11
28
27
24
34
44
50
(3.42)
and the specifics of this process are such that it proceeds through formation of autoionization states of forming atoms [174–176]. Table 3.12 contains the values of the
dissociative recombination coefficients at room temperature. A rough dependence
of this recombination coefficient αdis on the electron temperature Te at constant gas
temperature has the form
αdis ∼ Te−0.5 .
(3.43)
If a complex ion partakes in the recombination process, it is convenient to use
the 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 R0
leads to the recombination process (3.43). Then the rate of this process is given by
the formula [179]
√
2e2 2π
αdis = √
R0 ,
(3.44)
me Te
3.8 Attachment of Electrons to Molecules
95
where e is the electron charge, me is the electron mass, and the radius of
the sphere of strong electron–ion interaction R0 is equal usually to several Bohr
radii.
In a plasma with negative ions the charge recombination results in mutual neutralization 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 electron
temperature 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 autodetachment level, and subsequent decay of a forming negative ion in an autodetachment
state proceeds usually by dissociation of this negative ion [30, 180]. Since this autodetachment state is characterized by a large lifetime, the transfer of an excess
electron energy to atoms or molecules of a gas proceeds in the course of removal of
particles when their electron state becomes stable. Therefore, the process of electron
capture has a resonance character and results in the transition between two electron
terms, as shown in Fig. 3.12. The probability of electron capture as a function of the
electron energy is shown on the right in Fig. 3.12, which testifies to the resonance
character of this process.
Fig. 3.12. Electron terms of a molecule and its molecular ion, which are responsible for electron attachment to a molecule, and the variation of these terms proceeds along a coordinate
that determines this process (the distance between nuclei for a diatomic molecule). The probability of electron capture depending on the electron energy is represented at right
96
3 Elementary Processes Involving Atomic Particles
Table 3.13. The rate constants of electron attachment to halomethane molecules at room
temperature [181, 182]
Molecule
CH3 Cl
CH2 Cl2
CHCl3
CHFCl2
CH3 Br
CH2 Br2
CH3 I
CF4
CF3 Cl
Rate constant, cm3 /s
<2 ×10−15
5 ×10−12
4 ×10−9
4 ×10−12
7 ×10−12
6 ×10−8
1 ×10−7
<1 ×10−16
1 ×10−13
Molecule
CF2 Cl2
CFCl3
CF3 Br
CF2 Br2
CFBr3
CF3 I
CCl4
CCl3 Br
Rate constant, cm3 /s
1.6 × 10−9
1.9 × 10−7
1.4 × 10−8
2.6 × 10−7
5 × 10−9
1.9 × 10−7
3.5 × 10−7
6 × 10−8
In reality, the resonance dependence of the cross section of electron attachment on the electron energy is typical for diatomic molecules with a small number of autodetachment states of a forming negative ion. For complex molecules
with a large number of autodetachment states, the dependence of the cross section on the electron energy becomes smooth, and the electron attachment process
becomes possible at zero electron energy. We give in Table 3.13 the rate constants
of electron attachment to halomethane molecules at room temperature. As shown,
the rate constant of this process is nonzero at low electron energies, though the
position of an autodetachment level for electron capture is dependent of a certain molecule. Therefore, the rate constants of electron attachment differ by an order of magnitude for different halomethane molecules that have an identical structure.
The electron attachment process is of importance for electric breakdown of gases
or for gas discharge. A small concentration of electronegative molecules may give an
effective loss of electrons in a gas discharge plasma as a result of electron attachment
to molecules. This can prevent a gas from electric breakdown or change conditions
of stable gas discharge.
3.9 Processes in Air Plasma
Properties of an ionized gas that is not rare are determined by collision processes
involving electrons, ions, excited atoms, etc. and radiative processes. In particular,
this takes place in a gas discharge plasma where the dominant processes depend on
the composition and regime of a gas discharge plasma [162, 183]. In Table 3.14, we
give the rates of processes that proceed in a plasma of the Earth’s atmosphere at various altitudes. The rate constants of processes in an air plasma at room temperature
involving 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 room
temperature [14]. λ is the wavelength of emitting photon, τ is the radiative lifetime, k is the
rate 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
atoms
O(1 S) → O(1 D) + h̄ω
O(1 D) → O(3 P ) + h̄ω
τ = 0.8 s, λ = 558 nm
τ = 140 s λ = 630 nm
τ = 1.4 · 105 s, λ = 520 nm
τ = 6 · 104 s, λ = 520 nm
k = 2 × 10−7 cm3 /s
k = 2 × 10−7 cm3 /s
k = 4 × 10−7 cm3 /s
k = 2 × 10−6 cm3 /s
K = 7 × 10−26 cm6 /s
K = 1.6 × 10−25 cm6 /s
Recombination
Electron
attachment
Decomposition
of negative ion
Dissociation
Ion-molecular
reaction
Quenching
Chemical reaction
Three body
association
N(2 D5/2 ) → N(4 S) + h̄ω
N(2 D3/2 ) → N(4 S) + h̄ω
e + N+
2 →N+N
e + O+
2 →O+O
e + NO+ → N + O
e + N+
4 → N2 + N2
O− + O+
2 + N2 → O + O2 + N2
+
O−
2 + O2 + O2 → 3O2
++N
NO−
+
NO
2
2
→ NO + NO2 + N2
e + 2O2 → O−
2 + O2
e + O2 + N2 → O−
2 + N2
e + O2 → O− + O
O− + O3 → e + 2O2 + 2.6 eV
O−
2 + O2 → e + O3 + 0.62 eV
O− + O2 → e + O3 − 0.42 eV
e + O2 → 2O + e
O+ + N2 → NO+ + N + 1.1 eV
O+ + O2 → O+
2 + O + 1.5 eV
N+ + O2 → O+ + NO + 2.3 eV
+
N+
2 + O → NO + N + 3.2 eV
1
2O2 ( Δg ) → O2 + O2 (1 Σg+ )
O2 (1 Δg ) + O2 → 2O2
O2 (1 Σg+ ) + N2 → O2 + N2
N2 (A3 Σ + ) + O2 → N2 + O2
O(1 D) + O2 → O + O2
O(1 D) + N2 → O + N2
O(1 S) + O2 → O + O2
O(1 S) + O → O + O
O + O3 → 2O2
O + 2O2 → O3 + O2
O + O2 + N2 → O3 + N2
K = 1 × 10−25 cm6 /s
K = 3 × 10−30 cm6 /s
K = 1 × 10−31 cm6 /s
k = 2 × 10−10 cm3 /s,
ε = 6.5 eV
k = 3 × 10−10 cm3 /s
k = 3 × 10−10 cm3 /s
k < 1 × 10−12 cm3 /s
k = 2 × 10−10 cm3 /s,
ε > 6 eV
k = 6 × 10−13 cm3 /s
k = 2 × 10−11 cm3 /s
k = 2 × 10−11 cm3 /s
k = 1 × 10−10 cm3 /s
k = 2 × 10−17 cm3 /s
k = 2 × 10−18 cm3 /s
k = 2 × 10−17 cm3 /s
k = 4 × 10−12 cm3 /s
k = 5 × 10−11 cm3 /s
k = 6 × 10−11 cm3 /s
k = 3 × 10−13 cm3 /s
k = 7 × 10−12 cm3 /s
k = 7 × 10−14 cm3 /s
K = 7 × 10−34 cm6 /s
K = 6 × 10−34 cm6 /s
98
3 Elementary Processes Involving Atomic Particles
Table 3.15. The rate constants of ion-molecular reactions involving negative ions at room
temperature [30, 184] given in 10−11 cm3 /s
Type of process
Scheme of process
k
Charge exchange
Charge exchange
Charge exchange
Charge exchange
Charge exchange
Associative
decay
Associative
decay
Associative
decay
Associative
decay
Associative
decay
Associative
decay
O− + O 2 → O −
2 +O
O− + O 3 → O −
3 +O
−+O
+
O
→
O
O−
2
2
−
O−
2 + O3 → O3 + O2
NO− + O2 → O−
2 + NO
O − + O → O2 + e
O− + N → NO + e
O− + NO → NO2 + e
O− + N2 → N2 O + e
O − + O2 → O3 + e
O− + O3 → 2O2 + e
O − + O 2 (1 Δ g ) → O 3 + e
O−
2 + O → O3 + e
O−
2 + N → NO2 + e
O−
2 + O2 → 2O2 + e
O−
2 + N2 → O2 + N2 + e
1
O−
2 + O2 ( Δg ) → 2O2 + e
−
NO + NO → 2NO + e
10
80
33
35
74
20
30
14
10−3
5 × 10−4
30
30
30
40
2 × 10−7
2 × 10−5
20
0.6
4 Transport Phenomena in Gases
4.1 Transport Coefficients of Gases
Transport coefficients characterize the connection between fluxes and weak gradients of some quantities. The diffusion coefficient D for atoms or molecules of a gas
is the proportionality coefficient between the flux j of these atoms and the gradient
of their concentration c, that is,
j = −DN∇c,
(4.1)
where N is the total number density of gas atoms or molecules. If the concentration
of atoms of a given type is small (ci 1), i.e. this component is a small admixture
to 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 conductivity of a gas κ is defined as the proportionality coefficient between the thermal
flux q and the temperature gradient ∇T as
q = −κ∇T .
(4.3)
The viscosity coefficient η is the proportionality coefficient between the friction
force F per area unit of a moving gas and the gradient of a gas velocity. In the frame
of reference where the direction of the average gas velocity w is x and the average
velocity 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 is
defined as
∂wx
,
(4.4)
F = −η
∂z
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 temperatures is significantly larger than that of inelastic processes, the transport coefficients
in a gas are expressed through average cross sections of elastic collisions. The simple 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 numerical
parameter. The accuracy of the first Chapman–Enskog approximation is several percent. The diffusion coefficient D of a test atom or molecule in a gas is given in the
first Chapman–Enskog approximation [185, 186]
√
1 ∞ −t 2 ∗
μg 2
3 πT
e t σ (t) dt, t =
, σ ≡ Ω (1,1) (T ) =
.
(4.5)
D=
√
2 0
2T
8N 2μσ
Here T is the gas temperature expressed in energetic units, N is the number density
of gas atoms, μ is the reduced mass of a test atom and a gas atom, σ ∗ (g) is the
diffusion 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 Maxwell
distribution function.
The thermal conductivity coefficient κ in the first Chapman–Enskog approximation is given by [185, 186]
√
25 πT
(4.6)
κ=
√ ,
32σ2 m
where m is the mass of an atom or molecule, an average of the cross section of
elastic collision is determined by the following formula:
∞
(2,2)
σ2 ≡ Ω
(T ) =
t 2 exp(−t)σ (2) (t) dt,
0
μg 2
1 − cos2 ϑ dσ.
(4.7)
t=
, σ (2) (t) =
2T
The viscosity coefficient η of the first Chapman–Enskog approximation is determined by the formula [185, 186]
√
5 πT m
,
(4.8)
η=
24σ2
where the average cross section σ2 is given by formula (4.7). In this approximation
the coefficients of thermal conductivity and viscosity are connected by the relation
κ=
15
η.
4m
(4.9)
Formulas (4.5), (4.6) and (4.8) for the first Chapman–Enskog approximation are
simplified if particle collision is described by the hard sphere model (3.3). Then the
average cross sections are given by
σ = σ0 ;
σ2 =
2
σ0 ,
3
(4.10)
where σ0 = πR02 , 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):
√
√
√
75 πT
15 πT m
3 πT
;
κ=
.
(4.11)
η=
D= √
√ ;
48σ0
64σ0 m
8 2μNσ0
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 /s
and reduced to the normal number density
Gas
He
Ne
Ar
Kr
Xe
D, cm2 /s
1.6
0.45
0.16
0.084
0.048
Gas
H2
N2
O2
CO
D, cm2 /s
1.3
0.18
0.18
0.18
Gas
H2 O
CO2
NH3
CH4
D, cm2 /s
0.28
0.096
0.25
0.20
Table 4.2. The parameters of formula (4.12) with the diffusion coefficient D0 at temperature
T = 300 K that is reduced to the normal number density and is expressed in cm2 /s and the
parameter γ of formula (4.12) is given in parentheses
Atom, gas
He
Ne
Ar
Kr
Xe
He
1.66(1.73)
Ne
1.08(1.72)
0.52(1.69)
Ar
0.76(1.71)
0.32(1.72)
0.195(1.70)
Kr
0.68(1.67)
0.25(1.74)
0.15(1.70)
0.099(1.78)
Xe
0.56(1.66)
0.23(1.67)
0.115(1.74)
0.080(1.79)
0.059(1.76)
Table 4.3. The diffusion coefficient of metal atoms in inert gases at temperature 300 K, expressed in cm2 /s and reduced to the normal number density [187]
Atom, gas
Li
Na
K
Cu
Rb
Cs
Hg
He
0.54
0.50
0.54
0.68
0.39
0.34
0.46
Ne
0.29
0.31
0.24
0.25
0.21
0.22
0.21
Ar
0.28
0.18
0.22
0.11
0.14
0.13
0.11
Kr
0.24
0.14
0.11
0.077
0.11
0.080
0.072
Xe
0.20
0.12
0.087
0.059
0.088
0.057
0.056
The coefficients of self-diffusion given in Table 4.1 are reduced as usual to the
normal 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 number density of gas atoms. Let us approximate the temperature dependence of the
diffusion coefficient by
T γ
(4.12)
D = D0
300
where the temperature is expressed in K. The parameters of this formula are based
on the data [188, 189] for the temperature range T = 300–1500 K and are given
in Table 4.2. Note that, within the framework of the hard sphere model according
to formula (4.11), we have γ = 1.5 since the number density of atoms is inversely
proportional to the temperature. In addition to these data, the diffusion coefficients
of metal atoms in inert gases at temperature T = 500 K are given in Table 4.3.
102
4 Transport Phenomena in Gases
Table 4.4. Gas-kinetic cross sections for collisions of atoms or molecules σg expressed in
10−15 cm2
Colliding particles
He
Ne
Ar
Kr
Xe
H2
N2
O2
CO
CO2
He
1.3
Ne
1.5
1.8
Ar
2.1
2.6
3.7
Kr
2.4
3.0
4.2
4.8
Xe
2.7
3.3
5.0
5.7
6.8
H2
1.7
2.0
2.7
3.2
3.7
2.0
N2
2.3
2.4
3.8
4.4
5.0
2.8
3.9
O2
2.2
2.5
3.9
4.3
5.2
2.7
3.8
3.8
CO
2.2
2.7
4.0
4.4
5.0
2.9
3.7
3.7
4.0
CO2
2.7
3.1
4.4
4.9
6.1
3.4
4.9
4.4
4.9
7.5
Table 4.5. Thermal conductivity coefficients in gases κ expressed in 10−4 W/(cm K) [190]
T, K
H2
He
CH4
NH3
H2 O
Ne
CO
N2
Air
O2
Ar
CO2
Kr
Xe
100
6.7
7.2
–
–
–
2.23
0.84
0.96
0.95
0.92
0.66
–
–
–
200
13.1
11.5
2.17
1.53
–
3.67
1.72
1.83
1.83
1.83
1.26
0.94
0.65
0.39
300
18.3
15.1
3.41
2.47
–
4.89
2.49
2.59
2.62
2.66
1.77
1.66
1.00
0.58
400
22.6
18.4
4.88
6.70
2.63
6.01
3.16
3.27
3.28
3.30
2.22
2.43
1.26
0.74
600
30.5
25.0
8.22
6.70
4.59
7.97
4.40
4.46
4.69
4.73
3.07
4.07
1.75
1.05
800
37.8
30.4
–
–
7.03
9.71
5.54
5.48
5.73
5.89
3.74
5.51
2.21
1.35
1000
44.8
35.4
–
–
9.74
11.3
6.61
6.47
6.67
7.10
4.36
6.82
2.62
1.64
The gas-kinetic cross section is introduced within the framework of the hard
sphere model on the basis of the expression for the diffusion coefficient (4.11). The
value of the diffusion coefficient is used at room temperature. We have the following
connection between the gas-kinetic cross section σg and the diffusion coefficient D
0.47
σg =
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 of
colliding particles. The values of gas-kinetic cross sections σg obtained on the basis
of formula (4.13) are given in Table 4.4.
The values of thermal conductivity coefficients of inert gases are given in Table 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
H2
He
CH4
H2 O
Ne
CO
N2
Air
O2
Ar
CO2
Kr
Xe
100
4.21
9.77
–
–
14.8
–
6.88
7.11
7.64
8.30
–
–
–
200
6.81
15.4
7.75
–
24.1
12.7
12.9
13.2
14.8
16.0
9.4
–
–
300
8.96
19.6
11.1
–
31.8
17.7
17.8
18.5
20.7
22.7
14.9
25.6
23.3
400
10.8
23.8
14.1
13.2
38.8
21.8
22.0
23.0
25.8
28.9
19.4
33.1
30.8
600
14.2
31.4
19.3
21.4
50.6
28.6
29.1
30.6
34.4
38.9
27.3
45.7
43.6
800
17.3
38.2
–
29.5
60.8
34.3
34.9
37.0
41.5
47.4
33.8
54.7
54.7
1000
20.1
44.5
–
37.6
70.2
39.2
40.0
42.4
47.7
55.1
39.5
64.6
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 is
the heat capacity per atom or molecule that is cV = 3/2 for ideal gas, and m is the
mass 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 field
of strength E, we restrict by limiting cases of low and high electric fields. The limit
of low electric fields corresponds to the criterion
eEλ T ,
(4.15)
where λ = 1/(N σ ) is the mean free path of ions in a gas, σ is the cross section
of ion–atom scattering on large angles and N is the number density of atoms. We
assume that the ion and atom masses have the same order of magnitude. In this
limiting case the ion drift velocity, i.e. its average velocity, is proportional to the
electric field strength
w = EK,
(4.16)
and this is the definition of the ion mobility K in low electric fields when it is
independent of the electric field strength and the ion drift velocity is small compared
to its thermal velocity.
104
4 Transport Phenomena in Gases
The diffusion coefficient of a charged particle in a gas D is connected with the
ion 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 equilibrium
condition that leads to equality of the diffusion and drift fluxes. Note that, though
this relation is called the Einstein relation, it was derived by Townsend several years
before [194, 195] and Einstein used these results in the analysis of Brownian motion
of particles [21]. On the basis of formulas (4.5), (4.11), (4.17) we have for the ion
mobility in the first Chapman–Enskog approximation and within the framework of
the hard sphere model
√
3 πe
K= √
.
(4.18)
8 2μT Nσo
The diffusion cross section of ion–atom scattering under the polarization interaction potential (2.43) between them is close to the capture cross section and is
equal to [166, 196]
αe2
∗
σ (v) = (1 − cos ϑ) dσ = 2.21π
,
μg 2
i.e. it exceeds the capture cross section by about 10%. For this cross section, the ion
mobility reduced to the normal number density of gas atoms N = 2.69 × 1019 cm−3
is given by the Dalgarno formula [107, 196]
36 cm2
,
K=√
αμ 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 (a03 ). It is important that this mobility
does not depend on the temperature, and ion parameters are taken into account only
by the ion–atom reduced mass.
Table 4.7 contains the mobilities of alkali metal atoms in inert gases and nitrogen
at 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 inert
gases and nitrogen at room temperature [197, 198, 202]
Ion, gas
Li+
Na+
K+
Rb+
Cs+
He
22.9
22.8
21.5
20
18.3
Ne
10.6
8.2
7.4
6.5
6.0
Ar
4.6
3.1
2.7
2.3
2.1
Kr
3.7
2.2
1.83
1.45
1.3
Xe
2.8
1.7
1.35
1.0
0.91
H2
12.4
12.2
13.1
13.0
12.9
N2
4.15
2.85
2.53
2.28
2.2
4.2 Ion Drift in Gas in External Electric Field
105
Table 4.8. The ratio between experimental and theoretical mobilities of ions in rare gases
[166]
Ion, gas
Li+
Na+
K+
Rb+
Cs+
He
1.27
1.38
1.35
1.29
1.19
Ne
1.17
1.24
1.28
1.26
1.18
Ar
1.06
1.07
1.08
1.08
1.07
Kr
1.07
1.06
1.10
1.07
1.08
Xe
1.06
1.09
1.07
1.0
1.07
H2
1.02
1.13
1.14
1.14
1.15
N2
0.89
0.97
0.98
0.99
0.99
the accuracy of the Dalgarno formula (4.19) for the mobility of ions in a foreign
gas. Indeed, Table 4.8 contains the ratio of experimental mobilities of alkali ions
in inert gases and nitrogen at room temperature [197–202] to those calculated by
formula (4.19). One can see that experimental data exceeds theoretical data and the
statistical treatment gives the average ratio 1.15 ± 0.10. From this one can obtain
that the Dalgarno formula (4.19) may be used for a rough evaluation of the ion
mobility of ions. One can correct the Dalgarno formula by taking into account the
experimental data, and then we have
41 ± 4 cm2
,
K= √
αμ V · s
(4.20)
and the indicated accuracy (∼10%) is the statistical one. The real divergence from
formula (4.20) may be more. In particular, the ratio of the mobility of alkali metal
ions in helium to that given by formula (4.20) is equal to 1.13 ± 0.06. Thus, one
can suggest formula (4.20) for a rough evaluation of the ion mobility in gases. On
the basis of the Einstein relation and formula (4.20), we have for the ion diffusion
coefficient in a gas
1.0 ± 0.1 cm2
.
(4.21)
D= √
αμ
s
When atomic ions are moving in a parent atomic gas in an electric field, the
scattering 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 straightforward
trajectories, but charge exchange changes the trajectory of a charged particle so that
electron transfer to another atomic core leads to effective ion scattering. This takes
place, 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. Then
the cross section of resonant charge exchange determines the mobility of ions in a
parent gas.
Indeed, in the absence of elastic ion–atom scattering in the center-of-mass frame
of reference, the ion scattering angle is ϑ = π and the diffusion cross section of ion
scattering in the absence of elastic scattering is [203]
(4.22)
σ ∗ = (1 − cos ϑ) dσ = 2σres ,
106
4 Transport Phenomena in Gases
Fig. 4.1. The Sena effect. Atom and ion are moving along straightforward trajectories and
resonant charge exchange takes place, so that a valence electron transfers to another atomic
core. This leads to effective ion scattering
where σres is the cross section of the resonant charge exchange process. Then, assuming the resonant transfer cross section σres to be independent of the collision
velocity, we obtain from formula (4.18) for the ion mobility in a parent gas in the
first Chapman–Enskog approach [204]
0.331e
(4.23)
N T ma σres (2.2vT )
√
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 section
is taken. The second Chapman–Enskog approach [205] exceeds the ion mobility by
1/40 and gives for the ion mobility in a parent gas in the absence of elastic scattering
[166, 206]
0.341e
.
(4.24)
K= √
N T ma σres (2.1vT )
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]
KI =
K=√
√
cm2
1340
,
T ma σres (2.1vT ) V · s
(4.25)
where the temperature T is given in Kelvin, the atom and ion mass ma = M are
expressed in units of atomic mass and the resonant charge exchange cross section is
given in 10−15 cm2 ; the argument indicates the collision velocity that corresponds
to the collision energy 4.5 T in the laboratory frame of reference. Diagram Pt17
contains the values of the mobilities of atomic ions in parent gases at temperatures
T = 300 K, 800 K. The accuracy of this formula is approximately 10%.
Accounting for elastic ion–atom scattering due to the polarization ion–atom interaction, we obtain for the ion mobility in a parent gas at a zero electric field [166]
4.2 Ion Drift in Gas in External Electric Field
Pt17. Mobilities of atomic ions in parent gas
107
108
4 Transport Phenomena in Gases
Table 4.9. The contribution of elastic ion–atom scattering to the mobility of atomic ions in a
parent gas. K is the mobility without elastic scattering, K is a decrease of the mobility due
to elastic ion–atom scattering [166]
Ion, gas
He
Ne
Ar
Kr
Xe
K/K, %
5.8
7.8
11
6.0
9.2
Ion, gas
Li
Na
K
Rb
Cs
K0
K=
,
1 + x + x2
αe2 /T
x=
√
σres ( 7.5T /M)
K/K, %
12
8.2
8.9
7.7
7.5
(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 limits
of low and high temperatures. The contribution to the ion mobility in a parent gas of
elastic ion–atom scattering due to the polarization interaction is given in Table 4.9
for inert gases and alkali metal vapors [166]. Note that taking elastic scattering into
account leads to a decrease the ion mobility.
The diffusion coefficient of ions in a parent gas in zero field follows from the
Einstein relation (4.17) and formula (4.23) is given by
1
T
,
(4.27)
D = 0.12
√
M N σres ( 9T /M)
where the diffusion coefficient is expressed in cm2 /s and others are explained in
formula (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 the
following form if we ignore elastic ion–atom scattering and assume the cross section
of resonant charge exchange to be independent of the collision velocity
Mvx2
, vx > 0.
(4.29)
f (vx ) = C exp −
2eEλ
Here C is the normalization constant, and this formula gives for the ion drift velocity
wi and the average ion energy [113]
eEλ
2eEλ
Mvx2
,
=
.
(4.30)
ε=
wi = v x =
πM
2
2
Accounting for the velocity dependence for the cross section
√ of resonant charge
exchange, it is taken in formula (4.30) at the ion velocity 1.4 eEλ/M in the laboratory frame of reference (an atom is motionless). The diffusion ion coefficients in
4.2 Ion Drift in Gas in External Electric Field
109
the field direction D
and in the perpendicular direction to it D⊥ are given by [16]
D
= 0.137λw,
D⊥ =
D
Mw 2
eEλ
= 0.137
= 0.087
.
D⊥
T
T
Tλ
,
Mw
(4.31)
The ion drift velocity is given by formulas (4.15) and (4.28) in the limit of
low electric field strengths and by formula (4.30) in the limit of large electric field
strengths. One can construct the ion drift velocity in an intermediate range of electric
fields. Indeed, let us introduce the parameter
β=
eE
,
T N σres
(4.32)
and in the first approximation assume the cross section of resonant charge exchange
to be independent of the collision velocity. Let us represent the ion drift velocity in
the form
2T
Φ(β),
(4.33)
w=
m
and, according to formulas (4.23) and (4.30), the limiting dependencies for the function Φ(β) have the form
β
, β 1.
Φ(β) = 0.48β, β 1;
Φ(β) =
π
As follows from the solution of the kinetic equation [207, 208] for the ion distribution function on velocities, the reduced ion drift velocity Φ(β) may be approximated
by the dependence [100, 166]
Φ(β) =
0.48β
.
(1 + 0.22β 3/2 )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 on
the reduced electric field strength β (solid curve) in accordance with formula (4.34). Dotted
curves correspond to limiting cases
Formula
D = CKT
K = CD/T
√
D = C T /μ/(N σ1 )
√
K = C( T μN σ1 )−1
√
κ = C T /m/σ2
√
η = C T m/σ2
ξ = CE/(T N σ )
Number
1.
2.
3.
4.
5.
6.
7.
Coefficient C
8.617 × 10−5 cm2 /s
1 cm2 /s
11604 cm2 /(V s)
1 cm2 /(V s),
4.617 · 1021 cm2 /s
1.595 cm2 /s
171.8 cm2 /s
68.1115 cm2 /s
7338 cm2 /s
1.851 × 104 cm2 /(V s)
171.8 cm2 /(V s)
7.904 × 105 cm2 /(V s)
7338 cm2 /(V s)
1.743 × 104 W/(cm K)
7.443 × 105 W/(cm K)
5.591 × 10−5 g/(cm s)
1.160 × 1020
1 × 1016
Units
K in cm2 /(V s), T in K
K in cm2 /(V s), T in eV
D in cm2 /s, T in K
D in cm2 /s, T in eV
σ1 in Å2 , N in cm−3 , T in K, μ in a.u.m.
σ1 in Å2 , N = 2.687 × 1019 cm−3 , T in K, μ in a.u.m.
σ1 in Å2 , N = 2.687 × 1019 cm−3 , T in eV μ in a.u.m.
σ1 in Å2 , N = 2.687 × 1019 cm−3 , T in K, μ in e.u.m.
σ1 in Å2 , N = 2.687 × 1019 cm−3 , T in eV, μ in e.u.m.
σ1 in Å2 , N = 2.687 × 1019 cm−3 , T in K, μ in a.u.m.
σ1 in Å2 , N = 2.687 × 1019 cm−3 , T in eV, μ in a.u.m.
σ1 in Å2 , N = 2.687 × 1019 cm−3 , T in K, μ in e.u.m.
σ1 in Å2 , N = 2.687 × 1019 cm−3 , T in eV, μ in e.u.m.
T in K, m in a.u.m., σ2 in Å2
T in K, m in e.u.m., σ2 in Å2
T in K, m in a.u.m., σ2 in Å2
E in V/cm, T in K, σ in Å2 , N in cm−3
E in V/cm, T in eV, σ in Å2 , N in cm−3
Table 4.10. Conversion parameters for transport coefficients
110
4 Transport Phenomena in Gases
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 in
a gas D = KT /e, where D, K are the diffusion coefficient and mobility of a
charged particle, and T is the gas temperature.
2. The Einstein relation (4.17) for the mobility of a charged particle in a gas
K = 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 temperature, N is the number density of gas atoms or molecules, μ is the reduced mass
of a colliding particle and gas atom or molecule, and σ1 is the average cross
section 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 approximation
√
κ = 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; other
notations 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 N σ ),
where E is the electric field strength, T is the gas temperature, N is the number
density 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 action
of an external electric field is determined by electron–atom elastic scattering. The
cross section of electron–atom scattering at energies up to several eV is determined
by a finite number of collision momenta, and hence this process has a quantum
character. A sequent of this is the Ramsauer effect for electron scattering on atoms
with a negative electron scattering length. The parameters of this effect for inert
gas atoms Ar, Kr, Xe are given in Tables 3.1 and 3.2. Table 4.11 contains reduced
zero-field mobilities of electrons in inert gases at room temperature Ke N (Ke is
Table 4.11. Parameters of electron drift in inert gas atoms at zero field and room temperature
[163]
Gas
De Na ,1021 cm−1 s−1
Ke Na , 1023 (cm s V)−1
He
7.2
2.9
Ne
72
29
Ar
30
12
Kr
1.6
0.63
Xe
0.43
0.17
112
4 Transport Phenomena in Gases
the electron mobility, N is the number density of gas atoms) and the values of the
reduced diffusion coefficients of electrons De N (De is the diffusion coefficient of
electrons), 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 obtains
energy from the field and transfers it to atoms in elastic collisions until the electron
energy does not allow it to excite atoms. In each strong collision, an electron is
scattered on a large angle, so that its momentum changes significantly, whereas its
energy changes weakly. Therefore, the electron distribution function of velocities is
close to a spherically symmetric one. Hence, the electron drift velocity w and the
diffusion coefficient for an electron in a gas D⊥ in the direction perpendicular to the
field are expressed through the spherical distribution function of electrons, and this
connection has the form [209]
1 v2
eE 1 d v 3
,
D⊥ =
.
(4.35)
w=
3me v 2 dv ν
3
ν
Here brackets mean an average with the spherical distribution function of electron
velocities and the rate of electron elastic scattering in a gas is ν = N vσ ∗ (v), where
N 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 velocity
v, these formulas take the form
w=
eE
,
me ν
D⊥ =
v 2 .
ν
(4.36)
The data of Table 4.12 for the electron drift velocity in some atomic and molecular gases in an external electric field show that this case is not realistic. Note
that the drift velocity depends on the reduced electric field strength E/N, where
E 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
0.001
0.003
0.01
0.03
0.1
0.3
1
3
10
30
100
He
0.028
0.080
0.25
0.64
1.4
2.7
4.8
8.6
22
72
–
Ne
0.35
0.55
0.89
1.3
2.1
3.5
–
–
–
–
–
Ar
0.15
0.53
1.0
1.2
1.7
2.4
3.1
4.0
11
26
74
Kr
0.015
0.038
0.13
0.90
1.4
–
–
–
–
–
–
Xe
0.004
0.013
0.040
0.16
0.86
1.1
1.4
1.8
6.0
18
53
H2
0.020
0.052
0.17
0.46
1.4
3.2
6.2
10
19
37
130
N2
–
–
0.39
1.0
2.3
3.1
4.5
7.7
18
42
100
4.5 Diffusion of Excited Atoms in Gases
113
This character of electron processes when an electron takes energy from an external electric field and then transfers it to atoms as a result of elastic and inelastic
collisions, takes place in a gas discharge plasma. A gas discharge plasma results
from passage of electric current through a gas under the action of an external electric field. Collision of electrons with atoms in a gas discharge plasma establishes
a 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 and
molecules in a parent gas at room temperature.
Table 4.13. Diffusion coefficients of metastable atoms or molecules in parent gases at room
temperature reduced to the normal number density [30]
Metastable atom, molecule
D, cm2 /s
He(23 S)
0.59
0.52
0.45
0.20
0.20
0.067
0.071
0.039
0.043
0.024
0.020
0.24
0.19
0.20
0.19
He(21 S)
He(23 Σu )
Ne(3 P2 )
Ne(3 P0 )
Ar(3 P2 )
Ar(3 P0 )
Kr(3 P2 )
Kr(3 P0 )
Xe(3 P2 )
Xe(3 P0 )
N(2 D)
N(2 D)
N2 (A3 Σu )
O 2 (1 Δ g )
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 of
atoms or molecules. Conversion factors for this equation are given in Table 5.1. This
equation 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 the
gaseous constant R is equal to
R = 82.06
cm3 MPa
cm3 atm
= 8.314
.
mol K
mol K
Equation (5.2) describes a rare gas when an average distance between neighboring atoms or molecules N −1/3 significantly exceeds a distance R0 of a strong interaction between gas atoms or molecules U (R0 ) ∼ T , where U (R) is the interaction
potential of two atoms at a distance R, and T is a thermal energy. A subsequent
expansion over a small parameter N 1/3 R0 may lead to the van der Waals equation
a
(5.3)
p + 2 (V − b) = T
V
using two van der Waals constants, a and b. This equation may be spread to the critical 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
1.
Formula
N = Cp/T
2.
p = CN T
The proportionality factor C
7.339 × 1021 cm−3
9.657 × 1018 cm−3
1.036 × 10−19 Torr
Units
p in atm, T in K
p in Torr, T in K
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 critical point, the critical parameters [211] of this system of atoms are expressed through
the van der Waals parameters. The critical point of an atom system is determined by
the equations
2 ∂ p
∂p
= 0,
= 0,
(5.4)
∂V T
∂V 2 T
which allows us to express the critical parameters Vcr , pcr and Tcr through the parameters 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 the
framework of the van der Waals equation
ζ =
Tcr
8
= .
Vcr pcr
3
(5.6)
The accuracy of the van der Waals equation decreases with an increase of the gas
density. Its accuracy is about 30% at the critical point, as follows from comparison
of the data of Table 5.2 with formulas (5.5) and (5.6), where parameters of the van
der 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 molecular gases 3.49 ± 0.15 according to the data of Table 5.2. These values coincide
within the limits of their accuracy, but differ from that of formula (5.6) based on the
van der Waals equation.
Table 5.3 contains critical parameters for molecular systems consisting of molecules 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 molecular gases [2, 9, 10]
H2
He
N2
O2
F2
Ne
Cl2
Ar
Br2
Kr
J2
Xe
Rn
a
(105 MPa cm6 /mol2 )
0.245
0.0346
1.370
1.382
1.171
0.208
6.343
1.355
9.75
5.19
–
4.19
–
b
(cm3 /mol)
26.5
23.8
38.7
31.9
29
16.7
54.2
32
59.1
10.6
–
51.6
–
Tcr
(K)
32.97
5.19
126.2
154.6
144.1
44.4
416.9
150.9
588
209.4
819
289.8
377
pcr
(MPa)
1.293
0.227
3.39
5.043
5.172
2.76
7.99
4.90
10.3
5.50
(13)
5.84
6.28
Vcr
(cm3 /mol)
65
57
90
73
66
42
123
75
127
91
155
118
(150)
ζ = p TcrV
cr cr
3.26
3.33
3.44
3.49
3.51
3.18
3.52
3.41
3.74
3.48
–
3.50
–
5.2 Basic Properties of Ionized Gas
117
Table 5.3. The critical parameters for “round” molecules AF6 and the boiling point for condensed systems of these molecules at atmospheric pressure [212]
A
S
Mo
W
U
Tb , K
209.4
307.2
290.3
329.6
Tcr , K
318.7
473
444
505.8
pcr , MPa
3.77
4.75
4.34
4.66
Vcr , cm3 /mol
199
226
233
250
ζ = Tcr /(pcr Vcr )
3.53
3.66
3.65
3.61
Table 5.4. Critical parameters for some vapors and their boiling points [2, 9, 10]
Li
Na
P
S
K
As
Se
Rb
Cs
Hg
Tb , K
1615
1156
554
717.8
119.9
876
958
961
944
629.9
Tcr , K
3223
2573
994
1314
209.4
1673
1766
2093
1938
1750
pcr , MPa
67
35
–
20.7
5.50
–
27.2
16
9.4
172
Vcr , cm3 /mol
66
116
–
–
91
35
–
247
341
43
them is similar to interaction between atoms of inert gases. Correspondingly, the
combination of critical parameters ζ defined by formula (5.6) is 3.61 ± 0.06. This
value is close to that of the inert gases and molecular gases of Table 5.2, but differs
from 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 can
form bound atomic systems with strong interaction between atoms in which atoms
have lost their individuality. Interaction between atoms of these vapors differs from
that in the case of inert gases and similar systems. Therefore, the behavior of these
systems at the critical point is also different. In particular, according to the data of
Table 5.4, the parameter ζ = VcrTcrpcr for 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 interaction
of electrons and ions satisfies the gaseousness condition, that has the form
α≡
Ne e 6
1,
Te3
(5.7)
where Ne is the number density of electrons, and Te is the electron temperature
expressed in energetic units. According to this criterion, the interaction potential
118
5 Properties of Macroscopic Atomic Systems
Table 5.5. Conversion factors for plasma physics
Number Formula
Units
1.
Ne in cm−3 , T in eV
Ne in cm−3 , T in K
T in K, m in e.m.u.
T in K, m in e.m.u.
T in K, m in a.m.u.
T in eV, m in a.m.u.
Te in 1000 K
Ne in cm−3 , m in e.m.u.
Ne in cm−3 , m in a.m.u.
Ne in cm−3 , T in eV
Ne in cm−3 , T in K
m in a.m.u., l in cm, U in V
m in e.m.u., l in cm, U in V
m in a.m.u., l in cm, i in A/cm2
m in e.m.u., l in cm, i in A/cm2
m in a.m.u., U in V, i in A/cm2
m in e.m.u., U in V, i in A/cm2
2.
3.
4.
5.
6.
7.
8.
Proportionality
factor C
α = CNe /T 3
2.998 × 10−21
4.685 × 10−9
3/2
3/2
2.415 × 1015 cm−3
f = Cm T
3.019 × 1021 cm−3
1.879 × 1020 cm−3
2.349 × 1026 cm−3
9/2
K = C/Te
2.406 × 10−31 cm6 /s
√
5.642 × 104 s−1
ωp = C Ne /m
1322 s−1
√
525.3 cm
rD = C T /Ne
4.876 cm
5.447 × 10−8 A/cm2
i = CU 3/2 m−1/2 l −2
2.326 × 10−6 A/cm2
1/3
4/3
−2/3
69594 B
U = Cm l i
5697 V
l = Cm−1/4 U 3/4 i −1/2 2.3338 × 10−4 cm
1.525 × 10−3 cm
Explanations:
1. The ideality plasma parameter (5.7) α = Ne e6 /Te3
2. Preexponent of the Saha formula (5.9) ξ = [mT /(2π h̄2 )]3/2
3. The rate constant of three-body electron–ion recombination according to formulas (3.39)
1/2 9/2
)
and (3.40) K = 1.5 e10 /(me Te 4. The plasma frequency (5.14) ωp = 4π Ne e2 /me
2
5. The Debye–Hückel radius (5.13) rD =
T /(8π Ne e )
3/2
2
e U
6. The three-halves law (5.16) i = 9π
2m l 2 , where l is the distance between electrodes, U is the difference of electrode voltages, i is the maximum current density, e, m
is the particle charge and mass
2/3 ( 2m )1/3 l 4/3
7. The three-halves law in the form U = ( 9πi
e
2 )
3/4
2 )1/2 ( e )1/4 U
8. The three-halves law in the form l = ( 9π
2m
i 1/2
of two charged particles (electrons and ions) at an average distance between them
1/3
e2 Ne 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 established as a result of atom ionization by electron impact and three-body electron–ion
recombination according to the scheme
e + A ↔ 2e + A+ .
(5.8)
Then the relation between the number density of electrons Ne , ions Ni and atoms
Na is given by the Saha formula [14, 213]
5.2 Basic Properties of Ionized Gas
Ne Ni
ge gi me Te 3/2
J
.
=
exp −
Na
ga 2π h̄2
Te
119
(5.9)
Here ge , gi , ga are statistical weights of electron, ion and atom, respectively; me is
the electron mass, Te is the electron temperature, and J is the atom ionization potential.
We define below two basic parameters of an ionized gas that are determined
by a long-range Coulomb interaction between charged plasma particles (electrons
and ions). Since this interaction remains in the presence of neutral particles, these
properties become apparent also in a weakly ionized gas. If a weak electric field is
introduced in a plasma, it causes displacement of electrons and ions, which leads
to the screening of this field. Correspondingly, the electric field strength E on a
distance x from the plasma boundary is given by
x
,
(5.10)
E = E0 exp −
rD
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 a
charge e at a distance r between them equals
r
e2
exp −
U (r) =
,
(5.11)
r
rD
and the Debye–Hückel radius is given by formula [214]
1
1 −1/2
+
.
rD = 4πNe e2
Te
Ti
(5.12)
Here Te is the electron temperature, Ti is the ion temperature, Ne is the number
density of electrons or ions (a plasma is quasi-neutral). In the case of an equilibrium
plasma Te = Ti = T , this formula takes the form
T
rD =
,
(5.13)
8πNe e2
where Te = Ti = T . As is seen, the Debye–Hückel radius rD is an important plasma
parameter.
Take a plasma that occupies an infinite space and displace the plasma electrons
at some distance. Then an electric field arises that tends to return the electrons in the
initial positions. Because of inertia of electrons, their return to the initial positions
has the vibration character and the oscillation frequency ωp , the frequency of plasma
oscillations is equal to [215]
4πNe e2
,
(5.14)
ωp =
me
120
5 Properties of Macroscopic Atomic Systems
where me is the electron mass. The reciprocal value 1/ωp is a typical time of plasma
reaction 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 presence
of atoms and molecules in a plasma does not change collective plasma properties
that are determined by Coulomb interaction of charged particles because neutral
particles do not influence this interaction. Note that the ratio of the above parameters
rD /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 electrodes
with a distance l between them, and the voltage difference is U , the relation between
these parameters is given by the three-halves law [216, 217]
e U 3/2
2
,
(5.16)
i=
9π 2m l 2
where e is a particle charge and m is the mass of a charged particle. Conversion
factors 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 nanoparticles consisting of many atoms in a liquid drop. This liquid drop has a spherical
shape and its density coincides with that of a macroscopic system. The radius r of a
liquid drop containing n atoms is given by
r = rW n1/3 ,
(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 attachment to a cluster equals πr 2 ξ , where r is the cluster radius that significantly exceeds
the radius of action of atomic forces, and ξ is the probability of atom attachment to
the cluster after their contact. In this case the rate of cluster growth in its vapor is
equal to [13]
T
T
dn
2
2/3
= πr
N ξ = k0 ξ n N, k0 =
πr 2 .
(5.19)
dt
2πm
2πm W
5.3 Parameters and Rates of Processes Involving Nanoparticles
121
Table 5.6. Conversion factors for parameters and processes involving clusters or nanoparticles
Number
Formula
1.
2.
rW = Cm/ρ 1/3
n = C(r/rW )3
2 √T /m
k0 = CrW
Proportionality
factor C
0.7346 Å
4.189
4.5714 × 10−12 cm3 /s
3.
v = Cρr 2 /η
0.2179 cm/s
0.01178 cm/s
v = Cr 2 /η
4.
√
2 )
D0 = C T /m/(N rW
1.469 × 1021 cm2 /s
0.508 cm2 /s
54.69 cm2 /s
5.
√
2 )−1
K0 = C( T mN rW
1.364 × 1019 cm2 /(V s)
0.508 cm2 /(V s)
54.69 cm2 /(V s)
Units
m in a.u.m., ρ in g/cm3
r and rW in Å
rW in Å, T in K,
m in a.u.m.
r in µm, ρ in g/cm3 ,
η in 10−5 g/(cm s)
r in µm, ρ in g/cm3
η for air at p = 1 atm,
T = 300 K
rW in Å, N in cm−3 ,
T in K, m in a.u.m.
rW in Å,
N = 2.687 × 1019 cm−3 ,
T in K, m in a.u.m.
rW in Å,
N = 2.687 × 1019 cm−3 ,
T in eV, m in a.u.m.
rW in Å, N in cm−3 ,
T in K, m in a.u.m.
rW in Å,
N = 2.687 × 1019 cm−3 ,
T in K, m in a.u.m.
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 = 2ρgr 2 /(9η),
where g is the free fall acceleration, ρ is the cluster density, η is the viscosity of a matter
through which a cluster moves
4. The diffusion coefficient for a spherical cluster consisting of n atoms
2/3 , where in the first Chapman–Enskog approximation (4.5)
Dn = D√
0 /n
2 ), so that T is the gas temperature, N is the number
D0 = 3 2T /π m/(16N rW
density 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 =
where in the first Chapman–Enskog approximation (4.18) K0 =
K0 /n2/3 , √
2 2π mT ), and notations are given above
3e/(8N rW
122
5 Properties of Macroscopic Atomic Systems
Here N is the number density of atoms, T is the gas temperature, m is the atom
mass, and n is the number of cluster atoms. Note that the vapor pressure significantly
exceeds the saturated vapor pressure at a given temperature and the vapor may be
located in a buffer gas with a higher density. The values of the parameters rW and
k0 , 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 lanthanides
taken from [13]. The melting and boiling points are given for macroscopic metals.
Table 5.7 gives the parameters of molecular liquids consisting of diatomic molecules. Then the Wigner–Seits radius rW and the reduced rate constant k0 relate to
a molecule instead of an atom at room temperature T = 300 K. Here Tm , Tb are
the 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 = ε0 n − An2/3 .
(5.20)
The first term is the volume cluster energy and the second one is the surface cluster
energy, so that ε0 is the binding energy per atom for a bulk atom system, and A is
the specific surface cluster energy. The values of these energy parameters are given
in diagram Pt20. From this we have for the energy εn that is spent on one atom
liberation
dEn
2A
(5.21)
εn = −
= ε0 − 1/3 .
dn
3n
Values of the specific surface energy A in diagram Pt20 are expressed through the
surface tension of liquids which are taken from [2, 219, 220].
Let us include in the balance equation (5.18) cluster evaporation if the cluster
temperature is T . Within the framework of the liquid drop model for a cluster on the
basis of the principle of detailed balance, we have the following equation of cluster
growth instead of (5.18)
2A
dn
,
(5.22)
= k0 ξ N − Nsat (T ) exp
dt
3T n1/3
Table 5.7. Parameters of liquids and liquid drops consisting of diatomic molecules [2, 9, 10]
Element
H2
N2
O2
F2
Cl2
Br2
I2
Tm , K
14.01
63.3
54.8
53.5
172.2
265.9
386.7
Tb , K
20.28
77.4
90.2
85.0
239.2
331.9
457.5
ρliq , g/cm3
0.071
0.88
1.14
1.52
1.51
3.13 (293 K)
(4.9)
rW , Å
2.29
2.33
2.23
2.15
2.65
2.72
2.73
k0 , 10−11 cm3 /s
28
8.1
7.0
5.9
6.6
4.6
3.7
Pt18. Parameters of the liquid drop model for liquid metal clusters
5.3 Parameters and Rates of Processes Involving Nanoparticles
123
5 Properties of Macroscopic Atomic Systems
Pt19. Lanthanides
124
125
Pt20. Evaporation parameters of metal liquid clusters
5.3 Parameters and Rates of Processes Involving Nanoparticles
126
5 Properties of Macroscopic Atomic Systems
Table 5.8. Parameters of the pair interaction of inert gas atoms and liquid clusters of inert
gases
Re , Å
D, meV
Tm , K
Tb , K
a, Å
εsub , meV
A, meV
p0 , 104 atm
ε0 , meV
Hfus , meV
Ne
3.09
3.64
24.6
27.1
3.156
20
15.3
0.34
18.6
3.42
Ar
3.76
12.3
83.7
87.3
3.755
80
53
1.15
68
12.3
Kr
4.01
17.3
115.8
119.8
3.892
116
73
1.14
95
17.0
Xe
4.36
24.4
161.2
165.0
4.335
164
97
1.28
132
23.7
where Nsat is the saturated number density of atoms over a plane surface. The saturation vapor pressure psat over a plane surface is given by
ε0
psat = p0 · exp −
.
(5.23)
T
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 parameters of pair interaction of inert gas atoms: Re , the equilibrium distance for a
diatomic molecule, and D, the depth of the pair interaction potential, which were
found 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 is
the boiling point, a is the distance between nearest neighbors of the crystal at zero
temperature, εsub is the binding energy per atom for the crystal, and Hfus is the
fusion 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 atoms
when atoms conserve their individuality in bound systems of atoms, i.e. parameters
of macroscopic solids and liquids consisting of these atoms are determined by the
pair interaction potential of atoms. A simpler description of a condensed system of
atoms takes place when interaction between nearest neighbors dominates in these
systems. This allows us to construct any dimensional parameter of this condensed
system of bound atoms through three dimensional parameters: the atom mass m and
two 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]
a/Re
Tm /D
εsub /D
A/D
ε0 /D
Hfus /D
rW /Re
p0 Re3 /D
Ne
1.028
0.583
5.50
4.20
5.1
0.955
0.601
17
Ar
1.000
0.585
6.49
4.31
5.5
0.990
0.594
31
Kr
0.996
0.576
6.66
4.22
5.5
0.980
0.595
27
Xe
0.993
0.570
6.71
3.98
5.4
0.977
0.589
28
Average
1.004 ± 0.014
0.578 ± 0.006
6.35 ± 0.50
4.2 ± 0.1
5.4 ± 0.2
0.98 ± 0.01
0.595 ± 0.003
26 ± 6
between atoms that corresponds to the interaction potential minimum, and the minimum potential energy −D. If these conditions are fulfilled, the scaling law takes
place and, because of a short-range character of atom interaction in a condensed
system of bound atoms, the distance between nearest neighbors a in this condensed
system 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 lead
to identical reduced values of the same quantity. This fact is demonstrated in Table 5.9 for condensed inert gases and allows us to find parameters of one inert gas
through 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 condensed
identical molecular systems with a weak interaction (for example, AF4 , AF6 ).
We now consider condensed systems consisting of identical atoms. Diagram Pt21
gives the thermodynamic parameters of condensed systems of elements. In this diagram, the evaporation enthalpy Hev corresponds to a release of atoms (or molecules) from the surface of a macroscopic system, while the enthalpy of conversion
of a condensed system in an atomic gas or vapor Hgas respects to separation
of 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 temperature 298 K is represented also in diagram Pt21, as well as the entropy of the
atomic vapor state of this element at the indicated temperature.
Diagram Pt22 contains parameters of the crystal structure of a given element at
zero temperature. Carbon is an exclusion and has practically two stable structures at
low 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. Values
of the Debye temperature in diagrams Pt19, Pt22, that is, the parameter of the Debye
model for vibration excitation of crystal states, are taken from [228].
Metal parameters at room temperature, including the density, conductivity, thermal conductivity, and work function, are represented in diagram Pt23. According to
the definition, a metal is a conducting material inside which electrons are present
and are characterized by a large mean free path. According to the diagram Pt23, the
5 Properties of Macroscopic Atomic Systems
Pt21. Thermodynamic parameters of elements
128
Pt22. Parameters of crystal structures of elements at low temperatures
5.4 Parameters of Condensed Atomic Systems
129
5 Properties of Macroscopic Atomic Systems
Pt23. Metal parameters at room temperature
130
5.4 Parameters of Condensed Atomic Systems
131
ratio of the mean free path to the Wigner–Seits radius for the crystal lattice of metals
is tens and hundreds. In addition to this, diagram Pt19 contains the parameters of
liquid and solid rare-earth metals–lanthanides.
A deeper understanding of the metals’ nature follows from the analysis of their
properties 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 between
electrons. This model starts from the Drude model of independent classic electrons
[229, 230] and, though the momentum electron distribution in metals is more complex in reality [231], the model of a degenerate electron gas describes roughly the
behavior of electrons in metals. Then the parameter of this electron system is the
density and, at zero temperature, the electron distribution function on momenta corresponds 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 with
the electron number density Ne by the relation
εF =
(3π 2 Ne )2/3 h̄2
.
2me
(5.24)
This electron distribution accounts for the Pauli principle for particles with the spin
1/2, so that two electrons cannot be located in the same state. At non-zero temperature, the electron distribution is close to that of a degenerate electron gas if the
parameter
T
(5.25)
η=
εF
is small. Here T is the electron temperature expressed in energetic units. Table 5.10
gives the values of the Fermi energy εF , the electron velocity on the Fermi surface
vF , the Wigner–Seits radius rW , and the parameter η at room temperature for singlevalence metals, the atoms of which have one valence s-electron. Table 5.10 contains
also the values of the parameter
ξ=
e2
25/3
0.337
=
=
,
1/3
1/3
rW εF
3πa0 Ne
a0 Ne
(5.26)
that is, the ratio of the Coulomb interaction potential at an average distance between
the nearest electrons to the Fermi energy of electrons.
In Table 5.10, there are the reduced mean free paths of electrons λ/a for these
metals (λ is the mean free path of electrons at room temperature, a is the distance
between the nearest atoms of this metal), the electron effective mass m∗ and the
Debye temperature ΘD for these metals. The electron mobility Ke , the conductivity Σ and the thermal conductivity coefficient κ for these metals are given also in
Table 5.10. From these data it follows that metal electrons may be considered as a
degenerated 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 corresponds
to the mass of a free electron.
5 Properties of Macroscopic Atomic Systems
Pt24. Valence electrons of metals
132
5.4 Parameters of Condensed Atomic Systems
133
Table 5.10. Parameters of metals for elements of the first group of the periodical systems
with s-valence electrons at room temperature [5]. The crystal lattice has the body-centered
cubic (bcc) structure or the face-centered cubic (fcc) structure (a is the lattice constant, ρ is
the metal density)
Metal
Type of lattice
a, Å
ρ, g/cm3
Ne , 1022 cm−3
εF , eV
vF , 108 cm/s
η, 10−3
ξ
λ/a
ΘD , K
m∗ /me
Ke , cm2 /(V s)
Σ, 1016 s−1
κ, W/(cm K)
Li
bcc
3.51
0.534
4.6
4.7
1.3
5.5
1.8
38
370
1.40
18
9.7
0.85
Na
bcc
4.29
0.97
2.5
3.2
1.0
8.2
2.2
73
158
0.98
52
19
1.41
K
bcc
5.34
0.89
1.4
2.1
0.86
13
2.7
54
90
0.94
58
12
1.02
Cu
fcc
3.61
8.96
8.5
7.1
1.6
3.7
1.4
82
310
1.01
34
54
4.01
Rb
bcc
5.71
1.53
11
1.8
0.79
15
2.9
36
52
0.87
44
7.0
0.58
Ag
fcc
4.09
10.5
5.9
5.5
1.4
4.6
1.6
100
220
0.99
53
57
4.29
Cs
bcc
6.09
1.93
8.7
1.6
0.74
17
3.1
26
54
0.83
38
4.4
0.36
Au
fcc
4.08
19.3
5.9
5.5
1.4
4.6
1.6
62
185
0.99
32
40
3.17
In addition to the data in Table 5.10, we give similar data in diagram Pt24 for
some metals according to [2]. The Sommerfeld radius rS is determined from the
relation
4πrS3 Ne
= 1,
(5.27)
3
where Ne is the number density of electrons. Hence the Sommerfeld radius characterizes a size of a region that relates to one electron. If rS is small compared with
the lattice constant, the structure of the metal crystal is not of importance for metal
electron properties. If rS is small compared to the Bohr radius, the metallic plasma
is classical. As follows from the data of Table 5.10 and diagram Pt24, the indicated
values of these relations are comparable. This means that, though a metallic plasma
depends on the metal crystal lattice and is not classical, these assumptions may be
used for the metallic plasma as models.
Fig. A.1. Spectrum of hydrogen atom
A Atomic Spectra
Fig. A.2. Spectrum of helium atom
136
A Atomic Spectra
A Atomic Spectra
Fig. A.3. Spectrum of lithium atom
137
138
A Atomic Spectra
Fig. A.4. Spectrum of beryllium atom
A Atomic Spectra
Fig. A.5. Spectrum of boron atom
139
Fig. A.6. Spectrum of carbon atom
140
A Atomic Spectra
Fig. A.7. Spectrum of nitrogen atom
A Atomic Spectra
141
Fig. A.8. Spectrum of oxygen atom
142
A Atomic Spectra
Fig. A.9. Spectrum of fluorine atom
A Atomic Spectra
143
Fig. A.10. Spectrum of neon atom
144
A Atomic Spectra
A Atomic Spectra
Fig. A.11. Spectrum of sodium atom
145
Fig. A.12. Spectrum of magnesium atom
146
A Atomic Spectra
A Atomic Spectra
Fig. A.13. Spectrum of aluminium atom
147
Fig. A.14. Spectrum of silicon atom
148
A Atomic Spectra
Fig. A.15. Spectrum of phosphorus atom
A Atomic Spectra
149
Fig. A.16. Spectrum of sulfur atom
150
A Atomic Spectra
Fig. A.17. Spectrum of chlorine atom
A Atomic Spectra
151
152
A Atomic Spectra
Fig. A.18. Spectrum of potassium atom
A Atomic Spectra
Fig. A.19. Spectrum of copper atom
153
Fig. A.20. Spectrum of zinc atom
154
A Atomic Spectra
A Atomic Spectra
Fig. A.21. Spectrum of rubidium atom
155
Fig. A.22. Spectrum of strontium atom
156
A Atomic Spectra
A Atomic Spectra
Fig. A.23. Spectrum of silver atom
157
Fig. A.24. Spectrum of cadmium atom
158
A Atomic Spectra
A Atomic Spectra
Fig. A.25. Spectrum of caesium atom
159
Fig. A.26. Spectrum of barium atom
160
A Atomic Spectra
Fig. A.27. Spectrum of gold atom
A Atomic Spectra
161
Fig. A.28. Spectrum of mercury atom
162
A Atomic Spectra
References
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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-units
4. http://physics.nist.gov/cuu/units
5. 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
(Hemisphere, New York, 1987)
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Index
absorption coefficient 40
absorption cross section 40
abundance of elements 3
abundance of isotopes 4
affinity to hydrogen atom 65
affinity to oxygen atom 65
aggregate state of elements 128
air plasma, processes 97, 98
angle wave function 13
amplitude of scattering 73
angular atom momentum 11, 17
anharmonic constant of diatomic 47,
49–51
associative ionization 91
asymptotic atom parameters 22
atomization enthalpy 128
attachment of electron to molecule 95, 96
autodetaching state 76, 97
autoionizing state 91
balance equation for number density 76
binding energy of cluster atoms 124
boiling point of metals 122, 123
Boltzmann formula 41
broadening of spectral lines 39
capture cross section 79
CGS system of units 2
Chapman–Enskog approximation 100
closest approach distance 71
conductivity of metals 130
conversion factors for units 6
correlation between atomic and molecular
states 53
covalence chemical bond 46
critical parameters 116, 117
cross section of resonant charge exchange
40, 81
cross section of collision 71
Dalgarno formula 104
Debye–Hückel radius 119
Debye temperature 129, 133
degenerate electron gas 131
density of elements at room temperature
129
density of liquid clusters 122
density of metals 130
differential cross section 71, 72
diffusion coefficient 99
diffusion coefficient for atomic ion in parent
gas 107–109
diffusion coefficient of clusters 122
diffusion cross section 72, 73
diffusion of electrons in gases 111
diffusion of metastable atoms in gases 113
dissociation energy for diatomic 49
dissociation energy for negative diatomic
ion 51
dissociation energy for positive diatomic ion
50
dissociative recombination 93, 94
Doppler broadening of spectral lines 41
effective principle quantum number 15
Einstein coefficient 37
Einstein relation 104
electromagnetic (CGSM) system of units
2
electron affinity for atoms 31
electron affinity of diatomic molecules 64
electron drift in gases 111
electron–ion recombination 93
electron mobility in metals 133
172
Index
electron scattering length 47, 59, 73
electron term of diatomic molecule 47, 49
electron term of excited atom states 32
electron term of ground atom state 30, 32
electron term for negative diatomic ion 51
electron term of negative atomic ion 31
electron term for positive diatomic ion 50
electron wave function for hydrogen atom
11
electrostatic (CGSE) system of units 2
enthalpy of bulk atomization 128
entropy of atomic vapor state 128
equation of gas state 114
equilibrium distance for diatomic 48, 49
evaporation enthalpy 128
eveness of molecular state 53, 54
exchange interaction 15, 45
excimer molecules 46, 66
excitation cross section 77
excitation energy of atoms 32
Fermi energy of metal electrons 132
Fermi formula 47
fine structure splitting 14
first Townsend coefficient 87
free fall velocity of clusters 121
frequency distribution function of photons
39
fundamental physical constants 1
gaseous constant 115
gas discharge plasma 87, 113
gas-kinetic cross section 102
Grotrian diagrams 16, 18, 39, 135–162
hard sphere model 72
Hartree atomic units 9
helium atom 16, 136
helium-like ions 17
Hund empiric rule 23
Hund scheme of momentum coupling in
diatomic molecule 48
hydrogen atom 11, 135
hydrogen-like ions 11
ideality plasma parameter 117, 118
impact parameter of collision 71
ion drift velocity in parent gas 103, 108,
109
ion–ion chemical bond 46
ionization equilibrium 118
ionization potential of atomic ions 30
ionization potential of atoms 30
ionization potential of diatomic 49
isotopes 2
jj -scheme of momentum coupling
24, 25
lanthanides 124
Larmor frequency 10
lattice constant 129
lifetime of resonantly excited atom state
35, 42, 43
lifetime of radioactive isotopes 5
liquid drop model 120
long-range interaction 43
Lorenz broadening of spectral lines 41
LS-scheme of momentum coupling 23
mean free path 103
mean free path of electrons in metals
melting points of elements 130
metal thermal conductivity 130
mobility of atomic ions in parent gases
106, 107
mobility of charged clusters 121
mobility of ions in gases 105
mutual neutralization of ions 95
nanoparticles
120
one-electron approximation 15
oscillator strength for atom transition
37
34,
parameters of charge exchange cross section
82
parentage atom scheme 19
parity of molecule state 53, 54
Pashen notations 28
Pauli exclusion principle 15
Penning process 91
plasma frequency 119
polarizability of atom 36, 45
polarizability of molecule 36, 63
polarization capture 79
polarization interaction 45
potential curves of molecules 55–63
principle quantum number 11
Index
principle of detailed balance
78
quadrupole moment 44
quantum defect 15
quasi-statistic broadening of spectral lines
42
quenching rate constant 77
Racah coefficient 21
radial wave function 11, 13
radiative transitions in hydrogen atom 38
radiative transitions for heliumlike ions 39
Ramsauer effect 74
rate constant 76
reduced binding energy for bulk atoms
124, 125
reduced rate constant of cluster growth
120–122
resonant charge exchange 79
resonantly excited state 29, 34, 43
rotational constant 48
rotation energy for diatomic 47, 49
rotation constant for negative diatomic ion
51
rotation constant for positive diatomic ion
50
Rutherford formula 85
Rydberg states 15
Saha formula 118
scaling law 127
saturation vapor pressure 126
scattering phase 72
selection laws for radiation 39
self-diffusion coefficient 101
Sena effect 105
shell atom structure 18
shell of valence electrons for atom ground
state 31
173
shell of valence electrons for excited atom
states 32
shell of valence electrons for negative ion
31
SI units 2, 6
Sommerfeld radius 132, 133
specific surface energy of clusters 124,
125
spin–orbit interaction 13, 25
splitting of atom levels 35
standard atomic weights 2
stepwise atom ionization 93
stimulated emission cross section 40
system of atomic units 7
thermal conductivity coefficient 99
Thomson model for atom ionization by
electron impact 85
three body recombination process 93
three halves law 120
total cross section 41, 72
Townsend 6, 112
transport cross section 72, 73
types of crystal lattices 124, 129
van der Waals equation 115
van der Waals interaction constant 45
vibration energy for diatomic 47, 49
vibration energy for negative diatomic ion
51
vibration energy for positive diatomic ion
50
viscosity coefficient 99
wavelength of radiative transition 33, 37
Weiskopf radius 47
Wigner–Seits radius 120
work function for metals 130
zero-field mobility of electrons in gases
111
Springer Series on
ATOMIC , OPTICAL , AND PLASMA PHYSICS
Editors-in-Chief:
Professor G.W.F. Drake
Department of Physics, University of Windsor
401 Sunset, Windsor, Ontario N9B 3P4, Canada
Professor Dr. G. Ecker
Ruhr-Universität Bochum, Fakultät für Physik und Astronomie
Lehrstuhl Theoretische Physik I
Universitätsstrasse 150, 44801 Bochum, Germany
Professor Dr. H. Kleinpoppen, Emeritus
Stirling University, Stirling, UK, and
Fritz-Haber-Institut
Max-Planck-Gesellschaft
Faradayweg 4-6, 14195 Berlin, Germany
Editorial Board:
Professor W.E. Baylis
Professor B.R. Judd
Department of Physics, University of Windsor
401 Sunset, Windsor, Ontario N9B 3P4, Canada
Department of Physics
The Johns Hopkins University
Baltimore, MD 21218, USA
Professor Uwe Becker
Fritz-Haber-Institut
Max-Planck-Gesellschaft
Faradayweg 4–6, 14195 Berlin, Germany
Professor Philip G. Burke
Professor K.P. Kirby
Harvard-Smithsonian Center for Astrophysics
60 Garden Street, Cambridge, MA 02138, USA
Professor P. Lambropoulos, Ph.D.
Oak Ridge National Laboratory
Building 4500S MS6125
Oak Ridge, TN 37831, USA
Max-Planck-Institut für Quantenoptik
85748 Garching, Germany, and
Foundation for Research
and Technology – Hellas (F.O.R.T.H.),
Institute of Electronic Structure
and Laser (IESL),
University of Crete, PO Box 1527
Heraklion, Crete 71110, Greece
Professor M.R. Flannery
Professor G. Leuchs
Brook House, Norley Lane
Crowton, Northwich CW8 2RR, UK
Professor R.N. Compton
School of Physics
Georgia Institute of Technology
Atlanta, GA 30332-0430, USA
Friedrich-Alexander-Universität
Erlangen-Nürnberg
Lehrstuhl für Optik, Physikalisches Institut
Staudtstrasse 7/B2, 91058 Erlangen, Germany
Professor C.J. Joachain
Professor P. Meystre
Faculté des Sciences
Université Libre Bruxelles
Bvd du Triomphe, 1050 Bruxelles, Belgium
Optical Sciences Center
The University of Arizona
Tucson, AZ 85721, USA
Springer Series on
ATOMIC , OPTICAL , AND PLASMA PHYSICS
20 Electron Emission
in Heavy Ion–Atom Collision
By N. Stolterfoht, R.D. DuBois,
and R.D. Rivarola
21 Molecules and Their Spectroscopic
Properties
By S.V. Khristenko, A.I. Maslov,
and V.P. Shevelko
22 Physics of Highly Excited Atoms and Ions
By V.S. Lebedev and I.L. Beigman
23 Atomic Multielectron Processes
By V.P. Shevelko and H. Tawara
24 Guided-Wave-Produced Plasmas
By Yu.M. Aliev, H. Schlüter,
and A. Shivarova
25 Quantum Statistics of Nonideal Plasmas
By D. Kremp, M. Schlanges,
and W.-D. Kraeft
26 Atomic Physics with Heavy Ions
By H.F. Beyer and V.P. Shevelko
34 Laser Physics at Relativistic Intensities
By A.V. Borovsky, A.L. Galkin,
O.B. Shiryaev, T. Auguste
35 Many-Particle Quantum Dynamics
in Atomic and Molecular Fragmentation
Editors: J. Ullrich and V.P. Shevelko
36 Atom Tunneling Phenomena in Physics,
Chemistry and Biology
Editor: T. Miyazaki
37 Charged Particle Traps
Physics and Techniques
of Charged Particle Field Confinement
By V.N. Gheorghe, F.G. Major, G. Werth
38 Plasma Physics
and Controlled Nuclear Fusion
By K. Miyamoto
39 Plasma-Material Interaction
in Controlled Fusion
By D. Naujoks
27 Quantum Squeezing
By P.D. Drumond and Z. Ficek
40 Relativistic Quantum Theory
of Atoms and Molecules
Theory and Computation
By I.P. Grant
28 Atom, Molecule, and Cluster Beams I
Basic Theory, Production and Detection
of Thermal Energy
Beams By H. Pauly
41 Turbulent Particle-Laden Gas Flows
By A.Y. Varaksin
29 Polarization, Alignment and Orientation
in Atomic Collisions
By N. Andersen and K. Bartschat
30 Physics of Solid-State Laser Physics
By R.C. Powell
(Published in the former Series on Atomic,
Molecular, and Optical Physics)
31 Plasma Kinetics in Atmospheric Gases
By M. Capitelli, C.M. Ferreira,
B.F. Gordiets, A.I. Osipov
32 Atom, Molecule, and Cluster Beams II
Cluster Beams, Fast and Slow Beams,
Accessory Equipment and Applications
By H. Pauly
33 Atom Optics
By P. Meystre
42 Phase Transitions of Simple Systems
By B.M. Smirnov and S.R. Berry
43 Collisions of Charged Particles
with Molecules
By Y. Itikawa
44 Plasma Polarization Spectroscopy
Editors: T. Fujimoto and A. Iwamae
45 Emergent Non-Linear Phenomena
in Bose–Einstein Condensates
Theory and Experiment
Editors: P.G. Kevrekidis,
D.J. Frantzeskakis,
and R. Carretero-González
46 Angle and Spin Resolved Auger Emission
Theory and Applications to Atoms
and Molecules
By B. Lohmann