Sp.-V/AQuan/1999/10/07:14:19
Page 27
Chapter 3
Atoms and Molecules
Werner Däppen
3.1
3.1
Online Databases and Other Sources . . . . . . . . . .
27
3.2
Elements, Atomic Mass, and Solar-System Abundance
28
3.3
Excitation, Ionization, and Partition Functions . . . .
31
3.4
Ionization Potentials . . . . . . . . . . . . . . . . . . . .
35
3.5
Electron Affinities . . . . . . . . . . . . . . . . . . . . .
35
3.6
Atomic Cross Sections for Electronic Collisions . . .
35
3.7
Atomic Radii . . . . . . . . . . . . . . . . . . . . . . . .
43
3.8
Particles of Modern Physics . . . . . . . . . . . . . . .
44
3.9
Molecules . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.10
Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
ONLINE DATABASES AND OTHER SOURCES
The National Institute of Standards and Technology (NIST) gives access to extensive
physical and atomic data (http://physics.nist.gov).
The Plasma Laboratory of the Weizmann Institute (http://plasma-gate.weizmann.ac.il) and the Southwest Research Institute
(http://espsun.space.swri.edu/spacephysics/www.atomic.html) provide, besides their own data, many
useful links to other databases. For astrophysical applications, among the most extensive databases
are those of the Harvard–Smithsonian Center for Astrophysics (http://cfa-www.harvard.edu/amp/data)
(giving, e.g., searchable access to the data by R.L. Kurucz and R.L. Kelly) and of the Opacity Project
(http://astro.u-strasbg.fr/OP) (with monochromatic opacities, collision strengths, and other atomic
data). A further source of important data is the Iron Project (http://www.am.qub.ac.uk/projects/iron).
Gary Ferland’s Web Page (http://www.pa.uky.edu/gary/cloudy) has references to CLOUDY (“Photoionization Simulations for the Discriminating Astrophysicist”), which contains pointers to the atomic
27
Sp.-V/AQuan/1999/10/07:14:19
28 / 3
Page 28
ATOMS AND M OLECULES
databases they use and maintain (e.g., http://www.pa.uky.edu/verner/atom.html, “Atomic Data for Astrophysics”). The CHIANTI group (http://www.solar.nrl.navy.mil/chianti) has installed a database
with information suitable for extreme-UV applications. The Particle Data Group (http://pdg.lbl.com)
makes available periodically its newest releases of particle properties. Other sources of information
are the recent Atomic, Molecular, and Optical Physics Handbook [1], the results of the work of the
Collaborative Computational Project No. 7 (United Kingdom) [2], and the Handbook of Chemistry
and Physics [3].
3.2
ELEMENTS, ATOMIC MASS, AND SOLAR-SYSTEM ABUNDANCE
Atomic masses (weighted by the fractional abundances of the stable isotopes in normal terrestrial
composition [4]) are scaled to 12 C = 12.00. Standard values abridged to five significant digits are
given (from the International Union of Pure and Applied Chemistry (IUPAC); see [5]). For some
elements, atomic masses can be accurately measured to seven or more significant figures. IUPAC
regularly publishes these values irrespective of interest to any user. For many users, however, it is often
desirable that the published data remain valid over an extended period, which is helpful for textbooks
and numerical tables derived from atomic-mass data. IUPAC has recognized this need and approved
the use of the designation standard to its abridged atomic-mass table, with the hope that the quoted
values may survive for at least a decade.
The solar-system abundances (formerly denoted as cosmic abundances) are expressed logarithmically on a scale for which H is 12.00 dex. The intention is that they express cosmic abundance [6].
Thus, abundances are taken mainly from meteorites and the Sun’s photosphere. In both cases, values by
number are quoted. The agreement between meteoritic and solar data has improved remarkably since
the 1970s. Discrepancies have mostly gone away as the solar values—thanks especially to improved
transition probabilities and other atomic data—have become more accurate [6]. The two principal exceptions are the solar photospheric Li and Be abundances that are smaller than the meteoritic ones by
2.15 and 0.27 dex, respectively. The reason is that these elements are destroyed by nuclear reactions at
the bottom of the solar convection zone. For most other elements the agreement is better than ±0.04
dex (for this, and exceptions, see [6]). In the case of iron, a previous controversy has been solved: the
solar and meteoritic values agree now [7]. For details on isotopic abundances, see [1] and [4].
The group abundance ratios given in Table 3.1 are derived from Table 3.2. The H ratio is set to 100.
Table 3.1. Group abundance ratios.
Element group
Number
Mass
Stripped
electrons
H
He
C, N, O, Ne
Other
100
9.8
0.145
0.013
100
39
2.19
0.44
100
20
1.1
0.21
Total
109.96
141.63
121.3
The composition by mass [2] is as follows:
fraction of H, X
fraction of He, Y
fraction of other atoms, Z
0.707 ± 2.5%
0.274 ± 6%
0.018 9 ± 8.5%
Sp.-V/AQuan/1999/10/07:14:19
Page 29
3.2 E LEMENTS , ATOMIC M ASS , AND S OLAR -S YSTEM A BUNDANCE / 29
Mean atomic mass of cosmic material
Mean atomic mass per H atom
Mean atomic mass for fully ionized cosmic plasma
1.30
1.41
0.62
Table 3.2. Atomic masses and solar-system abundances.
Element
Symbol
[1]
Atomic
number
Hydrogen
Helium [3]
Lithium
Beryllium
Boron
H
He
Li
Be
B
1
2
3
4
5
Carbon
Nitrogen
Oxygen
Fluorine
Neon
C
N
O
F
Ne
6
7
8
9
10
Sodium
Magnesium
Aluminum
Silicon
Phosphorus
Na
Mg
Al
Si
P
Sulphur
Chlorine
Argon
Potassium
Calcium
Atomic
mass
1.007 9
4.002 6
6.941
9.012 2
10.811
Log abundance [2]
Meteoritic
12.00a
Solar
10.99a
3.31
1.42
2.8
12.00
10.99b
1.16
1.15
2.6c
12.011
14.007
15.999
18.998
20.180
8.56a
8.05a
8.93a
4.48
8.09
8.56
8.05
8.93
4.56
8.09d
11
12
13
14
15
22.990
24.305
26.982
28.086
30.974
6.31
7.58
6.48
7.55
5.57
6.33
7.58
6.47
7.55
5.45
S
Cl
Ar
K
Ca
16
17
18
19
20
32.066
35.453
39.948
39.098
40.078
7.27
5.27
6.56d
5.13
6.34
7.21
5.5
6.56d
5.12
6.36
Scandium
Titanium
Vanadium
Chromium
Manganese
Sc
Ti
V
Cr
Mn
21
22
23
24
25
44.956
47.88
50.942
51.996
54.938
3.09
4.93
4.02
5.68
5.53
3.10
4.99
4.00
5.67
5.39
Iron [2]
Cobalt
Nickel
Copper
Zinc
Fe
Co
Ni
Cu
Zn
26
27
28
29
30
55.847
58.933
58.693
63.546
65.39
7.51
4.91
6.25
4.27
4.65
7.54
4.92
6.25
4.21
4.60
Gallium
Germanium
Arsenic
Selenium
Bromine
Ga
Ge
As
Se
Br
31
32
33
34
35
69.723
72.61
74.922
78.96
79.904
3.13
3.63
2.37
3.35
2.63
2.88
3.41
Krypton
Rubidium
Strontium
Yttrium
Zirconium
Kr
Rb
Sr
Y
Zr
36
37
38
39
40
83.80
85.468
87.62
88.906
91.224
3.23
2.40
2.93
2.22
2.61
2.60
2.90
2.24
2.60
Niobium
Molybdenum
Technetium
Ruthenium
Rhodium
Nb
Mo
Tc
Ru
Rh
41
42
43
44
45
92.906
95.94
98.906
101.07
102.91
1.40
1.96
1.42
1.92
1.82
1.09
1.84
1.12
Sp.-V/AQuan/1999/10/07:14:19
30 / 3
Page 30
ATOMS AND M OLECULES
Table 3.2. (Continued.)
Element
Symbol
[1]
Atomic
number
Atomic
mass
Log abundance [2]
Meteoritic
Solar
Palladium
Silver
Cadmium
Indium
Tin
Pd
Ag
Cd
In
Sn
46
47
48
49
50
106.42
107.87
112.41
114.82
118.71
1.70
1.24
1.76
0.82
2.14
1.69
0.94c
1.86
1.66c
2.0
Antimony
Tellurium
Iodine
Xenon
Cesium
Sb
Te
I
Xe
Cs
51
52
53
54
55
121.76
127.60
126.90
131.29
132.91
1.04
2.24
1.51
2.23
1.12
1.0
Barium
Lanthanum
Cerium
Praseodymium
Neodymium
Ba
La
Ce
Pr
Nd
56
57
58
59
60
137.33
138.91
140.12
140.91
144.24
2.21
1.20
1.61
0.78
1.47
2.13
1.22
1.55
0.71
1.50
Promethium
Samarium
Europium
Gadolinium
Terbium
Pm
Sm
Eu
Gd
Tb
61
62
63
64
65
146.92
150.36
151.96
157.25
158.93
0.97
0.54
1.07
0.33
1.00
0.51
1.12
−0.1
Dysprosium
Holmium
Erbium
Thulium
Ytterbium
Dy
Ho
Er
Tm
Yb
66
67
68
69
70
162.50
164.93
167.26
168.93
170.04
1.15
0.50
0.95
0.13
0.95
1.1
0.26c
0.93
0.00c
1.08
Lutetium
Hafnium
Tantalum
Tungsten
Rhenium
Lu
Hf
Ta
W
Re
71
72
73
74
75
174.97
178.49
180.95
183.85
186.21
0.12
0.73
0.13
0.68
0.27
0.76c
0.88
Osmium
Iridium
Platinum
Gold
Mercury
Os
Ir
Pt
Au
Hg
76
77
78
79
80
190.2
192.22
195.08
196.97
200.59
1.38
1.37
1.68
0.83
1.09
1.45
1.35
1.8
1.01c
Thallium
Lead
Bismuth
Polonium
Astatine
Tl
Pb
Bi
Po
At
81
82
83
84
85
204.38
207.2
208.98
209.98
209.99
0.82
2.05
0.71
0.9c
1.85
Radon
Francium
Radium
Actinium
Thorium
Rn
Fr
Ra
Ac
Th
86
87
88
89
90
222.02
223.02
226.03
227.03
232.04
0.08
0.12
Protactinium
Uranium
Neptunium
Plutonium
Americium
Pa
U
Np
Pu
Am
91
92
93
94
95
231.04
238.03
237.05
239.05
241.06
−0.49
1.11c
< −0.45c
Sp.-V/AQuan/1999/10/07:14:19
Page 31
3.3 E XCITATION , I ONIZATION , AND PARTITION F UNCTIONS / 31
Table 3.2. (Continued.)
Element
Symbol
[1]
Atomic
number
Atomic
mass
Curium
Berkelium
Californium
Einsteinium
Fermium
Cm
Bk
Cf
Es
Fm
96
97
98
99
100
244.06
249.08
252.08
252.08
257.10
Mendelevium
Nobelium
Lawrencium
Md
No
Lr
101
102
103
258.10
259.10
262.11
Log abundance [2]
Meteoritic
Solar
Notes
a Based on solar data.
b Based on stellar observations and solar models [1, 3, 4].
c Uncertain.
d Based on other astronomical data.
References
1. IUPAC 1969, Comptes Rendus XXV Conference, p. 95
2. Anders, E., & Grevesse, N. 1989, Geochim. Cosmochim. Acta, 53, 197;
Grevesse, N., & Noels, A. 1993, in Origin and Evolution of the Elements,
edited by N. Prantzos, E. Vangioni, & M. Cassé (Cambridge University
Press, Cambridge), p. 15
3. Christensen-Dalsgaard, J., Däppen, W., & the GONG Team 1996, Science,
272, 1286
4. Biémont, E., Baudoux, M., Kurucz, R.L., Ansbacher, W., & Pinnington,
E.H. 1991, A&A, 249, 539
5. Kosovichev, A.G., Christensen-Dalsgaard, J., Däppen, W., Dziembowski,
W.A., Gough, D.O., & Thompson, M.J. 1992, MNRAS, 259, 536
3.3
3.3.1
EXCITATION, IONIZATION, AND PARTITION FUNCTIONS
Introduction
Finding the occupation of individual levels of atoms and ions and the fractions of ions of any given
chemical element in a plasma is a complex task. The difficulty arises from the interaction of the
plasma with the atoms. Therefore, in principle, the problems of modified atoms and of their statistical
occupation should be solved simultaneously and self-consistently. The typical task of quantumstatistical mechanics consists of the calculation of a density operator (ensemble) for the system of all
particles. The partition function, i.e., the trace over the density operator, not only gives the occupation
of all states, but it also leads to a thermodynamical potential.
It is evident that various approximations are necessary before this procedure can be carried out.
One such approximation consists of treating the motion of the heavy particles (nuclei, atoms, ions)
according to classical mechanics. Once the heavy particles are separated out, quantum-mechanical
electrons remain. In the treatment of electrons, we find a bifurcation into two distinct classes of
approach, the “chemical picture” and the “physical picture.” While in the more conventional chemical
picture, bound configurations (atoms, ions, and molecules) are introduced and treated as new and
independent species, only fundamental particles (electrons and nuclei) appear in the physical picture.
In the chemical picture, reactions between the various species occur. Thus the thermodynamical
equilibrium must be sought among the stoichiometrically allowed set of concentration variables by
means of a maximum entropy (or minimum free-energy) principle. In contrast, the physical picture
Sp.-V/AQuan/1999/10/07:14:19
32 / 3
Page 32
ATOMS AND M OLECULES
has the aesthetic advantage that there is no need for a minimax principle. The question of bound states
is dealt with implicitly through the Hamiltonian describing the interaction between the fundamental
particles.
It is obvious that these self-consistent approaches require extensive analytical and numerical work.
For a recent realization of the chemical-picture approach, see, e.g., [8–10]. For the physical picture, the
most detailed work so far was done as part of the OPAL opacity project [11–14]. In the OPAL project,
the physical picture was not only used to model excitation and ionization processes, but for the first time
also to yield the highly accurate thermodynamic quantities needed in computations of stellar models
[15]. The book by Ebeling et al. [16] contains further information and references on the physical
picture. The most recent addition to the set of stellar equations of state is based on the formalism
of the path integral in the framework of the Feynman–Kac representation. This formalism leads to a
virial expansion of the thermodynamic functions in the power of the total density of a Coulomb plasma
([17, 18], and references therein).
For many lower-density applications, especially stellar spectroscopy, adequate qualitative and
quantitative information can be extracted from simpler considerations, in which atoms are assumed to
have an unperturbed structure. In this case, excitation fractions are given by the Boltzmann factor, and
the ionization degree follows from the Saha equation, which is the mass-action law for the ionization
reaction. The Saha equation contains the internal partition functions for bound systems. A fundamental
theoretical flaw of this approximate approach is that isolated atoms would have infinite partition
functions because of their infinite number of excited states. Many heuristic recipes to truncate partition
functions exist. However, only the physical picture comes to a satisfactory solution, which then can
often be used to justify the intuitive concepts [19]. In many cases, neglecting all excited states, that is,
assuming ground-state-only internal partition functions, is a reasonable approximation. The following
simple treatment of excitation, ionization, and partition functions is, with reasonable care, still very
useful for many qualitative and semiquantitative astrophysical applications.
3.3.2
Approximate Methods and Results
For practical applications, a useful introduction to the statistical mechanics of plasmas is the book by
Eliezer et al. [20]. The number of atoms existing in various atomic levels 0, 1, 2, . . . when in thermal
equilibrium at temperature T is approximately described by the Boltzmann distribution
N2 /N1 = (g2 /g1 ) exp(−χ1,2 /kT ),
N2 /N = (g2 /U ) exp(−χ0,2 /kT ).
Numerically
log(N2 /N1 ) = log(g2 /g1 ) − χ12 (5040/T )
(χ12 in eV),
cm3 ,
where N is the total number of atoms per
N0 , N1 , and N2 are the numbers of atoms per cm3 in
the zero and higher levels, g0 , g1 , and g2 are the corresponding statistical weights, χ1,2 is the potential
difference between levels 1 and 2, and U is the partition function.
The degree of ionization in conditions of thermal equilibrium is given by the Saha equation
NY +1
UY +1 (2π m)3/2 (kT )5/2
Pe =
2
exp(−χY,Y +1 /kT ).
NY
UY
h3
Numerically
log
NY +1
Pe
NY
5040
= −χY,Y +1
+
T
5
2
2UY +1
log T − 0.4772 + log
UY
Sp.-V/AQuan/1999/10/07:14:19
Page 33
3.3 E XCITATION , I ONIZATION , AND PARTITION F UNCTIONS / 33
or
log
NY +1
Ne
NY
= −χY,Y +1 −
3
2
log + 20.9366 + log
2UY +1
UY
,
where NY and NY +1 are the numbers of atoms per cm3 in the Y and Y + 1 stages of ionization (Y = 1,
neutral; Y = 2nd, 1st ion; etc.), Ne is the number of electrons per cm3 , Pe is the electron pressure in
dyn cm−2 , χY,Y +1 the ionization potential in eV from the Y to the Y + 1 stage of ionization, = 5040
K/T , UY and UY +1 are the partition functions, and the factor 2 represents the statistical weight of an
electron.
The degree of ionization, when ionizations are caused by electron collisions and recombinations
are radiative, can be approximately given by
NY +1 /NY = S/α,
where the effect of both collisional ionizations from state of ionization Y + 1 and of recombinations
of Y + 2 in the abundance of ions in Y + 1 is neglected, and the possibility of multiple-ionization
events is excluded. In the formula, S is the collision ionization coefficient (such that S Ne NY = rate of
collisional ionization, see Sec. 3.6), and α is the recombination coefficient (such that α Ne NY +1 = rate
of recombination, see Chap. 5).
Detailed calculations of partition functions are given by Irwin [21] (atoms and molecules) and
Sauval and Tatum [22] (molecules). However, for the approximate purposes of this section, the
partition function may simply be regarded as the effective statistical weight of the atom or ion under
existing conditions of excitation. Except in extreme conditions it is approximately equal to the weight
of the lowest ground term. The ground term weight g0 is therefore given and this can normally be
extrapolated along the isoelectronic sequences to give the approximate partition function for any ion.
The partition functions, given in Table 3.3 in the form log U for = 1.0 and 0.5, are not intended
to include the concentration of terms close to each series limit. The part of the partition function
associated with these high-n terms is dependent on both T and Pe . This part is usually negligible unless
the atom concerned is mainly ionized in which case the high-n terms may be counted statistically with
the ion.
Lowering of χY,Y +1 in the Saha equation to allow for the merging of high-level spectrum lines
gives [23]
χY,Y +1 = 7.0 × 10−7 Ne Y 2/3 ,
1/3
with χ in eV and Ne in cm−3 , and where Y is the charge on the Y + 1 ion.
Table 3.3. Partition function [1–3].
Y =I
Y = II
log U
Element
g0
Y = III
log U
= 1.0
= 0.5
g0
= 1.0
= 0.5
g0
1
2
3
4
5
H
He
Li
Be
B
2
1
2
1
6
0.30
0.00
0.32
0.01
0.78
0.30
0.00
0.49
0.13
0.78
1
2
1
2
1
0.00
0.30
0.00
0.30
0.00
0.00
0.30
0.00
0.30
0.00
1
2
1
2
6
7
8
9
10
C
N
O
F
Ne
9
4
9
6
1
0.97
0.61
0.94
0.75
0.00
1.00
0.66
0.97
0.77
0.00
6
9
4
9
6
0.78
0.95
0.60
0.92
0.73
0.78
0.97
0.61
0.94
0.75
1
6
9
4
9
Sp.-V/AQuan/1999/10/07:14:19
Page 34
34 / 3
ATOMS AND M OLECULES
Table 3.3. (Continued.)
Y =I
Y = II
log U
Element
g0
Y = III
log U
= 1.0
= 0.5
g0
= 1.0
= 0.5
g0
11
12
13
14
15
Na
Mg
Al
Si
P
2
1
6
9
4
0.31
0.01
0.77
0.98
0.65
0.60
0.15
0.81
1.04
0.79
1
2
1
6
9
0.00
0.31
0.00
0.76
0.91
0.00
0.31
0.01
0.77
0.94
6
1
2
1
6
16
17
18
19
20
S
Cl
Ar
K
Ca
9
6
1
2
1
0.91
0.72
0.00
0.34
0.07
0.94
0.75
0.00
0.60
0.55
4
9
6
1
2
0.62
0.89
0.69
0.00
0.34
0.72
0.92
0.71
0.00
0.54
9
4
9
6
1
21
22
23
24
25
Sc
Ti
V
Cr
Mn
10
21
28
7
6
1.08
1.48
1.62
1.02
0.81
1.49
1.88
2.03
1.51
1.16
15
28
25
6
7
1.36
1.75
1.64
0.86
0.89
1.52
1.92
1.89
1.22
1.13
10
21
28
25
6
26
27
28
29
30
Fe
Co
Ni
Cu
Zn
25
28
21
2
1
1.43
1.52
1.47
0.36
0.00
1.74
1.76
1.60
0.58
0.03
30
21
10
1
2
1.63
1.46
1.02
0.01
0.30
1.80
1.66
1.28
0.18
0.30
25
28
21
10
1
31
32
34
36
37
38
39
Ga
Ge
Se
Kr
Rb
Sr
Y
6
9
9
1
2
1
10
0.73
0.91
0.83
0.00
0.36
0.10
1.08
0.77
1.01
0.89
0.00
0.7
0.70
1.50
1
6
4
6
1
2
1 + 15
0.00
0.64
0.00
0.70
0.62
0.00
0.34
1.18
0.66
0.00
0.53
1.41
2
1
9
9
6
1
10
40
48
50
56
57
70
82
Zr
Cd
Sn
Ba
La
Yb
Pb
21
1
9
1
10
1
9
1.53
0.00
0.73
0.36
1.41
0.02
0.26
1.99
0.02
0.88
0.92
1.85
0.21
0.54
28
2
6
2
21
2
6
1.66
0.30
0.52
0.62
1.47
0.30
0.32
1.91
0.30
0.61
0.85
1.71
0.31
0.40
21
1
1
1
10
1
References
1. Astrophysical Quantities, 1, §15; 2, §15
2. Cayrel, R., & Jugaku, J. 1963, Ann. d’Astrophys., 26, 495
3. Bolton, C.T. 1970, ApJ, 161, 1187
The degree of ionization in the material of stellar atmospheres is given in Table 3.4, relating gas
pressure Pg , electron pressure Pe , and temperature T . The data are averaged from [24] (rather high
heavy-element abundance) and [25] (rather low heavy-element abundance).
Sp.-V/AQuan/1999/10/07:14:19
Page 35
3.6 ATOMIC C ROSS S ECTIONS FOR E LECTRONIC C OLLISIONS / 35
Table 3.4. log Pg .
and T
log Pe
T (K)
−2
−1
0
1
2
3
4
5
3.4
0.1
50 400
0.2
25 200
0.4
12 600
0.6
8 400
0.8
6 300
1.0
5 040
1.2
4 200
1.4
3 600
−1.9
−0.8
+0.27
1.27
2.27
3.28
4.28
5.59
−1.8
−0.74
+0.29
1.30
2.30
3.30
4.31
5.30
−1.70
−0.70
+0.31
1.33
2.34
3.35
4.43
5.87
−1.67
−0.66
+0.35
1.47
2.98
4.87
6.84
8.66
−1.54
−0.01
+1.90
3.87
5.65
7.0
8.7
10.4
+0.78
2.57
3.9
5.2
6.7
8.3
10.0
11.8
+2.0
3.1
4.5
6.0
7.7
9.4
11.2
13.2
+2.4
3.9
5.3
6.7
8.5
10.4
12.4
14.4
IONIZATION POTENTIALS
Table 3.5 gives the energy in eV required to ionize each element to the next stage of ionization. I
(Y = 1) denotes the neutral atom, II the first ion, etc. Dividing lines between shells and subshells are
added to assist interpolation. Part of the data is based on an especially accurate compilation for selected
ions [6–20], made available by the National Institute of Standards and Technology (NIST, see Sec. 3.1
for online access). If the data are given in wave numbers, the currently recommended conversion factor
to energy is 1 eV = 8 065.541 cm−1 [26].
3.5
ELECTRON AFFINITIES
Electron affinity is the energy difference between the lowest state of the atom (or molecule or ion)
and the lowest state of the corresponding negative ion (see Table 3.6). It is positive for those atoms or
molecules that form stable negative ions. Regarding the astrophysically important H− , it was thought
earlier that a second stable state exists [27]. Later, however, it was proven rigorously that there is only
one stable state [28, 29].
3.6
ATOMIC CROSS SECTIONS FOR ELECTRONIC COLLISIONS
Definitions of symbols are presented below:
Q
v
πa02
Ne , Na , Ni
L = vQ
Ne L
Ne Na L
Pc
Atomic cross section [= Q(v)].
Precollision electron velocity.
Atomic unit cross section = 8.797 × 10−17 cm2 .
Electron, atom, ion densities (per cm3 ).
Collision rate for each atom per unit Ne .
Collision rate per atom (or ion).
Collision rate per cm3 .
Collisions encountered by an electron per cm at 0◦ C and 1 mm Hg.
pressure, then Q = 2.828 × 10−17 Pc = 0.321 5 πa02 Pc .
H
He
Li
Be
B
C
N
O
F
Ne
Na
Mg
Al
Si
P
S
Cl
Ar
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
13.598 44
24.587 41
5.391 72
9.322 63
8.298 03
11.260 30
14.534 14
13.618 06
17.422 82
21.564 54
5.139 08
7.646 24
5.985 77
8.151 69
10.486 69
10.360 01
12.967 64
15.759 62
4.340 66
6.113 16
6.561 44
6.828 2
6.746 3
6.766 64
7.434 02
7.902 4
7.881 0
7.639 8
7.726 38
9.394 05
I
54.417 78
75.640 18
18.211 16
25.154 84
24.383 32
29.601 3
35.117 30
34.970 82
40.963 28
47.286 4
15.035 28
18.828 56
16.345 85
19.769 4
23.337 9
23.814
27.629 67
31.63
11.871 72
12.799 67
13.575 5
14.66
16.485 7
15.639 99
16.187 8
17.083
18.168 84
20.292 40
17.964 40
II
122.454
153.897
37.931
47.888
47.449
54.936
62.708
63.45
71.620
80.144
28.448
33.493
30.203
34.79
39.61
40.74
45.806
50.913
24.757
27.492
29.311
30.96
33.668
30.652
33.50
35.19
36.841
39.723
III
217.713
259.366
64.492
77.472
77.413
87.140
97.12
98.91
109.265
119.99
45.142
51.444
47.222
53.465
59.81
60.91
67.27
73.489
43.267
46.71
49.16
51.2
54.8
51.3
54.9
55.2
59.4
IV
340.22
392.08
97.89
113.90
114.24
126.21
138.40
141.27
153.83
166.77
65.03
72.59
67.8
75.02
82.66
84.50
91.65
99.30
65.28
69.46
72.4
75.0
79.5
75.5
79.9
82.6
V
489.98
552.06
138.12
157.17
157.93
172.18
186.76
190.49
205.27
220.42
88.05
97.03
91.01
99.4
108.78
111.68
119.53
128.1
90.64
95.6
99.1
103
108
103
108
VI
667.03
739.29
185.19
207.28
208.50
225.02
241.76
246.49
263.57
280.95
114.20
124.32
117.56
127.2
138.0
140.8
150.6
161.18
119.20
124.98
131
134
139
136
VII
871.41
953.91
239.10
264.25
265.96
284.66
303.54
309.60
328.75
348.28
143.46
154.88
147.24
158.1
170.4
173.4
184.7
194.5
151.06
160
164
167
175
VIII
1 103.1
1 195.8
299.9
328.1
330.1
351.1
372.1
379.6
400.1
422.5
175.8
188.5
180.0
192.1
205.8
209.3
221.8
233.6
186.2
193
199
203
IX
1 362.2
1 465.1
367.5
398.8
401.4
424.4
447.5
455.6
478.7
503.8
211.3
225.2
215.9
230.5
244.4
248.3
262.1
276.2
224.6
232
238
X
1 648.7
1 761.8
442.0
476.4
479.5
504.8
529.3
539.0
564.7
591.9
249.8
265.1
255.1
270.7
286.0
290.2
305
321
266
274
XI
1 963
2 086
523
561
564
592
618
629
657
688
292
308
298
314
331
336
352
369
311
XII
2 304
2 438
612
652
657
686
715
727
757
788
336
355
344
361
379
384
401
412
XIII
2 673
2 817
707
750
756
787
818
831
863
896
384
404
392
411
430
435
454
XIV
36 / 3
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Atom
Stage of ionization
Table 3.5. Ionization potentials (electron volts) [1–20].
Sp.-V/AQuan/1999/10/07:14:19
Page 36
ATOMS AND M OLECULES
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
H
He
Li
Be
B
C
N
O
F
Ne
Na
Mg
Al
Si
P
S
Cl
Ar
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
Atom
3 070
3 224
809
855
862
895
927
941
975
1 011
435
457
444
464
484
490
XV
3 494
3 658
918
968
974
1 009
1 044
1 060
1 097
1 136
489
512
499
520
542
XVI
3 946
4 121
1 034
1 087
1 094
1 131
1 168
1 185
1 224
1 266
547
571
557
579
XVII
4 426
4 611
1 157
1 213
1 221
1 260
1 299
1 317
1 358
1 402
607
633
619
XVIII
4 934
5 129
1 288
1 346
1 355
1 396
1 437
1 456
1 500
1 546
671
698
XIX
5 470
5 675
1 425
1 486
1 496
1 539
1 582
1 602
1 648
1 698
738
XX
6 034
6 249
1 569
1 634
1 644
1 689
1 734
1 756
1 804
1 856
XXI
6 626
6 851
1 721
1 788
1 799
1 846
1 894
1 919
1 970
XXII
7 246
7 482
1 879
1 950
1 962
2 010
2 060
2 088
XXIII
Stage of ionization
Table 3.5. (Continued.)
7 895
8 141
2 045
2 119
2 131
2 182
2 234
XXIV
8 572
8 828
2 218
2 295
2 310
2 363
XXV
9 278
9 544
2 398
2 478
2 495
XXVI
10 030
10 280
2 560
2 660
XXVII
10 790
11 050
2 730
XXVIII
Sp.-V/AQuan/1999/10/07:14:19
Page 37
3.6 ATOMIC C ROSS S ECTIONS FOR E LECTRONIC C OLLISIONS / 37
Sp.-V/AQuan/1999/10/07:14:19
Page 38
38 / 3
ATOMS AND M OLECULES
Table 3.5. (Continued.)
Stage of ionization
Atom
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
Ga
Ge
As
Se
Br
Kr
Rb
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te
I
Xe
Cs
Ba
La
Ce
Pr
Nd
Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm
Yb
Lu
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
Tl
Pb
Bi
Po
At
I
II
III
IV
V
VI
VII
VIII
IX
X
5.999 30
7.900
9.815 2
9.752 38
11.813 81
13.999 61
4.177 13
5.694 84
6.217
6.633 90
6.758 85
7.092 43
7.28
7.360 50
7.458 90
8.336 9
7.576 24
8.993 67
5.786 36
7.343 81
8.64
9.009 6
10.451 26
12.129 87
3.893 9
5.211 70
5.577 0
.538 7
5.464
5.525 0
5.55
5.643 7
5.670 4
6.150 0
5.863 9
5.938 9
6.021 6
6.107 8
6.184 31
6.254 16
5.425 85
6.825 07
7.89
7.98
7.88
8.7
9.1
9.0
9.225 67
10.437 50
6.108 29
7.416 66
7.289
8.416 71
9.3
20.514
15.935
18.633
21.19
21.8
24.360
27.285
11.030
12.24
13.13
14.32
16.16
15.26
16.76
18.08
19.43
21.49
16.908
18.870
14.632
16.531
18.6
19.131
21.21
23.157
10.004
11.06
10.85
10.55
10.73
10.90
11.07
11.241
12.09
11.52
11.67
11.80
11.93
12.05
12.176
13.9
14.9
16
18
17
17
17
18.563
20.5
18.756
20.428
15.032
16.69
19
20
30.71
34.224
28.351
30.820
36
36.95
40
42.89
20.52
22.99
25.04
27.13
29.54
28.47
31.06
32.93
34.83
37.48
28.03
30.503
25.3
27.96
33
32.123
35
87
93.5
62.63
68.3
59.7
64.7
71.0
71.6
77.0
81.5
50.55
61.2
55
60
65
62
68
72
77
72.28
56
58.75
66
57
62
62
66
70
57.45
116
112
127.6
81.7
88.6
78.5
84.4
90.8
93
99
102.6
68
80
92
97
90
89
94
98
103
108
70.7
81
82
74
80
80
85
89
140
144
147
155.4
103.0
111.0
9.2
106
116
117
125
126.8
170
174
179
184
192.8
126
136
122.3
129
140
142
153
212
207
212
218
224
230.9
150
162
146.2
155
161
163
187
243
250
242
250
257
263
277.1
177
191
19.177
20.198
21.624
64
45.71
50.13
42.944
47.3
52.5
52.6
57
61.8
34.34
38.3
46.4
46
50
48
53
56
59
54.4
40.734
44.2
37.41
42
46
46
49
52
36.72
38.95
110
115
115
120
125
130
137
100
100
100
95
100
100
105
110
130
140
145
145
150
155
165
170
120
120
120
115
120
120
130
135
155
160
170
180
175
185
190
200
210
145
145
145
140
145
150
155
160
180
185
195
205
210
210
220
230
240
250
160
165
165
160
170
175
180
190
23.68
25.05
20.959
23.3
22
24
26
25
27
28
30
34.2
29.83
31.937
25.56
27
29
33.3
33
35
38
40
39
41
44
46
50.7
42.32
45.3
38
41
45
48
51
54
57
55
58
61
64
68.8
56.0
61
51
61
64
68
72
75
73
77
81
84
88.3
73
78
79
83
88
92
96
94
98
103
107
112
91
100
105
110
115
120
115
120
125
130
140
120
125
135
140
145
140
150
155
160
145
155
160
165
175
170
175
185
Sp.-V/AQuan/1999/10/07:14:19
Page 39
3.6 ATOMIC C ROSS S ECTIONS FOR E LECTRONIC C OLLISIONS / 39
Table 3.5. (Continued.)
Stage of ionization
Atom
86
87
88
89
90
91
92
93
94
95
Rn
Fr
Ra
Ac
Th
Pa
U
Np
Pu
Am
I
II
III
IV
V
10.748 50
4
5.278 92
5.17
6.08
5.89
6.194 05
6.265 7
6.06
5.993
21
22
10.147
12.1
11.5
29
33
34
20
20.0
44
43
46
49
28.8
55
59
58
62
65
VI
67
71
76
76
80
84
VII
VIII
IX
X
97
84
89
95
94
100
104
110
115
105
110
115
115
120
165
135
140
125
130
140
140
190
195
155
165
145
155
160
References
1. Astrophysical Quantities, 1, §16; 2, §16; 3, §16
2. Lotz, W. 1966, Ionisierungsenergien von Ionen H bis Ni (Inst. Plasmaphys, München)
3. Moore, C.E. 1970, Ionization Potentials, NSRDS-NBS 34, Washington
4. Finkelnberg, W., & Humbach, W. 1955, Naturwiss., 42, 35
5. Handbook of Chemistry and Physics, 77th ed. (CRC, Boca Raton, FL, 1996)
6. Martin, W.C. 1987, Phys. Rev. A, 36, 3575 (He I)
7. Martin, W.C., Kaufman, V., & Musgrove, A. 1993, J. Phys. Chem. Ref. Data, 22, 1179 (O II)
8. Martin, W.C., & Zalubas, R. 1981, J. Phys. Chem. Ref. Data, 10, 153 (Na I–XI)
9. Martin, W.C., & Zalubas, R. 1980, J. Phys. Chem. Ref. Data, 9, 1 (Mg I–XII)
10. Martin, W.C., & Zalubas, R. 1979, J. Phys. Chem. Ref. Data, 8, 817 (Al I–XIII)
11. Martin, W.C., & Zalubas, R. 1983, J. Phys. Chem. Ref. Data, 12, 323 (Si I–XIV)
12. Martin, W.C., Zalubas, R., & Musgrove, A. 1985, J. Phys. Chem. Ref. Data, 14, 751 (P I–XV)
13. Martin, W.C. Zalubas, R., & Musgrove, A. 1990, J. Phys. Chem. Ref. Data, 19, 821 (S I–XVI)
14. Sugar, J., & Corliss, C. 1985, J. Phys. Chem. Ref. Data, 14, Suppl. No. 2 (K, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni)
15. Sugar, J., & Musgrove, A. 1990, J. Phys. Chem. Ref. Data, 19, 527 (I–XXIX)
16. Sugar, J., & Musgrove, A. 1995, J. Phys. Chem. Ref. Data, 24, 1803 (Zn I–XXX)
17. Sugar, J., & Musgrove, A. 1993, J. Phys. Chem. Ref. Data, 22, 1213 (Ge I–XXXII)
18. Sugar, J., & Musgrove, A. 1991, J. Phys. Chem. Ref. Data, 20, 859 (Kr I–XXXVI)
19. Sugar, J., & Musgrove, A. 1988, J. Phys. Chem. Ref. Data, 17, 155 (Mo I–XLII)
20. Martin, W.C., Zalubas, R., & Hagan, L. 1978, Natl. Stand. Ref. Data Ser. (Natl. Bur. Stand., U.S.) 60 (Rare-Earth
Elements)
21. Cohen, E.R., & Taylor, B.N. 1988, J. Phys. Chem. Ref. Data, 17, 1795
Table 3.6. Electron affinities [1–2].
Atom
Electron
affinity (eV)
Atom
Electron
affinity (eV)
H
He
Li
Be
B
+0.754
−0.3
+0.618
−0.4
+0.277
Na
Mg
Al
Si
P
+0.479
−0.4
+0.441
+1.385
+0.747
C
N
O
O−
F
Ne
+1.263
−0.2
+1.461
−6.7
+3.401
−0.7
S
Cl
Br
I
K
Ca
+2.077
+3.612
+3.48
+3.17
+0.501
+0.018
Molecule
Electron
affinity (eV)
O2
O3
OH
SH
C2
C3
+0.451
+2.102 8
+1.827 67
+2.314
+3.269
+1.981
CN
NH2
NO
NO2
NO3
CH
+3.862
+0.771
+0.026
+2.273
+3.951
+1.238
References
1. Astrophysical Quantities, 1, §17; 2, §17; 3, §17
2. Handbook of Chemistry and Physics, 77th ed. (CRC, Boca Raton, FL, 1996)
Sp.-V/AQuan/1999/10/07:14:19
40 / 3
3.6.1
Page 40
ATOMS AND M OLECULES
Ionization Cross Section
The classical cross section of atoms for ionization by electrons [30] is
Q 1 = 4nπa02
1 χ
1−
,
χ
where χ is the ionization energy in rydbergs (Ry), the electronic energy before collision in Ry, and n
the number of optical electrons.
The general approximation for cross sections of atoms for ionization by electrons (see, [30–33]) is
Q 1 = nπa02
nπa02
1
F(Y,
/χ
)
=
q
χ
χ2
2
= 1.63 × 10−14 n(1/χeV
)(χ /)F(Y, /χ ),
where Y is the charge on the ionized atom (or next ion stage) and χeV is the ionization energy in
eV. The function F(Y, /χ ) is given and also q = (χ /)F(Y, /χ ), which is sometimes called the
reduced cross section in Table 3.7. The Y = 1 and Y = 2 values are from experiment and Y = ∞
from calculation. About ±10% accuracy may be expected for hydrogenic ions. In other cases ±0.3
dex may be expected. Other empirical forms have been suggested (see, e.g., [34–36]).
Table 3.7. Numerical functions F(Y, /χ ) and q(Y, /χ ).
/χ
1.0
1.2
1.5
2.0
3
5
10
F(classical) = 4(1 − χ /)
F(1, /χ )
F(2, /χ )
F(∞, /χ )
0.00
0.0
0.00
0.00
0.67
0.31
0.53
0.74
1.33
0.78
1.17
1.54
2.00
1.60
2.02
2.56
2.67
2.9
3.3
3.8
3.20
4.6
4.7
5.0
3.60
6.4
6.4
6.4
q(classical) = 4(χ /)(1 − χ /)
q(l, /χ )
q(2, /χ )
q(∞, /χ )
0.00
0.00
0.00
0.00
0.56
0.26
0.44
0.62
0.89
0.52
0.78
1.03
1.00
0.80
1.01
1.28
0.89
0.97
1.09
1.28
0.64
0.92
0.94
1.00
0.36
0.64
0.64
0.64
The maximum ionization cross section for the classical case is
Q max = nπa02 χ −2
at = 2χ .
The value of Q max is approximately the same in actual cases but the maximum occurs near = 4χ .
The rate of ionization by electrons (see [30–32]) is
L 1 = v Q1.
The neutral atom approximation (with kT < ionization energy) gives
−2 −5040χeV /T
L 1 = 1.1 × 10−8 nT 1/2 χeV
10
cm3 s−1 .
The coronal ion approximation (with kT < ionization energy) gives
−2 −5040χeV /T
L 1 = 2.1 × 10−8 nT 1/2 χeV
10
cm3 s−1 .
Sp.-V/AQuan/1999/10/07:14:19
Page 41
3.6 ATOMIC C ROSS S ECTIONS FOR E LECTRONIC C OLLISIONS / 41
3.6.2
Excitation Cross Section (Permitted Transitions)
An approximation for Q ex , the excitation cross section of an atom (see [30, 37]), is given. The
approximation applies fairly well when n ≥ 1 (notation of Chap. 5). For n = 0 the approximation
tends to be small:
f
8π
Q ex = √ πa02
b
W
3
= 1740πa02 λ2 (W/) f b
= 1.28 × 10−15 ( f /W )b cm2 ,
where f is the oscillator strength, W is the excitation energy in Ry (= 0.0912/λ with λ in µm), and is the electron energy before collision, also in Ry. See Table 3.8.
Table 3.8. Numerical factors b and bW/.
/W
1.0
1.2
1.5
2.0
3
5
10
30
100
b, neutral atoms
b, ions
0.00
0.20
0.03
0.20
0.06
0.20
0.11
0.20
0.21
0.24
0.33
0.33
0.56
0.56
0.98
0.98
1.33
1.33
bW/, neutral atoms
bW/, ions
0.00
0.20
0.03
0.17
0.04
0.13
0.06
0.10
0.07
0.08
0.07
0.07
0.06
0.06
0.03
0.03
0.01
0.01
The maximum excitation cross section is as follows:
• The neutral atom approximation gives
Q max = 125πa02 λ2 f
near = 3W.
• The ion approximation gives
Q max = 350πa02 λ2 f
near = W
(λ in µm).
The rate of excitation (see [34, 35, 37]) is
L = v Q ex = 17.0 × 10−4
f
T 1/2 W
eV
10−5040WeV /T P(W/kT ),
where WeV and W are the excitation energy in eV and in ergs (with 11 600WeV /kT = W/kT ) and
P(W/kT ) is tabulated from [37] (see Table 3.9).
Table 3.9. Numerical factors P(W/kT ) and W/kT .
P(W/kT )
W/kT
Neutral atoms
< 0.01
0.01
0.02
0.05
0.29E 1 (W/kT )a
1.16
0.96
0.70
Ions
1.16
0.98
0.74
Sp.-V/AQuan/1999/10/07:14:19
42 / 3
Page 42
ATOMS AND M OLECULES
Table 3.9. (Continued.)
P(W/kT )
W/kT
0.1
0.2
0.5
1
2
5
10
> 10
Neutral atoms
Ions
0.49
0.33
0.17
0.10
0.063
0.035
0.023
0.55
0.40
0.26
0.22
0.21
0.20
0.20
0.066/(W/kT )1/2
0.20
Note
a E ( ) is the first exponential integral.
1
The tabulated P(W/kT ) are too small when the total quantum number of Chap. 5 is unchanged.
The approximations quoted should be replaced by quantum calculations when available (see
[30, 38–40]). A Coulomb approximation for ions [41] gives b = geff (2L + 1)/g1 (L in Chap. 5).
The tabulations of geff , the effective Gaunt factor, range from 0.5 to 0.9.
3.6.3
Deexcitation Cross Sections
Deexcitation cross sections Q 21 are related to excitation cross sections Q 12 (2 being the upper level)
through
g2 2 Q 21 = g1 1 Q 12 ,
where 2 = 1 + W , and g2 and g1 are statistical weights.
The deexcitation rate L 21 and excitation rate L 12 are related by
g2 L 21 = g1 L 12 exp(W/kT ).
3.6.4
Excitation Cross Sections (Forbidden Transitions)
The collision strength for each line is defined by (see [33, 42])
Q f = π /g1 kν2 = πa02 /g1 =
h2
= 4.21/g1 v 2 ,
2
4π m g1 v 2
where kν /2π is the wave number of the incident electron (then kv2 in atomic units = in Ry), v is the
electron velocity, g1 is the statistical weight of the initial (lower) level, and Q f is the forbidden line
cross section for atoms in this level. Then 12 (excitation) = 21 (deexcitation).
3.6.5
Collision Strengths: Extensive Databases
Crude recipes to estimate the order of magnitude of collision strengths (for allowed and forbidden
transitions) can be found in older references [43]. In recent years, however, a wealth of accurate
Sp.-V/AQuan/1999/10/07:14:19
Page 43
3.7 ATOMIC R ADII / 43
collision strengths have been obtained for a very large number of transitions. They are based on
extensive UV and IR emission-line observations and on theoretical calculations. Data are available,
e.g., from the Opacity Project, the Iron Project, and the Harvard–Smithsonian Center for Astrophysics
(see Sec. 3.1 for information about online access of these sources).
3.6.6
Total Atomic Cross Section (Elastic and Inelastic)
An approximation for the total cross section is (see [31, 32, 44])
Q 180πa02 λ/ 1/2
(λ in µm, in Ry),
where λ is the wavelength of the strongest low-level lines.
3.6.7
Ionic Collision Cross Section
Cross section for collision deflection of at least a right angle (see [45])
Q = π(Y − 1)2 (e2 /mv 2 )2 = π(Y − 1)2 (e2 /2hc R)2
= πa02 (Y − 1)2 / 2 ( in Ry),
where Y − 1 is the ionic charge.
The effective ionic collision cross section is usually concerned with the more distant collision
involving deflections much less than a right angle. These increase the effective Q by a factor depending
logarithmically on the most distant collisions that enter the integration and also on the circumstances.
The factor is usually between 10 and 50 (see Sec. 3.10). We may write a general approximation:
Q(effective) 20πa02 (Y − 1)2 / 2 .
3.7
ATOMIC RADII
Atomic radii are defined through the closeness of approach of atoms in the formation of molecules
and crystals. The radius r so derived is approximately that of maximum radial density in the charge
distribution of neutral atoms (see Table 3.10). For ions the appropriate radius measures to the point
where the radial density falls to 10% of its maximum value. The atomic mass divided by the atomic
volume (4/3)πr 3 gives the density of the more compact solids. 2r is approximately the gas-kinetic
diameter of monoatomic molecules.
Table 3.10. Atomic radii [1–5].
Atom
H
He
Li
Be
B
r (Å)
Ion [3]
0.7
1.2
1.58
1.06
0.83
H−
r (Å)
1.8
Li+
Be2+
B3+
0.68
0.39
0.28
Atom
S
Cl
Ar
K
Ca
r (Å)
Ion [3]
1.05
1.02
1.6
2.37
1.97
S2−
Cl−
r (Å)
1.70
1.67
K+
Ca2+
1.52
1.14
Atom
Br
Kr
Rb
Sr
Ag
r (Å)
Ion [3]
r (Å)
1.2
1.82
2.54
2.3
1.44
Br−
1.82
Rb+
Sr2+
Ag+
1.66
1.32
1.29
Sp.-V/AQuan/1999/10/07:14:19
44 / 3
Page 44
ATOMS AND M OLECULES
Table 3.10. (Continued.)
Atom
r (Å)
Ion [3]
C
N
O
F
Ne
0.77
0.70
0.66
0.62
1.3
C4+
Na
Mg
Al
Si
P
1.91
1.62
1.43
1.09
1.08
Na+
Mg2+
Al3+
Si4+
P3−
r (Å)
0.22
1.92
1.26
1.19
N3−
O2−
F−
1.16
0.86
0.67
0.47
2.3
Atom
r (Å)
Ion [3]
Sc
Ti
V
Cr
Mn
1.64
1.46
1.39
1.28
1.26
Sc3+
r (Å)
Ti4+
V4+
0.89
0.75
0.61
Mn2+
0.81
Fe
Co
Ni
Cu
Zn
1.27
1.25
1.29
1.28
1.39
Fe2+
Co2+
Ni2+
Cu+
Zn2+
0.75
0.79
0.83
0.91
0.77
Atom
r (Å)
Ion [3]
r (Å)
Cd
Sn
I
Xe
Cs
1.6
1.62
1.4
2.00
2.73
Cd2+
Sn4+
I−
1.09
0.76
2.06
Cs+
1.81
Ba
Pt
Au
Hg
2.24
1.38
1.44
1.57
Ba2+
1.49
Au+
Hg2+
1.51
1.16
References
1. Astrophysical Quantities, 1, §19; 2, §19; 3, §19
2. Teatum, E., Gschneidner, K., & Waber, J. 1960, Los Alamos Scientific Laboratory, Report No. LA-2345
3. Shannon, R.D. 1976, Acta Cryst., A32, 751
4. Allen, F.H., Kennard, O., Watson, D.G., Brammer, L., Orpen, A.G., & Taylor, R. 1987, J. Chem. Soc. Perkin II,
S1
5. Alcock, N.W. 1990, Bonding and Structure: Structural Principles in Inorganic and Organic Chemistry, (Ellis
Horwood, New York)
3.8
PARTICLES OF MODERN PHYSICS
A representative selection of particles is given in Table 3.11. Hadrons include mesons, nucleons, and
baryons. Possible proton decay is not included. I denotes the isotopic spin, J the spin, and P the
parity. The lifetime is that in free space. In the column labeled “Decay” are given the main decay
products. The mean life τ for W and Z bosons is given as the linewidth (τ ≈ h).
Table 3.11. Selected particles of modern physics [1–3].
Name
Symbol
Charge
Mass
(amu)
I
JP
Mean life
(s)
1−
1
1
∞
= 2.1 GeV
= 2.5 GeV
eν , etc.
e+ e− , etc.
Decay
Bosons
Gauge bosons
Photon
W
Z
Mesons
π -mesons (pion)
K meson (kayon)
γ
W
Z
0
+1, −1
0
0.000
86.24
97.90
0, 1
π +, π −
π0
−
K+
0,K
+1, −1
0
+1, −1
0.149 84
0.144 90
0.530 15
1
1
1/2
0−
0−
0−
2.603 × 10−8
0.83 × 10−16
1.237 × 10−8
µν
γγ
µν, ππ 0
0
0.534 38
1/2
0−
0.892 × 10−10
π +π −, π 0π 0
0.534 38
1/2
0−
5.38 × 10−8
πeν, π µν, 3π 0
1/2
∞
∞
∞
∞
2.197 × 10−6
(3.4 ± 0.5) × 10−13
K0S
K0L
0
Fermions
Leptons
e Neutrino
µ Neutrino
τ Neutrino
Electron, Positron
µ meson (muon)
τ meson (tauon)
νe
νµ
ντ
e
µ
τ
0
0
0
−1, +1
−1, +1
−1, +1
< 5 × 10−8
< 5 × 10−4
< 0.2
0.000 548 6
0.113 4
1.915
1/2
1/2
1/2
1/2
1/2
eν ν̄
eν ν̄
Sp.-V/AQuan/1999/10/07:14:19
Page 45
3.9 M OLECULES / 45
Table 3.11. (Continued.)
Name
Symbol
Charge
Mass
(amu)
JP
Mean life
(s)
Decay
Nonstrange baryons
Proton
Neutron
p
n
+1, −1
0
1.007 275
1.008 664
1/2
1/2
1/2+
1/2+
∞
0.932 × 103
pe− ν
Strangeness-1 baryons
+
0
−
+
0
−
0
+1, −1
0
−1, +1
1.197 6
1.276 8
1.280 2
1.285 4
0
1
1
1
1/2+
1/2+
1/2+
1/2+
2.632 × 10−10
0.800 × 10−10
< 10−19
1.482 × 10−10
pπ − , nπ 0 , etc.
pπ 0 , nπ + , etc.
γ , etc.
nπ − , etc.
Strangeness-2 baryons
0
0
−
−
0
−1, +1
1.411 6
1.418 5
1/2
1/2
1/2+
1/2+
2.90 × 10−10
1.641 × 10−10
π 0 , etc.
π − , etc.
Strangeness-3 baryons
−
−
−1, +1
1.795
0
3/2+
0.819 × 10−10
K − , etc.
Nonstrange charmed baryons
c
c
−1, +1
2.450
0
1/2+
2.3 × 10−13
K − , etc.
I
Composite particles
Hydrogen (2 S1/2 )
Deuterium (2 S1/2 )
1H
Deuteron
α particle
D
α
2H
0
0
+1
+2
1.007 82
2.014 10
2.013 55
4.001 40
∞
∞
∞
∞
References
1. Astrophysical Quantities, 1, §20; 2, §20; 3, §20
2. Barnett, R.M. et al. 1996, Rev. Mod. Phys., 68, 611
3. Barnett, R.M. et al. 1996, Phys. Rev., D54, 1
3.9
MOLECULES
Some definitions follow:
NA , NB , NAB
m AB
r0
D0
g0
σ
v
Be , αe
E
ωe , ωe xe
IP
UA , UB
Q AB
I
Number of atoms A, B, and molecules AB per cm3 .
Reduced mass = m A m B /(m A + m B ).
Internuclear distance (lowest state).
Dissociation energy (lowest state).
Electronic statistical weight (lowest state), or
Multiplicity, = 2S + 1 for states, = 2(2S + 1) for other states.
= 1 for heteronuclear molecules, = 2 for homonuclear molecules.
Vibrational quantum number.
Rotational constants [46, 47].
Energy change = hcB = h 2 /8π 2 I = h 2 /8π 2 m ABre2 .
Vibrational constants.
Ionizational potential.
Atomic partition functions (Sec. 3.3).
Molecular partition function, = Q rot Q vib Q el , each term dimensionless.
Moment of inertia, = m ABre2 .
Sp.-V/AQuan/1999/10/07:14:19
Page 46
46 / 3
ATOMS AND M OLECULES
Molecular diameters (diatomic) are
3r0 3.4 Å.
Molecular dissociation is represented by
NA NB /NAB = (2π m AB kT / h 2 )3/2 e−D/kT UA UB /Q AB .
Numerically,
log(NA NB /NAB ) = 20.2735 +
3
2
log m AB +
3
2
log T − 5040D/T + log(UA UB /Q AB )
with m in amu, D in eV, N in cm−3 ,
Q rot = kT /σ hcBv = (T /1.439 K)σ Bv ,
Bv = Be − αe (v + 12 ),
1.439 K
Q vib =
[ωe v − ωe xe (v 2 + v)] ,
exp −
T
v
1.439 K
Q el =
Tel ,
gel exp −
T
el
with Bv , ωe , Tel (= electronic excitation energy) in cm−1 .
The main ground-level constants are given in Tables 3.12 and 3.13, but upper level constants
[46, 47] are required for dissociation calculations.
Table 3.12. Diatomic molecules [1–3].a
Be
(cm−1 )
αe
(cm−1 )
ωe
ωe x e
(cm−1 ) (cm−1 )
r0
(Å)
IP
(eV)
60.85
30.2
3.06
1.68
4401
2321
12.02
1.78
1.82
14.46
14.18
1.93
1.977
0.412
0.017
0.018
0.53
0.49
0.018
0.019
6.462
7.002
7.001
0.940
7.467
7.997
7.997
0.948
0.948
0.967
1.90
1.998
1.932
16.699
1.672
1.445
1.691
18.91
16.79
5.826
0.972
10.042
0.973
6.391
0.641
7.500
Molecule
g0
σ
D0
(eV)
m AB
(amu)
H2
H2 +
He2
BH
BO
C2
CH
CH+
CO
CO+
1
4
1
1
2
1
4
1
1
2
2
2
2
1
1
2
1
1
1
1
4.4781
2.6507
0b
3.42
8.28
6.296
3.465
4.085
11.092
8.338
0.504
0.504
2.002
0.923
6.452
6.003
0.930
0.930
6.856
6.859
CN
N2
N+
2
NH
NO
O2
O2 +
OH
OH+
MgH
2
1
2
3
4
3
4
4
3
2
1
2
2
1
1
2
2
1
1
1
7.76
9.759
8.713
3.47
6.497
5.116
6.663
4.392
5.09
1.34
AlH
AlO
SiH
1
2
4
1
1
1
3.06
5.27
3.06
0.741
1.052
15.426
2367
1886
1855
2859
2740
2170
2214
121
66.2
22.22
49.4
11.8
13.3
63.0
1.131
13.29
15.16
1.232
1.205
1.243
1.120
9.77
7.0
12.15
10.64
1.128
1.115
14.01
26.8
0.017
0.017
0.019
0.649
0.017
0.016
0.020
0.724
0.749
0.185
2068
2359
2207
3282
1904
1580
1905
3738
3113
1495
13.09
14.32
16.10
78.35
14.08
11.98
16.26
84.88
78.52
31.89
1.172
1.098
1.116
1.036
1.151
1.208
1.116
0.970
1.029
1.730
14.17
15.58
27.1
13.63
9.26
12.07
24.2
12.90
0.186
0.006
0.219
1683
979
2042
29.09
6.97
1.648
1.618
1.520
9.53
8.04
Sp.-V/AQuan/1999/10/07:14:19
Page 47
3.10 P LASMAS / 47
Table 3.12. (Continued.)
Molecule
g0
σ
SiO
SiN
SO
CaH
CaO
ScO
TiO
1
2
3
2
1
2
6
1
1
1
1
1
1
1
VO
CrO
FeO
YO
ZrO
LaO
4
1
1
1
1
1
1
2
6
2
D0
(eV)
αe
(cm−1 )
ωe
ωe x e
(cm−1 ) (cm−1 )
r0
(eV)
IP
0.727
0.731
0.721
4.276
0.445
0.513
0.535
0.005
0.006
0.006
0.097
0.003
0.003
0.003
1242
1151
1149
1298
732
965
1009
5.97
6.47
5.63
19.10
4.81
4.20
4.50
1.510
1.572
1.481
2.003
1.822
11.43
1.620
6.4
0.548
0.541
0.513
0.388
0.423
0.353
0.004
0.005
0.004
0.002
0.002
0.001
1011
898
965
861
970
812
4.86
6.75
8.71
2.93
4.90
2.22
1.589
1.615
8.2
1.790
1.712
1.825
4.95
m AB
(amu)
Be
(cm−1 )
8.26
4.5
5.359
1.70
4.8
6.96
6.87
10.177
9.332
10.661
0.983
11.423
11.797
11.994
6.4
4.4
4.20
7.29
7.85
8.23
12.173
12.229
12.438
13.556
13.579
14.343
10.29
5.86
Notes
a See Sec. 4.11 for further molecular data and references.
b The lowest electronic state supports no bound state. However, the ground-state energy (as a function of
nuclear separation) has a potential well. Its depth is De = 0.0009 eV.
References
1. Astrophysical Quantities, 1, §21; 2, §22; 3, §21
2. Herzberg, G. 1950, Spectra of Diatomic Molecules (Van Nostrand, New York)
3. Huber, K.P., & Herzberg, G. 1979, Molecular Spectra and Molecular Structure IV. Constants of Diatomic
Molecules (Van Nostrand, New York)
Table 3.13. Selected polyatomic molecules [1–2].
Molecule
IP
(eV)
D
(eV)
Diameter
(Å)
H2 O
N2 O
CO2
NH3
CH4
HCN
12.61
12.89
13.77
10.15
13.0
13.91
5.11
1.68
5.45
4.3
4.4
5.6
3.5
4.0
3.8
3.0
3.5
References
1. Astrophysical Quantities, 1, §21; 2, §22; 3, §21
2. Herzberg, G. 1966, Electronic Spectra of Polyatomic Molecules (Van Nostrand, New York)
3.10
PLASMAS
Some definitions follow:
Ne , Ni , Np , N
Zi
L
T, B, ρ
A
Electron, ion, proton, total heavy-particle densities.
Charge on i ion (denoted Yi − 1 in other sections).
Characteristic size (e.g., diameter) of plasma.
Temperature, magnetic field, density.
Mass in amu.
Sp.-V/AQuan/1999/10/07:14:19
Page 48
48 / 3
ATOMS AND M OLECULES
The
Debye length, electron screening, the distance from an ion over which Ne can differ appreciably
from i Ni Z i is
D = (kT /4π e2 Ne )1/2 = 6.92(T /Ne )1/2 cm,
with T in K and Ne in cm−3 .
The plasma oscillation frequency is
νpl = (N e2 /π m e )1/2 = 8.978 × 103 Ne s−2
1/2
(in cgs).
The gyrofrequency for electrons is
νgy = (e/2π m e c)B
= 2.7994 × 106 B s−1 ,
and for ions is
νgy = (Z e/2π m i c)B
= 1.535 × 103 Z i B/A s−1 ,
with B in G.
The gyroradius for electrons is
ae = m e v⊥ c/eB
= 5.69 × 10−8 v⊥ B cm
2.21 × 10−2 T 1/2 /B cm,
and for ions is
a1 = m i v⊥ c/Z i eB
= 1.036 × 10−4 v⊥ A/Z i B cm
0.945T 1/2 A1/2 /Z i B cm,
where v⊥ is the velocity normal to B.
The most probable thermal velocity for electrons is
v = (2kT /m e )1/2
= 5.506 × 105 T 1/2 cm/s,
and for atoms and ions is
v = (2kT /m)1/2
= 1.290 × 104 (T /A)1/2 cm/s.
√
For rms velocities increase v by the factor 3/2 = 1.225.
The velocity of sound is
vs = (γ kT /m)1/2 [(N + Ne )/N ]1/2 ,
comparable with thermal velocity.
The Alfvén speed (magnetohydrodynamic or hydromagnetic wave) is
vA = B/(4πρ)1/2 = 0.282B/ρ 1/2 .
Sp.-V/AQuan/1999/10/07:14:19
Page 49
3.10 P LASMAS / 49
The phase velocity is c(1 + 4πρc2 /B 2 )1/2 .
The electron drift velocity in crossed magnetic and electric fields is 108 E ⊥ /B cm/s, with E ⊥ in
V/cm and B in G.
The electron drift velocity in magnetic and gravitational fields is
m e gc/eB = 5.686 × 10−8 g/B cm/s,
with g in cm/s2 and B in G.
The collision radius p for right-angle deflection of electrons by an ion is
p0 = Z i e2 /m e ve2 12 Z 1 e2 /kT
= 8.3 × 10−4 Z 1 /T cm.
The corresponding collision cross section is
π p02 = 2.16 × 10−6 Z 12 T −2 cm2 .
The cross section for all electron collisions with an ion is
π p02 ln ,
with
d
ln = ln(d/c) =
p −1 d p
c
and where c is the minimum of p in circumstances and d is the maximum of p in circumstances.
c is the largest of
c1 = 8.3 × 10−4 Z 1 /T cm
or
c2 = 1.06 × 10−6 T −1/2 cm
from the right-angle definition
from electron size.
d is the smallest of
d1 = N −1/3 cm
or
from ion spacing
d2 = D = 6.9T 1/2 N −1/2
(the Debye length)
or
d3 = 1.8 × 105 T 1/2 /ν
for collisions giving free–free absorption of frequency ν radiation.
The most general approximation for is
ln = 9.00 + 3.45 log T − 1.15 log Ne .
The collision cross section for neutral atoms and molecules is 10−15 cm2 .
The collision frequency for electrons is N1 ve ×(cross section) = 2.5(ln )Ne T −3/2 Z i s−1 .
The collision frequency for ions with ions is 8 × 10−2 (ln )Ne A−1/2 T −3/2 Z 12 s−1 .
The mean free path of electrons among charged particles is 4.7 × 105 T 2 N1−1 N1−2 cm.
The mean free path of electrons among neutral particles is 1015 N −1 cm.
Sp.-V/AQuan/1999/10/07:14:19
Page 50
50 / 3
ATOMS AND M OLECULES
The electrical resistivity [48] is
η = 8 × 1012 (ln )T −3/2 (emu)
= 9 × 10−9 (ln )T −3/2 (esu),
applying when the energy gains during free path < kT .
The thermal conductivity [48–50] is 1.0 × 10−6 T 5/2 erg cm−1 s−1 K−1 .
The life of a magnetic field in a plasma is
τ = 4π L 2 /η
(η in emu)
= 1.5 × 10−12 L 2 (ln )−1 T 3/2 s.
For approximate parameters for some plasmas, see Table 3.14.
Table 3.14. Approximate parameters for some plasmas.a Values are logarithmic.
Interstellar. f
Definition
Quantity
Unit
log
log
log
log
log
cm
cm−3
cm−3
K
G
L
Ne
N
T
B
Intpl.c
Cor.d
Rev.e
7.0
5.5
11.0
3.0
−1.0
13.0
0.5
0.5
5.0
−5.0
10.0
8.0
8.0
6.0
0.0
7.0
12.5
16.5
3.7
0.0
19.5
−3.0
0.0
2.0
−5.0
19.5
0.0
0.0
4.0
−5.0
Ion.b
H Ig
H IIh
Plasma freq.
4.0 + 12 log Ne
s−1
6.8
4.2
8.0
10.2
2.5
4.0
Debye length
0.7 + 12 log T − 12 log Ne
cm
−0.6
3.0
−0.3
−3.6
3.2
2.7
Gyro freq.
Electron
Ion
6.4 + log B
3.2 + log B
s−1
s−1
5.4
2.2
1.4
−1.8
6.4
0.7
6.4
3.2
1.4
−1.8
1.4
−1.8
Collision freq.
Electron
1.7 + log Ne − 32 log T
s−1
2.2
−5.9
3.2
8.7
−1.8
−4.3
0.2 + log Ne − 32 log T
s−1
1.2
−7.4
−0.8
7.2
−5.8
−5.8
6.3 + 32 log T
esu
10.8
13.8
15.3
11.9
9.3
12.3
−14.6 + 32 log T
emu
−10.1
−7.1
−5.6
−9.0
−11.6
−8.6
Mean free path
Ion
Neutron
5.7 + 2 log T − log Ne
15.0 − log N
cm
cm
6.2
4.0
15.2
14.5
9.7
7.0
0.6
−1.5
12.7
15.0
13.7
15.0
Gyroradius
Electron
Proton
−1.7 + 12 log T − log B
0.0 + 12 log T − log B
cm
cm
0.8
2.5
5.8
7.5
1.3
3.0
0.1
1.8
4.3
6.0
5.3
7.0
Alfvén v
11.3 − 12 log N + log B
cm/s
7.5
6.1
7.3
5.1
7.8
6.3
Sound v
4.2 + 12 log T
cm/s
5.7
6.7
7.2
6.0
5.2
6.2
B decay
−13.1 + 2 log L + 32 log T
s
yr
5.4
−2.1
19.4
11.9
15.9
8.4
6.5
−1.0
29.9
22.4
31.9
24.4
Ion
Electrical
conductivity
Notes
a For spectral emission from high-temperature plasmas, see Chap. 14.
Sp.-V/AQuan/1999/10/07:14:19
Page 51
3.10 P LASMAS / 51
b Ion. denotes ionosphere.
c Intpl. denotes interplanetary space.
d Cor. denotes solar corona.
e Rev. denotes solar reversing layer.
f Interstellar denotes interstellar space.
g H I denotes the H I region.
h H II denotes the H II region.
ACKNOWLEDGMENT
The author was supported in part by Grant No. AST-9315112 of the National Science Foundation.
REFERENCES
1. Drake, G., editor, 1996, Atomic, Molecular, and Optical
Physics Handbook (AIP, New York)
2. Jeffery, C.S. 1996, QJRAS, 37, 39, and references
therein
3. Lide, D.R., editor-in-chief 1996, Handbook of Chemistry and Physics, 77th ed. (CRC, Boca Raton, FL)
4. DeLaeter, J.R., Heumann, K.G., & Rosman, K.J.R.
1991, Pure Appl. Chem., 63, 991, reprinted in J. Phys.
Chem. Ref. Data, 20, No. 6, 1327
5. DeLaeter, J.R., & Heumann, K.G. 1991, Pure Appl.
Chem., 63, 975, reprinted in J. Phys. Chem. Ref. Data,
20, No. 6, 1313
6. Anders, E., & Grevesse, N. 1989, Geochim. Cosmochim. Acta 53, 197; Grevesse, N., & Noels, A. 1993,
in Origin and Evolution of the Elements, edited by N.
Prantzos, E. Vangioni, and M . Cassé (Cambridge University Press, Cambridge), p. 15
7. Biémont, E., Baudoux, M., Kurucz, R.L., Ansbacher,
W., & Pinnington, E.H. 1991, A&A, 249, 539
8. Hummer, D.G., & Mihalas, D. 1988, ApJ, 331, 794
9. Mihalas, D., Däppen W., & Hummer, D.G. 1988, ApJ,
331, 815
10. Däppen, W., Anderson, L.S., & Mihalas, D. 1987, ApJ,
319, 195
11. Rogers, F.J. 1981, Phys. Rev. A, 24, 1531
12. Rogers, F.J. 1986, ApJ, 310, 723
13. Iglesias, C.A., & Rogers, F.J. 1991, ApJ, 371, L73
14. Rogers, F.J., & Iglesias, C.A. 1992, ApJS, 401, 361
15. Rogers, F.J., Swenson, F.J., & C.A. Iglesias 1996, ApJ,
456, 902
16. Ebeling, W., Förster, A., Fortov, V.E., Gryaznov, V.K.,
& Polishchuk, A.Ya. 1991, Thermodynamic Properties
of Hot Dense Plasmas (Teubner, Stuttgart)
17. Alastuey, A., & Perez, A. 1992, Europhys. Lett., 20, 19
18. Alastuey, A., & Perez, A. 1996, Phys. Rev. E 53, 5714
19. Iglesias, C.A., & Rogers, F.J. 1992, in Astrophysical
Opacities, edited by C. Mendoza and C. Zeippen (Revista Mexicana de Astronomı́a y Astrofı́sica, Mexico
City), Vol. 23, p. 161
20. Eliezer, S., Ghatak, A., & Hora, H. 1986, An Introduction to Equations of State: Theory and Applications
(Cambridge University Press, Cambridge)
21. Irwin, A.W. 1981, ApJS, 45, 621
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
Sauval, A.J., & Tatum, J.B. 1984, ApJS, 56, 193
Lochte-Holtgreven, W. 1958, Rep. Prog. Phys., 21, 312
Rosa, A. 1948, Z. Ap., 25, 1
Aller, L.H. 1961, Stellar Atmospheres, edited by J.
Greenstein (University of Chicago Press, Chicago.),
p. 232
Cohen, E.R., & Taylor, B.N. 1988, J. Phys. Chem. Ref.
Data, 17, 1795
Hyleraas, E. 1950, ApJ, 111, 209
Thirring, W. 1979, Lehrbuch der mathematischen
Physik, Vol. 3: Quantenmechanik von Atomen und
Molekülen (Springer, Berlin), p. 207
Hill, R.N. 1977, J. Math. Phys., 18, 2316
Bely, O., & Van Regemorter, H. 1970, ARA&A, 8, 329
Astrophysical Quantities, 1, §18
Astrophysical Quantities, 2, §18
Seaton, M.J. 1962, Atomic and Molecular Processes,
edited by D. R. Bates (Academic Press, New York),
p. 375
Lotz, W. 1967, ApJS, 14, 207
Sampson, D.H. 1969, ApJ, 155, 575
Bely, O., & Faucher, P. 1972, A&A, 18, 487
Van Regemorter, H. 1962, ApJ, 136, 906
Moiseiwitsch, B.L., & Smith, S.L. 1968, Rev. Mod.
Phys, 40, 238
Seaton, M.J. 1970, Trans. IAU, XIVA, 128 and references therein
Kieffer, L.J. 1969, JILA Rep. 7
Blaha, M. 1969, ApJ, 157, 473
Hebb, M.H., & Menzel, D.H. 1940, ApJ, 92, 408
Astrophysical Quantities, 3, §18, and references therein
Bederson, B., & Kieffer, L.J. 1971, Rev. Mod. Phys., 43,
601
Spitzer, L. 1956, Physics of Fully Ionized Gases (Interscience, New York)
Herzberg, G. 1950, Spectra of Diatomic Molecules (Van
Nostrand, New York)
Tatum, J.B. 1966, Pub. Dom. Ap. Obs., 13, 1
Spitzer, L. 1962, Physics of Fully Ionized Gases (Interscience, New York)
Astrophysical Quantities, 1; 2; 3, §22
Delcroix, A., & Lemaire, A. 1969, ApJ, 156, 787
Sp.-V/AQuan/1999/10/07:14:19
Page 52