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Chapter3

Atoms and Molecules

Werner Dappen

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

3.1 ONLINE DATABASES AND OTHER SOURCES

The National Institute of Standards and Technology (NIST) gives access to extensivephysical and atomic data (http://physics.nist.gov). The Plasma Laboratory of the Weiz-mann 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, manyuseful links to other databases. For astrophysical applications, among the most extensive databasesare 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 atomicdata). 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 (“Pho-toionization Simulations for the Discriminating Astrophysicist”), which contains pointers to the atomic

27

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28 / 3 ATOMS AND MOLECULES

databases they use and maintain (e.g., http://www.pa.uky.edu/verner/atom.html, “Atomic Data for As-trophysics” ). The CHIANTI group (http://www.solar.nrl.navy.mil/chianti) has installed a databasewith 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 informationare the recent Atomic, Molecular, and Optical Physics Handbook [1], the results of the work of theCollaborative Computational Project No. 7 (United Kingdom) [2], and the Handbook of Chemistryand Physics [3].

3.2 ELEMENTS, ATOMIC MASS, AND SOLAR-SYSTEM ABUNDANCE

Atomic masses (weighted by the fractional abundances of the stable isotopes in normal terrestrialcomposition [4]) are scaled to 12C = 12.00. Standard values abridged to five significant digits aregiven (from the International Union of Pure and Applied Chemistry (IUPAC); see [5]). For someelements, atomic masses can be accurately measured to seven or more significant figures. IUPACregularly publishes these values irrespective of interest to any user. For many users, however, it is oftendesirable that the published data remain valid over an extended period, which is helpful for textbooksand numerical tables derived from atomic-mass data. IUPAC has recognized this need and approvedthe use of the designation standard to its abridged atomic-mass table, with the hope that the quotedvalues may survive for at least a decade.

The solar-system abundances (formerly denoted as cosmic abundances) are expressed logarithmi-cally 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 bynumber are quoted. The agreement between meteoritic and solar data has improved remarkably sincethe 1970s. Discrepancies have mostly gone away as the solar values—thanks especially to improvedtransition probabilities and other atomic data—have become more accurate [6]. The two principal ex-ceptions are the solar photospheric Li and Be abundances that are smaller than the meteoritic ones by2.15 and 0.27 dex, respectively. The reason is that these elements are destroyed by nuclear reactions atthe bottom of the solar convection zone. For most other elements the agreement is better than ±0.04dex (for this, and exceptions, see [6]). In the case of iron, a previous controversy has been solved: thesolar 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.

StrippedElement group Number Mass electrons

H 100 100 100He 9.8 39 20C, N, O, Ne 0.145 2.19 1.1Other 0.013 0.44 0.21

Total 109.96 141.63 121.3

The composition by mass [2] is as follows:

fraction of H, X 0.707 ± 2.5%fraction of He, Y 0.274 ± 6%fraction of other atoms, Z 0.018 9 ± 8.5%

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3.2 ELEMENTS, ATOMIC MASS, AND SOLAR-SYSTEM ABUNDANCE / 29

Mean atomic mass of cosmic material 1.30Mean atomic mass per H atom 1.41Mean atomic mass for fully ionized cosmic plasma 0.62

Table 3.2. Atomic masses and solar-system abundances.

ElementSymbol

[1]Atomicnumber

Atomicmass

Log abundance [2]

Meteoritic Solar

Hydrogen H 1 1.007 9 12.00a 12.00Helium [3] He 2 4.002 6 10.99a 10.99b

Lithium Li 3 6.941 3.31 1.16Beryllium Be 4 9.012 2 1.42 1.15Boron B 5 10.811 2.8 2.6c

Carbon C 6 12.011 8.56a 8.56Nitrogen N 7 14.007 8.05a 8.05Oxygen O 8 15.999 8.93a 8.93Fluorine F 9 18.998 4.48 4.56Neon Ne 10 20.180 8.09 8.09d

Sodium Na 11 22.990 6.31 6.33Magnesium Mg 12 24.305 7.58 7.58Aluminum Al 13 26.982 6.48 6.47Silicon Si 14 28.086 7.55 7.55Phosphorus P 15 30.974 5.57 5.45

Sulphur S 16 32.066 7.27 7.21Chlorine Cl 17 35.453 5.27 5.5Argon Ar 18 39.948 6.56d 6.56d

Potassium K 19 39.098 5.13 5.12Calcium Ca 20 40.078 6.34 6.36

Scandium Sc 21 44.956 3.09 3.10Titanium Ti 22 47.88 4.93 4.99Vanadium V 23 50.942 4.02 4.00Chromium Cr 24 51.996 5.68 5.67Manganese Mn 25 54.938 5.53 5.39

Iron [2] Fe 26 55.847 7.51 7.54Cobalt Co 27 58.933 4.91 4.92Nickel Ni 28 58.693 6.25 6.25Copper Cu 29 63.546 4.27 4.21Zinc Zn 30 65.39 4.65 4.60

Gallium Ga 31 69.723 3.13 2.88Germanium Ge 32 72.61 3.63 3.41Arsenic As 33 74.922 2.37Selenium Se 34 78.96 3.35Bromine Br 35 79.904 2.63

Krypton Kr 36 83.80 3.23Rubidium Rb 37 85.468 2.40 2.60Strontium Sr 38 87.62 2.93 2.90Yttrium Y 39 88.906 2.22 2.24Zirconium Zr 40 91.224 2.61 2.60

Niobium Nb 41 92.906 1.40 1.42Molybdenum Mo 42 95.94 1.96 1.92Technetium Tc 43 98.906Ruthenium Ru 44 101.07 1.82 1.84Rhodium Rh 45 102.91 1.09 1.12

Sp.-V/AQuan/1999/10/07:14:19 Page 29


Table 3.2. (Continued.)

ElementSymbol

[1]Atomicnumber

Atomicmass

Log abundance [2]

Meteoritic Solar

Palladium Pd 46 106.42 1.70 1.69Silver Ag 47 107.87 1.24 0.94c

Cadmium Cd 48 112.41 1.76 1.86Indium In 49 114.82 0.82 1.66c

Tin Sn 50 118.71 2.14 2.0

Antimony Sb 51 121.76 1.04 1.0Tellurium Te 52 127.60 2.24Iodine I 53 126.90 1.51Xenon Xe 54 131.29 2.23Cesium Cs 55 132.91 1.12

Barium Ba 56 137.33 2.21 2.13Lanthanum La 57 138.91 1.20 1.22Cerium Ce 58 140.12 1.61 1.55Praseodymium Pr 59 140.91 0.78 0.71Neodymium Nd 60 144.24 1.47 1.50

Promethium Pm 61 146.92Samarium Sm 62 150.36 0.97 1.00Europium Eu 63 151.96 0.54 0.51Gadolinium Gd 64 157.25 1.07 1.12Terbium Tb 65 158.93 0.33 −0.1

Dysprosium Dy 66 162.50 1.15 1.1Holmium Ho 67 164.93 0.50 0.26c

Erbium Er 68 167.26 0.95 0.93Thulium Tm 69 168.93 0.13 0.00c

Ytterbium Yb 70 170.04 0.95 1.08

Lutetium Lu 71 174.97 0.12 0.76c

Hafnium Hf 72 178.49 0.73 0.88Tantalum Ta 73 180.95 0.13Tungsten W 74 183.85 0.68 1.11c

Rhenium Re 75 186.21 0.27

Osmium Os 76 190.2 1.38 1.45Iridium Ir 77 192.22 1.37 1.35Platinum Pt 78 195.08 1.68 1.8Gold Au 79 196.97 0.83 1.01c

Mercury Hg 80 200.59 1.09

Thallium Tl 81 204.38 0.82 0.9c

Lead Pb 82 207.2 2.05 1.85Bismuth Bi 83 208.98 0.71Polonium Po 84 209.98Astatine At 85 209.99

Radon Rn 86 222.02Francium Fr 87 223.02Radium Ra 88 226.03Actinium Ac 89 227.03Thorium Th 90 232.04 0.08 0.12

Protactinium Pa 91 231.04Uranium U 92 238.03 −0.49 < −0.45c

Neptunium Np 93 237.05Plutonium Pu 94 239.05Americium Am 95 241.06

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3.3 EXCITATION, IONIZATION, AND PARTITION FUNCTIONS / 31


ElementSymbol

[1]Atomicnumber

Atomicmass

Log abundance [2]

Meteoritic Solar

Curium Cm 96 244.06Berkelium Bk 97 249.08Californium Cf 98 252.08Einsteinium Es 99 252.08Fermium Fm 100 257.10

Mendelevium Md 101 258.10Nobelium No 102 259.10Lawrencium Lr 103 262.11

NotesaBased on solar data.bBased on stellar observations and solar models [1, 3, 4].cUncertain.d Based on other astronomical data.

References1. IUPAC 1969, Comptes Rendus XXV Conference, p. 952. 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. Casse (Cambridge UniversityPress, Cambridge), p. 15

3. Christensen-Dalsgaard, J., Dappen, W., & the GONG Team 1996, Science,272, 1286

4. Biemont, E., Baudoux, M., Kurucz, R.L., Ansbacher, W., & Pinnington,E.H. 1991, A&A, 249, 539

5. Kosovichev, A.G., Christensen-Dalsgaard, J., Dappen, W., Dziembowski,W.A., Gough, D.O., & Thompson, M.J. 1992, MNRAS, 259, 536

3.3 EXCITATION, IONIZATION, AND PARTITION FUNCTIONS

3.3.1 Introduction

Finding the occupation of individual levels of atoms and ions and the fractions of ions of any givenchemical element in a plasma is a complex task. The difficulty arises from the interaction of theplasma with the atoms. Therefore, in principle, the problems of modified atoms and of their statisticaloccupation should be solved simultaneously and self-consistently. The typical task of quantum-statistical mechanics consists of the calculation of a density operator (ensemble) for the system of allparticles. The partition function, i.e., the trace over the density operator, not only gives the occupationof 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-mechanicalelectrons remain. In the treatment of electrons, we find a bifurcation into two distinct classes ofapproach, the “chemical picture” and the “physical picture.” While in the more conventional chemicalpicture, bound configurations (atoms, ions, and molecules) are introduced and treated as new andindependent species, only fundamental particles (electrons and nuclei) appear in the physical picture.In the chemical picture, reactions between the various species occur. Thus the thermodynamicalequilibrium must be sought among the stoichiometrically allowed set of concentration variables bymeans of a maximum entropy (or minimum free-energy) principle. In contrast, the physical picture

Sp.-V/AQuan/1999/10/07:14:19 Page 31


has the aesthetic advantage that there is no need for a minimax principle. The question of bound statesis dealt with implicitly through the Hamiltonian describing the interaction between the fundamentalparticles.

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, themost 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 timealso 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 physicalpicture. The most recent addition to the set of stellar equations of state is based on the formalismof the path integral in the framework of the Feynman–Kac representation. This formalism leads to avirial 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 andquantitative information can be extracted from simpler considerations, in which atoms are assumed tohave an unperturbed structure. In this case, excitation fractions are given by the Boltzmann factor, andthe ionization degree follows from the Saha equation, which is the mass-action law for the ionizationreaction. The Saha equation contains the internal partition functions for bound systems. A fundamentaltheoretical flaw of this approximate approach is that isolated atoms would have infinite partitionfunctions because of their infinite number of excited states. Many heuristic recipes to truncate partitionfunctions exist. However, only the physical picture comes to a satisfactory solution, which then canoften 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 followingsimple treatment of excitation, ionization, and partition functions is, with reasonable care, still veryuseful 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 byEliezer et al. [20]. The number of atoms existing in various atomic levels 0, 1, 2, . . . when in thermalequilibrium 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 ).

Numericallylog(N2/N1) = log(g2/g1) − χ12(5040/T ) (χ12 in eV),

where N is the total number of atoms per cm3, N0, N1, and N2 are the numbers of atoms per cm3 inthe zero and higher levels, g0, g1, and g2 are the corresponding statistical weights, χ1,2 is the potentialdifference 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

NYPe = UY+1

UY2(2πm)3/2(kT )5/2

h3exp(−χY,Y+1/kT ).

Numerically

log

(NY+1

NYPe

)= −χY,Y+1

5040

T+ 5

2 log T − 0.4772 + log

(2UY+1

UY

)

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3.3 EXCITATION, IONIZATION, AND PARTITION FUNCTIONS / 33

or

log

(NY+1

NYNe

)= −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 indyn cm−2, χY,Y+1 the ionization potential in eV from the Y to the Y + 1 stage of ionization, = 5040K/T , UY and UY+1 are the partition functions, and the factor 2 represents the statistical weight of anelectron.

The degree of ionization, when ionizations are caused by electron collisions and recombinationsare 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 recombinationsof Y + 2 in the abundance of ions in Y + 1 is neglected, and the possibility of multiple-ionizationevents is excluded. In the formula, S is the collision ionization coefficient (such that SNe NY = rate ofcollisional ionization, see Sec. 3.6), and α is the recombination coefficient (such that αNe NY+1 = rateof recombination, see Chap. 5).

Detailed calculations of partition functions are given by Irwin [21] (atoms and molecules) andSauval and Tatum [22] (molecules). However, for the approximate purposes of this section, thepartition function may simply be regarded as the effective statistical weight of the atom or ion underexisting conditions of excitation. Except in extreme conditions it is approximately equal to the weightof the lowest ground term. The ground term weight g0 is therefore given and this can normally beextrapolated 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 intendedto include the concentration of terms close to each series limit. The part of the partition functionassociated with these high-n terms is dependent on both T and Pe. This part is usually negligible unlessthe atom concerned is mainly ionized in which case the high-n terms may be counted statistically withthe ion.

Lowering of χY,Y+1 in the Saha equation to allow for the merging of high-level spectrum linesgives [23]

�χY,Y+1 = 7.0 × 10−7 N 1/3e Y 2/3,

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].

Element

Y = I

log U

g0 = 1.0 = 0.5

Y = II

log U

g0 = 1.0 = 0.5

Y = III

g0

1 H 2 0.30 0.30 1 0.00 0.002 He 1 0.00 0.00 2 0.30 0.30 13 Li 2 0.32 0.49 1 0.00 0.00 24 Be 1 0.01 0.13 2 0.30 0.30 15 B 6 0.78 0.78 1 0.00 0.00 2

6 C 9 0.97 1.00 6 0.78 0.78 17 N 4 0.61 0.66 9 0.95 0.97 68 O 9 0.94 0.97 4 0.60 0.61 99 F 6 0.75 0.77 9 0.92 0.94 4

10 Ne 1 0.00 0.00 6 0.73 0.75 9

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Element

Y = I

log U

g0 = 1.0 = 0.5

Y = II

log U

g0 = 1.0 = 0.5

Y = III

g0

11 Na 2 0.31 0.60 1 0.00 0.00 612 Mg 1 0.01 0.15 2 0.31 0.31 113 Al 6 0.77 0.81 1 0.00 0.01 214 Si 9 0.98 1.04 6 0.76 0.77 115 P 4 0.65 0.79 9 0.91 0.94 6

16 S 9 0.91 0.94 4 0.62 0.72 917 Cl 6 0.72 0.75 9 0.89 0.92 418 Ar 1 0.00 0.00 6 0.69 0.71 919 K 2 0.34 0.60 1 0.00 0.00 620 Ca 1 0.07 0.55 2 0.34 0.54 1

21 Sc 10 1.08 1.49 15 1.36 1.52 1022 Ti 21 1.48 1.88 28 1.75 1.92 2123 V 28 1.62 2.03 25 1.64 1.89 2824 Cr 7 1.02 1.51 6 0.86 1.22 2525 Mn 6 0.81 1.16 7 0.89 1.13 6

26 Fe 25 1.43 1.74 30 1.63 1.80 2527 Co 28 1.52 1.76 21 1.46 1.66 2828 Ni 21 1.47 1.60 10 1.02 1.28 2129 Cu 2 0.36 0.58 1 0.01 0.18 1030 Zn 1 0.00 0.03 2 0.30 0.30 1

31 Ga 6 0.73 0.77 1 0.00 0.00 232 Ge 9 0.91 1.01 6 0.64 0.70 134 Se 9 0.83 0.89 4 936 Kr 1 0.00 0.00 6 0.62 0.66 937 Rb 2 0.36 0.7 1 0.00 0.00 638 Sr 1 0.10 0.70 2 0.34 0.53 139 Y 10 1.08 1.50 1 + 15 1.18 1.41 10

40 Zr 21 1.53 1.99 28 1.66 1.91 2148 Cd 1 0.00 0.02 2 0.30 0.30 150 Sn 9 0.73 0.88 6 0.52 0.61 156 Ba 1 0.36 0.92 2 0.62 0.85 157 La 10 1.41 1.85 21 1.47 1.71 1070 Yb 1 0.02 0.21 2 0.30 0.3182 Pb 9 0.26 0.54 6 0.32 0.40 1

References1. Astrophysical Quantities, 1, §15; 2, §152. Cayrel, R., & Jugaku, J. 1963, Ann. d’Astrophys., 26, 4953. Bolton, C.T. 1970, ApJ, 161, 1187

The degree of ionization in the material of stellar atmospheres is given in Table 3.4, relating gaspressure Pg, electron pressure Pe, and temperature T . The data are averaged from [24] (rather highheavy-element abundance) and [25] (rather low heavy-element abundance).

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3.6 ATOMIC CROSS SECTIONS FOR ELECTRONIC COLLISIONS / 35

Table 3.4. log Pg.

and T

0.1 0.2 0.4 0.6 0.8 1.0 1.2 1.4log Pe T (K) 50 400 25 200 12 600 8 400 6 300 5 040 4 200 3 600

−2 −1.9 −1.8 −1.70 −1.67 −1.54 +0.78 +2.0 +2.4−1 −0.8 −0.74 −0.70 −0.66 −0.01 2.57 3.1 3.9

0 +0.27 +0.29 +0.31 +0.35 +1.90 3.9 4.5 5.31 1.27 1.30 1.33 1.47 3.87 5.2 6.0 6.72 2.27 2.30 2.34 2.98 5.65 6.7 7.7 8.53 3.28 3.30 3.35 4.87 7.0 8.3 9.4 10.44 4.28 4.31 4.43 6.84 8.7 10.0 11.2 12.45 5.59 5.30 5.87 8.66 10.4 11.8 13.2 14.4

3.4 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 areadded to assist interpolation. Part of the data is based on an especially accurate compilation for selectedions [6–20], made available by the National Institute of Standards and Technology (NIST, see Sec. 3.1for online access). If the data are given in wave numbers, the currently recommended conversion factorto 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 ormolecules that form stable negative ions. Regarding the astrophysically important H−, it was thoughtearlier that a second stable state exists [27]. Later, however, it was proven rigorously that there is onlyone stable state [28, 29].

3.6 ATOMIC CROSS SECTIONS FOR ELECTRONIC COLLISIONS

Definitions of symbols are presented below:

Q Atomic cross section [= Q(v)].v Precollision electron velocity.πa2

0 Atomic unit cross section = 8.797 × 10−17 cm2.Ne, Na, Ni Electron, atom, ion densities (per cm3).L = vQ Collision rate for each atom per unit Ne.NeL Collision rate per atom (or ion).Ne NaL Collision rate per cm3.Pc 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 πa20 Pc.

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36 / 3 ATOMS AND MOLECULESTa

ble

3.5.

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(ele

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lts)

[1–2

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Stag

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172

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13.6

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17.4

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1995

3.91

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2863

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7.28

239.

101

195.

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211

Na

5.13

908

47.2

864

71.6

2098

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138.

4017

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5026

4.25

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91

465.

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7.64

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8.15

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592

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r15

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6227

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.74

59.8

175

.02

91.0

112

4.32

143.

4642

2.5

478.

753

9.0

618

686

756

19K

4.34

066

31.6

345

.806

60.9

182

.66

99.4

117.

5615

4.88

175.

850

3.8

564.

762

971

578

720

Ca

6.11

316

11.8

7172

50.9

1367

.27

84.5

010

8.78

127.

214

7.24

188.

521

1.3

591.

965

772

781

821

Sc6.

561

4412

.799

6724

.757

73.4

8991

.65

111.

6813

8.0

158.

118

0.0

225.

224

9.8

688

757

831

22T

i6.

828

213

.575

527

.492

43.2

6799

.30

119.

5314

0.8

170.

419

2.1

215.

926

5.1

292

788

863

23V

6.74

63

14.6

629

.311

46.7

165

.28

128.

115

0.6

173.

420

5.8

230.

525

5.1

308

336

896

24C

r6.

766

6416

.485

730

.96

49.1

669

.46

90.6

416

1.18

184.

720

9.3

244.

427

0.7

298

355

384

25M

n7.

434

0215

.639

9933

.668

51.2

72.4

95.6

119.

2019

4.5

221.

824

8.3

286.

031

434

440

426

Fe7.

902

416

.187

830

.652

54.8

75.0

99.1

124.

9815

1.06

233.

626

2.1

290.

233

136

139

227

Co

7.88

10

17.0

8333

.50

51.3

79.5

103

131

160

186.

227

6.2

305

336

379

411

28N

i7.

639

818

.168

8435

.19

54.9

75.5

108

134

164

193

224.

632

135

238

443

029

Cu

7.72

638

20.2

9240

36.8

4155

.279

.910

313

916

719

923

226

636

940

143

530

Zn

9.39

405

17.9

6440

39.7

2359

.482

.610

813

617

520

323

827

431

141

245

4

Sp.-V/AQuan/1999/10/07:14:19 Page 36

3.6 ATOMIC CROSS SECTIONS FOR ELECTRONIC COLLISIONS / 37Ta

ble

3.5.

(Con

tinu

ed.)

Stag

eof

ioni

zatio

n

Ato

mX

VX

VI

XV

IIX

VII

IX

IXX

XX

XI

XX

IIX

XII

IX

XIV

XX

VX

XV

IX

XV

IIX

XV

III

1H

2H

e3

Li

4B

e5

B6

C7

N8

O9

F10

Ne

11N

a12

Mg

13A

l14

Si15

P3

070

16S

322

43

494

17C

l80

93

658

394

618

Ar

855

918

412

14

426

19K

862

968

103

44

611

493

420

Ca

895

974

108

71

157

512

95

470

21Sc

927

100

91

094

121

31

288

567

56

034

22T

i94

11

044

113

11

221

134

61

425

624

96

626

23V

975

106

01

168

126

01

355

148

61

569

685

17

246

24C

r1

011

109

71

185

129

91

396

149

61

634

172

17

482

789

525

Mn

435

113

61

224

131

71

437

153

91

644

178

81

879

814

18

572

26Fe

457

489

126

61

358

145

61

582

168

91

799

195

02

045

882

89

278

27C

o44

451

254

71

402

150

01

602

173

41

846

196

22

119

221

89

544

1003

028

Ni

464

499

571

607

154

61

648

175

61

894

201

02

131

229

52

398

1028

010

790

29C

u48

452

055

763

367

11

698

180

41

919

206

02

182

231

02

478

256

011

050

30Z

n49

054

257

961

969

873

81

856

197

02

088

223

42

363

249

52

660

273

0

Sp.-V/AQuan/1999/10/07:14:19 Page 37



Stage of ionization

Atom I II III IV V VI VII VIII IX X

31 Ga 5.999 30 20.514 30.71 64 87 116 140 170 212 24332 Ge 7.900 15.935 34.224 45.71 93.5 112 144 174 207 25033 As 9.815 2 18.633 28.351 50.13 62.63 127.6 147 179 212 24234 Se 9.752 38 21.19 30.820 42.944 68.3 81.7 155.4 184 218 25035 Br 11.813 81 21.8 36 47.3 59.7 88.6 103.0 192.8 224 25736 Kr 13.999 61 24.360 36.95 52.5 64.7 78.5 111.0 126 230.9 26337 Rb 4.177 13 27.285 40 52.6 71.0 84.4 9.2 136 150 277.138 Sr 5.694 84 11.030 42.89 57 71.6 90.8 106 122.3 162 17739 Y 6.217 12.24 20.52 61.8 77.0 93 116 129 146.2 19140 Zr 6.633 90 13.13 22.99 34.34 81.5 99 117 140 15541 Nb 6.758 85 14.32 25.04 38.3 50.55 102.6 125 142 16142 Mo 7.092 43 16.16 27.13 46.4 61.2 68 126.8 153 16343 Tc 7.28 15.26 29.54 46 55 80 18744 Ru 7.360 50 16.76 28.47 50 60 9245 Rh 7.458 90 18.08 31.06 48 65 9746 Pd 8.336 9 19.43 32.93 53 62 90 110 130 155 18047 Ag 7.576 24 21.49 34.83 56 68 89 115 140 160 18548 Cd 8.993 67 16.908 37.48 59 72 94 115 145 170 19549 In 5.786 36 18.870 28.03 54.4 77 98 120 145 180 20550 Sn 7.343 81 14.632 30.503 40.734 72.28 103 125 150 175 21051 Sb 8.64 16.531 25.3 44.2 56 108 130 155 185 21052 Te 9.009 6 18.6 27.96 37.41 58.75 70.7 137 165 190 22053 I 10.451 26 19.131 33 42 66 81 100 170 200 23054 Xe 12.129 87 21.21 32.123 46 57 82 100 120 210 24055 Cs 3.893 9 23.157 35 46 62 74 100 120 145 25056 Ba 5.211 70 10.004 49 62 80 95 120 145 16057 La 5.577 0 11.06 19.177 52 66 80 100 115 145 16558 Ce .538 7 10.85 20.198 36.72 70 85 100 120 140 16559 Pr 5.464 10.55 21.624 38.95 57.45 89 105 120 145 16060 Nd 5.525 0 10.73 110 130 150 17061 Pm 5.55 10.90 135 155 17562 Sm 5.643 7 11.07 160 18063 Eu 5.670 4 11.241 19064 Gd 6.150 0 12.0965 Tb 5.863 9 11.5266 Dy 5.938 9 11.6767 Ho 6.021 6 11.8068 Er 6.107 8 11.9369 Tm 6.184 31 12.05 23.6870 Yb 6.254 16 12.176 25.0571 Lu 5.425 85 13.9 20.95972 Hf 6.825 07 14.9 23.3 33.373 Ta 7.89 16 22 33 4574 W 7.98 18 24 35 48 6175 Re 7.88 17 26 38 51 64 7976 Os 8.7 17 25 40 54 68 83 10077 Ir 9.1 17 27 39 57 72 88 105 12078 Pt 9.0 18.563 28 41 55 75 92 110 125 14579 Au 9.225 67 20.5 30 44 58 73 96 115 135 15580 Hg 10.437 50 18.756 34.2 46 61 77 94 120 140 16081 Tl 6.108 29 20.428 29.83 50.7 64 81 98 115 145 16582 Pb 7.416 66 15.032 31.937 42.32 68.8 84 103 120 140 17583 Bi 7.289 16.69 25.56 45.3 56.0 88.3 107 125 150 17084 Po 8.416 71 19 27 38 61 73 112 130 155 17585 At 9.3 20 29 41 51 78 91 140 160 185

Sp.-V/AQuan/1999/10/07:14:19 Page 38



Stage of ionization

Atom I II III IV V VI VII VIII IX X

86 Rn 10.748 50 21 29 44 55 67 97 110 165 19087 Fr 4 22 33 43 59 71 84 115 135 19588 Ra 5.278 92 10.147 34 46 58 76 89 105 140 15589 Ac 5.17 12.1 20 49 62 76 95 110 125 16590 Th 6.08 11.5 20.0 28.8 65 80 94 115 130 14591 Pa 5.89 84 100 115 140 15592 U 6.194 05 104 120 140 16093 Np 6.265 794 Pu 6.0695 Am 5.993

References1. Astrophysical Quantities, 1, §16; 2, §16; 3, §162. Lotz, W. 1966, Ionisierungsenergien von Ionen H bis Ni (Inst. Plasmaphys, Munchen)3. Moore, C.E. 1970, Ionization Potentials, NSRDS-NBS 34, Washington4. Finkelnberg, W., & Humbach, W. 1955, Naturwiss., 42, 355. 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].

Electron Electron ElectronAtom affinity (eV) Atom affinity (eV) Molecule affinity (eV)

H +0.754 Na +0.479 O2 +0.451He −0.3 Mg −0.4 O3 +2.102 8Li +0.618 Al +0.441 OH +1.827 67Be −0.4 Si +1.385 SH +2.314B +0.277 P +0.747 C2 +3.269

C3 +1.981C +1.263 S +2.077N −0.2 Cl +3.612 CN +3.862O +1.461 Br +3.48 NH2 +0.771O− −6.7 I +3.17 NO +0.026F +3.401 K +0.501 NO2 +2.273Ne −0.7 Ca +0.018 NO3 +3.951

CH +1.238

References1. Astrophysical Quantities, 1, §17; 2, §17; 3, §172. Handbook of Chemistry and Physics, 77th ed. (CRC, Boca Raton, FL, 1996)

Sp.-V/AQuan/1999/10/07:14:19 Page 39


3.6.1 Ionization Cross Section

The classical cross section of atoms for ionization by electrons [30] is

Q1 = 4nπa20

1

χε

(1 − χ

ε

),

where χ is the ionization energy in rydbergs (Ry), ε the electronic energy before collision in Ry, and nthe number of optical electrons.

The general approximation for cross sections of atoms for ionization by electrons (see, [30–33]) is

Q1 = nπa20

1

χεF(Y, ε/χ) = nπa2

0

χ2q

= 1.63 × 10−14n(1/χ2eV)(χ/ε)F(Y, ε/χ),

where Y is the charge on the ionized atom (or next ion stage) and χeV is the ionization energy ineV. The function F(Y, ε/χ) is given and also q = (χ/ε)F(Y, ε/χ), which is sometimes called thereduced 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 ±0.3dex may be expected. Other empirical forms have been suggested (see, e.g., [34–36]).

Table 3.7. Numerical functions F(Y, ε/χ) and q(Y, ε/χ).

ε/χ 1.0 1.2 1.5 2.0 3 5 10

F(classical) = 4(1 − χ/ε) 0.00 0.67 1.33 2.00 2.67 3.20 3.60F(1, ε/χ) 0.0 0.31 0.78 1.60 2.9 4.6 6.4F(2, ε/χ) 0.00 0.53 1.17 2.02 3.3 4.7 6.4F(∞, ε/χ) 0.00 0.74 1.54 2.56 3.8 5.0 6.4

q(classical) = 4(χ/ε)(1 − χ/ε) 0.00 0.56 0.89 1.00 0.89 0.64 0.36q(l, ε/χ) 0.00 0.26 0.52 0.80 0.97 0.92 0.64q(2, ε/χ) 0.00 0.44 0.78 1.01 1.09 0.94 0.64q(∞, ε/χ) 0.00 0.62 1.03 1.28 1.28 1.00 0.64

The maximum ionization cross section for the classical case is

Qmax = nπa20χ−2 at ε = 2χ.

The value of Qmax is approximately the same in actual cases but the maximum occurs near ε = 4χ .The rate of ionization by electrons (see [30–32]) is

L1 = vQ1.

The neutral atom approximation (with kT < ionization energy) gives

L1 = 1.1 × 10−8nT 1/2χ−2eV 10−5040χeV/T cm3 s−1.

The coronal ion approximation (with kT < ionization energy) gives

L1 = 2.1 × 10−8nT 1/2χ−2eV 10−5040χeV/T cm3 s−1.

Sp.-V/AQuan/1999/10/07:14:19 Page 40


3.6.2 Excitation Cross Section (Permitted Transitions)

An approximation for Qex, the excitation cross section of an atom (see [30, 37]), is given. Theapproximation applies fairly well when �n ≥ 1 (notation of Chap. 5). For �n = 0 the approximationtends to be small:

Qex = 8π√3πa2

0f

εWb

= 1740πa20λ2(W/ε) f b

= 1.28 × 10−15( f/εW )b cm2,

where f is the oscillator strength, W is the excitation energy in Ry (= 0.0912/λ with λ in µm), and ε

is the electron energy before collision, also in Ry. See Table 3.8.

Table 3.8. Numerical factors b and bW/ε.

ε/W 1.0 1.2 1.5 2.0 3 5 10 30 100

b, neutral atoms 0.00 0.03 0.06 0.11 0.21 0.33 0.56 0.98 1.33b, ions 0.20 0.20 0.20 0.20 0.24 0.33 0.56 0.98 1.33

bW/ε, neutral atoms 0.00 0.03 0.04 0.06 0.07 0.07 0.06 0.03 0.01bW/ε, ions 0.20 0.17 0.13 0.10 0.08 0.07 0.06 0.03 0.01

The maximum excitation cross section is as follows:

• The neutral atom approximation gives

Qmax = 125πa20λ2 f near ε = 3W.

• The ion approximation gives

Qmax = 350πa20λ2 f near ε = W (λ in µm).

The rate of excitation (see [34, 35, 37]) is

L = vQex = 17.0 × 10−4 f

T 1/2WeV10−5040WeV/T P(W/kT ),

where WeV and W are the excitation energy in eV and in ergs (with 11 600WeV/kT = W/kT ) andP(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 Ions

< 0.01 0.29E1(W/kT )a

0.01 1.16 1.160.02 0.96 0.980.05 0.70 0.74

Sp.-V/AQuan/1999/10/07:14:19 Page 41



P(W/kT )

W/kT Neutral atoms Ions

0.1 0.49 0.550.2 0.33 0.400.5 0.17 0.261 0.10 0.222 0.063 0.215 0.035 0.20

10 0.023 0.20

> 10 0.066/(W/kT )1/2 0.20

Notea E1( ) is the first exponential integral.

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 Q21 are related to excitation cross sections Q12 (2 being the upper level)through

g2ε2 Q21 = g1ε1 Q12,

where ε2 = ε1 + W , and g2 and g1 are statistical weights.The deexcitation rate L21 and excitation rate L12 are related by

g2L21 = g1L12 exp(W/kT ).

3.6.4 Excitation Cross Sections (Forbidden Transitions)

The collision strength � for each line is defined by (see [33, 42])

Qf = π�/g1k2ν = πa2

0�/g1ε

= h2

4πm2

�

g1v2= 4.21�/g1v

2,

where kν/2π is the wave number of the incident electron (then k2v in atomic units = ε in Ry), v is the

electron velocity, g1 is the statistical weight of the initial (lower) level, and Qf is the forbidden linecross 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 forbiddentransitions) can be found in older references [43]. In recent years, however, a wealth of accurate

Sp.-V/AQuan/1999/10/07:14:19 Page 42

3.7 ATOMIC RADII / 43

collision strengths have been obtained for a very large number of transitions. They are based onextensive 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 � 180πa20λ/ε1/2 (λ in µm, ε 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/mv2)2 = π(Y − 1)2(e2/2εhcR)2

= πa20(Y − 1)2/ε2 (ε 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 dependinglogarithmically 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) � 20πa20(Y − 1)2/ε2.

3.7 ATOMIC RADII

Atomic radii are defined through the closeness of approach of atoms in the formation of moleculesand crystals. The radius r so derived is approximately that of maximum radial density in the chargedistribution of neutral atoms (see Table 3.10). For ions the appropriate radius measures to the pointwhere the radial density falls to 10% of its maximum value. The atomic mass divided by the atomicvolume (4/3)πr3 gives the density of the more compact solids. 2r is approximately the gas-kineticdiameter of monoatomic molecules.

Table 3.10. Atomic radii [1–5].

Atom r (A) Ion [3] r (A) Atom r (A) Ion [3] r (A) Atom r (A) Ion [3] r (A)

H 0.7 H− 1.8 S 1.05 S2− 1.70 Br 1.2 Br− 1.82He 1.2 Cl 1.02 Cl− 1.67 Kr 1.82Li 1.58 Li+ 0.68 Ar 1.6 Rb 2.54 Rb+ 1.66Be 1.06 Be2+ 0.39 K 2.37 K+ 1.52 Sr 2.3 Sr2+ 1.32B 0.83 B3+ 0.28 Ca 1.97 Ca2+ 1.14 Ag 1.44 Ag+ 1.29

Sp.-V/AQuan/1999/10/07:14:19 Page 43



Atom r (A) Ion [3] r (A) Atom r (A) Ion [3] r (A) Atom r (A) Ion [3] r (A)

C 0.77 C4+ 0.22 Sc 1.64 Sc3+ 0.89 Cd 1.6 Cd2+ 1.09N 0.70 N3− 1.92 Ti 1.46 Ti4+ 0.75 Sn 1.62 Sn4+ 0.76O 0.66 O2− 1.26 V 1.39 V4+ 0.61 I 1.4 I− 2.06F 0.62 F− 1.19 Cr 1.28 Xe 2.00Ne 1.3 Mn 1.26 Mn2+ 0.81 Cs 2.73 Cs+ 1.81

Na 1.91 Na+ 1.16 Fe 1.27 Fe2+ 0.75 Ba 2.24 Ba2+ 1.49Mg 1.62 Mg2+ 0.86 Co 1.25 Co2+ 0.79 Pt 1.38Al 1.43 Al3+ 0.67 Ni 1.29 Ni2+ 0.83 Au 1.44 Au+ 1.51Si 1.09 Si4+ 0.47 Cu 1.28 Cu+ 0.91 Hg 1.57 Hg2+ 1.16P 1.08 P3− 2.3 Zn 1.39 Zn2+ 0.77

References1. Astrophysical Quantities, 1, §19; 2, §19; 3, §192. Teatum, E., Gschneidner, K., & Waber, J. 1960, Los Alamos Scientific Laboratory, Report No. LA-23453. Shannon, R.D. 1976, Acta Cryst., A32, 7514. Allen, F.H., Kennard, O., Watson, D.G., Brammer, L., Orpen, A.G., & Taylor, R. 1987, J. Chem. Soc. Perkin II,

S15. 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, andbaryons. Possible proton decay is not included. I denotes the isotopic spin, J the spin, and P theparity. The lifetime is that in free space. In the column labeled “Decay” are given the main decayproducts. 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].

Mass Mean lifeName Symbol Charge (amu) I J P (s) Decay

Bosons

Gauge bosonsPhoton γ 0 0.000 0, 1 1− ∞W W +1, −1 86.24 1 � = 2.1 GeV eν, etc.Z Z 0 97.90 1 � = 2.5 GeV e+e−, etc.

Mesonsπ-mesons (pion) π+, π− +1, −1 0.149 84 1 0− 2.603 × 10−8 µν

π0 0 0.144 90 1 0− 0.83 × 10−16 γ γ

K meson (kayon) K+0 , K− +1, −1 0.530 15 1/2 0− 1.237 × 10−8 µν, ππ0

K0S 0 0.534 38 1/2 0− 0.892 × 10−10 π+π−, π0π0

K0L 0 0.534 38 1/2 0− 5.38 × 10−8 πeν, πµν, 3π0

Fermions

Leptonse Neutrino νe 0 < 5 × 10−8 1/2 ∞µ Neutrino νµ 0 < 5 × 10−4 1/2 ∞τ Neutrino ντ 0 < 0.2 1/2 ∞Electron, Positron e −1, +1 0.000 548 6 1/2 ∞µ meson (muon) µ −1, +1 0.113 4 1/2 2.197 × 10−6 eνν

τ meson (tauon) τ −1, +1 1.915 1/2 (3.4 ± 0.5) × 10−13 eνν

Sp.-V/AQuan/1999/10/07:14:19 Page 44

3.9 MOLECULES / 45


Mass Mean lifeName Symbol Charge (amu) I J P (s) Decay

Nonstrange baryonsProton p +1, −1 1.007 275 1/2 1/2

+ ∞Neutron n 0 1.008 664 1/2 1/2

+ 0.932 × 103 pe−ν

Strangeness-1 baryons� � 0 1.197 6 0 1/2

+ 2.632 × 10−10 pπ−, nπ0, etc.�+ �+ +1, −1 1.276 8 1 1/2

+ 0.800 × 10−10 pπ0, nπ+, etc.�0 �0 0 1.280 2 1 1/2

+ < 10−19 �γ , etc.�− �− −1, +1 1.285 4 1 1/2

+ 1.482 × 10−10 nπ−, etc.

Strangeness-2 baryons�0 �0 0 1.411 6 1/2 1/2

+ 2.90 × 10−10 �π0, etc.�− �− −1, +1 1.418 5 1/2 1/2

+ 1.641 × 10−10 �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.

Composite particles

Hydrogen (2 S1/2) 1H 0 1.007 82 ∞Deuterium (2 S1/2) 2H 0 2.014 10 ∞Deuteron D +1 2.013 55 ∞α particle α +2 4.001 40 ∞

References1. Astrophysical Quantities, 1, §20; 2, §20; 3, §202. Barnett, R.M. et al. 1996, Rev. Mod. Phys., 68, 6113. Barnett, R.M. et al. 1996, Phys. Rev., D54, 1

3.9 MOLECULES

Some definitions follow:

NA, NB, NAB Number of atoms A, B, and molecules AB per cm3.mAB Reduced mass = mAmB/(mA + mB).

r0 Internuclear distance (lowest state).D0 Dissociation energy (lowest state).g0 Electronic statistical weight (lowest state), or

Multiplicity, = 2S + 1 for � states, = 2(2S + 1) for other states.σ = 1 for heteronuclear molecules, = 2 for homonuclear molecules.v Vibrational quantum number.Be, αe Rotational constants [46, 47].�E Energy change = hcB = h2/8π2 I = h2/8π2mABr2

e .

ωe, ωexe Vibrational constants.IP Ionizational potential.UA, UB Atomic partition functions (Sec. 3.3).QAB Molecular partition function, = Qrot Qvib Qel, each term dimensionless.I Moment of inertia, = mABr2

e .

Sp.-V/AQuan/1999/10/07:14:19 Page 45


Molecular diameters (diatomic) are

� 3r0 � 3.4 A.

Molecular dissociation is represented by

NA NB/NAB = (2πmABkT/h2)3/2e−D/kT UAUB/QAB.

Numerically,

log(NA NB/NAB) = 20.2735 + 32 log mAB + 3

2 log T − 5040D/T + log(UAUB/QAB)

with m in amu, D in eV, N in cm−3,

Qrot = kT/σhcBv = (T/1.439 K)σ Bv,

Bv = Be − αe(v + 12 ),

Qvib =∑

vexp

(−1.439 K

T[ωev − ωexe(v

2 + v)]

),

Qel =∑

el

gel exp

(−1.439 K

TTel

),

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

D0 mAB Be αe ωe ωexe r0 IPMolecule g0 σ (eV) (amu) (cm−1) (cm−1) (cm−1) (cm−1) (A) (eV)

H2 1 2 4.4781 0.504 60.85 3.06 4401 121 0.741 15.426H2

+ 4 2 2.6507 0.504 30.2 1.68 2321 66.2 1.052He2 1 2 0b 2.002 22.22BH 1 1 3.42 0.923 12.02 0.412 2367 49.4 1.232 9.77BO 2 1 8.28 6.452 1.78 0.017 1886 11.8 1.205 7.0C2 1 2 6.296 6.003 1.82 0.018 1855 13.3 1.243 12.15CH 4 1 3.465 0.930 14.46 0.53 2859 63.0 1.120 10.64CH+ 1 1 4.085 0.930 14.18 0.49 2740 1.131CO 1 1 11.092 6.856 1.93 0.018 2170 13.29 1.128 14.01CO+ 2 1 8.338 6.859 1.977 0.019 2214 15.16 1.115 26.8

CN 2 1 7.76 6.462 1.90 0.017 2068 13.09 1.172 14.17N2 1 2 9.759 7.002 1.998 0.017 2359 14.32 1.098 15.58N+

2 2 2 8.713 7.001 1.932 0.019 2207 16.10 1.116 27.1NH 3 1 3.47 0.940 16.699 0.649 3282 78.35 1.036 13.63NO 4 1 6.497 7.467 1.672 0.017 1904 14.08 1.151 9.26O2 3 2 5.116 7.997 1.445 0.016 1580 11.98 1.208 12.07O2

+ 4 2 6.663 7.997 1.691 0.020 1905 16.26 1.116 24.2OH 4 1 4.392 0.948 18.91 0.724 3738 84.88 0.970 12.90OH+ 3 1 5.09 0.948 16.79 0.749 3113 78.52 1.029MgH 2 1 1.34 0.967 5.826 0.185 1495 31.89 1.730

AlH 1 1 3.06 0.972 6.391 0.186 1683 29.09 1.648AlO 2 1 5.27 10.042 0.641 0.006 979 6.97 1.618 9.53SiH 4 1 3.06 0.973 7.500 0.219 2042 1.520 8.04

Sp.-V/AQuan/1999/10/07:14:19 Page 46

3.10 PLASMAS / 47


D0 mAB Be αe ωe ωexe r0 IPMolecule g0 σ (eV) (amu) (cm−1) (cm−1) (cm−1) (cm−1) (eV)

SiO 1 1 8.26 10.177 0.727 0.005 1242 5.97 1.510 11.43SiN 2 1 4.5 9.332 0.731 0.006 1151 6.47 1.572SO 3 1 5.359 10.661 0.721 0.006 1149 5.63 1.481 10.29CaH 2 1 1.70 0.983 4.276 0.097 1298 19.10 2.003 5.86CaO 1 1 4.8 11.423 0.445 0.003 732 4.81 1.822ScO 2 1 6.96 11.797 0.513 0.003 965 4.20TiO 6 1 6.87 11.994 0.535 0.003 1009 4.50 1.620 6.4

VO 4 1 6.4 12.173 0.548 0.004 1011 4.86 1.589CrO 1 4.4 12.229 0.541 0.005 898 6.75 1.615 8.2FeO 1 4.20 12.438 0.513 0.004 965 8.71YO 2 1 7.29 13.556 0.388 0.002 861 2.93 1.790ZrO 6 1 7.85 13.579 0.423 0.002 970 4.90 1.712LaO 2 1 8.23 14.343 0.353 0.001 812 2.22 1.825 4.95

NotesaSee Sec. 4.11 for further molecular data and references.bThe 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.

References1. Astrophysical Quantities, 1, §21; 2, §22; 3, §212. 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].

I P D DiameterMolecule (eV) (eV) (A)

H2O 12.61 5.11 3.5N2O 12.89 1.68 4.0CO2 13.77 5.45 3.8NH3 10.15 4.3 3.0CH4 13.0 4.4 3.5HCN 13.91 5.6

References1. Astrophysical Quantities, 1, §21; 2, §22; 3, §212. Herzberg, G. 1966, Electronic Spectra of Poly-

atomic Molecules (Van Nostrand, New York)

3.10 PLASMAS

Some definitions follow:

Ne, Ni, Np, N Electron, ion, proton, total heavy-particle densities.Z i Charge on i ion (denoted Yi − 1 in other sections).L Characteristic size (e.g., diameter) of plasma.T, B, ρ Temperature, magnetic field, density.

A Mass in amu.

Sp.-V/AQuan/1999/10/07:14:19 Page 47


The Debye length, electron screening, the distance from an ion over which Ne can differ appreciablyfrom

∑i Ni Z i is

D = (kT/4πe2 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 = (Ne2/πme)1/2 = 8.978 × 103 N 1/2

e s−2 (in cgs).

The gyrofrequency for electrons is

νgy = (e/2πmec)B

= 2.7994 × 106 B s−1,

and for ions isνgy = (Ze/2πmic)B

= 1.535 × 103 Z i B/A s−1,

with B in G.The gyroradius for electrons is

ae = mev⊥c/eB

= 5.69 × 10−8v⊥ B cm

� 2.21 × 10−2T 1/2/B cm,

and for ions isa1 = miv⊥c/Z ieB

= 1.036 × 10−4v⊥ 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/me)1/2

= 5.506 × 105T 1/2 cm/s,

and for atoms and ions isv = (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 Alfven speed (magnetohydrodynamic or hydromagnetic wave) is

vA = B/(4πρ)1/2 = 0.282B/ρ1/2.

Sp.-V/AQuan/1999/10/07:14:19 Page 48

3.10 PLASMAS / 49

The phase velocity is c(1 + 4πρc2/B2)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

megc/eB = 5.686 × 10−8g/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 ie2/mev

2e � 1

2 Z1e2/kT

= 8.3 × 10−4 Z1/T cm.

The corresponding collision cross section is

πp20 = 2.16 × 10−6 Z2

1 T −2 cm2.

The cross section for all electron collisions with an ion is

πp20 ln �,

with

ln � = ln(d/c) =∫ d

cp−1 dp

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 Z1/T cm from the right-angle definition

orc2 = 1.06 × 10−6T −1/2 cm from electron size.

d is the smallest of

d1 = N−1/3 cm from ion spacing

ord2 = D = 6.9T 1/2 N−1/2 (the Debye length)

ord3 = 1.8 × 105T 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 N1ve×(cross section) = 2.5(ln �)NeT −3/2 Z i s−1.The collision frequency for ions with ions is 8 × 10−2(ln �)Ne A−1/2T −3/2 Z2

1 s−1.The mean free path of electrons among charged particles is 4.7 × 105T 2 N−1

1 N−21 cm.

The mean free path of electrons among neutral particles is 1015 N−1 cm.

Sp.-V/AQuan/1999/10/07:14:19 Page 49


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 × 10−6T 5/2 erg cm−1 s−1 K−1.The life of a magnetic field in a plasma is

τ = 4π L2/η (η in emu)

= 1.5 × 10−12L2(ln �)−1T 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 Ion.b Intpl.c � Cor.d � Rev.e H Ig H IIh

log L cm 7.0 13.0 10.0 7.0 19.5 19.5log Ne cm−3 5.5 0.5 8.0 12.5 −3.0 0.0log N cm−3 11.0 0.5 8.0 16.5 0.0 0.0log T K 3.0 5.0 6.0 3.7 2.0 4.0log B G −1.0 −5.0 0.0 0.0 −5.0 −5.0

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 − 1

2 log Ne cm −0.6 3.0 −0.3 −3.6 3.2 2.7

Gyro freq.Electron 6.4 + log B s−1 5.4 1.4 6.4 6.4 1.4 1.4Ion 3.2 + log B s−1 2.2 −1.8 0.7 3.2 −1.8 −1.8

Collision freq.Electron 1.7 + log Ne − 3

2 log T s−1 2.2 −5.9 3.2 8.7 −1.8 −4.3

Ion 0.2 + log Ne − 32 log T s−1 1.2 −7.4 −0.8 7.2 −5.8 −5.8

Electrical 6.3 + 32 log T esu 10.8 13.8 15.3 11.9 9.3 12.3

conductivity−14.6 + 3

2 log T emu −10.1 −7.1 −5.6 −9.0 −11.6 −8.6

Mean free pathIon 5.7 + 2 log T − log Ne cm 6.2 15.2 9.7 0.6 12.7 13.7Neutron 15.0 − log N cm 4.0 14.5 7.0 −1.5 15.0 15.0

GyroradiusElectron −1.7 + 1

2 log T − log B cm 0.8 5.8 1.3 0.1 4.3 5.3Proton 0.0 + 1

2 log T − log B cm 2.5 7.5 3.0 1.8 6.0 7.0

Alfven 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 5.4 19.4 15.9 6.5 29.9 31.9

yr −2.1 11.9 8.4 −1.0 22.4 24.4

NotesaFor spectral emission from high-temperature plasmas, see Chap. 14.

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3.10 PLASMAS / 51

bIon. denotes ionosphere.cIntpl. denotes interplanetary space.d� Cor. denotes solar corona.e� Rev. denotes solar reversing layer.f Interstellar denotes interstellar space.gH I denotes the H I region.hH II denotes the H II region.

ACKNOWLEDGMENT

The author was supported in part by Grant No. AST-9315112 of the National Science Foundation.

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