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Energies and temperatures are in eV; all other units are cgs except where noted. Z is the charge state {Z — 0 refers to a neutral atom); the subscript e labels electrons. N refers to number density, n to principal quantum number. Asterisk superscripts on level population densities denote local thermodynamic equilibrium (LTE) values. Thus A/"n* is the LTE number density of atoms (or ions) in level n.

Characteristic atomic collision cross section:

(1)                                               nao2 = 8.80 X 10 —17 cm2.

Binding energy of outer electron in level labelled by quantum numbers n, I:

y 2 rp H

(n - Ai)2

where E^ = 13.6 eV is the hydrogen ionization energy and Ai = 0.751~5, I ^ 5, is the quantum defect.

Excitation and Decay

Cross section (Bethe approximation) for electron excitation by dipole allowed transition m —> n (Refs. 32, 33):

z .                                                                         ]_ 3 fn m Q (rt, 771) 2

(3)                                   (7ran = 2.36 X 10 ------------ ——------ cm ,

where /nm is the oscillator strength, g(n, m) is the Gaunt factor, e is the incident electron energy, and AEnrn = En Sm.

Electron excitation rate averaged over Maxwellian velocity distribution, Xmn = Ne{amnv) (Refs. 34, 35):

z.x              v                   1 g w in-5 fnm(g(n,m))Ne f AEnrn\ _1

(4)             Xmn = 1.6 X 10 -------------------------- —----- exp----------- —--- sec ,

A EnrnTl/2                                                                                 \ Te J

where (g(n,m)) denotes the thermal averaged Gaunt factor (generally 1 for atoms, rsj 0.2 for ions).

Rate for electron collisional deexcitation:

(5)                                                Ynrn — (N77i */Nn                   •

Here A/"m*/A/"n* = (gm/dn) exp(ASnm/Te) is the Boltzmann relation for level population densities, where gn is the statistical weight of level n.

Rate for spontaneous decay n —► m (Einstein A coefficient)


(6)                           Anm = 4.3 X 107(5fm/5fn)/mn(ASnm)2 sec 1.

Intensity emitted per unit volume from the transition n —> m in an optically thin plasma:

(7)                             I nm = 1-6 X 10 —19 Anrn Nn AEnrn watt/cm3.

Condition for steady state in a corona model:

(8)                                                  N0Ne(a0nv) = NnAn0,

where the ground state is labelled by a zero subscript. Hence for a transition n —► m in ions, where (g(n, 0)) ~ 0.2,

y _ . ^-25 fnmgoNeNo (AEnrn \ 3 ( AEno\ watt

(9)         Inrn = 5.1 X                    -------- exp^___j _

gmTf V

Ionization and Recombination

In a general time-dependent situation the number density of the charge state Z satisfies

z .                                  dN(Z)

- S(Z)N(Z) - ol(Z)N(Z) +S(Z - 1)N(Z - 1) + a(Z + 1)N(Z + 1)

Here S(Z) is the ionization rate. The recombination rate a(Z) has the form a(Z) = ar(Z) + NeOLs(Z), where ar and as are the radiative and three-body recombination rates, respectively.

Classical ionization cross-section36 for any atomic shell j

(11)                                       (n = 6 X 10~1Abjgj(x)/Uj2 cm2.

Here bj is the number of shell electrons; Uj is the binding energy of the ejected electron; x = e/Uj, where e is the incident electron energy; and g is a universal function with a minimum value gmin ~ 0.2 at x ~ 4.

Ionization from ion ground state, averaged over Maxwellian electron distribu­tion, for 0.02 < Te/E^ < 100 (Ref. 35):

(12)          S(Z) = 10"5---------- (Te/^oc) '--------------- exp                              cm3/sec,

V ^ V ^                                  (£7^)3/2(6.0+ Tc/£75)) P V Te ) '

where E^ is the ionization energy.

Electron-ion radiative recombination rate (e + N(Z) —► N(Z — 1) + his) for Te/Z2 < 400 eV (Ref. 37):

/ \ 1/2

1/1 / Uj x

(13)               OLr(Z) = 5.2 x 10 14Z



-1 e



3 /

cm /sec.


For 1 eV < Tej Z2 < 15 eV, this becomes approximately35

(14)                               ar(Z) = 2.7 X 10~13Z2Te~1/2 cm3/sec.

Collisional (three-body) recombination rate for singly ionized plasma:

(15)                                     as = 8.75 X 10-27Te-4'5 cm6/sec.

Photoionization cross section for ions in level n,l (short-wavelength limit):

(16)                             crph(n, I) = 1.64 x 10~16Z5/n3K7+21 cm2,

where K is the wavenumber in Rydbergs (1 Rydberg = 1.0974 x 105 cm-1),

Ionization Equilibrium Models

Saha equilibrium:39

, . NeN^(Z) 21 g?Te3/2 ( E^(n,l)\ (17) —' K \ = 6.0 X 1021 yi _e , exp            ' ; cm

Nn*(Z- 1                                                          g*-1                               Te J

where g^ is the statistical weight for level n of charge state Z and E^ (n is the ionization energy of the neutral atom initially in level                                                                                         given

Eq. (2).

In a steady state at high electron density,

NeN*(Z) S(Z - 1)


N*(Z-1)                                                                             a3

a function only of T. Conditions for LTE:39

(a)   Collisional and radiative excitation rates for a level n must satisfy

(19)                                                       ^nm ^ 10Anm.

(b)   Electron density must satisfy

(20)                             Ne > 7 X 1018Z7n~17/2(T/£^)1/2cm~3.

Steady state condition in corona model:

N(Z-l)                                                                              ar


N(Z) S(Z-l) Corona model is applicable if40

(22)                                     lO12^"1 < Ne < 1016Te7/2 cm-3

where tj is the ionization time.


N. B. Energies and temperatures are in eV; all other quantities are in cgs units except where noted. Z is the charge state (Z = 0 refers to a neutral atom); the subscript e labels electrons. N is number density.

Average radiative decay rate of a state with principal quantum number n is

(23)                             An = Anm = 1.6 X 1010Z4n~9/2 sec.


Natural linewidth (AE in eV):

(24)                                     AEAt = h = 4.14 x 10~15 eVsec,

where At is the lifetime of the line. Doppler width:

(25)                                          AA/A = 7.7 x 10~5(T//x)1/2,

where p is the mass of the emitting atom or ion scaled by the proton mass. Optical depth for a Doppler-broadened line:39

(26)       r = 3.52 x 10~13 fnrnX(Mc2/kT)1 /2NL = 5.4 x 10~9A(/i/T)1/2NL,

where /nm is the absorption oscillator strength, A is the wavelength, and L is the physical depth of the plasma; M, N, and T are the mass, number density, and temperature of the absorber; /j, is M divided by the proton mass. Optically thin means r < 1.

Resonance absorption cross section at center of line:

(27)                                      cr\=\c = 5.6 X 10~13A2/AA cm2.

Wien displacement law (wavelength of maximum black-body emission):

(28)                                           Amax = 2.50 X 10~5T_1 cm.

Radiation from the surface of a black body at temperature T:

(29)                                          W = 1.03 X 105T4 watt/cm2.

Bremsstrahlung from hydrogen-like plasma:


(30)                PBr = 1.69 X 10~32NeTe1/2 ^ \Z2N(Z)^ watt/


where the sum is over all ionization states Z. Bremsstrahlung optical depth:41

(31)                                     t — 5.0 X 10~38NeNiZ2gLT~7/2,

where ~g ~ 1.2 is an average Gaunt factor and L is the physical path length.

Inverse bremsstrahlung absorption coefficient42 for radiation of angular fre­quency uj:

(32)            k = 3.1 X 10 ~7Zne2 In AT~3/2cc;~2(1 - uj2p/uj2)~1/2 cm-1;

here A is the electron thermal velocity divided by V, where V is the larger of uj and up multiplied by the larger of Ze2/kT and fi/(mkT)1//2.





-1 e


Recombination (free-bound) radiation:

(33)           Pr = 1.69 x 10-32AreTe1/2 ^ \z2N(Z)

Cyclotron radiation26 in magnetic field B:

(34)                                  Pc = 6.21 x 10~28B2NeTe watt/cm3.

For NekTe = NikTi = B2/ 16tt (/3 = 1, isothermal plasma),26

(35)                                    Pc = 5.00 X 10~38AT2T2 watt/cm3.

Cyclotron radiation energy loss e-folding time for a single electron:

9.0 X 108B~2

(36)                                                tc « -------- ——--------- sec,

2.5 + 7

where 7 is the kinetic plus rest energy divided by the rest energy mc2. Number of cyclotron harmonics41 trapped in a medium of finite depth L:

(37)                                                    mtr = (57 (3BL)1/6, where (3 = 8ttNkT/B2.

Line radiation is given by summing Eq. (9) over all species in the plasma.


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