Continuum radiation from nonisothermal hydrogen plasmas

1. Quant. Specfrosc. Radial. Transfer. Vol. 12, pp. 1593-1608. Pergamon Press 1972. Printedin Great Britain

CONTINUUM

RADIATION FROM NONISOTHERMAL HYDROGEN PLASMAS H. F. NELSONand C. Y. WANG

Thermal Radiative Transfer Group, Department of Mechanical and Aerospace Engineering, University of Missouri-Rolla, Rolla, Missouri 65401, U.S.A. (Received 22 February 1972)

Ah&act--In this account of an investigation of the continuum radiative flux from nonisothermal stagnation shock layers composed of a hydrogen plasma, the general equations for the composition are derived and the Rankim+Hugoniot equations are simplified and solved to give the thermodynamic conditions in the shock layers. The radiative flux is calculated by considering ground to free state radiative transitions in atomic hydrogen for conditions roughly representing those of a space probe entering Jupiter’s atmosphere. The influence of Mach number, ambient density (altitude), spectral variation of the radiation absorption coefficient, shock layer thickness, excited state radiation, and temperature profiles are examined. The results show that not only the composition but also the equilibrium temperature behind the shock wave is essentially determined by considering only the ground state of atomic hydrogen. The excited state is largely responsible for the emitted radiation near the shock where the temperature is high, while the ground state produces the radiation in the cooler region of the shock layer near the body. In addition it is shown that the temperature profile is very important in determining the radiative flux reaching the body. The isothermal approximation not only overestimates the radiative flux but also influences the spectral distribution of the radiation reaching the body.

INTRODUCTION

HEATtransfer is a complex phenomenon

because pf the coupling of conduction, convection, and radiation. Radiative transfer in gases has become important to engineers in recent years because of the development of shock tubes and interest in planetary atmospheric entry. The complexity of the phenomenon has precluded precise treatment of every feature of the radiative fields. This study increases the understanding of radiative transfer in a shock layer through the development and analysis of a simple radiative-fluid-dynamic model. Its purpose is to show the influence of non-constant temperature profiles on the radiative tlux. The assumption of a specific temperature profile uncouples the fluid dynamics and the radiation; thus, the radiation can be investigated without the complicating influence of the fluid dynamics. Linear temperature profiles are assumed; however, the radiation is a function of the optical depth which is coupled nonlinearly to the temperature through the composition. Many investigations of the radiative transfer in shock layers have been published.“-‘) GOULARD~*~ considered the coupling between radiation and convection using a model that uncoupled the radiation from fluid dynamics. He showed that for most ballistic and earth re-entry problems the radiative flux is overestimated by considering an isothermal shock 1593

1594

H. F. N-N

and C. Y. WANG

layer; however, the radiation from an isothermal shock layer represents a first order approximation to the correct result. LASHER et cd.“) considered the energy radiated from an isothermal hydrogen plasma. They investigated both line and continuum processes and obtained results for temperatures between 10,000 and 40,000°K, pressures between 0.1 and 10 atm, and plasma thicknesses between 0.1 and 10 cm. They showed that the importance of the continuum radiation increased as the temperature and pressure increased. NELSON and GOLJLARD(‘~) considered line and continuum radiation from isothermal shock layers containing mixtures of hydrogen and helium for Jovian entry conditions. The radiative emission was calculated for shock layer temperatures between 15,000 and 33900°K for pure hydrogen cases and between 32,000 and 80,000”K for pure helium cases. Line radiation was generally more important than the continuum radiation at low ambient densities. As the ambient density increased, the continuum radiation became more important than the line radiation. MANDELL and CESS(’‘-13) considered radiative transfer in nonisothermal hydrogen plasmas for several cases: (1) radiative equilibrium, (2) pure radiation with uniform heat generation, (3) combined conduction and radiation, and (4) interaction of conduction, convection, and radiation. They assumed the Planck function to be a linear function of temperature. They used the exponential approximation and calculated both line and continuum radiation. Their results indicated that line radiation is important only in optically thin plasmas. NELSON and CROSBIE(‘~) considered the continuum radiative flux from a nonisothermal stagnation shock layer composed of an atomic gas Their results were derived by considering the ground to free state radiative transition. They used the principle of superposition to extend the results to include the excited state bound-free radiation. Their results emphasized the importance of the ionization edge location and the influence of the spectral shape of the bound-free radiative cross section. For the current study, the radiation flux to an entry vehicle for conditions approximating Jovian entry have been investigated. The Jovian atmosphere is assumed to be pure molecular hydrogen. Two atomic models for the atoms in the shock layer are considered. First, the atoms are assumed to have only one electronic state-the ground state--so that the plasma contains only ground state atoms, ion and electrons. Second, in order to investigate the influence of the excited states, the atom is allowed to have two electronic states-the ground and first excited state. The ions are assumed to exist in their ground electronic state. In addition, this study extends the previous work of NELSONand CR~SBIE(‘~) to include variable composition in the shock layer. PHYSICAL

MODEL

AND

GOVERNING

EQUATIONS

The shock layer model uncouples the fluid dynamics from the radiative transfer assuming a linear temperature profile. It assumes the shock layer to be composed of equilibrium atomic hydrogen plasma bounded by the shock wave (located a distance from the body) and the body. The maximum temperature (denoted by T,) is assumed occur at the shock as shown in Fig. 1. The linear temperature profiles are of the form: f

I

= Q+(l

-&)X

by an L

to

Continuum radiation from nonisothermal hydrogen plasmas

1595

where 0, is the nondimensional gas temperature at the body. The reference temperature, T,, and the density of the shock layer are obtained by solving Rankine+Hugoniot equations for a given ambient density and Mach number. The density of the shock layer is assumed to be constant.

Wall

Shock

FIG.1. Physical model.

The simplification of the radiation model and its flux calculation have been obtained previously by NELSONand C~osrun,~‘~~and the general details are described below. The shock layer model is based upon the following assumptions : (1) local thermodynamic and chemical equilibrium, (2) one-dimensional radiative energy transport, (3) negligible radiation emission from the body, (4) negligible precursor effects, and (5) no consideration of line radiation and influence of stimulated emission. The radiative flux at any position, r, in the shock layer can be written as m

F(r) = 2

Is

SI 0 0

e,,(t) Sgn(z - t)E(v)E,[E(v)lz- tl] dt dv,

(2)

where the optical depth and the optical thickness are defined as

r =

i/3(x’) dx’

(34

0

and L

T,

=

s

B(d) dx’.

0

W

1596

H. F.

NELSON and C. Y. WANG

The absorption coefficient has been assumed to be a separable function of position and frequency, i.e. k,(x) = CL(v) - /3(x)in this development. The Planck function is denoted by e,,(t), and E,(x) is the exponential integral of order 2. The function Rx) = 1

(4)

ociNi(x)

I

depends on the number density of the ith electronic level of atomic hydrogen, N,(x), and the absorption cross section at its ionization edge, aci. Thus, the evaluation of the radiative flux in the shock layer requires a knowledge of the composition, which is a function of the temperature profile. p(x) was treated as a constant in the previous work of NELSONand ~RosBIE.(‘~) The present study extends their work to include the variable /?. The equations for OEiare given in the Appendix. The frequency variation of the absorption coefficient is assumed to be a(v) = ; iii(V), i=O

where Ei(v)=O

forv
(6)

3 $(v)

=

;

for v 2 vi.

0 The ground state corresponds to i = 0, while the first excited state corresponds to i = 1. Defining a nondimensional frequency, 1 = hv/kT,, the dimensionless radiative flux to the body iso4) (7)

COMPOSITION In order to compute the absorption coefficient of the plasma, it is necessary to know the number densities of the atoms, ions and electrons which are present. The molecular hydrogen is assumed to be completely dissociated as it passes through the shock wave. The ionization process of atomic hydrogen is represented by X+H*,++e_+X

(8)

where X represents any species, p+ represents an ion, and e- represents an electron. The degree of ionization, a is the fraction of the atomic gas which is ionized and may be written aso’) a = $[(l +4/*)1/2-

11,

Continuum radiation from nonisothermal hydrogen plasmas

1597

where A = m,Z,*Z,*/(pZg);

(10)

A is the equilibrium constant of the above reaction, and Zl represents the complete partition function of species i. Also, Zp* = (Zi) exp[ - (I, + W/W)l,

(11)

Z;C, = (~)(Z~)exp[-D/(2kT)l,

(12)

z,*= 2(Ze).

(13)

The superscripts t and e are used to indicate the translational and electronic partition functions, respectively. The dissociation energy, D, for hydrogen is 4.476 eV. The symbol p represents the density of the plasma. The function A contains all the physical properties of the plasma as well as the thermodynamic properties of the shock layer. The degree of ionization is completely determined by A, and it is shown as a function of A in Fig 2. The degree of ionization for the plasma when the atoms are assumed to have only one electronic state (the ground state) is shown in Fig 3 as a function of temperature for seven densities. The results illustrate that for a constant temperature a increases as the plasma density decreases, because A is proportional to l/p and a is directly proportional to A. The degree of ionization for the plasma when the atoms are allowed to occupy only the ground state (n = 0) is slightly greater than when both the ground and first excited states are allowed (n = 1) because the electronic partition function increases due to the addition of the second electronic state. The difference in the degree of ionization for the cases n = 0 and n = 1 increases as density increases at constant temperature, as is shown in Fig. 3 by the dashed lines ; here, Aa = 100(~1,,~-a,,Ja~=~. The composition of the ionized gas may be easily calculated if the degree of ionization is known. From the definition of a, it follows that N, = (1 - a)N’

0.6 a

FIG. 2. The relation between a and A.

(14)

1598

H. F. NELSON and C. Y.

T.

WANG

OK

FIG. 3. Degree of ionization of hydrogen plasmas (-

a, - - - Aa).

and N, = N, = aN’,

(1%

where N’ is the initial number density of the nuclei. The composition of each electronic level atom may be found using the Boltzmann relationqo8-**’ (16)

NO = gON’U -4/G) and

Nl = $(l

H

-a) exp(-s,/kT),

where g, and g, are the degeneracies of the ground and first excited state atom respectively, and s1 is the energy of the first excited state, tZH. RANKINE-HUGONIOT

RELATIONS

The Rankine-Hugoniot equations relate the variables of state on either side of a shock front. They represent the conservation principles of mass, momentum, and energy, respectively, and can be written as(23) (18) PJJ. = PA7 p, + l&v,’ = p, + P& . 12-e L2 t.+*v. - $+2&Y

(19) (20)

Continuum radiation from nonisothermal hydrogen plasmas

1599

where 11is the gas velocity component normal to, and with respect to the shock. The subscripts a and r label the quantities in front and behind the shock, respectively. For a strong shock, ~~0,’ >>P,and iuz >> i, ; therefore, i, and Pawill be neglected. The enthalpy behind the shock is

(21) and the pressure behind the shock is

P,= kT,CNi= kT,e(l+or).

(22)

i

The calculation of the density and temperature behind the shock wave involves a system of five equations : equations (18)-(22), and five unknowns : p,, T,, u, and i,. The solution is found by an iteration process.“” The temperature behind a shock wave in hydrogen is shown in Fig. 4 as a function of Mach number for ambient conditions of T. = 300°K and p. = lo-’ to 10m6 g/cm3. The 36

8 /

-\

/

\

/

\-6

6

\ \ 30 I -7

6

25

4

\

I 30

‘1

I 40

I 35

2

I 45

5;6

MO FIG.

4. Rankine-Hugoniot solutions (hydrogen, T., = 3OO”K,-

r - - - AT).

atom is allowed to occupy only the ground electronic state. Figure 4 also shows the percentage change in the temperature behind the shock when both the ground and first excited states are considered; here, AT, = 100&,-T,=,)&,. The temperature behind the shock wave increases as the Mach number increases at constant ambient density because more kinetic energy becomes available to be converted to thermal

1600

H. F. NELWN and C. Y. WANG

energy. The temperature behind the shock wave, when only the ground state is considered (n = 0), is greater than that when both the ground and first excited state are allowed to exist (n = 1) at small Mach numbers because for n = 1 the electrons have an additional state to exist in. The reduction in temperature for n = 1 is proportional to the energy required to populate the excited state. The electrons which populate the excited state require energy (sl = jZH); thus, they lower the amount of energy available to increase temperature. When the Mach number is high, the n = 0 case produces slightly more electrons than the n = 1 case, because the n = 1 case allows some of the electrons to populate the excited state instead of becoming free. The energy [iZ,] is then left to increase temperature; consequently, the temperature for the n = 1 case becomes greater than it is for the n = 0 case at large Mach numbers. As ambient density increases, the temperature behind the shock wave increases, increasing the excited state population; thus, the excited state influences the temperature to a greater extent. However, the overall difference in temperature between the n = 0 and n = 1 cases is very small and never becomes greater than 0.1 per cent for the densities of interest. RADIATION

RESULTS

The radiation results were calculated for entry into a hydrogen atmosphere. The ambient density was varied between 10e9 and lop6 g/cm3. The shock layer thickness was assumed to be 1.0 cm. The ambient temperature was 300X, and the Mach number was varied between 25 and 50. These conditions approximate entry into the atmosphere of Jupiter. Ground state The optical thickness and flux results are shown as functions of the reference temperature, T,, in this section. The Mach number corresponding to a given T, can be found from Fig. 4. The minimum and maximum T, for a given ambient density shown in the following figures corresponds approximately to Mach numbers of 25 and 50, respectively. Figure 5 gives the optical thickness of the isothermal and nonisothermal shock layers for various ambient densities as a function of T,. The atoms in the shock layer are assumed to exist only in the ground state. As 8, decreases from unity, the optical thickness increases because in the cool region near the wall the ground state is highly populated. Note that the optical thickness varies directly (almost linearly) with the ambient density at constant T,. Figure 6 shows the influence of the ambient density, p,, on the flux for l.Ocm thick isothermal shock layers. The flux increases as p. increases from 10m9 to 10e6; however, the rate of increase becomes less rapid at large ambient densities. The flux increases because the optical thickness increases, increasing the emission. As pa increases, absorption becomes important and slows down the rate of increase of the flux. At an ambient density of 10m6, the plasma is very close to its optically thick limit. At large ambient densities, the flux increases continuously in the range of interest as T, increases because of the increase in the strength of the source function; while at small ambient densities, the flux first increases and then decreases because of the depopulation of the ground state.

Continuum radiation from nonisothennal

hydrogen plasmas

FIG. 5. Optical thickness of hydrogen shock layers (L = 1 cm, n = 0, T. = 3OO”K,. . . . . . . B,,,= 1.0, --- 8, = 0.5, 8, = 0.0).

10000

2OiJoo

I5000

25000

T, . *K FIG.6. Radiative flux to body as a function of T. at constant density and at constant Mach number (0, = 1.0, n = 0, T. = 3OO”K,L = 1 cm).

1601

1602

H. F. NUON and C. Y. WANG

Figure 7 shows flux to the body from a nonisothermal hydrogen plasma at 0, = 0.5 as a function of T,. If one considers a value of T,, the flux to the body is small at low ambient densities because the optical thickness is small. As p. increases, the optical thickness increases making the emission more important and increasing the flux. As p, increases further, absorption becomes important because of the increase in optical thickness and begins to reduce the flux. The flux goes through maximum between p4 = lo-* and lo-’ (near optical thicknesses of unity) as absorption becomes more important than emission. As pa increases beyond 10V6, the flux will decrease toward its optically thick limit.

\ 3

M;=25

10000

ZOCCG

15000

Tr .

25000

“K

FIG. 7. Radiative flux to body as a function of T, at constant density and at constant Mach number (0, = 0.5, n = 0, T. = 3WK, L = 1 cm).

Figure 8 shows the flux at (3, = 0 for various ambient densities as a function of T,. It shows the same trends as Fig. 7 except the optically thick limit for this case is zero. Lines of constant Mach number are also shown on Figs. 6-8. The influence of the temperature profile appears very dramatically if one considers a constant Mach number entry. In the isothermal case the radiative flux to the body continually increases during entry; whereas in the nonisothermal case the flux first increases and reaches a maximum at an ambient density of about lo-* g/cm’ and then decreases at the lower altitudes. These figures indicate that the isothermal case is very conservative with respect to heating loads and that the temperature profile is very important in determining the radiation reaching the body.

Continuum radiation from nonisothermal hydrogen plasmas

IS000

10000

Tr.

20000

25000

OK

FIG. 8. Radiative flux to body as a function of T. at constant density and at constant Mach number (e, = 0.0, n = 0, T. = 300x, L = 1 cm).

0-0006-

0~0002 -

25

30

I

I

40

35 Mo

12000 T ,l

, 14307

45

50 I 16506

“K

FIG. 9. Excited state optical thickness of hydrogen shock layem (t = 1 cm, I) = l;p, = 10-s g,/cm3, T. = 300°K).

1603

1604

H. F. NELSONand C. Y. WANG

Excited state The radiation from the first excited state was calculated for hydrogen at ambient density of lo-*. Superposing the excited state flux with that of the ground state gives the total flux. Figure 9 gives the excited state optical thickness as a function of the Mach number and/or T,. For the isothermal plasma, the excited state optical thickness increases as the Mach number increases at small Mach numbers because the excited state population increases due to the increase in shock layer density and temperature. As the Mach number increases further, the excited state optical thickness reaches its maximum and then decreases because the excited state population begins to decrease due to the high temperatures. For the nonisothermal cases, the excited state optical thickness increases continuously as the Mach number increases from 25 to 50. As the Mach number increases, the temperature increases; however, there is always a cooler region in the shock layer where the excited state is not depopulated, thus the optical thickness increases even if the temperature gets very large. Figure 10 shows the ground state flux and the total flux as a function of the Mach number for both the isothermal and nonisothermal cases. The total flux is obtained by adding the excited state flux to the ground state flux. The excited state flux is small but not negligible compared with the ground state flux. For the nonisothermal cases it comprises about 20 per cent of the total flux for 1 cm thick shock layers. The inlluence of shock layer thickness on the ground state flux, excited state flux, and their sum is shown in Figs. 11 and 12 for Mach 30 and 50 respectively. The B,,,= 1.0 curves

1

25

30

35

40

45

50

MLl

FIG. 10. Radiative flux from hydrogen plasmas (n = 1, P. = 10-s g/cm”, T. = 300X, L = 1 cm, - - - ground state, ground + excited).

Continuum radiation from nonisothermal hydrogen plasmas

\

I I02 L,

I 103

104

cm

FIG. 11. Radiative flux from hydrogen plasmas at M = 30(n= 1, T. = 3OO”K,p,, = 1.078 x 10e6 ground + excited). g/ems, T, = l6,053”K, . . . . . . ground state, - - - excited state, -

10-3

I o-2

10-l

IO

I

L.

102

103

104

cm

FIG. 12. Radiative flux from hydrogen plasmas at M = SO(n = 1, T. = 3OO”K,pa = 1.078 x lo-“ g/en?, T, = 25,53l”K,. . . . . . . ground state, - - - excited state, ground + excited).

1606

H. F. NELSONand C. Y. WANG

are the isothermal results, while those labeled 8, = 0.5 are the nonisothermal results. For these figures the ambient conditions are p. = 1.078 x 10m6g/cm3, TU= 3OO”K, and Pa = 1cmHg. The ground state flux is larger than the excited state flux for small values of L. As L increases, the ground state flux increases, reaches its maximum, and then decreases. It increases at small L because the shock layer is emission limited, and then at larger L it decreases because of self-absorption. The excited state flux behaves much the same way as does the ground state flux as L increases. The optical thickness of the excited state is much less than that of the ground state, hence the maximum in the excited state flux occurs at much larger L than it does for the ground state. In both cases the maximum flux reaches the body for the value of L which corresponds to an optical thickness of the respective process near unity. CONCLUSIONS

For nonisothermal shock layers in atomic hydrogen, the following conclusions can be made : 1. For Mach numbers less than 50, the population of atomic hydrogen atoms, electrons, and protons is essentially determined by considering only the ground electronic state of atomic hydrogen. 2. The solutions of the Rankine-Hugoniot equations are almost entirely determined by considering that the atomic hydrogen atoms exist entirely in their ground state for Mach numbers less than 50. 3. The radiative flux to the body is very sensitive to the temperature profile. The isothermal approximation is only justified if the temperature in the shock layer is approximately constant except for a very thin layer near the body. 4. The maximum flux reaching the body occurs for thicknesses such that the optical thickness of the process responsible for the radiation is near unity. 5. The exoited state flux becomes more important relative to the ground state flux as the shock layer thickness increases. The ground state radiation is more optically thick than the excited state radiation and becomes strongly self-absorbed as the shocklayer thickness increases. 6. The data given in Figs. 11 and 12 show that the total radiation flux reaching the body for Mach 30 and Mach 50 is approximately the same. At Mach 50 the maximum in the source function has shifted toward the ground state portion of the spectrum increasing the ground state emission while the higher temperatures have decreased its optical thickness. At the same time these effects have combined to decrease the efficiency of the excited state radiation. 7. The intent of this research was to investigate the influence of the temperature profile on stagnation radiative heating through the use of a simplified model. The results show that the radiation reaching the body for Jovian entry conditions is very sensitive to the temperature profile. The isothermal approximation not only overestimates the radiative heating, but also influences the spectral radiative distribution. Acknowledgements-The research for this paper was partially supported by a grant from the National Science Foundation (NSF Grant-10954).

Continuum radiation from nonisothermal hydrogen plasmas

1607

REFERENCES 1. D. B. OFJ_E,JQSRT 1,104 (1961). 2. J. H. CHIN and HEARNE,AI& .I. 2.1345 (1964). 3. W. A. PAGE, D. L. COMPTON,W. J. BORUCKI, D. L. CIFFONEand D. M. COOPER,AIAA Preprint 67-784 (1968). 4. H. H~~HIZAKIand L. E. LASHER,AIAA J. 6.1441 (1968). 5. M. THOMAS,AIAA Preprint 68-788 (1%8). 6. H. HO~HIZAKIand K. H. WILSON,AIAA J. 5,25 (1967). 7. M. E. TAUBER,J. Space-craft Rocket. 6, 1103 (1969). 8. R. GOULARD,JQSRT1,249 (1961). 9. L. E. LASHER,K. H. WILSONand R. GRIEF,JQSRT7,305 (1967). 10. H. F. NELSONand R. GOULARD,JQSRTS, 1351 (1968). 11. D. A. MANDELLand R. D. Cuss, JQSRT9,981 (1969). 12. D. A. MANDELL,JQSRT9,1553 (1969). 13. D. A. MANDELL,Ph.D. Thesis, State Univ. New York at Stony Brook (1968). 14. H. F. N-N and A. L. CROSBIE,AZAA J. 9,1929 (1971). 15. C. Y. WANG, M.Sc. Thesis, University of Missouri at Rolla, Rolla, MO. (1970). 16. C. D. HODGMAN (Ed.), Handbook of Chemistry und Physics, p. 1118. Chem. Rubber, Cleveland, Ohio (1963). 17. Ya B. ZELD~VICH and Yu P. RAIZER,Physics of Shock Wave and High Temperature Hydrodynamic Phenomena. Academic Press, New York (1966). 18. J. E. LAY, Thermodynumics. tier& Books, Columbus, Ohio (1963). 19. G. L. WEISSLER.Hand. Phvs. 21.337 (1961). 20. Om P. RUSTICI;E. I. FISHERand C. H. FULLER,J. opt. Sot. Am. 54,746 (1964). 21. C. E. MOORE,U. S. Nutn. Raw. Stand Circ. 467 (1949). 22. C. W. ALLEN,Astrophysical Quuntities, Athlone Press, England (1955). 23. F. A. GOLDSWORTHY, On the dynamics of an ionized gas, Progress in Aeronautical Science Vol. 1. Pergamon Press, Elmsford, New York (1961).

APPENDIX Absorption cross sections The spectral variation of the absorption coefficient is taken to be of the form of (v,,/v)~.For hydrogen hv, = 13.6 eV and hv, = 3.40 eV. The continuum absorption coefficient for ionization from the hydrogen nth electronic state is(“)

W, a31= NA,

(A-1)

where N, is the number density at nth state The continuum absorption cross section (bound-free), urn is given by Kramers’ formula.(i7) (I,

=

7.9x 10‘18~(

:)’ cm2

where hv, = I,z’/(n+

1)’

z=l n = 0,1,53,...

(A-3)

The absorption cross section at ionization edge, a, is 1. n=O k,(O, ~0) = N,c,,

(A-4

= B&t,,(v)

/IO(X)= 7.9 x lo-‘sk(l

%W = wv)3

-a)

(A-5)

v 2 vg.

M-6)

1608

H. F. NEUON and C. Y. WANG

2.n=l

W’, to) = NcP,, = B&)a&)

v 2 vg

&(x) = 7.9x lOP-J+-a) II If v 2 VI) %(V) = (%Iv)3 k,(L 0~))= N,o,,

= /&(~)a,(“)

(A-7) (A-8)

v 2 VI

&(x) = 2(7.9x lo-‘s)sexp(-s,/kT) H H v z VI al(v) = (v,/@

(A-9) (A- 10) (A-11)

where the atomic hydrogen electronic partition function is Z, = g,+g,

expj-s,/kT)

= 2+8exp(-E,/kT).

(A-12)