Spectral lines in hot dense matter

1. QUMI.

Spectmsc. Rod&.

Transfer. Vol. II, pp. 77-U.

Pergamon Press 1977.

Printed in Great Britain

SPECTRAL LINES IN HOT DENSE MATTERt Barn

F. ROZSNYAI

LawrenceLivermoreLaboratory,Universityof California,Livermore,CA 94550,U.S.A. (Received29 March 1976) Ah&act-An algorithmis presented for the computationof photoabsorptioncross sections at arbitrary temperatureand matterdensity.The “averageatom”modelis refinedto give an approximaateaccountfor the differentionization stages. The broadeningof spectral lines is accounted for in a simple approximation. Calculationsare presentedfor the berylliumandgermaniumplasmasin the frequencyregionof spectrallines. 1. INTRODUCTION THE SUBJECTof

photoabsorption or photoemission by hot matter in the frequency region of spectral lines has long been of interest in astrophysics. The appearance of high powered lasers and the experimental observation of X-ray line emission by laser produced plasmas has caused additional interest in the subject. Hydrogen and helium-like transitions for highly ionized, low-2 elements were reported by FELDMANet al.,“’ and transitions for heavier elements stripped to the L and M shells by laser heating were also observed and reported.‘24’ In the early laser experiments the plasma densities were low. In Ref. (l), the reported electron density derived from K satellite spectra was about 6 x lOI cme3. At such low densities, the identification of the observed spectral lines can easily be done by atomic structure calculations for the properly stripped ion. In a recent report, MALVEZZI”’et al. reported electron densities up to lo” cm-3 in laser produced Be plasma. At these densities, the effect of free electrons on the quantum-mechanical state of the ion should be taken into account. At electron densities of 10” - 10” cm-3, the effect of screening is not only a noticeable shift in the line positions but the oscillator strengths change up to a factor of two, compared to the predictions of an isolated ion model. In addition, the broadening of spectral lines becomes complicated compared to the predictions of the linear Stark broadening, which is most important at low densities. At high matter densities, the bound-free cross sections for the different shells also change compared with those of an isolated ion. The purpose of this study is to present a comprehensive treatment of the effect of finite matter density on the photoabsorption cross sections. The basis of calculations is a self-consistent temperature and density dependent “average atom” model@’extended to account for the different ionization stages. Although the calculations are restricted to photoabsorption cross sections, the conclusions are also valid for photoemission. Unless stated otherwise, all quantities are given in atomic units (1 a.u. = 27.204 eV). 2. THEORY

The concept of the “average atom” (AA) was suggested first by CHANDRASERHAR~in connection with stellar structure and stellar opacity. The basic idea of the AA is that the distribution of electrons over the bound and free states is given by the Fermi statistics. In that model, the bound and free electrons can be treated together in a self-consistent way. It can therefore be used to study the temperature and density dependence of the dominant atomic properties [see Ref. (6)]. The calculation of oscillator strengths and bound-free cross sections in the AA approximation was described in a previous paper.‘” The disadvantage of the AA model is in its average character. Features coming from detailed configuration accounting (DCA), such as positions and strengths of lines associated with different excitation and ionization stages, are not predictable because they are “averaged out”. To account for the different ionization stages, one can use Saha’s ionization equation. The form used in this paper, applicable to any degree of electron degeneracy and given by Cox,” is

F =y

exp {-[AE - CL- (j + l)/r,-,]/kT},

(1)

tWork performedunderthe auspicesof the U.S. EnergyResearch& DevelopmentAdministrationundercontractNo. W7405-Eng-48. 77

78

B.F.ROZSNYAI

where P, is the probability for the presence of an ion with j electrons missing, gi is the statistical weight of the ion, AE is the energy necessary to remove an electron, corrected by the density dependent quantity (j + l)/rO and p is the chemical potential or Fermi level given in the AA approximation. The symbol r. stands for the ion sphere radius determined by the matter density (r,’ = 3/47rp). The method of accounting for the different ionic states in this report is the following: For a high degree of ionization, when only hydrogen and helium-like ions exist, the self-consistent AA state is computed and using eqn (1) the relative probabilities for the hydrogen and helium-like states is obtained. The absolute probabilities can be computed by the normalization condition pHY+ 2PH. = N,, where N,, is the number of 1s electrons in the AA approximation. The single-particle levels for the hydrogen and helium-like ions are computed from those of the AA levels by first order perturbation calculations. The wave functions for the 1s hydrogen and helium-like ions are recomputed, whereas for the higher unoccupied states the AA wave functions are retained. For the case when more bound electrons exist, a self-consistent ion state is computed which is obtained from the AA state by truncating the noninteger electron occupational numbers for each shell to the nearest integer value. This in a sense produces an ion with the “highest probability” (HP) which is physically realistic, in contrast to the AA, which is fictitious. From the HP ion the energy levels of the next ionization state are computed by lirst order perturbation and the other higher ionized states are neglected. In terms of spectral lines, this procedure predicts the dominant lines and the next satellite lines with the highest probability. The photoabsorption cross section is given by a(h)

= ubb(ho) + Ub’(hW)+

us

(2)

where Iho is the photon energy and ubb, u bf and uff stand for the bound-bound, bound-free and free-free cross sections, respectively. The computation of the free-free cross sections is done in the Born-Elvert approximationoO’ and the computation of the bound-free cross section was discussed in detail in Ref. (8). The bound-bound cross section is given by (3) where (r is the fine structure constant, fil is the oscillator strength of a dipole transition i + j and bij is the line profile for the transition. The oscillator strengths and bound-free cross sections are calculated by replacing the AA electron populations of each shell by the proper ionic state populations multiplied by the probability of each ion. The computation of the line profiles is a major task and presently a simple approximation is given which is adaptable for large scale computations. Consideration is given to the Doppler, electron impact and ion or Stark broadenings, each of which predicts a different line profile. The main emphasis, in this report, is on the computation of the characteristic widths and it is discussed in Sections 3 and 4. The computation of line profiles is done in a simple approximation and is discussed in Section 5. 3,STARKBROADENING

The perturbing field of the heavy ions is treated in the quasi-static approximation. If the plasma density is low then the probability of a given strength of the quasi-static field is determined by the Holtsmark distribution. (‘I) On the other hand, if the plasma is dense then the screening by the free electrons reduce the probabilities of strong a perturbing field below their W) shows that when the plasma is so dense Holtsmark values. Detailed calculations by HOOPJIR that the Debye length is not much greater than the average distance between atoms, then the binary or “nearest neighbor” approximation of MARGENAU and LEWKP is valid. This last condition is satisfied in all calculations presented in this report.

19

Spectral lines in hot dense matter

Assuming that the perturbing ion with a net charge 2 is at a distance ri in the z direction from the radiating or absorbing atom, the perturbing potential is given by (4) where r is the position vector of the radiating or absorbing electron (Irl < 1~1)and F represents the “normal field strengths” for the multipole field (F,,, = Z x ~i-*-~). In this paper, the perturber-radiator (absorber) distance ri is taken as I;

=

r. + r,

POs rc),

where r. is the ion sphere radius mentioned before and r, is the core radius of the absorbing or radiating ion and it is computed from the self-consistent electron densities. For most practical purposes, the A = 1 and A = 2 terms (dipole and quadrupole) are sufficient and only these terms are considered. The A = 0 term is of no interest since it produces a constant shift for all levels. The matrix elements of eqn (4) are given by V*(nlm, n’l’m’) = -F~+,C*(lm,

1’m)mm’ x D,(nln’l’),

(5)

where the C’s are coefficients given by COND~Nand SHORTLEY”” and the D’s are radial integrals given by [see Ref. (8)] D,(nln’l’) =~~R”I(r)Q,,,(r)rA+2dr.

(6)

To solve the matrix equations for all states involves lengthy calculations;“” approximations are used as outlined below. The shift of an electron energy level E.~caused by the normal dipole field is calculated by = A&l, n’l+ l)+Ad(nl, n’l- l),

Adi

(7)

where Aa(nl, n’l’) is the shift caused by the admixture of the nearest n’l’(1’ = 12 1) state and the subscript d refers to the dipole field. The quantity Ar(nl, n’l’) is given by Ad(nl, n’l’) = T y

+ i [(q - q)’ + 4 B’(nl, n ‘1’)]“2,

(8)

where E)and l( are the larger and smaller of ln1and Q,, respectively, and v(nl, n’l’) is the dipole matrix averaged over the m and m ’ states given byt Vnl,

n’l’) = -FJWL

n’l’)

2

J

I,+ 1

J It21,+ 11t21,+ 2J11/2

(9)

with I( = min (11’). The effect of the quadrupole field is taken into account by calculating the spread over the m states of a level ln1given by 2

A4 = 3M2L, (21 + 3;(21_ 1), 1 >o,

(10)

where the subscript q refers to the quadrupole field and (r*>,,, is the expectation value of the r2 operator. Summarizing eqns (7)-(10), the half width of an nl - n’l’ transition line due to the normal Stark field in the nearest neighbor approximation is given by r.+(nl,n’l’) = (A,(nl)l+ (AJn’l’)) tEquat.ion (9) is exact

for s-p mixing.

(11)

B. F. ROZSNYAI

80

due to the dipole field, and r,(nl,n’l’)=~[A~(nl)+A,(n’[‘)1+~1A,(nl)-h,(n’l’)l

(12)

due to the quadrupole field. In eqn (12), the first term is associated with the Am = rl and the second with the Am = 0 transition. 4. ELECTRON

IMPACT BROADENING

The general formula for line broadening due to collisions between electrons and atoms or ions is given by BARANGER,“@

Ifi

-fi(fh$)(‘dR

II

(I”

(13)

where CT,“’ and a/” are the inelastic cross sections of the initial and final states connected by photoabsorpfion, fi and fi are the elastic scattering amplitudes in the initial and final states, respectively, u is the relative velocity of the colliding particles, n(v) is the number of electrons per unit volume in the velocity range u and u +du, the subscript implies averaging over the velocity distribution of electrons and ‘yli stands for the half-width of the spectral line. The calculation of 3/81using eqn (13) is very difkult and a number of approximations are therefore made: (1) It is assumed that the free electrons have a Maxwellian distribution. (2) The inelastic cross section is approximated by a formula for electron-impact ionization. (3) The contribution from the elastic scattering is calculated first for ls-2p transitions and is subsequently generalized. (A) Contribution from elastic scattering The difference of the two scattering amplitudes fi and fr is calculated in the Born approximation, I-fr(e)=fil(B)=-~~-‘~~rVi,(r)sinKrdr, 0

(14)

where IL2 = 2k2(l - cos e), K being the wave number of the incident electron and V,,(r) is the difference of the potentials in the initial and final states. It is assumed that the potentials are spherically symmetric, which is equivalent with averaging over different m states. For a Is-2p transition, one obtains 3 I rV,S_2P(f)= 1 + - a,r + 2,*r* + - Lu,?’ exp (-2cu,r) - [I+ (YJ] exp (-2a,r), 2 3 I

(19

where a, and a, are the exponents of the 2p and 1s radial wave functions, respectively. For hydrogenic orbitals rr, = 2, oP = Z/2. The insertion of eqn (15) into eqn (14) leads to a complicated formula, so instead of eqn (15) the following approximation is used: V(r) - W(r) = I3 exp l--k],

(16)

where B = 1.4[& -o.5cYp1,

A = [2B/A11’3, (17)

A=

_ V(r)? dr = _ W(r)r’ dr = 5 I0 I0

- 3.

Inserting eqn (16) into eqn (14) one obtains? 2A tBy choosing B and W(r) as given by eqns (17) and (16), the functions r’V(r) and r”W(r) agree within 1% at their maximum values.

81

Spectral lines in hot dense matter

and the total cross section is given by U(4) = $%*

12A’+ 86A*E+ 256~~ A4[A2+8e13 ’

(19)

where Q is the energy of the incident free electron. Equation (19) is still a difficult one to average over a Maxwellian distribution so U(E) is approximated byS

(20) Equation (20) can be averaged over Maxwellian distribution. The number of free electrons in the energy range Q, Q+de is given by n(r) dr = 2

VT

(kT)-3’2per”Zexp [-e/U],

where pL is the total density of free electrons and kT is the temperature in energy units. The . velocity is given by u = c[~E/~Ic~]“* and one obtains for the elastic contribution 1

cr'(~)on(~)

yL*p=20I

X

rhc[mc*]-“*(kT)-“*p,[B*/A ‘1

do =-“;iz

_ e[B2/6A2A4+e]-‘exp

= 23’24q

[-•/kT] de

PJcT)-~‘*[B~/A’] {kT -D exp (D/kT)E,(D/kT)},

(21)

where D = B2/6AZA4 and El(x) = J: (e-‘/t) dt, (gc/[mc*]“* = 1). After collecting all the constants, eqn (22) can be written as y&

= 0.59234$ Nr(kT)-3’2[B2/A2]{D exp (D)E,(-D)

+ kT},

(22)

where NI is the number of free electrons per average atom, M is the atomic mass number and p is the matter density in g/cm3. The generalization of eqn (22) to a nl + n’l’(f’ = 1 k 1) transition is made by the use of the relation valid for hydrogenic orbitals, al= I

5n2+1-31(1+1) C 2(r*L

I’* I

’

(23)

where the bracket ( ) indicates the mean value. In this paper, eqns (23), (17) and (22) are used to calculate the contributions to the line widths for nl + n’f’ (I’ = I + 1) transitions from elastic scattering. The mean values of the r* operator and N, are determined from the wave function and ionization state of the self-consistent average atom. (B) Contribution from inelastic scattering To get an estimate of the contribution to yij in eqn (13) from the inelastic parts, the following formula is used: a!,,(c) = 2.27 $J

n

In (E/IQ 011- exp (c/2.51~~111,

(24)

where abr is the electron impact ionization cross section of the shell with quantum numbers nl, l,,, is the energy level of the shell and N., the number of electrons occupying the shell. Formula (24) *Equation (20) impliesthat o(O)= u’(O) and Iii o(e) = lin a’(e) as c +a.

17. No.

82

B. F. ROZSNYAI

is analogous to that of Lorzon without the exponent in the bracket.? Averaging eqn (24) over the Maxwellian distribution of free electrons gives

where

Equation (25) takes into account the effect of ionization. The effect of electronic excitations is accounted for by replacing [a,[ with the lowest excitation energy for the nl shell. 5. THE LINE PROFILE

It is well known that the Doppler and electron impact broadenings predict Gaussian and Lorentzian profiles, respectively. The line profile predicted by the quasistatic Stark broadening of the nearest neighbor approximation is given byo8’

where A = 1,2, . . . for the dipole, quadrupole, etc. terms of eqn (9), Ao is the frequency shift caused by the normal field F,,, and o. is the frequency of the unperturbed line. The shape of the function given by eqn (26) is similar to the well known Holtsmark profile, with the maxima around o - o. - 0.8Aw and 0.6Ao for A = 1 and 2, respectively, whereas for the Holtsmark profile the maximum is at 0 - oO-- 1.6Aw.The half maximum values are around o - oO- 1.2Ao for h = 1 and 2. In this study, the line profile is computed by combining the electron impact and Stark widths to form a Lorentzian profile and subsequently using the Doppler width to form a Voigt profile. This approximation is justified by the fact that the electron impact broadening fills in the gap at the center of the ion broadening profile. The final Lorentz half-width for an nl + n’l’ transition is given by r,(nl, 11’1’)= r&d, n’l’)+

r,(nf, n’l’)+

y”(nl, n’l)+ y:+ y$,,,

(27)

where rd and r, are the dipole and quadrupole contributions of the ion field as given by eqns (11) and (12), and y” and y’” are the contributions from the elastic and inelastic electron scatterings. The Voigt profile is formed byo9’

@a) where K(x, y) =

I

+LT

exp (-t’)[y’+ JJ. $r -CC

(x - t)‘]-’ dt,

(28b)

y = 2 (In 2)“*,

(28~)

x = [(go - hwO)/rD](In ,)I”,

CW

and rD= frw0(2kT’In ~/MC*)“’ (Doppler half width). tThe factor in the bracket in eqn (24) improves the agreement of u’ with experimental data.

83

Spectral lines in hot dense matter

The Voigt profile satisfies the normalization condition

6. CALCULATIONS

(A) Beryllium plasma For a case study of hydrogen and helium-like states, the Be plasma was selected. The chosen temperature is kT = 21 eV; at this temperature and at a density of lOI atoms-cm-‘, the plasma is mainly hydrogenic. Table 1 shows some relations between the AA and DCA treatment. In the first column, the density is given and in the column labeled by AE the 1s - 2p excitation energies are given for the Be(IV), Be(II1) and AA States, respectively. The columns P1v and PlrI show the probabilities of the two ions in their ground states. The probabilities do not add up to 1 because of the presence of excited ionic states, which have not been considered. The data for the 1s - np transitions for the hydrogen and helium-like states are summarized in Tables 24. In the columns AE and f, the excitation energies and oscillator strengths are given. The different contributions to the line widths are shown in the next four columns. In the column ‘y,the half-widths due to the combined effects of the elastic and inelastic electron impact are shown; the half-widths due to the dipole and quadrupole Stark effect are shown in columns rd and r,, respectively, and the column rD shows the Doppler half widths. The oscillator strengths decrease with increasing matter density. This decrease is due to the screening by the free electrons and is in agreement with previous AA calculationP and also with the predictions of the Debye-Htickel model.“” The Table 1. Is -2p

p(a/cmY lOI 1020 lo*’ lO=

Table 2. Excitation

$

g X L9

ls-2p ls-3p Is-4p 1s -5p Is -6~ (ls)Z- ls2p (ls)Z- ls3p Us)*- ls4p Is’- ls5p (ls)*- ls6p

AE(Be(IV))

AE(Be(II1))

6.023 6.086 6.187 6.289

4.571 4.536 4.292 4.362

AEW) 6.017 5.389 4.940 4.641

&v

P nr

0.67 0.34 0.10 0.02

0.11 0.50 0.82 O.%

energies, oscillator strengths and line widths (au.) in a Be plasma at p = 10Lya/cm3 and kT = 21 eV. The numbers in parentheses indicate powers of ten AE

g

excitation energies and ionic probabilities in a beryllium plasma at kT = 21 eV at different densities

6.023 7.132 7.515 7.691 7.786 4.571 5.157 5.369 5.468 5.521

f 0.3%5 7.214(-2) 2.72q-2) 1.305(-2) 7.075(-3) 0.5510 0.1143 3.886(-2) 1.855(-2) 1.003(-2)

Y 1.306(-S) 9.502(-5) 2.221(-4) 4.362(-4) 7.872(-4) 2.32q-5) 1.689(-4) 3.949(-4) 7.755(-4) 1.399(-3)

r.i

r,

rD

8.115(-6) 5.392(-S) 1.868(-4) 4.818(-4) 1.054(-3) l&2(-5) 9.586(-S) 3.322(-4) 8.565(-4) 1.874(-3)

3.534(-4) 4.185(-4) 4.41q-4) 4.513(-4) 4.568(-4) 2.682(-4) 3.026(-4) 3.15q-4) 3.208(-4) 3.239(-4)

1.41q-3) 1.05q-2) 3.202(-2) 6.537(-2)

4.902(-5) 3.357(-4) 1.237(-3) 3.917(-2)

3.571(-4) 4.209(-4) 4.425(-4) 4.524(-4)

1.412(-4) 1.083(-2) 3.111(-2) 6.811(-2)

8.715(-5) 5.97q-4) 2.199(-3) 6.%3(-3) -

2.54-4) 2.948(-4) 3.707(-4) 3.161(-4) -

1.046(-3) 2.471(-3) 6.119(-3) 1.237(-2) 2.015(-2) 6.449(-6) 2.045(-3) 5.189(-3) 1.119(-2) 1.932(-2)

Table 3. As in Table 2 for p = Wa/cm’ AE

4.536 5.150 5.365 5.459

f 0.3056 7.072(-2) 2.563(-2) 9.6%(-3) 0.4556 0.1016 3.646(-2) 1.373(-2) -

Y

8.325(-5) 6.537(-4) 1.77q-3) 5.02q-3) 1.480(-4) 1.162(-3) 3.147(-3) 8.935(-3)

rd

B. F. ROZSNYAI

84

Table4. AsinTable2forp = lv’a/cm’ AE

ls-2p ls-3p ls-4p Is-5p i 1s -6~ (1s)z-1s2p 6 (ls)2- ls3p F; (ls)2-1s4p m w-ls5p 1 ls)‘-1s6p s s m

f

Y

rd

r,

rLl

6.187 7.240 -

0.2660 5.468(-2) -

1.11q-3) 1.433(-2) -

5.759(-3) 6.319(-2) -

5.317(-4) 4.485(-3) -

3.630(-4) 4.248(-4) -

4.292 5.060 -

0.3898 7.788(-2) -

1.983(-3) 2.548(-2) -

2.598(-3) 6.954(-2) -

9.45;-4) 7.974(-3) -

2.518(-4) 2.%8(-4) -

Table 5. Probabilities of configurations at kT = 144eV in a Ge plasma at diierent densities p(a/cm) 1OL9 loao 102’ KY*

HP Configuration (prob.)

Configuration (prob.)

(ls)2(2s)2(2p)~ (0.90) (lS)*(2s)y2py (0.99) (ls)‘(2s)*(Zp)“(3p)‘(3d)’ (0.58) (ls)*(2~)*(2p)~(3~)‘(3p)*(3d)~ (0.56)

(ls)*(2s)y2p)’ (0.10) (1sz)(2s)y2p)J (0.01) (1~)~(2~)~(2p)~(3p)’ (0.42) (1s)‘(2s)2(2p)6(3s)‘(3p)2(3d)’

(0.44)

transition energies for the helium like Be(II1) show red shifts in agreement with the predictions of the Debye-Hiickel model, whereas the hydrogen-like Be(IV) transitions show blue shifts, in agreement with experimental observations.‘22’ At low density, the line widths are dominated by the dipole Stark effect because the final states are not very different from hydrogenic states. An exception is the (1~)~- ls2p transition for Be(II1) where neither the initial nor the final state is hydrogenic. As the density increases, the relative magnitudes of the electron impact and quadrupole Stark contributions to the line widths become larger. The reason for this is twofold: tirst, the electron impact and quadrupole Stark widths are proportional to p, whereas the dipole Stark width is proportional to P~‘~; second, with increasing density, the states become less hydrogenic due to the screening by the free electrons. It should be noted that the electron impact widths are in about 50% agreement compared with those calculated by a formula given by ARMSTRONG et al.“’ The dipole Stark widths for Be(IV) agree within 30% with those of the more accurate calculations of Ref. (15). The total photoabsorption cross sections versus incident photon energies are illustrated in Figs. l-4. One can see that the Be(IV) lines are superimposed on the Be(II1) bound-free continuum. The free-free cross sections are also included but their cross sections are very small and not visible in the figures. 10'

7

1

t

I

,

I

I

I

,

1 100

BeIV

10“

N-0 2

D10-Z 10-3

1o-44j Fig. 1. Photoabsorption cross section versus incident photon energy of a Be plasma at p = 10” a/cm’ and kT - 21 eV.The arrowsindicatethe 1s photoionizationedges.

Spectral lines in hot dense matter 10*"77'j

10' : Be111 100 1

N-0

m D 10-l : 10-2:

10-3;

Fig. 2. As in Fig. I at p = 1020a/cm’.

Fig. 3. As in Fig. I at p = 102’a/cm”.

Y

10-j:

lo-4,

1o-54'

I

”

5

”

”

6

”

’

”

7

hw(a.u)

Fig. 4. As in Fii. I at p = Wa/cm3.

I "

8

86

B. F.

ROZSNYAI

102

n-2,

IIIa

n'=4,5...

I

100 N> T 10“

10-z

10

-3 4

6 ho(a.u)

Fig. 5. Photoabsorption cross section versus incident photon energy of a Ge plasma at p = lOI a/cm’ and kT = 144eV. The meaning of the marks are: la + 2p- 3s Ge(XXIII), Ib + 2p - 3s Ge(XXIX), 110+2p - 3d Ge(XXIII), Ilb +2p - 3d Ge(XXIV), Illa -2s -3p Ge(XXIII), IZlb 2s -3p Ge(XXIX).

lb

Fig. 6. As in Fig. 5 at p = lOtoa/cm’.

2p-3d -

n=z,

5

6

n*=4.5....

d 8

Fig. 7. Photoabsorption cross section of a germanium plasma at p = 102’a/cm’ and &‘I= 144eV. The lines are associated with the electronic configuration given in Table 5.

Spectral lines in hot dense matter

87

ho(a.u)

Fig. 8. As in Fig. 7 at p = 102’a/cm3.

(B) Germanium plasma The Ge plasma was selected to study transitions for medium 2 ions stripped to the L shell. The observed emission spectra of this plasma at low density (-10” a/cm3) were reported in Ref. (4). The selected temperature of the plasma is kT = 144eV. The electronic configurations and proabilities in parentheses at different matter densities are given in Table 5. The configurations and probabilities in the second column are those of the ion with “highest probability” (HP); as discussed in Section 2 and in the last column, the next ionization stage is indicated. If the probability of an ionic state is less than lo-*, then it is taken as zero. The probabilities are normalized to one. The photoabsorption cross sections at incident photon energies 40-80 a.u. are shown in Figs. 5-8. The spin-orbit separation of the lines is taken into account. For the sake of brevity, the author does not present tables for the germanium plasma but offers some explanations. At low density (1019a/cm3), the line positions are in agreement with the experimental observation reported in Ref. (4). At higher matter density, the lines show red shifts up to 68 eV at p = lo’* a/cm3. The n = 2 and n = 3 levels are not hydrogenic and, consequently, the broadening of lines connecting those states is dominated by the electron impact and quadrupole Stark effects. At low density, the broadening of lines for the n = 2n’ h 4 transitions is dominated by the dipole Stark effect. With increasing density, the relative magnitudes of the electron impact and quadrupole Stark contributions to the line widths increase and the upper levels gradually broaden into the continuum. 7. DISCUSSION The computational method presented in this paper provides a simple approximation to study the effects of density and temperature on spectral lines in a hot plasma. No attempt was made to resolve the different angular momentum states in the case of open-shell configurations. This can be accomplished in a straightforward manner by the use of Slater’s F integrals and it does not change the principal results of this study. The approximation of the line profiles by a Lorentzian shape at low densities may be crude. Work is underway to compute more accurate line profiles using convolution integrals.

Acknowledgement--The author is thankful to Dr. JONC. WEISHEIT~~T useful discussions during the course of this work. REFERENCES

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