Electronic energy-levels in dense plasmas

J. Quant. Spectrosc. Radiat. Transfer VoL 27, No, 3, pp. 345-357. 1982

0022-4073/82/030345-135030010

Printed in Great Britain.

Pergamon Press Ltd,

ELECTRONIC ENERGY-LEVELS IN DENSE PLASMAS+ RICHARD M. MORE University of California, Lawrence Livermore National Laboratory, Livermore, CA 94550. U.S.A.

(Received 22 April 1981) Abstract--Modern inertial-confinement fusion experiments subject matter to extreme physical conditions previously studied only in theoretical astrophysics. At very high plasma density, atomic energy states are significantly altered by electric fields of neighboring ions and by free electrons; the resulting phenomena of pressure ionization and continuum lowering may be analyzed with a sequence of models, each adding new subtleties to a complex picture. In this pap-r, we develop a simple parameterization of pressure ionization, discuss limitations of the Debye-Huckel model for plasma perturbations, and survey an approximate description of X-ray spectra based on the WKB approximation. WKB theory leads to a simple derivation of the screened hydrogenic model for plasma ionization and radiative properties. Electron eigenvalues are obtained from the total ion energy in agreement with Koopman's theorem, and the representation of spectral terms is improved by a new set of screening coefficients.

In recent years, inertial-confinement fusion research has stimulated increased interest in the physical properties of dense plasmas. Experiments with large pulsed lasers produce plasmas having densities in a range 0.01-20 g/cm 3 with temperatures as high as I keV. It is difficult to diagnose these plasmas because they are neither homogeneous nor static; however, modern diagnostic technology provides spatial resolution on the micron scale and time resolution in the picosecond range, and it is now possible to obtain space-and/or time-resolved x-ray line spectraJ The existence of this data and the importance of atomic processes in target dynamics appear to justify improvement of the fundamental theory of atomic structure in dense plasmas. In this paper, we address two related topics: first, pressure ionization, probably the main atomic phenomenon unique to dense plasmas, and second, energy-levels of complex ions in high-temperature plasmas. In both cases, we are seeking the simplest description that brings out the essential physics. 1. PRESSURE IONIZATION IN DENSE PLASMAS

It is well-known that highly-excited electronic states are easily perturbed by the influence of neighbor atoms. The hydrogenic partition function is a sum over the principal quantum number, i.e. {13.6 eV Z2] O~.=,~2n 2exp+k n2kT }. This sum diverges unless it is cut off or truncated at a finite quantum number. The physical justification for this truncation is that high-lying states have large orbit radii and consequently suffer large perturbations in the plasma enviroment. The device of limiting the summation to a finite number of terms is only a qualitative approach. In a dense plasma, the low-energy portion of the continuum does not have the free-electron form g(E) ~ E ~/2 and the highest discrete states are broadened in a complicated way (Section 2). Nevertheless, it is useful to develop a criterion to approximately locate the continuum boundary. The following three lengths are relevant to a discussion of pressure ionization: r, = orbit radius = aon2/Q,, Ro = ion-sphere radius = (3/4 "/7"/11)I/3, D ==Debye screening length = k/ k T/41rZ*2nte 2. +Work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. 345

346

RICHARDM. MORE

The orbit radius is determined by a Bohr formula containing an effective charge Q,. For inner-shell (core) electrons, Q, is determined by methods discussed in Section 3; for unoccupied excited states, Q, is essentially the ion charge state Z*. The radius re is a shell-averaged orbit radius (the l = 0 or l = 1 states extend to larger distances from the ion core); however, for qualitative purposes the Bohr formula is sufficiently accurate. The ion-sphere radius Ro is determined by the ion density nt in such a way that the average distance to the nearest neighbor ion is approximately 2 Ro. The Debye screening length D includes (only) ion screening, a choice appropriate to the case Z ,> 1. For partially ionized plasmas, the ion charge state is used in the definition of D. Two main criteria for pressure ionization are found in the literature. These criteria are obtained by comparing r, to Ro,2-4 viz.

n/-- 1.6×1024cm3\n 2)

(i)

and, by comparing rn to D, L8

n, ,.97x,O23cm-'

kT

(2)

The references are merely representative of recent papers using the two criteria; a considerable scholarly effort would be necessary to identify the earliest proponent of each viewpoint. Brush and Armstrong 9 quote almost 200 papers on pressure ionization and continuum lowering. In this paper, we compare the merits of Eqs. (i) and (2). Do the two criteria actually differ enough to warrant discussion? Figure 1 compares the criteria for aluminum at high temperature (T = 1 keV) where the LTE plasma is expected to be " fully ionized. The lines plotted give the highest allowed level (r/max) according to Eqs. (1) and (2). States above the upper line (r, = D) are pressure ionized according to either criterion; states below the lower line (r, = R0) may be assumed to remain discrete. The criteria disagree for states whose radii obey R0< r, < D ; at nt ~ 10~Scm 3, this is indeed a large number of quantum states (6 × 103). Only the criterion r, = R0 is relevant above a d e n s i t y - 8 × 1020 ions/cm -3 because the Debye-Huckel theory is completely untenable when D _-
5OO

I

I

I

I

2OO

Aluminum Z = 13

== ~ e v ) $ 1oo _E

5o

g

10

:

norit(lO.6 #)

1017 cm 3

1018

ncm(1.06 .u)

1019

1020

1021

1022 cm-3

Ion density n I

Fig. I. Pressureionizationfor aluminumplasma at temperature T = I keV. The lines correspondto Eqs. (I) and (2) of the text, and indicate the maximum allowed quantum number n as a function of ion number density nl.

Electronicenergy-levelsin denseplasmas

347

(1) Pressure ionization occurs because corresponding electron wavefunctions on adjacent atoms overlap, and this leads to a resonance broadening of atomic states into energy-bands. Overlap occurs when r. ~ Ro.

(3a)

(2) Pressure ionization occurs when the binding energy of an electron is exceeded by the (negative) electrostatic potential of a neighbor ion (evaluated at a distance half-way to the neighbor):

,2 _ 13.6 eV (Z,) n"

Z , e2 R0 '

(3b)

The r.h.s, of this equation is interpreted as the bottom of the continuous spectrum. For outer electrons, r, = aon2/Z * and Eq. (3b) is mathematically equivalent to Eq. (3a). (3) Pressure ionization can also occur when enough free electrons are compressed inside the orbit (radius = r~) to reduce its binding significantly. Assuming that Z* free electrons are uniformly distributed over the ion volume, the fractional number inside the orbit r~ is approximately ~Zfree ~

Z*(rn/go) 3.

(3C)

If 8Ztree-~Z*, the free electrons completely screen the ion core charge and there is no attraction or binding. If r, -~ Ro, this criterion is also automatically satisfied. As the plasma density is increased (e.g., at constant temperature), R0 decreases and successively deeper core states are pressure-ionized. Even the K-shell of low-Z atoms is liberated in the extreme case of white-dwarf interiors (p_-> 106g/cm3). For partially-ionized plasmas, Eq. (3) is also temperature-dependent through the ionization state Z*. If the temperature is raised at fixed ion density, Z* rises and r, decreases (especially for excited states) and the total number of allowed bound-states will increase. The alternative criterion of Eq. (2) could only be used in low-density plasmas where the Debye-Huckel model is valid (i.e., R0 '~ D). Equations (1) and (2) disagree for large orbits whose radii obey Ro ~ r. ~ D. These states are frequently studied by solving the Schroedinger equation for a Debye-Huckei potential.5-8 We find this model to be questionable, since a boundstate whose orbit obeys the inequality Ro ~ r, ~ D encircles many ions of the plasma. The Rydberg atorn~with such large orbits are very delicate and in a plasma they are subject to strong physical perturbations. The energy-levels are displaced by Debye screening of the ion potential. For small r,., the calculated shift is6 AE=+

Z *e2 e2ao,~ D -~pn'-l(l+l)]+...

(4)

Equation (4) is derived by treating the Debye screening as a perturbation to a hydrogenic eigenstate. By itself, Eq. (4) predicts a small red-shift of spectral lines. The electron energy and effective mass are changed by forward scattering by plasma particles) l This effect is not included in Eq. (4). Non-forward scattering by plasma particles leads to collisional destruction (lifetime) of the excited state. We can estimate this effect by comparing the orbit circumference 2~-r, to the usual electron mean free path L for elastic scattering evaluated at the Bohr velocity v, = Z* e2/hn:

2zrr._ 2~rr~ [47rZ*2e'n, l n A ) 67r[ro~ 3 L k ~ = \ E l lnA.

(5)

For electron states having R0 '~ r. '~ D, we find 2~rr./L ~> 1. The electrons are scattered many times be[ore they complete one orbit and it is therefore unphysical to calculate quantized energies in the Debye-Huckel potential.

348

RK'HAROM. MORE

In addition to this scattering, there is inhomogeneous line-broadening associated with thermal fluctuations of the adjacent plasma, leading to the well-known Inglis-Teller limit. ~2This effect causes bound states to hybridize when

E,+I - En ~ r,(Ze2/Ro2).

(6)

This condition is satisfied whenever r, _->Ro. It is easy to understand why states having r, > Ro are so strongly perturbed. For any instantaneous configuration of ions, the field at a fixed distance r from a given charge has the form E(r) = Zr--~er + ~ ~ Ze

(r - R,) + Ee(r),

where Ee is produced by electrons. The time average over positions of the electrons and other ions is the screened field obtained from the Debye-Huckel potential. For Prl ~ no, the average field is much less than the instantaneous field. Thus, the fluctuations of E(r) are much larger than the average value, and it is inappropriate to solve Schroedinger's equation considering only the small average field (i.e., Debye-Huckel potential). This problem is especially severe for weakly bound states near the continuum limit. For these reasons, we conclude that discrete eigenstates having R0"~ r, ¢ D do not exist in a plasma. The electronic energy spectrum near the continuum boundary is complicated because the electrons interact simultaneously with many atoms. Treatment of these states based on the Debye-Huckel potential, in particular Eq. (4), appears to be unjustified. Similar conclusions concerning Eq. (4) are reached by Nakayama and DeWitt ~3and Peacock. 7 These remarks refer to low-density plasma (Ro ~ D); at higher density one cannot use the Debye-Huckel potential in any case. 2. ELECTRON SPECTRUM NEAR THE CONTINUUM The description of pressure ionization by a simple criterion, such as Eqs. (1) or (3), merely serves to locate the density at which any particular bound state is strongly modified by plasma perturbations. In this section, we quote results from the literature of random solid-state systems in order to point out qualitative characteristics of electron states near the continuum in a dense plasma. ~4'~5 We will consider a sequence of models of increasing complexity. It is probably not necessary to perform numerical calculations of such difficult models in order to obtain an adequate description of inertial-fusion experiments, but the discussion may help to clarify conceptual questions about electron energy-levels in disordered systems. The first category of models employ .the central-field approximation: the ion is represented by a point charge fixed at the center of a spherical cavity in an otherwise uniform positive charge distribution. The electrons assume a self-consistent distribution n(r) in a potential V(r) determined by the total charge density. Thomas-Fermi theories yield calculations of n(r) by a semiclassical approximation; in a more elaborate central field model (Liberman~6), the electron density n(r) is obtained by summing thermally occupied solutions of the Dirac equation for the potential V(r). The central field models yield the result that deeply bound core states are discrete, but higher states move into the continuum as scattering resonances. The continuum density of states is calculated as

g(E) = cE ~/2+ 2 ~, (2 + 1)d~ rr t=o

dE'

where c = l/2zr 2 (2m/hZ) 3n, and 6~ is the scattering phase-shift for the /th partial wave. The density of states g(E) has a sequence of resonance peaks which correlate with the original bound states. As the spherical cavity is compressed (corresponding to higher densities), both bound states and resonances move to higher energies (relative to the continuum edge defined in

Electronicenergy-levelsin dense plasmas

349

terms of the asymptotic potential V ( r ~ ) ) . A resonance near the continuum boundary is relatively narrow (especially for large angular momentum l); higher resonances are broad and ultimately become indistinguishable from the continuum. In the central field model, pressure ionization is represented by a smooth change of a discrete boundstate into a narrow resonance. However, this is only the simplest model. A second category of theoretical model consists of a small cluster of nearby atoms in the plasma background. This case may be treated by multiple-scattering theory; one result is that the N-atom cluster has N distinct eigenvalues for each eigenstate of the single-center problem. ~s~7 The splittings depend upon the geometry of the cluster (especially upon the ion separations). It is possible to separate the additional contribution associated with N-atom clusters from that given by smaller clusters. This separation follows from the multiple-scattering series and yields a high-density cluster expansion for the density of states. It can be shown that coarse-grained features of the electronic density of states can be calculated from relatively small clusters, j7 A coarse-grained average density of states is defined as

g(E+i~):~ i

g(E')dE' (E - E') 2+ 712"

The integral on the right, a Lorentzian average of g(E') for energies near E, is equivalent to analytic continuation of g(E) to complex energies E + irr Small clusters strongly dominate g(E + i~) for large values of rt, while the spectrum for small values of rt is sensitive to larger clusters. In the limit of large clusters (N ~ ~) required for 77-+0, one obtains an irregular or fluctuation spectrum unless the potentials produced by the ions have special properties such as spatial (crystalline) order. As a third category of theoretical models, we can consider an infinite system of ion core potentials (still held at fixed positions). If these are arranged in a periodic crystalline array, one finds the usual energy bands, with one band generated by each atomic state. Bands arising from excited states with r, > Ro overlap and hybridize; the resulting energy-bands have sharply defined band-edges where g(E) ~ ( E - Eo). I/2 A disordered infinite system of atoms may have either positional disorder (as in a plasma or fluid phase) or compositional disorder (as occurs when the atoms are not in identical ionization states). There are two general approaches for disordered systems. A self-energy method such as the coherent-potential approximation ~8 is most appropriate when the fluctuations of the poten' tial are small compared to an electron's kinetic energy. In this case one obtains energy bands with a complex wave-number representing a finite mean-free path. There are sharp band edges, and normally the mean free path is very small near the band edge. There exist a limited number of calculations performed by "exact" methods, especially for one-dimensional disordered potentials. 14'~5"~9-2~These methods yield a very different spectrum consisting of bands of propagating states surrounded by band-tail regions containing trapped or localized states attached to strong fluctuations of the local configuration (e.g., to a pair of unusually near neighbors). One-dimensional systems are anomalous in having no propagating states. This complicated structure (i.e., propagating band + localized band-tail) is repeated for each eigenstate of the parent atom, and again there is the possibility of hybridization of the band states. To describe a plasma one must add an additional important complication: the plasma potential fluctuates in time, both through the spatial motion of ions and through changes in their ionization states (the free electron charge density also fluctuates on a more rapid time-scale). These dynamical effects are well-known in low-density line-broadening theory. The usual line-broadening theory is based upon a second-order treatment of the plasma perturbation, and this is accurate only for the deeply bound states. For high densities, we expect the dynamical effects to reduce the complexity of the fluctuation spectrum of the infinite random system. In the dynamical problem, there are probably no truly localized states; there is only a temporary QSRT Vol. 27, No. 3-.-J

350

RICHARDM. MORE

trapping for the duration of the corresponding fluctuation. We do not have appropriate parameters to measure the relative importance of static and dynamic perturbations for states near the series limits. The remarks of this section should serve as a counterweight to the simplified discussion of Section 1. It is clear that dense plasma spectroscopy offers a rich ground for theoretical study, and a complete understanding of the continuum boundary could not be obtained without very prolonged research. For the case of deeply bound states, the plasma perturbations are small compared to electronic binding energies, and the usual theory of line broadening provides an adequate description. 3. WKB TREETMENT OF COMPLEX IONS We now shift our attention to the more practical problem of describing the X-ray spectrum associated with deeply bound (core) states. From the work of Sommerfeld,22 it is known that the semiclassical WKB theory is able to describe the main regularities of X-ray spectra. The WKB theory has practical value because it gives closed-form algebraic expressions for various quantities of interest. We will extend the theory to treat ions and show that it provides a justification for the widely used screened hydrogenic spectral model.22-26 The WKB energy-quantization condition is

nr+~ f) q(r)dr= Tr(')

nr=0,1,2 . . . .

(7)

where n, is the number of oscillations (nodes) in the radial wave-function related to the principal quantum number n by nr = n - l - 1. The wave-number q(r) is defined by

q(r)=( E+eV(r)h2/ m2

(1+;2)2)

(8)

q(r) vanishes at inner and outer turning points r~, r2; V(r) is the electrostatic potential. We may assume V(r) is available from a separate selfconsistent field calculation. Equation (7) determines electron energy levels E = E,.~ in the nonrelativistic WKB approximation. The radial wave-function R,j(r) is R.,l(r) = ~

C.t

(~-+ f'q(r')dr') rl

(9)

sin 4

The normalization C,.I of the wave-function is approximately determined by replacing the sin2 by I/2:

1=

1 JR.,(r)l z d r ~- -~ IC.tJ" Jr, q(r)

(10)

Averages of any function fir) are evaluated by a similar approximation. For hydrogen-like atoms with purely Coulomb potentials, these formulas can be evaluated by contour integration; the results are extended to the arbitrary ion by assuming that each electron experiences an effective charge Q,. For each radius r, a screened charge Z(r) is determined from the electrostatic potential by

(11)

V(r) = Z ( r ) e

r

Another effective charge

Q(r) is defined by the electric field at radius E(r) = - d-~ d V-= ~Q(r)e

r: (12)

351

Electronicenergy-levelsin denseplasmas 10

[

~

I

•

I

~

i

'

I --0

~o~O ~ ~ ° -

~-ls

Liberman SCF-~

~ 4

~:~""

2 ~ o /

L2p

~TFD-WKB

/

LAJ

0.1

0.2

0.4 0.6 Temperature(keV)

0.8

1.0

Fig. 2. A comparison of temperature-dependent eigenvalues for iron (Fe) at fixed density 7.59g/cm 3 and varying temperatures. The points are obtained from Liberman's relativistic self-consistent field theory, the curves from the theory of Section 3.

The relation of Z(r) and O(r) is 27

Q(r) = Z(r)- rZ'(r),

(13)

where Z' = dZ/dr. Q(r) is the nuclear charge minus inner screening associated with electrons at r' _-
Z(r). For the example given in Table 1, Z and Q differ by about 30%. The distinction between these effective charges is especially important in the calculation of radiative processesY For each electron shell defined by principal quantum number n, the charge Z(r) is expanded in a Taylor's series around an average orbit radius r,, giving V(r) = eZ'. + Q.e 4 6Z.(r)e r

(14)

r

Here Q. = Q(r.) and Z . ' = Z'(r.). The outer screening produces a constant potential eZ.'; the second term is a Coulomb potential including only inner screening. The residual non-Coulomb potential is described by 8Z~(r).

352

RICHARDM. MORE

In order to simplify the formulas, we represent the energy level E..t in terms of a wave-vector K, E,; = En° --

K2

(15)

Here, E. ° e2Zn ' is the potential produced by outer electrons. As a first approximation, we neglect 6Z~ and work with =

-

qo(r) = ( - K2-~ 2Q"

(l+1/2)2f/2

(16)

t/oF

ao = h2/me 2 is the Bohr radius. This approximation is most accurate for circular orbits (l = n - 1) which have a minimum extension in radius, qo(r) is zero at points r, °, r2°. The turning points are close together for l = n - 1, and it is reasonable to select r, as (rj°+ r2°)/2. This choice gives: n2

r. = a,, Q~-~

(17)

We next apply the quantization condition Eq. (7). The result can be simplified by defining the quantum defect A,t

A,,t=lf,.iZq(r')dr'-lflZ~'q°(r')dr

'

(18)

The integral involving qo(r) is /,r2o°q°(r)dr= ¢r(a0--~-(1 + 1/2))

(19)

Combining this with Eq. (18), we find the WKB approximation to the energy levels:

E.i = E. ° -

Q"" e2

2ao(n - A.i)2

(20)

where E, °= - e 2 Z n ' is an energy shift associated with outer screening, Q, = effective charge including only inner screening, and A I = quantum defect produced by interactions with electrons interior to r,; the quantum defect is zero if the potential is exactly hydrogenic. The WKB theory described here reproduces the main regularities of X-ray spectral terms. 22 For example, the term splitting of irregular or screening doublets is 1"i = E,~ - E,,,,_, which is proportional to (A.~-A.H). It is easily shown from Eq. (11) that A . t - n/Q. so that Tr ~ Q./n 2 which is correct. For the regular or spin doublets, TR = E..u_t+l/2 - E.,u=t_~/z. One can evaluate this spin-orbit splitting using a WKB matrix-element given in Table 2. The result is TR=2

n

(21)

which is again approximately correct. To match the experimental X-ray spectra, it is important that the inner-screened charge Q, appear in these splittings (as implied by the WKB model). -'2 In Table 1, the quantities Q,, Z,, E, ° and A,I are given for iron plasma at 7.59g/cm 2 and 200eV. The values are generated using the finite-temperature TFD theory for the atomic potential V(r). They are compared with energy-levels obtained from the relativistic selfconsistent field model developed by Liberman? ~ The energy-levels E~ are accurate to about 10%.

Electronic energy-levels in dense plasma,s

353

Table I. TFD/WKB energy-levels (iron, p = 7.59g/cm 3 and T = 200eV). The last column contains relativistic self-consistent field eigenvalues calculated by D. Liberman (Ref. 16). r State

Qn

Zn

En£ ' keV

i i i

keY

En£ j (Liberman)

r

is

22.97

I 25.52

i

2s

-7.086

i

i

-7.284

! 21.95

16.98

-0.896 i I i

2p

3S

-7.206

0.0063

17.193

i1.376

-0.144

0.0968

-1.071

-1.063

0.444

-0.976

-0.961, 0.948

i i 0.285

-0.244

3p

0.205

0.213

-0.206, 0.203

3d

0.069

0.167

0.154, 0.153

-0.238

In order to calculate physical averages using WKB wave-functions, it is necessary to evaluate various radial integrals. The required integrals are obtained by contour integration. For example, choosing a contour C which encloses the branch cut of q°(r) in a clockwise sense, we find f q°(r) dr Io(Z) = ~ r-Z .' c

2~-K. +

~, I,(~ =

. (r

-

2~'(l + 1/2) + 2~riq°(Z) Z

dr Z)q°(r)

2 rr 27ri K. ~ q°(Z)

-

= d~ dr 2~'i 5, • r(r - Z)q°(r) - Zq°(Z)

(22)

Table 2. Expectation values of the uantities listed in the first column. Exact nonrelativistie expectation

WKB Approximation

a0[

--

n2 _

(i

2Q n

r

values

+7 ~-)

atoms

a0 ~n 2 - 9.(9. + i)] ~-~

2 2 a0n [5 1 1 2Q2n n 2 - 3 (£+ 3)

2

for h y d r o g e n i c

2 2 a0~n [5n2 + 1 - 39.(9.+1] 2Q2

Qn 2 a0 n

a0 n

2

1 r2

2 Qn 2 a0

1 1 n 3 (9. + ~)

a 02

1 1 n3(9. + ~)

1

3 Qn a 03

1 1 3 n 3 (9. + ~)

3 a0

1 n39.(9. + ~) (9. + i)

4 Qn

1 2 3n2 - (9. + 2)

r

3

1

4

r4

-p 4 8m3c 2

a0

1

2n 5 (9. + ~)

4Q_

5

4 a0

(~2Qne42 /

I!__3_e__~ __3 2a0n4

k4

1

8

o/

3n 2 _ £( £ + ii 2 n5 (9.+~)(9.+i) (9.+~) 9.,9.

2e I 2aon4

£ +

f 2)

354

RICHARDM. MORE

Z is a complex variable with units of length. The integrals are connected by algebraic formulas s u c h as

I,(Z) - I,(0) = Z I2(Z). Useful results are obtained by expanding the integrals in Laurent series about Z = 0 and Z = ~; for example 2

dr

_

, r2q°(r)

~

2 dr

7r (/~Si/2)

l~c dr _ 7rQ

,

The first of these is proportional to /2(0) and the second is obtained from the second term ( - Z -2) in the Laurent series of I2(Z) about Z = ~. The ratio of these integrals gives the WKB expectation value of 1/r2. Other useful averages are given in Table 2 (many of these formulas are exact for hydrogen atoms).

4. SCREENED HYDROGENIC IONIZATION MODEL

The WKB theory gives a very clear justification for the screened hydrogenic ionization model developed by Mayer25 and Lokke and Grasberger 26 (see also Refs. 23, 24). This model provides a useful description of atomic ionization and emission properties which can be adapted to a more complete simulation of plasma dynamics. In the screened hydrogenic model, the atomic configuration is described by giving the populations P, of shells defined by principal quantum number n. The electron eigenvalues E, are calculated by a Bohr formula extended to include inner and outer screening corrections. The screening is calculated by a linear approximation using a fixed set of screening coefficients. The energy eigenvalue E, for electrons of the n th shell is taken to be E. = E."

Q"2e2 - 2aon 2

(23)

Subshell splittings E,.~- E,,r produced either by quantum defects h./ or by spin-orbit interactions are neglected; this approximation is acceptable for calculation of ionization but must be refined for calculation of an emission spectra. The orbit radius r, is taken to be aon2/Q,. The screened nuclear charge Q, depends on the electron populations P,, of shells inside radius r,. As a simple approximation, this dependence is represented by a linear formula: 1

Q. = Z - ~ o'(n. re)P.. - ~ o ( n . n)P.

(24)

rn
P, =0,1,2 ..... 2n 2 is the population of the nth shell and Z is the nuclear charge. The coefficient o-(n,m) describes screening of the nth shell by the mth. A satisfactory approximation for the outer screening is obtained by assuming

o 1 e2 E. = ~ 7. o'(n, n)P. + ~ r~ n

e2P,.

o-(m, n).

(25)

This form is justified by noting that eP,,/r,, is the potential inside a spherical shell of charge eP,, having radius r,,. The screening coefficients o-(n, m) describe the spatial extent of the charge distribution; i.e., the n = 3 shell is not entirely inside the n = 4 shell.

Electronicenergy-levelsin dense plasmas

355

The total ion energy E~o. is not simply the sum of one-electron eigenvalues E~, because that sum counts the electron-electron interactions twice. Instead, we have

Eios=2 U e ~ = - ~1

EnP n -

(26)

Uee

P. < ~ . l e V ( r ) - ZrJ ~ " >

(27)

The electron-electron interaction energy Uee is readily evaluated by using formulas from the WKB model (Table 2), with the result

[ - Q.: e2~ Eion = ,~ \ 2aon 2] P,.

(28)

This equation counts the interaction of each shell with the nucleus and with other electrons inside r,. From this equation one can establish the consistency condition

cggion

(29)

Qn2e2 t- E. ° = E.. OP, = - ----~" 2aon

This equation is the equivalent of Koopman's theorem for the model Hamiltonian defined by Eqs. (23-28). Equation (29) is important to the achievement of thermodynamic consistency when the theory is applied to high-temperature plasmas. 24 Any generalization of the screening model to include quantum defects must endeavor to preserve the consistency constraint expressed by Eq. (29). 5. SCREENING COEFFICIENTS The linear screening model of Eqs. (23, 24,25) predicts ionization potentials for all the elements in terms of the 10-by-10 matrix of screening constants. Mayer 25 used first-order perturbation theory applied to hydrogenic wave-functions to calculate a set screening constants 6~(n, i). These screening constants are sufficiently accurate for highly stripped ions, but less suitable for nearly neutral species. To test Mayer's screening constants, it is useful to compare the predicted ionization potentials with a large data-base generated by J. Scofield using the Hartree-Fock-Siater theory. 28 The data base includes 800 ionization potentials for 30 elements. The rms fractional deviation of the calculated ionization potentials from the Scofield data base exceeds 100%--an average error of h factor of two in ionization potential. The largest errors occur for the low ionization states of high-Z elements (see Table 3). Since the screening constants enter the theory as a fixed set of parameters we are able to improve the model by adjusting the screening constants. The formal structure of the average Table 3. Firstfew ionizationpotentialsof gold. Ionization Stage

Hartree-Fock-Slater theory (numbers provided by Dr. J. Scofield)

Screening Model New screening constants from Table 4 (eV)

Screening Model Original H. Mayer screening constant (Ref. 25)

9.2

13.4

69.8

20.5

23.0

82.6

32.3

33.1

95.7

46.2

43.6

109.3

61.1

54.6

123.4

(eV)

RICHARD M. MoRE

356

Table 4. Newscreening constants ~(n, m) 1

2

3

4

5

6

10

0.9999

0.9999

0.9990

0.9999

0.9999

0,9920

0.9999

0.3125

0.9380

0.9840

0.9954

0.9970

0.9Q70

0.9990

0.999

0.2345

0.6038

0.9040

0.9722

0.9979

0.9880

0.9900

0.1093

0.4018

0.6800

0.9155

0.9796

0.9820

0.9860

0.9900

0.0622

0.2430

0.5150

0.7100

0.9200

0.9600

0.9750

0.9830

0.9860

0.9900

0.0399

0.1597

0.3527

0.5880

0.7320

0.8300

0.9000

0.9500

0.9700

0.9800

0.7248

0.8300

0.9000

0.9500

0.9700

0.6098

0.7374

0.8300

0.9000

0.9500

0.0277

0.1098

0.2455

0.4267

0.5764

0°0204

0.0808

0.1811

0.3184

0.4592

0.0156

0.0624

0.1392

0.2457

0.3711

0.5062

0.6355

0.7441

0.8300

0.9000

0.0123

0.0493

0.1102

0.1948

0.2994

0.4222

0.5444

0.6558

0.7553

0.8300

0.0100

0.0400

0.0900

0.1584

0.2450

0.3492

0.4655

0.5760

0.6723

0.7612

atom model guarantees that this adjustment will not effect the thermodynamic consistency of the resulting theory. Using a least-squares optimization procedure, we have developed a new matrix of screening constants (Table 4). The rms deviation of the new ionization potentials from the Scofield data-base is reduced to 25%; this appears to be the best that can be done without adopting a more realistic description of sub-shell splittings. It has been verified that the new screening matrix does not predict negative ionization potentials for any element. The summary characteristics of the new screening constants are as follows: • They yield a factor of 4 reduction in average error of ionization potentials. • The new matrix retains the important symmetry property: IZcr(l, J) = J2cr(J, I). • The matrix elements are smooth monotone functions of I, J. • The largest alteration in any screening constant from the Mayer value is less than 10% Using the new screening coefficients, we find the following effective charges for Fe at p = 7.59 g / c m 3, T = 200 eV: QI = 25.7

Q2 = 21.9

Q3 = 16.7

Q4=

15.3.

These are similar to the TFD/WKB values of Table 1. As a final check on the new ionization coefficients, we compare the first few ionization potentials of gold in Table 3. The Mayer coefficients give ionization potentials which are much too large; in a low-temperature portion of a laser target this would lead to a serious underestimate of the degree of ionization. The new screening coefficients are considerably more satisfactory.

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357

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