Non-LTE opacity calculations with n − l splitting for radiative hydrodynamic codes

J. Quant Spectrosc. Radtot. Transfer Vol. 58, No. 4-6, pp. 791-802, 1997 0 1997 Published by Elsewer Science Ltd. All rights reserved

Pergamon

Pnnted m Great Britain

PII: SOO224073(97)000%8

0022~4073/97 $19.00 + 0.00

NON-LTE OPACITY CALCULATIONS WITH n - 1 SPLITTING FOR RADIATIVE HYDRODYNAMIC CODES A. MIRONE,

J.C. GAUTHIER,

F. GILLERON

and

C. CHENAIS-POPOVICS Laboratolre pour I’Utihsation des Lasers Intenses, UMR 100 du CNRS, Ecole Polytechnique, 9 1128 Palaiseau Cedex, France (Received I I February 1997)

Abstract-We present a new method for calculating opacities in a non-LTE plasma, based on modeling of the one-electron atomic potential in the framework of the “average atom” approach. In this model, the collisional-radiative equations of an average ion are solved and the resulting screened charges are used to reconstruct the one-electron atomic potential. Average atom wave functions, oscillator strengths and Slater integrals are calculated by quantum mechanics in the reconstructed potential. Dipolar matrix elements have improved values compared to hydrogenic formulas, especially for orbitals close to the ion core. Configuration accounting is done using the independent electron approximation and perturbation theory. Our method, compared to previous LTE average atom calculations gives realistic values of plasma opacities, especially in the soft x-ray range where the An = 0 transitions play an important role. A comparison of radiative hydrodynamics simulations performed with opacities calculated in the n and n - 1 representation shows significant differences. 0 1997 Published by Elsevier Science Ltd. All rights reserved

1. INTRODUCTION In the interaction of high power lasers with a high- or medium-Z solid target, an important role is played by the radiation heat wave (RHW) generated by the diffusion of radiation into the cold material.’ The radiative wave propagating into the target as well as the x-ray emission from the plasma blowoff represent a considerable fraction of the deposited laser energy. In the RHW, cooling by radiative heat conduction competes with heating by electron conduction and radiation transport from the critical density layer. Indeed, at times later than peak laser intensity, the target material may be subdivided into a dense, optically thick re-emission zone, behind the RHW front, and a hot low density conversion layer, optically thin for the x-rays. In the later region, microreversibility may be seriously broken and one has to take into account NonLocal-Thermodynamic-Equilibrium (non-LTE) emissivities and opacities to obtain acceptable predictions. In order to describe correctly this region, we must consider both radiative and collisional phenomena at once.2 The usual approach to solve efficiently-with little computer timethe collisional-radiative (CR) equations is within the framework of the hydrogenic “average atom” (AA) with screened charges.3-6 Such a model is numerically very convenient but, if applied to opacity modeling, fails to describe some important spectral features, especially the An = 0 transitions, which play a significant role in the soft x-ray range. We show in this work how to reconstruct, on the basis of the screened charges values, some atomic structure properties which are not taken into account in the original model. We assume in our calculations that:

1. T, e 7l e 7h where the 7’s stand for excitation, ionization and hydrodynamic characteristic time scales, respectively. For this reason, we restrict our considerations to the steady-state solution of the rate equations. Time-dependent AA calculations in the non-LTE regime are yet possible.7 2. Stimulated emission and absorption effects are neglected in the CR equations. However, the present model can be made potentially radiation dependent by weighting the radiative rates by photon escape probabilities.* 791

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The refinement of spectral properties is made after solving the collisional-radiative equations. This is consistent with assumption 2, because radiation field effects have been neglected. Assumption 1 has already been made by other authors.‘*” The effect of assumption 2 on the state of ionization of the plasma has been discussed at length by Rose.” A method has been recently proposed’2 to deal with the inclusion of the radiation field in non-LTE ionization models. Our code takes into account approximately the non-LTE physics of the laser-plasma corona and joins smoothly LTE in the dense part of the plasma where collisions with the thermal distribution of the free electrons play the main role. Assumptions 1 and 2 are the basis for a minimal non-LTE code. A more correct treatment of the problem would have been to perform timedependent ionization calculations and to couple self-consistently the non-LTE radiation field to the distribution of the average atom “ionic states”. This would require time-consuming calculations of the opacities and emissivities “on-line” in the hydrodynamic radiative hydrocode. This is clearly outside the scope of this paper which deals primarily with an improvement of the atomic physics “ingredients” (energies, oscillator strengths) of the average atom model. This is done without using I-splitted screening constants.13 The goal of our approach to the non-LTE plasma radiative properties is to provide non-computer-intensive calculations of the opacities and emissivities with reasonable accuracy for the position and strength of the transition features. In Section 2 we present the basics of the model and in Section 3 we present computational results. Conclusions are given in Section 4. 2. DESCRIPTION

OF THE

MODEL

2.1. From the hydrogenic average atom with screened charges to the reconstructed potential model The ions in a plasma can exhibit an enormous number of electronic excited levels. The simplest method to describe an ion of a given charge state is to consider mono-particle electron bound states. In this theoretical frame, each electronic configuration is specified by distributing the electrons over the bound states. For high- or medium-Z elements, even this simple approach gives a considerable number of different configurations. In order to further simplify the computational task, one can do the calculations for just one ion: the most representative one, the Average Ion (AI). The AI is defined, in analogy with mean field techniques in statistical mechanics, by the occupation probabilities of its bound states. The detailed configurations are taken into account, after calculation of the AI, by first-order perturbation theory. Neglecting orbital relaxation,14 the detailed configuration properties are written as simple functions of occupation numbers and AI Slater’s integrals. The AI occupation probabilities are calculated self-consistently with energies, wave functions and the external environment. Under LTE conditions, one needs just to know the temperature and the chemical potential. In our non-LTE calculation, we solve the CR equations for the screened charges of the ion. In this model, the electron energy depends only I5 on the principal quantum number n through the screened charge Qn defined by:

Qn= Z - CSnj2j2p, J

where Z is the nuclear charge and P, is the occupation probability of the nth shell. The factors are empirical screening constants.16 The screening by free electrons and the energy broadening into bands for the outermost electrons are also accounted in the complete treatment. Expressing electron energies and oscillator strengths by simple hydrogenic formulas, the screened-charge model can be applied to the resolution of the collisional radiative equations. Due to its analytical simplicity, the screened-charge model is numerically very convenient, but it is yet too simple to match some important radiative properties. In order to improve the opacity calculations, we propose a method to reconstruct the one-electron atomic potential. Our driving idea is that the electric field at the shell radius r, is given by: S,i

-g(rJ

=

(Qn - 4qhd3) r2 n

(2)

where pel is the free electron density. A potential V satisfying the above equation is easily found

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by a piecewise Thomas-Fermi method. The potential V is univocally determined by the above conditions and within each interval [rhrr + i] by the Poisson’s equation:

where FljZ is the Fermi-Dirac integral and pL,is the chemical potential in [rI,ri + ,] which is a function of i (we are under non-LTE conditions) and whose corresponding V(r) satisfies Equation (2). 2.2. Energies and wave functions in the reconstructed potential ion By solving the Schriidinger equation in the reconstructed potential, we can now get transition energies and wave functions. The degeneracy in the angular momentum number is removed because V departs from the Coulomb potential. Another important improvement concerns the dipolar matrix elements. The wave functions in the reconstructed potential are sensitive to the overall features of the potential while the hydrogenic formulas relies on a local approximation to a l/r potential. Table 1 shows how certain matrix elements are depending on the overall features of the potential. The table refers to a Iron plasma (Z = 26) at p = 10e3 g/cc and T = 20 eV corresponding to an AI having a charge Z*=7.4. The transition (n = 5,l = 4) -+ (n = 6,l = 5) is well described by hydrogenic formulas. The transition (n = 2,/ = 1) -+ (n = 3,l = 2) involve orbitals much closer to the ion core where the hydrogenic approximation is not valid. Figure 1 shows what happens: in the latter case, the difference between the dipolar element values depends on the behavior of the 2p functions in the 0 < r < 1 region. The hydrogenic model describes the orbital at r 2: 0.5 by a screened charge Qs = Qz while in this region, since we are close to the 3d maximum, we have Qs = Qs. 2.3. Occupation probabilities in the reconstructed potential ion

The reconstructed potential method gives energies and matrix elements but not the occupation probability for each nl “level”. Since our starting point model, the screened-charge AA model5 makes no distinction between levels having the same n, we face now the problem of assigning to each level nf its occupation probability. In order to solve the electron distribution problem without rejecting the screened-charge AA model, we need one additional assumption. We assume that the populations of the states of a given shell nl, are in thermodynamic equilibrium among themselves. This is a reasonable approximation for the plasma that we study, for which electronic collisions play the major role in redistributing the populations of the levels involved in the An = 0 transitions. Moreover, this assumption is coherent with our initial assumption 1; it has been discussed in previous papers2,17 and it has a similar range of validity than the quasisteady-state approximation.’ This hypothesis can be checked’* by verifying that the ratio of the radiative transition rate R to the collisional rate C: -R = 1.34 x

1013 T ;(rrnyl

C

(4)

e

is very much smaller than one for typical An = 0 transitions having a sufficiently low /iw. From the occupation probability P, and the energy E,, of the shell n in the screened-charge model, we can estimate” the chemical potential p,, associated to the shell n by: P, =

1 1 + e(G-&)lkT

Table 1. Comparison of dlpolar matrix elements from the hydrogenic model and the reconstructed potential model in Fe (p = 10S3g/cc, T = 20 eV). Transition (n = 5,1 = 4) + (n = 6,l = 5) (n = 2,1 = 1) + (n = 3,1 = 2)

(Ijl’lRI$)*Schradinger 11.64 0.028

($‘IR1$)2Hydrogenic formulae 11.7 0.019

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-r

2.0' dipole element I.5 E E s! 1 1.0 x ._ 2 0.5

0.0

1.5 r (au)

2.0

2.5

3.0

Fig. I, Comparison between wave functions and dipolar matrix elements from the reconstructed potential model and the hydrogenic model for the 2p -r 3d transition in Fe (p = 10m3g/cc,T = 20 eV). The wave functions, in the reconstructed potential, have larger extensions than the hydrogenic ones for large r.

where T is the electronic temperature. Providing that the ratio defined in Equation (4) is much smaller than one, the occupation probabilities in the reconstructed ion P,[ can now be calculated by replacing, in Equation (5), the energies I?, with the En, of the reconstructed model. This assumes that the chemical potential ,unrdefined by:

is equal to pnL,.This is approximately true when the energy separation of the 1 “levels” is much smaller than the temperature. This limitation, particularly stringent at low electron temperatures, can be relaxed by solving the collisional-radiative equations of the screened-charge AA model over all the nl configurations instead of the average n configurations. Work is in progress to implement such a model in our code. For the present time, we note that one of the benefits of this assumption is that in the limit of the LTE screened-charge ion we obtain an LTE reconstructed potential atom. This is a very important point because we want that the calculated opacities and emissivities converge smoothly to LTE values at high densities. 2.4. Configuration accounting A detailed configuration is defined by a set of integer occupation numbers {nJ, one number for each shell nJj. The configurations having the highest probabilities are the ones whose occupation numbers are closest to the AI ones, i.e. 2 x (21 + 1) x P,i. The calculation of configuration energies is estimated by using perturbation theory by means of a Taylor series expansion of the total energy to second order in the shell populations.*’ The ion shell energy is given by:

where the indexes run over the shells nl, ni and 61 are the occupation numbers of the shell j in the detailed configuration and in the AI respectively, Kj is the kinetic energy, & is the electronnucleus interaction and Eij describes the electrostatic interactions between shells.

Non-LTE

The calculation of the 2 x (2Zi+ 1) states can be obtained by inter-shell interaction,

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calculations

195

Ev is made assuming that the ni electrons are uniformly distributed over of the shell nf. The Eii are a simple combination of Slater’s integrals that spherical harmonics expansion of the inter-electronic potential. For the we get:

(where k is the order in the expansion) while for i = j we get, taking into account the Pauli principle for initial and final configurations: (9) We write the configuration probability P({ni}) as the product of the usual binomial formula for independent electrons, times a correlation term: P((%l) =

Pb x PC

(10)

The binomial factor is Pb =

np:l(l -f,,)d,-“,

(11)

J

where the index j runs over the shells nl, pj is the occupation probability of the shell j in the AI and 4 is the shell degeneracy 2 x (2/,+ 1). While the binomial distribution is valid within the independent electrons approximation both at LTE and at non-LTE conditions, we do not have a general formula to describe correlation effects at non-LTE conditions. We decide anyway to adopt the LTE correlation formula:2’ P, = exp(-AE/kT); where T is the electronic temperature

(12)

and AE is:

AE = cEu(nJ

- l?;)(n, - iii).

(13)

The inclusion of correlation effects in the occupation probability calculation is very important. Neglecting the correlation effects means larger spectral features and a wider charge-state distribution. The choice of an LTE correlation factor for a non-LTE code is questionable, but it has the advantage to give the right emissivities and opacities when plasma conditions approach LTE. Moreover, non-LTE conditions occur usually in the higher temperature zones, where the exponent of the correlation term gets smaller.

2.5. Line broadening The line broadening is calculated as the sum of three contributions: Doppler broadening, electron impact broadening and Stark broadening. The Doppler broadening for the plasmas that we usually consider is small compared to the latter two contributions and it is calculated assuming equality between the ion and electron temperatures. The electron impact broadening is modeled, following Refs22~23,by a Lorentzian whose width Wimp,, is: 3.2 x Winqxm

=

1o-22

n4k&m-31dTkW Z2ao

where pet is the electron density and a0 the Bohr radius. The ion Stark broadening is calculated by the usual simple formula22724 multiplied by a factor F(T) to take approximately into account strong correlation conditions:

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(32 - I(1 + I)) WSmrk

=

W-1

R2

(15)

where 1 is the angular quantum number, R is the ionic radius and F is the cutoff function for strongly correlated plasmas:

where P is the correlation parameter. 2.6. Emissivity The total plasma opacity is the difference between absorption and stimulated emission. For an LTE plasma it is easy to calculate the stimulated emission contribution. Applying the detailed balance relation, we get for the total opacity: k’ = k( 1 - exp(-ho/kT

))

This formula is not applicable to non-LTE situations for bound-bound the following formula for the contribution of a transition i --f:

kJ+,f= ki-+f 1 (

(17) transitions. We apply

(4 - 4 + l)(n/ + 1)Pb n&$ - nr )P, )

where P, and Pb are the probabilities of the initial and final configurations, d and n are the degeneracy and number of electrons of the transition shells. In order to calculate the right emissivity, particular care must be taken when treating the line broadening. The perturbers (free electrons and ions) are usually at LTE. The wings of transition lines must be multiplied, when calculating emissivity, by exp(-AE/kr) where T is the electronic temperature. This is quite important when calculating emissivity: neglecting such a factor would give unrealistically high contributions to the emissivity from the upper wings of transition lines.

3. RESULTS

3.1. Opacity and emissivity calculations In Fig. 2, we show opacity calculations for a Germanium plasma at T = 50 eV (Fig. 2a) and at T = 100 eV (Fig. 2b) and p = 10M2g/cm3,obtained with our reconstructed-potential model at LTE. Our results are compared with STA2’ calculations using the same plasma parameters. The agreement is satisfactory for the position of the inter-shell radiative features and for the absolute value of the opacities. However, the apparent line broadening is clearly underestimated in our model. This is not only due to the very crude estimate of the line profile used in our code but to the very limited number of configurations used, by construction, in our nl shell description. Our results at T = 100 eV have been also compared with more sophisticated codes like OPAL.26 Results have been presented as case “Ge3” in the last Work’Op opacity code comparison workshop.27,28 The agreement is good especially for the M shell (n = 3 --) n = 4) features around 350 eV. Spectral opacities differ slightly for the width of the An = 0 transitions for which one could obtain a better description by using detailed term structure (DTA)26 and for the n = 2 -+ n = 3 features around 1200 eV for which one could obtain more accurate results by including orbital relaxation.14 In Fig. 3, we show non-LTE opacity calculations for Germanium at p = 5 x 10m3g/cm3 (Fig. 3a) and p = 5 x 104g/cm3 (Fig. 3b) and T = 100 eV. Results are shown for the two models: the screened-charge AI model and the reconstructed-potential model. Spectral opacities have a similar shape in both models because, as noted before, we use the results of the screenedcharge AI model as an input to the shell occupation probabilities in the reconstructed-potential model. The most noticeable differences are the appearance of An = 0 transitions in the 3090 eV range, which are taken into account in our model contrary to the hydrogenic AI model, and the different oscillator strengths around 350 eV. The later difference arises from the

191

Non-LTE opacity calculations

a) ----

STA model this model

L shell

lo5 3 L-z 6 s 0 g lo4

lo3 1

I

I

1

500

1000

1500

2000

photon energy (eV)

-

500

this model

1000

1500

2000

photon energy (eV) Fig. 2. (a) LTE opacity of Germanium at T = 50 eV, p = 10-2g/cc. (b) LTE opacity of Germanium at T = lOOeV, p = IO-*g/cc. The solid line is the result of our model. The dashed line is the result of STA calculations by A. Bar Shalom.

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et al

lo*

lo* lo3 photonenergy(eV) Fig.

3. Non-LTE opacity of Germamum Solid line, reconstructed-potential

10' 10'

104

lo* lo3 photonenergy(N)

lo4

at T = 100 eV. (a) p = 5 x 10-3g/cc. (b) p = 5 x 104g/cc. AI model; dotted line, hydrogenic AI model.

improved treatment of the wave functions in our model (see Section 2.2). The shift towards lower photon energies of the M-shell absorption at p = 5 x 10” (Z* = 16.2) with respect to the case p = 5 x lop3 (Z* = 17.3) is due to the lower average charge. At T = 100 eV, one could expect the maximum contribution to the emissivity to be localized around ho x 3kT and the emissivity at lower photon energy to be small. However, it turns out that a non negligible contribution to non-LTE emissivity comes from the X-UV range (ho < 100 eV) as we can see in Fig. 4. This corresponds to An = 0 transitions with n = 3 and n = 4. Line broadening is sufficiently strong to merge all relevant transitions into a broad, single peak. We note that the difference between the screened-charge AI model and the reconstructedpotential model is again larger for lower material densities around /io x 300 eV because, as a result of non-LTE effects, the n = 3 shell is more populated.

‘3i

1o'O

IO’O

c .E ;

.E 3

r;

2. 0

s .z? .L i2 .E

109

2

.-3

E

lo8

1077 0

log

lo8

10'

200

400

600

600

photonenergy(eV) Fig. 4. Non-LTE emissivity of Germanium Solid line, reconstructed-potential

1000

,066

0

200

400

600

600

photonenergy(eV) at 7’ = 100 eV. (a) p = 5 x IOm3g/cc. (b) p = 5 x 10Ag/cc. AI model: dotted line, hydrogenic AI model.

1000

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3.2. Hydrodynamic simulations We show in Figs 5-7 the results of radiative hydrodynamics simulations performed with the diffusion approximation, multi-group code MULTI.29 This code uses a LTE equation of state derived from the Sesame tables.30 The spectral range has been subdivided into 80 spectral groups chosen to take into account properly the radiation transport properties in the different energy ranges. Group energy bins were closely spaced in the sub-kilovolt range and more widely spaced at higher energies. Figures 5 and 6 refer to a Germanium target 3 pm thick irradiated by a 500 ps Gaussian laser pulse at 1 = 0.53 pm. Figure 7 refers to a Molybdenum target 1 pm thick irradiated by a Gaussian Planckian radiation drive with a 60 eV characteristic temperature. We show in Fig. 5 the hydrodynamic effects of the radiation losses under non-LTE conditions. The curves show the spatial profile of the electron temperature at the maximum (t = 600 ps) of the laser pulse with an intensity of 1013W/cm2. Non-LTE and LTE opacities and emissivities have been used for comparison. The non-LTE simulation show a hotter corona, a less pronounced radiative heat wave and more preheating in the target bulk than the LTE one. A comparison between the results of simulations performed with hydrogenic and reconstructed-potential non-LTE opacities and emissivities is shown in Fig. 6. We have plotted the electron temperature at the maximum of the laser pulse (I = 2 x 10’2W/cm2)) as a function of position. The reconstructed-potential model predicts a colder corona due to the higher emissivity in the soft x-ray range and, as a result, a radiative heat wave penetrating deeper into the target due to the different opacities in the two models in the high energy photon range and a hotter reemission zone due to the higher opacity in the soft x-ray range (see Fig. 3). Finally, we have performed a simulation of the interaction of a Planckian radiation drive, with a black-body radiation temperature of 60 eV, on a Molybdenum target of 1 urn thickness. Results are shown in Fig. 7 where we have plotted the density and the temperature at the maximum of the radiation drive pulse at t = 500 ps. Due to lower soft x-ray opacities obtained with the screened-charge AI model, the penetration depth of the 60 eV radiation is larger, when compared with the results of the reconstructed-potential model. Accordingly, the decompression of

4 3

t.

2-

10 8-

non-LTE . .. . . . . LTE

5432-

1 j& 0.001

, , , , , , ,,, 0.01

, , , ( , , ,,,

, , , , , , ((,

0.1

1

-4

position (vm) Fig. 5. Electron temperature in Germanium at the maximum of the drive as a function of position for LTE and non-LTE conditions. The reconstructed-potential opacities and emissivities have been used in both simulations. Laser drive I = lO”W/cm*, r = 500 ps, coming from the left.

800

A. Mirone et al 11,

I

I

“““(“‘.

, *.-*

160”

,.** .i

,

,

,,,,,

,

,

,,,,,,

,

,

r

-..*

: :: -_-- hydrogenic model mconst. potential model

60 40 20 1 0.001

0.01

0.1

_.

1

position (pm) Fig. 6. Electron temperature in Germanium at the maximum of the drive as a function of positron for non-LTE conditions. The hydrogenic model (dotted line) and the reconstructed-potential model (solid line) are compared. Laser drive I = 2 x 10”W/cm2, r = 500 ps, coming from the left.

10

60

1

60

0.1 .._. -

% 8

hydrogenic modef reconst. potential model

0.01

0.001 0.6

0.8

1.0

WN Fig. 7. Electron temperature (left scale) and density (right scale) in Molybdenum at the maximum of the drive as a function of position for non-LTE conditions. The hydrogenic model (dotted line) and the reconstructed-potential model (solid line) are compared. Radiation drive I”, = 60 eV, r = 500 ps, coming from the left.

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the dense target region occurs over a larger depth. These differences underline the urgent need for experimental guidance in this area. Work is in progress at Lawrence Livermore National Laboratory,31 Max-Planck Institute for Quantum Optics3* and Laboratoire d’utilisation des Lasers Intenses to address this particular issue. 4. CONCLUSION We have presented a new model for calculating opacities and emissivities in a non-LTE plasma, based on the modeling of the Average Ion radiative and collisional properties. In the proposed model, the degeneracy in the angular momentum quantum number is removed by reconstructing the one-electron potential from the screened charges of the Average Ion. Dipolar matrix elements and Slater integrals are calculated from the Average Ion wave functions and have improved value with respect to hydrogenic formulas, especially for the orbitals close to the ion core. The correct treatment of An = 0 transitions and the improved values of oscillator strengths allows to generate opacity tables which, compared to hydrogenic model calculations, constitute a more reliable basis for simulations of soft x-ray radiative transfer. Acknowledgements-We would like to thank J. Meyer-ter-Vehn, Klaus Eidmann, and S. Hiiller for then help in the use of the MULTI code. One of us (JCG) is deeply indebted to A. Bar Shalom for kindly providing STA calculations. This work has been partly supported by the Centre National de la Recherche Scientifique and by the EC/HCM contract CHGECT930046 (Access to Large Facilities).

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4. More, R. M., Physics of Dense Plasmas, Report UCRL-84991, Lawrence Livermore National Laboratory, unpublished (198 1). 5. Chenais-Popovics, C., Gauthier, J. C., Puech, F., Renaudin, P. and Visconti, L., Int. Syrnp. on Optical Applied Science and Engineering. Application of laser plasma radiation, S.P.I.E., 1993, 2015, 220. 6. Shi-Chang, L., Guo-Xing, H. and Ze-Qing, W., JQSRT, 1995, 54, 257. 7. Bayle, S., Ph.D. Thesis, Orsay University, unpublished (1991). 8. Sobolev, V. V., Sov. Astron. Astrophys. J., 1957, 1, 678. 9. Busquet, M., Raucourt, J. P. and Gauthier, J. C., JQSRT, 1995, 54, 81. 10. Itoh, M., Yabe, T. and Kiyokawa, S., Phys. Rev. A., 1987, 35, 233. 11. Rose, S. J., JQSRT, 1995, 54, 333. 12. Busquet, M., Phys. Fluids, 1993, B5, 4191. 13. Perrot, F., Physica Scripta, 1989, 39, 332. 14. Rose, S., JQSRT, 1995, 53, 39. 15. Zimmerman, G. B. and More, R. M., JQSRT, 1980, 23, 517. 16. More, R. M., JQSRT, 1982, 27, 345. 17. Gauthier, J.-C., Geindre, J.-P., Grandjouan, N. and Virmont, J., J. Phys. D: Appl. Phys., 1983, 15, 32. 18. Busquet, M., C.E.A. Limeil-Valenton, private communication (1996). 19. Takabe, H. and Nishikawa, T., JQSRT, 1994, 51, 379. 20. Liberrnan, D. A., in 6th APS Topical Conference on Atomic Processes in high temperature plasmas, Santa Fe (1987). 21. Faussurier, G., Ph.D. Thesis, Ecole Polytechnique, unpublished (1996). 22. Nardi, E. and Zinnamon, Z., Phys. Rev., 1979, 20, 1197. 23. Rickert, A., Ph.D. Thesis, Max-Planck Institute fiir Quantenoptik, Garching, MPQ 175 Internal Report (1993). 24. Rickert, A. and Meyer-Ter-Vehn, J., Laser and Particle Beams, 1990, 8, 715. 25. Bar-Shalom, A., Oreg, J. and Goldstein, W. H., JQSRT, 1994, 51, 27. 26. Iglesias, C. A., Rogers, F. J. and Wilson, B. G., ApJ, 1992, 397, 717. 27. Rickert, A., Eidmann, K., Meyer-ter-Vehn, J., Serduke, F. J. D. and Iglesias, C., WorkOp-111:94, MaxPlanck Institute fiir Quantenoptik, Garching, MPQ 204 Internal Report (1995). 28. Rickert, A., JQSRT, 1995, 54, 325. 29. Ramis, R., Schmalz, R. F. and Meyer-ter-Vehn, J., Comput. Phys. Commun., 1988, 49, 475. 30. Holian, K. S., Los Alamos National Laboratory Report No. UCID-18754-82-2, Los Alamos, unpublished (1982). 31. Bauer, J. D., Back, C. A., Castor, J. I., Dykema, P. G., Hammel, B. A., Lee, R. W. and Nash, J. K., Phys. Rev. E, 1995, 52, 6736.

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32. Merdji, H., Eidmann, K., Chenais-Popovics, C., Winhart, G., Gauthier, J. C., Mirone, A. and Iglesias, C. A., JQSRT, 1997, 58, 773. 33. Merdji, H., MiDala, T., Gilleron, F., Chenais-Popovics, C., Gauthier, J. C., Renandin, P., Gary, S. and Bruneau, J., JQSRT, 1997, 58, 783.