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High Energy Density Physics 6 (2010) 311–317

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High Energy Density Physics

journal homepage: www.elsevier .com/locate /hedp

XUV absorption by solid-density aluminum

Carlos A. Iglesias*

Lawrence Livermore National Laboratory, PO Box 808, Livermore, CA 94550, USA

a r t i c l e i n f o

Article history:Received 5 October 2009Received in revised form6 January 2010Accepted 6 January 2010Available online 18 January 2010

Keywords:Warm dense matterInverse bremsstrahlungOpacity

* Tel.: þ1 925 422 7252; fax: þ1 925 423 7228.E-mail address: iglesias1@llnl.gov

1574-1818/$ – see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.hedp.2010.01.004

a b s t r a c t

An inverse bremsstrahlung model for plasmas and simple metals that approximates the cold, solid Alexperimental data below the L-edge is applied to matter conditions relevant to XUV laser applications.The model involves an all-order calculation using a semi-analytical effective electron-ion interaction. Thepredicted increases in XUV absorption with rising temperature occur via two effects: increased avail-ability of final states from reduced electron degeneracy and a stronger electron-ion interaction fromreduced screening.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Recently Vinko et al. [1] (hereafter VGW) observed that free-electron lasers offer an excellent opportunity to study the opacity ofwarm dense Al. One complication is eliminated since the heating isfast and hydrodynamic motion does not play an important role.Furthermore, since inverse bremsstrahlung dominates the XUV Alopacity below the L-edge, the experiments would isolate a singleabsorption process.

Several authors have studied photon absorption by electronfluids. [1,2] Nevertheless motivated by the renewed interest, Alabsorption calculations relevant to free-electron laser applicationsusing a simple model are presented. Inverse bremsstrahlung isbriefly reviewed in Sec. 2 and the model, which is based on stan-dard concepts, is introduced in Sec. 3. Comparisons to experimentaldata are presented in Sec. 4 with a discussion in Sec. 5. Comparisonsto the VGW results are provided in Sec. 6 and conclusions in the lastsection.

2. Inverse bremsstrahlung

The non-relativistic, thermally averaged inverse bremsstrahlungextinction coefficient of a photon with energy Zu by an isolatedelectron-ion pair is often written in the form [3]

aIBðuÞ ¼ gðu; TÞaKðuÞ (2.1)

ll rights reserved.

with T the temperature in energy units. The thermally averagedGaunt factor, g(u,T), accounts for quantum mechanical correctionsto the classical Kramers result, aK(u) [4].

2.1. Collective effects

Eq. (2.1) assumes that electron-ion pairs are independent.Collective phenomena, however, alter the absorption by screeningthe electron-ion interaction [5–8]. Furthermore, degeneracymodifies the thermal average so that [8]

gðu; TÞ ¼ffiffiffiffipp

2I1=2ðhÞ

ZNo

d3

Tfh�3

T

��1� fh

�3þ Zu

T

��gð3;uÞ (2.1.1)

where the Fermi-Dirac distribution and Fermi integrals are definedas

fhðxÞ ¼�

1þ ex�h��1

(2.1.2)

ImðhÞ ¼1

Gð1þ mÞ

ZNo

dx xmfhðxÞ (2.1.3)

with h the chemical potential divided by T and G(x) the Gammafunction [9].

The Gaunt factor g(3,u) with 3 the initial electron energy isusually computed in the one-electron picture. The calculations areoften performed in the dipole approximation and partial wavedecomposition leads to [3]

C.A. Iglesias / High Energy Density Physics 6 (2010) 311–317312

gð3;uÞ ¼ffiffiffi3pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip XN

[n

M2[;[�1ð3;uÞ þM2

[;[þ1ð3;uÞo

2p 3ð3þ ZuÞ [¼1

(2.1.4)

The dipole matrix element is most readily evaluated in the‘‘acceleration’’ form,

M[;[0ð3;uÞ ¼ZNo

drj3[ðrÞdVsðrÞ

drj3þZu;[0ðrÞ (2.1.5)

where j3[ðrÞ is the electron wave function with energy 3 and orbitalangular momentum [, moving in the spherically symmetricpotential Vs(r) with outgoing wave boundary conditions.

The static potential is determined by an average charge distri-bution of the ion. For example,

VsðrÞ ¼ VbðrÞ �Ze2

re�r=l (2.1.6)

with Ze the net ion charge, l a many-body screening length, andVb(r) a short-ranged, frozen-core potential due to bound electronsso that Vs(r / 0) ¼ �ZNe2/r with ZNe the nuclear charge.

2.2. Dispersion and multiple collisions

In addition, there are dispersion and multiple collisions that canimpact the absorption for photon frequencies near and below theplasmon resonance [7,10]. The Drude model approximates theseeffects with a complex refractive index of the form [10]

N2ðuÞ ¼ 1�u2

p

u½uþ inðuÞ� (2.2.1)

where v(u) is a collision frequency and up ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4pe2n=m

pwith n

and m the electron free number density and mass, respectively. Theresulting absorption and index of refraction are [10]

aDðuÞ ¼2u

cImNðuÞ ¼

u2p

u2 þ n2ðuÞnðuÞ

nðuÞc (2.2.2)

nðuÞ ¼ ReNðuÞ

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu�

u2 � u2p þ n2

�þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ n2

�u2 � u2

p

�2þu2n2

r2uu2 þ n2

vuuut

(2.2.3)

where c is the speed of light in vacuum.

2.3. Born approximation

Systematic derivations of the inverse bremsstrahlung cross-section in a fluctuating medium lead to second-order expressionsof the energy dependent Gaunt factor [5–7],

gð2Þð3;uÞ ¼ffiffiffi3p

p

Zkþk�

dqq

SðqÞ F2ðqÞjDðq;uÞj2

(2.3.1)

where F(q) is simply related to the Fourier transform of theunscreened electron-ion interaction,

FðqÞ ¼ q2

4pe2

Zd r!ei q!$ r!VðrÞ (2.3.2)

and

Z2k2�

2m¼ 23þ Zu� 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð3þ ZuÞ

p(2.3.3)

define the maximum and minimum wave number momentumtransfer. This expression for the energy dependent Gaunt factorincludes dynamic screening and ion–ion correlations through theplasma dielectric function, D(q,u), and the ion structure factor, S(q),respectively.

The results in Eq. (2.1.4) simplify in the Born approximationto [3]

gBornð3;uÞ ¼ffiffiffi3p

p

Zkþk�

dqq

F2s ðqÞ ¼

ffiffiffi3p

p

Zkþk�

dqq

F2b ðqÞ þ Z2gl

Bornð3;uÞ

(2.3.4)

with Fs(q) and Fb(q) related to the Fourier transform of Vs(r) andVb(r), respectively, and

glBornð3;uÞ ¼

ffiffiffi3p

2p

(ln

"1þ l2k2

þ

1þ l2k2�

#þ 1

1þ l2k2þ� 1

1þ l2k2�

)

(2.3.5)

where Eq. 2.1.6 was used to get the last line of Eq. (2.3.4). Onedifference between Eqs. (2.3.1) and (2.3.4) is dynamic rather thanstatic screening of the interaction. The second difference is theabsence of ion–ion correlations in the latter. Both are considered inSect 5.

3. Proposed model

An approximation for the collision frequency can be obtained bycomparing Eq. (2.1) and Eq. (2.2.2) for large photon frequencies forwhich n(u) z 1 and v(u)� u, to yield

nðuÞc¼ u2

u2pgðu; TÞaKðuÞ (3.1)

The proposed ad hoc model for the photon absorption assumesthe form in Eq. (2.2.2) corrected for stimulated emission,

aðuÞ ¼�

1� e�Zu=T� u2

p

u2 þ n2ðuÞnðuÞ

nðuÞc (3.2)

with n(u) and n(u) in Eqs. (3.1) and (2.2.3). The model incorporatesmany of the essential phenomena in inverse bremsstrahlung forplasmas and simple metals [11].

The input to the model is the static potential. Here, the form inEq. (2.1.6) is kept by using an analytic potential designed toreproduced atomic data of isolated ions with accuracy comparableto single configuration, relativistic, self-consistent-field calcula-tions [12]. Thus, the potential reasonably describes the averagecharge distribution of a nucleus plus bound electrons. The many-body screening of the long-ranged Coulomb tail from the net ioncharge assumes

l2 ¼ T4pe2ne

I1=2ðhÞI�1=2ðhÞ

(3.3)

where l is a generalized electron Debye length [13]. The degeneracyparameter is obtained assuming an ideal electron gas,

nl3T ¼ 2I1=2ðhÞ (3.4)

where lT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pZ2=mT

qis the thermal de Broglie wavelength.

Table 1Degeneracy h from Eq. (3.4), screening parameter l from Eq. (3.3), and value of Gh(u)in Eq. (4.2.1) where u ¼ Zu/T for solid-density Al at several temperatures andZu ¼ 30 eV.

T [eV] h l[ao] Gh(u)

0.025 467.2 0.918 0.05470.1 115.9 0.923 0.1091 11.58 0.924 0.3432 5.676 0.934 0.4764 2.579 0.981 0.6277 1.038 1.088 0.72510 0.2745 1.204 0.765

C.A. Iglesias / High Energy Density Physics 6 (2010) 311–317 313

4. Results

The L-shell electrons in thermal equilibrium, solid-density Al arenot expected to ionize significantly until the temperature is abovew10 eV [1] The model electron-ion interaction is then given byZ ¼ 3 and frozen-core potential,

VbðrÞ ¼ �e2

r

h2e�r=lK þ 8e�r=lL

i(4.1)

where lk ¼ 0.1009ao and lL ¼ 0.375ao with ao the Bohr radius [12].

4.1. Cold aluminum

Absorption data for cold, solid Al is commonly quoted from twosources [14,15] that disagree below the L-edge. Recent measure-ments, however, agree with the earlier data; thus, it is taken as themore accurate. [1,16] The Al photon absorption from the presentmodel at T ¼ 0.025 eV and measurements [14] are presented inFig. 1 (the calculation with Vb(r) ¼ 0 is discussed in Sec. 5 and thosefrom VGW in Sect. 6). The degeneracy and screening parameter inthe calculation are provided in Table 1 and the matrix elements inEq. (2.1.5) were computed using the phase-amplitude method. [17]For these short-ranged potentials, however, the Green’s functionapproach [18] should not be significantly more time consuming.The figure shows reasonable agreement between the presentmodel and experimental data (e.g.; earlier comparisons show largerdiscrepancies [1,19]).

Index of refraction calculations for cold, solid Al are displayed inFig. 2 showing good agreement with the measurements [20,21].The figure includes the high frequency approximation (collisionlessplasma, v ¼ 0),

noðuÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

u2p

u2

s(4.1.1)

valid for u > up and in reasonable agreement with the model abovethe plasma frequency. The figure also explains the sharp increase inabsorption below the plasma frequency as a consequence of thedecrease in the index of refraction. Finally, note the discrepancy at

Fig. 1. Photon absorption cross-section versus photon energy for cold, solid Al from thepresent model, semi-analytic and MD calculations in VGW [1], and experimental data[14] with an L-edge at Zu z 73 eV.

small photon energies due in part to neglected ion–ion correlationsdiscussed in Sec. 5.

4.2. Warm aluminum

Given the reasonable success of the model in reproducing thecold Al data, it is tempting to try warmer conditions accessible withfree-electron laser experiments. The ratios of absorption calcula-tions at several temperatures to the T ¼ 0.025 eV result are plottedin Fig. 3 with input degeneracy and screening parameter given inTable 1. The figure shows decreased absorption for u < up caused inpart by the behavior of the index of refraction. For example, dis-played in Fig. 2 are calculations for solid-density Al at T ¼ 10 eVshowing the temperature behavior of n(u). The enhanced absorp-tion for u > up with increasing temperature is due to changes inscreening length and degeneracy.

The longer screening length strengthens the electron-ioninteraction increasing the collision frequency (shape resonancescan produce exceptions [18]). Note that although there is a netincrease in l with increasing temperature, there are competingeffects between the explicit and implicit temperature dependencein Eq. (3.3). The former clearly increases l while the latter decreasesl (the ratio of Fermi functions in Eq. (3.3) goes to 1 for h��1 andincrease linearly with h for h [ 1).

The increases in absorption by the reduced electron degeneracyfollow from the thermal average. Consider the weighting functionin Eq. (2.1.1),

Fig. 2. Index of refraction versus photon energy for solid-density Al: present model atT ¼ 0.025 eV (solid) and T ¼ 10 eV (short-dash), cold experimental data (circles), [20,21]and collisionless limit no(u) (long-dash).

Fig. 3. Photon absorption for solid-density Al at various temperatures from presentmodel versus photon energy. The results are normalized to the T ¼ 0.025 eV modelcalculations.

Fig. 4. Thermally averaged collision frequency versus photon energy from presentmodel using all-order and Born approximation for solid-density Al at T ¼ 0.025 eV and10 eV. Also, all-order calculations for T ¼ 10 eV with l corresponding to theT ¼ 0.025 eV value.

Fig. 5. Photon absorption cross-section versus photon energy for cold, solid Li and Nafrom present model (solid) and experimental data (circles). [14] The data is not plottedbeyond the first photon ionization threshold.

C.A. Iglesias / High Energy Density Physics 6 (2010) 311–317314

GhðuÞ ¼ffiffiffiffipp

2I1=2ðhÞ

ZNo

dxfhðxÞh1� fhðxþ uÞ

i

¼ffiffiffipp

21� e�u

I1=2ðhÞ

ln�

1þ eh

1þ eh�u

�(4.2.1)

The range of integration in Eq. (4.2.1) is controlled by the degen-eracy. The extreme degeneracy limit clearly shows this effect whereGh(u) increases linearly with u ¼ Zu/T until u ¼ h at which point itbecomes constant. Table 1 shows the decrease in electron degen-eracy with rising temperature and consequent increase in Gh(u) forZu ¼ 30eV.

Displayed in Fig. 4 are calculations of v(u) for solid-density Al atT ¼ 0.025 and 10 eV. To estimate the separate impact of the twoeffects (increased l and decreased h) a calculation at T¼ 10 eV usingthe same effective electron-ion interaction as the cold case (i.e.;artificially adjust the screening length to the cold result) is alsodisplayed. The figure shows the largest increase in v(u) fromreduced degeneracy at small u, which is expected since the smallerthe photon energy the more degeneracy restricts the available finalstates. The increase from a larger l is approximately u independent.

Also displayed in Fig. 4 is the Born approximation to the collisionfrequency using Eq. (2.3.4). It considerably overestimates the all-order calculations for either T ¼ 0.025 or 10 eV. Furthermore, theBorn results underestimate the temperature dependence of v(u).Clearly, weak-scattering theory fails for the electron-ion interactionin the present model.

4.3. Cold Li and Na

Although Al is the prototypical simple metal with tightly bound,weakly polarizable core electrons, the model can be readily appliedto other elements and conditions since the input is an effectiveelectron-ion interaction defined by the material conditions. In Fig. 5comparisons are made to solid Li and Na experimental data [14].The frozen core interaction has the same functional form as in Eq.(4.1) except that the fitting parameters reproduce atomic data forthe valence electron in neutral Li and Na [12] and the screeninglength is adjusted to the appropriate free-electron density. Thefigure shows reasonable agreement of the present model withexperimental data for Li and Na for u > up.

5. Approximations

Although the reasonable agreement with the cold experimentaldata is tantalizing, the present model makes several ad hocapproximations. It is possible, however, to estimate errors due toseveral neglected or approximated physical processes.

5.1. Electron screening

The present model not only neglects dynamic screening, but italso assumes an exponentially decaying interaction. To test theapproximation, the thermally averaged collision frequency calcu-lations using g(2)(3,u) in Eq. (2.3.1) with S(q) ¼ 1 and a Coulombpotential screened by the RPA D(q,u) are compared to calculationswith gl

Bornð3;uÞ in Eq. (2.3.5). In addition, the results usinggl

Bornð3;uÞ are corrected by the index of refraction to account for

C.A. Iglesias / High Energy Density Physics 6 (2010) 311–317 315

dispersive behavior present in D(q,u). Their ratio is plotted in Fig. 6for solid-density Al at T ¼ 0.1 eV showing that for u > up the staticapproximation and dynamic screening agree reasonably well.

5.2. Ion–ion correlations

A neglected process in the present model is the interference ofscattered waves from nearby ions. This phenomenon can be readilyincluded through the ion structure factor in the Born approxima-tion [5–8]. To estimate their impact, the collision frequency iscomputed in the Born approximation using the static screeninglength in Eq. (3.3) with an approximation for S(q). The structurefactor was obtained using the Percus–Yevick equation for hard-spheres [22,23] with a packing fraction of 0.45 [24]. The ratio of theresults with and without ion–ion correlations for solid-density Al atT ¼ 0.1 eV is plotted in Fig. 6 showing w10% correction over thephoton energies of interest. Increasing the ion temperature leads tobroadening of the features in S(q) [24] further reducing theimportance of ion correlations.

The small impact of ion–ion correlations on the collisionfrequency for u > up follows from the limits on the momentumtransfer integration. For example, at high degeneracy the thermalaverage in Eq. (2.1.1) is weighted towards values near the chemicalpotential. Assuming Al at solid-density and T ¼ 0.1 eV, then for3 ¼ hT z 12 eV and Zu¼Zupz15eV the limits of integration in Eq.(2.3.1) are

k�z0:94 �A�1

and kþz4:5 �A�1

(5.2.1)

This range encompasses the first peak and part of the second in thestructure factor. [24] Consequently, for u > up ion–ion correlationsshould not significantly affect the photon absorption. Conversely,for u � up the ion–ion correlations would impact the results.

5.3. Frozen-core potential

The model assumes a frozen-core potential to describe thelocalized bound electrons. At small frequencies the polarization ofthe bound electrons is negligible. At higher photon energies,however, the dynamic response of the core can become important

Fig. 6. Ratio of thermally averaged electron-ion collision frequencies versus photonenergy for solid-density Al at T ¼ 0.1 eV: Born approximation with dynamic and staticscreening (solid), Born approximation with PY, hard-sphere S(q) [22–24] and S(q) ¼ 1(dash).

[19]. That is, the assumed core potential is adequate to describeelectron scattering from Alþ3 at large impact parameters, but it isonly approximate for the core penetrating collisions required forabsorption at photon energies approaching the ionization of theL-shell electrons.

The relative contribution of the frozen-core potential is dis-played in Fig. 1 by an absorption calculation using the presentmodel but setting Vb(r) ¼ 0 that shows considerable largerdisagreement with the experimental data than the full potentialresult. The discrepancy between the present model and experi-mental data together with the model variation when settingVb(r) ¼ 0 suggest that simulating the dynamic response of thebound electrons in the present model could improve the agreementwith the cold experimental data [19].

6. Vinko et al. [1] calculations

Recently VGW [1] presented photon absorption calculationsfor solid-density Al. Briefly, the VGW semi-analytical calculationsare based on Ron and Tzoar [6], which is a weak-scatteringapproximation with thermal average in Eq. (2.1.1) corrected forstimulated emission (see Appendix). Their calculations used theempty-core electron-ion interaction screened by the randomphase approximation (RPA) or local field corrected (LFC) dielectricfunction plus an ion structure factor for the crystal or one suitablefor liquids [1].

The VGW results are displayed in Fig. 1. The figure shows thattheir molecular dynamics (MD) simulations agree with the coldexperimental data (and the present model) only at low frequencies.The semi-analytic VGW-RPA and MD results agree for u � 15 eV(limit of available semi-analytical VGW calculations) and there isimprovement compared to the measurements with the VGW-LFCsemi-analytic model.

The temperature dependence of the solid-density Al absorptionat constant photon energy in Fig. 7 shows different behaviors fromVGW-LFC and the present model. To emphasize the temperaturedependence, each calculation in Fig. 7 was normalized to itsrespective value at room temperature. In an effort to ascertain thesource of the discrepancy, additional calculations using thefollowing expressions are displayed in Fig. 7,

Fig. 7. Photon absorption for solid-density Al at Zu ¼ 30 eV as a function of temper-ature: Present model (solid), a1(u) (dot-dash), a2(u) (short-dash), and VGW-LFC [1](long-dash). Each calculation is normalized to their respective T ¼ 0.025 eV value.

C.A. Iglesias / High Energy Density Physics 6 (2010) 311–317316

a1ðuÞ�¼�

1� e�Zu=T� u2

p2

�nðuÞ

(6.1)

a2ðuÞ u c nBornðuÞ

Both a1(u) and a2(u) use the electron-ion interaction from thepresent model and both neglect dispersion and multiple collisions.The difference is that a1(u) is all-order in the interaction whilea2(u) makes the Born approximation.

The similarities between a1(u) and the present model show thatdispersion and multiple collisions have, as expected [7], smallimpact well above the plasma frequency. The rough similaritiesbetween a2(u) and VGW-LFC suggest that the weak-scatteringapproximation may partly explain the discrepancy in temperaturebehavior between VGW-LFC and the present model. In evaluatingthese comparisons recall that the temperature behavior of thesemi-analytical VGW models agrees with the MD simulations [1].Bear in mind, however, that although computer simulations areoften used to validate theoretical models, in the present case theMD results disagree significantly with the cold Al data in the XUVregion and the source of the discrepancies may not be wellunderstood.

It is interesting to compare the perspective of the twoapproaches. The semi-analytical VGW model is based on thepseudo-potential method [25] designed to reproduce solid-statephenomena. Such a description may not accurately describe exci-tations well above the Fermi energy for which a quasi-particlepicture is incomplete (very short quasi-particle lifetimes). Thispotential shortcoming may partly explain the considerable steeperslope of the VGW photon absorption calculations compared to thecold data. In contrast, the present model neglects solid-stateaspects and instead emphasizes energetic processes more closelyassociated with binary collisions here corrected for many-bodyeffects. Consequently, the present model may not adequatelydescribe properties of the solid involving thermal excitations.

7. Conclusion

A model for the photon absorption of plasmas and simple metalswas proposed. The approach is based on a Drude picture where theelectron-ion collision frequency is obtained from the usual inversebremsstrahlung all-order formula corrected for degeneracy andmany-body screening. The results are in reasonable agreementwith the cold, solid Al measurements of the photon absorption andindex of refraction. The contribution of the atomic core portion ofthe electron-ion interaction to the absorption was found to besignificant. Furthermore, it is possible that the neglected dynamicresponse of the bound electrons is partly responsible for residualerrors. As expected [5,7,8], estimates of ion–ion correlations anddynamic screening effects produced only small corrections to theXUV absorption. It was also found that in the XUV range the Bornapproximation fails for the strong electron-ion interaction in themodel.

The proposed model displayed an enhancement in the XUVabsorption with rising temperature due to increases in the many-body screening length and a reduction in the electron degeneracy.Note, however, that differences in the absorption temperaturedependence exist between the present model and the Vinko et al.calculations [1]. Measurements of the XUV absorption by solid-density Al would enhance the fundamental understanding of warmdense matter and help resolve these theoretical discrepancies.

Finally, the proposed model is not specific to solid-density Al.The input is a semi-analytical electron-ion interaction defined bythe material conditions. Consequently, the model can be used ina wide range of applications. For example, the model is in reason-able agreement with cold, solid Li and Na experimental photon

absorption data and was previously implemented in the OPAL andTOPAZ opacity codes [26–28].

Acknowledgments

Thanks are due to Sam M. Vinko for providing their calculationsin tabular form. This work performed under the auspices of the U.S.Department of Energy by Lawrence Livermore National Laboratoryunder Contract DE-AC52-07NA27344.

Appendix. Ron-Tzoar [6] and Born approximation

Ron and Tzoar give the absorption extinction coefficient fora plasma or simple metal in the form [1,6]

aðuÞ ¼ n6p2m2u3c

ZNo

dqq6~V2ðqÞjDðq;uÞj2

SðqÞImfDðq;uÞ � Dðq;0Þg

(A.1)

This expression can be rewritten using the dielectric function [6]

Dðq;uÞ ¼ 2Z

d p!

ð2pÞ3nj p!j � n

j p!þ q!jZuþ E

j p!j � Ej p!þ q!j þ ix

¼ 2Z

d p!

ð2pÞ3nj p!� q!=2j

� nj p!þ q!=2j

Zu� Z2 p!$ q!=mþ ix(A.2)

where the second line involves a change of variables and analyticcontinuation has x / 0þ. Then

aðuÞ ¼ ne2

3p2m2u3c

Zd p!

ZNo

dqq4~V2ðqÞjDðq;uÞj2

SðqÞ

nnj p!� q!=2j

� nj p!þ q!=2j

od

Zu� Z2

mp!$ q!

!(A.3)

The one-electron Fermi distribution and energy are, respectively,

np ¼ fh

�Ep

T

�(A.4)

with fh(x)in Eq. (2.1.2) and

Ep ¼Z2p2

2m(A.5)

To proceed, use the energy conserving delta-function in Eq. (A.3) towrite,

nj p!� q!=2j

� nj p!þ q!=2j

¼�

1� e�Zu=T�

nj p!� q!=2j�

1� nj p!þ q!=2j

�(A.6)

Then, substituting Eq. (A.6) into Eq. (A.3), performing the solidangle integration, changing variables p! ¼ p!þ q!=2, and intro-ducing x ¼ Z2p2/2mT and u ¼ Zu/T, yields

aðuÞ ¼ 16ne6

3Z4u3c

�1� e�u

�ZNo

dxfhðxÞh1

� fhðxþ uÞi Zkþ

k�

dqq

SðqÞ F2ðqÞjDðq;uÞj2

(A.7)

C.A. Iglesias / High Energy Density Physics 6 (2010) 311–317 317

with F(q) and k� defined in Eqs. (2.3.2) and (2.3.3). The second-order expression in Eq. (A.7) explicitly displays the stimulatedemission correction and the thermal average over initial particleplus final holes and is identical to the Born approximation from the‘‘inverse bremsstrahlung’’ approach [5,7,8].

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