High Energy Density Physics 3 (2007) 109e114www.elsevier.com/locate/hedp

Hybrid atomic models for spectroscopic plasma diagnostics

S.B. Hansen a,*, J. Bauche b, C. Bauche-Arnoult b, M.F. Gu c,1

a Lawrence Livermore National Laboratory, P.O. Box 808, L-473, Livermore, CA 94550, USAb Laboratoire Aime Cotton, Campus d’Orsay, Batiment 505, 91405 Orsay, France

c Kavli Institute for Particle Astrophysics and Cosmology and Department of Physics, Stanford University, CA 94305, USA

Available online 20 February 2007

Abstract

We propose a hybrid approach to treating atomic structure and rates in collisional-radiative models, combining the completeness of highlyaveraged models with the accuracy of detailed models. The hybrid scheme supplements a small subset of coronally accessible fine structurelevels with a complete set of configuration- and superconfiguration-averaged levels and produces spectra composed of transitions amonga mix of fine structure and relativistic configuration-averaged levels. Convenient expressions are given for obtaining rates between the fine struc-ture and averaged levels and a technique for propagating configuration interaction from the fine structure calculations to configuration averagesis described. We present results from a trial hybrid model of germanium which demonstrate the accuracy of the hybrid model for charge statedistributions and spectra.Published by Elsevier Ltd.

Keywords: Collisional-radiative; Non-LTE; Spectroscopic

1. Introduction

Collisional-radiative spectroscopic models are valuablediagnostic tools for a wide variety of plasmas. Their reliabilityis limited by two competing factors: (1) the detail and preci-sion of the energy levels and rates used in the underlyingatomic model, and (2) the completeness of that energy levelstructure and level coupling. The importance of the first com-ponent is clear: Emission wavelengths, intensities, and thelevel populations obtained by solving a coupled set of colli-sional-radiative rate equations can be no more accurate thanthe underlying atomic data. The second component, complete-ness, is no less important. At low and moderate densities, ac-curate calculations require the inclusion of a large number ofAuger channels and high Rydberg states to properly describeionization balances and radiative cascades to metastablelevels. At high densities, continuum lowering may reducethe maximum quantum number needed for accuracy, but can

* Corresponding author.

E-mail address: hansen50@llnl.gov (S.B. Hansen).1 Present address: Lawrence Livermore National Laboratory, P.O. Box 808,

L-473, Livermore, CA 94550, USA.

1574-1818/$ - see front matter Published by Elsevier Ltd.

doi:10.1016/j.hedp.2007.02.032

also greatly increase the number of accessible configurationsin complex ions.

Highly complete models tend to rely on statistical atomicdata and, while they perform well at high densities where col-lisions enforce local thermodynamic equilibrium (LTE), theycan neither account for the kinetic effects of low-lying meta-stable levels nor reliably reproduce the spectra of closed-shellions at low and moderate densities. In contrast, highly detailedmodels based on fine structure atomic data are reliable nearclosed-shell ions and at low densities but become intractablefor complex ions. To bridge the gap between these two cate-gories of collisional-radiative models, we propose a hybridapproach to the level structure in which a small subset of de-tailed levels is augmented by a complete set of configuration-and superconfiguration-averaged levels.

The level structure of the proposed hybrid model isdescribed in the following section. In Section 3, we describevarious ways to obtain rates between hybrid levels. At thispoint, the hybrid model will give reliable charge state distribu-tions and level populations with all the advantages of both de-tailed and averaged models. In Section 4, we discuss how onecan also obtain spectroscopic-quality emission or absorptionspectra by reconstituting highly averaged levels into relativistic

110 S.B. Hansen et al. / High Energy Density Physics 3 (2007) 109e114

configuration averages and approximating configuration inter-action effects in transition arrays using information containedin the fine structure level subset. Finally, in Section 5, we presentresults from a trial hybrid model of germanium.

2. Proposed hybrid model level structure

We begin with the observations that X-ray emission fromclosed-shell ions and from even complex ions in the low-densitycoronal regime [1] is often very well described by models witha fairly restricted set of fine structure levels directly accessiblevia single-electron transitions from the ground configuration.Such models work well whenever collisional excitation fromthe ground state is the dominant population mechanism andspontaneous decay is the dominant depopulation mechanism,with two important caveats: First, reliable coronal modelsmust include configurations up to a sufficiently large quantumnumber (usually n¼ nvalenceþz5 is sufficient [2]) to ade-quately describe radiative cascades into metastable levels,which give rise to the familiar density-sensitive diagnostic emis-sion lines [3]. Second, accurate calculations of charge state dis-tributions in the coronal regime require the inclusion of manydielectronic recombination channels, often through doubly ex-cited levels not accessible from the ground configuration. Inpractice, these recombination channels are often approximatedusing convenient tables given in the literature [4,5].

Since any model making claims for spectroscopic accuracymust be able to describe high resolution spectra from wellcharacterized, low-density plasma sources, we set as theskeleton of our hybrid model a fairly restricted set of finestructure levels belonging to configurations which are directlyaccessible via single-electron transitions from the ground state.Using the 15-electron P-like ion as an example, we have:

� ground configuration:1s22s22p63s23p3

� single excitations from the valence shell to n� nvalenceþ 21s22s22p63s3p4

1s22s22p63s23p23d1s22s22p63s23p24[1s22s22p63s23p25[1s22s22p63s3p33d1s22s22p63s3p34[1s22s22p63s3p35[� single excitations from the first inner shell to n� nvalenceþ 1

1s22s22p53s23p4

1s22s22p53s23p33d1s22s22p53s23p34[1s22s2p63s23p4

1s22s2p63s23p33d1s22s2p63s23p34[

A model based on this set of configurations, which we shallcall coronal, will give a reasonable description of majorresonance lines from low-density sources such as tokamaks,electron beam ion traps, and many astrophysical plasmas.

Coronal models become unreliable at moderate- or high-densi-ties or in the presence ofa thermal radiation field, where collisionalexcitation from the ground level is no longer the dominant excita-tion mechanism and excited configurations can have populationsthat approach or exceed (because of their large statistical weights)the population of the ground configuration. Large populations insingly excited configurations in turn open new channels to multi-ply excited configurations. For example, the coronal configura-tions of the P-like ion described above would not account forstrong dipole transitions from 1s22s22p63s 3p23d2 to 1s22s22p63s3p23d 4f, which may be significant in moderate- or high-densityplasmas.A reliable, general-purpose model therefore must includea more extensive set of configurations. We specify these additionalconfigurations using Layzer’s complexes defined according to theoccupation numbers of the n shells [6] with notation (n)N (n0)N0.denoting the full complement of possible configurations withina superconfiguration (SC).

The number and type of superconfigurations required fora reasonably complete model have been investigated by Peyr-usse et al. in a previous work [7], where it was determined thatincluding SCs with energies up to three times the ionizationpotential should be adequate for thermal plasmas. If non-ther-mal electrons are present, excitation from the inner shellsshould also be included.

For this work, we consider the following modest set ofsuperconfigurations, using the P-like ion again as an example:

� all configurations in the ground SC:(1)2(2)8(3)5

� single excitations from the valence shell to n� nvalenceþ 4(1)2(2)8(3)4(4)1

(1)2(2)8(3)4(5)1

(1)2(2)8(3)4(6)1

(1)2(2)8(3)4(7)1

� single excitations from the first inner shell to n�nvalenceþ 2(1)2(2)7(3)6

(1)2(2)7(3)5(4)1

(1)2(2)7(3)5(5)1

For closed-shell ions, we also include double excitationsfrom the valence shell to n¼ nvalenceþ 1. For general models,more extensive doubly and even triply excited SCs might beadded to the list above to ensure completeness.

Fig. 1 shows the number of fine structure levels and config-urations in ions from H-like to Ar-like that result from this setof superconfigurations. Since we must solve a system of N� Ncoupled rate equations, where N is the total number of levels,it is clear that the number of fine structure levels becomesintractable as soon as we have more than two or threeelectrons in the M shell, The number of configurations,however, remains tractable, as does the number of coronalfine structure levels defined above.

Thus, we arrive at our proposal for the hybrid level struc-ture: from a set of SCs with sufficient completeness to de-scribe the general non-LTE problem, we replace a subset ofcoronal configurations with fine structure levels. In this way,

111S.B. Hansen et al. / High Energy Density Physics 3 (2007) 109e114

we combine the accuracy of the coronal approach for low den-sities and closed-shell ions with the completeness of an aver-aged-level approach at moderate and high densities, whilerestricting the computational effort required to a tractable setof levels, as shown in Fig. 1.

While this scheme can be used with any atomic data source,we have based our investigations on data from the fully rela-tivistic flexible atomic code (FAC) code [5]. FAC can operatein two modes, giving atomic structure and rate data for eitherfine structure levels or relativistic configuration averages. Inthe hybrid model level scheme, the highly accurate calcula-tions of fine structure levels and rates are rapid because weinclude only a restricted set of levels. Calculations of thelarger, more complete set of relativistic configurations andrates are also rapid, since they include only restricted config-uration interaction and bypass a great deal of the angularmomentum algebra required for fine structure levels.

Once we have obtained a complete set of relativistic config-urations (RC) and rates from the FAC code, we replace thesubset of relativistic coronal configurations with fine structurelevels, then average the remaining RCs into non-relativisticconfigurations with energies given by:

EC ¼XRC

ðgRCERCÞ�X

RC

gRC ð1Þ

Here, RC is an index running over all relativistic configura-tions in the non-relativistic configuration C.

Finally, following the results of Ref. [8], if there are morethan q z 500 non-relativistic configurations in any superconfi-guration, we average them into a single SC with energy:

ESC ¼X

C

ðgCECÞ�X

C

gC ð2Þ

where the index C runs over all configurations in the supercon-figuration SC.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18number of electrons

Fine Structure levels

Configurations

nu

mb

er o

f levels

CoronalFine Structure levels

ProposedHybrid

107

103

101

105

Fig. 1. Number of fine structure levels, coronal fine structure levels, configu-

rations, and hybrid model levels in the configuration sets described in the text

for ions from H-like to Ar-like.

3. Calculation of rates

Along with the calculation of the hybrid level structure de-scribed above, we obtain from FAC a complete set of ratesamong relativistic configurations (RC / RC0) supplementedby a complete subset of rates among fine structure levels(FS / FS0). Since we retain all of the fine structure levels,we must also retain all of the FS / FS0 rates. We obtain con-figuration-to-configuration rates by first integrating the RC /RC0 cross sections over electron and photon distribution func-tions, and then averaging the RC / RC0 rates through:

Rate�C/C0

�¼XRC0

PRC gRCRateðRC/RC0ÞP

RC gRC

ð3Þ

Configuration to superconfiguration rates are given by:

Rate�C/SC0

�¼X

C0RateðC/C0Þ ð4Þ

with the indices RC0 and C0 running over their parent ensem-bles. Note that the averaging given in Eqs. (3) and (4) and theaveraging continuing below does not have to be statistical: itcould include a Boltzmann factor with the real temperatureor an effective temperature obtained following the numericalprocedure used in the code MOST [9], the code RADIOM[10], or using the analytic form given in Ref. [11].

The final question is how to treat the rates between finestructure levels and configurations. There are three possibleapproaches: We might calculate an entire set of fine structurelevels and all rates coupling the coronal subset with the whole,and then average the FS / FS0 rates to obtain FS / C0 rates:

Rate�FS/C0

�¼XFS0

RateðFS/FS0Þ ð5Þ

This, however, obviates a substantial part of the efficiency ofthe hybrid scheme. The second approach is to obtain approx-imate FS / C0 rates by statistically decomposing the availableC / C0 rates from Eq. (3), ensuring that on reversal we regainC / C0 rates which are averages over the initial levels andsums of the final levels of the FS / FS0 rates:

RatestatðFS/C0Þ ¼ RateðC/C0Þ ð6Þ

and

Ratestat

�C/FS0

�¼ gFS0P

FS0 gFS0Rate

�C/C0

�ð7Þ

with FS0 running over all fine structure levels in C0.This statistical approach is reasonable, but we can better

preserve the individual character of the fine structure levelsby defining an additional factor f for the fine structure to con-figuration rates using coefficients of Slater integrals [12], suchthat RateðCaJ/C0Þ ¼ f RatestatðFS/C0Þ. Let C¼(n[)N(n0[0)N0þ1 and C0 ¼ (n[)Nþ1(n0[0)N0 denote the initial andfinal non-relativistic configurations, respectively, and aJ be

112 S.B. Hansen et al. / High Energy Density Physics 3 (2007) 109e114

the fine structure term of configuration C. For electric dipoleradiative transitions, we find:

f E1 ¼ 1

2[>

�ðN0 þ 1Þð2[0 þ 1Þ[>þC

�G1; aJ

�� ð4[þ 2Þð4[0 þ 2Þð4[�Nþ 2ÞðN0 þ 1Þ

ð8Þ

where C(G1;aJ ) is the coefficient of the Slater integral G1(n[,n0[0;aJ ) in the expansion of the total interelectronic-repulsionenergy of the aJ level in terms of the Slater integrals and [> isthe larger of [ and [0. This expression is valid even for transi-tions with one or more spectator electrons in open subshells,and may be applied to collisional excitation and de-excitationin the Van Regemorter and Mewe approximations [13,14]. Itmay also be applied to photonionization and collisional ioniza-tion and their inverse processes, because these are all monoe-lectronic processes. This generalization is linked with themultipolar expansion of the inverse distance between a freeelectron and a bound electron [15]. The comparison given inFig. 2 shows remarkably good agreement between fullyrelativistic calculations from FAC and Eq. (8).

In a similar way, one can define f factors for electric quad-rupole transitions. For such transitions, there are three cases,depending on the values of [ and [0:

f E2[0¼[ ¼

�N0 þ 1

�C�G2; A

��C

�G2; aJ

��N0 þ 1

�C�G2; A

��C

�G2; B

� ð9Þ

with

C�G2; A

�¼�[

�[þ 1

���2[� 1

��2[þ 3

�

and

C�G2; B

�¼ C

�G2; A

�N�N0 þ 1

���4[þ 2

�:

f E2[0¼[þ2 ¼

�N0 þ 1

�C�G2; D

��C

�G2; aJ

��N0 þ 1

�C�G2; D

��C

�G2; E

� ð10Þ

with

0.4

0.6

0.8

1.0

1.2

1.4

1.6

transition

f E

1 facto

rs

analyticfully relativisticstatistical

Fig. 2. Comparison of f E1 factors for Rate( p2d 4aJ / p3d 3). The analytic for-

mulation given in Eq. (8) matches the fully relativistic level-to-level factors

very well.

C�G2; D

�¼�3

�[þ 1

��[þ 2

��2�2[þ 3

��2[þ 5

�

and

C�G2; E

�¼ C

�G2; D

�N�N0 þ 1

���4[þ 2

�:

The case [0 ¼ [� 2 can be obtained from Eq. (10) afterexchanging n[ and n0[0, and changing [ into [0, N into(4[0 � N0 þ 1), and N0 into (4[� Nþ 1).

Having obtained this hybrid level structure and a completeset of hybrid rates, the standard collisional-radiative rate ma-trix can be solved to obtain reliable populations and chargestate distributions: Since the hybrid model retains some finestructure detail, it accounts for the important effects of ladderionization from highly populated metastable levels at lowdensities. It describes the populations of those metastablesreliably, since it includes a complete set of excited levelsfrom which radiative cascades can populate metastable levels.And since we can afford a fairly complete set of doublyexcited configurations using the hybrid scheme, we can alsoaccount for many of the important Auger channels that mightonly be approximated in standard low-density models.

4. Calculation of spectra

So far we have presented a hybrid model which should givereliable populations and charge state distributions from thecoronal to the LTE regimes. We wish also, however, to havea diagnostic tool, and for this we need more than a small sub-set of fine structure levels complemented by configurationsand superconfigurations. Previous investigations have shownthat transition arrays between relativistic configurations area reasonable representation of emission and absorption oncomplex ions [16]. Since our original data set was based onrelativistic configurations, and since FAC provides the neces-sary transition array energies and widths (followingRef. [17]), we can return to that level of detail with ease bysimply reconstituting the relativistic configurations as individ-ual levels.

This can be done most simply statistically by:

XRC ¼�gRCXC

��XRC

gRC ð11Þ

where XC is the population of the non-relativistic configurationand the index RC runs over all relativistic configurationswithin the configuration C. But, as with the rates, a Boltzmannfactor including an effective temperature might be introducedhere, either from a numerical calculation following Ref. [9](which could be iterated with the populations) or analyticallyfollowing Ref. [11]. We may also be able to preserve the non-statistical properties of the relativistic configurations by track-ing various line formation processes, as suggested by Rosmejin Ref. [18].

We emphasize that there is no risk of double-countingemission from fine structure levels and the relativistic config-urations, since we have discarded all of the relativistic

113S.B. Hansen et al. / High Energy Density Physics 3 (2007) 109e114

configurations duplicated by fine structure levels in the set ofcoronal configurations when constructing our energy levelscheme.

Even with reliable populations and transition widths, theemission features arising between relativistic configurationswill not be as accurate as those between fine structure levels,since configuration interaction (CI) is included only withineach configuration in the relativistic configuration mode ofthe FAC atomic structure code. Because configuration interac-tion can have significant effects on both transition energies andstrengths [19e21], its absence in transitions among the RClevels could degrade the diagnostic utility of spectra fromthe hybrid model.

A solution is to appeal to the configuration interaction in-formation encoded in the fine structure levels, which includeCI effects among all of the coronal configurations. Since thecoronal configurations are calculated twice (once as finestructure levels and once as relativistic configurations), wecan directly compare the transition energies and strengthsbetween fine structure levels in coronal configurations andthe (previously discarded) transition energies and strengthsof the relativistic configurations in the coronal set. The dif-ference between the two calculations can be propagated toall nlj/n0l0j0 transitions regardless of the arrangement ofspectator electrons in the relativistic configurations. In thisway, we can approximately extend the accuracy of the finestructure calculations to the averaged levels of the hybridmodel.

5. Results from hybrid Ge model

We have implemented the hybrid model described above inthe collisional-radiative code SCRAM [2] based on FAC datafor H- through Ar-like ions of Germanium. To test theaccuracy of the hybrid model, we have made three additionalmodels comprising the set of levels described schematically inSection 2:

� A complete set of relativistic configurations for H- throughAr-like Ge.� A fully detailed, fine structure model of O- through Na-

like Ge.� A fully detailed, fine structure model of H- through Be-

like Ge.

Fig. 3 shows the calculated average ion charge as a functionof temperature from the hybrid model and compares it to theresults of some models from the third NLTE code comparisonworkshop [22]. This figure shows that the hybrid model givesvery reasonable results, when compared with a variety of othercodes, over a wide range of hZi and is at least as accurate asthe model based on solely on relativistic configurations.

We find that the hybrid model is in fact more accurate thanthe straight relativistic configuration model by comparing thepredictions of the two models for charge state distributions andlevel populations of the closed-shell Ne-like ion with theresults of the fully detailed, fine structure model of O- through

Na-like Ge. The comparison given in Fig. 4(a) shows that therelativistic configuration model predicts a lower population forthe F-like ion at a temperature of 600 eV and a density of1020 cm�3 than the hybrid and fine structure models, whichagree quite well. The reason for the failure of the relativisticconfiguration model is shown in Fig. 4(b): since the RC modelaverages over the metastable levels in the (2[)7(3[) configura-tions, it underpredicts their populations and cannot accountproperly for ladder ionization from Ne-like to F-like Ge.This underprediction of the (2[)7(3[) populations will also

12

14

16

18

20

22

24

100 200 300 400 500 600T

e (eV)

av

era

ge

io

n c

ha

rg

e <

Z>

RC modelHybridother codes

ne = 1017cm-3

Fig. 3. Comparison of average ion charge calculations for Ge at

ne¼ 1017 cm�3 from the hybrid model of the present work, a model based

on relativistic configurations, and other codes (from Ref. [22]).

1000 1500 2000 2500

Fine structureRC modelHybrid

energy (eV)

X/g

(2l)73l

0.0

0.2

0.4

0.6

8 9 10 11number of electrons

Io

n fractio

n

(a)

(b)

Fine StructureRC modelHybrid

10-3

10-5

10-7(2l)

74l

(2l)7nl

Fig. 4. (a) Shows the charge state distribution of Ge ions at Te¼ 600 eV and

ne¼ 1017 cm�3 as calculated by the hybrid model and models based on rela-

tivistic configurations and detailed fine structure levels. (b) Shows that the hy-

brid model agrees with the detailed models, in its predictions for the

populations of metastable states in the (2[)7(3[) configurations, which the

RC model underpredicts.

114 S.B. Hansen et al. / High Energy Density Physics 3 (2007) 109e114

lead to an underprediction of the 3[� 2[ resonance line inten-sities in the Ne-like spectrum.

Finally, we test our proposed approximate extension of con-figuration interaction effects from the fine structure coronalconfigurations to the non-coronal relativistic configurations.Fig. 5 shows a comparison of Li-like K-shell satellite emissionat a temperature of 8 keV and an electron density of 1014 cm�3

The low-density, few-electron case should be a challengingregion for the statistical component of the hybrid model. Inthis ion, configuration interaction shifts the 2pe1s transitionsof the detailed level model by z20 eV. In the top figure, no CIcorrection has been applied to the non-coronal relativistic con-figurations of the hybrid model, and the shape of the satellitefeature does not conform to that of the detailed fine structuremodel. When a CI correction is applied to the non-coronaltransitions, using information obtained by comparing finestructure and relativistic configuration transition energieswithin the coronal configurations, the non-coronal transitionsare shifted to agree quite well with the emission feature ofthe fully detailed model.

6. Conclusions

We have described a novel scheme for combining theadvantages of fine structure and averaged models for use in

Fine structurecoronal (FS)Hybrid - no CI correction

10100 10150 10200 10250 10300energy (eV)

In

ten

sity (arb

. u

nits)

In

ten

sity (arb

. u

nits) Fine structure

coronal (FS)Hybrid with CI correction

Fig. 5. Comparison of detailed fine structure model emission spectra of Li-like

Ge at Te¼ 8 keV and ne¼ 1014 cm�3 with the predictions of the hybrid model

with and without approximate configuration interaction corrections to its rel-

ativistic configurations. Emission from only the coronal configurations of

the hybrid model are given by thin solid lines.

collisional-radiative modeling and spectroscopic diagnostics,and have demonstrated the quality of its results throughcomparisons with standard relativistic configuration and finestructure models.

Acknowledgements

The work of SH was performed under the auspices of theU.S. Department of Energy by University of CaliforniaLawrence Livermore National Laboratory under contract No.W-7405-Eng-48. Part of this work was done during the stayof JB and CBA at the Lawrence Livermore National Labora-tory. They would like to thank the people of V Division fortheir warm hospitality.

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