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High Energy Density Physics 6 (2010) 399e405

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High Energy Density Physics

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

Robust algorithm for computing quasi-static stark broadening of spectral lines

Carlos A. Iglesias*, Vijay SonnadLawrence Livermore National Laboratories, P.O. Box 808, Livermore, CA 94550, USA

a r t i c l e i n f o

Article history:Received 17 March 2010Received in revised form14 April 2010Accepted 24 April 2010Available online 4 May 2010

Keywords:Line shapeStark broadeningHessenberg

* Corresponding author.E-mail address: iglesias1@llnl.gov (C.A. Iglesias).

1574-1818/$ e see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.hedp.2010.04.002

a b s t r a c t

A method previously developed to solve large-scale linear systems is applied to quasi-static Stark linebroadening calculations. The method is formally exact, numerically stable, and allows optimization of theintegration over the quasi-static field to assure numerical accuracy. Furthermore, the method iscomputationally efficient compared to the conventional approach, particularly for complex problemsinvolving large matrices.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

In standard Stark broadening theory of spectral lines, the per-turbing ions are assumed to have negligible motion during theaverage lifetime of the radiating states [1,2]. In this theory an atomor ion (henceforth the radiator) emits or absorbs photons ina statistical fluctuating electric field produced by the slow-movingions. Thus, the line shape involves averaging the field induced statemixing and energy shifts over the probability distribution for thequasi-static field [3,4]. In the absence of external fields, the lineshape can be written in the form [1e4],

IðuÞ ¼ZN0

d3 Pð3ÞJðu; 3Þ (1.1)

where Zu is the photon energy, P(3) is the probability of findinga Stark field of magnitude 3 at the radiator, and J(u;3) represents theline shape due to electroneradiator interactions in the presence ofthis field.

The numerical evaluation of the line shape replaces the integralin Eq. (1.1) with the sum

IðuÞ ¼XN3

i¼1

wiPð3iÞJðu; 3iÞ (1.2)

ll rights reserved.

where N3 is the number of field points and wi are weightsappropriate to the integration scheme. The calculation of Eq. (1.2)poses a challenge when selecting the field mesh. Firstly, for theintegration to yield accurate results the mesh must resolve featuresin J(u;3). Since the structure of J(u;3) is usually not known a priori,a large number of field points are required. Secondly, the conven-tional method uses an explicit matrix inversion at each frequencyand field [5e8]. This is computationally expensive for largematrices, scaling asm3 operations for amatrix of sizem [9], so thereis a premium for defining a field mesh with a large number ofpoints.

Often the practical resolution of these contending needs isa fixed mesh that has sparsely spaced values compromising theaccuracy of the results. When such compromises are made in lineshape calculations of multi-electron systems involving largematrices [10,11], the complexity of the spectrum may make itdifficult to identify possible errors. Alternatively, an adaptivequadrature scheme can provide numerical accuracy but at anexorbitant computational cost.

The computational challenge above has compelled the intro-duction of approximations, such as no lower-upper state inter-actions and neglect of interference terms (diagonal electronwidthoperator) [5e7], in order to reduce the size of the matrices formulti-electron radiator calculations. The errors from someapproximations cannot always be quantified and are justified bycomparisons to more complete calculations [7]. Clearly, a methodto compute the standard line shape reliably and efficientlywithout introducing further approximations would provevaluable.

C.A. Iglesias, V. Sonnad / High Energy Density Physics 6 (2010) 399e405400

The purpose here is to apply an approach previously developedfor large-scale linear systems [12,13] to quasi-static Stark linebroadening. The method is exact in principle, numerically stable,and reveals the resonant structure of J(u;3) allowing optimizationof the numerical integration over the field. Furthermore, theproposedmethod is more efficient than the conventional approach,in particular for large matrices, making adaptive quadratureschemes more affordable.

The discussion begins with a brief introduction to Stark linebroadening theory in Section 2. The description of two relatedmethods are given in Sections 3 and 4, respectively. The firstmethod relies on more familiar techniques and, although not rec-ommended, serves to illustrate the basic ideas. The second andpreferred method is not only numerically stable, but also moreefficient for large problems. To demonstrate the method a simplenumerical example is provided in Section 5 with conclusionsoffered in the last section.

2. Stark broadening theory

The field-dependent line shape function appearing in standardStark-broadening theory in Eq. (1.1) can be written in the form[1,2,5e8]

Jðu; 3Þ ¼ �p�1Im Trf d!$Rðu; 3Þr d!g ¼ �p�1Im Trf d!$D!ðu; 3Þg

(2.1)

where Tr denotes a trace over the radiator internal states with r andd!

the radiator density matrix and dipole operators, respectively.The resolvent is defined by its inverse

R�1ðu; 3Þ ¼ AðuÞ � 3B (2.2)

where the field independent contribution to the line shape is givenby

AðuÞ ¼ Du� FðuÞ (2.3)

DuY ¼ ½u� H; Y � (2.4)

and the coefficient to the quasi-static field is

BY ¼ Z�1½dz; Y � (2.5)

Here, Y is an arbitrary operator in the radiator subspace, [/,/]denotes a commutator, H is the Hamiltonian for the internal statesof the radiator, and the electric field has been chosen along the z-axis. The “width and shift” operator, F(u), contains the plasmaelectron contribution to the relaxation processes of the radiatorstates [1,2].

The evaluation of the line shape requires radiator energy levelsand reduced dipole matrix elements usually obtained from atomicdata calculations [14e16]. Using these data, together with thedescription for electron collisional excitations, the matrices aboveare constructed. The calculation usually proceeds with an explicitmatrix inversion to obtain R(u;3) [5e8]. Thus, for each block of sizem the total count of complex number operations at a givenfrequency is

NizN3m3 (2.6)

Incidentally, Eq. (2.1) introduced the vector D!ðu; 3Þ that satisfies the

system of equations

R�1ðu; 3ÞD!ðu; 3Þ ¼ r d!

(2.7)

and is amenable to a variety of techniques. For example, solution ofEq. (2.7) by LU factorization of R�1(u;3) is w3 times faster thanexplicit inversion to obtain R(u;3) [9].

The form of R�1(u;3) in Eq. (2.2) is suitable for techniquesdeveloped for large-scale linear systems [12,13] where the quasi-static field is the independent variable. Two approaches are dis-cussed below. The first uses the eigen decomposition and illustratesthe advantages of these methods. The second relies on the Hes-senberg decomposition, which for large matrices is preferred sinceit is computationally faster and stable compared to the eigenvectorapproach.

3. Eigen decomposition

This method relies on the eigen decomposition of Eq. (2.2), butnot as previously done (see Appendix A). Instead, the approachexploits the functional form in Eq. (2.2) to extract the fielddependence of J(u;3) in an efficient manner.

3.1. Method

For each m�m diagonal block, rewrite the trace in the field-dependent line shape function, (in the following the u dependenceis suppressed for brevity)

Trnd!$½A� 3B��1r d

!o¼

�d!*�t

$½I � 3C��1A�1r d!

(3.1.1)

where the trace was performed and the notation implies matrixalgebra in “line space” [1,2]. Then I is the identity matrix andsuperscript t denotes transpose in line space, while superscript *

denotes Hermitian conjugate in the radiator subspace. The eigendecomposition of the matrix C¼ A�1B leads to

Trnd!$½A� 3B��1r d

!o¼

�d!*�t

P$½I � 3L��1P�1A�1r d!

(3.1.2)

where L is a diagonal matrix with entries {l1,l2,.,lm} that are theeigenvalues of C with

C ¼ PLP�1 (3.1.3)

and the columns of P are the eigenvectors. It is stressed that since Cis not Hermitian, the transformation in Eq. (3.1.3) is not unitary;that is, P�1s Ph, where the superscript h denotes the Hermitianconjugate in line space.

After straightforward manipulation of Eq. (3.1.2), the field-dependent line shape function can be written as sums over zeroand non-zero eigenvalues,

Jðu; 3Þ ¼ �p�1ImXmj¼1

fj þ igj1� lj3

¼ J0 þ p�1X

j ðljs0Þ

fjs00j þ gj�3� s0j

��3� s0j

�2þs002j(3.1.4)

where the contribution from vanishing eigenvalues is given by

Jo ¼ �p�1ImX

j ðlj ¼0Þ

h�d!*�t

Pij$hP�1A�1r d

!ij

(3.1.5)

the weights of the components in the dispersion formula are

�fjgj

�¼

�ReIm

�nl�1j

h�d!*�t

Pij$hP�1A�1r d

!ij

o(3.1.6)

C.A. Iglesias, V. Sonnad / High Energy Density Physics 6 (2010) 399e405 401

and

l�1j ¼ sj ¼ s0j þ is00j (3.1.7)

defines the poles of J(u;3) in the 3 complex plane.

3.2. The A(u) matrix

The eigen decomposition requires that A�1(u) exist. This isalways the case since A(u) is a non-singularmatrix representing theline shape in the absence of a quasi-static Stark field with non-zeroeigenvalues. Furthermore, the interference terms are oftenneglected making A(u) diagonal [5e8]. In any case, all explicitinversions are avoided by solving the field independent linearequations at each frequency

AC ¼ B (3.2.1)

A u! ¼ r d!

(3.2.2)

P z! ¼ u! (3.2.3)

by LU factorization or other techniques. For example, Eqs. (3.2.1)and (3.2.2) may be efficiently solved with sparse matrix iterativeschemes [17].

3.3. Efficient Stark field integration

An advantage of the method is the dispersion formula in Eq.(3.1.4) that defines the structure of J(u;3); sj

0locates the center and

sj00the half width at half maximum of the jth resonance in J(u;3).

Thus, the integration over the Stark field can be optimized. Forexample, the minimal set of points necessary to resolve thedistribution function P(3) is, if necessary, supplemented with pointsto resolve J(u;3) obtained from knowledge of the poles. As analternative, it may be possible to use adaptive quadrature to eval-uate the integral (see Sections 3.4 and 4.2)

3.4. Operation count

The operation count determines the relative costs of an algo-rithm. Although the matrices A and B are typically sparse, it isexpedient to ignore this property to obtain conservative estimatesof the relative computational efficiencies. In the following, allcounts assume operations performed with complex numbers.

The eigen decomposition requiresw27m3 operations, which areconsiderably more operations than a single matrix inversion [9]. Onthe other hand, only m operations are necessary to solve Eq. (3.1.2)at each field value. In addition, the solution of Eqs. (3.2.1)e(3.2.3)requires two LU factorizations with mþ 2 right hand sidesleading to w2m3 operations. Thus, the total operations of theeigenvector decomposition at each frequency is

Nvz29m3 þ N3m (3.4.1)

to be comparedwith Eq. (2.6). Note that since the cost of computingJ(u;3) at any given field value is of order m operations, it may befeasible to use an adaptive quadrature algorithm rather than opti-mizing the field mesh to perform the integration.

4. Hessenberg method

The eigen decomposition is a relatively expensive procedurethat can suffer from numerical instability for non-Hermitian

matrices [9]. It is possible, however, to take advantage of the ideasin Section 3 using the numerically stable Hessenberg decomposi-tion together with the QR algorithm [9,12,13].

4.1. Hessenberg decomposition

The Hessenberg decomposition is defined as [9]

C ¼ QWQh (4.1.1)

where Q is a unitary matrix and W is an upper Hessenberg matrix(i.e.; Wij¼ 0 for i> jþ 1). Following manipulations similar to thoseleading to Eq. (3.1.1) yields

�d!*�t

$½A� 3B��1r d! ¼

�d!*�t

Q$½1� 3W��1QhA�1r d!

(4.1.2)

and the procedure again avoids explicit inversion by using Eqs.(3.2.1) and (3.2.2).

4.2. Operation count

The Hessenberg decomposition together with the QR algorithmto obtain the unitary transformation requires w5m3 operations. Inaddition, obtaining the eigenvalues involves additional w5m3

operations [9]. On the other hand, the solution of Eq. (4.1.2) usingLU factorization of a Hessenberg matrix takes m2 operations ratherthan m as in the eigenvector approach [13]. Thus, the total opera-tion count at each frequency, including solving Eqs. (3.2.1) and(3.2.2), is

Nwz12m3 þ N3m2 (4.2.1)

which is less than Nv in Eq. (3.4.1) for m�N3/17.Comparing the operational count of the Hessenberg method to

explicit inversion yields from Eqs. (2.6) and (4.2.1)

Nw

Niz12þ N3

mN3

/N3�m

12N3

< 1 (4.2.2)

since typically N3> 12 [10,11]. Note that the ratio in Eq. (4.2.2)would be increased by a factor of 3 when comparing to a solutionof Eq. (2.7) by LU factorization. In that case the Hessenbergdecomposition approach may be approximately comparable inspeed for not very large N3; however, there is the benefit in theHessenberg method of advance knowledge concerning the field-dependent resonances.

It is interesting to consider using an efficient and reliableadaptive quadrature algorithm to perform the integration over thefield. To obtain the eigenvalues of the matrix C, or equivalently thepoles of J(u;3), requires w5m3 operations. This cost could be avoi-ded so that the operation count now is given by

~Nwz7m3 þ ~N3m2 (4.2.3)

where ~N3 is determined by the adaptive quadrature routine. Themerits of this approach depend on the size of the matrix comparedto the relative values of N3 and ~N3 to achieve the desired accuracy inthe numerical integration.

4.3. Numerical issues

Although the matrix A(u) is non-singular, it may be ill condi-tioned and solving Eqs. (3.2.1) and (3.2.2) could lead to inaccurateresults. As an alternative, the generalized Schur decomposition canbe applied to matrices A(u) and B. Even though B can be a singularmatrix, this is a numerically stable algorithm requiring w66m3

Fig. 2. J(o)(Du;3) for the shifted Stark components of the hydrogen Lyman-a line atconditions of Fig. 1 versus Stark field in units of the Holtsmark field 3H [3] atDu¼ 0.001 eV. The solid dots represent calculations on a fixed field mesh.

C.A. Iglesias, V. Sonnad / High Energy Density Physics 6 (2010) 399e405402

operations [9]. This is a computationally expensive procedure,nevertheless, it may be the best alternative for numerically difficultcases. Note that the generalized Schur decomposition provides theeigenvalues and requires order m2 operations to solve the simul-taneous equations at each field value. Incidentally, the Schurdecomposition can be applied to the matrix C(u) leading to similarresults as those in Section 4.1. Although it is also numerically stable,the Schur decomposition requires about twice the number ofoperations as the Hessenberg method including eigenvalues[9,12,13].

5. Numerical example

It is well known that the far line wing of one-electron radiatorsinvolves large Stark field values. In line shape calculationscompromises are often made in order to minimize the numericaleffort by using a widely spaced field mesh in the tail of the distri-bution [10,11]. As a simple example amenable to analyticalmanipulations, results for the Stark shifted components of thehydrogen Lyman-a line are plotted in Fig. 1 where the details of thecalculation are provided in Appendix B. Briefly, in one case theintegration over the microfield distribution uses a fixed field meshwhile the second calculation has a flexible mesh that resolves theresonance in J(u;3) at each frequency. Fig. 1 clearly shows that theflexible mesh produces a smooth line wing while the fixed meshyields large oscillations beyond a certain detuning frequency.

The results in Fig. 1 are readily explained by considering theresonant structure of J(u;3). A plot of J(u;3) at Du¼ 0.001 eV in Fig. 2shows that the fixed mesh resolves the resonant feature at thisfrequency. Therefore, the resulting profile near line center isnumerically accurate. At much larger detuning frequencies,however, the fixed mesh fails to resolve the resonance as shown inFigs. 3 and 4. Consequently, the fixed mesh integration can over-estimate or underestimate the correct value leading to the oscil-latory behavior in Fig. 1.

The numerical example is intentionally simple and clearlyillustrates the advantages of determining the resonant structure of

Fig. 1. Stark shifted component contribution to the hydrogen Lyman-a profile atT¼ 1 eV and Ne¼ 1016 cm�3 versus detuning frequency. Calculations using a fixedmesh (solid) and with a flexible field mesh (dash).

the field-dependent line shape. By using the Hessenberg method inSection 4 it is possible to add judiciously field points and achieveaccurate integration without unduly increasing the computationaleffort. Note that the added points may include extension of theintegration range to include features that otherwise would bemissed using a fixed field mesh. This last point is relevant evenwhen using an adaptive quadrature scheme so that importantcontributions to the integral are not omitted. Obviously, thecomputational efficiency of the proposed method would be betterrealized for large matrices.

Fig. 3. Same as Fig. 2 at Du¼ 0.00475 eV. The dashed line indicates the error using thetrapezoidal rule with the fixed mesh.

Fig. 4. Same as Fig. 3 at Du¼ 0.00525 eV.

C.A. Iglesias, V. Sonnad / High Energy Density Physics 6 (2010) 399e405 403

6. Conclusion

A robust numerical algorithm for calculating quasi-static Starkbroadening of spectral lines without introducing approximationswas proposed. The method evaluates the field-dependent lineshape using the Hessenberg decomposition of a specially con-structed field independent matrix requiring w7m3 operations fora matrix of sizem. Then to compute the field-dependent line shapeat each field value involves solving a system of simultaneous linearequations requiring order m2 operations compared to order m3 inthe conventional approach using explicit matrix inversion. There-fore, the proposed method is more computationally efficient thanthe conventional approach, in particular for complex problemsinvolving large matrices. Incidentally, successful theoreticalapproaches to incorporate ion dynamics, an important correctionto the standard theory, use the quasi-static line shape as input thatcan dominate the computational effort [18,19]; thus, these modelscould benefit from the present method.

An interesting feature of the proposedmethod is identification ofthe resonant structure in the field-dependent line shape requiringadditional w5m3 operations to obtain the eigenvalues of the fieldindependent Hessenberg matrix. Thus, it is possible to optimize thefield mesh at each frequency when performing the integration overthe microfield distribution leading to improved numerical accuracywhile minimizing the computational effort. Alternatively, themethod is sufficiently efficient that using a reliable adaptive quad-rature algorithm to perform the integration may be feasible.

The discussion relied on the eigen decomposition of a fieldindependent matrix to illustrate advantages of the presentapproach. Although this more familiar decomposition may appearas an attractive method, it can be numerically unstable for non-Hermitian matrices involved in line broadening theory. Since theintent is to provide a robust algorithm to calculate quasi-static lineshapes, the eigen decomposition is not recommended.

Acknowledgments

It is a pleasure to thank Richard W. Lee for valuable discussions.This work was performed under the auspices of the U.S.

Department of Energy by Lawrence Livermore National Laboratoryunder Contract DE-AC52-07NA27344.

Appendix A. Eigen decomposition

The eigen decomposition is not new to line shape theory and hasbeen applied previously in both formal and practical developments.Two examples are given below that help distinguish these earlierapproaches from the present method.

A.1. Decomposition of resolvent

It follows from Eq. (2.1) that the field dependent line shape canbe written in the form

Jðu; 3ÞfIm�utRðu; 3Þv� (A.1.1)

and an eigen decomposition such that

Uðu; 3ÞhRðu; 3Þ�1�uI ¼ Pðu; 3ÞSðu; 3ÞP�1ðu; 3Þ (A.1.2)

with S(u;3) a diagonal matrix with the eigenvalues of U(u;3) yields

Jðu; 3ÞfImnutPðu; 3Þ½uI þ Sðu; 3Þ��1P�1ðu; 3Þv

o(A.1.3)

involving inversion of a diagonal matrix.The result in Eq. (A.1.3) is formally exact but has not lead to

practical applications. Specifically, the eigen decomposition of U

(u;3) requires w13m3 for a matrix of size m (includes an optimisticfactor of 1/2 to account for symmetry) and the procedure can beunstable for non-Hermitian matrices [9]. (Line broadening theoryusually includes relaxation processes that yield non-Hermitianmatrices.) Furthermore, the procedure must be repeated for eachfield value and any frequencies where F(u) displays significantfrequency dependence. Consequently, without some approxima-tion or numerical technique the method remains a formalmanipulation.

A.2. Decomposition of Stark Hamiltonian

As an alternative, it is possible to perform an eigenvectordecomposition of the real, symmetric matrix contribution repre-senting the radiator in the external Stark field [7]

H5I � I5H* � 3B ¼ Mð3ÞEð3ÞMtð3Þ (A.2.1)

where 5 denotes the Kronecker product of the initial and finalstates subspaces of the radiator that generate the line spacerepresentation and E(3) is a diagonal matrix with the energyeigenvalues. Then, Eq. (A.1.1) can be written as

Jðu; 3ÞfIm�utMð3Þ

huI þ Eð3Þ � i~Fðu; 3Þ

i�1Mtð3Þ�1v

�(A.2.2)

The resulting expression does not contain a diagonal matrix since itinvolves the width and shift operator transformed to the newrepresentation,

~Fðu; 3Þ ¼ Mtð3ÞFðuÞMð3Þ (A.2.3)

and in general requires 2m3 operations.The practical advantage of this approach is realized when the

off-diagonal elements of F(u) are assumed to vanish. Thenapproximating ~Fðu; 3Þ by its diagonal part reduces the operationcount in Eq. (A.2.3) to order m2. This approximation is presumedvalid in cases where the homogeneous broadening is smallcompared to the ion Stark broadening. Since M(3) is frequency

C.A. Iglesias, V. Sonnad / High Energy Density Physics 6 (2010) 399e405404

independent, the approximation leads to substantial savings incomputational effort [7].

Unfortunately, in cases where the off-diagonal terms of F(u)are not negligible the operation count for the diagonal elements of~Fðu; 3Þ in Eq. (A.2.3) is of order m3. More importantly, it is notclear that approximating ~Fðu; 3Þ by only the diagonal terms canreproduce subtle interference effects present even for large Starkfields [20].

A.3. Present method

The present method does not follow either of the approachesabove. Instead, as shown in Sections 3 and 4, it exploits the func-tional form of the field dependence in J(u;3) by rewriting Eq. (A.1.1)in the form

Jðu; 3ÞfImnut ½AðuÞ � 3B��1v

o¼ Im

nut ½I � 3CðuÞ��1A�1ðuÞv

o(A.3.1)

where C(u)¼ A�1(u)B. The functional form in Eq. (A.3.1) isamenable to methods used to solve large-scale linear systemsreliably and efficiently without introducing approximations[12,13].

Interestingly, A(u) is a non-Hermitian matrix that in general hasnon-zero off-diagonal elements. Thus, it may appear advantageous,since B is a real symmetric matrix, to introduce instead C�1(u)¼B�1A(u) in Eq. (A.3.1). Unfortunately, the matrix B can be singularwhere A(u) is never singular. Note that C(u) being singular does notaffect the present method.

Appendix B. Calculation of Lyman-a line

The Lyman-a spectral line describes a radiative transitionbetween principal quantum numbers n¼ 2 and n¼ 1 in thehydrogen atom. In the present calculations all interactionsbetween states with different principal quantum numbers areneglected simplifying the algebra. The resulting matrix involvingthe upper 4 states reduces to block diagonal form: a 2� 2 blockplus 2 (1�1) blocks. Therefore, analytical expression can bereadily obtained.

B.1. Line shape function

The line shape function can be written in the form,

IðuÞ ¼ S3

n2Ið1ÞðuÞ þ Ið0ÞðuÞ

o(B.1.1)

where S is the total line strength. The first term is the areanormalized profile for the unshifted 2p states with magneticquantum number jmj ¼ 1 that are not affected by the quasi-staticStark field,

Ið1ÞðuÞ ¼ gðuÞ=pDuþ g2ðuÞ (B.1.2)

with g(u) the electron impact width (line shifts are neglected forsimplicity). The second term is the contribution from the 2s and 2pstates with m¼ 0 that couple through the Stark field,

Ið0ÞðuÞ ¼ZNo

d3Pð3ÞJð0Þðu; 3Þ (B.1.3)

with

Jð0Þðu; 3Þ ¼ �p�1Im

(Duþ 3igðuÞ

)

½Duþ igðuÞ�½Duþ 3igðuÞ� þ ½3eao3=Z�2

(B.1.4)

where ao is the Bohr radius and e the elemental electric charge.

B.2. Poles of J(0)(u;3)

The present method can be applied to the Lyman-a example. Forthe m¼ 0 block, Eqs. (2.3)e(2.5) yield

AðuÞ ¼�Duþ 3ig 0

0 Duþ ig

(B.2.1)

and

B ¼ 3eaoZ

�0 11 0

(B.2.2)

leading to

CðuÞ ¼ 3eaoZ�1

Du2 � 3g2 þ 4iDug

�0 ðDuþ igÞ

ðDuþ 3igÞ 0

(B.2.3)

The eigenvalues of C(u) are given by

lðuÞ ¼ � 3eaoZ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðDuþ igÞðDuþ 3igÞ

p (B.2.4)

Thus, from Eq. (3.1.7) the poles of J(0)(u;3) are located at

sðuÞ ¼ � -

3eao

n�Du2 � 3g2

�2þ16Du2g2o1=4

��cos

�qðuÞ2

�þ isin

�qðuÞ2

��ðB:2:5Þ

with

qðuÞ ¼ ArghDu2 � 3g2 þ 4iDug

i(B.2.6)

As an example consider Du¼ 0, then q¼�p leading tos ¼ �iZg=

ffiffiffi3

peao.

B.3. Calculations

The calculations used a standard approximation for the width,g(u), [21] and the necessary reduced dipole matrix elements areavailable [22]. For simplicity, the integration assumed the Holts-mark microfield distribution [3] and a trapezoidal rule. The fixedmesh integration used N3¼ 60 points in a predetermined grid [23].No systematic effort was made to optimize the mesh. Instead, theflexible mesh complemented the fixed grid with 7 uniformlyspaced points centered at the peak of the resonance with spacingset equal to Im s(u). Both the location and width of the resonanceare obtained from Eqs. (B.2.5) and (B.2.6). This flexible meshreduced the oscillations in the line wing to less than �5%compared to �100% with the fixed mesh.

References

[1] M. Baranger, in: D. Bates (Ed.), Atomic and Molecular Processes, AcademicPress, New York, 1962.

[2] H.R. Griem, Spectral Line Broadening by Plasma. Academic, New York, 1974.[3] J. Holtsmark, Ann.Phys. (Leipzig) 7 (1919) 578.[4] H. Margeneau, M. Lewis, Rev. Mod. Phys. 31 (1959) 569.[5] L.A. Woltz, C.F. Hooper, Phys. Rev. 38 (1988) 4766.[6] R.C. Mancini, et al., Comput. Phys. Commun. 63 (1991) 314.[7] A. Calisti, et al., Phys. Rev. A42 (1990) 5433.

C.A. Iglesias, V. Sonnad / High Energy Density Physics 6 (2010) 399e405 405

[8] P.A. Loboda, et al., Laser & Particle Phys. 18 (2000) 275.[9] G.H. Golub, C.F. Loan, Matrix Computations. John Hopkins Press, 1996.

[10] R.W. Lee (private communication).[11] R. Mancini (private communication).[12] A. Laub, IEEE Trans. Automat. Control 26 (1981) 407.[13] C.F. Loan, Math. Program. Study 18 (1982) 102.[14] R.D. Cowan, The Theory of Atomic Structure and Spectra. University of Cal-

ifornia Press, Berkeley, 1981.[15] A. Bar-Shalom, M. Klapisch, Comput. Phys. Commun. 50 (1988) 375.

[16] F.A. Parpia, et al., Comput. Phys. Commun. 94 (1996) 249.[17] R.W. Freund, N. Nachtigal, ACM Trans. Math. Softw. 22 (1996) 46.[18] D.B. Boercker, C.A. Iglesias, J.W. Dufty, Phys. Rev. A36 (1987) 2254.[19] A. Calisti, et al., Phys. Rev. E81 (2010) 016406-1.[20] C.A. Iglesias, doi: 10.1016/j.hedp.2010.10.007, in press.[21] H.R. Griem, M. Blaha, P.C. Kepple, Phys. Rev. A19 (1979) 2421.[22] B.W. Shore, D.H. Menzel, ApJS 12 (1965) 187.[23] I.I. Sobelman, L.A. Vainshtein, E.A. Yukov, Excitation of Atoms and Broadening

of Spectral Lines. Springer-Verlag, Berlin, 1981, p. 278.