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Addendum to “Equation of state of classical Coulomb plasma mixtures”

A. Y. Potekhin,1,2,*G. Chabrier,2,†A. I. Chugunov,1H. E. DeWitt,3and F. J. Rogers3

1Ioffe Physical-Technical Institute, 194021 St. Petersburg, Russia

2Ecole Normale Supérieure de Lyon, CRAL, UMR CNRS 5574, 69364 Lyon Cedex 07, France

3Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, California 94550, USA

?Received 25 June 2009; revised manuscript received 11 September 2009; published 9 October 2009?

Recently developed analytic approximation for the equation of state of fully ionized nonideal electron-ion

plasma mixtures ?A. Y. Potekhin, G. Chabrier, and F. J. Rogers, Phys. Rev. E 79, 016411 ?2009??, which covers

the transition between the weak and strong Coulomb coupling regimes and reproduces numerical results

obtained in the hypernetted-chain ?HNC? approximation, is modified in order to fit the small deviations from

the linear mixing in the strong-coupling regime, revealed by recent Monte Carlo simulations. In addition, a

mixing rule is proposed for the regime of weak coupling, which generalizes post-Debye density corrections to

the case of mixtures and numerically agrees with the HNC approximation in that regime.

DOI: 10.1103/PhysRevE.80.047401 PACS number?s?: 52.25.Kn, 05.70.Ce, 52.27.Gr

I. INTRODUCTION

The high accuracy of the linear-mixing rule ?LMR? for

multicomponent strongly coupled Coulomb plasmas has

been confirmed in a number of papers ?1–7?. Nevertheless,

the accuracy of modern Monte Carlo ?MC? calculations al-

lows one to reveal certain deviations from the LMR for the

Coulomb energy U of binary ionic mixtures ?BIM?. On the

other hand, for weakly coupled plasmas, the Debye-Hückel

?hereafter DH? formula is applicable instead of the LMR.

Several terms in the density expansion of U beyond the DH

approximation were obtained by Abe ?8? and by Cohen and

Murphy ?9? ?hereafter ACM? in the one-component plasma

?OCP? case.

In Ref. ?3?, deviations from the LMR for BIM were stud-

ied in the hypernetted-chain ?HNC? approximation and fitted

by Padé approximants. In Ref. ?4?, the LMR was confirmed

by HNC method for polarizable background of partially de-

generate electrons. In Ref. ?5?, deviations from the LMR for

strongly coupled BIM were studied using both HNC and MC

techniques. The corrections to the LMR for U were found to

be on the same order of magnitude for HNC and MC but

numerically different; in particular, it does not depend on the

mean ion Coulomb coupling parameter ? according to HNC

results but decreases as a function of ? in MC simulations.

These results were confirmed in Ref. ?7?, where an analytic

fit to the calculated corrections was suggested. The fitting

formulas of Refs. ?3,7? are applicable only at ??1; in par-

ticular, they do not reproduce the DH limit at ?→0 ?besides,

the fit parameters in ?3? are given only for five fixed ionic

charge ratios from 2 to 8?.

In Ref. ?10?, HNC calculations of BIM and three-

component ionic mixtures ?TIMs? were performed in a wide

range of values of ?, charge ratios, and partial densities of

the ion components, and a parametric formula was suggested

to fit the fractional differences between the LMR and calcu-

lated plasma energies at any ? in liquid multicomponent

plasmas. It recovers the DH formula at ??1 and gives a

vanishing fractional difference from the LMR at ??1.

However, in the regime of strong coupling, the accuracy

of the HNC method ?typically a few parts in 1000, for U? is

not sufficient to reproduce the values of the energies of mix-

tures at the precision level needed to study deviations from

the LMR ?see, e.g., ?5??. Indeed, according to Refs. ?3,5,7?

these deviations are typically on the order of a few ??10−3

−10−2?kT per ion ?where k is the Boltzmann constant?, while

U?−?kT per ion at ??1.

In this Brief Report, we suggest two improvements for

analytic treatment of ion mixtures. First, we introduce a mix-

ing rule for weakly coupled plasmas, which provides an ex-

tension of the ACM formula to the case of ion mixtures and

agrees with HNC results up to the values of the Coulomb

coupling parameter ??0.1 ?whereas, the DH approximation

becomes inaccurate at ??0.01?. Second, using MC simula-

tions of strongly coupled liquid BIM, supplementary to those

already published in ?5–7?, we suggest a modified version of

the formula ?10?, which maintains the accuracy of the previ-

ous fit at intermediate and weak coupling, but delivers con-

sistency with the MC data for strongly coupled Coulomb

liquids.

In Sec. II we introduce basic notations and formulas; in

Sec. III we propose a mixing rule applicable at weak cou-

pling; in Sec. IV we present a fitting formula for the internal

energy of mixtures, applicable in the entire domain of ?

values for weakly and strongly coupled classical Coulomb

gases and liquids; and in Sec. V we summarize the results.

II. BASIC EQUATIONS

Let nebe the electron number density and njthe number

density of ion species with charge numbers Zj?j=1,2,...?.

The total number density of ions is nions=?jnj. The electric

neutrality implies ne=?Z?nions. Here and hereafter, the angu-

lar brackets denote averaging with statistical weights propor-

tional to nj

?Z? ??

j

xjZj,where xj?

nj

nions

.

?1?

*Electronic address: palex@astro.ioffe.ru

†Electronic address: chabrier@ens-lyon.fr

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©2009 The American Physical Society047401-1

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The strength of the Coulomb interaction of ion species j is

characterized by the Coulomb coupling parameter defined ?in

cgs units? as ?j=?Zje?2/ajkT=?eZj

ion sphere radius, ?e?e2/aekT, and ae??4?ne/3?−1/3. In

other words, partial coupling parameters ?jand ion sphere

radii ajare defined to be those of the OCP of ions of the jth

kind at the same electron density ne, as in the considered

multicomponent plasma. The Coulomb coupling in the mix-

ture of different ions is conventionally characterized by the

average coupling parameter ?=?e?Z5/3?.

A common approximation for the Coulomb contribution

to the internal energy of a strongly coupled ion mixture is the

LMR

5/3, where aj=aeZj

1/3is the

uLM??? =?

j

xju??j,xj= 1?,

?2?

where u?U/NionskT is the reduced Coulomb energy, Nionsis

the total number of all ions, and the subscript “LM” denotes

the linear-mixing approximation. Obviously, the LMR has

the same form for the Coulomb contribution to the reduced

Coulomb free energy f?F/NionskT.

When the Coulomb interaction is sufficiently weak com-

pared to the thermal energy, then the DH approximation can

be applied uDH=?q2?/kTrD, where ?q2? is the mean-squared

charge ofthe considered

=?kT/4?nions?q2??1/2is the Debye radius. For the model of

ions in the “rigid” electron background, applicable if the

electrons are extremely strongly degenerate, ?q2?=e2?Z2?,

whereas in the case of completely nondegenerate electrons,

using our definition ?1? of averaging over the ion species and

taking into account the neutrality condition, we have ?q2?

=e2??Z2?+?Z??.

In this Brief Report, we consider the model of rigid elec-

tron background, but the extension to the case of compress-

ible background is possible by adjusting the parameter ? in

Eq. ?9? below, according to the expression for ?q2?. In Ref.

?10?, this extension was shown to be compatible with nu-

merical HNC data ?4? for ion mixtures with allowance for

electron polarization.

mixtureand

rD

III. WEAKLY COUPLED ION MIXTURES

For a OCP at ??1, a cluster expansion yields ?8,9?

u = −?3

2?3/2− 3?3?3

− ?9/2?1.687 5?3 ln ? − 0.235 11? + ¯ ,

where CE=0.577 21... is the Euler constant. Here, the first

term is the DH energy.

In order to generalize this expression to the case of mul-

ticomponent Coulomb plasmas, let us write the OCP energy

in the form

8ln?3?? +CE

2

−1

3?

?3?

u??? = ?u ˜?a/rD?,

?4?

where a is the ion sphere radius for the OCP, and u ˜ is the

Coulomb energy per ion in units of ?eZ?2/a ?u ˜=−0.9 in the

ion sphere model ?11??. Then the following relation holds in

the DH approximation for multicomponent plasmas:

u =?

j

xj?ju ˜??j?,

?j?aj

rD

=?3?e

?Z2?

?Z?Zj

1/3.

?5?

Let us assume that relation ?5? can be applied also to the

higher-order corrections beyond DH. In this case, according

to Eqs. ?3? and ?4?, in the ACM approximation

2− ?4?1

− ?7?1

u ˜??? = −?

4ln ? − 0.014 908 5?

8ln??? − 0.073 69?.

?6?

Since ???? for a fixed composition, f can be obtained from

u by integration, which yields

f =?

j

xj?jf˜??j?,

?7?

where

f˜??? = −?

3−?4

12?ln ? − 0.226 3? − ?7?0.027 78 ln ?

− 0.019 46?.

?8?

In Figs. 1–4, deviations from the LMR ?u?u−uLM, cal-

culated according to Eqs. ?5? and ?6?, are plotted by long-

dashed lines and compared to the DH formula ?short-dashed

lines? and the HNC data ?crosses?. We see that the suggested

approximation ?5? agrees with the data to much higher ?

values than the DH approximation.

FIG. 1. ?Color online? Correction to the LMR ?u=u−uLMas a

function of ? for BIM with Z2/Z1=2, x2=0.2. HNC ?crosses? and

MC ?dots? data are compared to the DH approximation ?short-

dashed lines?, the modified ACM approximation ?5? ?long-dashed

lines?, the fit from ?10? ?dot-dashed lines?, and the present fit ?9?

?solid lines?.

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IV. COULOMB LIQUIDS AT ARBITRARY COUPLING

In order to find an analytic approximation for the correc-

tion to the LMR in the largest possible interval of ? for ion

gases and liquids, we have selected from the numerical HNC

data ?10? the subset related to ??1, which counts 161 dif-

ferent combinations of x2, Z2, and ? in BIM and 54 combi-

nations of x2, Z2, x3, Z3, and ? in TIM ?assuming Z1=1?,

supplemented this HNC data by numerical MC data for BIM

at ??1 ?94 combinations of x2, Z2, and ??, and looked for

an analytic formula, which provides a reasonable compro-

mise between simplicity and accuracy for representing this

data. The MC data have been partly taken from the previous

work ?5–7? and partly obtained by new MC simulations us-

ing the same computer code as before. Our fitting formula

for the addition to the reduced free energy f=F/NionskT, rela-

tive to the LMR prediction fLM, reads as

?f ? f − fLM=?e

3/2?Z5/2?

?3

?

?1 + a????1 + b????,

?9?

where ? is determined by the difference between the LMR

and DH formula at ?→0 ?exactly as in Ref. ?10??

? = 1 −

?Z2?3/2

?Z?1/2?Z5/2?,

?10a?

for rigid electron background model, and

? =?Z?Z + 1?3/2?

?Z5/2?

−??Z2? + ?Z??3/2

?Z?1/2?Z5/2?

,

?10b?

for polarizable background. The expression ?10b? for ? is

exact in the limit of nondegenerate electrons, but its use in

Eq. ?9? provides a satisfactory agreement with numerical

data ?4? obtained with allowance for the polarizability of

partially degenerate electron gas ?see ?10??.

The fit parameters a, b, and ? are chosen so as to mini-

mize the mean-square difference between the fit and the data

for ?u/uLMat ??1 and for ?u at ??1, while the power

index ? is defined so as to quench the increase of ?f at ?

→?. These parameters depend on the plasma composition as

follows:

a =2.6? + 14?3

1 − ?

b = 0.0117??Z2?

,

? =?Z?2/5

?Z2?1/5,

?11?

?Z?2?

2

a,

? =

3

2?− 1.

?12?

FIG. 2. ?Color online? ?u=u−uLMas a function of ? for BIM

with Z2/Z1=2, x2=0.05. Here crosses ?HNC1? correspond to ?u

obtained from the HNC data using the OCP fit from ?12? for calcu-

lation of uLM, and asterisks ?HNC2? correspond to ?u from Ref. ?5?,

where both u and uLMare based on the HNC results. Dots ?MC1?

correspond to ?u calculated from the recent MC data for u, and uLM

calculated from the OCP fit ?12?, while circles ?MC2? represent MC

data ?5? for ?u.

FIG. 3. ?Color online? The same as in Fig. 1 but for Z2/Z1=5

and x2=0.05.

FIG. 4. ?Color online? The same as in Fig. 1 but for Z2/Z1=8

and two values of x2: 0.01 and 0.1 ?marked near the dots?.

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The numerical difference of Eq. ?9? from the formula in

Ref. ?10? is small at ??1, but at ??1 the correction to the

LMR prediction for the reduced internal energy

=?3

?u = ????f?

??

2−

a???

1 + a??−b????

1 + b????f

?13?

now decreases at large ? in agreement with the MC results.

Moreover, Eq. ?13? describes most of the data with much

higher accuracy than the fit to ?u suggested in Ref. ?7? for

BIM at ??1.

A comparison of the numerical HNC data for ?f and ?u

and MC data for ?u to Eq. ?9? and to the previous fit ?10?

shows that the present fit has nearly the same accuracy as the

previous one for BIM at ??1 ?slightly worse for small

?u/u, slightly better for larger ?u/u?, but it is generally

better for TIM at ??1 and substantially better for BIM at

??1. Examples of ? dependences of ?u are shown in Figs.

1–4, where the dot-dashed lines correspond to the older fit

and the solid lines to the present fit. The modification of the

fit at small ? values proves to be negligible, which has been

checked by comparison of fractional differences between the

Coulomb part of the free energy and the LMR prediction, as

in Ref. ?10?; whereas the modification at large ? can be

significant, as confirmed by Figs. 1–4.

V. CONCLUSIONS

We have reconsidered free and internal energies of classi-

cal ionic mixtures in the liquid state, taking into account the

results of HNC calculations in the regime of weak and mod-

erate Coulomb coupling and MC simulations at strong cou-

pling, and proposed two analytic approximations for such

mixtures: the mixing rule ?5?, which works well at the Cou-

lomb coupling parameter ??1, and the analytic fitting for-

mula ?9?, which is along with its derivative ?13? applicable at

any values of ?.

ACKNOWLEDGMENTS

The work of A.I.C. and A.Y.P. was partially supported by

the Rosnauka Grant No. NSh-2600.2008.2 and the RFBR

Grant No. 08-02-00837. The work of H.E.D. and F.J.R. was

partially performed under the auspices of the U.S. Depart-

ment of Energy at Lawrence Livermore National Laboratory

under Contract No. DE-AC52-07NA27344.

?1? J. P. Hansen and P. Vieillefosse, Phys. Rev. Lett. 37, 391

?1976?.

?2? J. P. Hansen, G. M. Torrie, and P. Vieillefosse, Phys. Rev. A

16, 2153 ?1977?.

?3? B. Brami, J. P. Hansen, and F. Joly, Physica 95A, 505 ?1979?.

?4? G. Chabrier and N. W. Ashcroft, Phys. Rev. A 42, 2284

?1990?.

?5? H. DeWitt, W. Slattery, and G. Chabrier, Physica B 228, 21

?1996?.

?6? H. E. DeWitt and W. Slattery, Contrib. Plasma Phys. 39, 97

?1999?.

?7? H. E. DeWitt and W. Slattery, Contrib. Plasma Phys. 43, 279

?2003?.

?8? R. Abe, Prog. Theor. Phys. 21, 475 ?1959?.

?9? E. G. D. Cohen and T. J. Murphy, Phys. Fluids 12, 1404

?1969?.

?10? A. Y. Potekhin, G. Chabrier, and F. J. Rogers, Phys. Rev. E 79,

016411 ?2009?.

?11? E. E. Salpeter, Aust. J. Phys. 7, 373 ?1954?.

?12? A. Y. Potekhin and G. Chabrier, Phys. Rev. E 62, 8554 ?2000?.

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