Equation of state of classical Coulomb plasma mixtures

0.25 MB·English

by  A. Y. Potekhin

Equation of state of classical Coulomb plasma mixtures
Alexander Y. Potekhin∗ and Gilles Chabrier†
Ecole Normale Sup´erieure de Lyon, CRAL (UMR CNRS No. 5574), 69364 Lyon Cedex 07, France
Forrest J. Rogers
Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA 94550, USA
(Dated: January 8, 2009)
Wedevelopanalyticapproximationsofthermodynamicfunctionsoffullyionizednonidealelectron-
ion plasma mixtures. In the regime of strong Coulomb coupling, we use our previously developed
9
analytic approximations for the free energy of one-component plasmas with rigid and polarizable
0
0 electron background and apply the linear mixing rule (LMR). Other thermodynamic functions are
2 obtainedthroughanalyticderivationofthisfreeenergy. Inordertoobtainananalyticapproximation
for the intermediate coupling and transition to the Debye-Hu¨ckel limit, we perform hypernetted-
n
chain calculations of the free energy, internal energy, and pressure for mixtures of different ion
a
species and introduce a correction to the LMR, which allows a smooth transition from strong to
J
weak Coulomb coupling in agreement with thenumerical results.
8

PACSnumbers: 52.25.Kn,05.70.Ce,52.27.Gr
]
h
p
I. INTRODUCTION II. PLASMA PARAMETERS
-
m
s Letn betheelectronnumberdensityandn thenum-
e j
a
berdensityofionspeciesj=1,2,...,withmassandcharge
l
p We study the equation of state (EOS) of fully ionized numbersA andZ ,respectively. Thetotalnumberden-
j j
. nonideal electron-ion plasmas (EIP). In a previous work sity of ions is n = n . The electric neutrality
s ion j j
c [1, 2], hypernetted chain (HNC) calculations were per- implies n = Z n . Here and hereafter the brackets
e ion
ysi fcoarlmcueldatiaonnds oafnEalIyPticcofnotraminuilnage awesirneglperoiopnossepdecfioers.EFOoSr h...i denote avheriaging oPf the type hfi= jxjfj, where
x n /n .
h mixtures ofdifferentionspecies,the EOSwascalculated j ≡ j ion P
The state of a free electron gas is determined by the
p
usingthelinearmixingrule(LMR),whosehighaccuracy
electron number density n and temperature T. Instead
[ e
atΓ>1waspreviouslyconfirmedinanumberofstudies ofn itisconvenienttointroducethe dimensionlessden-
e
2 [3,4,5,6,7]. However,theLMRisinaccurateforweakly sity parameter r = a /a , where a = (4πn )−1/3 and
v coupledplasmas. Someconsequencesofitsviolationwere s e 0 e 3 e
a is the Bohr radius.
4 studied by Nadyozhin and Yudin [8], who showed that 0
Atstellardensitiesitisconvenienttouse,insteadofr ,
4 s
43 tahtemdoiffdeerreanteceCsobueltowmebencothueplliinngea(r0.a1n.d nΓon.lin1e)acramnisxhinifgt twhheerreelaptFiv=ity~p(a3rπa2mneet)e1r/3[1i0s]txhreele=lecptFro/nmeFcer=m0i.m01o4mrs−en1-,
the nuclear statistical equilibrium at the final stage of a
. tum. TheFermikineticenergyisǫ =c (m c)2+p2
12 stellar gravitationalcollapse. mec2, and the Fermi temperatureFequals TF e ǫF/kBF=−
p ≡
8 T (γ 1), where T m c2/k = 5.93 109 K,
r r r e B
− ≡ ×
0 γ 1+x2 , and k is the Boltzmann constant.
: Inthispaper,weperformHNC calculationsofthe free r ≡ rel B
v energy,internalenergy,andpressureformixtures ofvar- Thepions are nonrelativistic in most applications. The
i strength of the Coulomb interaction of ion species j is
X ious kinds of ions in the weak, intermediate, and strong
characterizedby the Coulomb coupling parameter,
coupling regimes, and suggest an analytic correction to
r
a the LMR. Γ =(Z e)2/(a k T)=Γ Z5/3, (1)
j j j B e j
where a = a Z1/3 is the ion sphere radius and Γ
In Sec. II we define the basic plasma parameters. In j e j e ≡
e2/(a k T). In a multicomponent plasma, it is useful to
Sec.IIIwecalculatetheEOSofionmixturesandpropose e B
introducethemeanion-couplingparameterΓ=Γ Z5/3
ananalyticformulaforthe EOSofmulticomponentEIP, eh i
[3]. At a melting temperature T , corresponding to Γ
applicable at any Γ values. The summary is given in m ≈
175(e.g.,[2]),theplasmafreezesintoaCoulombcrystal.
Sec. IV.
An important scale length is the thermal de Broglie
wavelength λ =(2π~2/m k T)1/2, where m is the ion
j j B j
mass. The electron thermal length λ is given by the
e
same expression with m replaced by m .
∗Also at the Ioffe Physical-Technical Institute, 194021 St. Peters- j e
burg,Russia;Electronicaddress: [email protected]ffe.ru The quantum effects on ion motion become impor-
†Electronicaddress: [email protected] tant at T Tp, where Tp ~ωp/kB and ωp =
≪ ≡
2
4πe2n Z2/m 1/2 is the ion plasma frequency. analytic approximation [1]. The electron-gas contribu-
ion
In this paper we consider only the classical Coulomb tionsto χT, CV,andS tendto zeroatT TF. Related
(cid:0)liquid, whi(cid:10)ch imp(cid:11)li(cid:1)es T T and T >T . numericalproblemsandtheir curewillbe≪discussedelse-
p m
≫ where [12].
III. EQUATION OF STATE
B. Nonideal plasmas containing one type of ions
Assuming commutativity of the kinetic and potential
operators and separation of the traces of the electronic
and ionic parts of the Hamiltonian, the total Helmholtz Let us recall fit formulae for nonideal EIP containing
free energy F can be conveniently written as a single kind of ions.
a. Electron exchange and correlation. Electron-
F =Fion+F(e)+F +F +F , (2) electron (exchange-correlation) effects were studied by
id id ee ii ie
manyauthors. For the reasonsexplainedinRefs. [1, 12],
where Fion and F(e) denote the ideal free energy of ions we adopt the fit to fee Fee/(NekBT) presented in
id id ≡
and electrons, and the last three terms represent an ex- Ref. [16].
cess free energy arising from the electron-electron, ion- b. One-component plasma. The internal energy of
ion, and ion-electron interactions, respectively. the liquid one-component plasma (OCP) at any values
ThepressureP,theinternalenergyU,andtheentropy of Γ is given by [2]
S of an ensemble of fixed number of plasma particles
in volume V can be obtained using the thermodynamic A A B Γ2 B Γ2
rUel=atiFon+sTPS.=The−s(e∂cFon/d∂-Vor)Tde,rthSer=mod−y(n∂aFm/i∂cTfu)Vnc,tiaonnds uii =Γ3/2(cid:18)√A21+Γ + 1+3Γ(cid:19)+ B21+Γ + B43+Γ2,
(5)
are derived by differentiating these first-order ones. The
where u U /k TN , and
ii ii B ion
decomposition (2) induces the analogous decomposition ≡
ofP,U,S,theheatcapacityC =(∂S/∂lnT) ,andthe
V V
logarithmic pressure derivatives χT = (∂lnP/∂lnT)V A3 =−√3/2−A1/ A2 (6)
and χ = (∂lnP/∂lnV) . Other second-order func-
ρ − T p
tions can be expressed through these by Maxwell rela- ensures the correcttransitionto the Debye-Hu¨ckellimit.
tions (e.g., [11]). The parameters A = 0.907347, A = 0.62849, B =
1 2 1
0.0045, B = 170, B =− 8.4 10−5, and B = 0.0037
2 3 4
− ×
allowonetoreproducethebestavailableMCsimulations
A. Ideal electron-ion plasmas of liquid OCP at 1 Γ 190 [13] with an accuracy
≤ ≤
matching the numerical MC noise.
The free energy of a gas of N = n V nonrelativistic From Eq. (5) one obtains the analytic expression for
j j
classical ions of jth kind is f F /k TN byintegration,andthentheCoulomb
ii ii B ion
≡
contributions to the other thermodynamic functions by
F(j) =N k T ln(n λ3/g ) 1 , (3) differentiation [2].
id j B j j j −
c. Electron polarization. Electron polarization in
(cid:2) (cid:3)
where gj is the spin multiplicity. The total free energy Coulomb liquid was studied by perturbation [14, 15]
is given by the sum Fiidon = jFi(dj). Analogous sums and HNC [1, 2, 4] techniques. The results for fie ≡
give U, S, P, and CV. Since Eq. (3) contains nj under Fie/NionkBT have been fitted by the expression [2]
P
logarithm, these sums for F and S naturally include the
entropy of mixing Smix =−kB jNjlnxj. f = Γ cDH√Γe+cTFaΓνeg1(rs,Γe)g3(xrel). (7)
The free energy of the electron gas is given by ie − e 1+ b√Γ +ag (r ,Γ )Γν/r γ−1
P e 2 s e e s r
F(e) =µ N P(e)V, (4) (cid:2) (cid:3)
id e e− id The coefficients cDH, cTF, a, b, ν and functions
g (r ,Γ )andg (x )parametricallydependontheion
whereµ istheelectronchemicalpotential. Thepressure 1,2 s e 3 rel
e charge Z. Here the coefficients c and c are not free
P(e) andthe numberdensityn =N /V arefunctionsof DH TF
id e e fit parameters, because
µ andT,whichcanbe writtenthroughthe Fermi-Dirac
e
integrals I (χ ,τ), where χ = µ /k T and ν = 1/2,
3/2, and 5ν/2.e In Ref. [1] wee gaveeanBalytic approxima- cDH =(Z/√3)[(1+Z)3/2 1 Z3/2] (8)
− −
tions for the Fermi-Dirac integrals, based on fits [9] to
electron-positron thermodynamic functions. The chemi- ensures the transition of the excess free energy to the
cal potential at a givendensity canbe found either from Debye-Hu¨ckel limit at small Γ, and c at large Z is
TF
a numerical inversion of function n (χ ,T) or using the given by the Thomas-Fermi theory [10].
e e
3
FIG. 1: Fractional difference between the Coulomb part of FIG. 2: Fractional differences between the nonideal part of
the internal energy (Uii) in different approximations and the thefreeenergyduetotheion-ionandion-electroninteractions
LMRpredictionasafunctionoftheaverageionCoulombcou- (Fii+Fie)indifferentapproximationsandtheLMRprediction
plingparameterΓforbinarymixturesofionswithZ2/Z1 =2 for a mixture of ions with Z1 = 1 and Z2 = 8 and x2 =
and x2 = 1−x1 = 0.05 and 0.2 in the rigid background. 1−x1=0.25intherigidandpolarizableelectronbackground,
Dashed lines (DH): Debye-Hu¨ckel formula, dot-dashed lines as functions of Γ. Dashed lines: DH, dot-dashed lines: CLM
(CLM): corrected linear mixing [8]; dots: MC results of [8]; asterisks: HNC results of Ref. [4]; solid lines: present fit.
Ref. [6] for x2 =0.05; crosses: HNCresults of Ref. [6]; trian-
gles: present HNCresults; solid lines: present fit.
authors found that the LMR remains accurate when the
electron response is taken into account in the inter-ionic
C. Nonideal mixtures of ions potential, as long as the Coulomb coupling is strong
(Γ>1).
Acommonapproximationfortheexcess(nonideal)free On the other hand, the LMR is invalid at Γ 1.
≪
energy of the strongly coupled ion mixture is the LMR, Indeed, in this case the Debye-Hu¨ckel theory gives
fLM(Γ) x f (Γ ,x =1), (9) fDH = Γ3/2/√3, fDH =fDHζDH, (10)
ex ≈ j ex j j ee − e ii ee ii
j
X and for the EIP
where superscript “LM” denotes the linear-mixing ap-
proximation, and all Γj correspond to the same Γe (as- feDxH =fiDiH+fiDeH+hZifeDeH =feDeHζeDipH, (11)
suming that the pressure is given almost totally by the
where
strongly degenerate electrons): Γ = ΓZ5/3/ Z5/3 . In
j j h i
Eq. (9), f is the reduced nonideal part of the free Z2 3/2 ( Z2 + Z )3/2
ex ζDH = h i , ζDH = h i h i . (12)
energy: fex = fii for the “rigid” (uniform) charge- ii Z 1/2 eip Z 1/2
neutralizing electron background and f = f +f + h i h i
ex ii ie
Z f for the polarizable background However,the LMR at Γ 1 gives another result:
j ee ≪
The highaccuracyofEq.(9) forbinary ionic mixtures
fLM fDHζLM, fLM fDHζLM, (13)
in the rigid background was first demonstrated by cal- ii ∼ ee ii ex ∼ ee eip
culations in the HNC approximation [3] and confirmed
where
later by MC simulations (e.g., [5, 6, 7]).
The validity of the LMR in the case of an ionic mix- ζLM = Z5/2 , ζLM = Z(Z +1)3/2 . (14)
ture immersed in a polarizable finite-temperature elec- ii h i eip h i
tron backgroundhas been examined by Hansen et al. [3] Nadyozhin and Yudin [8] considered several possible
in the first-order thermodynamic perturbation approxi- modifications of the LMR at intermediate Γ and found
mation and by Chabrier and Ashcroft [4] by solving the that such modification can appreciably shift the statis-
HNC equationswith effective screenedpotentials. These tical nuclear equilibrium at the conditions typical of the
4
FIG. 3: Fractional difference between the Coulomb part of FIG. 4: Fractional difference between the Coulomb part of
theinternalenergy indifferentapproximationsandtheLMR thefreeenergyandtheLMRpredictionforbinaryionicmix-
predictionforabinarymixtureofionswithZ2/Z1 =8inthe tures with Z2/Z1 =12 and Z2/Z1 =16, and with x2 =0.01,
rigidbackground. Dashedlines(DH):Debye-Hu¨ckelformula, 0.05, and 0.2. Dashed lines (DH): Debye-Hu¨ckel formula,
symbols: presentHNCresults; solid lines: present fit. Differ- symbols: present HNC results; solid lines: present fit for
entsymbolscorrespondtodifferentx2 values: 0.01(squares), Z2/Z1 = 16; long-dash-dot lines: present fit for Z2/Z1 = 12.
0.05 (solid triangles), 0.2 (dots), 0.25 (asterisks), 0.5 (empty Different symbols correspond to different combinations of x2
triangles), and 0.7 (empty circles). and Z2/Z1 values.
final stage of a stellar gravitational collapse. They con-
Ref. [3] ∆u/u and ∆f/f are negative at any Γ). Addi-
sidered the rigid background and advocated a modifica-
tional modifications of the coefficient A in Eq. (9) also
2
tion of every term in Eq. (9) by multiplying the leading
do not solve the problem.
fit coefficient at small Γ by factor d = Z2 /Z Z .
j h i jh i An alternative to the CLM, named “complex mixing”
It corresponds to replacing √3/2 in Eq. (6) by d √3/2
p j inRef.[8],maintainsthe correctsignof∆f and∆u,but
(and a possible simultaneous change of A ). For a com-
2 leads to still larger absolute values of these corrections
pressible background an analogous modification implies
(i.e., still slower recoveryof the LMR) at large Γ.
additionally replacement of Eq. (8) by
In order to find a more accurate approximation, we
have performed HNC calculations of F , U and P for
Z Z2 3/2 Z2 3/2 ii ii ii
c(j) = j h i +1 h i 1 . (15) binaryionicmixturesintherigidbackgroundforabroad
DH √3 "(cid:18) hZi (cid:19) − hZi3/2 − # Γ range and various values of the charge number ratios
Z /Z and fractional abundances x = 1 x . Some of
2 1 2 1
−
Hereafter modifications of this type will be called cor- the results are shown by triangles in Figs. 1 and 2. In
rected linear mixing (CLM). agreement with the previous studies (e.g., [3]), our nu-
The result of such modifications is shown by dot- merical results show monotonically decreasing fractional
dashed lines in Figs. 1 and 2, where we plot the ra- deviations from the LMR with increasing Γ. The re-
tios ∆u /uLM and ∆f /fLM as functions of Γ. Here sults agree to at least 5 digits with those published in
ii ii ex ex
∆f f fLM and ∆u u uLM are the deviations of [6] (crosses in Fig. 1). At Γ & 1, HNC results tend to a
≡ − ≡ −
the reducedfree andinternal energies,respectively, from constantresidualwithin1%,whichisduetotheintrinsic
the LMR. In Fig. 1 the electron response is neglected inaccuracy of the HNC approximation for strongly cou-
(rigid background). We see that, for example, at Γ 10 pled plasmas because of the lack of the bridge functions
≈
the CLM prescription gives ∆u and ∆f corrections of in the diagrammatic representation of this approxima-
about1%inFig.1andseveralpercentsinFig.2,whereas tion. To prove this statement, in Fig. 1 we plot by dots
they must be much smaller according to the MC results the values of ∆u /uLM from MC simulations [6]. The
ii ii
[6,7]. Moreover,∆u and∆f in the CLMapproximation latter simulations give tiny deviations from the LMR at
have the incorrect sign at Γ& 1 (note that according to Γ&3, which are invisible in the figure scale.
5
FIG. 5: Fractional difference between the Coulomb part of FIG. 6: Fractional difference between the Coulomb part of
the internal energy and the LMR prediction for binary ionic thefreeenergyandtheLMRpredictionformixturesofthree
mixtures at Γ1 = 0.03 as function of the ion charge ratio (solid lines) and two (dot-dashed and dotted lines) types of
Z2/Z1. Dashed lines (DH): Debye-Hu¨ckel formula, symbols: ions with charge ratios Zj/Z1 =2, 8, and 12. Different sym-
presentHNCresults;solidlines: presentfit. Differentsymbols bols correspond to HNC results with different combinations
correspond to different x2 values: x2 =0.01 (triangles), 0.05 of xj and Zj. Solid lines: Z2/Z1 = 8 or 12 (Z2 is marked
(dots), and 0.2 (asterisks). near the curves, assuming Z1 = 1), x2 = 0.2; Z3/Z1 = 12,
x3 =0.01. Dot-dashed lines: the same but without the third
kindofions(x3=0). Dottedline: binarymixturewithcharge
ratio Z3/Z1 =2 (x2=0) and numberfraction x3=0.2.
AsterisksinFig.2correspondtotheHNCcalculations
for mixtures in the polarizable electron background [4],
which give qualitatively the same results as the calcula- respectively,
tions for the rigid background. The second dot-dashed
curve in this figure shows CLM for the polarizable back- 3 abcΓb
∆u = ∆f, (17)
ground, according to Eq. (15). 2 − 1+aΓb
(cid:18) (cid:19)
A correction to the linear mixing rule, which exactly abcΓb 1 ab2cΓb
recoverstheDebyelimitatΓ 0andtheLMRatΓ 1, ∆c = ∆u ∆f. (18)
→ ≫ 1+aΓb − 2 − (1+aΓb)2
and which agrees with the HNC data, can be expressed (cid:18) (cid:19)
by the following analytic fitting formula: ThisapproximationhasbeencomparedwithourHNC
calculations for binary ionic mixtures in the rigid elec-
ζDH ζLM tron background at Z2/Z1 = 2, 5, 8, 12, and 16, with
∆f =fDH − , (16) x =1 x rangingfrom0.01to0.7. Someofthe results
ee (1+aΓb)c 2 − 1
are shown in the figures. In all figures the Debye-Hu¨ckel
approximation is drawn by dashed lines for comparison.
where ζ = ζii or ζeip for the rigid or polarizable back- In Figs. 1 and 2, discussed above, and in Figs. 3, 5 our
ground, respectively, and the parameters a, b, and c de- approximation is drawn by solid lines. Figure 3 shows
pend on plasma composition as follows: the fractional difference between the different results for
internal energy and the LMR approximation for binary
2.2δ+17δ4 ζLM ζDH ionic mixtures with Z2/Z1 = 8 and x2 = 0.01, 0.05, 0.2,
a = , δ = − ,
1 b Z5/2 0.25, 0.5, and 0.7 (marked near the curves). In Fig. 4
− h i we show the case of the highest considered chargeasym-
b = d−0.2, c=1+d/6, d= Z2 / Z 2.
h i h i metries: long-dash-dotlinesshowtheapproximation(16)
forZ /Z =12andsolidlinesforZ /Z =16. Finally,in
2 1 2 1
Thanks to the simple form of this formula, its deriva- Fig. 5 we show a dependence of ∆f /fLM on the charge
ii ii
tivesarealsorathersimple. Forexample,thecorrections ratio Z /Z at Γ = 0.03. Here different symbols corre-
2 1 1
to the reduced internal energy and heat capacity read, spond to the HNC results for different values of x .
2
6
We have also performed calculations for mixtures of
ions of three different types on the rigid electron back-
ground and compared the results with Eq. (16). The re-
sultsofthe comparisonareshowninFigs.6and7. Solid
linesshowthedifferenceoftheCoulombfreeenergyfrom
the LMR according to Eq. (16) for 3-component mix-
tures; for comparison, dot-dashed and dotted lines are
plotted for 2-component mixtures with the same charge
ratios;symbols representthe HNC results. In all consid-
eredcases,addingathirdcomponenttoabinarymixture
increasesthe deviationsofEq.(16)fromHNC resultsby
lessthanafactorof1.5. Weconcludethattheagreement
between the fit and numerical results remains satisfac-
tory.
IV. CONCLUSIONS
We have performed HNC calculations of the free
energy, internal energy, and pressure for various ionic
mixtures with different fractional abundances of the ion
species in a broad range of Z and Γ values. We have
constructed an analytic approximation to the deviation
fromtheLMR,whichrecoverstheDebye-Hu¨ckelformula
FIG.7: SameasinFig.6,butfordifferentchargeratiosand for multicomponent plasmas at Γ 1 and the LMR
≪
fractional abundances. Solid lines: Z2/Z1 =2and5(marked at Γ 1, and which describes our calculations at any
≫
near the curves), x2 = 0.05; Z3/Z1 = 12, x3 = 0.01. Dot- Γ values, as well as the HNC and MC results for the
dashed lines: the same but with x3 = 0. Dotted line: the internal energy of plasma mixtures, available in the
same with x2 =0 (x3=0.01). literature, with an accuracy better than 2%.
The difference between the HNC results and formula
(16)forf lieswithin0.013andwithin1%. Theinternal
ex
Acknowledgments
energy calculated using the analytic derivative of the fit
(16) deviates from the HNC results by not more than
0.017andnotmorethan1.5%. Thesemaximaldeviations The work of G.C. and A.Y.P. was partially supported
areattainedfortheextremelyasymmetricmixtureswith by the CNRS French-Russian Grant No. PICS 3202.
Z /Z =16. The work of A.Y.P. was partially supported by the Ros-
2 1
The analytic formula (16) has also been compared to nauka Grant NSh-2600.2008.2and the RFBR Grant 08-
available HNC results for binary ionic mixtures in the 02-00837. The work of F.J.R. was partially performed
polarizable electron background [4] and found to be sat- under the auspices of the U.S. Department of Energy
isfactorywithinthe accuracyofthe latter results(anex- byLawrenceLivermoreNationalLaboratoryunderCon-
ample is shown in Fig. 2). tract DE-AC52-07NA27344.
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