PHASE SEPARATION OF IONIC MIXTURES

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PHASE SEPARATION OF IONIC MIXTURES

J. Hansen

To cite this version:

J. Hansen. PHASE SEPARATION OF IONIC MIXTURES. Journal de Physique Colloques, 1980, 41

(C2), pp.C2-43-C2-52. �10.1051/jphyscol:1980207�. �jpa-00219799�

PHASE SEPARATION OF IONIC MIXTURES *

Abstract.- Recent calculations of the phase diagram of binary ionic mixtures and al- loys in a uniform background of degenerate electrons are critically reviewed. The limits of weak and strong electron screening are treated separately and are found to yield similar critical parameters. The dependence of the critical temperature on pressure and ionic charge ratio is explicitly investigated, in particular for ratios much larger than one. Among astrophysical applications, the possible segregation of

iron from hydrogen under solar conditions is briefly considered.

1. Introduction.- Except under extreme con- ditions, stellar matter is made up of ions, belonging to several fully of partially ioni- zed atomic species, and of unbound (nearly free) electrons. At sufficiently high densi- ties and temperatures the multi-component io- nic fluid can be essentially described by classical Statistical Mechanics, while the electron gas behaves like a degenerate, and possibly relativistic, Fermi gas. Due to the Pauli principle, the electronic and ionic fluids can, to a first approximation, be con- sidered as de-coupled.

The present paper deals with such highly com- pressed Coulombic systems, and reviews some of the recent theoretical results which have been obtained for the thermodynamic and structural properties of binary (two atomic species) ionic fluids. An important theoreti- cal issue, which may have far-reaching astro- physical consequences, is the miscibility of two ionic species in a background of elec- trons. Since the pioneering work of Steven- son [l] on mixtures of protons and a-parti- d e s in the interior of Jupiter, it is now

well-established that binary ionic mixtures phase separate at sufficiently low tempera- tures, under a wide variety of physical conditions [2j.

Some astrophysical situations where a pre- cise knowledge of the thermodynamics and miscibility of ionic mixtures may be very

important include the deep interior of the major planets, containing essentially Hy- drogen and Helium ; white dwarfs and neu- trons star crusts, where the highest densi- ties of Coulombic matter are reached ; and the interior of the Sun, where the physical conditions may lead to the segregation of highly ionized Iron [ 3 ] .

2. Physical conditions and parameters.- Consider a mixture of n ionic species in a volume n at a temperature T. Let N , Z and A be the number of ions, the ionic valence and the atomic mass number,of species v

(1 < v < n) ; if N = Z N is the total num- v v

ber of ions, the number density is p = N/fi, while the number concentrations are x =

v N /N, and the mean ionic valence is defined as :

Due to charge neutrality, the total number of electrons is

4 Paper presented at the C.N.R.S. Intertio-

nal Colloquium on the Physics of Dense Matter, Paris, September 1979.

Equipe associee au C.N.R.S.

JOURNAL DE PHYSIQUE Colloque C2, supplément au n° 3, Tome 41, mars 1980, page C2-43

J . P . Hansen

Laboratoire de Physique Théorique des Liquides

Université Pierre et Marie Curie, 4, Place Jussieu - 7S2S0 Paris Cedex 05, France.

Résumé.- Nous p r é s e n t o n s une revue c r i t i q u e des c a l c u l s r é c e n t s du diagramme d e s p h a s e s de mélanges ou a l l i a g e s i o n i q u e s dans un fond c o n t i n u d ' é l e c t r o n s d é g é n é r é s . Les l i m i t e s d ' ë c r a n t a g e é l e c t r o n i q u e f a i b l e e t f o r t s o n t t r a i t é e s s é p a r é m e n t , mais

f o u r n i s s e n t d e s p a r a m è t r e s c r i t i q u e s s i m i l a i r e s . La v a r i a t i o n de l a t e m p é r a t u r e c r i - t i q u e avec l a p r e s s i o n e t l e r a p p o r t des c h a r g e s i o n i q u e s e s t é t u d i é e e x p l i c i t e m e n t , en p a r t i c u l i e r pour des r a p p o r t s g r a n d s d e v a n t u n . Parmi l e s a p p l i c a t i o n s a s t r o p h y - s i q u e s , nous c o n s i d é r o n s b r i è v e m e n t l a p o s s i b i l i t é de l a d é m i x t i o n du f e r e t de l ' h y d r o g è n e i o n i s é à l ' i n t é r i e u r du s o l e i l .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980207

JOURNAL DE PHYSIQUE

N' = $ N ~

z \ , = N Z

each species, Av, is much shorter than the ion-sphere radius, i.e. if :

and the charge density is p' = N1/3 = Z p . An equilibrium state of the mixture is cha-

= h

racterized by the values of n

+

^{2 indepen-}

-

_a a(2nMVkgT)

'4

^{= 0-0586 v2-'/6 (5)}

dent thermodynamic variables which, for fi- ^A v ^Z xed N or N', can be chosen as 3, T and a set

of n-1 concentrations xv. Alternarively we where M v is the mass of species

v.

shall choose the following set of n

+

² di- The electrons, on the other hand, are dege- mensionless parameters : rs,

r

^{and xl,}

....

nerate if the temperature is much lower

rs and

r

are defined as follows. The than the Fermi temperature TF, i.e. :

xn-l

mean ion sphere radius, i.e. the radius of

a sphere which, on the average, contains one T <<TF = 6 x10 5 K ion, is related tG the number density by : ²

, r~

a = (3/4ap) 1/3

They are, moreover, relativistic if :

similary the electron sphere radius is rela- 1 1, ted to -charge density through :

a = (3/4npg) 1/3 = ./21/3

and the dimensionless density parameter rs i..e. if rs _<

-

_{; 2} here me is the electron is then defined as the ratio of a' over the rest mass and a = e /(hc) the fine struc-

ture constant. The equation of state of the Bohr radius a. :

degenerate electron gas is well known. In the non-relativistic regime, and for rs _< 1,

.-.

(1) T < TF, it is accurately given by the NoziSres-Pines [5] formula plus the leading

Next we define the dimensionless coupling temperature :

constants or

r'

pe =

[+ ^-

^{+ 0.0103 + 0.18}

2 2 '/3

r ^{= A .}

^rl

=e= r~

⁼ 31.56 x lo4 s Is s

akgT alkBT rs X T (K)

l o '

" rg T' (K)

1

m a r j x ~ < c 1 An effective coupling constant for the ionic ^_I

fluid is obtained by multiplying

r

^{by some} In the opposite, ultra-relativistic regime effective valence squared. In the strong (xF>>l), the equation of state is "softer"

coupling limit, we expect the Salpeter ion- ^{( Q} l;/r4) s and accurately given by the F e m i sphere model [4]to give a good description term :

of the ibnic equation of state ; in this ap-

proximation' the excess (non ideal) internal e' = me e' c 5 (<

-

^xi)

12n2 h3 ⁽⁹⁾

energy per ion is given by :

( 3 )

The coupling between the ionic fluid and This immediately suggests the following de- the electron gas (i.e.. the "uniform back-

finition of the effective coupling constant ground") is characterized by the electron

- -

screening length h which measures the ex-

,113 5/3

r e

=

z

= z z =

r r

z ^5/3 ( 4 ) tent to which the electron gas is polarized by the ionic charge distribution and hence The ions obey classical Mechanics screens the ion-ion interaction, In the if the de BrogIie thermal wave-length of

sically meaningful mean valence. Hence it is natural to distinguish a weak screening re- 22/3 ~ ' 6 1 5 4 gime, at high densities (rS<< 1) from a

- -

; XF >> 1

3 2/3 0.112

z

¹¹³

z

^{113 (lo)} strong screening regime at lower densities

(rs= 1). The two regimes will be treated on In the non-relativistic regime, XTF/a is a somewhat different footing. Some typical seen to increase with increasing density, values of the preceding parameters are given whereas in the ultra-relativistic regime, in table 1 for three very different astro-

XTF scales as the interionic spacing, while physical situations. _{. . ..} being appreciably larger than a for any phy-

Table 1 :

Table 1 : Typical values of characteristic physical parameters in three astrophysical situa- tions.

At sufficiently strong coupling, the ionic ories of mixtures, it is not un-reasonable fluid is expected to crystallize. While to adopt as a "rule of thumbs" that an alloy the fluid-solid coexistence curve is melts whenever

reff

= 155. A glance at ta- well known for a one component ionic fluid ble 1 shows that this simple rule implies [6], virtually nothing is known in the case that "cold" white dwarfs and neutron star

of alloys, In the context of one-fluid the- crusts are solid.

Neutron star crust

He

-

^Fe

Xl = 0.5 XZ = 0.5

1o1O 8 x

lo8

460

I

10l7 1.4 2.2

18

!

_I

White dwarf

C - 0

x 1 = 0.5 x 2 = 0.5

lo8 4

-

⁸

3

-

³

200

-

²⁰

3 10l3 0.36

-

^0.12

3.

4.

f

Composition

,om ( g r ~ m - ~ r s

T (K)

T/TF reff P (Pibar) Al/a A ~ ~ / a

%F

Interior of Jupiter

H

-

^He

XI = 0.9 X2 = 0.1

5 0.85

1.2 x lo-2

45 4

o

0.36 0.7 1.6 x

C2-46 JOURNAL DE PHYSIQUE

3. The regime of strong electron screeninq.- where Fid is the free energy of an ideal A perturbation expansion in the ion electron mixture Of nOn-interacting Fex is the interaction leads to the following expres- excess (non ideal) part of the ionic free sion for the potential energy of N ions, a- energy and e is the free energy the hO- Verag& over the electron coordinates[7] : mOgeneOus gas, which

is easily calculated from the equation of VN (;iN,

-

¹ state (8) or (9)

.

The non-trivial term is

-

-

_2Rk3 ^{' l}

M

_{k z} ^{[ p l b}

p ~ g -

^Niz] Fex. Since the screened ion-ion interaction

- -

[11, is short range, it is not unreasonable to . ^I

+ -

1 ^{C 1} calculate it by a variational principle ba- 2Q +

K sed on the Gibbs-Bogoliubov inequality [8],

higher order terms choosing a mixture of hard spheres of dia-

where meters al and a2 as a reference system 19,

101 (we restrict ourselves to binary mixtu- res for clarity) ; according to this ine-' is the kth Fourier component of the micros- quality ^:

copic ionic charge density, and ~,(k) is F e x < B (0 (Ul,4) iF1.i) (ulIu2)

the static dielectric function of the ex (16)

electron gas. The first term on the r.h.s.

of eq. (11) is the ionic potential energy in a rigid uniform background (the "Madelung"

term), the second order term describes the.

linear response of the electron gas to a gi- ven, ionic charge distribution, while the

higher order terms, which are not explici- tly reproduced here, correspond to the non- linear response of the electrons.

For illustrative purposes, assume that ze (k) is replaced by its Thomas-Fermi appro- ximation :

where F(') is the excess free energy of the ex

reference system (denoted by the superscript

,

^{and F)::} is the first order term in thermodynamic perturbation theory [8] :

with Q N =

-

XINZl e2kTF/2 ²

K

E ~ F(k) = 1

+

^TF _{2 (13)} The statistical average of VN is taken over k the reference system (hard sphere) ensemble where kTF = l/h,jjF (cf- eq. (10)). The first it is expressible in terms of the three par-

(0) (0)

two terms in (11) are then

easel^

cast in tial pair distribution functions, gll

,

^g12

the form : (r), g:;) (r)

,

of the hard sphere mixture.

where cPN is a structure-independent cons- tant. The effective ion-ion air interac- tion is seen to be short range, due to e- lectron screening. The higher order terms in eq. (11) would lead to three and more- body interactions between ions.

The total Helmholtz free energy of the plas ma consists then of three terms :

One of the reasons for choosing such a mixtu- re as a reference system, is that approxima- te, but very accurate, expressions for the free energy (entropy) F ( and the g(') (r)

vv are known from an analytic solution [ll] of the coupled PY integral equations [8] ; the integrals in (17) can then-, in fact, be cal- culated analytically[9,10]. The r.h.s. of eq.

(16)is finally minimized with respect to ul and a2 for a given thermodynamic state (Q,T, xl)

,

yielding an estimate for Fex.

(15) To investigate a possible phase separation at constant pressure, we must switch from the Helmholtz to the Gibbs free energy :

G(P,T,xl) = F

+

^P R

and consider the Gibbs free energy of mixing

For fixed P,T, there will be a phase separa- tion for concentrations such that :

The critical concentration and temperature, above which the mixutre is always stable, are determined by the two equations :

The calculation sketched above has in fact been carried out for mixtures of protons and a-particles by Firey and Ashcroft [lo]. The earlier calculation by Stevenson Pjin- cluded the following refinements over the former, simplified model :

a) a "realistic" dielectric function ~,(k), due to Hubbard [12], which includes ex- change and correlation effects, was used.

b) Terms up to third order in the ion-elec- tron interaction were included in eq.(ll) leading to effective three-body interac- tions between ions.

c) Higher order terms in thermodynamic per- turbation theory were included, to im- prove the estimate of Fex based on eq.

(16) [13,8].

d) The h quantum correction of the Wigner- 2 Kirkwood expansion [8] was added to the

free energy of the ions.

The essential results of Stevenson's calcu- lation can be summarized as follows. A mix- ture of protons and a-particles in a back- ground of polarizable, degenerate electrons phase separates below temperatures of the order of 10 4 K into Hydrogen-rich and Helium rich phases ; the critical temperature Tc decreases with increasing pressure ; at P = 60 Mbar (roughly the condition iiside Jupi-

ter ) , T,= 8500 K ; in the region of com- plete miscibility, Stevenson's calculation yields a sizeable volume non-additivity, i.

e, the mixture occupies systematically

less

volume than the pure phases. Quantitative- ly, if the relative volume difference at constant pressure and temperature is defi- ned as :

Stevenson finds that AQ/R is negative, and of the order of a few percent. His equa- tion of state data for low He concentra- tions (xH& 0.1) have been confirmed by ex- plicit Monte Carlo computations Of De Witt and Hubbard [14].

A difficulty arises if one considers physi- cal conditions which are typical of the in- terior of Jupiter and Saturn ; under pressu- res expected in the deep interior of these planets (P ranging from several Mbars to se- veral tens of Mbars), hydrogen is certainly completely pressure-ionized ; Helium however is only expected to become metallic around P = 70 Mbars [15]. Although the perturbation series leading to eq.(ll) may very well con- verge at considerably lower pressures, He- lium is expected to remain essentially

=-

mic below

-

P

-

10 Mbars. The solubility of a- tomic Helium in metallic Hydrogen has been estimated by Stevenson [16] who finds that atomic Helium at low pressures (P $ 10 Mbars) is even less soluble in metallic Hydrogen than metallic Helium at higher pressures.

'This result disagrees with a simple model calculation by Pollock and Alder [17] which constitutes probably an over-simplification of a rather complex physical situation.

The possible existence of a miscibility gap in the solid phase is a much more difficult problem, because of a number of additional complications,like the relative sfability of

-

various lattice structures [18], or the exis.

tence of an order-disorder transition in the

!alloy. The only existing calculation is for la H-He alloy, by StraUS et el. [19], The free

c2-48 JOURNAL DE PHYSIQUE

energy consists again of an electronic con- tribution, which remains unchanged, and an ionic contribution which the authors separa- te into static and vibrational parts :

They then introduce a "randomly disordered alloy" (r.d.a.) as a reference system ; in the r.d.a. all correlations between the occu- pations of neighbouring lattice sites are neglected ; more precisely the probability that a given lattice site be occupied by an ion of species v is just equal to the concen.

tration xv, and is in no way affected by the nature of the ions occupying the neighbou- ring sites. The free energy (22) is then re- arranged in the form :

L J

The first term on the r.h.s. is the expec- tation value of the static lattice energy calculated for the r.d.a. reference system the second term is treated in the quasi-che mica1 (or Fowler-Guggenheim [2 0 1 ) approxi- mation, which is a generalized mean-field theory ; the vibrational free energy of the r.d.a. is calculated in the harmonic latti- ce approximation, while the last term is neglected. The resulting free energy for H- He mixtures leads to a complicated phase diagram exhibiting complete phase separa- tion below a pressure-dependent temperature and a critical temperature much higher than in the fluid phase (Tc> 20 000 K at P = 90 Mbars ! ) . Although this work yields a strong indication in favour of an enhanced miscibility gap, the results should be ac- cepted with caution, in view of the fairly large number of poorly controlled approxi- :mations which have to be mad% in this dif-

ficult calculation.

6.

5 . -

In the high density (rs<<l) limit, the e- lectron screening length greatly exceeds the inter-ionic spacing and it is then rea-

sonable to neglect electron screening com- pletely in a first approximation. This a- mounts to retaining only the first term on the r.h.s. of eq,(ll) and hence to consider as a reference system a binary ionic mixtu- re in a rigid uniform background, corres- ponding to the limit ce(k) = 1 for all wave numbers. We then deal with a well-defined model, which is a straightforward generali- zation of the widely studied "one component plasma" (OCP)

,

or classical "Jellium" [21].

Due to the simple scale invariance of the Coulonb potential, the dimensionless ex- cess thermodynamic properties of such a bi- nary ionic mixture depend only on inde- pendent variables, which we choose to be x1 and.

r

(or equivalently T

'

⁾ , The equation of state for this model has been accurately determined from numerical 'solQtionseof the coupled "hypernetted chainW(HNC) integral equations for the three pair distribution functions 122,233 and "exact" Monte Carlo

(MC) . simulations [23]

,'

for ionic charge ra- tios Z2/Z1 = 2 and 3, The MC data show that the thermodynamics determined from the approximate HNC theory are very reliable, since they differ by less than 1% from the

"exact" results over the whole fluid range, The main result, summarizing the extensive numerical data, is that the dimensionless ex.

cess thermodynamic properties at constant temperature and charge density (i.e. at cons- tant I?') are, to an excellent approximation, linear combinations of the corresponding pro- perties for the

=

(one component) phases.

In particular the reduced excess free energy is well represented by :

where fo

( r )

is the reducedexcess free energy of the OCP and Tv = T'

. " : z

The deviation from th,Ls simple linear law :

which is just the reduced excess free energy of mixing at constant

r ' ,

turns out to repre-

sent less than 0.1% of f (r' ,xl)

,

except in the weak coupling limit ( r l < 1). It is inte- resting to note that the simple ion-sphere model, eq.(3), satisfies the linear relation

(24) exactly. The latter is easily generali- zable to mixtures of more than two ionic spe- cies. Since an accurate, semi-empirical, e- quation of state is available for the OCP[~~]

eq. (24) yields a' simple analytic expression for the thermodynamic properties of binary ionic mixtures in a rigid, uniform back- ground. The lowest order (sh 2 ) quantum cor- rection to the free energy takes on the par- ticularly simple form [23] :

contribution (cf. eq. (8) or (9) ) to the ex- cess free energy (24). The phase diagram is finally constructed by switching to the Gib- bs free energy at constant pressure. It is worth stressing that, although the free ener- gy of mixing (25) is very small, it has a

non-negligible influence on the precise lo- cation of the critical point, Typical re- sults for a,mixture of protons and a-parti- cles (Z2/Z1 = 2) are compared in table 2 to the corresponding results of Stevenson ; this comparison shows that electron scree- ning effects tend to increase the critical temperature for a given pressure, a result which may appear to be somewhat contrary to

f E

a

^{= 1}

r '

2 ^{xlZlme + x Z m} ₍₂₆₎ ^{intuit ion.}

'

^{8 r s [ M1}

2 2 e l

^M2

The total free enerav is obtained by adding the ideal gas term, f and the electron gas

Table 2 :

Table 2 : Comparison between critical parameters for a mixture of protons and a-particles in a rigid electron background and in a respon- ding (polarizable) electron background.

P (Mbar)

3 0 9 0

Clearly Tc decreases with increasing pres- sions of the reduced excess free energy a- sure. On the basis of the ion sphe- round xl = 0 and x2 = 0 :

re model, Pollock and Alder [17] have esti- r S

0.92 0.77

mated that the miscibility gap vanishes (Tc

-+ 0 ) at P-lo8 Mbar for a metallic Hydrogen-

~ e l i u m mixture.

The simple linear relation (24) is very ac- curate.forrelatively low charge ratios (Z2/

Z1 ,$, 3), but is expected to fail for very dissymmetric mixtures (Z2/Z1 >>1)

.

^{A syste-}

matic procedure, which generalizes eq.(24) in the case of high charge ratios, has been recently put foreward by Brami et al.121.

It is based on first order Taylor expan- The slopes pl and p2 are exactly expressi- ble in terms of the chemical potentials of Rigid background F23,21

Tc ( K )

7000 6000

Responding background [ll Xlc

0.71 0.72

Tc 9000 8000

X 1 ~ 0.65 0.65

I

C2-50 JOURNAL DE PHYSIQUE

the two species in the infinitely dilute solutions (i.e. xl or x 2 + 0) of one spe- cies in the pure fluid of the other. In the HNC approximation, these chemical poten- tials are easely calculated from the pair distribution functions, The linear rela- tion (24) amounts to setting pl

=

^{p2 P :}

R f (T2)

-

^fo (rl). The HNC calculations in-- dicate that systematically pl>p >p2. The linear law (24) was hence replaced by a il

simple Pad6 approximant at constant T' :

where the four coefficients a,b,c, and d we- re determined, for each r", by the two Tay- lor expansions (27a) and (27b). The procedu- re has been tested by explicit HNC computa- tions at finite concentrations and turns out to yield free energies with errors of less than one part in lo5 for charge ratios as high as 24 : 1 ! Some critical parameters calculated on the basis of the free energy

(28) are summarized in table 3. It is worth Table 3 :

Table 3 : Typical critical parameters for a variety of binary ionic mixtures in a rigid elec- tron background. T is the pressure in ato- mic units (e2/a40) ; T = 1 corresponds to P = 294.2 Mbars.

noting that the relative volume change on mi.

xing, defined in eq.(21) is always negative, but siqnif icantly smaller than Stevenson' s result for a screened Hydrogen-Helium mixtu- re ; moreover ~ A Q ~ / Q seems to decrease with increasing oharge ratio Z2/Z1. In fact, in the more dissymmetric cases, an additive vo- lume law (setting An,= 0) would be an excel- lent approximation, yielding accurate ther- modynamics of the mixture in terms of those of the pure phases.

A typical phase diagram is shown in figure 1, for a mixture of fully ionized Carbon and nearly fully ionized Iron ( F ~ ~ ~ * ) . Simple analytic expressions for the critical tem- perature and concentration as a function of

z1

and Z2 have been derived by steienson

[25f on the basis of the ion-sphere model.

Although the model does not allow a deter- mination of the pressure-dependence of Ihe critical parameters, the results are in surprisingly good agreement with the high pressure limit of the more elaborate calcu- lation by Brami et al, [2]. On the other hand a calculation by Pollock and Alder[l7]

based on a "one fluid approximation" with an effective charge given by eq,(4), yields critical temperatures for H-He mixtures which are about a factor two too high.

Contact can be made with the strong scree- ning regime via thermodynamic perturbation theory, by considering the ionic mixture in a rigid background as a reference system, and the second and higher-order terms in eq,(ll) as a perturbation. The advantage over the hard sphere reference system consi'

dered in section 3 is that the perturbation becomes vanishingly small in the high den- sity (rs+O) limit. In this limit the first order correction to the free energy can be expanded in powers of qTF, and the leading term turns out to be proportional to qTF,iSe 2 to rs[26,23], in agreement with the MC data od De Witt and Hubbard [I$]. The main result of the perturbation calculations for mixtu- res[23] is that the first order correction to the free energy is less than 10% of the total ionic free energy up to rs

-

^{1, while}

the screening correction to the pressure is even smaller (about 1%). Morebver the scree ning corrections do not modify the phase diagrams for H-He and H-Li mixtures drasti- cally [23], even when electron screening effects are expected to be important (rs n

l), in agreement with the comparisbn made in table 2.

5. A speculation : limited solubility of iron in the center of the sun.- Recently Pollock and Alder [ 3 ] have suggested that the solar neutrino dilemma might be resol- ved, if Iron had limited solubility in Hy- drogen under solar interior conditions,

6+ 24+

F i g . 1 . - Phase diagram for a C -Fe mixture i n a r i g i d uniform background for three pressures

(IT = l o 3 , l o 4 and 105 atomic u n i t s ) ; based on r e s u l t s from r e f . f2].

since a segregation of Iron would lead to decrease of the opacity in the center of the Sun. The conditions inside the Sun are expec- ted to correspond to a temperature of about 1.5 x 10 7 K, a pressure of about lo5 Mbar and a density parameter rs 2 0.5. Under such con- ditions Iron is expected to retain its 1 s electrons (with an ionization potential of about 8 KeV), while the 2s and 2p electrons

(with an ionization potential of about 1.5 keV) are only partially retained. The un- bound electrons behave nearly classically, since T/TF = 4 under the previously mentio- ned conditions. ~ s s u m i n ~ cosmic abundance of Iron, the effective ionic coupling constant is of the order of 0.05, which corresponds to relatively weak coupling. Consequently Pollock and Alder have used Debye-Hiickel the- ory with the leading Abe [27] corrections to calculate the free energy of mixtures of pro- tons and Fen+ (with n = 20 or 24) and cons- truct the phase diagram. This calculation leads to a predicted critical temperature which is somewhat higher than solar interior conditions and would hence be compatible with a limited solubility of Iron under such con- ditions. However a more accurate calculation based on the HNC integral equation, to treat the strongly coupledIron-rich phase correctly leads to the following critical temperatures at P = 5 x lo4 blbar : T = 4.8 x 10 6 K, for n = 24 and Tc = 3.7 x 10 K, for n 6 = 20 [28]; ^' these temperatures lie well below solar inte rior temperatures, leading to complete solu- bility of Iron.

These calculations neglect electron scree- ning effects, an approximation which is pro- bably less justified than at lower temperatu- res, because a non-degenerate electron gas is expected to screen the interaction between ions more efficiently. More realistic HNC calculations for a three-component fluid of iron, protons and semi-classical electrons are under way, using effective pair poten- tial [29] which take Quantum diffraction and symmetry effects into account. A similar cal- culation, based on the weak-coupJing Debye- HUckel theory, is being reportes by C.

Deutsch at this conference.

Acknowledgements : The author is indebted to B. Brami and F. Joly for their efficient help

c2-5'2 JOURNAL DE PHYSIQUE in preparing this lecture.

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