Improved Debye Hückel theory for one-and multicomponent plasmas

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Improved Debye Hückel theory for one-and multicomponent plasmas

P. Vieillefosse

To cite this version:

P. Vieillefosse. Improved Debye Hückel theory for one-and multicomponent plasmas. Journal de

Physique, 1981, 42 (5), pp.723-733. �10.1051/jphys:01981004205072300�. �jpa-00209058�

Improved Debye ^Hückel theory for one-and multicomponent plasmas

P. Vieillefosse

Laboratoire de Physique Théorique ^des Liquides (*), Université P. et M. Curie, 4, place Jussieu, 75230 Paris cedex 05, ^France

(Rep le 25 novembre 1980, accepté ^le 20 janvier 1981)

Résumé.

La théorie bien

de Debye ^Hückel appliquée

plasmas classiques neutralisés _par

fond continu n’est valide que lorsque ^le couplage ^est suffisamment faible. La raison essentielle de ceci est que l’approxi-

mation de champ moyen

laquelle ^est fondée la théorie de Debye ^Hückel néglige ^les corrélations entre les ions du nuage d’écrantage. ^{Nous montrons ici comment il est} possible ^{de tenir} compte de la contribution de ces corré- lations à l’énergie ^interne

modifiant de façon adéquate la formule donnant l’énergie

^{fonction du} champ

moyen. L’énergie ainsi obtenue

bien les deux bons comportements ^{limites en} couplages ^{faible et fort. De} plus,

en

couplage ^fort, la linéarité dans les concentrations de l’énergie interne des plasmas ^à plusieurs composants ^est montrée et

lien

la condition d’écrantage parfait ^est établi. Enfin nous présentons

^modèle simple ^de

démixtion sous forte pression ^des mélanges ^binaires

présence d’électrons.

Abstract.

When applied ^{to classical} plasmas ^which

neutralized by

^uniform background, ^{the well known} Debye ^Hückel theory ^is only valid in the weak coupling limit. The main

for this limitation is that the

field approximation, upon which the Debye ^Hückel theory ^is based, neglects correlations between ions of the

screening cloud. Here

show how it is possible ^{to take into account} the correlation contributions to the internal energy by properly modifying the formula which determines the internal _energy

function of the mean field.

The internal energy

obtain correctly ^{exhibits the two} asymptotic behaviours in the weak and strong coupling

limits. Moreover, ^at strong coupling,

get the well known linear law for the internal _{energy of} multicomponent plasmas

^{function of the} concentrations, ^{and its} relationship ^{to the} perfect screening ^{condition is} proved.

Finally,

^show

simple ^{model for} phase separation ^of binary mixtures neutralized by electrons under high

pressure.

Classification Physics ^Abstracts

05.20

52.25

Introduction. - Dense fully ^{ionized matter has}

been extensively studied for several _{years. A} simple

and accurate model is the classical one-component

plasma (OCP). ^The point ions interact via purely

coulombic forces and a uniform and rigid background,

which represents ^the degenerate electron gas, ^ensures overall charge neutrality. ^The strength ^{of the} coupling

is defined by the dimensionless parameter ^{r which} is essentially the ratio of the potential energy of two ions separated by ^the characteristic mean distance to their mean kinetic _energy.

In the weak coupling limit, thermodynamical

functions can be analytically ^obtained by ^{two theories} (i) ^The Debye ^{Hfckel one} [1] (DH1 which is based on a mean field approximation ^and gives ^{accurate results}

for r lower than 0.01 when the Boltzmann factor is linearized and _up to F - 1 without linearization [2] ; (ii) ^{the exact Abe} expansion [3], ^{which is} rapidly diverg-

(*) Equipe ^associ6e

^C.N.R.S.

ing ^{for r} larger ^{than 0.3} because of its asymptotic

character.

When the coupling ^is stronger, ^more powerful

methods must be used. First, ^{there are} purely ^numeri-

cal ones like Monte Carlo [4] (MC) ^and molecular dynamics (MD) computer simulations [5] ^which yield,

of course, exact results. In addition to their own

interest, these results allow ^one to test theories the accuracy of which cannot otherwise be directly ^evaluate

ed. Among the theories which give results close to the exact ones, the hypernetted ^chain equations [6] (HNC)

and the mean spherical approximation [7] (MSA)

must be mentioned.

Unfortunately, all these theories require lengthy

numerical calculations. Moreover, going ^{from one,}

to several components ^{leads to serious} complications.

Here we present ^{an extension of the DH} theory, ^called improved ^DH theory (IDH) ^{in the} following, ^which

is also a mean field approximation. ^The equation

to solve is identical to the nonlinearized DH one. On the other hand, ^the expression for the internal _energy

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01981004205072300

is modified in order to take into account the correla- tions between ions of the screening ^{cloud which are} neglected by ^{the mean field} approximation. The result-

ing expressions correctly exhibits the leading ^terms

-

(J3/2) ^{r3/2 and - (9/10) r} in the weak and strong coupling ^limits, respectively. ^{The maximum deviation}

from exact results is of the order of 5 % ^{for T close to}

one.

Going ^{from one to several} (two ^or more) compo- nents presents ^no difficulty because the mean poten- tials around ions of different species ^{are not correlat-}

ed. Moreover, ^{it is indeed} found that the excess energy is remarkably linear in the concentrations for suffi-

ciently strong coupling [2] ^{and the} relationship ^{of this} linearity ^{with the} perfect screening condition is clearly

shown.

In the first part, ^the basis of this theory ^{is discussed}

for the OCP _case; some exact results and expansions

are given and results obtained in the one-component

case are extended to the multicomponent ^{case. We also} present ^a simple ^{model of} phase separation ^of binary

mixtures neutralized by ^a degenerate ^electron gas in the strong coupling limit; ^{in this} calculation, ^{the linear}

law for the ion _energy is used.

Some partial results have been published ^else-

where [8].

1. One-component plasma.

^{Here we consider the} simplest system ^of particles interacting ^{via Coulomb} forces, ^{i.e. a set of N ions of} charge ^{ze, of mass m in a}

box of volume V ; the numerical density ^{is p}

N/V.

A uniform and rigid background neutralizes the

charge of the ions so that the thermodynamic ^limit

is well defined. The ions are supposed ^classical (non quantum) and the interactions are purely ^coulombic.

The Hamiltonian is the sum of three parts,

which are respectively the kinetic energy, the potential

energy of the ions and the potential energy of the

background.

As the potential ^{is a} homogeneous ^{function of the}

coordinates of the ions, ^{the excess free} energy takes the form

where k is the Boltzmann constant, ^{T the} temperature and T(F) ^{is some} function of the coupling parameter

For r « 1, the gas is nearly perfect. ^{On the other} hand,

when the ions are strongly coupled, ^the system ^beco-

mes solid [9] (F - 155).

The various thermodynamic ^excess quantities ^are easily derived from IF(F) :

Sex is the entropy per particle, eex the energy per parti- cle, Pex the pressure and Mex the chemical potential.

The knowledge ^{of the excess internal energy is}

sufficient to determine all thermodynamic quantities

and we shall be essentially interested in this quantity

because approximations ^{can be more} easily ^{made for}

it.

Fixing ^{an ion at the} origin ^of coordinates, ^we obtain the excess internal energy per particle in the form :

where p(r) ^{is the exact mean} density ^of particles

around the central ion Of course, p(r) depends only

on the distance r II ^{and tends to} p as 11 r II goes ^to

infinity, 7¡)*(r) ^{is the exact mean} potential ^{due to all} charges except the central ion; the total potential 7¡)(r) ^is equal ^{to the sum} 7¡)*(r) ⁺ ze/r.

1.1 MEAN FIELD APPROXIMATION.

The Debye

Hfckel theory [1], ^{which is} principally ^{known under}

its linearized form, amounts to assuming ^{that each}

ion of the cloud around the central ion interacts with the others only by ^{a mean} potential cp(r) ^{which is} produced by ^the equilibrium distribution p(r) ^{of the}

ions assumed to form a perfect gas in the external field cp(r) ; ^i.e. p(r) ^is given by the Boltzmann distribu- tion

whereas the potential cp(r) ^is determined by

where the background is taken into account.

Such an approximation clearly neglects ^{the corre-}

lations between the ions of the cloud, and the distri- bution p(r) ^{as well as the} potential g(r) ^are approximate

values of p(r) ^and (j)(r) (p(r) ^{is not} equal ^to

p exp {

zT(r) 1).

Some exact results following ^from equations (6)

and (7) will be shown later. Here we only ^mention

three properties, ^{the first one} of which is well known :

By taking 1/2 ze(p*(O) ^{for the excess} energy ^we do not make a good approximation, except ^{in the case} of small F values where the correlations between the ions of the cloud are weak and give ^{terms of} higher

order in F. Moreover, ^{we know that} Jieex ^is essentially given by - (9/10) ^{F when F is} large [4]. ^{The ions} sphere ^model [10] gives this value correctly ^and

shows us how the excess energy ^must be calculated.

In the strong-coupling limit, the cloud is likened to a sphere of radius A

rp, in which the only ^{ion is}

the central one and outside of which the ions have a

uniform density p. The excess energy is taken as the total electrostatic energy of such ^a distribution with the energies of the central ion and of the background

inside the sphere giving contributions to fleex equal

to - (3/4) ^{T and -} (3/20) ^F respectively. ^{As we shall}

see later, ^the equations (6) ^and (7) exhibit, ^{when r}

is large enough, ^{the same} distribution of ions

(where ^Y is the Heaviside function). These remarks suggest ^{to us that the} potential energy of the charge

distribution ze { p(r) - p I ^{must be added to the}

central ion _energy 1/2 zecp*(O). However, ^{it must be} carefully noted that this supplementary ^{term is to be}

divided amongst the n ions inside the screening

cloud. The number n is not well defined; ^{it can be}

estimated as p(4/3) ^{nA 3}

(Alrp)3, where A is the

screening length, ^{i.e. the} length ^{at which} (p tends to zero. Thus, ^{we are led to the} expression

At large ^{values of} r, ^{A is} equal to rp ^and

At small values of r, ^{A is} equal ^to rpl.,,13-F ^{and the}

second term in (9) gives ^a negative correction, pro-

portional ^to F’, ^{which is} qualitatively ^correct.

However, ^the expression (9) ^{is not} satisfactory

because of the poor definition of n (n is defined within

a numerical factor, except ^{in the} strong-coupling limit). ^Another way ^to divide the cloud _energy amongst the ions is to take for _eex the work spent ^to polarize

the cloud, ^{i.e. the work} done when the charge ^{of the}

central ion is switched ^on from zero to ze with the other ions keeping ^{a constant} charge ^ze. If the central

charge ^{is Aze} (0 K h K 1), ^the potential cp ;.(r) ^{and the} density pi(r) satisfy ^the equations :

The excess energy which we write _as,

takes ^{the following form} by taking (10) ^and (11) ^into account

where (p(r) ^and p(r) ^are solutions of (6) ^and (7) (1).

Such an expression correctly gives ^{the two} limiting

behaviours of peex : - (/2) ^{r3/2 as r goes to zero}

and - (9/10) ^{F when F is} large compared ^{to one}

Another expression ^of fleex ^{can be} given by taking (7)

into account :

Peex’ ^{as a} function of p(rx is minimum for

The expression (12) ^of eex ^must be compared ^to

that one which gives ^{the excess chemical} poten- tial [11] :

where, it is recalled that 7P ^{is the exact mean} potential (calculated ^{with the exact} distribution p(r)). ^{In the}

weak coupling limit, cp and 7P ^are close; ^as cp). and 7P).

(1) ^{This is obtained} by integrating by ^parts. The calculation is

very similar ^to that

of ppex in the HNC approximation [12].

are linear in À. then, peex ^is equal ^to pJlex. This result

can easily be deduced from (4) ^with

On the other side, (p and 7P ^are different in the strong coupling limit; fie,, - - (9/10) ^{F leads to}

As an example ^{it can} be noticed that :

1

Another comparison ^can also be made with the HNC equations. ^{The mean} density around the central ion is written in the form

where the mean forces potential 0 is the solution of the equations :

The internal energy takes the form :

which looks _very much like formula (13). ^{The che-}

mical potential, given by (16) ^{is obtained with the} help of the HNC equations [12] ^{as :}

As r tends _{to zero,} v(r) ^is equal ^to 1/r except ^{in a} small region around the origin; 0(r) is then close to p(r). Moreover, ^the following equation ^{can be}

written :

When r is large compared ^{to one, the} region ^where v(r) is different from 1/r ^{is no} longer negligible ^and

(p becomes different from _cpo Keeping only ^the leading

terms in (21) ^and (22), ^we get :

from which we deduce

These results have to be compared ^to (17) ^and (8).

The proof ^{of the} validity ^of expression (12) ^{for the}

internal _energy ultimately lies in the good numerical

accuracy of the results when compared ^{to those}

obtained by ^Monte Carlo calculations [4] ^or by ^the

solution of HNC equations [6]. The difference is in the worst case (r - 1), of the order of 5 %.

1.2 EXACT RESULTS FOR THE DEBYE HUCKEL EQUA- TIONS.

We introduce the following dimensionless

quantities :

The screening factor satisfies the following equation :

the solution of which can be written

The derivative at the origin §’(0) ^{must be chosen so}

that §(x) ^{tends to zero when x} goes ^to infinity.

According ^to (27), 0" and 0 ^{have the same} sign

so §’(0) ^{must be} negative; ^{in the} opposite ^case,

§ ^would indefinitely increase from one. Hence,

§(x) ^decreases initially; if 0’(x) ^{went to zero before} §(x),

then beyond ^this point t/J(x) ^would indefinitely ^increase

while if t/J(x) ^{went to zero before} §’(x), ^then beyond

this point O(x) ^would indefinitely decrease. There-

fore, 0 ^and 41’ ^{must be zero at the same} point (x = ^oo,

in fact) ^{and the} function 0 monotonically ^decreases

from one at x

0 to zero at x

oo. g(x) ^{is mono-}

tically increasing ^{and does not exhibit} oscillations.

The integral ^{in the} equation (28) ^is positive ^so the,

function

must be positive; y(x) ^{has a minimum} (at

which has also to be positive. ^We deduce from this the following inequality :

For _very large ^{r, the} integral ⁱⁿ (28) ^{is zero as} long as 0 ^{is not} of the order of 1/T ; ^i.e. § ^is equal ^to

y ^as long as y is not of the order of 1/r. Thus, ^the

minimum value of y(x) ^is of the order of 11F ^and

§’(0) ^{tends to} its lower bound with a difference of the order of 11F ^{when r becomes} large.

The ^excess _energy given by (13) takes the form :

We show in appendices ^{A and B} how the solution of (27) ^{can be} obtained in the two limits of weak and strong coupling; ^the following expansions ^{are found}

where _y is the Euler constant, and,

The weak coupling expansion ^{must be} compared

to the exact expansion ^{of Abe} [3]

Numerical comparison (Table I) shows that the

expansion (32) ^{is more accurate than the} exact one, due to the poor convergence of the latter.

The strong coupling expansion correctly ^contains

the leading term, linear in r, ^which corresponds ^to

the electrostatic part of the energy. On the other hand, ^the second term, which corresponds ^{to the}

thermal part of the energy, does ^not increase with r

as the results from the Monte Carlo calculations [4]

or from HNC [6] ^{and Mean} Spherical Approxima-

tion [7] ^indicate.

Table I.

Comparison of - peex(r) for ^{the OCP,}

calculated by ^{various methods :}

a) ^IDH expansions (32) ^and (33) ^{in the weak and} strong coupling limits, respectively.

b) ^Abe expansion (34).

c) ^IDH (31) ^numerical calculation.

d) ^HNC Springer ^et al. for ^{F 50 and} Ng for T > ^100.

e) ^{MC Hansen.}

1. 3 NUMERICAL RESULTS.

The numerical inte-

gration ^of equation (27) by ^formula (28) ^is very easy and can be done on a pocket calculator. In table I,

we compare these results ^to those of Monte Carlo and HNC calculations. The maximum relative devia- tion is obtained for r close to one where it reaches 5 %.

2. Ionic mixtures.

Here we look at mixtures of several species of ions which have charges ^{of the}

same sign. ^Each species ⁱ is characterized by ^its

number of ions Ni, ^the charge zi e and the mass mi of the ions. The numerical density pi and _p, the charge density po and the concentrations _ci and yi ^{are defined} by ^the equations :

where

and V is the volume of the system. ^The charge ^of

the ions is neutralized by ^{a uniform} background ^of charge density - po ^e. Interactions are assumed to be purely ^coulombic.

Two length units,

and two coupling parameters

are introduced; the notation f ^{indicates the mean}

value of a quantity f ^{over the} concentrations :

The excess free _energy is written here in the form :

The entropy per particle Sex

SE.IN, ^the energy per

particle ^eex

Eex/N, the pressure Pex ^{and the chemical} potentials Pie. ^are given by the formulae

2.1 DH _EQUATION FOR MIXTURES.

We apply ^to

the ionic mixtures the formalism that has been

developed for the OCP in the first part.

The density ^{of a} species j ^{around an ion i is written :}

where x

r/rpo. ^{The screening factor §;(x) is solu-}

tion of the differential equation

Here two remarks must be made :

-

The functions ql, ^which correspond ^{to different} species, ^each satisfy ^a separate differential equation;

going ^{from one to several} components ^{does not} make difficulties and leads only ^to complications ^of writing

at the utmost.

-

Generally, gij will be different from gji; ^{this is} related to the previously mentioned fact that /(r)

does not stand for the exact mean density ^of ions j

around an ion i (except in the weak coupling limit).

The internal excess energy per particle is obtained in the same way ^as in the OCP case

The first term is the usual definition of the _energy while the second occurs as the result of the correlations bet-

ween the ions in the screening ^cloud.

The functions ^are always monotically decreasing and their derivatives at the origin satisfy

and tend to their lower bound when To ^is large compared ^{to one.}

The weak and strong coupling expansions ^{of the OCP are} easily generalized

where _y is the Euler constant.

An important point is that the excess energy per particle ^{becomes a linear} function of the concentrations,

at constant To, ^when Fo ^is large enough compared ^{to one. The} leading ^{term in} (46) ^{can be obtained} by ^a simple

calculation which ^{shows the reason} for the linearity.

It is easily ^{seen that the} leading ^{term comes from the two} following contributions :

Integrating by parts, ^{we obtain}

At large Fo, zi t/J;’lx ^{tends to a} step function 3 Y(xo - x),

the length xo of which is determined by ^the perfect screening condition which can be written

A straight forward calculation of (48) yields

for f3eex. Therefore, ^the linearity ^of fle,,. in ^{the concen-}

trations is closely ^{related to the} perfect screening

condition.

Numerical integration ^{of the} equations ^confirms

the linearity ^of f3eex ^{as a function of the concentra-}

tions at constant ro :

The approximation is all the more valid when To ^is larger ^{and the} charges ^{are not too} different. For a

binary ^mixture Z1

1, Z2

2, the maximum relative deviation is 4 x 10-2 at To ^{= 0.01, 5 x 10-3 at} to = 1.0, ^{and 5 x 10-6 at} Fo

^{100. At} Fo

10.0,

this deviation increases from 2 ^x 10-4 for _Z2 = 2,

to 3 ^x 10-4 for _z2

3 and to 6 x 10-4 for _Z2

10.

Some results are given ^{in tables II and III.} Finally, f3eex(r 0’ { ci }) ^is always ^found (for binary mixtures)

to be larger ^{than when} calculated from the linear law (50).

Comparison with the HNC and MC results [6, 4]

is satisfactory, the deviation being ^{of the same order}

of magnitude ^as for the OCP case. It must be noticed that each calculation leads ^to the linear law with a

very high accuracy; this agrees with the fact that the linear law is related to the perfect screening ^condi-

tion.

2.2 APPLICATION TO THE MISCIBILITY PROBLEM.

Here we apply ^the previous ^{results to the} problem

of the miscibility ^{of a} binary ionic mixture in the presence of electrons. We limit ourselves ^to high

pressures (i.e. large densities) ^{so that the} coupling

Table II.

Comparison of - peex(r) for ^{a mixture}

Z1

1, Z2

2, calculated by HNC, ^{IDH and IDH} expansions ((45) for ^r

^{0.1 and} (46) for ^r

10).

r’B...

Table III.

Excess internal _{energy -} peex ^{and excess}

internal _energy of mixing

calculated by ^IDH for To

0.1 and three values of Z2

(z 1

1).

between ions and electrons would be negligible. ^A

similar calculation has already been done by ^Ste-

venson [13], ^{but with a} perfect ^{gas term} unfortunately

wrong.

For sufficiently high pressures, the free energy of the ionic mixture in the presence of electrons is the

sum of the perfect gas free _energy of the ions, ^{of the}

excess free _energy of the ions and of the zero tempe-

rature free _energy of the electrons :

The excess free energy is taken in the form

The free energy of electrons is given by the well known

expansion :

where _ao is the electron Bohr radius, rso

rpolao ^and

the constants 7, 6 ^{and e are} equal ^to 1.105, - ^0.458

and 0.0311, respectively.

To the lowest order in the variable

where P is the _pressure, the Gibbs free _energy of

mixing is obtained as :

This term does not depend ^{on the} pressure, i.e. the

phase diagram T(cl) ^becomes independent ^{of P}

when the _pressure is high enough. ^{An exact calcula-}

tion with the expression (55) ^{shows that} the mixture is unstable and separates ^{into two} phases ^{below the} critical temperature T c

which of course tends to zero as Z1 tends ^to z2.

The critical temperatures ^{are 2.7 x} 103 K for _Z1 = 1, z2 =2 ^{and 1.1 x 104 K for} Z1

1, z2

3.

The critical concentration is given by

Calculation of the Gibbs free _energy of mixing ^at

the next order in _r,,o yields ^a correction to the critical temperature (56) ^{which is} positive ^and proportional

to rSo (i.e. P - 1/5) ^{and a} correction to the critical concentration (57) which has the same sign ^as Zl -Z2 and is also proportional ^to r,, ,o :

Comparison ^{with exact results} [2, 14] ^{shows that}

the agreement ^is very good for the critical concen-

tration (better ^{than 1} %). The critical temperature ^is of the right ^{order of} magnitude but is lower than the

«

exact » values by ^{50 to 100} %. ^The difference results from the deviation from the linear law.

3. Conclusion.

The mean field theory ^of DH, which is accurate in the weak coupling limit, rapidly

breaks down with increasing coupling strength ^because

it neglects correlations between ions of the screening

cloud around each ion. We have shown how it is

possible ^{to take these} effects into account by ^some

sort of renormalization of the _energy and to obtain correct results even at strong coupling. Nevertheless,

this modification of the _energy, which is simpler ^than

the mean potential ^one (e.g. in the HNC equations),

has the disadvantage ^{of not} giving ^the pair ^distribu-

tion functions correctly.

Moreover, ^{there is} difficulty ⁱⁿ generalizing ^this method, ^to potentials ^{which are not} purely ^coulombic.

For example, ^{when the ionic} potentials ^{are screened} by the electrons by ^{means of a} static dielectric func- tion [15], ^{one obtains, in the} strong coupling limit,

a negative correction to the ions _energy which is

correctly proportional ^to r q?F’ ^where qTF is the Thomas-Fermi wavenumber, ^{but with a} coefficient which is twice the correct value (for ^z

1). Perhaps

it is possible ^to make these results better. This would be _very useful in the study ^of phase separation ^of

screened ionic mixtures.

Acknowledgments.

^The author thanks B. Bernu for valuable discussion and is indebted to J. P. Hansen and R. Mazighi ^{for a careful} reading ^{of the manus-} cript.

Appendix ^A.

^{Here we look at the limit F « 1.} Introducing ^the quantities

equation (27) takes the form

The perfect screening condition, which follows from (A. 2), is written :

and the excess energy (31) ^{becomes :}

When the parameter ^{a tends} to zero, it would be tempting ^to expand ^the exponentials in power of a. Unfortu-

nately, this leads to divergences ^{at small u even in first} order; ^{hence we make a cut-off at the} point ^u = çrx, ç being of the order of unity. By correctly summing up terms, j disappears ^{from final results.}

For u > çrx, ^the equation (A. 2) ^{can be solved} by expanding ^the exponentials ; the solution that vanishes

at large ^u depends ^{on an} arbitrary ^constant which is determined with the help ^of equation (A. 3).

To lowest order, ^{we have} the well known result of the linearized theory

The supplementary ^{terms which come} from the second integral ⁱⁿ (A. 4) ^{do not} contribute to this order.

To first order, t/1(1) is the solution for u > çex ^of

and can be written in the form

cp(u) ^{is a} monotonically decreasing ^function; g(O)

Log ^{3 and} g(u) - 2/3 ^{u at} large ^u. By taking ^the equa- tion (A. 3) ^{to first} order, ^we determine the constant A. It is easily ^seen that the domain u ça, ^{where we can}

substitute e - u . - 1 for 0(u), gives ^a contribution which is proportional ^to OC2 the integrals ^{over the domain}

u > çrx, ^where 0(u) = BM) ⁺ 4/(’)(u), ^{can be extended to.O u oo without errors to} this order and finally

we get :

The domain u çrx gives ^a contribution to the internal _energy which is proportional ^{to a2 and here we can}

substitute 1 for ik(u)

By neglecting ^{terms of} higher ^{order, we} obtain for the domain u > a :

It is necessary ^{to sum} up the terms a" t/J(O)n which in fact give contributions to order a2 ; ^the following integrals

are expanded asymptotically

and only keeping ^terms of order lower or equal ^to a2, the contribution of the domain u > çrx ^{is obtained}

where _y

0.577 2 is the Euler constant. ç disappears ^{from the sum of} (A. 9) ^and (A. .12) by taking ^{into account}

the identities :

Finally peex is obtained to order a2

Appendix ^B.

^{In this} appendix ^{we consider the} limit T > 1 ; ^the function g very quickly increases from 0 to 1 near the point where y (29) ^is minimum, ^{i.e. x N 1.} As t/J is of order lit and t/J" ^{of order} unity, ^{the variation}

distance of g ^is proportional ^to 1/Jl’ ^{and §’} is of order 1/Jl’ ; ^{then g, g’ and g" are of order} 1, Jl’ ^{and r, res-}

pectively.

In the equation ^satisfied by g

we can neglect ^{the term 2} gg’, which is small compared ^{to the others,}

As x has disappeared from the differential equation, ^the function g is defined to within ^a translation and one

supplementary condition from (B.1) ^or (27) ^{must be} assigned; ^{we take the} perfect screening ^{condition wich fol-}

lows from (27) :

Equation (B. 2) ^{can be} integrated in the form :

For g -+ ⁰ and g -+ 1, ^{the solutions are} approximately

where _x, and x2 ^{are constants} (N 1).

From (31), ^{the internal} energy ^can be written

with the help ^{of the} equation :

which follows from (27).

If we neglect ^terms which tend to zero as r increases (according ^{to the} approximation ^{that we have made at}

the beginning ^{of this} appendix), ^the integral of g log g ⁱⁿ (B. 6), ^{which is} proportional ^to r - 1/2, is ^missed.

Introducing ^x

^{1 +} (u ⁺ uo)/-,/-3-F ^{and assigning the} supplementary condition that defines _uo

we obtain from (B. 3) ^and (B. 4)

and finally ^the expansion for the internal _energy is given by

References

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