THERMODYNAMICAL FUNCTIONS FOR DENSE MULTICOMPONENT PLASMAS

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THERMODYNAMICAL FUNCTIONS FOR DENSE MULTICOMPONENT PLASMAS

M. Gombert, C. Deutsch, H. Minoo

To cite this version:

M. Gombert, C. Deutsch, H. Minoo. THERMODYNAMICAL FUNCTIONS FOR DENSE MUL- TICOMPONENT PLASMAS. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-695-C7-696.

�10.1051/jphyscol:19797337�. �jpa-00219330�

JOURNAL DE PHYSIQUE CoZZoque C7, suppZIment au n07, Tome 40, JuiZZet 1979, page C7- 695

THERMODYNAMICAL FUNCTIONS FOR DENSE MUTICOMPONENT PLASMAS

M.M. Gornbert,

C.

Deutsch and Ht Minoo.

Laboratoire de Physique des Gaz e t des PZasms X

,

Betiment 212, UniversttS Paris-Sud, Centre df0rsav, 91405 Ozsay, Cedez, France.

'1;aboratoire associd au C. N. R. S.

To study plasma thermodynamics, we need an ef- fective potential u (r) for the interaction bet-

i j

ween a particle of species i and a particle of spe- cies j. This potential takes in account the quantum effects at small relative For the thermodynamical calculations, we use simple analyti?

cal expressions :

uijCzizj

-

e2 ^{(I-exp(-6 L ) )} for unlike particles,

xi j

,..

2 ^{( 1 )}

2 e ~ o g 2 d Z

U. 11 .=zi ~ ( l - e x p ( - ( ~

-

^-)r))for like particles, z.e

B

ii

B

¹¹²

with : X . . * ( - )

,

pij= reduced mass.

1 J Pij

These potentials yield back the exact value for r=o and ~=o(~).~og(2) takes account exchange effect of the two particles(spin=1/2). We know of a better appro~imation(~)which reads :

where P(r) is a polynomial. For the electron-proton potential a very good approximation is :

9 "

2 A~ 2 A~

P(r)=A-(--

-

^+r-A(-

-

^{-)r2, with : A=}

- a

%2 52 2

plasma parameter A=e B/X and a quantum parameter 2 0

XIAD, where XD is the Debye length(4).~e use the expansion with respect to A in term of nodal graphs (5)

.

To describe the thermodynamics of multicompo- nent plasma, we use a matrix language(6). The Debye like potential,Vij(r), is shown in the next figure in which the sums run over all species,and straight lines uQk(r) are the interactions(temperature de- pendent) between two particles.

There is a maximum relative error of 0.01 at 10 6 K, . . ^{n k . n n k}

7 8

0.003 at 10 K and less than 0.001 at 10K. With the

~;v..(r);-+ckLal

^{I I}

+cc1/.\J ⁺

k = l k=7 kl

,

form(l), there is a relative error of 0.1 (near r=X). (Debye chain1

The approximations (1) or (2) always imply e 2 p<<X

+...

k:i 1=1 m=l (kTrl.Ry.). These techniques of effective potentials

are adequate for X<<d(mean distance between parti- There is the relation : $(k)= S(k). [I

+

^{$(k) .~]-l}

cles). We take into account only of the exchange of (3)

two particles.

=("! ) = (

^I

^tl)

We started to study the thermodynamics with the With :

%.

3

%

simpler form(]) for u(r). In fact, in a plasma made u In"'Unn .?nn of electrons and several ion species,we consider _,I,

(uij and ?ij denote the Fourier transforms of u

the potential between two ions to be classical and i j

and V ) we use the form(1) only for a potential between an i j

/VBC 1 0

J

electron and an ion or between two electrons. c = (

' . 1.

Now, we can study the plasma thermodynamics,

\ o

P BC

1

compute the correlation functions gij (r)and the ca- n

C. is the i species concentration, pCi is the nume- nonical thermodynamical functions-in term of the rical density of species i.The relation (3) is si-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797337

milar to the well known one for a one component plasma.

For an electron-proton plasma, with the potentials:

2 r

JZ

² r

JG

u (r)= _ee 5 (I-exp(- -)),u _eP =

-

^%(]-exp(- _r ^-))

and u (r)= eL/r (3= 5ee, uee(r) does not take PP

account symetry effects), we obtain, for instance :

with : q = b/& AD << 1.

For a classical electron-proton plasma (u(r) is the coulombic potential), if the number of Debye chains arriving to a nodal point is odd, the graph

van is he^'^).

This is not right when temperature dependent potentials are needed. This is right only for the coulombic contribution to the graph but there is an additional quantum contribution. We are currently considering the electron-proton plasma and also the electron-proton-iron ion plasma at least a first order in A. We shall be able to give results in a short time.

(1) A.A. BARKER, J. Chem. Phys.,

3,

1751 (1971);

(2) B. DAVIES and R.G. STORER, Phys. Rev.,

171,

150 (1968).

(3) MINOO

'

s communication.

(4) C. DEUTSCH and M.M. GOMBERT, J. Math. Phys., 17, 1077 (1976).

-

M.M. GOMBERT et C. DEUTSCH, J. de Physique (supl.),

2,

C1-184 (1978).

(5) F.E. SALPETER, Ann. Phys.,

2,

183 (1958).

(6) EIVIND HIIS HAUGE, J. Chem. Phys.,

44,

²²⁴⁹

(1 966).

(7) C. DEUTSCH, Phys. Rev.,

E ,

909 (1978).