^{1}

^{2}

^{3}

^{*}

^{2}

The objective of this paper is to calculate the third virial coefficient in Hartree approximation, Hartree-Fock approximation and the MontrollWard contribution of plasma byusing the Green’s function technique in terms of the interaction parameter

The thermodynamic functions are of great interest for understanding the properties of plasma such as the excess free energy and the pressure. An accurate description of the pressure, volume and temperature (P-V-T) behavior represents one of the most important goals of statistical thermodynamics. Besides simulations, the equilibrium behavior can be determined either from the equation of state (EOS) or from virial expansion. The properties and the behavior of many particle systems with Coulomb interactions are essentially determined by the long range character of the forces between the charged particles. Therefore, systems with Coulomb interactions are of special interest and importance in statistical physics [

Many authors have calculated the excess free energy, Ebeling et al. [

The model under consideration is the two-component plasma (TCP); i.e. neutral system of point like particles of positive and negative charges which antisymmetric with respect to the charges and therefore symmetrical with respect to the densities. Also, we used the one component plasma model (the model of identical point charges immersed in a uniform background, while the continuous charge density of the background is chosen to be equal and opposite to the average charge density of the point charges, so that the system as a whole is electrically neutral) for example the electron gas and therefore symmetrical with respect to the mass.

This paper is organized as follows: In Section 2, we present the Green’s function. In Section 3, we calculate the excess free energy until the third virial coefficient for one and two component plasma in quantum form. Also, we calulate the general formula of the third virial coefficient in Hartree-Fock approximation. Finally, in Section 4, we calculate the pressure for one and two component plasma until the third virial coefficient.

The n-particle Green’s function is defined by [

where,

is the time-ordering operator and with is the vector of location, is time, is -projection of spin.

So we can defined the greater and lesser Green’s function by

In the Hartree and the Hartree-Fock approximation the two particle Green’s function is given by

Or the variational representation [

where the polarization function

and is the effective potential.

Also the three particle Green’s function is defined by [

The excess free energy corresponds to the part of the free energy change in the real system that arises from interactions among ions [

where

(11)

and are the mean interaction potential for two and three particle respectively.

After performing the integration with respect to in Equation (8) we can get

where the second virial coefficient is given by [

(13)

and

Now we will calculate the third virial coefficient; for Coulomb systems it is useful to apply, instead of, the screened potential in Eq.14 for three particle which is given by the following form

Then we can get the Eq.14 in the form

where,

(18)

and

(19)

where are the binary potential, the triplet potential, the triplet screened potential, the triplet polarization function, one particle Green’s function, the two particle Green’s function in Hartree approximation, the three particle Green’s function, the triplet Hartree term, the triplet Hartree-Fock, the triplet screened respectively.

By substituting from Eq.5 and 7 into Eq.17 then the Hartree term of the quantum third virial coefficient becomes

By taking the inverse Fourier transformation of the above equation and making use of the Wigner distribution function [

and using Equation (20) we obtain

where the number density

Eq.23 is vanishes for example for one component plasma such as the electron gas.

The polarization function is defined random phase approximation by

By substituting from Eq.25 into Eq.18 we can get the Hartree-Fock term of the quantum third virial as follow

The inverse Fourier transform of this equation with the help of the Eq.22 gives

(27)

where, and are Fermi functions and the Fourier transform of Coulomb potential which is defined by

with is the dielectric constant where are the vacuum and the relative dielectric constant.

We assume that

(29)

By expanding and in powers of and using the spherical polar coordinates (see Appdenix A) then

(30)

Then

let

The calculation of analytically, gives

By substituting from Eq.33 into Eq.31 then we get

where, and are the Fermi integral, the volume, the spin projection, the degeneracy parameter, the chemical potential, the Boltzmann’s constant and the absolute temperature respectively.

Then we can written the third virial coefficient in the following form

(35)

By substituting from Eqs.6 and 7 into Eqs.19 we can rewrite the screened quantum third virial until first three terms in the following form

by using the pair potential and pair polarization function then

Taking the inverse Fourier transformation of Eq.37 and using Eq.20, where

is the Fourier transform of binary potential then we get in the weakly degenerate case or low degeneracy limit and the case of high temperature limit or low density in the following form

where is the inverse Debye length and denotes the confluent hypergeometric function.

The analytical calculation of the integral from Eq.38 is evaluated by solving this integral by parts and using Gamma functions in the regions of small (low densities, high temperatures) then we have

By substituting from Eqs.13, 23, 35 and 39 into Eq.12 we get the excess free energy until the third virial coefficient for one component plasma as follow

where is the quantum virial function which given by [

Similarly, we can write the excess free energy until the third virial coefficient for two component plasma as follow

Following the method of effective potentials developed by [

where is the ideal pressure and

By substituting from Eq.40 into Eq.42 we can get the equation of state until third virial coefficient for one component plasma;

Also, by substituting from Eq.41 into Eq.42 we get the equation of state until third virial coefficient for two component plasma

In the numerical calculation, we let

To our knowledge there is no paper to calculate the third virial coefficient by using Green’s function technique until now; this paper is the first paper to calculate the third virial coefficient in Hartree, Hartree-Fock approximation and the Montroll-Ward contribution by using the Green’s function technique, and used it to calculate the quantum thermodynamic functions. In past the potential was used as the mean potential for two particles only so their results were until the second virial coefficient, but in this paper we used the potential as the sum of the mean potential of two and three particles so our results were evaluated until the third virial coefficient. Also the quantum thermodynamic functions until the third virial coefficient which are calculated by using the binary Slater sum are near at the classical limit only; they used the potential as the pair potential only and neglected the triplet potential so there results Is not exactly correct results. We considered only the thermal equilibrium plasma in the case of one and two component plasma by using Green’s function method. We obtained the general formula of the third virial coefficient in Hartree-Fock approximation analytically (Eq.34).

As shown in Figures 1-3, we plotted the comparison between the excess free energy until the second virial coefficient for one and two component plasma of Ebeling et al. [

(a) one component plasma (b) two component plasma

^{2}.

(a) one component plasma (b) two component plasma

^{4}.

(a) one component plasma (b) two component plasma

^{6}.

(a) one component plasma (b) two component plasma

^{2}.

(a) one component plasma (b) two component plasma

^{4}.

al. [

By expanding and in powers of then

where,

by substituting by, we get

by using the integration by parts we get without Dirichlet formula

we have the integral region for the inner bounds, if we look at the curves which we want to solve in terms of we find that and finally we have

which can be written as