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Astron. Astrophys. 364, 157-164 (2000)

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3. Coulomb coupling

The accuracy of the physical description demands a large range of applicability of the formalism for the EOS. Thus the formalism should be valid, e.g., from the central part of the Sun, with a temperature [FORMULA] and a density [FORMULA], to the solar surface with [FORMULA] and [FORMULA]. Since [FORMULA] at the centre and [FORMULA] near the surface, the solar plasma cannot strictly be regarded as being in the weakly coupled state; the polarization and quantum effects of the electrons play significant roles in determining the plasma properties. Although the Debye-Hückel approximation provides the correct limit for [FORMULA], it overestimates the Coulomb effects when the coupling becomes significant at moderately small [FORMULA]. An accurate representation of the excess free energy due to Coulomb coupling may be obtained by going beyond the Debye-Hückel approximation, including the following three modifications:

  • effects of exchange and correlation interactions between electrons,

  • electron finite-temperature (finite-[FORMULA]) effects,

  • screening effects of the degenerate electrons.

For a weakly coupled plasma, the local-field effects between particles can be neglected (Ichimaru 1982). The Coulomb interactions can be treated within the linear screening theory. Under this condition, the total Hamiltonian H of the two-component plasma can be separated into (Ichimaru et al. 1987):

[EQUATION]

where [FORMULA] and [FORMULA] represent the Helmholtz free energy of the uniform electron background and the kinetic energy of the ions; [FORMULA] is the Fourier component of the ion charge number fluctuations; and the dielectric function [FORMULA] ([FORMULA]) is the static screening function of the electron fluid. For the ionic mixture in a rigid electron background, [FORMULA]. Thus, the Coulomb term of the free energy can be written as

[EQUATION]

where the quantities labeled [FORMULA], [FORMULA] and [FORMULA] refer to the contributions corresponding to the electron-electron interaction, ion-ion interaction, and ion-electron interaction, respectively. It is convenient to introduce dimensionless quantities for the free energy

[EQUATION]

and for the internal energy

[EQUATION]

The Coulomb free energy can be obtained from integration of the internal energy with respect to the coupling constant (Tanaka et al. 1985a; Ichimaru et al. 1987)

[EQUATION]

By using dimensionless form defined above, Eq. (25) is expressed as

[EQUATION]

or

[EQUATION]

where [FORMULA] denote the number fraction of ions and electrons, respectively, and [FORMULA] is the total number of charged particles.

A model for the H-He mixture requires a knowledge of the interaction between the charged hydrogen and helium species; however, the non-ideal free energy of the mixture can be expressed with high accuracy by the so-called linear mixing rule in terms of the free energy of the pure phases (Hansen et al. 1977). Taking hydrogen as "1" and helium as "2", the excess free energy due to Coulomb coupling is written as

[EQUATION]

where [FORMULA] represents the number fraction of ions of species j in the total ionic configurations. The electron-electron interaction term [FORMULA] is a function of [FORMULA] and [FORMULA], which are related the number of the degenerate electrons.

The contributions to [FORMULA], have been studied by various procedures by solving a set hypernetted-chain (HNC) or Monte Carlo simulations (Brami et al. 1979; Tanaka et al. 1985a, 1985b; Ichimaru et al. 1987; Ebeling 1990; DeWitt et al. 1996; Stolzmann & Blöcker 1996; Chabrier & Potekhin 1998). In this paper, we make use of fitting formulae for dealing with the weakly coupled plasmas.

3.1. Electron-electron interaction

For the sake of simplicity, we describe an interacting electron system in terms of an effective single-particle problem. Considering the contribution from the sum of ring-diagrams to the free energy, the exchange-correlation free energy of electrons [FORMULA] is calculated numerically according to (Fetter & Walecka 1971; Tanaka et al. 1985a):

[EQUATION]

where

[EQUATION]

and [FORMULA] is the quantum-mechanical Fermi-Dirac distribution

[EQUATION]

with [FORMULA] and [FORMULA]. Eq. (30), based on the random-phase approximation (RPA), gives good results in calculating the Coulomb coupling for a weakly coupled plasma because the kinetic energy is dominant.

In order to derive the equation of state, we must find an analytic formula for interaction energy which is sufficiently accurate to enable the necessary integration and differentiation. The formula should include limiting conditions for [FORMULA]

  1. [FORMULA]. When [FORMULA] is kept at a finite value, the electrons thus form an unpolarizable negatively charged background to the ions. The contribution [FORMULA] of the electrons to the interaction energy is given by the Hartree-Fock value

    [EQUATION]

  2. In the weak-coupling limit [FORMULA], the lowest-order Hartree-Fock exchange energy of the electrons is the dominant contribution to [FORMULA], so that

    [EQUATION]

    where an accurate fitting formula for [FORMULA] has been obtained by Perrot & Dharma-wardana (1984) as

    [EQUATION]

    In the classical limit [FORMULA], [FORMULA], and the second term in Eq. (33) proportional to [FORMULA] becomes the leading contribution in the weak coupling regime; this term can be approximately evaluated in the Debye-Hückel theory as [FORMULA].

Considering these limiting cases, we adopt an approximate expression for the contribution arising from the electron-electron interaction. Tanaka et al. (1985a); Tanaka et al. (1985b) and Tanaka & Ichimaru (1989) computed the interaction energy of the finite-temperature electron liquids in the RPA and the Singwi-Tosi-Land-Sjölander approximation (STLS) (Singwi et al. 1968). The values of [FORMULA] obtained in the RPA and the STLS approximations may provide accurate estimates of the exact free energy [FORMULA] at finite temperatures as an interpolation between the classical and degenerate limits. A fitting formula may be constructed which parametrizes the computed values, valid when [FORMULA]; not only can this be applied to the weak coupling region ([FORMULA]), but it also yields the Debye-Hückel limiting law in the classical limit ([FORMULA]):

[EQUATION]

where

[EQUATION]

Eq. (35) reproduces the RPA values and the STLS values within 1% for [FORMULA]. The excess free energy [FORMULA] can be derived from the above expression with the aid of Eq. (26):

[EQUATION]

The Coulomb pressure [FORMULA] can be obtained from Eqs (35) and (37) as

[EQUATION]

where

[EQUATION]

3.2. Ion-ion interaction

All the thermodynamic functions of classical ions in a uniform (rigid) electron background can be expressed as functions of the single parameter [FORMULA], where [FORMULA] denotes the coupling parameter of ions of species j, An accurate analytic fit of the internal energy of ions must recover the Debye-Hückel limit [FORMULA]. For the ion-ion interaction, we adopt a simplified internal energy formula proposed by Chabrier & Potekhin (1998) in the framework of the N-body HNC theory, which accurately reproduces the Debye-Hückel value for [FORMULA] and provides a smooth transition from [FORMULA] to [FORMULA]:

[EQUATION]

where the fitting parameters are [FORMULA], [FORMULA] and [FORMULA].

The excess free energy [FORMULA] for ions of species j arising from the contribution of ion-ion interaction can be derived from the above expression with the aid of Eq. (26):

[EQUATION]

For a H-He mixture, the total ion-ion contribution to the free energy is given by the linear-mixing formula Eq. (29) to good accuracy

[EQUATION]

The corresponding changes in pressure due to ion-ion interaction is obtained by using Eq. (14):

[EQUATION]

3.3. Ion-electron interaction

When the Fermi degeneracy of the electrons is weak, we can assume classical statistics both for the electrons and for the ions. We consider here a modified Coulomb interaction rather than the bare Coulomb potential through which purely classical particles would interact. Suppose that a point charge [FORMULA] is introduced into the thermal uniform plasma. We allow the plasma to settle down to a steady state after the charge is introduced. Now the electron and ion densities will be determined by the Maxwell-Boltzmann distribution, so that the changes in density are given by the Boltzmann factors:

[EQUATION]

[EQUATION]

Since the unperturbed state has zero charge density, so that [FORMULA], the electrostatic potential [FORMULA] is related to the charge density through the Poisson-Boltzmann equation, which can be written as (Sturrock 1994; Brüggen & Gough 1997):

[EQUATION]

where [FORMULA] is the position of the ion j, and [FORMULA] is the Dirac delta function. Furthermore, linearization yields

[EQUATION]

If [FORMULA] is a weak potential then expansion of the exponentials in Eq. (46) leads to

[EQUATION]

In solving Eq. (48) one must require that both [FORMULA] and the electric field [FORMULA] be continuous across the exclusion sphere [FORMULA] and that [FORMULA] vanish as [FORMULA]. In the exterior region, [FORMULA], the screened potential of a given ion j is given by the extend Debye-Hückel theory with hard-core correction (Lee & Fisher 1996; Levin & Fisher 1996):

[EQUATION]

with

[EQUATION]

Since Eq. (49) is the solution of Eq. (48), the net electron density in the cloud around ion j is given by

[EQUATION]

or alternatively

[EQUATION]

and the ion density [FORMULA] is

[EQUATION]

The ion-electron interaction energy for ions of species j is obtained by (Shaviv & Shaviv 1996):

[EQUATION]

On the basis of the spherically symmetric approximation, substituting Eqs (52) and (53) into Eq. (54), the dimensionless form of Eq. (54) is given by

[EQUATION]

By using Eq. (4), Eq. (55) becomes

[EQUATION]

In order to consider screening effects of the degenerate electrons, Eq. (9) can be rewritten as

[EQUATION]

where

[EQUATION]

is a correction for degenerate electrons. In the non-degenerate limit, [FORMULA].

By using Eqs (3), (4) and (57), we obtain the relation

[EQUATION]

With the aid of Eqs (26), (29) and (59), the excess free energy [FORMULA] due to the ion-electron interaction for the weak coupling limit is approximately given by the linear-mixing rule

[EQUATION]

The corresponding change in pressure due to screening effects of degenerate electrons is obtained by using Eq. (14)

[EQUATION]

Finally, the Coulomb pressure [FORMULA] is given by

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: December 15, 2000
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