SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


Astron. Astrophys. 364, 879-886 (2000)

Previous Section Next Section Title Page Table of Contents

2. Free energy model for a H-He mixture

Since the main matter constituting the Sun is a mixture of hydrogen and helium, we need an expression for the free energy including the contribution of at least six chemical species with free and bound state, i.e., electrons, ions [FORMULA], and neutral particles [FORMULA]. The total free energy F can be written as the sum of four terms (Saumon et al. 1995; Potekhin et al. 1999):

[EQUATION]

where [FORMULA] is the ideal-gas contribution of the particles; [FORMULA], the contribution from the uniform electron gas; [FORMULA], the radiation term of free energy; [FORMULA], the excess free energy arising from nonideal effects.

The pressure P, enthalpy H and the electron chemical potential [FORMULA] can be given by differentiating the free energy with respect to either V and T, at fixed [FORMULA] and [FORMULA], or [FORMULA], at fixed V and T, respectively:

[EQUATION]

[EQUATION]

and

[EQUATION]

2.1. Plasma parameters

The behaviour of a H-He mixture manifests nonideal effects of interacting charged particles. Degeneracy of electron, Coulomb coupling and pressure ionization are important features of the plasma. To describe nonideal effects of the plasma, it is convenient to introduce dimensionless parameters to characterize the plasma. We consider the plasma consisting of [FORMULA] ions of species [FORMULA], with charges [FORMULA], [FORMULA] neutral particles of species [FORMULA], and [FORMULA] electrons in a volume V at a temperature T. The mean ionic charge is defined as

[EQUATION]

In the multicomponent plasma, the averaged Coulomb coupling parameters for ions and for electrons are given by (Ichimaru et al. 1987; Stolzmann & Böcher 1996):

[EQUATION]

and

[EQUATION]

where

[EQUATION]

is the mean ion-sphere radius, and [FORMULA] is the Boltzmann constant. The total number of ions is [FORMULA]. The charge neutrality condition requires [FORMULA].

The dimensionless density parameter of electrons [FORMULA] and the degree of Fermi degeneracy [FORMULA] are described as, respectively

[EQUATION]

and

[EQUATION]

where [FORMULA] is the mass of an electron.

2.2. Ideal part of the free energy

The main purpose of this work is the investigation of the thermodynamic functions of the plasma. We stress that charged and neutral particles behave classically. In this paper, we limit ourselves to the consideration of one-component plasma (OCP) where the particles are regarded as point particles while the electrons are assumed to form a uniform background of neutralizing space charges. Thus [FORMULA] and [FORMULA] can be given by Maxwell-Boltzmann statistic and Fermi-Dirac integrals. The ideal free energy of classical particles can be written as

[EQUATION]

with the thermal de-Broglie wavelength of particles

[EQUATION]

where k runs over five species [FORMULA]; [FORMULA] is the statistical weights both in bound and free states; the corresponding energies [FORMULA] of ionization for five species have values [FORMULA] and -78.98 in electron volts.

For partially degenerate electrons, the ideal part of the free energy for electrons is

[EQUATION]

where [FORMULA] is the pressure of the ideal Fermi gas. The pressure [FORMULA] and electron density [FORMULA], in turn, are functions of chemical potential [FORMULA] and temperature T, which can be given by

[EQUATION]

and

[EQUATION]

where the degeneracy parameter is defined as [FORMULA], and

[EQUATION]

is the usual nonrelativistic Fermi-Dirac integral. The chemical potential is obtained from the relationship

[EQUATION]

Furthermore, [FORMULA] can be expressed in useful analytic formula (Ichimaru & Kitamura 1996):

[EQUATION]

with [FORMULA], [FORMULA], and [FORMULA], as a function of the degeneracy parameter [FORMULA].

The third term on the right of Eq. (1), i.e., radiative term [FORMULA], is

[EQUATION]

where [FORMULA] and c are the Stefan-Boltzmann constant and speed of light, respectively.

The excess free energy, [FORMULA], from the contribution of nonideal effects, will be discussed in Sect. 3.

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
helpdesk.link@springer.de