2. Theoretical schemes
2.1. Parameters of plasma for a H-He mixture
To describe non-ideal effects of the plasma, it is convenient to introduce several dimensionless parameters to characterize the plasma. We consider a H-He plasma mixture consisting of electrons and ions of species , which we label with the index , with charges , masses in a volume V. If the abundances by mass are X and Y for H and He with density and temperature T, the number of nuclei for H and He are given by
where is Avogadro's number, and and are the atomic weights. The averaged nuclear charge is defined as
For the ion system, the Wigner-Seitz radius defined as
is the so-called ion-sphere radius which measures the mean interionic distance; here the total number of ions is determined by the electroneutrality condition, and is the number of free electrons.
The averaged Coulomb coupling constant of the ions, which describes the strength of the Coulomb coupling, is given by (Ichimaru et al. 1987):
where denotes the classical Coulomb coupling constant of the electrons, which is written as
where is the Boltzmann constant.
A typical dimensionless density parameter characterizing the system of electrons is:
where is the mass of an electron. The parameter is the Wigner-Seitz radius of the electrons in units of the Bohr radius and depends only on the number of electrons .
The most evident quantum-mechanical characteristic length is the thermal de-Broglie wavelength of particles
The degree of Fermi degeneracy of electrons, which is measured by the ratio of the temperature T to the Fermi temperature (Iyetomi & Ichimaru 1986), is described as
Another important quantity is the Debye shielding length for a mixture of ions and electrons, which is defined as
with the charge average
where is the reciprocal of the Debye shielding length , and j runs over all ion species in the plasma, so that denotes the number of ions of species j.
For a classical system of charged particles the potential energy is of the order of , and the weakly non-ideal condition is reduced to the inequality
It will be useful to evaluate the number of charged particles within the Debye sphere. This number is equal to
where n is the total number density of the plasma (note that full ionization is assumed). This extremely important inequality says that there is more than one charged particle within the Debye sphere. The closer to being ideal is the plasma, the greater is the number of particles within the Debye sphere. In the Debye sphere each charged particle is in a self-consistent plasma field. This means that the long-range character of Coulomb interactions makes particles interact with each other simultaneously even in a weakly non-ideal plasma. Thus, the Debye-Hückel theory is valid under solar interior conditions. However, the simple Debye-Hückel theory does not include any quantum-statistical effect on the electrons.
2.2. Free energy model for a H-He mixture
The total free energy F of a fully ionized H-He mixture consisting of ions and electrons can be written as (Chabrier & Potekhin 1998):
where denote the ideal free energy of ions and electrons respectively, and denotes the excess free energy due to Coulomb coupling.
The pressure P, the entropy S and the chemical potentials µ can be obtained by differentiation of the free energy F with respect to either V and T, at fixed , or with respect to and , at fixed V and T, respectively:
Here we restrict ourselves to conditions where the ions are regarded as point charges while the electrons are assumed to form a uniform background of neutralizing space charges. Thus and are given by Maxwell-Boltzmann statistics and Fermi-Dirac integrals. The ideal free energy of classical ions, neglecting their spin statistics, can be written as
For partially degenerate electrons, the ideal free energy of the quantum electron is
where is the pressure of the ideal electron Fermi gas. The pressure and electron density , in turn, are functions of the chemical potential and temperature T, given by:
(For simplicity, we assumed that the electrons can be treated as non-relativistic.) Here , and the Fermi-Dirac integrals are defined by
The chemical potential is related to the Fermi-degeneracy parameter
© European Southern Observatory (ESO) 2000
Online publication: December 15, 2000