## 2. Theoretical schemes## 2.1. Parameters of plasma for a H-He mixtureTo 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 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 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 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 ## 2.2. Free energy model for a H-He mixtureThe total free energy where denote the ideal free energy of ions and electrons respectively, and denotes the excess free energy due to Coulomb coupling. The pressure 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
and (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 |