## 3. Non-ideal effectsThe excess free energy arising from nonideal effects is written as where denotes Coulomb coupling, and takes account of the contributions from interactions of neutral species with electrons, ions and other neutral species. Two physical processes, i.e., Coulomb correction of charged particles and pressure ionization of neutral species, have been taken into account in this paper. Owing to the complexity of such physical processes, we only consider simplified and approximate models. ## 3.1. Improvement of Coulomb couplingThe interaction contributions of Coulomb coupling, as presented in
the work of Bi et al. (2000), allow fast access to different
thermodynamic properties in wide ranges of density and temperature.
The Debye-Hückel approximation overestimates Coulomb effects when
the coupling becomes significant at moderately coupling value.
Therefore, the present improvements are made on the basis of the
available analytical formulae for the quantum exchange effect of
electrons at finite temperature, The Coulomb term of free energy can be expressed as the sum of the separate contributions, electron-electron, ion-ion and ion-electron interaction, that is, where the dimensionless form of excess free energy in Eq. (21) can be written by the linear-mixing rule (Ichimaru & Kitamura 1996): where represents the number
fraction of ions of species where is the dimensionless form of the Coulomb internal energy. ## 3.1.1. Electron-electron interactionFor the electron-electron contribution, we adopt the expression of interaction energy proposed by Bi et al. (2000), which is based on the results of quantum-statistic calculations of electrons at the finite temperature: The excess free energy arising from electron-electron interaction is then obtained by performing integration: In the weak coupling limit (), the values of asymptically approach the lowest-order exchange energy, i.e., Hartree-Fock energy. In the classical and weak coupling limits, the Debye-Hückel value then becomes the leading contribution to the electron-electron interaction. Considering those boundary conditions mentioned above, Eq. (26) reproduces the RPA values (Fetter & Walecka 1971; Tanaka et al. 1985) and the STLS values (Singwi et al. 1968) within 1% for . ## 3.1.2. Ion-ion interactionFor the ion-ion interaction, we adopt a simplified internal energy
formula for ions of species where denotes coupling parameter
of ions of species Neglecting the interactions between the charged hydrogen and helium species, the total internal energy and the excess free energy arising from the contribution of ion-ion interaction can be given by the linear-mixing rule at good accuracy, and ## 3.1.3. Ion-electron interactionIf we assume that ion-electron interaction is weak, in which the Poisson-Boltzmann equation for the electrostatic potential can be linearized, the screened OCP model for the description of the thermodynamic properties of the two-component ion-electron plasma can be adopted. When the Fermi degeneracy of the electrons is also weak, we can farther assume that electrons and ions obey Maxwell -Boltzmann statistics. For ion-electron screening effect, we adopt the internal energy formula given by Bi et al. (2000) on the basis of the extended Debye-Hückel theory with hard-core correction, namely where the function is defined as and the reciprocal of the Debye shielding length is given by Here is a correction for degenerate electrons. In the non-degenerate limit, . By using Eqs. (6), (8) and (33), after some operations, we have a relation With the aid of Eqs. (5) and (35), the total internal energy from the contribution of ion-electron interaction is given by the liner-mixing rule, so that Thus, the excess free energy due to ion-electron interaction is obtained by performing integration in Eq. (23) under weakly coupled limit, as As a result, corresponding Coulomb corrections to the thermodynamic functions, i.e., the pressure , the electron chemical potential and the enthalpy , can be calculated from and where the Coulomb correction of the internal energy is given by ## 3.2. Approximate treatment of pressure ionizationThe nonideal part of the atomic free energy , which takes account of the contributions from interactions of an atom with surrounding particles, can be given by Here the first term denotes the neutral-neutral interactions based on the hard-sphere excluded-volume treatment. This excluded volume contribution is the lowest-order approximation to the total configuration of the free energy. The second term represents the neutral-charged interaction. In the plasma with high temperatures (), pressure ionization of atoms is predominantly based on Stark-ionization theory. For the cooler surface of the Sun, however, neutral atoms dominate the interaction between interparticles. Rather sophisticated physical models for the interactions based on advanced physics theory are discussed in Saumon & Chabrier (1991; 1992). For astrophysical application and research, we only adopt an approximate model for the complicated physical processes of pressure ionization to produce the essential thermodynamic features. The free energy of the hard-sphere mixture is represented by the occupation probabilities according to the definition given by Potekhin et al. (1999): with the occupation probability formalism given approximately by Luo (1997) where is the total number of
atomic nuclei. A detailed treatment of the neutral-charged interaction requires complicated and time-consuming calculation. Pressure ionization is predominantly a volume effect. Owing to the destruction of relatively loosely bound states by interactions of both charged and neutral particles in the plasma, the particles are jammed closely together, bound electron orbitals filling too large a volume fail to survive, and the electrons migrate from atom to atom. The electrons in a bound state are so called "acting" electrons. We consider that "acting" electron in a bound state can move freely with respect to a particle in bound, thus making the gross simplification that the "acting" electrons depend only on the density of free electrons. For simplicity, we adopt the expression for by EFF (1973) and Christensen-Dalsgaard (1977), namely where and are constants, and is the total number of electrons per unit mass, in bound or free Here As a result, the contribution from pressure ionization to the pressure , the electron chemical potential and the enthalpy can be formulated in terms of and where is the number of electrons
per unit volume. The hard-sphere term of Eqs. (48) and (50) is
explicit functions of © European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |