3. Non-ideal effects
The 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 coupling
The 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, N-body semi-analytic theory for ions, and the extended Debye-Hückel theory with hard-core correction for ion-electron interaction under the weakly coupled and weakly degenerate conditions. These modifications not only can be applied to the weak coupling region, but can also yield the Debye-Hückel limiting law in the classical limit.
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 i in the total ionic configurations . Since , we may approximately regard the electron-electron and ion-ion interaction terms as being the functions of only and , respectively. In general, the excess free energy due to Coulomb coupling is calculated through the coupling constant integration of the internal energy (Ichimaru et al. 1987), namely
where is the dimensionless form of the Coulomb internal energy.
3.1.1. Electron-electron interaction
For 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 interaction
For the ion-ion interaction, we adopt a simplified internal energy formula for ions of species i proposed by Chabrier & Potekhin (1998), which accurately reproduces the Debye-Hückel value for and provides a smoother transition from to :
where denotes coupling parameter of ions of species i. The fitting parameters in Eq. (27) are , , and . It should be noted that is expressed as functions of the only parameter in a uniform electron background for the classical ions. The excess free energy of ions of species i is calculated according to Eq. (23), that is
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,
3.1.3. Ion-electron interaction
If 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
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
Thus, the excess free energy due to ion-electron interaction is obtained by performing integration in Eq. (23) under weakly coupled limit, as
where the Coulomb correction of the internal energy is given by
3.2. Approximate treatment of pressure ionization
The 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. j includes three configurations, i.e., ; is an effective radius of species j. The radii of atomic and singly ionized helium are taken to be , and , respectively, with Bohr radius .
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 X, Y and Z are the mass abundance of , and elements beyond hydrogen and helium, respectively. is Avogadro's number. and are the atomic weights. Eq. (46) ensures that the effect of pressure ionization on the thermodynamic functions becomes zero as .
As a result, the contribution from pressure ionization to the pressure , the electron chemical potential and the enthalpy can be formulated in terms of
where is the number of electrons per unit volume. The hard-sphere term of Eqs. (48) and (50) is explicit functions of T, , and . Here and denote the number of hydrogen atoms and helium atoms, respectively. The factor is introduced to take account of the fact that the interactions of particles have actually been counted twice for each atom in the sum.
© European Southern Observatory (ESO) 2000
Online publication: January 29, 2001