C

Ne ke

3af tr

4n xr3

The correlation corrections in the relativistic gas are negligible, so that one can set Fxc ^^ Fx in this case. In general, the exchange and correlation corrections are of little importance in neutron star matter as long as the electrons are relativistic.

2.4.4 Electron polarization in ion liquid

In §§ 2.3.2-2.3.3 we considered the Coulomb plasma of ions with the rigid electron background. This approximation is justified in many applications, especially in dense layers of neutron star envelopes (§ 2.2.1, Table 2.1). However, in some cases the polarization ion-electron (ie) interaction associated with deviations from the rigid electron background can be noticeable (for example, in a neutron star atmosphere, where the density is relatively low, § 2.5.2). In § 2.4.2 we have already considered the electron polarization effect in the Thomas-Fermi approximation (valid for high-Z ions). Here we present a more accurate treatment of the problem.

2.4.4 a Theory

In a plasma of ions and electrons, the ion interactions are partly screened by the electron polarization. As long as the polarization interaction is weak compared to the kinetic energy of the electrons (for instance, at Ze2/ae ^ kBTF, for degenerate electrons), this interaction can be treated within the linear response theory. Under these conditions, the exact Hamiltonian of the electron-ion plasma can be separated into a Hamiltonian for a fluid of ions screened by electrons and a Hamiltonian for a rigid electron background, the so-called "jelly" Hamiltonian (Galam & Hansen, 1976).

At rs ^ 1 and T ^ TF, the polarization corrections due to the electron screening can be obtained in the zero-temperature random-phase approximation [Jancovici 1962; cf. § 2.1.2]. If rs < 1 and T < TF, the nonlinear effects due to the local field correction (2.21) and a finite electron degeneracy should be taken into account. The electrons can be treated by the DFT in various approximations or by the Monte Carlo technique (e.g., Pierleoni et al. 1994 and references therein). The Monte Carlo simulations are the most rigorous ones, but require excessive computer resources.

The screening contribution can also be evaluated by the method of effective potentials (see, e.g., Ichimaru et al. 1987). In this method, the bare Coulomb potential in the expression for the electrostatic energy U is replaced by the effective potential screened by electrons. Its Fourier transform is

where e(k) is the static screening function of the electron gas [Eq. (2.18)] or fluid [Eq. (2.21)].

A numerical scheme based on the effective potential technique and the HNC equations was developed and applied to the hydrogen plasma by Chabrier (1990) who considered an arbitrary electron degeneracy and a broad range of r, from the strong-coupling limit r » 1 to the Debye-Hiickel limit r ^ 1. Chabrier & Potekhin (1998) extended these calculations to ion charges up to Z = 26.

2.4.4 b Analytic formulae

The numerical HNC results were fitted by the following expression (Chabrier & Potekhin, 1998; Potekhin & Chabrier, 2000)

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