Equation of state in metals and cold stars: Evaluation of statistical models

The Thomas-Fermi-Dirac (TFD) statistical model for electronic structure is refined by Weizsäcker gradient and correlation energy corrections. The resulting TFD-λWc model is then used to calculate the nonrelativistic, cold matter equation of state (EOS) in the Wigner-Seitz spherical cell approximation. The correction terms are important mainly at low densities near matter equilibrium (i.e., zero pressure). Inclusion of the gradient term removes many of the unphysical features of the TFD model. Results are summarized for several elements of astrophysical and theoretical importance including ₆¹²C, ₁₂²⁴Mg, ₁₃²⁷Al, ₂₆⁵⁶Fe, and ₉₂²³⁶U. Nonspherical lattice structure effects are then incorporated using a muffin-tin potential model and are found to be small except at very low density. An accurate EOS near matter equilibrium requires a full quantum-mechanical treatment of the electrons to handle shell-structure effects. Some recent self-consistent band structure calculations based on the Kohn-Sham method of solving the many-body Schrödinger equation are summarized. The corresponding EOS for selected solids is used, together with other empirical EOSs and experimental data, to evaluate our computationally less involved TFD-AWc EOS. A simple, semiempirical model based on a pseudopotential provides physical understanding of solids near matter equilibrium and is used to calculate the EOS for low pressures. We also construct and critically evaluate a hybrid model which attempts to integrate shell-structure effects with a Thomas-Fermi statistical treatment of the valence electrons. Finally, we calculate the mass-radius relation of a low-mass white dwarf model using the TFD-λWc EOS. We find that the maximum radius of a carbon white dwarf is R/R_⊙ = 3.9 × 10^-2 at a mass of M/M_⊙ = 2.4 × 10^-3.