Theoretical High-Pressure Equations of State including Correlation Energy

Abstract

The correlation energy of the electron gas is treated as a perturbation on the total free energy computed for the zero-temperature Thomas-Fermi-Dirac (TFD) model of a solid. The contribution to the free energy is computed on the basis of two extreme assumptions: (i) The correlation energy is considered to be a local function of position inside the Wigner-Seitz sphere; and (ii) correlation forces are sufficiently long range so that the correlation part of the free energy is determined only by the average electron density. It is argued that (ii) is the more valid approach in the region of normal metallic densities. In terms of the parameter $s{a}_{0}$, the radius of a sphere occupied by a single electron, an interpolation formula is developed for the correlation energy in the intermediate region between $s\ensuremath{\ll}1$ (degenerate electron gas) and $s\ensuremath{\sim}60$ (electron lattice-fluid phase transition). At extremely high densities, a series expansion in $s$ is used to obtain an analytic expression for the electron pressure up to terms of order ${s}^{3}$. For these high densities, prescriptions (i) and (ii) are equivalent. At lower densities, numerical results are presented for an assortment of elements for the TFD model, and TFD with correlations based on (i) and (ii). These are compared with shock-wave and seismic data, and the average correlation contribution computed from (ii) is found to be in better agreement than the other two procedures. A semiempirical formula is presented, which fits the numerical data to better than 0.5% for all pressures and atomic numbers, and reduces to the correct expression in the high-density limit. From this semiempirical formula, pressure-density curves are obtained for a variety of elements and minerals which have been considered as likely constituents of the core and mantle of the earth. A table of effective atomic numbers, to be used in the semiempirical formula, is given for the minerals.