Energy, Specific Heat, and Magnetic Properties of the Low-Density Electron Gas

Abstract

A perturbation expansion in powers of ${{r}_{s}}^{\ensuremath{-}\frac{1}{2}}$ has been used to investigate the ground-state energy of a dilute electron gas, the result being, in rydberg units per particle, $E=\ensuremath{-}\frac{1.792}{{r}_{s}}+\frac{2.66}{{{r}_{s}}^{\frac{3}{2}}}+\frac{b}{{{r}_{s}}^{2}}+O(\frac{1}{{{r}_{s}}^{\frac{5}{2}}})+$terms falling off exponentially with ${{r}_{s}}^{\frac{1}{2}}$. The dimensionless parameter ${r}_{s}$ is the radius of the unit sphere in Bohr radii. The term in ${{r}_{s}}^{\ensuremath{-}1}$ is the energy of a body-centered cubic lattice of electrons as calculated by Fuchs; the ${{r}_{s}}^{\ensuremath{-}\frac{3}{2}}$ term is the zero-point vibrational energy of the lattice, as obtained from a calculation of the normal modes, the result differing only by a small amount from the values estimated by Wigner; and $b{{r}_{s}}^{\ensuremath{-}2}$ is the first-order effect of anharmonicities in the vibration. The constant $b$ has been estimated, its magnitude being smaller than unity.The vibrational part of the specific heat has been calculated, and a first-order approximation has been obtained for the exponential terms in the energy. Part of this energy comes from exchange, which leads to the result that, except for very low densities (${r}_{s}\ensuremath{\gtrsim}270$), the electron spins are antiferromagnetically aligned. An order of magnitude for the N\'eel temperature has been calculated.