A perturbation expansion in powers of ${r_s}^{-\frac{1}{2}}$ has been used to investigate the ground-state energy of a dilute electron gas, the result being, in rydberg units per particle, $E=-\frac{1.792}{r_s}+\frac{2.66}{{r_s}^{\frac{3}{2}}}+\frac{b}{{r_s}^2}+O\left(\frac{1}{{r_s}^{\frac{5}{2}}}\right)+$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 ${r_s}^{-1\mathrm{}}$ is the energy of a body-centered cubic lattice of electrons as calculated by Fuchs; the ${r_s}^{-\frac{3}{2}}$ 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 $b{r_s}^{-2\mathrm{}}$ 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 ($r_s\gtrsim 270$), the electron spins are antiferromagnetically aligned. An order of magnitude for the Néel temperature has been calculated.

• Received 14 April 1960

DOI:https://doi.org/10.1103/PhysRev.122.1437