The "homogeneous electron gas", which consists of a large number of electrons in a uniform background, is a very important model system which is still yielding insights into many-electron phenomena. Wigner first predicted that a system of electrons in a uniform background would crystallize at low densities. Localizing electrons around lattice sites increases their kinetic energy, but at sufficiently low densities the reduction in their interaction energy is always greater. Two-dimensional Wigner crystals have been created on droplets of liquid helium.

Drummond et al. calculated the energy of the three-dimensional Wigner crystal as a function of density using the diffusion quantum Monte Carlo (DMC) method, and compared their results with the DMC data for the low-density fluid phase of Zong et al.. At zero temperature the stable phase is the one with the lowest energy. Hence we were able to determine the crystallisation density of the homogeneous electron gas. In particular, the Wigner crystal phase was found to be more stable than the fluid phase at densities lower than r_s=106±1.

DMC energy of the fluid and Wigner crystal phases of the three-dimensional electron gas as a function of density. Fluid data from Zong et al., crystal data from Drummond et al.