Note on the Electron Lattice

The question of the stability of Wigner's electron lattice, which is the ground state of a dilute electron gas, is considered. To determine the density range in which the electron lattice becomes unstable (coming from lower densities) a simple stability criterion is formulated. It is based on the notion that, in order for a solid structure to exist, the single particle potential should exhibit a localized well with at least one bound state. The disappearance of bound states with increasing density, marking the onset of melting in the model, is related to the possibility of defect formation. In this fashion an upper and a lower limit for the melting density can be established, which are found to occur at densities corresponding to $r_s\approx 47$ and $r_s\approx 100$, respectively ($r_s$ is the radius of the unit sphere in Bohr units). Consequently, melting of the electron lattice is expected to take place at densities much lower than estimated previously.