Lattice vibrations and elastic constants of three- and two-dimensional quantal Wigner crystals near melting

The phonon dispersion relations and the elastic constants are evaluated for Wigner electron crystals near the critical coupling strength for melting in the fully quantal regime at zero temperature. The structures considered are the body-centred and face-centred cubic lattices in dimensionality D = 3 and the triangular lattice in D = 2. The calculations are based on a density functional approach requiring as input the linear density response function of the fluid phase at freezing and the Debye-Waller factor of the crystal at melting. These are known from quantal Monte Carlo simulations both for D = 3 and for D = 2. Comparison with earlier results of harmonic calculations shows appreciable softening from anharmonicity, which is mainly associated with the exchange and correlation contributions to the effective force constants of the quantal crystal near melting. Mechanical stability of the body-centred cubic electron crystal at melting is demonstrated through a self-consistent calculation of the lattice vibrations and the mean square particle displacement entering the Debye-Waller factor, as well as by calculations of the elastic constants using the methods of long waves and of homogeneous deformations. Finally, a relationship is displayed between phonon dispersion curves in the triangular Wigner crystal near melting and plasmon excitations in the two-dimensional electron fluid near freezing.