Some static and dynamical properties of a two-dimensional Wigner crystal

The static ground-state energy of a two-dimensional Wigner crystal has been obtained for each of the five two-dimensional Bravais lattices. At constant electron number density the hexagonal lattice has the lowest energy. Phonon dispersion curves have been calculated for wave vectors along the symmetry directions in the first Brillouin zone for the hexagonal lattice. In the long-wavelength limit one of the two branches of the dispersion relation vanishes with vanishing two-dimensional wave vector $\overrightarrow{\mathrm{q}}$ as $q$, the second as $q^{\frac{1}{2}}$. The coefficient of $q$ in the former branch is pure imaginary for certain directions of propagation in the square lattice, implying a dynamical instability of this lattice; the hexagonal lattice is stable. The vibrational zero-point energy and low-temperature thermodynamic functions have been obtained for the hexagonal lattice. The dielectric susceptibility tensor of a two-dimensional Wigner crystal ${\chi }_{\alpha \beta }\left(\overrightarrow{\mathrm{q}}\right)$ has been determined in the long-wavelength limit, in the presence of a static magnetic field perpendicular to the crystal, and the result has been used to obtain the dispersion relation for plasma oscillations in the electron crystal.