Phonon dispersion of crystalline solids from the density-functional theory of freezing

Phonon dispersion in solids is usually calculated starting from model interaction potentials. In this paper, we discuss an approach for calculating phonon energies in a crystalline solid close to melting, using the density-functional theory of freezing. This theory uses the (measured or calculated) direct correlation functions of the corresponding liquid close to freezing as input parameters. To illustrate the method we calculate phonon dispersion of solid argon close to its triple point, using two sets of experimental structure-factor data for liquid argon. We then discuss the phonon dispersion for a model solid with Lennard-Jones interaction potential, using a fit to the computer simulation results on the structure factor of the Lennard-Jones fluid. We also calculate the force constants of the solid from the liquid structure-factor data. Our calculation uses a parametrization of the solid density as a periodic isotropic Gaussian distribution centered at the corresponding crystal lattice sites, an approximation used by several authors previously in a variety of other contexts. The results that we obtain are qualitatively reasonable, but quantitatively they do not agree very well with the experimentally measured values: For example, at the zone boundaries our calculated phonon energies are larger by a factor of about 1.5. We discuss the reasons for the discrepancy, and how it can be overcome by improved calculations.