Calculation of Dynamical Surface Properties of Noble-Gas Crystals. I. The Quasiharmonic Approximation

Abstract

Various dynamical surface properties have been calculated for noble-gas crystals in the quasiharmonic approximation. The calculations were carried out for slab-shaped fcc crystals in which the atoms interact through a Lennard-Jones potential. Changes in the force constants near the surface caused by changes in the interplanar spacings (static displacements) have been taken into account. Mean-square amplitudes and mean-square velocities of vibration were calculated for the (100), (111), and (110) surfaces. The surface specific heat, which was calculated for the (100) surface, is found to be positive and to vary as ${T}^{2}$ at low temperatures. So-called dynamic displacements, which are the displacements of the mean atomic positions in a vibrating crystal with free surfaces from the mean positions in the bulk, were calculated for the (100) surface. They are found to increase rapidly with temperature, indicating that thermal expansion is considerably greater at the surface than in the bulk. Some qualitative features of the static displacements for the (100), (111), and (110) surfaces are pointed out and explained. It is shown that the dynamical matrix for a general slab-shaped crystal can be reduced to a real symmetric matrix of the same size; this fact greatly facilitates calculations based on such models. The fact that the mean-square amplitudes diverge at finite temperatures for infinite slab-shaped crystals is discussed; it is found that accurate calculations based on slab-shaped models are possible despite this feature.