Theory of Surface Modes of Vibration in Two- and Three-Dimensional Crystal Lattices

Abstract

Theoretical expressions have been developed for the frequencies and displacements of the normal modes of vibration for two- and three-dimensional alternating diatomic lattices with free boundaries. Only square and cubic lattices are considered. Nearest-neighbor Hooke's law forces having both longitudinal and transverse components are assumed. The results have been obtained both by a perturbation method in which the ratio of the transverse and longitudinal force constants is treated as a small quantity and by a Green's function method. The use of the free boundary condition leads to the existence of surface modes of vibration in which the displacement amplitude is relatively large for a light atom on a boundary and decreases roughly exponentially toward the interior of the lattice. A band of surface mode frequencies lies in the "forbidden" gap between the acoustical and optical branches.