D2 - LOCALIZED MODES AT EXTENDED DEFECTS IN CRYSTALS

Abstract

Perturbations in the homogeneity of a crystal can give rise to localized modes of vibration. We have discussed the simplest cases of extended defects, namely planes of impurity atoms with special directions (001, 011, etc.) in a simple cubic crystal with nearest neighbour interaction. Extended defects, such as planes of impurity atoms, will have localized modes with exponentially decreasing amplitudes in the direction perpendicular to the planes and wave-like character in directions parallel to them. The different modes of localized vibrations have been analyzed group-theoretically. It comes out that there will be in general an acoustical and an optical branch of localized modes for a plane defect, the occurrence of which and the frequencies relative to the band of the ideal lattice frequencies depend on the defect-parameters (changes in mass and force constants). In the limit of vanishing coupling between defect plane and “host” lattice we get a free surface, which has been considered by Wallis et al. This limiting case has only an acoustical branch, which is identical with the Rayleigh-surface modes for long waves (provided the force-constants fulfill the conditions allowing localized states at all). Also lines of defects can have two branches of modes. The details depend as in other cases on the defect-parameters.