Effect of Boundary Surfaces on Localized Mode Frequencies of a Crystal Containing a Mass Defect

The vibrational frequencies of the localized modes due to a mass defect in a simple cubic lattice with boundary surfaces have been studied. The perfect crystal is treated in the harmonic approximation with nearest- and next-nearest-neighbor central-force interactions between atoms. The most general problem is solved formally, and a detailed treatment is presented for the case of a very small mass of the defect atom compared to that of the host atom. The detailed calculations are performed in the case where a condition is imposed on the force constants of the crystal which leads to elastic isotropy in the long-wavelength limit. A pair of boundary surfaces is created mathematically be setting to zero all interatomic forces which cross a given plane. It is found that two of the three localized modes caused by the defect are degenerate. The nondegenerate mode corresponds to motion of the impurity atom normal to the surface, and the degenerate ones correspond to motions in two perpendicular directions parallel to the surface. For a light defect in the crystal surface and a ratio of its mass to the host atom's mass of $\frac{1}{5}$, the frequency of the nonedegenerate mode is 71% of the bulk value, i.e., the frequency of a localized mode corresponding to the same defect in the absence of a free surface; the frequency of the degenerate modes is 91% of the bulk value. The localized-mode vibrational frequency when the defect atom is in the first layer below the surface differs from the bulk value by less than 0.02% of the latter value.