Asymptotic Description of Localized Lattice Modes and Low-Frequency Resonances

Point imperfections in crystals may produce localized modes of vibration with frequencies not allowed in the perfect crystal, or resonances with frequencies within the perfect-crystal frequency bands. In general, such frequencies depend both on parameters characteristic of the defect and on Green's functions characteristic of the dynamical properties of the perfect lattice. In the limits of frequencies, both very high and very low with respect to the maximum perfect-lattice frequency, simple equations, giving physical insight into the nature of the defect mode, are developed, and the results are compared with recent calculations on specific models and with experiment. It is shown that for a variety of situations the defect-mode properties are, to first order, very directly related to the mass and force-constant parameters in the center only.