Localized Modes and Cell-Model Limit in the Crystal Impurity Problem

Some aspects of the crystal impurity problem are again studied. The validity of the cell-model limit of a localized mode is discussed, and the extension of the cell model to nonasymptotic frequencies is presented. The condition $M^{\prime }<\frac{1}{2}M$ is shown to be sufficient to guarantee localized modes in the presence of mass defects. (The condition $M^{\prime }<M$ is only necessary.) Qualitative differences between mass and force-constant changes are discussed.