Vibrations of Disordered Solids

An important portion of the vibrational spectrum of an infinite disordered solid is attributed to "localized" vibrations. These can be investigated in detail through the study of the vibrations of very small solids: It is argued that any vibrational mode found in a small solid that has the property that atomic amplitudes are very small near the boundary will also be found, with at most slight perturbation, in a large solid. It is observed that such modes appear whenever a group containing light atoms occurs and is bounded by a sufficient number of heavy atoms; the number that is "sufficient" depends somewhat on force model, mass ratio, and dimensionality, but is generally small. The density of these local modes in the spectrum of the large crystal can then be computed from the probability of the occurrence in the large crystal of the configuration of masses that produces it. For the case of the one-dimensional binary alloy, the major features of the earlier direct numerical computation (Dean) can be reproduced and understood by this simple method. Similar features (peaks) are predicted for two- and three-dimensional structures, but their intensity compared to the continuous part of the spectrum is shown to be smaller.