Localization and propagation of phonons in crystals with heavy impurities

The energy spectrum of a harmonic lattice with heavy impurities is studied within the framework of the scalar model approximation. A detailed analysis of the dynamical properties of the system is made in the frequency range close to the quasi-localized mode. It is shown that the dwell effect for the phonon diffusion coefficient occurs provided the lifetime of the quasi-localized mode becomes of the order of the phonon lifetime. The diffusion coefficient manifests a drastic reduction (the dwell effect) under the same condition necessary for the peak in the heat capacity to be observed. It is shown that the concentration of the impurities needed for observing this effect depends on the impurity mass. For higher concentration of the impurities, the energy gap is shown to appear in the frequency range dose to the quasi-local mode. In addition, localized modes are shown to exist in the frequency ranges adjoined to the energy gap.