Resonance Scattering of Lattice Waves by Isotopes

Abstract

An investigation is made of the scattering of lattice waves by isotopes in a Bravais lattice by studying classical time-independent equations of motion for the displacement of atoms from their equilibrium positions. Expressions for the cross section or the intensity, of a scattered wave by an isolated isotope and by a pair of isotopes are obtained from the solutions of the equations of motion. It is shown that a resonance scattering occurs when the difference between the mass of an atom of the host crystal and that of an isotope exceeds a certain critical value. The Rayleigh scattering formula obtained by Pomeranchuk and Klemens is shown to be valid only when the mass difference is small. As a model, a simple cubic lattice with the nearest neighbour interaction is studied in detail, and the scattering cross section as well as the intensity of the scattered wave is evaluated explicitly as a function of the frequency and the mass difference. It is suggested that (1) a very heavy isotope causes a sharp resonance scattering in the low frequency region, and (2) in the high frequency region a resonance scattering occurs only when the mass of the isotope is so small that it causes a localized mode.