Phonon free path in an isotopic mixture

A time-dependent perturbation theory is developed for the classical laws of motion of a crystal lattice. This is applied to the case of a cubic crystal which is chemically pure but contains various isotopic constituents. The random variations in the masses of neighboring atoms scatter plane wave phonons. This can be described by a phonon mean free path, I. A formula l=a/3.3e is obtained, where 1= mean free path, a= lattice spacing, ε=((mn-m)/m)rm8; mn being the mass of the nth atom, and m the average mass. This result is in agreement with an independent quantum-mechanical calculation which is sketched.