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, $l$. A formula $l=\frac{a}{3.3\epsilon }$ is obtained, where $l=\mathrm{mean}\mathrm{free}\mathrm{parth}$, $a=\mathrm{lattice}\mathrm{spacing}$, $\epsilon ={〈\frac{(m_n-m)}{m}〉}_{\mathrm{rms}}$; $m_n$ being the mass of the $n\mathrm{th}$ atom, and $m$ the average mass. This result is in agreement with an independent quantum-mechanical calculation which is sketched.