Thermal Conductivity of Perfect Dielectric Crystals in the Absence of Umklapp Processes

Abstract

The thermal conductivity of a perfect but finite crystal is investigated at low temperatures when Umklapp processes may be neglected. The sample is taken to be large compared with the mean free path due to momentum conserving phonon-phonon processes. A mean free time approximation is used for the phonon distribution function and hydrodynamic equations for the phonon gas are derived. Heat flow is shown to consist of two contributions - one is the usual (diffusion-like) heat flow and the other is due to a drift motion of the phonon gas. A temperature difference between the ends of a long cylinder with a rough surface will lead to a Poiseuille flow of the phonon gas. An abrupt change of temperature in the immediate vicinity of a surface through which heat is flowing is obtained.