Theory of Heat Conduction in Rare-Gas Crystals

Abstract

The thermal resistance due to three-phonon interactions has been calculated numerically for large perfect crystals of neon, argon, krypton, and xenon. These crystals have been approximated by a model crystal having a face-centered cubic structure, one atom in each primitive cell, and central forces acting only between nearest neighbors. Data on the interatomic forces are the only parameters used in the calculation. The thermal conductivities calculated for neon, argon, and krypton agree satisfactorily with experiment for temperatures above one-fourth of the Debye temperature. The discrepancy at lower temperatures is ascribed to the effects of surfaces and defects, which are not taken into account in the calculation. An explicit procedure is given to enable the reader to calculate readily the thermal conductivity of any crystal that may be approximated with the previously mentioned model crystal and for which data on the interatomic forces are available. The basis of the calculations is a formula for the thermal conductivity that is essentially the same as the formula proposed by Leibfried and Schl\"omann. However, the formula is obtained by a new method that gives insight into its validity. The present work supports quantitatively and cogently the theory of heat conduction originated by Peierls. In particular, umklapp processes play the leading role.