Thermal Conductivity of LiF and NaF and the Ziman Limit

Abstract

A model calculation of lattice thermal conductivity is presented and applied to measurements on LiF and NaF crystals of high purity. The treatment is in the spirit of a Callaway analysis, but at a somwhat more fundamental level: The Ziman variational principle for thermal conductivity derived from the phonon Boltzmann equation is used, with the phonon distribution function approximated by a displaced Planck part plus another term reducing the deviation from equilibrium for high-frequency phonons. An isotropic Debye approximation for the phonon spectra of LiF and NaF gives a good fit to the conductivity data, with only two semi-adjustable parameters (Gr\"uneisen constant and a zone-edge longitudinal phonon frequency) for the anharmonic contribution. The most important feature of the calculation is the failure of the thermal conductivity to approach the Ziman limit of resistanceless phonon-phonon $N$ processes. This is due to the important role played by high-frequency phonons in thermal conduction. Even for an infinite perfect crystal at arbitrarily low temperatures, the Ziman limit underestimates the conductivity by at least 50%. If this prediction is correct, it is not a peculiarity of LiF and NaF alone, and should be of importance for the theory of second-sound propagation in insulators.