Spatial Variation of Phonon Distribution in Thermal Conduction

Abstract

Thermal conduction is usually studied for a homogeneous solid in which there is a uniform temperature gradient dT/dx, which is sufficiently small that the linear relation Q = -KdT/dx is valid, where Q is the heat current and K is the thermal conductivity. However there is increasing interest in inhomogeneous situations in which heat is injected into a solid from a heater or flows across an interface between two solids as in Kapitza conduction [1]. We shall only consider insulating solids in which heat conduction is by lattice phonons. Moreover, at liquid helium temperatures, where such experiments are often performed, phonons are principally elastically scattered by defects or boundaries. The existence of a local phonon temperature cannot then be assumed since inelastic processes are necessary to equilibrate phonons of different frequencies.