Scattering of Phonons by Elastic Strain Fields and the Thermal Resistance of Dislocations

Abstract

A theory of the scattering of phonons by static elastic strain fields is presented. It is found that the Fourier component of the strain field plays a role similar to that of the potential in the external field approximation. All quantities (except the strain field) in the formulas obtained refer to specifically atomic characteristics, allowing in principle the examination of the influence or crystal structure, interatomic potentials, etc., and also scattering between different polarization modes. A Boltzmann equation is found.The results of the theory are used to estimate the low temperature thermal resistance (in nonconductors) due to dislocations. This is done by finding a relaxation time $\ensuremath{\tau}$; with some simplifying assumptions one finds for an edge dislocation ${\ensuremath{\tau}}^{\ensuremath{-}1}=A\ensuremath{\sigma}{[\mathrm{ln}(n{b}^{\ensuremath{-}1}{\ensuremath{\sigma}}^{\ensuremath{-}\frac{1}{2}})]}^{2}q$. In this equation $\ensuremath{\sigma}$ is the density of dislocations, $n$ is the average number in a slip plane, $b$ is the Burgers vector, $q$ is the wave vector of the phonon, $A$ is a constant. This result differs from that obtained previously by Klemens by essentially the presence of the logarithm. This latter factor seems to be essential in explaining the experimental observations of Sproull et al. that the thermal resistance due to dislocations in LiF is three orders of magnitude greater than predicted by Klemens. For a screw dislocation ${\ensuremath{\tau}}^{\ensuremath{-}1}$ lacks the logarithm term so that the scattering is much smaller than for an edge dislocation.