Lattice Thermal Resistivity due to Dense Arrays of Dislocations

The scattering of phonons by dislocations is modified if the strain field of dislocations is cut off at a finite range. The lattice thermal resistivity of a dense dislocation array is calculated on the assumption that the cutoff radius is comparable to the average distance between dislocations. Phonons of very long wavelength are only weakly scattered, and phonon-electron scattering must be invoked to obtain a finite thermal resistivity. The lattice thermal conductivity of deformed alloys departs from the usual $T^2$ dependence. On a reduced plot the shape of the conductivity curve depends on the ratio of the lattice conductivity in the annealed state to that in the deformed state. Curves are given for two values of this parameter. In a typical case of a heavily deformed copper alloy, significant deviations from the $T^2$ dependence should occur at liquid-helium-three temperatures. The shape of the curves could be used to estimate dislocation densities independently of estimates requiring knowledge of the strength of the interaction between phonons and dislocations.