Topological interaction of phonons with dislocations and disclinations. II. The scattering problem

Scattering of acoustic phonons by linear defects is studied in the continuum limit of elasticity theory and the scattering matrix induced by the change in the phase of a phonon that passes round a defect in a closed contour is calculated. It is shown that on the background of a screw dislocation and for negative Frank angle of a disclination phonon modes containing components with kinetic angular momentum $\mu$ satisfying the inequality $0<|\mu|<1$ are singular, namely, near the defect line such components can increase unboundedly as $\rho^\mu$, where $\rho$ is the distance to the line of the defect. In the presence of singular modes, the curvature of the gauge group $G=SO(3)\rhd T(3)$, which is concentrated on the defects, leads to transitions between different polarizations. The topological interaction plays a leading role in the case when the phonon wavelength is much greater than the scattering length corresponding to scattering by the short-range potential of the defect core and, thus, it is most important in problems with long-range correlations.