Multiple scattering of elastic waves by pinned dislocation segments in a continuum

Abstract

The coherent propagation of elastic waves in a solid filled with a random distribution of pinned dislocation segments is studied to all orders in perturbation theory. It is shown that, within the independent scattering approximation, the perturbation series that generates the mass operator is a geometric series that can thus be formally summed. A divergent quantity is shown to be renormalizable to zero at low frequencies. At higher frequencies said quantity can be expressed in terms of a cut-off with dimensions of length, related to the dislocation length, and physical quantities can be computed in terms of two parameters, to be determined by experiment. The approach used in this problem is compared and contrasted with the scattering of de Broglie waves by delta-function potentials as described by the Schrödinger equation.

Scattering of phonons by quantum-dislocation segments in an elastic continuum

“…The details of these computations are presented in Appendix A. These results are consistent with those of previous works 19,21,22 in the large wavelength limit kL 1, and furthermore completely define the scattering amplitude.…”

“…As expected 22 , the real part of (37) vanishes as ω → 0 faster than the imaginary part, which goes as ω 3 at low frequencies. Furthermore, it is manifestly finite at all frequencies, as every piece of the integrand is bounded.…”

“…The results contained herein establish, to the best of the authors' knowledge, the first time that such a theory incorporating the Peach-Koehler force has been developed. In addition to that, for the quadratic action we established the finiteness of the theory at all wavelengths, avoiding the need to engage in renormalization procedures previously used in classical settings 22 .…”

“…It is given by the well known Peach Koehler force: b i is the i-th component of the Burgers vector, σ ij is the elastic stress tensor, evaluated at the current position of the dislocation line, and the surface δS describes the region bounded by the string and its equilibrium position. As mentioned in the Introduction, the classical scattering of phonons by dislocations using this interaction term has already been dealt with 21,22 .…”

“…A comparatively recent result is the computation of the scattering cross section for classical elastic waves by pinned dislocation segments 19 , as well as by prismatic dislocation loops 20 . Use of the formalism developed therein, together with multiple scattering theory, has enabled the computation of an effective, complex, index of refraction for the propagation of coherent elastic waves in a medium filled with randomly distributed and oriented, pinned dislocation segments 21,22 . In turn, this has led to novel characterization tools to describe the plasticity of metals and alloys [23][24][25] .…”