The velocity of acoustic phonons in crystals with edge dislocations and point impurities

The velocity of longitudinal long-wave phonons in crystals with a small number of randomly distributed static edge dislocations was shown to be the same as in ideal crystals. The correction to the velocity of phonons is determined by their interaction with the field of lattice deformations caused by the presence of dislocations. The field of deformations of dislocation displacements is long-range. In addition, these deformations experience sign reversal when the sign of the coordinate perpendicular to the dislocation line and Burgers vector changes. It follows that a crystal contracts or expands in this direction because of the interaction of phonons with dislocations as the defect deformation sign alters. Such an anisotropic volume (and density) change over the whole crystal should however be balanced somehow. For this reason, the density of samples with edge dislocations should remain constant, and the correction δc to the velocity of phonons should be zero. The procedure for averaging that gives δc = 0 is based on the independence of this macro value from the orientation of phonon momemtum incident on randomly distributed dislocations, the unaveraged correction being a diverging value because of the long-range character of the field of defect deformations. For comparison, the correction to the velocity of phonons in crystals with point impurities was calculated. Lattice deformation caused by such defects is isotropic. For this reason, the mean density of crystals with point impurities and the velocity of phonons in them are different from those in ideal crystals. All calculations were performed using the Dyson equation derived on the basis of the Keldysh diagram technique.