Dislocation theory of vibrational excitations in small-angle grain boundaries

We study vibrational excitations in the system of arrayed dislocations composing the interfacial misfit structure of the small-angle grain boundary. Coupled oscillations of dislocation lines in the strain field of the interface are analyzed by taking into account the Peierls energy and forces of line tension. Within the framework of isotropic elasticity theory we calculate the effective force constants of harmonic springs which establish the correlations between vibrating dislocations. It is shown that these long-range force constants reorganize the spectrum of individual dislocation lines and create a band of collective vibrations. We find that in dynamics of arrayed dislocations a key role is acquired by non-Debye effects associated with a distinct frequency scale omega(B) similar to theta omega(D) much less than omega(D) (theta is the angle of misorientation between the grains, and omega(D) is the Debye frequency). The concept of this low frequency is used to characterize the properties of the anisotropic vibrational band formed by the grain boundary.