Phonon Scattering by Line Defects

Abstract

The lattice dynamics of a solid containing small density-of-line defects has been discussed in the lattice model of Montroll and Potts. The lattice is assumed to be simple cubic. Changes in the mass and force constants along and perpendicular to the line defect are considered. The line-defect symmetry has been exploited for simplifying the phonon-scattering $T$ matrix. The symmetry configuration in which the line defect moves gives rise to different types of local and resonance modes in two host-line-defect systems. In systems of type I, $\epsilon -\xi =0\mathrm{}$, where $\epsilon$ and $\xi$ denote the relative changes in mass and force constants along the line defect, respectively. In such systems, incomplete bands of acoustic localized modes and scattering resonances occur. The lowest localized mode lies at zero frequency, whereas the lowest resonance mode occurs at a higher frequency depending upon the perturbation parameters. The widths of these incomplete bands depend upon the strength of the perturbation. At low temperatures, resonance modes might not influence the usual phonon scattering. In a more general type of solid (systems of type II), the perturbation on the line, $\epsilon -\xi$, does not vanish. Complete bands of localized and resonance acoustic modes appear in these systems. The lowest localized or resonance mode occurs at zero frequency. Resonance phonon scattering will be observed in type-II systems. The contribution of the phonon-resonance scattering by line defects in a solid is about 20% at temperatures of the order of ${10}^{-2\mathrm{}}\Theta$, if there exists resonance scattering due to dislocation motion of the kind treated by Granato. Direct observation of resonance scattering by line defects is possible in bcc metals because there is no dislocation motion in these solids. The specific heat of a solid containing a small density of dislocations has been calculated. It consists of two terms: one linear and one cubic in the temperature. The linear term dominates at quite low temperatures ($T<\mathrm{}0.\mathrm{}2$ °K), whereas the cubic term is accessible to observation only in highly deformed solids.