Isotope scattering of dispersive phonons in Ge

With the use of a lattice-dynamical model which reproduces observed phonon dispersion curves very accurately, the isotope scattering of near-zone-boundary acoustic phonons in Ge is investigated theoretically. We show that the lattice dispersion which becomes appreciable in the THz-frequency region leads to the phonon relaxation time that decreases more rapidly than ${\omega }^{-4\mathrm{}}$. This deviation from the ${\omega }^{-4\mathrm{}}$ dependence (the ${\omega }^{-4\mathrm{}}$ behavior is predicted by the continuum elasticity theory) amounts to about 20% at 1 THz and is enhanced heavily at the zone boundaries. Calculations are based on the lowest-order perturbation theory. Higherorder corrections are estimated as well and shown to be negligibly small.