Influence of isotopic disorder on phonon frequencies and phonon linewidths of an anharmonic crystal

Based on a perturbation theory using Matsubara functions and a diagrammatic treatment, it is shown that the phonon linewidths of an anharmonic crystal with isotopic impurities include two different contributions, a disorder-induced temperature-independent part, caused by harmonic scattering processes of phonons at isotopic impurities, and a disorder-influenced temperature-dependent part, caused by anharmonic decay processes. Whereas in case of an anharmonic ideal crystal it is usual to restrict the corresponding self-energy on diagonal elements only, this is no longer sufficient in the presence of impurities. For that reason it is shown that only nondiagonal elements of the effective vertex function of the harmonic phonon propagator contribute to the disorder-induced contribution to the phonon linewidth, whereas the corresponding harmonic and the anharmonic frequency shifts are determined by diagonal and nondiagonal elements as well. As an application, numerical results of the disorder-induced temperature-independent as well as the disorder-influenced temperature-dependent contribution to the phonon linewidths of a diatomic linear chain with a single mass defect are presented. {copyright} {ital 1996} {ital The American Physical Society}