Theory of phonon-phonon interaction in anharmonic crystals with randomly distributed isotopic impurities.

In the expansion of well known results of dynamic properties of anharmonic ideal crystals, a perturbation theory using Matsubara functions and diagrammatic treatment has been carried out to describe the phonon-phonon interaction in anharmonic crystals with randomly distributed isotopic impurities. Taking explicitly into account anharmonic contributions of the potential expansion as well as mass changes between impurities and host-lattice particles, it is shown that the anharmonic phonon propagator can be written as a sum of terms, which are added up within a Dyson equation. In contrast to the results obtained recently using the formalism of the double-time retarded Green functions, our proper self-energy of the nonaveraged crystal includes explicitly the influence of the isotopic disorder. As an application, a diatomic linear chain with a single mass defect was chosen and numerical results of the impurity-influenced two-phonon density of states as well as the anharmonic damping function are presented.