Theory of anharmonic processes in crystals with isotopic impurities.

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A theory of the influence of isotopic impurities on phonon frequencies and bandwidths, based on the formalism of the retarded Green's functions, is presented. The theory takes into account both harmonic and anharmonic processes in the crystal and yields a proper self-energy that includes impurity-activated contributions depending on the concentration of the impurities and on a parameter \ensuremath{\lambda} defined as the ratio of the mass difference to the mass of the impurity atoms. The theory is developed up to second order in the parameter \ensuremath{\lambda} and in the concentration of impurities. Central to the theory is the technique of averaging the anharmonic-phonon propagators over the ensemble of impurity configurations. This gives rise to a proper self-energy that includes harmonic and anharmonic contributions to both the phonon shift and damping. The most important result is that processes linear in the quartic anharmonicity, that in the pure crystal contribute only to the phonon shift, in the presence of isotopic impurities give rise also to a contribution to the damping.