THEORY OF ANHARMONIC PROCESSES IN CRYSTALS WITH ISOTOPIC IMPURITIES

A theory of the influence of isotopic impurities on phonon frequencies and bandwidths, based on the formalism of the retarded Green's functi ons, is presented. The theory takes into account both harmonic and anh armonic processes in the crystal and yields a proper self-energy that includes impurity-activated contributions depending on the concentrati on of the impurities and on a parameter 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 lambda and in the concen tration of impurities. Central to the theory is the technique of avera ging the anharmonic-phonon propagators over the ensemble of impurity c onfigurations. This gives rise to a proper self-energy that includes h armonic and anharmonic contributions to both the phonon shift and damp ing. 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 con tribution to the damping.