A Green's Function Method for the Frequency Spectrum of an Isotopically Disordered Lattice

A systematic investigation is made of the frequency spectrum of an isotopically disordered lattice on the basis of method of Green's function. From the momentum and coordinate representations of Green's function, formal expressions for the frequency spectrum are obtained. When the number of isotopes is few, an exact analytical expression for the frequency spectrum is obtained from solutions of simultaneous equations satisfied by its coordinate representation, and the results are applied to one- and two-impurity problems. When a large number of isotopes is distributed at random, it is first shown that with use of the results for the few-impurity problem the frequency spectrum can be expanded in terms of the concentration of isotopes. Next, with the aid of a diagrammatic analysis, perturbation methods are presented for simultaneous equations satisfied by the momentum representation and for those satisfied by the coordinate representation, of Green's function. By these calculations, it is established that the frequency spectrum is characterized by a shift and a damping of the spectrum of the regular lattice. Approximate analytical expressions for the frequency spectrum are obtained by employing several mathematically simplified procedures, and the concept of virtual crystal is examined. Various formal results are applied to one- and three-dimensional lattices. The criterion for the appearance of an impurity band, the shape of the main continuum of the band, etc. are studied. In the latter case, it is shown that a very heavy isotope causes a hump in the curve of the frequency spectrum in the low frequency region.