Random Impurity Problem

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Abstract
The problem of determining the vibrational properties of a random distribution of defects in an otherwise perfect lattice has most conveniently been treated by Green's-function techniques. The quantities of physical interest, such as the phonon density of states and the infrared or Raman spectra, are obtained from an average Green's function which has previously been obtained by means of a diagrammatic technique due to Langer. The purpose of this paper is to show that the average Green's function, or alternatively the "proper self-energy," for the random impurity problem can be obtained by a differential method which is conceptually much simpler than the diagrammatic technique. For a number of specific examples, the results previously obtained are rederived. With the differential method the relation between the single-defect problem and the random crystal problem becomes very clear, even for crystal systems with many phonon branches or for defects in the force constants as well as in the mass. The particular significance of this is that it provides an immediate qualitative explanation for the diverse experimental results on mixed crystals. The method is also amenable to numerical calculations and can be used to obtain quantitative results.