A Note on the Lattice Vibration with Glass-Like Disorder

Abstract

The general formulation of the lattice vibration with glass-like disorder has not so far been given. The main difficulty is to choose a proper unperturbed system with which some iteration procedure may be started. For the one-dimensional lattice, however, useful models have been constructed and investigated. For instance Dean has proposed as a model a linear chain with force constants having a continuous probability distribution and has presented a· experiment by calculating its frequency spectrum through a direct numerical method.l) A Green's function method has been applied to Dean's model by Hidley,2l and it has been shown that much less computation based on the Green's function method can recover most of the results of computer experiment fairly accurately. Since the approximation adopted in Hindley's self-consistent treatment is nothing more than that used in other related problems (e. g., essentially the same as that of the Matsubara-Toyozawa method3l), one might hope that the Green's function method can provide a simple mathematical tool for discussing the glasslike disorder even in three dimensions, if, as Hindley suggested, the problem of disorder in structure can reduce to that of disorder in force constants by means of a suitable transformation. Unfortunately this is not the case, and even the mathematical realization of the three-dimensional glasslike disorder is not easy. Therefore as a first step let us define a weak structure disorder such that a one-to-one correspondence can be set up betweencfhe atoms in a regular lattice and those in a disordered lattice. In this note we shall point out that (i) a simple extension of the Hindley theory to more complicated systems with many internal degrees of freedom and of higher dimensions will give a fairly good result in calculating the phonon spectrum, and that (ii) in the course of the general formulation for weak structure disorder, argument can be much simplified by introducing effective complex force constants. These force constants are determined selfconsistently within the framework of the theory ~nd facilitate us to analy~e phonons iri a disordered lattice in terms of usual concepts used in a regular lattice. We shall illustrate our statement by using as an example a diatomic linear chain with glass-like disorder. The lattice dynamics of a regular diatomic linear chain can be described by a two-dimensional Green's function matrix D 0= {D~,} which satisfies