Vibrations of a Chain with Glass-Like Disorder

Abstract

The vibrations of a monatomic chain with harmonic nearest-neighbor interactions are studied, the force constants having a continuous probability distribution. A Green's function is defined by means of the classical equations of motion, and expressed in terms of the exact normal modes of the system. It is indicated how the system may be quantized, and the physical interpretation of the Green's function and its spectral weight function are discussed, using a quantum-mechanical definition. A perturbation series is derived, and it is shown how the terms may be represented by diagrams. It is found that the lowest order approximation leads to a divergence in the self-energy, and a self-consistent approximation is described, which effectively sums an infinite number of terms in the series and which removes this divergence. The approximation neglects multiple scattering at a single force constant. The frequency spectrum is calculated in this approximation and shown to be in very good agreement with the results of Dean, who studied the same problem by a direct numerical method. The complex frequencies corresponding to plane-wave excitations are calculated as a function of wave number. The method is applicable to some three-dimensional models.