Structure of the vibrational spectrum of a two-component disordered chain with arbitrary composition

Abstract

A detailed calculation and analysis of the vibrational spectrum of long disordered linear chains with arbitrary ratios of the concentrations of two components are performed in the isotopic approximation using Dean’s method. Special attention is focused on the case where light atoms predominate. It is shown that a large number of resonance features corresponding to vibrational modes of short chains of light atoms with clamped ends arise in the continuous spectrum. The widths of the dips formed are linear functions of the concentration of the heavy atoms, and the depths of the dips are independent of the concentration and are determined by the ratio of the atomic masses. As this ratio increases, starting with two, the dips systematically transform into forbidden bands whose widths are also linear functions of the concentration of the heavy atoms. New conditions for opening of such bands are found and analyzed, and it is shown that they cannot fall within the spectrum of an ideal chain of heavy atoms. The transition from a quasi-continuous spectrum with a series of dips to a hierarchical structure of local vibrations, which occurs at low concentrations of the light atoms, is traced.