Vibrations of glass-like disordered chains

Abstract

Monatomic and alternating diatomic chains, with nearest neighbour harmonic interactions, are considered; in each case the force constants are governed by a probability distribution function which is continuous and bounded in a well-defined region and zero elsewhere. It is shown that the lattice modes are, in general, spatially localized. In the case of the diatomic chain the condition for a band gap is derived and the positions of the spectral branches are found. Spectra are calculated explicitly for monatomic and diatomic chains of a certain class, and typical displacement eigenvectors for the monatomic chain are depicted.