The vibrational spectra of disordered chains

Abstract

The dynamical systems discussed here are long chains of particles with randomly distributed masses; both nearest- and next-nearest-neighbour harmonic forces are taken into account. It is shown that in order to find the spectrum of frequencies of vibration the distribution of eigenvalues of a large matrix of special type must be determined, and an accurate and rapid method of computing this distribution is described. By this means the vibrational spectra for several degrees of next-nearest-neighbour interactions are found; they are shown to lend support to a theory relating their main features to similar features in the spectra of almost regular chains possessing defect modes.