Vibrational Frequency Spectra of Disordered Lattices I. The Linear Chain

摘要

Using the Moment-Tract method we have computed the vibrational frequency-spectrum of a randomly disordered, two-component, isotopic linear chain for a wide range of the concentrations of the two kinds of particles and of the mass ratio. With the aid of the theory of random walks on lattices we have obtained a combinatorial expression for the even moments of the spectrum and have used it to calculate the moments up to μ20.nIn addition we have been able to treat exactly the particular case of a chain in which the mass of one of the isotopic constituents becomes infinite, and the results of this calculation shed light on the form of the spectra for lattices with large but finite mass ratios for the two constituents. The spectra are characterized by the disappearance of the square-root singularity at the maximum frequency which is found in ordered one-dimensional lattices, and by the appearance of impurity bands, and the nature of these is discussed.