Vibration Frequency Spectra of Disordered Lattices. II. Spectra of Disordered One-Dimensional Lattices
By using a combination of the moment-trace method and a new method, the "delta-function" method, the vibrational frequency spectrum of a randomly disordered, two-component, isotopic, linear chain has been computed for a wide range of the concentrations of the two kinds of particles and of their mass ratios. In addition the particular case of a chain in which the mass of one of the isotopic constituents becomes infinite can be treated exactly, and the results of this analysis 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, the nature of which is discussed. Finally, the zero-point energy of a randomly disordered lattice is calculated and compared with the zero-point energy of an ordered lattice and of the separated phases.