Lattice vibrations of one-dimensional disordered systems

Abstract The study of the effects of defects on the vibrational properties of solids is almost as old as the study of lattice dynamics in crystals. In fact, the effect of certain kinds of defects on the normal–mode vibrational frequencies of a system of particles was first investigated theoretically in the 1890s by Lord Rayleigh. Even earlier, the result that a light atom in a monatomic linear chain gives rise to a nonpropagating mode of vibration was contained in the work of Sir William Rowan Hamilton published in 1841. However, because of the simplifying feature in the geometry of the crystalline solids, elegant theorems were developed so that the studies of normal modes and thermodynamic properties of solids with periodic structures soon became the main stream of research in lattice dynamics. On the other hand, the importance of understanding the dynamic properties of disordered systems is underscored by the fact that the materials we deal with d o not have the simplicity and regularity found in the physical systems which have received most of our attention. The theoretical understanding of the atomic vibrations in disordered systems has unfortunately been very slow in developing. The main difficulty arises directly from the lack of periodicity in the structures of the disordered systems. This loss of translational symmetry brings about a situation whereby the theorems developed for the crystalline solids can no longer be applied. The result is that the concepts which have proven fruitful in the study of the dynamics of periodic lattice cannot provide the same kind of useful information in the disordered systems. For example, because some of the normal modes of vibration in the disordered systems are highly localized spatially, there can be no well–defined wave vector or k-value. Hence the concept of the dispersion curve which has been one of the most important tools in the description of lattice vibration in periodic systems does not even hold in the disordered systems.