The Disordered Lattice Problem: A Review

Abstract

Introduction. In the eighteenth century a mathematician would probably have been familiar with the latest advances and problems in physics. In the twentieth century, due to the increasing tendency to specialize in both physics and mathematics the problem of communication is a real one, notwithstanding the obvious mutual benefits which result from exchanges between the two sciences. It is the purpose of this paper to describe a mathematical problem of some interest in current investigations of solid state physics which does not seem to have received any previous publicity in mathematical circles. This problem, that of determining the vibrational properties of a disordered lattice of masses and springs, has not been solved in a satisfactory fashion. Since the main difficulties are mathematical it is our hope that this paper will interest mathematicians who would not ordinarily be acquainted with solid state physics. We begin by giving some of the preliminaries necessary to a discussion of the problem. The heat capacity at constant volume, Cv, is defined as the amount of energy needed to raise the temperature of a solid by one degree centigrade. If E is the energy, C, equals ( 3E/T)X , where T is the absolute temperature and the subscript v refers to the fact that the volume of the solid is held constant during this process. A modern theory of the heat capacity of solids was originated by Einstein [1] in 1907 through the application of the Planck quantum theory. Einstein approximated a crystal by an assembly of 3N independent harmonic oscillators each vibrating with the same frequency P. The energy of such an ensemble is given by