Dynamic structure factor of a disordered harmonic solid

A model system is considered in which particles undergo small vibrations about an irregular array of equilibrium positions. The system is assumed to be dynamically described by a set of harmonic normal modes. Wave-number- and frequency-dependent correlation functions for this system are analyzed. An approximation due to Hubbard and Beeby is rephrased in terms of memory functions where its properties and deficiencies become clearer. The phonon expansion of the dynamic structure factor S(k,ω) is then considered. As in a crystal, the one-phonon term is proportional to the displacement-displacement correlation function ω-2φ(k,ω) but the peaks in this correlation function are no longer sharp since the normal modes no longer have wave vector as a good quantum number. This effect is illustrated by an exact calculation of φ(k,ω) for a one-dimensional model: the harmonic chain with nearest-neighbor forces, and fixed mass and spring constant, but random equilibrium positions. For a plausible Gaussian distribution of equilibrium positions, φ(k,ω) has a very simple analytic form. Physically it illustrates the superposition of normal-mode frequencies corresponding to a spatially periodic initial disturbance. The quasicrystalline approximation of Takeno and Goda, obtained from the second frequency moment of φ(k,ω), is shown to overestimate the normal-mode frequencies in this model system, and thus give too small a density of states at low frequencies.