Dynamic structure factor of linear harmonic chain – a recurrence relation approach

An approximation for the dynamic structure factor S ( q, ω ) of the linear harmonic chain with periodic boundary conditions is obtained from a recurrence relation approach. It is compared with (numerically) exact results for S ( q, ω ). Explicit expressions for the moments of the structure factor are obtained from the short time expansion up to order n = 12. For the long-wavelength as well as the deep-inelastic limit, simple expressions are derived permitting to some extent analytic results for S ( q, ω ). Introducing a linear regression, an approximation scheme is defined which shows excellent agreement with the numerical results.