Multiphonon scattering formula of dynamic structure factors for classical Debye solids

A semianalytic formula of the dynamic structure factor S(k,ω) for classical Debye solids over the entire wave-number (k) and frequency (ω) range is constructed by taking into account multiphonon thermal diffuse scattering up to infinite order. The formula adopts Gaussian approximations to the spatial and time decay of the multiphonon part of the displacement correlation function. Numerical illustrations for isotropic polycrystals reveal that, as k increases, sharp peaks due to one-phonon normal scattering in the hydrodynamic regime (k→0) are replaced by diffuse spectra consisting of umklapp scattering and multiphonon continuum; approach toward the ideal-gas spectra in the large-k limit is proven from analytic properties of the multiphonon term. When k coincides with a Bragg reflection point, total thermal diffuse scattering S_{TDS}(k,ω) exhibits a 1/ω divergence as ω→0, which in turn gives rise to a logarithmic enhancement of the corresponding static structure factor S_{TDS}(k). Overall accuracy of the theory is confirmed through the exact zeroth-order frequency-moment sum rule between S_{TDS}(k,ω) and S_{TDS}(k); agreement with the second-order sum rule is shown to be satisfactory except for the vicinity of the Debye cutoff region.