Temperature diffuse scattering of nanocrystals

The effects of thermal vibrations on X-ray powder diffraction patterns are discussed. Special considerations for extremely small crystallites are described, including the occurrence of surface and edge vibrational modes, and a restriction on the maximum phonon wavelength. In doing so, a complete temperature diffuse scattering (TDS) model is presented, which includes the influence of these features on: the Debye-Waller parameter; first-order TDS; and higher-order TDS terms. The importance of using an accurate TDS representation is studied as a function of temperature and crystallite size. It is found that a misrepresentation of the TDS for small crystallites can lead to an error in the determined Debye-Waller parameter on the order of 20-40% and a slight overestimation of the peak broadening. While the presented theory is primarily developed considering X-ray scattering, the same expressions are expected to describe the TDS in faster-than-sound neutron powder diffraction measurements.