Time-Dependent Dynamical Diffraction Theory for Phonon-Type Distortions

A rigorous derivation of the dynamical theory of X-ray diffraction from crystals with “optical phonon”-type distortions is presented. The result is a compact, surprisingly simple equation with a strong formal similarity to the well known Takagi-Taupin equation, with the latter included as a special case. Time-dependence is explicitly retained, and thus the analysis is applicable to situations where the crystal is modified on time-scales comparable with that for the X-rays to traverse an extinction depth, as well as being valid for the analysis of diffraction from disturbances that travel at close to luminal velocities, such as polaritons. A comparison is made between the influence of coherent acoustic and optical phonons on the diffraction of X-rays. Numerical and perturbative analytical solutions of the generalised Takagi-Taupin equation are presented in the presence of such phonons.