Finite-element simulation for crystals with surface undulations based on Takagi–Taupin theory

A finite-element method based on a weak formulation of the Takagi–Taupin equations was adopted to study the X-ray diffraction of crystals with surface undulations. A general diffraction geometry was simulated to investigate the diffraction features caused by the surface undulations. The numerical results reveal that the effects of surface undulations on Bragg diffraction are local for those limited in the low-frequency range and physically result from the refractive effect, brought on by the variation of the local asymmetry angle. Thus, a formula based on the local perfect flat crystal approximation was introduced to efficiently evaluate the change in the direction of the diffracted wave caused by surface undulations.