General method to calculate the elastic deformation and X-ray diffraction properties of bent crystal wafers

Toroidally and spherically bent single crystals are widely employed as optical elements in hard X-ray spectrometry at synchrotron and free-electron laser light sources, and in laboratory-scale instruments. To achieve optimal spectrometer performance, a solid theoretical understanding of the diffraction properties of such crystals is essential. In this work, a general method to calculate the internal stress and strain fields of toroidally bent crystals and how to apply it to predict their diffraction properties is presented. Solutions are derived and discussed for circular and rectangular spherically bent wafers due to their prevalence in contemporary instrumentation.