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

Curved single crystals are widely employed in spectrometer designs in the hard X-ray regime. Due to their large solid angle coverage and focusing properties, toroidally bent crystals are extremely useful in applications where the output of photons is low. Spherically bent crystals, a subgroup of toroidally bent crystals, particularly have found their way in many instruments at synchtrotrons and free electron laser lightsource end-stations but also in the re-emerging field of high-resolution laboratory-scale X-ray spectroscopy. A solid theoretical understanding of the diffraction properties of such crystals is essential when aiming for optimal spectrometer performance. In this work, we present 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. Solutions are derived and discussed for circular and rectangular spherically bent wafers due to their prevalence in contemporary instrumentation.