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Curved single crystals are widely employed in spectrometer designs in thehard X-ray regime. Due to their large solid angle coverage and focusingproperties, toroidally bent crystals are extremely useful in applications wherethe output of photons is low. Spherically bent crystals, a subgroup oftoroidally bent crystals, particularly have found their way in many instrumentsat synchtrotrons and free electron laser lightsource end-stations but also inthe re-emerging field of high-resolution laboratory-scale X-ray spectroscopy. Asolid theoretical understanding of the diffraction properties of such crystalsis essential when aiming for optimal spectrometer performance. In this work, wepresent a general method to calculate the internal stress and strain fields oftoroidally bent crystals and how to apply it to predict their diffractionproperties. Solutions are derived and discussed for circular and rectangularspherically bent wafers due to their prevalence in contemporaryinstrumentation.

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