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The use of strongly bent crystals in spectrometers for pulses of a hard x-rayfree-electron laser is explored theoretically. Diffraction is calculated inboth dynamical and kinematical theories. It is shown that diffraction can betreated kinematically when the bending radius is small compared to the criticalradius given by the ratio of the Bragg-case extinction length for the actualreflection to the Darwin width of this reflection. As a result, the spectralresolution is limited by the crystal thickness, rather than the extinctionlength, and can become better than the resolution of a planar dynamicallydiffracting crystal. As an example, we demonstrate that spectra of the 12 keVpulses can be resolved in 440 reflection from a 20 micron thick diamond crystalbent to a radius of 10 cm.

X-ray diffraction from strongly bent crystals and spectroscopy of XFEL pulses