  The use of strongly bent crystals in spectrometers for pulses of a hard x-ray
free-electron laser is explored theoretically. Diffraction is calculated in
both dynamical and kinematical theories. It is shown that diffraction can be
treated kinematically when the bending radius is small compared to the critical
radius given by the ratio of the Bragg-case extinction length for the actual
reflection to the Darwin width of this reflection. As a result, the spectral
resolution is limited by the crystal thickness, rather than the extinction
length, and can become better than the resolution of a planar dynamically
diffracting crystal. As an example, we demonstrate that spectra of the 12 keV
pulses can be resolved in 440 reflection from a 20 micron thick diamond crystal
bent to a radius of 10 cm.
