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

## TL;DR

This paper investigates the use of strongly bent crystals in X-ray spectrometers for XFEL pulses, showing that kinematic diffraction can improve spectral resolution beyond traditional limits.

## Contribution

It introduces a theoretical framework for using strongly bent crystals in XFEL spectroscopy, demonstrating enhanced spectral resolution capabilities.

## Key findings

- Spectral resolution limited by crystal thickness, not extinction length.
- Kinematic diffraction applicable for small bending radii.
- Resolution of 12 keV pulses in 440 reflection from a 20 μm diamond crystal.

## Abstract

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.