**Date Published:** January 01, 2020

**Publisher:** International Union of Crystallography

**Author(s):** Vladimir M. Kaganer, Ilia Petrov, Liubov Samoylova.

http://doi.org/10.1107/S2053273319014347

**Abstract**

**A strongly bent crystal diffracts kinematically when the bending radius is small compared with the critical radius given by the ratio of the extinction length to the Darwin width of the reflection. Under these conditions, the spectral resolution of the X-ray free-electron laser pulse is limited by the crystal thickness and can be better than under dynamical diffraction conditions.**

**Partial Text**

Bent single crystals are commonly used as the X-ray optic elements for beam conditioning as well as the analysers for X-ray spectroscopy. The dynamical diffraction from bent crystals has been a topic of numerous studies over decades (Penning & Polder, 1961 ▸; Kato, 1964 ▸; Bonse, 1964 ▸; Chukhovskii & Petrashen’, 1977 ▸; Chukhovskii et al., 1978 ▸; Kalman & Weissmann, 1983 ▸; Gronkowski & Malgrange, 1984 ▸; Chukhovskii & Malgrange, 1989 ▸; Gronkowski, 1991 ▸; Honkanen et al., 2018 ▸).

For numerical estimates in this section, we consider, as a reference example, the symmetric Bragg reflection 440 of X-rays with energy E = 12 keV (wavelength λ = 1.03 Å) from a D = 20 µm-thick diamond crystal bent to a radius R = 10 cm.

Equations in the previous section do not include an angular deviation of the incident wave from the Bragg condition and are restricted with the limit . To avoid these restrictions and also allow a coherent superposition of waves with different wavelengths, we use a more general expression for the kinematical diffraction amplitude as an integral over the scattering plane of the crystal,We restrict ourselves in this section to the Fraunhofer diffraction. The wavevector is the deviation of the scattering vector from the reciprocal-lattice vector . We have for the wave of the reference wavelength incident on the crystal exactly at the Bragg angle corresponding to that wavelength and reflected at the Bragg angle. The components of the scattering vector depend on the angular deviations of both incident and scattered waves, and on the deviation of the length of the wavevector in the incident spectrum from the reference wavevector (since scattering is elastic, the lengths of the wavevectors of the incident and the scattered waves coincide). Explicit expressions for and are derived in Appendix B. It is convenient, for the purpose of comparison of the incident and the diffracted spectra of an XFEL pulse, to represent the diffracted intensity in an energy spectrum by considering the scattering angle as a Bragg angle for the respective wavevector . The components of the scattering vector expressed through the angular deviation of the incident beam and the wavevector deviations are given by equation (40). In particular, an XFEL pulse can be described as a coherent superposition of plane waves with different wavelengths propagating in the same direction. With the crystal oriented at the Bragg angle for the reference wavelength (), we get

In this section, we consider the finite-distance free-space propagation of the wave diffracted by a bent crystal. This allows us to establish the applicability limits of the Fraunhofer approximation used in the previous section and evaluate corrections due to a finite distance from the bent crystal to a detector.

The spectra in the self-amplified spontaneous emission (SASE) mode of the European XFEL have been generated with the simulation code FAST (Saldin et al., 1999 ▸), which provides a 2D distribution of electric field in real space at the exit of the undulator for each moment of time for various parameters of the electron bunch charge and the undulator. Simulation results are stored in an in-house database (Manetti et al., 2019 ▸). The spectra are simulated for the electron energy 14 GeV, photon energy 12.4 keV, and the active undulator length corresponding to the saturation length, the point with the maximum brightness, for a given electron bunch charge (Schneidmiller & Yurkov, 2014 ▸).

X-ray diffraction from a bent single crystal can be treated kinematically when the bending radius is small compared with the critical radius given by the ratio of the Bragg-case extinction length for the actual reflection to the Darwin width of this reflection. The critical radius varies, depending on the X-ray energy, the crystal and the reflection chosen, from centimetres to metres.

Source:

http://doi.org/10.1107/S2053273319014347