X-ray diffraction from strongly bent crystals and spectroscopy of X-ray free-electron laser pulses

Abstract

The use of strongly bent crystals in spectrometers for pulses of a hard x-ray freeelectron 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 µm thick diamond crystal bent to a radius of 10 cm.

Resolution of a bent-crystal spectrometer for X-ray free-electron laser pulses: diamond versus silicon

“…The wavevector k and the scattering angle can be replaced in this term with sufficient accuracy with the reference values " k k and " . Calculation of the integral (10) with this additional phase factor has been performed in our previous work (Kaganer et al, 2020). After evaluation of the integral over x in the infinite limits, we arrive at a modified equation ( 15),…”

“…where is the wavelength, and the Bragg law 2d sin " ¼ is used. Kaganer et al (2020) considered this parameter in the case ¼ 0 (and hence b ¼ 0) and stated that the Fraunhofer limit is reached when the diameter of the first Fresnel zone ðLÞ 1=2 exceeds the crystal thickness seen from the direction of the diffracted beam, D cos " . In the general case b 6 ¼ 0, both contributions tob b need to be taken into account, to provide the desired condition jb bj < $ 2.…”

“…We consider its diffraction on a bent crystal in the kinematical (first Born) approximation. The applicability limits of this approximation were established by Kaganer et al (2020) and are discussed in Section 5. We follow the description of the first Born approximation by Born & Wolf (2019, Section 13.1.2), but keep explicitly the time exponent expðÀi!tÞ, which is usually implicit when considering diffraction of a monochromatic wave, and keep the integration over the frequencies !.…”

“…Recently, we have given a detailed theoretical description of the X-ray diffraction at strongly bent crystals (Kaganer et al, 2020). We found that, because of the strong bending, the orientation of the crystal with respect to the incident X-ray wave deviates from the Bragg condition within the Darwin width Á B in an interval of distances that is small compared with the extinction length Ã.…”

X-ray diffraction reveals the amount of strain and homogeneity of extremely bent single nanowires

“…Owing to the significantly stronger bending as compared to previous studies, our experiment required a modification of the diffraction setup along with a new scheme of data presentation. Moreover, current approaches of X-ray theory are limited to bending radii of the order of above 10 cm (Serrano et al, 2008;Kaganer et al, 2020) and no theory exists so far that is applicable to systems with such small bending radius. Using micro-focused X-ray beams, we study the bending of individual NWs and develop a suitable X-ray diffraction theory based on the kinematical approximation.…”