Abstract
Toroidally and spherically bent single crystals are widely employed as optical elements in hard X-ray spectrometry at synchrotron and free-electron laser light sources, and in laboratory-scale instruments. To achieve optimal spectrometer performance, a solid theoretical understanding of the diffraction properties of such crystals is essential. In this work, a general method to calculate the internal stress and strain fields of toroidally bent crystals and how to apply it to predict their diffraction properties is presented. Solutions are derived and discussed for circular and rectangular spherically bent wafers due to their prevalence in contemporary instrumentation.
Highlights
Crystal analysers are the heart of most contemporary mid-tohigh energy resolution X-ray spectrometers in the hard X-ray regime (Suortti & Schulze, 1995; Yamaoka et al, 1998)
With spherically bent crystal analysers (SBCA) one can efficiently cover and analyse photons collected over a large solid angle
It is no wonder that many inelastic X-ray scattering (IXS) and X-ray emission spectroscopy (XES) endstations at synchrotron and free-electron laser light sources worldwide, such as SOLEIL (Ablett et al, 2019), ESRF (Kvashnina & Scheinost, 2016; Huotari et al, 2017; Moretti Sala et al, 2018), APS (Fister et al, 2006), Spring-8 (Cai, 2004; Ishii et al, 2013), SSRF (Duan et al, 2016), SLS (Kleymenov et al, 2011), SSRL (Sokaras et al, 2012) and DESY (Welter et al, 2005), utilize SBCAs in their instrument designs
Summary
Crystal analysers are the heart of most contemporary mid-tohigh energy resolution X-ray spectrometers in the hard X-ray regime (Suortti & Schulze, 1995; Yamaoka et al, 1998). The original derivation relies on many geometrical features and symmetries which can not be generalized to toroidal bending or other types of crystal shapes, such as rectangular ones used, for example, in recently introduced strip-bent analysers designed to minimized the influence of the in-plane stress (Rovezzi et al, 2017). We present a general framework to calculate internal stress and strain fields and diffracted X-ray intensities of an arbitrarily shaped, toroidally bent crystal wafer. The procedure is utilized to derive stress and strain expressions for isotropic and anisotropic circular and rectangular toroidally bent crystals due to their prevalence in the contemporary instrumentation scene The models and their properties are discussed in detail and the accuracy of the predicted diffracted X-ray intensities is validated by comparison with experimental data.
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