Bent perfect crystals as X-ray focusing polychromators in symmetric Laue geometry

Abstract

The focusing properties of cylindrically bent crystals in symmetric Laue geometry are discussed using the formalism of Fresnel diffraction and the analytical solution of the Takagi-Taupin equations for a point source on the entrance surface. The existence of a focal shift in the dynamical focusing effect is pointed out and discussed. The present theoretical framework is applied to experiments performed at the energy-dispersive X-ray absorption spectroscopy beamline of the European Synchrotron Radiation Facility concerning the position and the size of the focal spot obtained from a polychromatic source at a large distance from the bent crystal.

Dynamical focusing by bent, asymmetrically cut perfect crystals in Laue geometry

“…Absorption is taken into account. As expected from previous works (Kushnir & Suvorov, 1982;Guigay et al, 2013), we obtain focusing at two distances in the symmetric case, if the crystal curvature is sufficiently strong. With a certain amount of asymmetry, a single focusing distance is observed, with increased focusing efficiency.…”

“…It is found that the focusing performance can indeed be strongly improved by weak asymmetry, as compared to the case of symmetric geometry. We intend to continue the present analysis by considering the important case of a polychromatic source instead of the monochromatic source considered in the present work; for this purpose, it will be necessary to combine the present analysis with the condition for polychromatic focusing which is cos 2 =q À cos 1 =q ¼ 2=R (Chukhovskii & Krisch, 1992;Mocella et al, 2004Mocella et al, , 2008Guigay et al, 2013) and to discuss the influence of the dependence on the X-ray energy of the lateral displacement x c pointed out in the present investigation. In the case of a symmetric reflection, according to equation (34) with s ¼ 0, x c ' Àðt sin B Þ=2, if q and R cos B are much smaller than p; the lateral displacement is consequently approximately proportional to the X-ray energy.…”

Accurate Bond Lengths to Hydrogen Atoms from Single‐Crystal X‐ray Diffraction by Including Estimated Hydrogen ADPs and Comparison to Neutron and QM/MM Benchmarks

“…When heavy atoms dominate scattering, the frozen core approximation of the Hansen–Coppens multipole model leads to a decreasing suitability73 for improving a structural model compared to the IAM. Therefore the invariom approach initially focused on amino acids52 and then organic compounds in general 23. The invariom approach could in principle be combined with other scattering‐factor models, for example, the quantum‐chemical density model used in HAR or other alternatives like the virtual‐spherical‐charges model 74.…”

“…To ensure a consistent model, aspherical scattering factors of the invariom database23 were used instead. This procedure also avoids any possible ambiguity for molecules crystallizing in non‐centrosymmetric space groups 49.…”

“…On the experimental side the supramolecular‐synthon‐based fragments approach SBFA20 and the experimental library multipolar atom model ELMAM221, 22 are both based on averaged experimental multipole parameters from high‐resolution diffraction experiments for chemical environments considered to be similar. Theoretical databases are the generalized invariom database GID,23 which is based on DFT geometry‐optimized model compounds using empirical rules of chemical similarity, and the University at Buffalo Databank UBDB2011,24, 25 which relies on DFT single‐point computations of experimental input structures with similarity deduced from statistical treatment. Our hope was that including crystal‐field effects13 would be the solution.…”