Model of sagittally bent silicon crystals diffracting in the Laue mode: Effects of elastic anisotropy on the rocking-curve widths (abstract)

Abstract

Sagittal focusing of synchrotron x rays with asymmetric Laue crystals has been proposed and demonstrated [Z. Zhong, C. C. Kao, D. P. Siddons, and J. B. Hastings, J. Appl. Crystallogr., Pts I&II (submitted)]. At high x-ray energies, sagittal focusing by a Laue crystal is preferred because of the small extent of the beam’s footprint on such a crystal, and the ability to use the inverse-Cauchois geometry in the meridional plane to improve energy resolution. Reflectivity curves of sagittally bent Laue crystals were measured at x rays energies from 15 to 70 keV. A model for the diffraction properties of sagittally bent Laue crystals, which takes into account the anisotropy in the elastic property of the crystals, was developed. Analytical formulas were derived for the rocking-curve width of such crystals, including both the contribution from the lattice angle change and the d-spacing change due to sagittal bending, across the thickness of the crystal. The contribution of the orientation-dependent elastic compliance S23 is as important as S13 in the specific case of sagittally bent Laue crystals. The widths of the rocking curves, calculated using the analytical model, were compared with measurements on crystals of two different orientations (111) and (100). Good agreements were found. An interesting sign reversal in the slope of the lattice angle versus depth curve between the (111)- and (100)-oriented crystals is predicted by the model, as a result of the highly anisotropic properties of silicon crystal. Results of depth-resolved rocking-curve measurements and high-resolution diffraction, designed to confirm these theoretical findings, are discussed.