Anisotropy and the optimal aspect ratio of a bent crystal

Abstract

A thin bent rectangular crystal plate can be used as the second crystal in a double-crystal monochromator to focus sagittally a wide synchrotron X-ray beam. When bent in the sagittal direction, the plate also bends in the tangential direction (so-called anticlastic bending), which can adversely affect the diffracted beam. It is known that the undesirable anticlastic deformation can be minimized if the plate aspect ratio (length to width) is optimized. The present study investigates the influence of crystal material anisotropy and shows that ignoring it in the design of a sagittally bent crystal can lead to a significant underestimation of anticlastic deformation. For example, analysis shows that the optimal aspect ratio for a simply supported bent Si(220) plate ranges from 2.13 to 2.82 depending on the in-plane orientation of the cut rectangular plate. Treating silicon as isotropic with material properties in the literature leads to a plate design with an optimal aspect ratio of 2.36 and a plate with significant anticlastic bending.