Two-dimensional focusing X-ray optics: application of anisotropic elasticity theory for characterization of bent crystals

Anisotropic elasticity theory is applied to calculations of the one- and two-dimensional curved crystals widely employed in X-ray optics. The expression for the displacement vector involved in the process of X-ray Bragg diffraction by a curved crystal is derived. The derivation takes into account the anisotropic elasticity effects for a crystal plate subjected to pure bending. Expressions for the angular deviation from the Bragg angle along the crystal surface are obtained for cylindrically, spherically and toroidally bent crystal geometries. The relationship between the radii of curvature of the crystal and the crystal's compliance tensor is discussed. It is shown that, for a given crystal, the occurrence of the anticlastic curvature effect directly depends on the crystal orientation. Bent silicon crystal plates with (200), (111) and (220) orientations are used to illustrate the influence of the anisotropic elasticity effects on the shape of the diffracting region on the crystal's surface.