Anisotropic Elasticity Corrections for Reflection Efficiency and X-ray Standing-Wave Patterns using Bent Crystals

Anisotropic elasticity corrections are taken into account to evaluate the reflection efficiency I int and the X-ray standing-wave patterns (XSWP) P(Δθ,Φ) for bent crystals, where Δθ is the angular coordinate associated with the conventional rocking curve and the phase Φ=hr p is related to the position with respect to the crystalline lattice of the adsorbed atoms on the crystal surface, contributing to the XSWP. Analytical expressions are derived for the uniform strain-gradient parameter B=1/4(∂ 2 /∂ s0 ∂ sh ) [hu(r)] governing the peculiarities of the Bragg diffraction within elastically bent crystals, where h is the reflecting vector, u(r) is the displacement vector and s 0 and s h are the dimensionless coordinates along the incident and diffracted waves, respectively. The cases of the so-called free and forced bending in the Johann and von Hamos geometries are considered. The results of the anisotropic elasticity corrections depending on the crystal-surface orientation are presented for bent silicon (111) and quartz (10.0) and (10.1) orientations.