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 = ¼ (∂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 s₀ 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.) orientations.