Darwin's approach to X-ray diffraction on lateral crystalline structures

Abstract

Darwin's dynamical theory of X-ray diffraction is extended to the case of lateral (i.e., having a finite length in the lateral direction) crystalline structures. This approach allows one to calculate rocking curves as well as reciprocal-space maps for lateral crystalline structures having a rectangular cross section. Numerical modelling is performed for these structures with different lateral sizes. It is shown that the kinematical approximation is valid for thick crystalline structures having a small length in the lateral direction.

Simulation of X-ray diffraction in a cylindrical crystal

“…1(b) and 1(c)]. If the crystal has small lateral size, although its thickness is greater than the length of the primary extinction, diffraction can be kinematical (Punegov et al, 2014(Punegov et al, , 2016. This is also true for spatially restricted X-ray beams with small slit sizes (Punegov et al, 2017).…”

“…Note that the points (m; n) of this network do not correspond to the nodes of the crystal lattice. Taking into account dynamical scattering for the reflected wave S and transmitted wave T, the following recurrence relations can be written (Punegov et al, 2014):…”

“…Under the above-specified boundary conditions, the propagation of X-ray waves, both in vacuum and inside of the cylindrical crystal, can be described by equations (1), taking into account the fact that in the vacuum r 0;h 0. Implementing the procedure of recurrent calculations described by Punegov et al (2014), we introduce restrictions related to the vacuum-crystal interface. An X-ray wave propagating in a vacuum is written as T m…”

“…The calculation procedure should be supplemented by the boundary conditions for the reflected wave: S 0 n ¼ 0 and S m N z ¼ 0. The intensity of the X-ray reflection wave is found from the relation (Punegov et al, 2014)…”

“…At present, triple-crystal diffractometry (Iida & Kohra, 1979;Punegov, 2015) and Bragg coherent diffraction imaging (Robinson et al, 2001;Pavlov et al, 2017Pavlov et al, , 2018Shabalin et al, 2017) are widely applied to the study of crystalline structures. Calculations of rocking curves and reciprocal space maps RSMs) for lateral crystals of rectangular cross section have been performed using the Takagi equations (Kolosov & Punegov, 2005) and two-dimensional recurrence relations (Punegov et al, 2014(Punegov et al, , 2016. For a lateral imperfect crystal, a deterministic version of Bragg coherent diffraction imaging in the kinematical approximation has been considered (Pavlov et al, , 2018.…”