Primary Extinction and Absorption: a Theoretical Approach Based on the Takagi-Taupin Equations. Application to Spherical Crystals

Abstract

The primary-extinction problem for X-ray diffraction by perfect crystals is treated using the Becker-Coppens iterative procedure within the Takagi-Taupin equations. An analytical approximation for the primary-extinction factor yp describing both the effects of the X-ray multiple scattering and the absorption processes within the perfect crystal of an arbitrary shape is derived. The solution differs from the known expressions given by Zachariasen and Becker & Coppens on the basis of the HamiltonDarwin intensity transfer equations and in the limiting case of a non-absorbing crystal it concurs with the KatoBecker formula found in the Laue approximation of the dynamical theory. The theoretical results are consistent with experimental data of a number of reflections of Ge and Si single-crystal spheres measured at X-ray wavelengths A = 0.56, 0.71 and 1.54 A with a laboratory CAD-4 and a Huber four-circle diffractometer at HASYLAB, DESY, Hamburg, Germany. Two novel features are discussed. First, it is shown that by neglecting the X-ray absorption effect the calculated extinction factor yp is close to the value given by the Becker--Coppens formula. Second, it was found that for absorbing spherical crystals with /zR > 1 absorption effects cannot be treated separately from the primaryextinction phenomenon because of imaginary dispersion corrections to the atomic form factors. The experimental data are fitted to the Becker-Coppens and present theoretical models. The best fits are found to relate to the present model and produce relatively low R factors of 3 to 6% for the Bragg intensities measured in the cases of Si and Ge spherical crystals.

Bragg–Laue X-ray dynamical diffraction on perfect and deformed lateral crystalline structures

“…Usually, Takagi's equations are used to simulate dynamical X-ray diffraction on lateral crystalline structures (Becker, 1977;Becker & Dunstetter, 1984;Olekhnovich & Olekhnovich, 1980;Saldin, 1982;Chukhovskii et al, 1998;Thorkildsen & Larsen, 1999b). However, these approaches do not allow ISSN 1600-5767 # 2016 International Union of Crystallography one to simulate reciprocal space maps (RSMs).…”

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

“…However, using Takagi's equations to simulate dynamical X-ray diffraction on lateral crystalline structures (Becker, 1977;Becker & Dunstetter, 1984;Olekhnovich & Olekhnovich, 1980;Saldin, 1982;Chukhovskii et al, 1998;Thorkildsen & Larsen, 1999a) is a time-consuming procedure that hinders the use of dynamical diffraction to solve inverse problems for such structures. It is timely to explore possibilities offered by simple algebraic Darwin recurrence relations in application to lateral crystalline structures.…”

Fortran90 sources of the subroutines ofUMWEG. III. TheUMWEG-specific subroutines

“…The two subroutines provide the primary extinction correction for a particular mosaic block radius and Bragg angle calculated according to Chukhovskii et al (1998). The two subroutines provide the primary extinction correction for a particular mosaic block radius and Bragg angle calculated according to Chukhovskii et al (1998).…”

“…CHUK and CHUK1. The two subroutines provide the primary extinction correction for a particular mosaic block radius and Bragg angle calculated according to Chukhovskii et al (1998). Because of the shortcomings of this correction discussed in Ro-00b-x2.3.2 and Ro-00b-x2.3.3, the primary extinction correction provided by these two subroutines is used with restrictions for single diffraction only (Ro-00b-x3).…”