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

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 Hamilton-Darwin intensity transfer equations and in the limiting case of a non-absorbing crystal it concurs with the Kato-Becker 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 wave-lengths 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 primary-extinction 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. 1