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 y_p 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 wavelengths λ = 0.56, 0.71 and 1.54 Å 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 y_p is close to the value given by the Becker–Coppens formula. Second, it was found that for absorbing spherical crystals with μR ≥ 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.