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 crysta ls is treated using the Becker-Coppens iterative procedure within the Takagi-Taupin equations. An analytical approximation for the primary-e xtinction factor y(p) describing both the effects of the X-ray multipl e scattering and the absorption processes within the perfect crystal o f an arbitrary shape is derived. The solution differs from the known e xpressions given by Zachariasen and Becker & Coppens on the basis of t he Hamilton-Darwin intensity transfer equations and in the limiting ca se of a non-absorbing crystal it concurs with the Kato-Becker formula found in the Laue approximation of the dynamical theory. The theoretic al results are consistent with experimental data of a number of reflec tions of Ge and Si single-crystal spheres measured at X-ray wavelength s lambda = 0.56, 0.71 and 1.54 Angstrom with a laboratory CAD-4 and a Huber four-circle diffractometer at HASYLAB, DESY, Hamburg, Germany. T wo 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 mu R greater than or equal to 1 absorption effects cannot be treated separately from the p rimary-extinction phenomenon because of imaginary dispersion correctio ns to the atomic form factors. The experimental data are fitted to the Becker-Coppens and present theoretical models. The best fits are foun d to relate to the present model and produce relatively low IZ factors of 3 to 6% for the Bragg intensities measured in the cases of Si and Ge spherical crystals.