Bragg Reflection in Mosaic Crystals. I. General Solution of the Darwin Equations

The Darwin equations, which describe the multiple Bragg reflection of X-rays or neutrons in a mosaic crystal slab, have previously been solved only for special cases. Here, the complete and exact analytical solution of these equations is obtained for both the Bragg case (reflection geometry) and the Laue case (transmission geometry) with the help of a computer algebra program and it is shown that the resulting general expressions for both the reflectivity R and the transmissivity T can each be expressed in a compact form. It is found, for example, that for a mosaic crystal anomalous absorption occurs only in the Bragg case and not in the Laue case. This is in contrast to the dynamical theory of diffraction, which applies to an ideally perfect crystal, where anomalous absorption (due to the Borrmann effect) is found in both Laue and Bragg cases. With this new general expression for R, the Fankuchen gain is calculated for a crystal of finite thickness, taking correctly into account the effects of both absorption and secondary extinction. General expressions for the optimum crystal thickness are also obtained for both Bragg and Laue cases. In a companion paper, these general results are applied to a detailed numerical calculation of the reflecting properties of various neutron monochromator crystals.