X-ray Bragg reflexion from perfect crystals – dependence on geometrical factors with particular regard to the zero-level-of-interaction limit

Abstract
The variation of X-ray Bragg reflexion properties of centro-symmetric perfect crystals (both absorbing and non-absorbing) with thickness and degree of asymmetry of the reflexion is explored systematically by direct numerical evaluation of the dynamical theory. In particular, it is shown that well-defined universal limits exist where the integrated reflectivity of a perfect crystal (i. e. dynamical theory) approaches asymptotically that for an ideally imperfect crystal (i. e. kinematical approximation) of the same material under the same diffraction conditions. That is, in these limits the level of extinction goes to zero with zero gradient when plotted against an appropriate parameter. The first case occurs when the crystal thickness tends to zero, while the second case occurs when the degree of asymmetry tends toward the asymmetric limits. In each case it is shown that the level of interaction, as indicated for example by the maximum reflectivity, also goes to zero. If absorption is small it is found that the integrated reflectivity for a finite crystal can exceed that for the corresponding semi-infinite crystal, a particular example being the difference between the Ewald and Darwin results which occur for zero and negligible absorption respectively. If absorption and anomalous dispersion are both large, then for some range of parameter values the integrated reflectivity for a perfect crystal can exceed that for an ideally mosaic crystal leading to the phenomenon of negative extinction. A cautionary message arising from the present investigations relates to the dubious practical value of theoretical results for variation of diffraction properties with asymmetry, when these are based on the assumption of zero absorption.