Dynamical X-ray diffraction from imperfect crystals in the Bragg case—extinction and the asymmetric limits

Abstract

The variation of X-ray Bragg-reflexion properties of uniformly bent and mosaic-imperfect crystals is systematically explored within the framework of the TakagiTaupin equations by using spherical-wave boundary conditions. For simplicity, consideration is restricted to centrosymmetric crystals with zero anomalous dispersion, although the methods used are quite general. For uniformly bent crystals, the diffraction properties are explored as functions of the bending radius, and the asymmetry parameter, β = cot θ B tanα, where θ B is the Bragg angle and α is the asymmetry angle. For mosaic crystals, the diffraction properties are explored as functions of (i) the block size (assumed uniform), l , (ii) the standard deviation of the mosaic-block tilt distribution, α b , (iii) the standard deviation of the mosaic shift distribution, α c , and (iv) the asymmetry parameter, β. From these results, strong evidence is obtained for the universal nature of the asymmetric limits as zero-extinction (kinematical) limits, and, moreover, that the limits are attained in such a manner that the imperfect-crystal result (for intensity along the surface and therefore also for the integrated reflectivity) first tends asymptotically to the corresponding dynamical-theory result for a perfect crystal, before finally tending to the kinematical value in the limit. The relevance of the present work to the conventional approach to the extinction problem is discussed, and various limiting cases of the general mosaic-block model (g.m.b.m.) are explored. In particular, pure primary and secondary extinction are considered. On the problem of understanding the diffraction behaviour in the asymmetric limits, an examination of the results suggests that the conventional classification scheme for types of extinction is incomplete and that a third type of secondary extinction, at least, should be added to the classification scheme. This third type arises owing to angular spreading-out of the diffracted beam as it passes through the crystal, and the possibility of its occurrence is excluded in the formulation of the Darwin transfer equations.