A Universal Treatment of X-ray and Neutron Diffraction in Crystals. II. Extinction

Abstract

Based on the formalism for calculating the integrated reflection power ratio of a plane mosaic crystal by using three dimensionless parameters as described in paper I [Hu (1997). Acta Cryst. A53, 0–0], exact and universal expressions for the secondary-extinction factors in X-ray and neutron crystallography are developed that can be applied to reflections of all possible values of extinction factor, reflection symmetry and the absorption-to-scattering cross-section ratio of the crystal. The representation by three parameters gives a clear and definite physical meaning to the concept of extinction. The theory has been extended to treat the extinction of a spherical crystal, and the striking difference in the evaluated secondary-extinction factor between the equivalent single-plate and the exact method in the spherical-crystal treatment under θB = 0° is explained. As a demonstration of the feasibility of using these expressions, the diffraction data for LiF and MgO crystal plates measured by Lawrence [Acta Cryst. (1972), A28, 400–404; (1973), A29, 208–210] are reanalyzed by this method. All the reflections including the strongest ones (Yo down to 0.026) are reanalyzed simultaneously with single-valued particle size and mosaic spread as fitting parameters and allowing for primary extinction if necessary. The results (R-factor = 0.014 and 0.053 for LiF and MgO, respectively) are unprecedentedly good. Furthermore, in disagreement with Lawrence, the extinction of LiF is found to be of secondary type and in the case of MgO both primary and secondary extinction should be considered. The analysis also shows that the formula Y ~ YpYp is valid only for very weak extinctions and that the Hamilton–Darwin equations are valid in a range much broader than previously anticipated.