Cylindrically averaged diffraction by distorted lattices

Abstract

Theory is developed for calculating cylindrically averaged diffraction from an ensemble of finite two-dimensional distorted lattices. The two-dimensional lattices are distorted in three dimensions, and the correlations between displacements of different lattice sites are described by an imposed correlation field. Expressions for diffraction by lattices with circularly symmetric statistics are derived that permit efficient numerical calculation of cylindrically averaged intensities. The motivation for this investigation is to develop a model for describing disorder in, and diffraction by, polycrystalline fibres in which the constituent crystallites are randomly rotated about the fibre axis, and the molecules in the crystallites are subject to correlated displacements. The approach adopted, however, is also applicable to systems without rotational disorder. The diffraction properties of distorted lattices are investigated by numerical calculation. The profiles and distribution of sharp `Bragg' reflections throughout reciprocal space are examined as functions of site variances, correlation lengths and crystallite size. The results show that certain distinct combinations of these parameters (different crystallite sizes and kinds of disorder) can lead to diffraction patterns that are almost identical. The results have implications for the analysis of disorder and molecular structures in polycrystalline fibres using X-ray diffraction analysis.