A capillary specimen aberration for describing X-ray powder diffraction line profiles for convergent, divergent and parallel beam geometries

X-ray powder diffraction patterns of cylindrical capillary specimens have substantially different peak positions, shapes and intensities relative to patterns from flat specimens. These aberrations vary in a complex manner with diffraction angle and instrument geometry. This paper describes a fast numerical procedure that accurately describes the capillary aberration in the equatorial plane for convergent focusing, divergent and parallel beam instrument geometries. Axial divergence effects are ignored and only a cross section of the capillary, a disc, is considered; it is assumed that axial divergence effects can be described using an additional correction that is independent of the disc correction. Significantly, the present implementation uses the TOPAS-Academic aberration approximation technique of averaging nearby aberrations in 2? space to approximate in-between aberrations, which results in no more than ~30 disc aberrations calculated over the entire 2? range, even when the diffraction pattern comprises thousands of peaks. Finally, the disc aberration is convoluted with the emission profile and other instrument and specimen aberrations in a Rietveld refinement sense, allowing for refinement on the specimen's absorption coefficient and capillary diameter, as well as the instrument focal length. Large differences between refined and expected values give insight into instrument alignment