Use of radial symmetry for the calculation of cylindrical absorption coefficients and optimal capillary loadings

The problem of numerically evaluating absorption correction factors for cylindrical samples has been revisited using a treatment that fully takes advantage of the sample symmetry. It is shown that the path lengths for all points within the sample at all possible diffraction angles can be trivially determined once the angle-dependent distance distribution for a single line of points is calculated. This provides advantages both in computational efficiency and in gaining an intuitive understanding of the effects of absorption on the diffraction data. A matrix of absorption coefficients calculated for μR products between 0 and 20 for diffraction angles θD of 0-90° were used to examine the influence of (1) capillary diameter and (2) sample density on the overall scattered intensity as a function of diffraction angle, where μ is the linear absorption coefficient for the sample and R is the capillary radius. On the basis of this analysis, the optimal sample loading for a capillary experiment to maximize diffraction at angles of 0-50° is in general expected to be achieved when the maximum radius capillary compatible with the beam is used and when the sample density is adjusted to be 3/(4μR) of its original density.