Describing small-angle scattering profiles by a limited set of intensities

Small-angle scattering (SAS) probes the size and shape of particles at low resolution through the analysis of the scattering of X-rays or neutrons passing through a solution of particles. One approach to extracting structural information from SAS data is the indirect Fourier transform (IFT). The IFT approach parameterizes the real-space pair distribution function [P(r)] of a particle using a set of basis functions, which simultaneously determines the scattering profile [I(q)] using corresponding reciprocal-space basis functions. This article presents an extension of an IFT algorithm proposed by Moore [J. Appl. Cryst. (1980), 13, 168-175] which used a trigonometric series to describe the basis functions, where the real-space and reciprocal-space basis functions are Fourier mates. An equation is presented relating the Moore coefficients to the intensities of the SAS profile at specific positions, as well as a series of new equations that describe the size and shape parameters of a particle from this distinct set of intensity values. An analytical real-space regularizer is derived to smooth the P(r) curve and ameliorate systematic deviations caused by series termination. Regularization is commonly used in IFT methods though not described in Moore's original approach, which is particularly susceptible to such effects. The algorithm is provided as a script, denss.f it_data.py, as part of the DENSS software package for SAS, which includes both command line and interactive graphical interfaces. Results of the program using experimental data show that it is as accurate as, and often more accurate than, existing tools.