A computational tool for symbolic derivation of the small angle scattering from complex composite structures

Abstract

Analysis of small angle scattering (SAS) data requires intensive modelling to infer and characterize the structures present in a sample. This iterative improvement of models is a time consuming process. Here we present the Scattering Equation Builder (SEB), a C++ library that derives exact analytic expressions for the form factor of complex composite structures. The user writes a small program that specifies how sub-units should be linked to form a composite structure and calls SEB to obtain an expression for the form factor. SEB supports e.g. Gaussian polymer chains and loops, thin rods and circles, solid spheres, spherical shells and cylinders, and many different options for how these can be linked together. In the present paper, we present the formalism behind SEB, and give simple case studies such as block-copolymers with different types of linkage and more complex examples such as a random walk model of $100$ linked sub-units, dendrimers, polymers and rods attached to surfaces of geometric objects, and finally the scattering from a linear chain of 5 stars, where each star is build by four diblock copolymers. These examples illustrate how SEB can be used to develop complex models and hence reduce the cost of analyzing SAS data.Comment: 18 pages, 13 figure