Generalizing small-angle scattering form factors with linear transformations

Nanostructure characterization using small-angle scattering is often performed by iteratively fitting a scattering model to experimental data. These scattering models are usually derived in part from the form factors of the expected shapes of the particles. Most small-angle-scattering pattern-fitting software is well equipped with form factor libraries for high-symmetry models, yet there is more limited support for distortions to these ideals that are more typically found in nature. Here, a means of generalizing high-symmetry form factors to these lower-symmetry cases via linear transformations is introduced, significantly expanding the range of form factors available to researchers. These linear transformations are composed of a series of scaling, shear, rotation and inversion operations, enabling particle distortions to be understood in a straightforward and intuitive way. This approach is expected to be especially useful for in situ studies of nanostructure growth where anisotropic structures change continuously and large data sets must be analysed.