Small-Angle Scattering of Interacting Particles. I. Basic Principles of a Global Evaluation Technique

Indirect Fourier transformation is a well established method for the evaluation of small-angle scattering data. This technique, however, is restricted to dilute solutions, as for higher concentrations particle interactions can no longer be neglected. As the scattering intensity contains intra- and interparticle scattering contributions, the evaluation of scattering data is no longer just the solution of a linear weighted least-squares problem because the scattering intensity can, under certain conditions, be written as the product of the particle form factor and the so-called structure factor, which leads to a highly non-linear problem. In this paper a global evaluation technique including the structure factor is presented so that it is possible to determine the form factor and the structure factor simultaneously. This technique can be understood as a generalized version of the indirect Fourier transformation method. Like in the indirect Fourier transformation, there are no models or no analytical restrictions used for the form factor, and the structure factor is parameterized with up to four parameters for a given interaction model. A simultaneous determination of these two functions is possible due to the different analytical behavior of these functions, which also leads in most cases to the existence of a global minimum in the parameter surface. An algorithm to solve this nonlinear least-squares problem has been developed and applied to simulated data for a variety of different uncharged systems.