Determination of size distribution from small-angle scattering data for systems with effective hard-sphere interactions

Methods for the free-form determination of size distributions for systems with hard-sphere interactions are described. An approximation, called the local monodisperse approximation, is introduced. Model calculations show that this approximation gives relatively small errors even at relatively high polydispersities and large volume fractions. The size distributions are determined by least-squares methods with smoothness and non-negativity constraints. The local monodisperse approximation leads to normal equations that are linear in the amplitude of the size distribution. This is used when solving the least-squares problem: only the two effective parameters describing the interference effects are treated as nonlinear parameters in an external optimization routine. The parameters describing the size distribution are determined by a linear least-squares method. The size distribution is also determined using the nonlinear equations from the calculation of the scattering intensity in the Percus–Yevick approximation. For this, a nonlinear least-squares routine with a smoothness constraint and a non-negativity constraint is used. Both approaches are tested by analysis of simulated examples calculated by the analytical expressions in the Percus–Yevick approximation. Finally, the methods are applied to two sets of experimental data from silica particles and from δ′ precipitates in an Al–Li alloy. For the simulated examples, good agreement is found with the input distributions. For the experimental examples, the results agree with the expected and known properties of the samples.