Simple inversion formula for the small-angle X-ray scattering intensity from polydisperse systems of spheres

A practical inversion method to calculate the size distribution of colloidal homogeneous particles from small-angle scattering experiments is presented. It is based on the analysis of the deviations of the scattering signal from the Porod law. The resulting inversion formula provides a reliable way to assess complex size distributions such as power-law, multimodal or very broad distributions. It is particularly suitable for colloidal dispersions that have a substantial fraction of very small particles. These cases correspond to large deviations from the Porod law over a broad range of scattering vector values, q. Shannon's theorem gives a way to obtain the maximum amount of information concerning the size distribution, without requiring an arbitrary extrapolation of the data beyond the available q range. It is demonstrated that this protocol yields a calculated distribution of particle sizes that is stable.