The chemical structures of the phases of CaCO₃ samples undergoing a calcination process, as well as the volume fractions and matter densities of those phases, are determined by the Porod invariant behaviour.

This work presents a novel method for the analysis of small-angle scattering data from a highly aligned domain structure based on a modification of the well known Porod law. The analysis approach allows the range of accessible length scales to be extended from 1 µm to up to 40 µm using a conventional small-angle neutron scattering setup.

A new Guinier-Porod empirical model is introduced to fit small-angle scattering data from spherical and nonspherical objects such as rods or platelets. This model is used to fit small-angle scattering data from a Pluronic solution that sequentially forms unimers, spherical micelles, cylindrical micelles and lamellar micelles on heating.

The application of the anisotropic Porod law is illustrated for the case of ellipsoidal particles.

The kinetics of precipitation of CaCO₃ particles in the presence of polystyrene sulfonate was studied by small- and wide-angle X-ray scattering. Employment of the Porod invariant and the Hurd–Flower model provides essential information on the kinetics.

The Python extension CDEF is a suitable evaluation tool for experimentalists to calculate single-particle small-angle X-ray scattering profiles with satisfactory accuracy, as shown by comparison with common well-known analytical form factors. Using different complex cubic models, a direct comparison between CDEF and the already established SPONGE with respect to the size distribution of imperfect Au nanocubes showed a clear agreement of the results.

The high scattering power of a sample at low angles may lead to significant secondary scattering contributions in the measured small- and ultra-small-angle X-ray scattering patterns. A correction procedure is presented for the removal of the excess intensity and to improve the dynamic range of the measurement.

An indirect Fourier transform method is presented which describes a solution scattering profile from a reduced set of intensities. Equations are derived to fit the experimental profile using least squares and to calculate commonly used size and shape parameters directly from the reduced set of intensities, along with associated uncertainties. An analytical equation is derived enabling regularization of the real-space pair distribution function. Convenient software is provided to perform all described calculations.

The freely available open-source GSAS-II package now contains modules for the analysis of small angle X-ray scattering data. This includes processing of two-dimensional images to create one-dimensional patterns, analysis of the size distribution and modelling of the one-dimensional data with a combination of various scattering components, and slit smearing corrections can be applied where needed.

The results of small-angle and ultra-small-angle X-ray scattering on porous CaCO₃ microparticles of pulverulent vaterite made by a conventional chemical route and by using supercritical CO₂ are presented.