On two methods of determination of particle size distribution functions by means of small-angle X-ray scattering

The total X-ray intensity as a function of h (h is the radial coordinate in reciprocal space), scattered by an isotropic system of particles of equal shapes but of different sizes R, can, under certain conditions, be expressed as an integral over the particle size distribution function D(R), multiplied by a common single-particle function of hR which can be calculated from the assumed particle shape. In the first method D(R) is calculated from this relation by the method of least squares, in which values of D at a limited number of particle sizes are the unknowns. To avoid oscillations in the D curve, constraints are imposed on the D values. The proper weight to be assigned to these constraints must be determined by trial and error. The method has been adapted to suit various assumptions and requirements as to the shape of the particles, the type of distribution function to be calculated, and experimental conditions (slit or pinhole focusing). The second method is essentially the one described by Schmidt, Weil & Brill [X-ray & Electron Methods of Analysis, pp. 86-100. (1968), New York: Plenum], which, however, is adapted to the use of slit-smeared intensities. Both methods may give rise to artefacts in the calculated distribution functions in the range of the smallest particle sizes, which are sensitive to the setting of the various parameters and to experimental errors. However, the position and shape of the main maxima can usually be determined quite well. The agreement between the results obtained by the two methods is satisfactory.