Particle size distribution from small-angle X-ray scattering data

An improved transform technique has been developed [Gerber (1983). Thesis, Wilhelm-Pieck-University Rostock, German Democratic Republic] for calculating the particle size distribution N(R) for spherical particles with radii R from small-angle X-ray scattering data. This method permits N(R) to be calculated from analytical expressions that were derived for point collimation and for infinitely long slit collimation. A special procedure has been introduced in order to reduce termination errors. The technique described and those developed by Schmidt [Brill, Weil & Schmidt (1968). J. Colloid Interface Sci. 27, 479–492], Vonk [J. Appl. Cryst. (1976), 9, 433–440] and Glatter [J. Appl. Cryst. (1980), 13, 7–11] were used for calculating particle size distributions from theoretical scattering curves and from an experimental scattering curve of suspended SiO₂ particles (Ludox). The results obtained by the different techniques were compared, and reasonable results are given by all methods employed. The accuracy of the size distributions calculated by the improved method is somewhat higher than that obtained by Schmidt's transform technique. With Glatter's procedure, the deviations from the exact distributions are comparable to those from this improved transform technique, but the use of Glatter's program requires a large computer, whereas the new method has the advantage of being suitable for a small computer. Vonk's program also requires a large computer, and the deviations obtained are larger than those produced by other methods. The experimental scattering curve of the Ludox sample was also evaluated by assuming a log-normal distribution for the particles. The parameters μ and σ of this function were determined from a set of small-angle X-ray scattering structural parameters. The resulting log-normal distribution is significantly different from the size distribution calculated by our method.