Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method

Small-angle scattering data from polydisperse systems can be evaluated under the assumption that all particles have the same shape and that the size distribution depends only on one linear size parameter R. The shape of the particles is assumed to be known a priori. The corresponding size distribution function for the number of particles Dn(R) or for the volume Dv(R) can be computed from the smeared, unsmoothed scattering data by the indirect transformation method restricting the range of definition of the D(R) functions to a finite range Rmin ≤ Rmax. Rmin may be equal to zero, Rmax is limited by the sampling theorem of the Fourier transformation. The resolution in real space is given by the distance of the knots of the B spline functions approximating the size distribution function. The propagated statistical error band in real space can be computed using the inverted matrix of the normal equations. The method gives satisfactory results even in those cases where the shape of the particles is not known exactly, and is superior to analytical methods if the termination effect is critical.