A general analytical method for calculating particle-dimension distributions from scattering data

A method has been developed for using the Titchmarsh transform to calculate the dimension distribution function Q(a) for particles with a characteristic dimension a (such as the length or the diameter) from the measured small-angle X-ray scattering intensity I(h) for a polydisperse system of independently scattering particles, where h = 4πλ⁻¹ sin (θ/2), λ is the wavelength and θ is the scattering angle. This technique permits the dimension distribution to be calculated for any particle shape for which the intensity l_a(h) for a single particle with characteristic dimension a is proportional to [J_v(ha)]² where J_v(x) is the Bessel function of the first kind and order v. Techniques for computing Q(a) from the characteristic function γ(a) are discussed, and a method is given for calculating Q(a) directly from the measured scattering intensity F(h) for collimating slits with arbitrary length and negligible width. This calculation avoids the errors which may be produced by collimation corrections. The behavior of I(h) and F(h) for large h is discussed.