Interpretation of small-angle scattering curves proportional to a negative power of the scattering vector

The intensity of the small-angle X-ray and neutron scattering from a polydisperse system of randomly oriented independently scattering particles is shown to be proportional to h^−α for all values of the scattering vector h when the distribution of particle dimensions is proportional to r^{−(2d + 1 − α)}, where h = 4πλ⁻¹ sin(θ/2);θ is the scattering angle; λ is the wavelength; r is the maximum dimension of a particle; and d is the number of dimensions of the particles. The value of α lies in the interval 0 < α < ω, where ω = 4, 2, and 1 for d = 3, 2, and 1 respectively. This relationship between the scattered intensity and the particle-dimension distribution does not depend on the shape of the particles in the polydisperse system, provided that the particle-shape distribution is independent of the distribution of particle dimensions.