Interpretation and Utility of the Moments of Small-Angle X-Ray Scattering Distributions

Small angle x-ray scattering has been proven to be a valuable method for accessing structural information below the spatial resolution limit implied by direct imaging. Here, we theoretically derive the relation that links the subpixel differential phase signal provided by the sample to the moments of scattering distributions accessible by refraction sensitive x-ray imaging techniques. As an important special case we explain the scatter or dark-field contrast in terms of the sample's phase signal. Further, we establish that, for binary phase objects, the nth moment scales with the difference of the refractive index decrement to the power of n. Finally, we experimentally demonstrate the utility of the moments by quantitatively determining the particle sizes of a range of powders with a laboratory-based setup.