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 $n$th 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.

Figure 1

Generic representation of an imaging system sensitive to small angle x-ray scattering. The shaded box represents optical elements for several methods: an analyzer crystal for ABI, two gratings for GI, two masks for EI, and absent optics (but large $z$) for CXS.

Figure 2

Kurtosis of the model sample [Eq. (27)] as a function of the structure size parameter $\sigma$. Horizontal lines are the experimentally determined average kurtosises of different powders and the right-hand branch of the model’s kurtosis was chosen for the determination of the powder’s size parameters.

Figure 3

Calibration for particle size determination by means of the moments of scattering distributions provided by EI-SAXS. The linear regression (dashed line) shows a very strong correlation ($R^2=0.92$, $p=0.009$) between the size parameter $\sigma$ of the powders from EI-SAXS (Fig. 2) and their actual mean particle sizes as measured with microtomography.