A generalized quantitative interpretation of dark-field contrast for highly concentrated microsphere suspensions

Abstract

In X-ray grating interferometry, dark-field contrast arises due to partial extinction of the detected interference fringes. This is also called visibility reduction and is attributed to small-angle scattering from unresolved structures in the imaged object. In recent years, analytical quantitative frameworks of dark-field contrast have been developed for highly diluted monodisperse microsphere suspensions with maximum 6% volume fraction. These frameworks assume that scattering particles are separated by large enough distances, which make any interparticle scattering interference negligible. In this paper, we start from the small-angle scattering intensity equation and, by linking Fourier and real-space, we introduce the structure factor and thus extend the analytical and experimental quantitative interpretation of dark-field contrast, for a range of suspensions with volume fractions reaching 40%. The structure factor accounts for interparticle scattering interference. Without introducing any additional fitting parameters, we successfully predict the experimental values measured at the TOMCAT beamline, Swiss Light Source. Finally, we apply this theoretical framework to an experiment probing a range of system correlation lengths by acquiring dark-field images at different energies. This proposed method has the potential to be applied in single-shot-mode using a polychromatic X-ray tube setup and a single-photon-counting energy-resolving detector.

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

“…This will be done for two different models of scattering distributions. First, we will use the power spectrum of the complex wave amplitude as a model for the scattering distribution, which is a well-established fact for CXS and related techniques [2] and a reasonable approximation for GI and ABI [23][24][25][26]. Second, we will use the histogram over refraction angles as a model for the scattering distribution and we will argue that this applies for the incoherent illumination conditions utilized by EI.…”

“…Please note that the kurtosis is independent of δ and, thus, of the powder's specific material as can be seen by combining Eqs. (20) and (26). The experimentally determined kurtosis for each powder is indicated as a dashed horizontal line in Fig.…”

“…By following an analogous approach, the proposed method should enable quantitative particle size determination with subpixel scattering contrasts also with other phase-based imaging methods (e.g., GI or ABI). The robustness of this approach is made evident by the utilization of a laboratory-based x-ray source with a large spectrum as well as a wide range of powder materials and shapes that featured a significant spread of cross-sectional particle sizes [38], where other published approaches are limited to monodisperse microspheres [24,26]. Further, the strong correlation justifies the choice for the model sample [Eq.…”

Statistical uncertainty in the dark-field and transmission signal of grating interferometry

“…Dark-field images (DFIs) obtained by grating interferometry with X-rays (xGI) 1 and neutrons (nGI) 2,3 enable the extraction of spatial correlation functions in each pixel that yield information on structures well below the spatial resolution of the imaging system when altering the probed autocorrelation length of the setup. [4][5][6][7] This paves the way for quantifying the retrieved changes of the DFI signal with varying autocorrelation length and allows the interpretation as the local autocorrelation function of the system in each pixel. 8 In order to understand the accuracy of the data, the intrinsic uncertainty of the transmission image (TI) and the dark-field signal in grating interferometry has to be evaluated to make interpretations reliable.…”

Dual phase grating interferometer for tunable dark-field sensitivity

“…Additionally, this might not be feasible depending on the length of the interferometer compared to the analysed sample. A different approach makes use of the change of the wavelength k; 23 however, implementing such an approach on an X-ray tube setup would require spectral detectors that still are not widely available. In order to tackle all these limitations, an interferometer that can change its correlation length in a robust way without major geometrical changes would be beneficial.…”