Small-angle X-ray scattering: characterization of cubic Au nanoparticles using Debye's scattering formula

Abstract

A versatile software package in the form of a Python extension, named CDEF (computing Debye's scattering formula for extraordinary form factors), is proposed to calculate approximate scattering profiles of arbitrarily shaped nanoparticles for small-angle X-ray scattering (SAXS). CDEF generates a quasi-randomly distributed point cloud in the desired particle shape and then applies the open-source software DEBYER for efficient evaluation of Debye's scattering formula to calculate the SAXS pattern (https://github.com/j-from-b/CDEF). If self-correlation of the scattering signal is not omitted, the quasi-random distribution provides faster convergence compared with a true-random distribution of the scatterers, especially at higher momentum transfer. The usage of the software is demonstrated for the evaluation of scattering data of Au nanocubes with rounded edges, which were measured at the four-crystal monochromator beamline of PTB at the synchrotron radiation facility BESSY II in Berlin. The implementation is fast enough to run on a single desktop computer and perform model fits within minutes. The accuracy of the method was analyzed by comparison with analytically known form factors and verified with another implementation, the SPONGE, based on a similar principle with fewer approximations. Additionally, the SPONGE coupled to McSAS3 allows one to retrieve information on the uncertainty of the size distribution using a Monte Carlo uncertainty estimation algorithm.

Metrological Protocols for Reaching Reliable and SI-Traceable Size Results for Multi-Modal and Complexly Shaped Reference Nanoparticles

“…A detailed description and discussion of Figure 11 can be found in Ref [ 1 ]. For example, the authors also show that a spherical model function would not fit the experimental data.…”

“…The optical path through the sample thickness was determined from the transmission through the Fluorinert layer with a known attenuation coefficient [ 27 ]. For the fitting of the averaged SAXS curve, a model must be used, e.g., Debye’s scattering equation [ 1 ].…”

“…SAXS is an ensemble method that measures the differential X-ray scattering cross-section of a sample, i.e., the sum of the scattering signals of all NPs present in the sample as a function of the scattering angle. Fitting the total signal with an appropriate model function allows the extraction of the parameters of the underlying model, including the assumed size distribution [ 1 ].…”

“…The sizes obtained with SAXS, as an ensemble method, were compared with those given by microscopy, which is a local and counting method. Specific algorithms have been developed in nPSize to model complex shapes of NPs such as bipyramids (nPSize03) and nanocubes (nPSize04) [ 1 ].…”

Quantification of PEFC Catalyst Layer Saturation via In Silico, Ex Situ, and In Situ Small-Angle X-ray Scattering

“…Another alternative is to simulate the scattering of an object described by its outer envelope using point cloud scattering. 43,44 For quantifying the change in electron density of the component, a model-independent invariant calculation is typically used. 45 Multi-phase structures (meaning that a material with a number of phases >2) scattering length densities within an order of magnitude can be difficult to interpret.…”