Clustering characteristic diffraction vectors in 4-D STEM data sets from overlapping structures in nanocrystalline and amorphous materials

We describe a method for identifying and clustering diffraction vectors in four-dimensional (4-D) scanning transmission electron microscopy data to determine characteristic diffraction patterns from overlapping structures in projection. First, the data is convolved with a 4-D kernel, then diffraction vectors are identified and clustered using both density-based clustering and a metric that emphasizes rotational symmetries. The method works well for both crystalline and amorphous samples and in high- and low-dose experiments. A simulated dataset of overlapping aluminum nanocrystals provides performance metrics as a function of Poisson noise and the number of overlapping structures. Experimental data from an aluminum nanocrystal sample shows similar performance. For an amorphous Pd_77.5Cu₆Si_16.5 thin film, experiments measuring glassy structure show strong evidence of 4- and 6-fold symmetry structures. A significant background arises from the diffraction of overlapping structures. Quantifying this background helps to separate contributions from single, rotationally symmetric structures vs. apparent symmetries arising from overlapping structures in projection.