Statistical measures of angular correlations in amorphous materials from electron nano-diffraction in the scanning/transmission electron microscope

Crystallography employing conventional large-volume diffraction has enabled the firm connections between structure and properties and structure and function that have solved many of the most difficult problems in materials science and biology. Disordered materials possess a large variety of local structural arrangements and pose a special challenge for crystallography. Often the local structures and symmetries that are responsible for observed phenomena in structurally complex disordered materials cannot be distinguished from conventional diffraction alone. In this article, we review analytical approaches for understanding local structure and symmetry from angular correlations in limited-volume diffraction patterns of amorphous materials, with a special focus on electron nano-diffraction. We discuss how these angular correlations can be interpreted in the context of dense, disordered, three-dimensional materials probed in a projection geometry and highlight the experimental challenges and considerations. New developments in this field are described whereby these angular correlations are statistically analyzed to probe the symmetry and variety of local structures, transformed to a real-space function that contains the 2-, 3- and 4-body particle correlations, and employed to develop reverse Monte Carlo models with more realistic higher-order correlations.