Correlating Local Chemical and Structural Order Using Geographic Information Systems-Based Spatial Statistics

Analysis of nanoscale short-range chemical and/or structural order via (scanning) transmission electron microscopy (S/TEM) imaging is fundamentally limited by projection of the three dimensional sample, which averages informational along the beam direction. Extracting statistically significant spatial correlations between the structure and chemistry determined from these two-dimensional datasets thus remains challenging. Here, we apply methods commonly used in Geographic Information Systems (GIS) to determine the spatial correlation between measures of local chemistry and structure from atomic-resolution STEM imaging of a compositionally complex relaxor, Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_{3}$ (PMN). The approach is used to determine the type of ordering present and to quantify the spatial variation of chemical order, oxygen octahedral distortions, and oxygen octahedral tilts. The extent of autocorrelation and inter-feature correlation among these short-range ordered regions are then evaluated through a spatial covariance analysis, showing correlation as a function of distance. The results demonstrate that integrating GIS tools for analyzing microscopy datasets can serve to unravel subtle relationships among chemical and structural features in complex materials that can be hidden when ignoring their spatial distributions.