Objective identification of local spatial structure for material characterization

Objective tools for characterizing materials at the atomic level are often difficult to develop because of the size or structure of the data. Atom probe tomography (APT) is a measurement tool that maps the location and type of atoms in materials in three-dimensions (3D), producing data sets with potentially billions of observations. In this work, we present a set of spatial statistics methods developed to test the null hypotheses of no global spatial association; no local spatial association; and no local spatial cross-correlation and apply these for the first time to APT data. The empirical and modeled covariogram and Moran's I can be used to study the global structure of a spatially referenced atomic element. The local indicator of spatial association (LISA) identifies volumes where high levels of values (hot spots) or low levels of values (cold spots) of elemental clustering exist. The local indicator of spatial cross-correlation (LISC) reports where simultaneously high levels or low levels of two atomic elements occur. For each test statistic at each location, an associated p -value is produced that can be used to weigh the evidence in favor of spatial clustering. The size of APT data sets presents some challenges, so the effect of weight functions and neighborhood selection on the computation and significance of the test statistics are discussed, and the issue of multiple statistical testing is also considered. These methods are illustrated using an APT data set with atomic percentages reported in voxels binned to 1 nm3.