Three dimensional cluster analysis for atom probe tomography using Ripley’s K-function and machine learning

The size and structure of spatial molecular and atomic clustering can significantly impact material properties and is therefore important to accurately quantify. Ripley’s K-function (K(r)), a measure of spatial correlation, can be used to perform such quantification when the material system of interest can be represented as a marked point pattern. This work demonstrates how machine learning models based on K (r)-derived metrics can accurately estimate cluster size and intra-cluster density in simulated three dimensional (3D) point patterns containing spherical clusters of varying size; over 90% of model estimates for cluster size and intra-cluster density fall within 11% and 18% error of the true values, respectively. These K (r)-based size and density estimates are then applied to an experimental APT reconstruction to characterize MgZn clusters in a 7000 series aluminum alloy. Here we find that the estimates are more accurate, consistent, and robust to user interaction than estimates from the popular maximum separation algorithm. Using K (r) and machine learning to measure clustering is an accurate and repeatable way to quantify this important material attribute.