Enumeration and Classification of Anomaly/Peak Bases in Two-Dimensional Intensity Histograms. Application of Graph Theory to Crystallographic Data Imaging

Abstract

Bragg-event peaks, spikes, intensity streaks and other anomalies generate abnormal regions in two-dimensional intensity histograms from area-detector images. Examination of the shapes of these regions can contribute to the identification of the types of phenomena that generated them. The points that define the anomaly bases, when connected with imaginary lines, form unique graphs. Individual graphs, in turn, can be enumerated by employing graph-theoretical notation and the graph shapes classified. The number of lines in any given graph can also be determined by summing the degrees of the graph points and dividing by two. The ratio of the number of lines per point is a direct indication of the shape of the anomaly base. Long linear and curved shapes, like those associated with intensity streaks and powder rings, will have small lines-per-point ratios, while compact round, square or oval shapes, similar to those belonging to Bragg-event peaks, will have larger lines-per-point ratios. For any given number of points (Np ), for any given graph, the minimum number of lines (q) will equal Np − 1, while the maximum number of lines (q max, Np ) is determined from a round-shaped graph. A graph-shape parameter (GS) can thus be defined as (q − Np − 1)/(q max. Np − Np − 1), where a value near one indicates a round graph shape and a value near zero indicates a linear graph shape. The application of graph-theoretical techniques to anomaly bases can thus provide insight into the nature of the intensities distributed throughout the two-dimensional crystallographic data image.