Computing the 3D Radial Distribution Function from Particle Positions: An Advanced Analytic Approach

The radial distribution function, g(r), is ubiquitously used to analyze the internal structure of particulate systems. However, experimentally derived particle coordinates are always confined to a finite sample volume. This poses a particular challenge on computing g(r): Once the radial distance, r, extends beyond the sample boundaries in at least one dimension, substantial deviations from the true g(r) function can occur. State of the art algorithms for g(r) mitigate this issue for instance by using artificial periodic boundary conditions. However, ignoring the finite nature of the sample volume distorts g(r) significantly. Here, we present a simple, analytic algorithm for the computation of g(r) in finite samples. No additional assumptions about the sample are required. The key idea is to use an analytic solution for the intersection volume between a spherical shell and the sample volume. In addition, we discovered a natural upper bound for the radial distance that only depends on sample size and shape. This analytic approach will prove to be invaluable for the quantitative analysis of the increasing amount of experimentally derived tomography data.