Solid-angle based nearest-neighbor algorithm adapted for systems with low coordination number

TL;DR

This work targets the challenge of identifying nearest neighbors in systems with low coordination numbers, where the parameter-free SANN method overcounts neighbors. It introduces a parameter-free geometric correction, the inscribed circle trick (mSANN), backed by a 2D extension via an existence-uniqueness formulation and a 3D analogue, to reduce overcounting without sacrificing the original method’s benefits. Across crystalline, quasicrystalline, and coexistence phases in both two and three dimensions, mSANN shows robust neighbor identification, correctly reproducing coordination numbers in low-CN lattices and preserving tiling topology in complex environments. Performance benchmarks reveal that, while Voronoi can be faster for small systems, parallelized mSANN outperforms Voronoi for larger systems, offering a practical, parameter-free, and scalable tool for on-the-fly structural analysis in heterogeneous materials.

Abstract

Nearest-neighbor identification is central to the analysis of local structure in condensed matter systems. The solid-angle-based nearest-neighbor (SANN) algorithm is widely used offering a parameter-free and computationally efficient alternative to cutoff- or Voronoi-based methods. Unfortunately, however, in systems with low coordination numbers, SANN tends to identify many particles as neighbors that are outside the nearest neighbor shell. Here, we propose a solution to this problem. Specifically, we propose a geometric modification, the ``inscribed circle modification'', that resolves systematic overcounting in low-coordination lattices without introducing free parameters. We benchmark the modified algorithm (mSANN) against Voronoi and the original SANN algorithm in crystalline, quasicrystalline, and heterogeneous systems, and demonstrate that it provides robust and low-cost neighbor identification across both two and three dimensions.

Figures (12)

• Figure 1: Angle $\theta_{ij}$ associated with the chord passing through neighbor $j$. $R^{(m)}_i$ denotes the shell radius of particle $i$ with $m$ neighbors, and $r_{ij}$ is the distance between $i$ and $j$.
• Figure 2: Interpretation of Eq. \ref{['eq:sann']} in two dimensions. Local rotations around particle $i$ map neighbor positions onto the midpoints of a cyclic polygon, with $R_i$ as the radius of the circumscribed circle.
• Figure 3: Inscribed and circumscribed circles of the first four regular polygons.
• Figure 4: SANN applied to an ideal honeycomb lattice. The outer circle (circumscribed) leads to overcounting, while the inscribed circle yields the correct CN. Arithmetic and geometric mean circles (dashed and dot-dashed) both eliminate overcounting.
• Figure 5: Comparison of nearest-neighbor identification in 2D lattices using Voronoi, SANN, and mSANN. Dots represent particles and line segments connect the neighbors. Configurations include correlated thermal noise.