Source-linked AI summary
VoroTypes: automated enumeration of Voronoi cell topologies for structural identification of complex crystals in atomistic simulations
Emanuel A. Lazar, Chris H. Rycroft
TL;DR
Per-atom identification methods built around fixed reference structures leave many complex crystals without a matching reference. VoroTypes enumerates perturbation-accessible Voronoi cell families from crystal structures and exports VoroTop filters; the paper reports exact, per-sublattice filters, while noting that highly degenerate cases can exceed practical limits.
Problem
Common reference-based identification methods cover only a fixed set of structures, leaving many complex crystals without a matching reference and prompting ad hoc indicators.
Source: §1 · p. 1Method
VoroTypes enumerates the complete perturbation-accessible Voronoi cell family for each Wyckoff site and writes a VoroTop-compatible filter.
Source: §1 · p. 1, §2.1 · p. 2Results
The ω-titanium filter contains 390 site-labeled entries covering 388 distinct vectors and distinguishes its two titanium sublattices through unshared cell types.
Source: §3.1 · p. 4Takeaways & Limitations
The filters provide per-sublattice structural identification from the unit cell, including for sites of the same chemical element.
Source: §3.1 · p. 4, §4 · p. 5Enumeration cost grows with vertex degeneracy, and some structures exceed practical time limits; intermediate subdivisions are omitted from all-mode filters.
Source: §2.4 · p. 4
Abstract
from arXiv · showhide
VoroTypes enumerates, for an arbitrary crystal structure, the complete family of Voronoi cell topologies that can arise under small generic perturbations of atomic positions, and exports it as a topological filter for the VoroTop framework. This extends per-atom structural identification in molecular dynamics simulations, previously limited to a handful of simple reference structures, to complex crystals such as Frank--Kasper and Laves phases, with per-sublattice resolution. The package takes a standard crystallographic description (a CIF or POSCAR file) as input, applies exact symmetry reduction and validity pruning to accelerate the enumeration, and produces filters that load directly into the standalone VoroTop program or into OVITO. A library of pre-computed filters covers over a thousand structures from the Materials Project.
1. Motivation and significance
Per-atom structural identification commonly relies on a fixed set of reference structures, leaving researchers to use ad hoc indicators for many complex crystals. VoroTypes computes site-specific families of perturbation-accessible Voronoi topologies and exports them as VoroTop filters.
1. Motivation and significance
- Common neighbor analysis and polyhedral template matching use fixed reference structures, while many Frank–Kasper, Laves, oxide, and intermetallic crystals lack a matching reference.Researchers therefore resort to indicators tuned to individual simulations.Source: §1 · p. 1
- Voronoi topology provides a structure-agnostic fingerprint, but thermal perturbations let equivalent sites exhibit a characteristic family of cell types rather than one topology.VoroTop matches each cell’s Weinberg vector against a recorded family; before VoroTypes, its filters were limited to BCC, FCC, and HCP because full families required structure-specific analysis.Source: §1 · p. 1
- VoroTypes deterministically enumerates the complete perturbation-accessible Voronoi topology family for each Wyckoff site and writes a VoroTop-compatible filter.The companion paper develops the mathematical foundations and enumeration algorithm [15].Source: §1 · p. 1
- In ω-titanium, the figure illustrates distinct perturbation-family members for both Wyckoff sites, including cell faces absent from the ideal 2d-site cell.The displayed examples are two of eight 1a-family members and two of 382 2d-family members.Source: Figure 1 · p. 2
2. Software description
VoroTypes takes crystallographic structures through an exact enumeration pipeline and produces filters for per-site classification. It offers input and enumeration options, validates filters against stochastic sampling, and has practical limits for highly degenerate geometries.
2.1. Software architecture
- The four-stage pipeline identifies degenerate vertices, enumerates their regular-triangulation resolutions, prunes geometrically inconsistent combinations, and identifies surviving cells by Weinberg vectors.A symmetry-reduced depth-first search prunes combinations during construction; at ω-titanium’s 2d site, it constructs 83,785 canonical representatives of 16.8 million nominal combinations.Source: §2.1 · pp. 2–3
2.2. Software functionalities
- CIF or POSCAR inputs support three nested enumeration modes, optional symmetrization and near-gap thresholds, and filters with separate type indices for inequivalent sites.For CoWO4, a near-gap threshold grows the family from 6 to 20 cell types; the output filter loads into OVITO workflows.Source: §2.2 · p. 3
2.3. Validation against stochastic sampling
- At σ = 5 × 10−4 of nearest-neighbor distance, perfect containment holds for 90.5% of structures against primary filters, 97.9% against default filters, and 98.0% against all-mode filters.These rates cover 1,025 structures completed in all three modes; at one-tenth the perturbation amplitude, 20 of 22 default-filter exceptions reach perfect containment.Source: §2.3 · pp. 3–4
2.4. Scope and limitations
- Enumeration is exact but can exceed practical limits as vertex degeneracy grows; users can fall back to stochastic sampling, and small near-equidistant gaps may require a nonzero threshold.The all-types mode omits intermediate subdivisions; this affects fewer than 1% of library structures’ all-mode filters and no default-mode filter.Source: §2.4 · p. 4
3. Illustrative examples
The ω-Ti and MgCu2 examples show VoroTypes handling both large, coupled families of cell topologies and minimal filters with one type per sublattice. In MgCu2, the generated filter identifies the grain boundary across temperatures approaching melting.
3.1. A coupled, degenerate structure: ω-Ti
- For ω-Ti’s 2d site, 22,092 candidates survive full-cell validation and resolve to 382 cell types; both site families are enumerated in 29 s on one laptop core.Only 262,144 of 16,777,216 nominal resolution combinations are geometrically consistent; the 1a site has eight types.Source: §3.1 · p. 4
- The ω-Ti filter certifies 390 site-labeled entries as exhaustive, with a minimum relative gap of 0.34, and distinguishes the two titanium sublattices without chemical differences.The companion paper reports that this filter resolves an eight-grain polycrystalline sample’s grain-boundary network up to 95% of the ω →β transition [15].Source: §3.1 · p. 4
3.2. A complete workflow: grain-boundary identification in a Laves phase
- For MgCu2, each Wyckoff site has one stable topology, so a two-Weinberg-vector filter suffices with a minimum relative gap of 0.50.The two topologies are a 16-face Mg-8a cell and a pentagonal-dodecahedral Cu-16d cell.Source: §3.2 · p. 4
- Bulk identification stays above 95% through 0.7 Tm and is 77.9% at 0.95 Tm, while the spherical grain boundary remains the dominant connected component.The 525,884-atom MgCu2 bicrystal was classified with the two-vector filter; the boundary remains identifiable in the temperature-slice figure.Source: §3.2 · pp. 4–5, Figure 2 · p. 5
- Without a C15 reference, CNA tags much of the icosahedral Cu sublattice but loses essentially every Mg atom, Ackland–Jones similarly loses Mg, PTM abstains, and CSP distributions overlap at 0.5 Tm.Ackland–Jones recovers more of the Cu sublattice than CNA.Source: §3.2 · p. 5
- The examples span opposite enumeration regimes—two cell types for MgCu2 and hundreds for ω-Ti—yet the same command handles both.For MgCu2, enumeration certifies that two vectors exhaust the family; ω-Ti’s hundreds of types arise from millions of coupled resolution combinations.Source: §3.2 · p. 5
4. Impact
VoroTypes makes topological filters available for complex crystals in existing analysis workflows, extending structural identification beyond the small set of reference structures supported by common methods. It also resolves crystallographic sublattices and provides a pre-computed filter library, while not replacing Laves-specific defect analyses.
Impact
- Filters generated by VoroTypes load into OVITO’s built-in VoroTop modifier, extending its topological defect analyses from FCC, BCC, and HCP metals to complex intermetallics without additional installation.The companion paper demonstrates the workflow for the 30-atom, five-site σ phase of CrFe and polycrystalline ω-Ti [15].Source: §4 · p. 5
- VoroTypes automatically provides C15 identification from a unit cell, replacing the purpose-built analysis previously needed for that capability while resolving both sublattices per atom.LaCA also classifies Laves-specific defects and deformation mechanisms, which this work does not consider.Source: §4 · p. 5
- Because filters assign a type to each topologically distinct site, they identify an atom’s crystallographic site even when sites are chemically identical, as in ω-Ti.The authors describe this as a per-atom capability not provided by any other automated method they are aware of [15].Source: §4 · p. 5
- A distributed library provides pre-computed filters for over a thousand Materials Project structures; entries beyond package complexity limits are marked incomplete, leaving stochastic sampling as the remaining route.The library’s systematic sweep is validated against stochastic sampling.Source: §4 · p. 5
5. Conclusions
VoroTypes makes complete topological filters computable on demand for arbitrary crystal structures and usable in existing analysis pipelines. The conclusion points to extending the enumeration and applying it to planar systems and ideal defect environments.
5. Conclusions
- Complete topological filters, previously available only for BCC, FCC, and HCP through structure-specific analysis, can be computed on demand for arbitrary crystals and used in existing analysis pipelines.Future development could extend per-vertex enumeration to full regular subdivisions, closing the all-mode gap, and migrate inner loops to a compiled core.Source: §5 · pp. 5–6
- The enumeration machinery could also be applied to planar particle systems and to vacancy or interstitial environments, enabling point defects to be identified and tracked through per-atom workflows.The proposed defect filters would apply the same workflow used for crystal phases rather than inferring defects from clusters of unclassified atoms [15].Source: §5 · p. 6
S1. Cluster-size sensitivity for the MgCu2 example
The “other” atoms form a strongly bimodal cluster-size distribution, making the chosen cluster threshold largely immaterial at low-to-moderate temperatures. At high temperatures, thermal broadening enlarges the grain-boundary cluster and lowers the bulk identification rate.
S1. Cluster-size sensitivity for the MgCu2 example
- Voronoi-face adjacency defines clusters without a free connectivity parameter, while the size threshold remains a free parameter.Two atoms are adjacent when their Voronoi cells share a face.Source: S1. Cluster-size sensitivity for the… · p. 8
- The giant grain-boundary cluster (size ≥10,000) is separated from remaining clusters—none above size 103 at most temperatures—by an essentially empty band.The size-1 and size-2 bins are empty at every temperature, while adjacent topology shifts keep the smallest clusters at size ≥3.Source: S1. Cluster-size sensitivity for the… · p. 8, Table S1 · p. 8
- Up to 0.6 Tm, bulk identification rates vary by at most 0.75 percentage points across thresholds 10–1000, and by under 0.2 points up to 0.5 Tm.At 0.8 Tm the spread reaches 6.7 points as thermal scatter fills intermediate cluster sizes; the adopted threshold is 100.Source: S1. Cluster-size sensitivity for the… · p. 8
- At ≥0.8 Tm, thermal fluctuations near the boundary bring adjacent bulk atoms into the grain-boundary component, which grows from roughly 17,000 atoms at 0.1 Tm to 116,000 at 0.95 Tm.The authors interpret the high-temperature identification-rate drop as physical boundary broadening, not a methodological artifact.Source: S1. Cluster-size sensitivity for the… · p. 8
S2. Conventional methods on the MgCu2 bicrystal
On the MgCu2 bicrystal, conventional methods lack a C15 reference and either classify only part of the structure or abstain. VoroTop’s two-vector filter identifies both sublattices and resolves the grain boundary across the studied temperature range.
S2. Conventional methods on the MgCu2 bicrystal
- VoroTop’s two-vector C15 filter distinguishes the Mg-8a and Cu-16d sublattices and resolves the spherical grain across 0.1–0.95 Tm, unlike the conventional methods tested.The conventional methods lack a C15 reference and do not identify the phase or isolate the grain as a connected defect.Source: S2. Conventional methods on the MgCu2… · pp. 8, 10
- CNA classifies 63.4% of atoms as ICO at 0.1 Tm and 27.0% at 0.5 Tm, identifying Cu-16d but assigning Mg-8a and the grain boundary to “other”.The Mg-8a Friauf polyhedron, a structurally distinctive feature of the Laves phase, is missed.Source: S2. Conventional methods on the MgCu2… · p. 9
- At 0.5 Tm, CSP’s broadened distributions overlap, so it shows the grain boundary as intermediate values but cannot classify individual atoms as Cu, Mg, or boundary.CSP is a continuous per-atom metric rather than a structural classification.Source: S2. Conventional methods on the MgCu2… · p. 9, Figure S1 · p. 10
- Ackland–Jones tags 74% of Cu-16d but only 0.02% of Mg-8a as ICO; 97% of Mg atoms remain “other”.It recovers more of the Cu sublattice than CNA but does not recognize the Mg-8a Friauf polyhedron.Source: S2. Conventional methods on the MgCu2… · p. 9
- Default PTM labels all atoms “other”; enabling its optional icosahedral template recovers 108,766 Cu-16d atoms at 0.5 Tm but still misses Mg-8a.Neither the Mg-8a Friauf polyhedron nor the Cu-16d icosahedron is in PTM’s default template set.Source: Table S3 · p. 9, S2. Conventional methods on the MgCu2… · p. 9