Structural identification in general crystalline systems via Voronoi cell topology

1 Introduction

• Structural identification supports measurements of defect densities, phase transformations, grain-boundary structure, and crystalline order under extreme conditions.Atomistic simulations track atomic positions and interactions over physically relevant timescales.Source: §1 · pp. 1–2
• Common structural-identification methods classify only a limited set of structures, leaving many others unidentified or assigned to a catch-all or nearest built-in type.Examples include Frank–Kasper phases, Laves compounds, multi-sublattice oxides, and intermetallics with large unit cells; researchers may resort to system-specific indicators.Source: §1 · p. 2
• Voronoi cell topology distinguishes FCC and HCP environments whose cells share the same Voronoi index because their faces connect differently.The Voronoi index counts face types but does not determine the precise connectivity of faces, edges, and vertices.Source: §1 · p. 3
• Voronoi topology classifies an atom by whether its cell belongs to the family characteristic of its crystal structure under small perturbations.The approach requires no cutoffs, is independent of interatomic potentials, and accommodates thermal noise.Source: §1 · p. 3
• The algorithm deterministically enumerates every cell topology reachable under sufficiently small generic perturbations of an arbitrary crystal structure.It uses Voronoi–Delaunay duality to resolve degenerate vertices through regular triangulations, then combines resolutions using site symmetry.Source: §1 · p. 3
• Each topology is encoded by a Weinberg vector, and the resulting lists serve as topological filters for matching atoms in finite-temperature simulations.The paper provides VoroTypes and pre-computed filters for a range of crystal structures.Source: §1 · p. 4

2.1 Voronoi cell topology and the question of stability

• After perturbation, triangular-lattice cells remain hexagons, while square, Kagome, and graphene cells span 4–8, 4–10, and 3–12 edges, respectively.These different responses motivate enumerating the cell types accessible under small perturbations.Source: §2.1 · p. 5, Figure 2 · p. 6

2.2 Generic and degenerate Voronoi vertices

• Generic Voronoi vertices are stable, whereas degenerate vertices split under perturbation; higher multiplicity k permits more resolutions.The examples range from two resolutions at square-lattice k = 4 vertices to many more at k = 6 Kagome and graphene vertices.Source: §2.2 · pp. 5–6

2.3 How a degenerate vertex resolves

• At a degenerate vertex, a resolution may promote any subset of the k − 3 partial neighbors into new edge-sharing neighbors of the central cell.The triangular lattice has no partial neighbors; the square has four, while Kagome and graphene have six and nine per cell, respectively.Source: §2.3 · pp. 7–8

2.4 Combining resolutions: enumerating the family

• The square cell has 16 resolution combinations but only five distinct topologies after site-symmetry equivalences; the 6-edge type can arise from two distinct patterns.For Kagome and graphene, independent vertex choices combine, then symmetry and coincident topologies reduce the raw combinations to a finite, lattice-specific family.Source: §2.4 · pp. 8–9

2.5 Crystals with multiple inequivalent sites

• Crystals with inequivalent Wyckoff sites have separate perturbation-accessible families: Lieb corner cells are stable, while edge-midpoint cells develop multiple types.The section notes that sigma-phase CrFe and ω-Ti also contain inequivalent sites, with all sigma-phase sites stable and both ω-Ti sites producing multiple types.Source: §2.5 · p. 9, Figure 3 · p. 9

2.6 From two dimensions to three

• In three dimensions, the same resolution-combination approach applies, but flat four-particle subsets and coupling between degenerate vertices add possible resolutions and constrain combinations.Unlike in two dimensions, not every combination of vertex resolutions necessarily corresponds to an actual Voronoi cell.Source: §2.6 · p. 10

3.1 Voronoi cells and vertices in three dimensions

• In three dimensions, vertices with k > 4 equidistant particles and edges shared by four or more cells are degenerate.BCC cells have only generic vertices and remain truncated octahedra under sufficiently small perturbations; FCC and HCP have both generic and degenerate vertices.Source: §3.1 · p. 11

3.2 The Weinberg vector

• The Weinberg vector uniquely identifies a convex polyhedron’s combinatorial type through its face, edge, and vertex connectivity.Unlike the p-vector, it distinguishes FCC and HCP cells, which share a p-vector.Source: §3.2 · p. 12

3.3 Resolution of a degenerate vertex in three dimensions

• A resolution type is a combinatorially distinct way to resolve a degenerate vertex, including distinctions in new-face shapes and how they fit together.At a degenerate vertex, perturbation splits the higher-valence vertex into generic four-fold vertices; which partial neighbors become face-sharing depends on perturbation direction.Source: §3.3 · p. 13
• Primary resolutions persist as perturbation amplitude σ approaches zero, whereas secondary resolutions occur at finite σ but vanish in that limit.Zero-volume tetrahedra may yield either kind, depending on the geometry of the equidistant particles.Source: §3.3 · pp. 13–14
• The simple-cubic k = 8 vertex has 256 primary resolutions, out of 544 total resolutions.The 256 primary resolutions arise from 32 local forms multiplied by the two resolutions of each of three degenerate edges leaving the vertex.Source: §3.3 · p. 15

3.4 Enumerating the family

• The enumeration combines distinct resolutions at each degenerate vertex, prunes combinations inconsistent across shared edges, collapses site-symmetry equivalents, and labels cells by Weinberg vector.It returns a finite list of vectors for each Wyckoff site.Source: §3.4 · pp. 15–16
• FCC has 2,815 distinct cell types in the default enumeration mode, reduced from 117,649 combinations of seven resolutions at each of six degenerate vertices.The six vertices are pairwise nonadjacent, so every combination is valid; site symmetry and Weinberg-vector classification collapse the raw count.Source: §3.4 · p. 16
• The primary, default, and all modes form nested families; for FCC they yield 44, 2,815, and 6,294 cell types, respectively.Default is suited to finite-temperature simulations; all includes unresolved vertices, relevant when exact coincidences persist in zero-temperature structures.Source: §3.4 · p. 16

4.1 Coupling between adjacent degenerate vertices

• Resolutions at adjacent vertices can conflict when their shared Voronoi edge is degenerate, so valid combinations must be checked across that edge.Such coupling can dominate the family’s combinatorics.Source: §4.1 · p. 17
• Validity-pruning reduces ω-2d combinations from over 16 million candidates to 262,144, yielding 382 distinct Weinberg types across 106 p-vectors.At each of six degenerate edges, only 8 of 16 paired resolutions are compatible; the resulting family contrasts with 8 cell types at ω-1a.Source: §4.1 · pp. 17–18

4.2 Near-equidistant neighbors and structural noise

• Symmetrizing database coordinates restores symmetry-enforced equidistances, but does not correct distortions confined to a symmetric subregion.A gap threshold can treat near-equidistant particles as borderline participants and enumerate topologies reachable under perturbations up to the specified tolerance.Source: §4.2 · pp. 18–19

4.3 Complexity limits

• Enumeration can be intractable when many high-multiplicity vertices combine: simple cubic requires about 10^19 primary-mode and 10^22 default-mode combinations before symmetry reduction.Restricting to primary resolutions helps some structures but not simple cubic; stochastic sampling is an alternative, though the applications use analytical enumeration only.Source: §4.3 · p. 19

5 Applications

• The applications construct complete topology families and filters for structures outside conventional reference sets, with contrasting challenges in enumeration and classification.Sigma-phase CrFe has generic Voronoi vertices; ω-Ti exercises degenerate-vertex resolution and validity pruning.Source: §5 · p. 20

5.1 Sigma phase CrFe: a Frank–Kasper phase

• VoroTop identifies 97.6% of atoms as sigma phase after minimization and 96.9% after annealing at 300 K, while the grain boundary remains resolved.The annealed boundary roughens but remains a clearly resolved shell against the crystalline bulk.Source: Figure 7 · p. 21, §5.1 · p. 21
• For sigma-phase CrFe, VoroTop distinguishes three cell topologies across five Wyckoff sites, whereas conventional methods fail to identify the phase.CNA and PTM label many atoms as icosahedral or “other,” while Ackland–Jones assigns BCC or icosahedral labels.Source: §5.1 · pp. 20, 22

5.2 ω-Ti polycrystal: degenerate-vertex resolution and samespecies sublattice identification

• At 0.95 Tωβ, 69.2% of atoms remain bulk-classified, and the grain-boundary network remains resolved; VoroTop distinguishes ω-Ti’s same-species 1a/2d sites.The sample is studied at 1112 K, only 5% below the ω → β transition.Source: §5.2 · pp. 22, 24, 25, Figure 8 · p. 24
• Shared ω-Ti cell topologies bias the measured 1a:2d ratio upward, but do not affect bulk-versus-grain-boundary classification.Two Weinberg vectors are reachable from either site, and shared types increase in prevalence at higher temperatures.Source: §5.2 · p. 25

5.3 A library of Voronoi cell families

• Of 1,300 sampled structures, 1,054 complete enumeration within 300 s per structure; about 73% are topologically stable, while others yield multiple cell types.Families for degenerate structures range from a handful to nearly 20,000 cell types.Source: §5.3 · p. 26

• Classifying the 204,523-atom polycrystal snapshot takes about 4 s of CPU time and under one second of wall-clock time on a multicore laptop.Per-particle Voronoi construction and Weinberg-vector computation account for roughly equal cost; filter lookup adds negligible overhead, and independent computations parallelize with near-perfect speedup.Source: §6 · p. 27
• A filter matches the exact geometry of its reference structure, so different representations of a nominal crystal can yield different Weinberg-vector sets.For AlCu, two relaxed structures share 3 of 6 Weinberg types under strict enumeration, but both yield the same 13 cell types at gap threshold τ = 0.01.Source: §6 · p. 27
• Topology identifies structural type rather than chemical species, so distinguishing isostructural compounds requires chemical descriptors.Across library structure pairs, 83% have no cell type in common and 5% have identical filters.Source: §6 · p. 27
• Topology does not always distinguish inequivalent sites: ω-Ti 1a and 2d share two cell topologies.An atom realizing either shared topology cannot be assigned to a sublattice by topology alone.Source: §6 · p. 27
• The enumeration also applies without modification to ideal vacancy, interstitial, and substitutional environments, although this extension is not pursued in the applications.Each such site has its own perturbation-accessible topology family and can have a filter for direct identification in simulation.Source: §6 · p. 28
• Secondary topology types requiring m zero-volume tetrahedra occur with frequency of order σ^m, while primary-type leading frequencies are independent of σ.Computing leading coefficients and using measured topology statistics to estimate effective perturbation magnitude and temperature are proposed as future work.Source: §6 · p. 28

• The algorithm deterministically enumerates all perturbation-accessible Voronoi cell topologies for arbitrary crystal structures, removing the main practical barrier to general structural identification.It decomposes enumeration into local resolution problems at degenerate vertices, using site symmetry and validity pruning to keep computation fast for most encountered structures.Source: §7 · p. 28
• Applications demonstrate effective identification in Frank–Kasper phases and other structures with multiple inequivalent sites, where existing automated methods offer little or no classification.The method is purely geometric, requires no interatomic potential or templates, and accommodates thermal fluctuations in finite-temperature simulations.Source: §7 · p. 28

• VoroTypes and its pre-computed filter library are available online, while the molecular-dynamics configurations and input scripts are available from the authors upon reasonable request.VoroTypes is listed at https://github.com/VoroTop/VoroTypes and the filter files at https://www.vorotop.org.Source: Data availability statement · p. 29

• For each particle in the basis, the algorithm treats it as the central particle; symmetry-related particles share a family of cell types, while inequivalent Wyckoff sites generally produce distinct families.Families from symmetry-equivalent sites are automatically merged in the output.Source: A Algorithm · p. 35
• Stage 1 constructs a supercell and Delaunay triangulation, then identifies the central particle’s Voronoi vertices and classifies them as generic or degenerate.Source: A Algorithm · p. 35
• Stage 2 enumerates the ways each degenerate vertex can split into generic vertices, each realizing a subset of that vertex’s partial neighbors.Source: A Algorithm · p. 35
• Stage 3 combines vertex resolutions, discards incompatible combinations, and uses crystallographic site symmetry to retain one representative per equivalence class.Source: A Algorithm · p. 35
• Stage 4 reconstructs each resulting Voronoi cell and identifies its combinatorial type through its Weinberg vector.The procedure is summarized in Algorithm 1, with further sections detailing enumeration modes and Stage 3 symmetry reduction.Source: A Algorithm · p. 35

A.1 Voronoi analysis

• The supercell uses periodic images with N_j = 3 in each direction, satisfying the stated boundary-independence bound for every structure considered.The bound depends on the maximal circumradius of Delaunay tetrahedra incident to the central particle.Source: §A.1 · p. 35
• A Delaunay triangulation supplies the central particle’s incident tetrahedra and their circumcenters, which are used to locate Voronoi vertices.At degenerate vertices, multiple tetrahedra may cover the region, while the triangulation returns only one choice.Source: §A.1 · p. 35
• Near-coincident circumcenters are clustered by adaptive gap detection, then particle-distance gaps identify the full equidistant set and its multiplicity k.The resulting point configuration alone is used for subsequent per-vertex analysis; vertices with k = 4 are generic and those with k > 4 are degenerate.Source: §A.1 · pp. 36–37
• Tetrahedra from generic vertices form a fixed part of the Delaunay star because their four-particle circumspheres remain unchanged under sufficiently small perturbations.These vertices have k = 4.Source: §A.1 · pp. 36–37

A.2 Per-vertex resolution enumeration

• Each distinct star of the central particle across triangulations of the convex hull defines a resolution type, characterized by its tetrahedra, neighbors, and contributed face valences.For the small co-spherical configurations considered, k ≤ 8, every face-matching triangulation is stated to be realizable under some small perturbation.Source: §A.2 · p. 37
• A valid triangulation matches each boundary face to one tetrahedron and each internal triangular face to zero or two tetrahedra.The search selects candidate tetrahedra formed from four particles in the co-spherical set.Source: §A.2 · pp. 37–38
• The volume bound is essential because face matching alone admits over-covering complexes built from zero-volume tetrahedra.At the simple-cubic k = 8 vertex, face matching accepts 3,002,678 closed complexes, but only 11,648 satisfy the volume bound.Source: §A.2 · p. 38
• Flat tetrahedra must remain candidates because perturbations breaking coplanarity can make them valid, and six of nine octahedral FCC triangulations use one.The octahedral configuration occurs at each FCC degenerate vertex.Source: §A.2 · p. 38
• Near-equidistant particles trigger enumeration across subsets of the neighbor set, making the resulting family monotone in the threshold and insensitive to the core-boundary split.The implementation caps the set at seven neighbors per vertex; no reported enumeration is affected because completed structures have at most eight near-equidistant particles including the central particle.Source: §A.2 · pp. 38–39

A.3 Combining resolutions across vertices

• The algorithm combines the fixed generic tetrahedra with one local resolution at each degenerate vertex to enumerate candidate global Delaunay stars.For n degenerate vertices with m_i resolutions each, it considers combinations across the per-vertex resolution sets.Source: §A.3 · p. 39
• At the ω-2d site, six degenerate-vertex pairs share degenerate edges, and many combinations of their resolutions are incompatible.The compatibility test rejects no pairs at the FCC, HCP, fluorite Ca, or ω-1a sites.Source: §A.3 · pp. 39–40
• Adjacent degenerate vertices can have coupled choices across degenerate Voronoi edges, so pairwise compatibility checks prune combinations that overfill shared triangular faces.The compatibility test rejects a face in three or more tetrahedra; it is integrated into a symmetry-canonical depth-first search.Source: §A.3 · pp. 39–40
• Cell construction rejects combinations that survive pairwise pruning but leave a shared face underfilled: 61,693 of 83,785 canonical ω-2d combinations are rejected late.No combinations are rejected late at FCC, HCP, fluorite Ca, or ω-1a sites.Source: §A.3 · p. 40

A.4 Cell construction and identification

• Each Voronoi face is dual to a Delaunay edge, with vertices from circumcenters and adjacency from tetrahedra sharing a face containing that edge.Tracing the adjacency cycle gives the face’s ordered vertex sequence.Source: §A.4 · pp. 40–41
• A face whose adjacency graph is not a single cycle marks a malformed cell, so the corresponding combination is discarded.Unresolved degenerate vertices are grouped and consecutive face entries with the same group identifier are collapsed to merge them back into a higher-valence vertex.Source: §A.4 · p. 41
• Consistent face orientation is propagated across shared edges, after which the Weinberg vector provides a unique identifier for the cell’s combinatorial type.Malformed cells that pass the face check are discarded if the traversal finds duplicate or missing directed edges.Source: §A.4 · p. 41

A.5 Enumeration modes

• Default includes all fully resolved types, including resolutions with zero-volume tetrahedra, and suits finite-temperature analysis because primary and secondary resolutions occur with nonzero frequency at T > 0.The listed representative structures have no borderline atoms at τ = 0.005, so subset enumeration does not change their type counts.Source: §A.5 · p. 42, Table 1 · p. 43
• All adds an unresolved option at each degenerate vertex to default’s fully resolved types, so it can match more cells but is more permissive and computationally expensive.For FCC, the combinations increase from 76^3 = 117,649 in default to 86^3 = 262,144 in all.Source: §A.5 · p. 42
• Primary admits only resolutions reached with positive frequency under arbitrarily small generic perturbations, making it appropriate for T → 0 analyses or when default enumeration is intractable.For simple cubic, default has 544 resolutions per each of 8 degenerate vertices, while primary has 256 per vertex and still yields about 10^19 combinations per cell.Source: §A.5 · pp. 41–42
• All omits intermediate regular subdivisions, leaving its filters incomplete for seven of 1,025 completed structures with prismatic or cubic vertices; default remains complete for all 1,025 under sufficiently small perturbations.The omitted topologies arise at boundary-face vertices; for finite-temperature identification, the default filter covers them.Source: §A.5 · p. 43
• At very small perturbation amplitudes, sampled cells can have nearly coincident vertices reported as one and fall outside the all filter.This is a numerical-resolution caveat for sampled data.Source: §A.5 · p. 44

A.6 Symmetry reduction

• Combinations related by site symmetry produce identical Voronoi cells and Weinberg vectors, so the algorithm need enumerate only one representative from each symmetry class.Site symmetry is obtained from the detected space group; without one, enumeration proceeds without symmetry reduction, changing runtime but not the resulting type family.Source: §A.6 · p. 44
• A depth-first orderly-generation search builds combinations vertex by vertex, pruning partial assignments that are already non-canonical and using pairwise compatibility to prune invalid resolutions.The pairwise test catches only one of two ways a combination can fail, so some candidates are rejected later during cell construction.Source: §A.6 · p. 44
• For ω-2d, symmetry and pairwise pruning reduce 16,777,216 nominal combinations to 83,785 canonical combinations, yielding 382 distinct cell types after invalid cells are rejected.Cell construction discards 61,693 of the 83,785; the remaining 22,092 valid canonical combinations expand by symmetry to 262,144 valid combinations.Source: §A.6 · p. 44
• For FCC, site symmetry reduces 117,649 combinations to 2,821 canonical representatives, while HCP reduces the same total to 9,940 representatives.These correspond to 42-fold and nearly 12-fold reductions, respectively; the representatives yield 2,815 FCC and 9,556 HCP distinct Weinberg types.Source: §A.6 · pp. 44–45

S1.1 Supercell sufficiency

• For supercell sufficiency, the image extent depends on the largest incident Delaunay circumradius and perpendicular unit-cell face spacing; oblique cells may require more than three images.Circumcenter clustering uses a jump exceeding 100 times the distance spread, with a minimum spread of 10−8 and a below-jump distance cutoff of 10−3; the equidistant-set test uses the same jump rule after including the four nearest particles.Source: S1.1 Supercell sufficiency · p. 46, S1.2 Circumcenter-clustering tolerances · pp. 46–47

S1.3 Orderly generation of canonical combinations

• Symmetry-related resolution combinations produce identical cell topologies, so orderly depth-first generation keeps one canonical representative and prunes partial assignments already shown non-canonical.At completion, the number of symmetry operations divided by the number fixing a combination gives its symmetry-class multiplicity; the chosen ordering changes representatives but not the resulting cell-type list.Source: S1.3 Orderly generation of canonical… · pp. 47–48

S2.1 Summary statistics

• The default-mode library contains 1,054 structures spanning all seven crystal systems and 91 space groups, with 77,644 unique Weinberg vectors.In 72.9% of structures, every Wyckoff site has one topology; the median structure has 3 cell types, while the maximum is 19,775. Triclinic structures show multi-type sites in 29% of cases, often from near-equidistant atoms; cubic structures are least likely, partly because many degenerate candidates did not complete enumeration.Source: S2.1 Summary statistics · pp. 49–50

S2.2 Weinberg vector sharing

• The most widely shared Weinberg vector, BCC’s truncated octahedron, appears in 338 of 1,054 structures (32.1%).Overall, 54,725 of 77,644 unique vectors occur in exactly one structure, while 83% of structure pairs have no vector in common.Source: S2.2 Weinberg vector sharing · p. 50

S3 Near-equidistant neighbor validation

• At σrel = 5 × 10−4, default-filter containment was perfect for 1,003 of 1,025 structures (97.9%), rising to 986 structures at σrel = 5 × 10−5.At the smaller amplitude, 20 of 22 structures imperfect against the default filter at the reference amplitude became perfect; the worst rose from 69.6% to 100%. Small-σ numerical resolution can produce partially resolved cells, so the all filter is appropriate near zero perturbation; increasing τ accommodates larger thermal displacements.Source: S3 Near-equidistant neighbor validation · pp. 51–52

S4 Primary resolutions: criterion and validation

• A resolution with a zero-volume tetrahedron is secondary when equidistant particles lie on both sides of its plane, and primary otherwise.Sampling supported the criterion: FCC secondary-star frequency decreased with σ, while ω-Ti 2d resolutions appeared at stable frequencies. Among 1,300 structures, 267 had primary families containing zero-volume tetrahedra; at the ω-Ti 2d site, only 13 of 382 primary cell types came from resolutions without them.Source: S4 Primary resolutions: criterion and… · p. 54

S5.1 BCC

• BCC has only generic Voronoi vertices, so its cell remains the truncated octahedron under sufficiently small perturbations.The cell has 24 vertices per particle and 14 face-sharing neighbors.Source: S5.1 BCC · p. 55

S5.2 FCC

• FCC’s six degenerate vertices yield 2,815 distinct Weinberg types from 117,649 valid combinations, with cell face counts ranging from 12 to 18.Each vertex has seven resolution types; all resolution pairs are compatible. In all mode, including the unresolved option at each vertex, the total is 262,144 combinations and 6,294 types.Source: S5.2 FCC · pp. 55–56

S5.3 HCP

• HCP’s all-enumeration mode yields 86 = 262,144 valid combinations and 21,611 distinct Weinberg types, 3.4× the FCC count because of lower site symmetry.The ordinary combination counts match FCC, with every combination valid.Source: S5.3 HCP · p. 56

S5.5 Fluorite

• The fluorite Ca site has 64 valid combinations yielding six 8-faced cell types, while the F-site space of about 10^18 combinations exceeds the algorithm’s reach.The F site has 4 degenerate vertices with k = 8 and 12 with k = 5.Source: S5.5 Fluorite · pp. 57–58

S5.6 ω phase

• In the ω phase, all 64 combinations at the 1a site are valid and yield eight 14-faced types; validity pruning reduces 412 nominal 2d combinations to 382 distinct types.At 2d, resolutions at degenerate vertices are coupled through shared atoms along degenerate edges.Source: S5.6 ω phase · p. 58, Table 5 · p. 59

S6 Cell types common to the two ω-Ti sites

• The shared (0, 4, 4, 6) type identifies a 1a site with probability 0.82, versus 0.39 for the truncated octahedron; shared types cause roughly 3.5–4% net mislabeling at 0.1 and 0.5 Tωβ.At higher temperatures, the truncated octahedron’s increasing share likely reflects BCC-like local environments, which the filter cannot distinguish from ω.Source: S6 Cell types common to the two ω-Ti… · pp. 59–60

S7 Conventional methods on the ω-Ti polycrystal

• PTM’s result depends on its RMSD cutoff: at 0.10 it labels 98.80% of atoms other, while with no cutoff it labels roughly 50% HCP and 40% BCC.Neither cutoff enables PTM to identify ω-Ti because ω is not among its templates.Source: S7 Conventional methods on the ω-Ti… · pp. 61–62, Table 6 · p. 62
• At 585 K, CNA labels 97.85% of atoms other, PTM at RMSD 0.15 labels 69.09% other, and Ackland–Jones assigns 49.07% HCP and 29.27% BCC without an ω reference.The methods do not distinguish ω from genuine BCC or HCP material or separate grain boundaries from the bulk.Source: S7 Conventional methods on the ω-Ti… · pp. 61–62, Table 7 · p. 63