PROBE-006: WIGNER-SEITZ CELL & INTERSTITIAL VOID RESONANCE — LITERATURE DATA ================================================================================ Date: 2026-03-28 Status: LITERATURE REVIEW — supporting data for HPC-030 Purpose: Catalog published DATA on how void geometry in crystal lattices affects electromagnetic, phononic, and material properties. Context: HPC-030 tests Wigner-Seitz cells as EM cavities. This probe collects existing evidence that void shape controls physics. ================================================================================ I. CARBON SOLUBILITY: THE DEFINITIVE VOID GEOMETRY CASE ================================================================================ The clearest demonstration that void GEOMETRY (not total void volume) controls material properties. Jiang & Carter (2003), Physical Review B 67, 214103. "Carbon dissolution and diffusion in ferrite and austenite from first principles" DATA: ┌──────────────────────────────────────────────────────────────────────┐ │ Property │ BCC (ferrite) │ FCC (austenite) │ ├──────────────────────────────────────────────────────────────────────┤ │ Lattice parameter │ 0.286 nm │ 0.357 nm │ │ Octahedral void radius / R │ 0.155 │ 0.414 │ │ Tetrahedral void radius / R │ 0.291 │ 0.225 │ │ Total void space │ 32% │ 26% │ │ Oct sites per unit cell │ 6 │ 4 │ │ Tet sites per unit cell │ 12 │ 8 │ │ Total interstitial sites │ 18 │ 12 │ │ Max carbon solubility (wt%) │ 0.022% │ 2.14% │ │ C solution enthalpy (oct) │ +0.74 eV │ -0.17 eV │ │ C migration barrier │ 0.86 eV │ 0.99 eV │ │ C migration (exp) │ 0.87 eV │ (matches) │ └──────────────────────────────────────────────────────────────────────┘ KEY: BCC has MORE void space (32% vs 26%) and MORE sites (18 vs 12), yet carbon solubility is 100x LOWER. The octahedral void is 2.7x smaller in BCC (0.155R vs 0.414R). Individual void GEOMETRY wins over total void volume. ADDITIONAL: Solution enthalpy FLIPS SIGN — endothermic in BCC (+0.74 eV), exothermic in FCC (-0.17 eV). Same host element (Fe), same solute (C), different void shape → opposite thermodynamics. II. HYDROGEN DIFFUSION: 10 ORDERS OF MAGNITUDE FROM VOID SHAPE ================================================================================ Hirata, Iikubo et al. (2018), Metallurgical & Materials Transactions A 49, 5015. "First-Principles Study on Hydrogen Diffusivity in BCC, FCC, and HCP Iron" DATA: ┌──────────────────────────────────────────────────────────────────────┐ │ Property │ BCC-Fe │ FCC-Fe │ ├──────────────────────────────────────────────────────────────────────┤ │ Preferred H site │ Tetrahedral │ Octahedral │ │ H solution energy │ 0.23 eV │ ~0.5 eV │ │ Migration barrier │ 0.087-0.15 eV │ ~0.5 eV │ │ Diffusion pre-factor │ 1.379e-4 cm²/s │ 3.22e-4 cm²/s │ │ D at room temp (approx) │ ~10⁻⁵ cm²/s │ ~10⁻¹⁶ cm²/s │ └──────────────────────────────────────────────────────────────────────┘ KEY: 10 ORDERS OF MAGNITUDE difference in H diffusivity. BCC tetrahedral sites are smaller (lower binding, 0.23 eV) and closer together (shorter jump). FCC octahedral sites are larger (higher binding) and further apart. Void SHAPE determines which site H occupies; void SPACING determines migration barrier. III. ELECTRON LOCALIZATION IN INTERSTITIAL VOIDS ================================================================================ Rousseau & Ashcroft (2008), Physical Review Letters 101, 046407. "Interstitial Electronic Localization" - Array of impenetrable spheres in BCC and FCC lattices - As compression increases, valence electrons LOCALIZE in interstitial regions - Electronic densities develop interstitially-centered maxima - Electrons preferentially localize BETWEEN atoms, not on them Storm, Racioppi, Duff, McHardy, Zurek & McMahon (2025), Communications Materials. "Experimental signatures of interstitial electron density in transparent dense sodium" - FIRST EXPERIMENTAL EVIDENCE of interstitial electron localization - Single-crystal X-ray diffraction at 223 GPa (hP4 sodium) - p and d states hybridize, pointing toward CENTER OF LATTICE INTERSTICES - Charge density accumulation at CENTER of interstitial cavities (confirmed by ELF) - Non-nuclear maximum (NNM) at void center = electride state - Criterion: ELF > 0.7 at interstitial sites identifies electride KEY: Electrons concentrate at the GEOMETRIC CENTER of crystal voids. This is direct experimental evidence that void geometry traps charge. IV. ELECTRON DENSITY TOPOLOGY: BCC vs FCC ================================================================================ Quantum Theory of Atoms in Molecules (QTAIM) topological analysis: BCC: 8 bond paths to nearest neighbors. Cage critical point at CENTER of octahedral hole. Ring critical point at CENTER of tetrahedral hole. FCC: 12 bond paths to nearest neighbors. Cage critical points in BOTH tetrahedral and octahedral holes. The critical point structure maps directly to the void geometry. V. PHOTONIC/PHONONIC BAND GAPS: VOID SHAPE AS CONTROL PARAMETER ================================================================================ A. Corbett et al. (2021), Nature Communications 12, 2543. "The diversity of three-dimensional photonic crystals" - 150,000+ band calculations across thousands of crystal templates - 351 templates with complete band gaps (~300 previously unreported) - Band gaps most frequent in FCC and BCC lattices - "A nearly spherical Brillouin zone is neither necessary nor sufficient" - Full dataset: glotzerlab.engin.umich.edu/photonics/ B. Kuang, Hou & Liu (2004), Physics Letters A 332. "Effects of shapes and symmetries of scatterers on phononic band gap" - MAXIMUM band gap occurs when scatterer shape MATCHES coordination polygon - Hexagonal rods in triangular lattice = best - Square rods in square lattice = best - KEY: shape matching between void and lattice maximizes band gap C. arXiv 2508.03045 (2025) "Simultaneous photonic and phononic bandgaps... circular-to-triangular air gap holes" - Silicon hexagonal lattice, continuous circle → triangle void transformation - Photonic gap tunability: 49.7% for circle-to-triangle vs 8.6% for circles only - Void SHAPE is 5.8x more powerful than void SIZE for tuning band gaps D. Warmuth, Wormser & Korner (2017), Scientific Reports 7, 3843. - First experimental verification of 3D phononic band gaps in single-phase material - Ti-6Al-4V, selective electron beam melting - Band gap position and extension tuned by geometric design of unit cell VI. WIGNER-SEITZ CELL FOAMS AS PHOTONIC CRYSTALS ================================================================================ Klatt, Torquato et al. (2019), PNAS 116(47), 23480. "Phoamtonic designs yield sizeable 3D photonic band gaps" DATA (dielectric contrast ε = 13): ┌──────────────────────────────────────────────────────────────────────┐ │ Foam Structure │ Based On │ Max Band Gap │ ├──────────────────────────────────────────────────────────────────────┤ │ Kelvin │ Truncated octahedron │ 7.7% │ │ │ (= BCC Wigner-Seitz) │ │ │ C15 │ Frank-Kasper polyhedra │ 13% │ │ Weaire-Phelan │ A15 crystal relaxation │ 16.9% │ └──────────────────────────────────────────────────────────────────────┘ KEY: The truncated octahedron (BCC WS cell) as a foam produces a 7.7% photonic band gap. Different Wigner-Seitz geometries → different gaps. Supplementary data: Zenodo 3401635. VII. DIRECT WS CELL → MATERIAL PROPERTY CORRELATION ================================================================================ Journal of Magnetism and Magnetic Materials (1996). "The importance of the Wigner-Seitz cell volume in determining the cobalt site occupancy in Nd₂Fe₁₄B" - Cobalt-57 emission Mössbauer spectroscopy - Co preferential occupancy at 6 Fe crystallographic sites shows VIRTUALLY PERFECT LINEAR CORRELATION with WS cell volume of each site - Metallic radius of Co + WS cell volume controls site substitution - DIRECT MEASURED connection: WS cell geometry → observable property VIII. DFT STUDIES: SOLUTE BEHAVIOR DEPENDS ON VOID GEOMETRY ================================================================================ Ahlawat, Srinivasu & Choudhury (2019), Computational Materials Science 170. - C, N, B, O in BCC-Fe vs FCC-Fe (DFT) - BCC: C, N, O favor octahedral; B favors substitutional - FCC: ALL solute atoms occupy octahedral sites - Vacancy trapping stronger in BCC than FCC - Covalent bond formation for C observed ONLY in BCC (within vacancy clusters) Zeng, Li & Bai (2018), Computational Materials Science 144, 232-247. - Machine learning on 94 binary impurity-host systems (R² = 0.9) - Predicted 554 new diffusion activation energies across 54 metals - Dominant parameter: ELASTIC STRAIN ENERGY (= geometric mismatch between atom and void). Void geometry is the primary predictor. IX. RELEVANCE TO HPC-030 AND CIPHER ================================================================================ The published data consistently shows: 1. Void GEOMETRY (not volume) controls solubility (100x C difference) 2. Void GEOMETRY controls diffusion (10 orders of magnitude H difference) 3. Electrons LOCALIZE at void centers (experimental, 2025) 4. Void SHAPE is 5.8x more powerful than void SIZE for band gap tuning 5. Wigner-Seitz cell geometry correlates linearly with site occupancy 6. Truncated octahedron (BCC) and rhombic dodecahedron (FCC) produce different photonic band gaps in foam crystals HPC-030 asks: do these Wigner-Seitz cells show different EM concentration patterns when treated as resonant cavities? The literature says the geometry matters enormously. No one has directly measured concentration vs angular deficit in Wigner-Seitz cells as EM cavities. If HPC-030 shows BCC = uniform, FCC = selective, HCP = anisotropic, it provides the MECHANISM connecting cipher archetype to material property: the internal cavity resonance of the Wigner-Seitz cell. SOURCES ================================================================================ [1] Jiang & Carter 2003, Phys. Rev. B 67, 214103 [2] Hirata et al. 2018, Metall. Mater. Trans. A 49, 5015 [3] Rousseau & Ashcroft 2008, Phys. Rev. Lett. 101, 046407 [4] Storm et al. 2025, Commun. Mater. [5] Corbett et al. 2021, Nat. Commun. 12, 2543 [6] Kuang et al. 2004, Phys. Lett. A 332 [7] arXiv 2508.03045 (2025) [8] Warmuth et al. 2017, Sci. Rep. 7, 3843 [9] Klatt & Torquato 2019, PNAS 116(47), 23480 [10] J. Magn. Magn. Mater. 1996 (Nd₂Fe₁₄B WS cell volume) [11] Ahlawat et al. 2019, Comput. Mater. Sci. 170, 109167 [12] Zeng et al. 2018, Comput. Mater. Sci. 144, 232-247 X. EM FIELD ENHANCEMENT IN POLYHEDRAL GEOMETRIES (PLASMONICS DATA) ================================================================================ NOTE: No published data exists on truncated octahedron or rhombic dodecahedron as RF/microwave cavity resonators. The closest data comes from plasmonic nanoparticle studies, which measure near-field enhancement. A. Noginov, Barnakov et al. (2019), J. Phys. Chem. C 123, 11833-11839. "Estimating Near Electric Field of Polyhedral Gold Nanoparticles" Geometries: cube, rhombic dodecahedron, pentagonal bipyramid, octahedron, sphere RANKING (near-field |E/E0|): Oct > PentBipyr > Cube > RhomDodec > Sphere Octahedral Au NPs: ~10x higher E-field than sphere of comparable size B. Butet, Bachelier et al. — Gold nanocube near-field mapping Experimental + FDTD measurement |E/E0|² at nanocube corners: 600 ± 140 (measured) Strongest at corners, weakest on faces C. Polyhedral Au@SiO2@Au Core-Shell (2022), J. Phys. Chem. C Tet/Oct cores vs spherical: 100-1000x stronger local E-field Enhancement bandwidth: 100-200 nm (polyhedral) vs 50-60 nm (sphere) D. Rycenga et al. (2011), Inorganic Chemistry 50, 8106-8111. Au rhombic dodecahedra (32 nm edge): SERS detection 10⁻⁸ M Au cubes and octahedra: SERS detection 10⁻⁷ M Rhombic dodecahedron = BEST of three shapes for SERS E. Octahedral tip-to-tip supercrystals (2025), Nature Communications Vertices of octahedral NPs produce enhanced near-field focusing Tip-to-tip configuration optimizes vertex concentration XI. MEIXNER EDGE CONDITION — MATHEMATICAL FRAMEWORK ================================================================================ For a perfectly conducting wedge with exterior angle α: E_perp ~ r^τ where τ = π / α Specific values: ┌──────────────────────────────────────────────────────────────────────┐ │ Exterior Angle α │ Exponent τ │ Physical Geometry │ ├──────────────────────────────────────────────────────────────────────┤ │ 2π (360°, knife edge) │ 0.5 │ Thin screen edge │ │ 3π/2 (270°) │ 2/3 │ 90° exterior corner │ │ π (180°, flat) │ 1.0 │ No singularity │ └──────────────────────────────────────────────────────────────────────┘ For a PEC cone (Van Bladel, 1985, IEEE Trans. AP 33, 893): E ~ R^(ν-1) where ν from P_ν(cos θ₀) = 0 Needle limit: ν → 0 proportionally to 1/ln(θ_max) Costabel, Dauge, Nicaise (1999, 2000): "Singularities of electromagnetic fields in polyhedral domains" ESAIM M2AN 33(3), 627-649; Arch. Rational Mech. Anal. 151(3), 221-276. Edge exponents = 2D interface problem singularity exponents Three types of singularities (types 1, 2, 3) Most rigorous treatment of Maxwell in polyhedral geometry KEY GAP: This mathematical framework has NEVER been applied to a systematic polyhedral cavity sweep. Our HPC-024/026/027 angular deficit sweep is the first systematic test of geometry → EM concentration. XII. CONFIRMED LITERATURE GAPS — UNCHARTED TERRITORY ================================================================================ The following searches returned ZERO relevant results: 1. Truncated octahedron as RF/microwave cavity — NO PAPERS 2. Rhombic dodecahedron as RF/microwave cavity — NO PAPERS 3. Systematic EM mode comparison across Platonic/Archimedean solids — NO PAPERS 4. Wigner-Seitz cell as isolated EM cavity resonator — NO PAPERS 5. Angular deficit as predictor of EM concentration — NO PAPERS 6. Icosahedral or dodecahedral cavity resonator — NO PAPERS The entire field of polyhedral EM cavity resonance (as distinct from nanoparticle plasmonics) appears to be UNPUBLISHED TERRITORY. HPC-030 (Wigner-Seitz cell resonance) and HPC-024/026/027 (angular deficit sweep) are exploring genuinely new ground. XIII. DIMENSIONAL PROGRESSION OF VOID GEOMETRY ================================================================================ VOID FRACTION BY DIMENSION (densest known lattice packings): ┌──────────────────────────────────────────────────────────────────────┐ │ Dim │ Packing dens. │ Void fraction │ Kiss # │ Void types │ Voronoi│ ├──────────────────────────────────────────────────────────────────────┤ │ 1D │ 1.000 │ 0% │ 2 │ 0 │ segment│ │ 2D │ 0.9069 │ 9.3% │ 6 │ 1 │ hexagon│ │ 3D │ 0.7405 │ 26.0% │ 12 │ 2 │ rh.dod.│ │ 4D │ 0.6169 │ 38.3% │ 24 │ 3+ │ 24-cell│ │ 8D │ 0.2537 │ 74.6% │ 240 │ 2+ │ E8 poly│ │ 24D │ 0.00019 │ 99.98% │196,560 │ 307 │ Leech │ └──────────────────────────────────────────────────────────────────────┘ Three measurable trends with dimension: 1. Void fraction INCREASES: 0% → 9.3% → 26% → 38.3% → 74.6% → 99.98% 2. Void TYPE COMPLEXITY increases: 0 → 1 → 2 → 3+ → 307 3. Percolation threshold DECREASES: 0.593 → 0.312 → 0.197 → 0.141 (void networks connect at lower filling fractions) CRITICAL: The 24-cell is the ONLY dimension where the densest lattice Voronoi cell is a regular polytope unique to that dimension. The 3D rhombic dodecahedron is NOT regular. The 2D hexagon IS regular but not unique to 2D. The 24-cell exists ONLY in 4D. Source: Conway & Sloane, Sphere Packings, Lattices and Groups (1999) XIV. THE 24-CELL AS 4D WIGNER-SEITZ CELL ================================================================================ Regular 24-cell {3,4,3}: Vertices: 24, Edges: 96, Faces: 96 (all equilateral triangles) Cells: 24 (all regular octahedra, 3 to an edge) Vertex figure: Cube Dihedral angle: 120° Symmetry: F4 (order 1152) Self-dual: YES (unique among regular convex 4-polytopes) As Voronoi cell (24-cell honeycomb {3,4,3,3}): Each cell shares octahedral face with 24 nearest neighbors Each cell shares single vertex with 24 next-nearest neighbors Total neighbors: 48 GAP: No published crystallographic-style catalog of interstitial void shapes within the 24-cell honeycomb. The mathematical framework exists (Koca et al. 2018, Acta Cryst. A 74(5):499-511) for Delaunay cell construction, but the void-by-void analysis is OPEN. XV. QUASICRYSTAL VOIDS — DIMENSIONAL BRIDGE ================================================================================ Beeli, Godecke & Luck (1998), Phil. Mag. Lett. 78, 339-348. "Highly faceted growth shape of microvoids in icosahedral Al-Mn-Pd" - Microvoids exhibit ICOSAHEDRAL FACETING — pentagonal symmetry - Forbidden in periodic crystals; reflects higher-dimensional parent Positron annihilation studies (1999), Phys. Rev. B 59, 6712. - Every Al-based icosahedral quasicrystal has dense structural vacancies - Vacant cluster centers linked to STABILITY (not defects) Quasicrystal → parent dimension projection: Icosahedral QC = projection from 6D periodic lattice Decagonal QC = projection from 5D Dodecagonal QC = projection from 4D (D4 root lattice connection!) Photonic properties (Man et al. 2005, Nature 436, 993): Icosahedral QC: gap center frequencies closer together, widths more uniform than periodic crystals → more isotropic band gaps. Higher symmetry from parent dimension → better energy trapping. XVI. PERCOLATION THRESHOLDS — VOID NETWORK CONNECTIVITY ================================================================================ Site percolation thresholds (lattice type): ┌──────────────────────────────────────────────────────────────────────┐ │ Lattice │ Coordination │ Site p_c │ Bond p_c │ ├──────────────────────────────────────────────────────────────────────┤ │ 2D honeycomb │ 3 │ 0.697 │ 0.653 │ │ 2D square │ 4 │ 0.593 │ 0.500 (exact) │ │ 2D triangular │ 6 │ 0.500 │ 0.347 │ │ 3D simple cubic │ 6 │ 0.312 │ 0.249 │ │ 3D BCC │ 8 │ 0.246 │ 0.180 │ │ 3D FCC │ 12 │ 0.198 │ 0.120 │ │ 4D hypercubic │ 8 │ 0.197 │ 0.160 │ │ 5D hypercubic │ 10 │ 0.141 │ 0.118 │ └──────────────────────────────────────────────────────────────────────┘ Void percolation (continuum, Rintoul & Torquato 1997): 3D monodisperse overlapping spheres: φ_c = 0.0301 ± 0.0003 Source: Xun et al. 2020, Phys. Rev. Research 2, 013067 ADDITIONAL SOURCES ================================================================================ [13] Noginov et al. 2019, J. Phys. Chem. C 123, 11833 [14] Butet, Bachelier et al. — HAL hal-00777236v1 [15] J. Phys. Chem. C 2022 (polyhedral core-shell) [16] Rycenga et al. 2011, Inorg. Chem. 50, 8106 [17] Nature Communications 2025 (octahedral supercrystals) [18] Costabel, Dauge, Nicaise 1999, ESAIM M2AN 33(3), 627 [19] Costabel, Dauge, Nicaise 2000, Arch. Rat. Mech. Anal. 151(3), 221 [20] Van Bladel 1985, IEEE Trans. AP 33, 893 [21] Conway & Sloane, Sphere Packings, Lattices and Groups (1999) [22] Koca et al. 2018, Acta Cryst. A 74(5), 499 [23] Beeli et al. 1998, Phil. Mag. Lett. 78, 339 [24] Phys. Rev. B 59, 6712 (1999) [25] Man et al. 2005, Nature 436, 993 [26] Xun et al. 2020, Phys. Rev. Research 2, 013067 [27] Rintoul & Torquato 1997 OUTPUT-AGNOSTIC. DATA SHOWS WHAT IT SHOWS. ================================================================================