From phase diagram to tensor component — the seven mechanistic layers turn why? into a SPARQL query
A materials ontology that records that BNT-BT reaches d33 ≈ 580 pC/N at its morphotropic phase boundary stores a fact. OCO additionally stores the explanation: Bi-6s² lone-pair stereo-activity → anomalous Born charges → soft mode at the Brillouin-zone centre → R3c↔P4mm domain coexistence → V_O^•• / Mn_Ti” defect-dipole chemistry → Hall-Petch grain-boundary mediation. The seven mechanistic layers make this chain traversable — as a SPARQL query, not a literature search.
What you get
- Seven explanation layers as an orthogonal axis: symmetry · energy/DFT · thermo/CALPHAD · kinetics · microstructure · defect chemistry · bonding — generic skeleton for all crystalline ionic oxides, concretised on the BNT-BT pilot.
- Neumann tensor engine algorithmically generates 5,920 reified constraints across all 32 crystallographic point groups — validated against Nye (1985) Tab. 9 + IEEE Standard 176.
- Phase-state coupling via SPARQL: the active point group of a sample is derived from (T, x) — a BNT-BT sample “knows” in the database whether it is currently ferroelectric.
- External caches keep the TBox compact: ~155,000 Materials Project DFT entries, 1,934 IUCr bond-valence parameters, 1,731 Wyckoff positions, 497 Shannon ionic radii, 91 Pauling electronegativities — version-pinned, without inflating the OWL hierarchy.
The seven mechanistic layers
Each layer answers one fundamental materials-science question. They apply generically to all crystalline ionic oxides — the central material class for functional ceramics (ferroelectrics, magnetics, ion conductors, high-T_c superconductors). BNT-BT is the pilot on which the skeleton was concretised, but not the reason it has seven layers.
| # | Layer | Question answered | OCO module · external caches |
|---|---|---|---|
| 1 | Symmetry | Which crystal structure, which symmetry operations? Point-group theory (Neumann’s principle) governs which tensor components are allowed; subgroup relations drive phase transitions; Wyckoff positions give dopant and defect sites. | oco-symmetry (11 cl) on oco-crystal + oco-tensor · pyxtal Wyckoff (1,731), IUCr BVP (1,934) |
| 2 | Energy / DFT | Which electronic structure, which phonon modes, which Born charges? First-principles ground for intrinsic properties without empirical fitting; soft-mode analysis identifies phase-transition mechanisms. | oco-energy-dft (33 cl, 7 prop) · Materials Project (~155,000) |
| 3 | Thermo / CALPHAD | Which phases are stable at (T, p, x)? Sublattice models and Redlich-Kister polynomials govern Gibbs-energy surfaces; sintering windows, miscibility gaps, and MPB locations follow. | oco-thermo (25 cl: MorphotropicPhaseBoundary, ChemicalPotentialDiagram) atop oco-phase |
| 4 | Kinetics | How fast do diffusion, reaction, sintering, switching, aging proceed? Fick, Arrhenius, JMAK, Hillert grain growth, Cahn-Hilliard spinodal decomposition, domain-wall mobility. | oco-kinetics (kinetic models as reified-constraint pattern) |
| 5 | Microstructure | Which grain size, texture, grain-boundary character? Polycrystalline averaging via Voigt-Reuss-Hill, Mori-Tanaka, Hashin-Shtrikman bounds; Hall-Petch coupling to mechanical response. | oco-microstructure (grain statistics, averaging schemes, texture components) |
| 6 | Defect chemistry | Which point defects, in which concentrations, with which charge compensation? Kröger-Vink notation, Brouwer diagrams, donor / acceptor / iso-valent / amphoteric doping. | oco-defect (Kröger-Vink + Brouwer) |
| 7 | Bonding chemistry | Which electron configuration, which hybridisation, which bonding character (ionic / covalent / metallic)? Pauling rules, HSAB classification, crystal-field theory, lone-pair stereo-activity. | oco-bonding (Shannon radii 497, Pauling electronegativities 91) |
The explanation chain is auditable: seven SHACL NodeShapes verify that cross-layer annotations are complete — a morphotropic phase boundary (layer 3) must reference a domain-state set (layer 1); a soft mode (layer 2) must reference a subgroup relation (layer 1); a Brouwer diagram (layer 6) must reference a chemical-potential diagram (layer 3); and so on. Deliberately sh:Warning severity, because cross-layer annotation grows evolutionarily but should attract modeller attention.
End-to-end example: BNT-BT at the morphotropic phase boundary
The BNT-BT pilot ABox carries concrete instances on each of the seven layers with explicit cross-layer annotations. A sample query “why does BNT-BT achieve d33 ≈ 580 pC/N at the MPB?” traverses the following chain:
| Layer | Concrete anchor in the BNT-BT pilot |
|---|---|
| 7 — Bonding | Bi-6s² lone-pair stereo-activity, documented via Pauling electronegativity differences |
| 2 — Energy/DFT | Born effective charge Z*33(Ti) ≈ +7.2, derived from PBEsol-PAW calculation (8×8×8 Monkhorst-Pack, 600 eV cutoff) — causally linked to the lone-pair excitation from layer 7 |
| 1 — Symmetry | Soft mode at the Γ point, linked to the R3c→P4mm subgroup relation |
| 3 — Thermo/CALPHAD | Morphotropic phase boundary at xBT ≈ 0.06, linked to the R3c/P4mm/Cm domain-state set |
| 6 — Defect chemistry | V_O^•• / Mn_Ti” defect-dipole chemistry that influences switching behaviour (Mn-acceptor pinning of domain-wall mobility) |
| 4 — Kinetics | Aging-kinetics model linked to the defect-dipole chemistry from layer 6 |
| 5 — Microstructure | Hall-Petch parameterisation σ0 ≈ 3 GPa, kHP ≈ 0.8 MPa√m at grain size d = 2 µm — linked to grain-boundary classification from layer 6 |
This is not “list of mechanisms” as documentation, but a reasoning chain with evaluable SPARQL links. A user asking the question receives the provenance annotation along with it (PBEsol functional, pseudopotential, mesh, cutoff) — reproducible layer by layer.
Neumann engine and phase-state coupling
Two specific reasoning components elevate OCO architecturally above a pure vocabulary collection:
- Neumann tensor engine: 5,920 reified
oco:NeumannConstraintinstances, one per (tensor class × point group) pair. Algorithmically generated across all 32 crystallographic point groups × ~185 relevant tensor classes. validated against Nye Tab. 9 + IEEE 176. Answers “which tensor components are allowed in point group X?” without a lookup table. - Phase-state coupling: Sample → Region → point-group inference via SPARQL — the temperature- and composition-dependent active phase state is derived from the phase-diagram vocabulary. So far populated for BNT-BT and NiCuZn ferrite. Answers “which point group applies to this sample at the current temperature?”.
External caches — depth without TBox inflation
Reference-data corpora that would either make a flat ontology’s TBox unworkably large or stay outside it entirely are connected to OCO as version-pinned local caches. The manifest bridge/external_versions.yaml records the upstream SHA per cache.
| Cache | Volume | Source | Used by layer |
|---|---|---|---|
| Pyxtal Wyckoff positions | 1,731 positions across 230 space groups | pyxtal library | 1 — Symmetry |
| IUCr bond-valence parameters | 1,934 parameters (Brown 2020) | IUCr | 1 — Symmetry / 7 — Bonding |
| Materials Project DFT corpus | ~155,000 computed entries | Materials Project (LBNL) | 2 — Energy/DFT |
| Shannon ionic radii | 497 values with quality flags | Shannon (1976) compilation | 7 — Bonding |
| Pauling electronegativities | 91 values | Pauling scale | 7 — Bonding |
Three places where the materials depth becomes most visible
Coupled-Effect Family
32 classesPiezoelectricity, pyroelectricity, electrostriction, magnetostriction, magnetoelectric, thermoelectric, Verdet, Cotton-Mouton effects — plus 24 diagonal and cross effects after Nye 1985.
170 cross-axioms link each effect to the crystallographic point group it requires. Reasoning over “can this material exhibit pyroelectricity?” becomes a SPARQL query.
Kröger-Vink Defect Chemistry
71 classesComplete defect notation for ionic solids (V_O••, Mg_Si”). Brouwer-diagram modelling of defect equilibria.
Material-specific: for metallurgy this module would be replaced with Burgers-vector dislocation notation — that is exactly the logic of L2 swappability.
Newnham Connectivity
12 classesComposite topology (0-0, 0-3, 1-3, 2-2, 3-3, …) for two-phase functional ceramic composites, after Newnham et al. 1978.
Small but architecturally critical: connectivity notation is what makes reasoning over piezoelectric composites possible at all.
Extension to further ceramic families
The seven-layer skeleton is designed as generic for crystalline ionic oxides. A documented framework describes how to extend the skeleton to further ceramic families without changing the layer count.
| Family | Examples | Layer changes | Effort |
|---|---|---|---|
| Further ionic oxides | TiO₂, ZrO₂, PZT, BaTiO₃ | L3 vocabulary only | small (1-2 days) |
| Other ionic anion classes | Si₃N₄, AlN, CaF₂, ZnS, HAP | layer 6 L2 anion variant | small-medium (2-4 days) |
| Covalent ceramics | SiC, BN, B₄C, diamond | layer 6 + 7 parallel hierarchies (bond defects, sp³ hybridisation) | medium (1 week) |
| Metallic ceramics | TiB₂, ZrB₂, HfB₂, WC, MoSi₂ | layer 2/6/7 parallel (Fermi surface, dislocations, MetallicBond) | medium (1 week) |
| Composites (CMC) | SiC/SiC, C/C, Al₂O₃/ZrO₂, FGM | layer 5 dominant (fibre/matrix interface as first-class) | medium (1 week) |
| Layered / 2D | MAX phases, MXenes | layer 1/5/7 parallel (2D space groups, nano-laminates, vdW) | medium (1 week) |
| Glass / amorphous | silicate, phosphate, chalcogenide glass, ZBLAN | layer 1 sister skeleton (short-range order), layer 4 GlassTransition, layer 7 NetworkFormer/Modifier | large (~3 weeks) |
Example competency questions
Eight of the 163 published CQs that are particularly relevant for materials and ceramics researchers.
Which tensor components are allowed in point group 6mm for the piezoelectric tensor?
oco-symmetry · oco-tensor · executable SPARQLWhich phases are stable at (T = 300 K, x_BT = 0.06) in the BNT-BT system?
oco-thermo · oco-phase · executable SPARQLWhich Born charge is anomalously enhanced by Bi-6s² lone-pair stereo-activity?
oco-energy-dft · oco-bondingWhich defect-dipole mechanisms explain aging in BNT-BT actuators?
oco-defect · oco-kineticsWhich Hall-Petch parameters (σ0, kHP) fit a grain size of 2 µm?
oco-microstructureWhich Wyckoff position carries the Mn acceptor in the R3c phase?
oco-symmetry · oco-defect · pyxtal cacheWhich sintering route leads to morphological texturing in the polycrystal?
oco-process · oco-microstructureWhat bond-valence value is expected for TiO₆ polyhedra in the P4mm phase?
oco-localstructure · oco-bonding · IUCr BVP cacheRelation to the OCO distribution
The material / L2 / L3 / SHACL content (all layers 1–7, Neumann engine, phase-state coupling, BNT-BT pilot ABox, extension framework) sits in the proprietary tier of the OCO distribution. The L0 bridges to PMDco, QUDT, EMMO-Crystallography, CIF Core, IUCr, pyxtal, Materials Project are freely available under CC-BY 4.0. Tensor roots, role individuals, cross-axioms and the pilot ABox are published under CC-BY-SA 4.0. Full integrated materials stack via the bundle “OCO Full Stack — with Reasoning (+L3)” or a dedicated materials profile on request. → Distribution & licence architecture