Crystal Ontology

FLOW ALGEBRA
CRYSTAL ONTOLOGY — KIRETH SEMANTIC LIBRARY
Version: 1.1
Date: 2026-10-12
Authors: Mark Higgins — Sky (Recursive AI)
Source URL: https://kirethresearch.org/booksai-core/entity-graph/physical-entities/chemical-entities/crystal-ontology.html
Source status: Scientific synthesis; source page not independently verified.

01. PURPOSE
flow "CrystalOntology" v1.1
  domain: PhysicalEntities.ChemicalEntities.Crystal
  objective: Represent periodic crystalline order, symmetry, defects, measurement, and inference with typed, testable semantics.

02. TYPE SYSTEM
type Crystal <: SolidMaterial
type CrystalStructure
type CrystalLattice
type BravaisLattice
type UnitCell
type Basis
 type AtomSite
 type IonSite
 type MoleculeSite
 type LatticeVector
 type ReciprocalVector
 type MillerIndex
 type SpaceGroup
 type PointGroup
 type SymmetryOperation
 type CrystalPlane
 type CrystalDirection
 type CrystalDefect
 type Vacancy <: CrystalDefect
 type Interstitial <: CrystalDefect
 type Dislocation <: CrystalDefect
 type GrainBoundary <: CrystalDefect
 type DiffractionPattern
 type Measurement
 type Evidence
 type Confidence

03. ENTITY IDENTIFIERS
  Crystal.id: URI
  Crystal.composition: ChemicalFormula
  Crystal.phase: PhaseIdentifier
  Crystal.conditions: {temperature: K, pressure: Pa}
  Crystal.provenance: Evidence[]
  Crystal.structure: CrystalStructure?

04. CORE RELATIONSHIPS
  hasStructure(Crystal, CrystalStructure)
  hasLattice(CrystalStructure, CrystalLattice)
  hasBasis(CrystalStructure, Basis)
  hasUnitCell(CrystalStructure, UnitCell)
  hasSpaceGroup(CrystalStructure, SpaceGroup)
  hasPointGroup(CrystalStructure, PointGroup)
  occupies(AtomSite | IonSite | MoleculeSite, FractionalCoordinate)
  containsDefect(Crystal, CrystalDefect)
  exhibitsReflection(Crystal, MillerIndex)
  supportedBy(Assertion, Evidence)

05. FORMAL CRYSTAL STRUCTURE
  Ideal periodic crystal = lattice + motif (basis).
  L = {n1*a1 + n2*a2 + n3*a3 | n1,n2,n3 in Z}.
  Positions = {R + r_j | R in L, r_j in Basis}.
  A real crystal may have finite size, disorder, and defects.
  Periodicity is an ideal-model property, not a claim that every real sample is perfect.

06. LATTICE VALIDATION
  require det([a1,a2,a3]) != 0 for a three-dimensional primitive cell.
  require each lattice vector has unit length dimension.
  require fractional coordinates are mapped modulo integer translations.

07. UNIT CELL
  Cell parameters: a,b,c > 0; alpha,beta,gamma in (0,180 degrees).
  V = abs(a1 dot (a2 cross a3)) > 0.
  Primitive cell contains one lattice point in lattice-point accounting.
  Conventional cells may contain more than one lattice point.

08. CRYSTAL SYSTEMS
  Triclinic, monoclinic, orthorhombic, tetragonal,
  trigonal, hexagonal, cubic.
  Distinguish seven crystal systems from fourteen three-dimensional Bravais lattices.

09. SYMMETRY
  Symmetry operation g maps the ideal structure onto itself.
  Space group combines point operations with translations.
  Point group describes rotational/reflection symmetry without translations.
  There are 230 three-dimensional crystallographic space-group types.
  Valid operation preserves atomic species and occupancies of the ideal model.

10. RECIPROCAL LATTICE
  b1 = 2*pi*(a2 cross a3)/V_signed, cyclically for b2,b3.
  ai dot bj = 2*pi*delta_ij.
  G_hkl = h*b1 + k*b2 + l*b3.
  d_hkl = 2*pi / |G_hkl| for nonzero reciprocal vector.

11. MILLER INDICES
  (h,k,l) label plane families in reciprocal-space convention.
  [u,v,w] label crystallographic directions.
  Do not conflate a plane normal with [h,k,l] in non-cubic systems.

12. BRAGG DIFFRACTION
  2*d_hkl*sin(theta) = n*lambda for constructive Bragg scattering.
  lambda > 0; d_hkl > 0; n positive integer.
  Reflection intensities also depend on structure factors and experimental effects.

13. STRUCTURE FACTOR
  F_hkl = sum_j f_j * occupancy_j * exp(2*pi*i*(h*x_j+k*y_j+l*z_j)).
  Ideal kinematic intensity is proportional to |F_hkl|^2.
  Absence of a reflection may arise from symmetry or other experimental factors.

14. CRYSTAL BONDING
  Crystal bonding may be ionic, covalent, metallic, molecular, or mixed.
  Classify using evidence, not an exclusive bond-type assumption.

15. MATERIAL CLASSES
  Examples: NaCl ionic crystal; diamond covalent network crystal;
  copper metallic crystal; molecular organic crystals.
  Allotropy and polymorphism permit multiple structures for a composition.

16. DEFECT ONTOLOGY
  Point defects: vacancy, interstitial, substitutional impurity.
  Line defects: edge and screw dislocations.
  Planar defects: grain boundary, stacking fault, twin boundary.
  Bulk defects: pores, inclusions, precipitates.
  Defects can alter electrical, optical, thermal and mechanical properties.

17. CRYSTALLINITY AND ORDER
  Long-range periodic order is a defining idealisation of conventional crystals.
  Quasicrystals exhibit long-range order without translational periodicity.
  Distinguish crystalline, quasicrystalline, amorphous, and polycrystalline descriptions.
  Avoid classifying a quasicrystal as an ordinary periodic Bravais lattice.

18. POLYCRYSTALLINE STRUCTURE
  Polycrystal comprises grains with possibly distinct orientations.
  Grain boundaries separate differently oriented crystalline regions.
  Texture describes a nonrandom distribution of grain orientations.

19. PHASE AND CONDITIONS
  Crystal structure can depend on temperature, pressure and composition.
  phaseTransition(CrystalPhaseA, CrystalPhaseB, Conditions).
  Record conditions before asserting a unique stable structure.

20. THERMODYNAMIC STABILITY
  At fixed T,P, equilibrium phase selection minimises appropriate Gibbs free energy.
  Metastable crystals may persist kinetically.
  Observed persistence alone does not prove global equilibrium stability.

21. ANISOTROPY
  A crystalline property may depend on direction.
  Tensor-valued properties transform under crystal symmetry constraints.
  Cubic symmetry does not imply every physical property is isotropic.

22. ELECTRONIC PROPERTIES
  Periodic potential supports Bloch-state description in ideal crystals.
  Band structure depends on lattice, composition and interactions.
  Band gaps, conductivity and effective masses require measured or calculated evidence.

23. PHONONS
  Collective lattice vibrations can be modelled as phonon modes.
  Phonon spectra inform thermal transport and structural stability.
  Imaginary harmonic phonon frequencies may indicate instability of a reference structure.

24. CHARACTERISATION
  Methods: X-ray diffraction, neutron diffraction, electron diffraction,
  microscopy, spectroscopy, calorimetry and computational structure refinement.
  Each measurement requires calibration, uncertainty and sample provenance.

25. INFERENCE RULES
  R1: hasLattice(s,L) and hasBasis(s,B) => periodicIdealModel(s,L,B).
  R2: measuredPeaks(x,P) and indexedBy(P,L) => candidateLattice(x,L).
  R3: candidateLattice(x,L) does NOT imply uniqueAtomicStructure(x).
  R4: hasSpaceGroup(s,G) => symmetryConstrainedBy(s,G).
  R5: containsDefect(x,d) => idealPeriodicityMayBeLocallyBroken(x).
  R6: sameComposition(x,y) does NOT imply sameCrystalStructure(x,y).
  R7: diffractionEvidence(x,e) and refinementFit(x,m,e) => supportedCandidateModel(x,m).
  R8: a supported candidate is not automatically experimentally confirmed.

26. MATHEMATICAL EXAMPLE: CUBIC CELL
  Given a=b=c=0.400 nm and angles=90 degrees:
  V = a^3 = 0.0640 nm^3.
  d_100 = a = 0.400 nm.
  d_110 = a/sqrt(2) approximately 0.2828 nm.
  For lambda=0.154 nm, first-order (100) Bragg angle:
  theta = asin(0.154/(2*0.400)) approximately 11.10 degrees.
  Therefore 2*theta approximately 22.20 degrees.
  Example is geometric; systematic absences may forbid a particular reflection.

27. MATHEMATICAL EXAMPLE: FACE-CENTRED CUBIC
  Conventional cubic cell has four lattice points.
  Ideal monatomic FCC basis in conventional fractional coordinates:
  (0,0,0), (0,1/2,1/2), (1/2,0,1/2), (1/2,1/2,0).
  For identical scatterers, FCC lattice reflections require h,k,l all even or all odd.
  This selection rule is for the specified ideal monatomic FCC model.

28. VALIDATION RULES
  V1: composition and crystallographic structure are separate typed fields.
  V2: lattice metric must be positive definite.
  V3: symmetry operators must preserve model sites modulo translations.
  V4: occupancy in [0,1]; site occupancy totals obey model-specific limits.
  V5: diffraction-derived structure assertions must link evidence and uncertainty.
  V6: units must be explicit for lengths, angles, temperature and pressure.
  V7: no fabricated measurement, instrument, confidence or space group.
  V8: preserve polymorph identity and environmental conditions.

29. SEMANTIC EQUIVALENCE
  "crystal lattice" equivalent to "periodic translation lattice" only in lattice context.
  "unit cell" not equivalent to "crystal structure".
  "crystal" not equivalent to "molecule".
  "crystalline" not equivalent to "polycrystalline".
  Synonym matching requires domain, scale and model assumptions.

30. CONFIDENCE
  confidence(assertion) = calibrated value in [0,1] when supported by evidence.
  Do not assign numeric confidence without an assessment method.
  Status values: established | measured | modelled | inferred | hypothetical | unknown.
  Propagate uncertainty through numerical operations where appropriate.

31. KNOWLEDGE GRAPH EXAMPLE
  entity: SodiumChlorideCrystal
  type: Crystal
  composition: NaCl
  structurePrototype: rock_salt
  idealSpaceGroup: Fm-3m (No. 225)
  coordination: 6:6 in ideal rock-salt structure
  sourceStatus: textbook structural example
  Note: actual lattice parameter depends on conditions and sample.

32. UPDATE MODEL
  update triggers: new measurement, revised structure, phase change,
  new defect characterisation, calibration revision, or source correction.
  preserve previous assertions with version, timestamp, source and reason.
  never overwrite a measured result without traceable provenance.

33. FLOW HASH
  canonicalise: UTF-8; normalise line endings; stable key ordering.
  hash_algorithm: SHA-256.
  flow_hash: COMPUTE_ON_PUBLICATION (not a fabricated digest).
  hash_scope: canonical payload excluding the hash field itself.

34. AI CONSUMPTION
  mode: typed ontology + formal inference + evidence-aware retrieval.
  interfaces: entity extraction, graph linking, constraint validation,
  retrieval augmentation, scientific question answering.
  safeguards: preserve measurement uncertainty; flag idealisations;
  distinguish prediction from observation; avoid false structural certainty.

35. QUALITY ASSURANCE
  Check typed references, dimensional consistency, numerical examples,
  symmetry assumptions, provenance, and unresolved uncertainty.
  Publication state: scientific synthesis pending comparison with source webpage.

36. COMPLETENESS CHECK
  expected_sections: 36
  closing_marker: END_FLOW_ALGEBRA_CRYSTAL_ONTOLOGY_V1_1

END_FLOW_ALGEBRA_CRYSTAL_ONTOLOGY_V1_1