Algorithms

Mathematical details of tricor’s structure-generation pipeline, ordered to match the natural flow of a build: measure the reference crystal, lay down a Voronoi-tiled initial cell, optionally walk grain orientations into a better-aligned basin, FIRE-quench with the spring network, and (optionally) synthesise a target g3 distribution against which the supercell can be compared.

• Three-Body Distribution (g3)
  • Definition
  • Reduced coordinates
  • Coordination shell target
• Supercell generation
  • Voronoi grain construction
• Orientation refinement
  • Why orient grains before FIRE
  • Algorithm
  • Pair-distance score
  • Multi-species grains
  • What gets tuned
  • Cost of the search vs the FIRE that follows
  • Where to read the code
• Static relaxation
  • Energy
  • Force terms
  • Bond topology
  • Integration (FIRE-style)
  • Per-atom cost (trajectory colour scale)
  • See also
• Large-cell shortcut
  • When the shortcut beats FIRE
  • bond_relax: attract-to-peak + repel-from-hard-core
  • enforce_hard_core: pure projection
  • Pseudocode
  • Where to read the code
  • See also
• Target g3 Construction
  • Construction pipeline
  • Parameters
  • Usage
  • The g2 pair distribution
• Glossary
  • Disorder regimes
  • Three-body distribution
  • Coordination shell target
  • Grain construction
  • Shell relaxation
  • Orientation refinement
  • Composite shell targets

Stage	Page	What it covers
1. Measure	Three-body distribution	g3 + g2 histogram, reduced coordinates, CoordinationShellTarget extraction.
2. Build	Supercell generation	Voronoi grain construction: seeds → tessellation → master block tiling → cell filling → overlap removal → close-pair push → optional thermal jitter.
3. Align grains	Orientation refinement	Per-grain SO(3) coordinate search that picks rotation perturbations minimising a topology-free pair-distance score against the local environment, before the FIRE quench runs.
4. Refine springs	Static relaxation	FIRE-style gradient descent on the bond + angle + repulsion (+ restraint) springs. This is what Supercell.generate() calls internally.
4’. Large-cell shortcut	Large-cell shortcut	cKDTree bond_relax + enforce_hard_core sweeps that replace most of FIRE at 100³ Å and larger. O(N log N) per iteration.
5. Compare	Target g3 construction	Blur + blend a measured crystalline g3 toward the random limit to produce a comparison target.
Reference	Glossary	Disorder regimes, shell-target fields, force-term nomenclature.