On this page
A physics-informed approach to interstitial site identification and dopant placement in crystalline materials.
This module provides tools for computationally doping crystal structures with interstitial atoms. It uses Voronoi tessellation to identify void spaces in the crystal lattice, then ranks potential interstitial sites based on geometric, chemical, and energetic criteria.
The implementation is designed for high-throughput screening of doped materials where approximate placement is sufficient, rather than for precise defect formation energy calculations (which would require DFT relaxation).
The core idea is that Voronoi vertices in a crystal structure represent points equidistant from multiple atoms—natural candidates for interstitial sites.
Periodic Boundary Conditions (PBC):
To correctly handle periodicity, we replicate the unit cell into a 3×3×3 supercell before computing the Voronoi tessellation, then filter vertices back to the original cell.
Each Voronoi vertex is characterized by:
Property | Description |
|---|---|
Void radius | Distance to nearest atom minus that atom's effective radius |
Coordination number | Number of atoms within 1.3× the nearest-neighbor distance |
Site geometry | Tetrahedral, octahedral, or general (based on coordination analysis) |
Nearest neighbors | Species and distances of surrounding atoms |
Sites are scored and ranked using a multi-factor quality function:
Where:
Q_size: Preference for void/dopant radius ratio in [1.0, 1.5] (ideal fit)
Q_geometry: Bonus for well-defined tetrahedral (+4) or octahedral (+3.5) sites
Q_coordination: Preference for 4-fold or 6-fold coordination
Q_chemistry: Electronegativity-based compatibility scoring
Q_uniformity: Bonus for uniform neighbor distances (low coefficient of variation)
Q_order: Bonus for high order parameter scores
Dopants are placed iteratively, selecting the best available site at each step while enforcing:
Minimum host-dopant separation: Based on sum of ionic/atomic radii
Minimum dopant-dopant separation: Configurable (default 3.0 Å) to ensure uniform distribution
The module uses a hierarchy of radius sources:
Ionic radius (preferred for interstitial chemistry)
Atomic radius (fallback)
Default 1.0 Å (if neither available)
The true void radius accounts for the size of surrounding atoms:
Where:
d_nearest = distance to nearest host atom
r_host = effective radius of nearest host atom
This ensures we're measuring the actual available space, not just the geometric distance.
The coordination shell is defined as all atoms within 1.3× the nearest-neighbor distance. This tolerance accounts for:
Thermal vibrations
Minor structural distortions
Non-ideal geometries in real materials
We use Pauling electronegativity differences (ΔX) as a proxy for bonding compatibility:
ΔX Range | Interpretation | Score |
|---|---|---|
0.4 – 1.8 | Favorable ionic/covalent bonding | +1.0 |
< 0.4 | Metallic-like (acceptable) | +0.5 |
> 2.5 | Highly ionic (may cause strain) | −0.5 |
Limitation: This is a simplified heuristic. True chemical compatibility depends on oxidation states, coordination preferences, and local electronic structure.
Identified when:
Coordination number = 4
Normalized distance variance < 0.02 (i.e., 4 equidistant neighbors)
In FCC metals, tetrahedral sites are located at (¼, ¼, ¼) and equivalent positions.
Identified when:
Coordination number = 6
Normalized distance variance < 0.02 (i.e., 6 equidistant neighbors)
In FCC metals, octahedral sites are at (½, 0, 0) and equivalent positions.
In BCC metals, octahedral sites are smaller and located at edge midpoints.
All other coordination environments. These may still be valid interstitial positions but lack the symmetry of tetrahedral/octahedral sites.
The minimum distance between a dopant and any host atom is:
Where α is the min_separation_factor (default 0.8).
α = 1.0: No overlap of atomic spheres
α = 0.8: Allows ~20% "overlap" (accounts for bonding, lattice relaxation)
α < 0.8: Increasingly compressed; may require DFT relaxation
A configurable minimum distance (default 3.0 Å) between dopant atoms ensures:
Dopants don't cluster
More uniform distribution in the structure
Avoids dopant-dopant interactions that might destabilize the structure
When placing multiple dopants, we optimize for spatial uniformity by combining:
Site quality score (geometric/chemical preference)
Spatial score (minimum distance to already-placed dopants)
The factor 0.2 balances site quality against uniform distribution.
✅ High-throughput screening: Quickly generates candidate doped structures
✅ Physics-informed placement: Uses geometric and chemical heuristics
✅ Handles periodicity: Proper PBC treatment via supercell replication
✅ Reproducible: Deterministic with fixed random seed
❌ Calculate formation energies: Requires DFT (use VASP, Quantum ESPRESSO, etc.)
❌ Structural relaxation: Positions are unrelaxed; always relax with DFT
❌ Charge state analysis: Assumes neutral dopants
❌ Defect interactions: No treatment of defect-defect coupling beyond separation distance
❌ Finite-size corrections: For dilute defects, use appropriate supercell sizes
Always relax structures with DFT before calculating properties
Validate site types against known interstitial positions for your crystal structure
Use appropriate supercell sizes (typically 3×3×3 or larger for dilute defects)
Check convergence of properties with respect to supercell size
Consider symmetry-inequivalent sites separately for defect formation energy calculations
pymatgen: Crystal structure handling, atomic data
scipy: Voronoi tessellation, KDTree for spatial queries
numpy: Numerical operations
Class | Purpose |
|---|---|
| Voronoi-based void detection and characterization |
| Data container for site properties |
| Validates separation constraints with PBC |
KDTree used for O(log n) nearest-neighbor queries
Spatial hashing for O(1) duplicate site detection
max_voronoi_sites parameter limits analysis for large cells
Unit cell analysis → supercell replication (more efficient than analyzing full supercell)
Corrected void radius calculation (subtracts host atom radius)
Added chemical compatibility scoring via electronegativity
Improved geometry classification with coordination shell analysis
Made random seed configurable for reproducibility
Initial implementation with basic Voronoi analysis
Simple distance-based site filtering
Interstitial Doping is a tool that helps place extra atoms inside crystal structures. It uses a physics-informed approach to find likely interstitial sites with Voronoi tessellation, and then ranks these sites by how well they fit the dopant atom and how favorable the surrounding chemistry is. The method works in periodic crystals by expanding the cell into a small supercell, performing the analysis, and then mapping the results back to the original structure. It characterizes each potential site by void size, coordination, geometry, and nearby atoms, and it scores them to guide dopant placement. Dopants are added one by one while maintaining minimum distances to hosts and to other dopants. This is designed for fast, high‑throughput screening and does not perform energy calculations or structural relaxations; users should relax all structures with DFT afterward.