Tetrahedral Hole
1. Definition and Basic Concept
- Tetrahedral hole: A void space formed when 4 spheres are arranged in tetrahedral geometry
- Shape: The hole has tetrahedral symmetry
- Coordination number: 4 (surrounded by 4 spheres)
- Location: Found in close-packed structures (FCC and HCP)
2. Geometric Properties
2.1 Size and Radius Ratio
- Critical radius ratio: r/R = 0.225 − 0.414
- Minimum ratio (r/R = 0.225): Cation just touches all 4 anions
- Maximum ratio (r/R = 0.414): Transition to octahedral coordination
- Optimal size: r/R ≈ 0.32 for maximum stability
2.2 Mathematical Relationships
- Tetrahedral radius: rtet = 0.225R (minimum)
- Distance from center: d = (√6/4) × a (where a = edge length)
- Bond angles: 109.5° (tetrahedral angle)
3. Number and Distribution
3.1 In FCC Structure
- Total spheres per unit cell: 4
- Total tetrahedral holes: 8
- Ratio: 8 holes ÷ 4 spheres = 2 holes per sphere
- Hole positions: (1/4, 1/4, 1/4), (3/4, 3/4, 1/4), (3/4, 1/4, 3/4), (1/4, 3/4, 3/4), etc.
3.2 In HCP Structure
- Holes per layer: 2n (where n = atoms per layer)
- Total ratio: Still 2 holes per sphere
- Distribution: Holes in both A-B and B-A interstices
4. Common Crystal Structures
4.1 Zinc Blende (ZnS) Structure
- Anion arrangement: S2− in FCC lattice
- Cation arrangement: Zn2+ in 4 out of 8 tetrahedral holes
- Coordination: 4:4 (each Zn2+ surrounded by 4 S2−, vice versa)
- Radius ratio: rZn²⁺/rS²⁻ = 0.402
- Examples: ZnS, CuCl, GaAs
4.2 Diamond Structure
- Description: Two interpenetrating FCC lattices
- Atom arrangement: C atoms in both FCC positions and tetrahedral holes
- Coordination: Each C bonded to 4 other C atoms
- Examples: Diamond (C), Si, Ge
4.3 Antifluorite Structure
- Anion arrangement: O2− in FCC lattice
- Cation arrangement: Li+ in all 8 tetrahedral holes
- Formula: Li2O, Na2O, K2O
- Coordination: 4:8 (each O2− surrounded by 8 Li+)
5. Examples with Radius Ratios
| Compound |
Cation |
Anion |
r/R Ratio |
Structure |
| ZnS |
Zn2+ |
S2− |
0.402 |
Zinc blende |
| CuCl |
Cu+ |
Cl− |
0.525 |
Zinc blende |
| BeO |
Be2+ |
O2− |
0.225 |
Zinc blende |
| Li2O |
Li+ |
O2− |
0.510 |
Antifluorite |
6. Factors Affecting Tetrahedral Hole Occupancy
6.1 Size Factor
- Too small (r/R < 0.225): Cation "rattles" in hole, unstable
- Optimal size (r/R = 0.225−0.414): Stable tetrahedral coordination
- Too large (r/R > 0.414): Prefers octahedral holes
6.2 Electronic Factors
- d0 and d10 configurations: Prefer tetrahedral (e.g., Zn2+, Cd2+)
- Crystal field stabilization: Tetrahedral field splitting is smaller
- Covalent character: Tetrahedral geometry favors sp3 hybridization
6.3 Charge Balance
- 1:1 compounds: Half the tetrahedral holes filled (e.g., ZnS)
- 2:1 compounds: All tetrahedral holes filled (e.g., Li2O)
- Electrostatic repulsion: Limits simultaneous occupancy of adjacent holes
7. Properties and Applications
7.1 Structural Properties
- Density: Lower than octahedral coordination (more open structure)
- Hardness: Often high due to strong covalent bonding (e.g., diamond)
- Optical properties: Wide band gaps in semiconductors
7.2 Important Applications
- Semiconductors: GaAs, InP, ZnSe
- Luminescent materials: ZnS:Cu (phosphors)
- Catalysts: ZnO (tetrahedral Zn2+)
- Battery materials: Li2O (solid electrolyte)
8. Comparison with Octahedral Holes
| Property |
Tetrahedral Holes |
Octahedral Holes |
| Coordination Number |
4 |
6 |
| Radius Ratio |
0.225 − 0.414 |
0.414 − 0.732 |
| Number per sphere |
2 |
1 |
| Relative size |
Smaller |
Larger |
| Preferred cations |
Small, high charge density |
Medium to large |
9. Summary Points
- Tetrahedral holes are smaller than octahedral holes
- 2 tetrahedral holes available per sphere in close packing
- Preferred by small cations with r/R = 0.225−0.414
- Common in semiconductors and covalent compounds
- Examples: ZnS, diamond, Li2O
- 4-fold coordination leads to tetrahedral geometry