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Topic: Cubic Hole animation

Octahedral Holes

1. Definition and Basic Concept

  • Octahedral hole: A void space formed when 6 spheres are arranged in octahedral geometry
  • Shape: The hole has octahedral symmetry (square bipyramid)
  • Coordination number: 6 (surrounded by 6 spheres)
  • Location: Found in close-packed structures (FCC and HCP)
  • Alternative name: Six-coordinate holes

2. Geometric Properties

2.1 Size and Radius Ratio

  • Critical radius ratio: r/R = 0.414 − 0.732
  • Minimum ratio (r/R = 0.414): Cation just touches all 6 anions
  • Maximum ratio (r/R = 0.732): Transition to cubic coordination
  • Optimal size: r/R ≈ 0.57 for maximum stability

2.2 Mathematical Relationships

  • Octahedral radius: roct = 0.414R (minimum)
  • Distance from center to vertex: d = a/√2 (where a = edge length)
  • Bond angles: 90° (octahedral geometry)
  • Hole size comparison: roct/rtet = 0.414/0.225 = 1.84

3. Number and Distribution

3.1 In FCC Structure

  • Total spheres per unit cell: 4
  • Total octahedral holes: 4
  • Ratio: 4 holes ÷ 4 spheres = 1 hole per sphere
  • Hole positions:
    • Body center: (1/2, 1/2, 1/2)
    • Edge centers: (1/2, 0, 1/2), (0, 1/2, 1/2), (1/2, 1/2, 0)

3.2 In HCP Structure

  • Holes per layer: n (where n = atoms per layer)
  • Total ratio: Still 1 hole per sphere
  • Distribution: Holes between alternate layers (A-B-A or A-C-A)

4. Common Crystal Structures

4.1 Rock Salt (NaCl) Structure

  • Anion arrangement: Cl− in FCC lattice
  • Cation arrangement: Na+ in all 4 octahedral holes
  • Coordination: 6:6 (each Na+ surrounded by 6 Cl−, vice versa)
  • Radius ratio: rNa+/rCl− = 0.525
  • Examples: NaCl, MgO, CaO, FeO

4.2 Fluorite (CaF2) Structure

  • Cation arrangement: Ca2+ in FCC lattice
  • Anion arrangement: F− in all 8 tetrahedral holes
  • Note: This is NOT octahedral coordination
  • Formula: CaF2, SrF2, BaF2
  • Coordination: 8:4 (each Ca2+ surrounded by 8 F−)

4.3 Rutile (TiO2) Structure

  • Description: Distorted octahedral coordination
  • Cation arrangement: Ti4+ in octahedral holes
  • Anion arrangement: O2− in distorted close packing
  • Coordination: 6:3 (each Ti4+ surrounded by 6 O2−)
  • Examples: TiO2, SnO2, MnO2

4.4 Corundum (Al2O3) Structure

  • Anion arrangement: O2− in HCP lattice
  • Cation arrangement: Al3+ in 2/3 of octahedral holes
  • Coordination: 6:4 (each Al3+ surrounded by 6 O2−)
  • Examples: Al2O3, Cr2O3, Fe2O3

5. Examples with Radius Ratios

Compound Cation Anion r/R Ratio Structure Coordination
NaCl Na+ Cl− 0.525 Rock salt 6:6
MgO Mg2+ O2− 0.47 Rock salt 6:6
CaO Ca2+ O2− 0.71 Rock salt 6:6
FeO Fe2+ O2− 0.51 Rock salt 6:6
TiO2 Ti4+ O2− 0.42 Rutile 6:3

6. Factors Affecting Octahedral Hole Occupancy

6.1 Size Factor

  • Too small (r/R < 0.414): Prefers tetrahedral coordination
  • Optimal size (r/R = 0.414−0.732): Stable octahedral coordination
  • Too large (r/R > 0.732): Prefers cubic coordination
  • Perfect fit: When cation just touches all 6 anions

6.2 Electronic Factors

  • Crystal field stabilization energy (CFSE):
    • d4, d5, d6, d7 configurations favor octahedral
    • High spin vs low spin complexes
  • Ligand field effects: Strong field ligands favor octahedral
  • Jahn-Teller distortion: d4 and d9 may distort octahedral geometry

6.3 Charge and Electronegativity

  • Higher charges: Favor higher coordination numbers
  • Ionic character: Pure ionic compounds prefer octahedral
  • Polarization effects: May cause distortion from ideal octahedral

7. Crystal Field Theory in Octahedral Holes

7.1 d-Orbital Splitting

  • eg orbitals: dx2−y2, dz2 (higher energy)
  • t2g orbitals: dxy, dxz, dyz (lower energy)
  • Crystal field splitting: Δoct = E(eg) − E(t2g)
  • Pairing energy: P (energy to pair electrons in same orbital)

7.2 High Spin vs Low Spin

Configuration High Spin Low Spin Condition
d4 t2g3 eg1 t2g4 Δoct > P
d5 t2g3 eg2 t2g5 Δoct > P
d6 t2g4 eg2 t2g6 Δoct > P
d7 t2g5 eg2 t2g6 eg1 Δoct > P

8. Properties and Applications

8.1 Structural Properties

  • Density: Higher than tetrahedral coordination (more compact)
  • Ionic conductivity: Often good due to close packing
  • Mechanical properties: Generally hard and brittle
  • Thermal stability: High melting points for ionic compounds

8.2 Important Applications

  • Ceramics: MgO, Al2O3 (refractory materials)
  • Electronics: NaCl (ionic conductors)
  • Catalysts: TiO2 (photocatalysis)
  • Pigments: Cr2O3 (green), Fe2O3 (red)
  • Abrasives: Al2O3 (corundum, sapphire)

9. Defects in Octahedral Structures

9.1 Common Defects

  • Schottky defects: Paired cation-anion vacancies
  • Frenkel defects: Cation displacement to interstitial site
  • Non-stoichiometry: Excess/deficiency of one component
  • Solid solutions: Substitution of similar-sized ions

9.2 Examples of Defective Structures

  • Wüstite (Fe1−xO): Fe2+ deficiency compensated by Fe3+
  • Rochelle salt: Non-stoichiometric NaCl with water
  • Solid solutions: (Mg,Fe)O, (Ca,Sr)O

10. Comparison: Tetrahedral vs Octahedral Holes

Property Tetrahedral Holes Octahedral Holes
Coordination Number 4 6
Radius Ratio Range 0.225 − 0.414 0.414 − 0.732
Number per sphere 2 1
Relative size Smaller (r = 0.225R) Larger (r = 0.414R)
Geometry Tetrahedral (109.5°) Octahedral (90°)
Preferred cations Small, high charge density Medium to large size
Crystal field splitting Δtet = 4/9 Δoct Δoct (reference)
Common examples ZnS, SiO2, diamond NaCl, MgO, Al2O3

11. Advanced Topics

11.1 Distorted Octahedral Geometries

  • Jahn-Teller distortion: Elongation/compression along one axis
  • Trigonal distortion: Compression along [111] direction
  • Tetragonal distortion: Common in d4 and d9 complexes

11.2 Solid State Chemistry Applications

  • Intercalation compounds: Li+ in layered oxides
  • Superionic conductors: Fast ion transport in octahedral sites
  • Magnetic materials: Spin interactions in octahedral complexes

12. Summary Points

  • Octahedral holes are larger than tetrahedral holes
  • 1 octahedral hole available per sphere in close packing
  • Preferred by medium-large cations with r/R = 0.414−0.732
  • Most common coordination in ionic compounds
  • Examples: NaCl, MgO, Al2O3
  • 6-fold coordination leads to octahedral geometry
  • Crystal field effects important for transition metals
  • Found in most ceramic and refractory materials