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Topic: Cubic Hole animation

Cubic Hole

Cubic Holes - Complete Notes

1. Definition and Basic Concept

  • Cubic hole: A void space formed when 8 spheres are arranged in cubic geometry
  • Shape: The hole has cubic symmetry
  • Coordination number: 8 (surrounded by 8 spheres)
  • Location: Found in simple cubic and body-centered cubic structures
  • Alternative names: Eight-coordinate holes, cubic sites

2. Geometric Properties

2.1 Size and Radius Ratio

  • Critical radius ratio: r/R = 0.732 − 1.000
  • Minimum ratio (r/R = 0.732): Cation just touches all 8 anions
  • Maximum ratio (r/R = 1.000): Cation and anion have equal size
  • Optimal size: r/R ≈ 0.85 for maximum stability

2.2 Mathematical Relationships

  • Cubic radius: rcubic = 0.732R (minimum)
  • Distance from center to vertex: d = (a√3)/2 (where a = edge length)
  • Bond angles: 90° and 70.5° (cubic geometry)
  • Hole size comparison: rcubic/roct = 0.732/0.414 = 1.77

3. Number and Distribution

3.1 In Simple Cubic Structure

  • Total spheres per unit cell: 1
  • Total cubic holes: 1
  • Ratio: 1 hole ÷ 1 sphere = 1 hole per sphere
  • Hole position: Body center at (1/2, 1/2, 1/2)

3.2 In Body-Centered Cubic (BCC)

  • Spheres per unit cell: 2
  • Cubic holes: Limited due to atom at body center
  • Alternative sites: Smaller tetrahedral and octahedral holes

3.3 In Face-Centered Cubic (FCC)

  • No true cubic holes: Structure too densely packed
  • Coordination: 12 (close-packed arrangement)
  • Available holes: Only tetrahedral and octahedral

4. Common Crystal Structures

4.1 Cesium Chloride (CsCl) Structure

  • Anion arrangement: Cl− in simple cubic lattice
  • Cation arrangement: Cs+ in the cubic hole (body center)
  • Coordination: 8:8 (each Cs+ surrounded by 8 Cl−, vice versa)
  • Radius ratio: rCs+/rCl− = 0.934
  • Examples: CsCl, CsBr, CsI

4.2 Calcium Fluoride (CaF2) - Modified Cubic

  • Cation arrangement: Ca2+ in FCC lattice
  • Anion arrangement: F− in tetrahedral holes (not cubic)
  • Coordination: 8:4 (each Ca2+ has 8 F− neighbors)
  • Note: Ca2+ has pseudo-cubic coordination
  • Examples: CaF2, SrF2, BaF2

4.3 Perovskite (ABO3) Structure

  • A-cation: Large cation in cubic coordination (12 neighbors)
  • B-cation: Small cation in octahedral coordination
  • Anions: O2− forming the framework
  • Examples: CaTiO3, BaTiO3, SrTiO3

5. Examples with Radius Ratios

Compound Large Cation Anion r/R Ratio Structure Coordination
CsCl Cs+ Cl− 0.934 Cesium chloride 8:8
CsBr Cs+ Br− 0.876 Cesium chloride 8:8
CsI Cs+ I− 0.778 Cesium chloride 8:8
NH4Cl NH4+ Cl− 0.832 Cesium chloride 8:8
TlCl Tl+ Cl− 0.832 Cesium chloride 8:8

6. Factors Affecting Cubic Hole Occupancy

6.1 Size Factor

  • Too small (r/R < 0.732): Prefers octahedral coordination
  • Optimal size (r/R = 0.732−1.000): Stable cubic coordination
  • Equal size (r/R = 1.000): Maximum packing efficiency
  • Perfect fit: When large cation efficiently fills cubic void

6.2 Electronic Factors

  • Large, low-charge cations: Cs+, Rb+, NH4+
  • Minimal crystal field effects: Due to high coordination number
  • Ionic character: Pure ionic bonding favored
  • Polarization: Large cations with low charge density

6.3 Packing Efficiency

  • Simple cubic packing: Only 52% space filling
  • Low coordination: Each sphere touches only 6 neighbors
  • Rare in nature: Most elements prefer denser packing
  • Stability: Only stable for very specific size ratios

7. Structural Variations

7.1 Distorted Cubic Coordination

  • Pseudo-cubic: Slight deviations from perfect cubic symmetry
  • Compressed cubic: Flattening along one axis
  • Elongated cubic: Stretching along one direction
  • Trigonal distortion: Compression along [111] direction

7.2 Mixed Coordination Environments

  • Perovskite structure: A-site in 12-coordinate cubic environment
  • Spinel structure: Some cations in distorted cubic sites
  • Garnet structure: Large cations in 8-coordinate sites

8. Properties and Applications

8.1 Structural Properties

  • Low density: Due to inefficient packing
  • High coordination: 8-fold coordination provides stability
  • Ionic conductivity: Large cavities allow ion migration
  • Phase transitions: Often transform under pressure/temperature

8.2 Important Applications

  • Ionic conductors: CsI (solid electrolytes)
  • Scintillators: CsI:Tl (radiation detection)
  • Optical materials: CaF2 (fluorite, lenses)
  • Ferroelectrics: BaTiO3 (perovskite)
  • Catalysts: Perovskite oxides for various reactions

9. Coordination Chemistry

9.1 Eight-Coordinate Geometries

  • Cubic: 8 atoms at cube vertices
  • Square antiprismatic: Two square planes rotated 45°
  • Dodecahedral: Distorted cube with triangular faces
  • Bicapped trigonal prismatic: 6 + 2 arrangement

9.2 Examples in Coordination Complexes

Complex Central Ion Ligands Geometry
[Mo(CN)8]4− Mo4+ 8 CN− Square antiprismatic
[TaF8]3− Ta5+ 8 F− Square antiprismatic
[Zr(ox)4]4− Zr4+ 4 oxalate Dodecahedral

10. Phase Transitions and Polymorphism

10.1 Pressure-Induced Transitions

  • CsCl structure: Transforms to NaCl structure under pressure
  • Coordination change: 8:8 → 6:6 coordination
  • Volume reduction: Higher density at high pressure
  • Examples: RbCl, CsBr show similar transitions

10.2 Temperature Effects

  • Thermal expansion: Large cavities accommodate expansion
  • Order-disorder transitions: NH4+ rotation in NH4Cl
  • Ferroelectric transitions: BaTiO3 cubic ↔ tetragonal

11. Defects and Non-stoichiometry

11.1 Common Defects

  • Schottky defects: Paired cation-anion vacancies
  • Substitutional defects: Size-matched ion substitution
  • Interstitial defects: Extra ions in large cavities
  • Anti-site defects: Cation-anion site exchange

11.2 Solid Solutions

  • (Cs,Rb)Cl: Complete solid solution
  • (Ca,Sr)F2: Fluorite solid solutions
  • (Ba,Sr)TiO3: Perovskite solid solutions

12. Comparison with Other Coordination Numbers

Property Tetrahedral (4) Octahedral (6) Cubic (8)
Coordination Number 4 6 8
Radius Ratio Range 0.225 − 0.414 0.414 − 0.732 0.732 − 1.000
Geometry Tetrahedral Octahedral Cubic
Bond Angles 109.5° 90° 90°, 70.5°
Hole Size (relative) Smallest Medium Largest
Preferred cations Small, high charge Medium size Large, low charge
Common examples ZnS, SiO2 NaCl, MgO CsCl, CaF2
Stability High (covalent) High (ionic) Moderate (size-dependent)