Hexagonal Close Packing (HCP)
Definition and Basic Structure
Hexagonal Close Packing (HCP) is one of the two most efficient ways to pack identical spheres in three dimensions, achieving a packing efficiency of 74.05%. In HCP, spheres are arranged in a pattern where every third layer is identical, creating an ABAB... stacking sequence.
Key Characteristics
- Stacking Sequence: ABAB... (alternating layers)
- Coordination Number: 12 (each sphere touches 12 others)
- Packing Efficiency: 74.05% (π/3√2 ≈ 0.7405)
- Unit Cell: Hexagonal with lattice parameters a, b, and c
- Axial Ratio: c/a = √(8/3) ≈ 1.633 for ideal HCP
Unit Cell Parameters
| Parameter |
Value |
Description |
| Lattice Parameters |
a = b ≠ c |
Hexagonal symmetry |
| Angles |
α = β = 90°, γ = 120° |
Hexagonal angles |
| Atoms per Unit Cell |
2 |
Effective atoms in unit cell |
| c/a Ratio (Ideal) |
1.633 |
√(8/3) for perfect spheres |
| Coordination Number |
12 |
Nearest neighbors |
| Packing Efficiency |
74.05% |
Same as FCC |
Interstitial Sites
Hole Types and Coordination
Octahedral Holes
Number: 2 per unit cell
Coordination: 6
Size: rhole/rsphere = 0.414
Location: Between layers
Tetrahedral Holes
Number: 4 per unit cell
Coordination: 4
Size: rhole/rsphere = 0.225
Location: Within layers
Mathematical Relationships
Geometric Calculations
Volume of Unit Cell:
V = (3√3/2) × a² × c
Atomic Packing Factor:
APF = (Number of atoms × Vatom) / Vunit cell = π/(3√2) ≈ 0.7405
Nearest Neighbor Distance:
dnn = a (in basal plane and between layers)
Density Calculations
Density:
ρ = (n × M) / (NA × Vunit cell)
where n = 2 atoms per unit cell, M = atomic mass, NA = Avogadro's number
HCP vs FCC Comparison
Hexagonal Close Packing (HCP)
- Stacking: ABAB...
- Symmetry: Hexagonal
- Unit Cell: Hexagonal
- Atoms/Cell: 2
- c/a Ratio: 1.633
- Examples: Mg, Zn, Ti, Co
Face-Centered Cubic (FCC)
- Stacking: ABCABC...
- Symmetry: Cubic
- Unit Cell: Cubic
- Atoms/Cell: 4
- c/a Ratio: N/A (cubic)
- Examples: Cu, Au, Ag, Al
Common HCP Metals and Their Properties
| Metal |
a (Å) |
c (Å) |
c/a Ratio |
Density (g/cm³) |
Properties |
| Mg |
3.21 |
5.21 |
1.624 |
1.74 |
Lightweight, reactive |
| Zn |
2.66 |
4.95 |
1.861 |
7.14 |
Corrosion resistant |
| Ti |
2.95 |
4.68 |
1.587 |
4.51 |
High strength-to-weight |
| Co |
2.51 |
4.07 |
1.622 |
8.90 |
Magnetic, hard |
| Cd |
2.98 |
5.62 |
1.886 |
8.65 |
Toxic, soft |
| Be |
2.29 |
3.58 |
1.563 |
1.85 |
Lightweight, toxic |
HCP in Ionic Compounds
HCP arrangement is commonly found in ionic compounds where anions form the close-packed lattice:
Common HCP-based Ionic Structures:
ZnS (Wurtzite)
ZnO
BeO
SiC
AlN
GaN
CdS
CdSe
Wurtzite Structure (ZnS)
- Anion Arrangement: S²⁻ ions in HCP
- Cation Occupation: Zn²⁺ in half of tetrahedral holes
- Coordination: 4:4 (Zn²⁺:S²⁻)
- Characteristics: Polar structure, piezoelectric
Slip Systems and Deformation
Slip Systems in HCP
| Slip System |
Slip Direction |
Slip Plane |
Number of Systems |
Activity |
| Basal |
⟨112̄0⟩ |
(0001) |
3 |
Primary (easy) |
| Prismatic |
⟨112̄0⟩ |
{101̄0} |
3 |
Secondary |
| Pyramidal |
⟨112̄0⟩ |
{101̄1} |
6 |
Difficult |
| Pyramidal |
⟨c+a⟩ |
{112̄2} |
12 |
Very difficult |
Twinning in HCP
HCP metals readily undergo twinning due to limited slip systems:
- Common Twin Planes: {101̄2}, {101̄1}, {112̄1}
- Twin Direction: ⟨101̄1⟩
- Significance: Accommodates deformation when slip is restricted
- Examples: Mg, Zn show prominent twinning
Key Points to Remember:
• HCP and FCC have identical packing efficiency (74.05%)
• HCP has fewer slip systems, making it less ductile than FCC
• The c/a ratio deviation from ideal (1.633) affects properties
• HCP metals often show anisotropic properties due to hexagonal symmetry
• Twinning is more common in HCP than in FCC structures
• Many important engineering materials adopt HCP structure (Ti, Mg, Zn)