Hexagonal Close Packed (HCP) Structure
Master ABAB stacking, coordination number, and the mathematics of the HCP unit cell.
Nature loves efficiency. When atoms (treated as identical hard spheres) pack together in a 3D space, they seek arrangements that minimize empty volume. One of the two most efficient ways to achieve this is the Hexagonal Close Packed (HCP) structure. Common metals like Magnesium ($Mg$), Zinc ($Zn$), and Titanium ($Ti$) crystallize in this form.
1. Formation: The ABAB... Stacking Sequence
The HCP lattice is generated by stacking 2D hexagonal close-packed layers one over another.
- Layer A: The first layer is a standard 2D hexagonal close-packed layer.
- Layer B: The spheres of the second layer are placed in the depressions (voids) of Layer A.
- The Crucial Third Layer: To form HCP, the spheres of the third layer are placed directly over the tetrahedral voids of the second layer. Because they cover tetrahedral voids, the spheres of the third layer perfectly align vertically with the spheres of the first layer (Layer A).
2. Critical Parameters of the HCP Unit Cell
The unit cell of HCP is a hexagonal prism. Understanding the exact geometry and atomic contributions inside this prism is guaranteed to yield marks in competitive exams.
A. Rank / Number of Atoms per Unit Cell ($Z$)
Let's calculate the effective number of atoms ($Z$) belonging specifically to one hexagonal unit cell:
- 12 Corner Atoms: Shared by 6 adjacent hexagons. Contribution = $12 \times \frac{1}{6} = \mathbf{2}$
- 2 Face-Centered Atoms: (One on top face, one on bottom face). Shared by 2 hexagons. Contribution = $2 \times \frac{1}{2} = \mathbf{1}$
- 3 Body-Centered Atoms: Located entirely within the body of the unit cell (forming the 'B' layer). Contribution = $3 \times 1 = \mathbf{3}$
B. Coordination Number
The coordination number is the number of nearest neighbors touching a specific atom. For an atom in the central 'A' layer of an HCP lattice:
- It touches 6 atoms in its own plane.
- It touches 3 atoms in the plane directly above it.
- It touches 3 atoms in the plane directly below it.
C. Packing Efficiency
HCP is one of the most efficient ways to pack spheres. Like FCC (Cubic Close Packing), it minimizes the empty void space.
(Leaving exactly 26% of the volume as empty void space).
3. Voids in the HCP Lattice
Any close-packed structure contains two types of 3D voids: Tetrahedral and Octahedral. A fundamental rule of solid state chemistry connects the number of voids directly to the number of atoms in the lattice ($N$).
For an HCP Unit Cell ($N = Z = 6$):
Number of Octahedral Voids = $N = \mathbf{6}$
Number of Tetrahedral Voids = $2N = \mathbf{12}$
This means there are a total of 18 voids within a single HCP unit cell.
In an ideal HCP unit cell constructed of perfectly hard spheres, the ratio of the height of the hexagonal prism ($c$) to the edge length of the base ($a$) is a mathematical constant.
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