# Hexagonal Close Packing

Solid materials show specific crystal structures. They can be classified as crystalline and amorphous on the basis of arrangement of their constituent particles. A crystalline solid consists of a large number of small crystals, each of them having a definite characteristic geometrical shape. In these crystals, arrangement of constituent particles is in order and repetitive in three dimensions. We can accurately predict the arrangement of particles of another region, if we know the arrangement of particles in one region of the crystal. Thus, a crystal has a long range order which means that there is a regular pattern of arrangement of particles which repeats itself periodically over the entire crystal.

A smallest group of constituent particles which has overall symmetry of the crystal and by repetition of it in three dimensions, the entire lattice can be built is called unit cell of that crystal.

In the three-dimensional crystal structure, unit cell is characterized by –

• Its dimensions along the three edges a, b and c. These edges may or may not be mutually perpendicular.

•  Angles between the edges, α (between b and c), β (between a and c) and γ (between a and b). Thus, a unit cell is characterized by six parameters a, b, c, α , β and γ.

### Hexagonal Close Packing

Hexagonal close packing, or hcp in short, is one of the two lattice structures which are able to achieve the highest packing density of ~74%, the other being face centered cubic (fcc) structure. This packing structure is found in metals such as zinc, cadmium, cobalt and titanium.

### Hexagonal Close Pack Structure

Hexagonal close packing structure consists of alternating layers of spheres or atoms arranged in a hexagon, with one additional atom at the center. Another layer of atoms is sandwiched between these two hexagonal layers which is in a triangular form and the atoms of this layer fill the tetrahedral holes created by the top and bottom layers. Such a structure can be pictorially represented as below:

As you can see in the image above, the lattice structure is essentially an alternative pattern of layers ‘a’ and ‘b’. The series for this lattice structure can also be written as …..-a-b-a-b-a-b-…..and so on.

### Hcp Structure Unit Cell

We have shown the hcp structure diagram in the previous section. In this section, we will discuss the hcp unit cell in a little more detail. Hcp unit cell can be represented as :

In this unit cell, the atoms in the middle layer are not shared with any other unit cell, while the atoms in upper and bottom layers are shared with adjacent unit cells. The central sphere or atom in top and bottom layers is shared with 1 other unit cell, hence they only contribute ½ to one unit cell. The atoms at the hexagonal vertices in top and bottom layers are shared with 5 other unit cells, hence they contribute just 1/6 to one unit cell. Using this understanding, we can compute the number of atoms per unit cell in a hcp structure as below:

Number of atoms fully contributing to the unit cell = 3 (all in the middle layer)

Number of atoms contributing ½ to the unit cell = 2 (1 in top and 1 in bottom layer, respectively)

Number of atoms contributing 1/6 to the unit cell = 12 (6 in top and 6 in bottom layer, respectively)

Total number of atoms in the unit cell = 3 + ½ x 2 + 1/6 x 12 = 3+1+2= 6

Hence, one unit cell of hcp structure comprises 6 atoms. The coordination number, i.e. number of atoms surrounding the central atom, for hcp structure is 12. The coordination number can also be visualized as below:

Volume of Hcp Unit Cell=  $\frac{3\sqrt{3}a^{2}c}{2}$
The ratio between the space occupied by spheres and empty space in a hcp structure is approximately 74%:26%. The amount of space occupied is given by the formula $\frac{π}{3\sqrt{2}}$ which approximately translates to 0.74048.