Closed Packing in Solids

In the solid state, atoms, ions, or molecules are packed together to occupy the minimum possible space. This arrangement is known as closed packing. There are two main types of closed packing in three dimensions: Hexagonal Close Packing (HCP) and Cubic Close Packing (CCP), also known as Face-Centered Cubic (FCC) packing. These packings are the most efficient ways to fill space using spheres of equal size.

Hexagonal Close Packing (HCP)

In HCP, each sphere is surrounded by 12 other spheres. Six spheres are arranged in a hexagon around a central sphere in one layer, and three spheres are in the gaps above the first layer, and three below, making a total of 12.

Coordination Number

The coordination number in HCP is 12, which means each sphere is in contact with 12 other spheres.

Layers Arrangement

The layers in HCP are arranged in an ABAB pattern. This means that the third layer is directly above the first layer, and the fourth layer is directly above the second layer, and so on.

Efficiency

The packing efficiency of HCP is about 74%, which is the fraction of the volume in a crystal that is occupied by the spheres.

Cubic Close Packing (CCP) or Face-Centered Cubic (FCC)

In CCP or FCC, each sphere is also surrounded by 12 others. The arrangement is similar to HCP, but the stacking sequence is different.

Coordination Number

The coordination number in CCP is also 12.

Layers Arrangement

The layers in CCP are arranged in an ABCABC pattern. This means that the third layer is not directly above the first layer, creating a three-layer repeating pattern.

Efficiency

The packing efficiency of CCP is also about 74%, the same as HCP.

Differences between HCP and CCP

Feature	HCP	CCP
Coordination Number	12	12
Layers Arrangement	ABAB...	ABCABC...
Packing Efficiency	~74%	~74%
Unit Cell	Hexagonal	Cubic
Examples	Zinc, Magnesium	Copper, Gold, Silver

Formulas Related to Closed Packing

Packing Efficiency

The packing efficiency (PE) is the fraction of volume occupied by the spheres in a unit cell.

$$ PE = \frac{\text{Volume occupied by spheres}}{\text{Total volume of the unit cell}} \times 100\% $$

For both HCP and CCP:

$$ PE = \frac{\pi}{3\sqrt{2}} \approx 74.05\% $$

Number of Atoms per Unit Cell

For CCP (FCC):

$$ \text{Number of atoms per unit cell} = 4 $$

For HCP:

$$ \text{Number of atoms per unit cell} = 6 $$

Atomic Packing Factor (APF)

The atomic packing factor is the volume of atoms in a unit cell divided by the volume of the unit cell.

$$ APF = \frac{\text{Volume of atoms}}{\text{Volume of unit cell}} $$

For both HCP and CCP:

$$ APF = \frac{\pi}{3\sqrt{2}} \approx 0.7405 $$

Examples

Example 1: Coordination Number

In both HCP and CCP structures, each atom is in contact with 12 other atoms. This can be visualized by placing an atom at the center of a cluster and counting the number of atoms that directly touch it.

Example 2: Packing Efficiency

The packing efficiency of both HCP and CCP is approximately 74%. This means that in a crystal lattice, 74% of the space is filled with atoms, while the remaining 26% is empty space.

Example 3: Unit Cells

The unit cell of HCP is a hexagonal prism, while the unit cell of CCP is a cube with atoms at each corner and an atom at the center of each face.

By understanding closed packing, we can predict the properties of materials, such as density, stability, and how they might interact with other substances. This knowledge is crucial for material science, chemistry, and various engineering fields.