Atomic Packing Factor

In crystallography , the atomic packing factor (APF), also known as packing efficiency or packing fraction, represents the fraction of volume in a crystal structure that is occupied by its constituent particles. This dimensionless quantity, always less than unity, provides critical insight into the spatial arrangement of atoms within a crystalline solid. By convention, atoms are treated as rigid spheres, with their radii defined such that they do not overlap. For one-component crystals—those containing only a single type of particle—the APF is mathematically expressed as:

[ \mathrm{APF} = \frac{N_{\mathrm{particle}} V_{\mathrm{particle}}}{V_{\text{unit cell}}} ]

where:

( N_{\mathrm{particle}} ) is the number of particles in the unit cell,
( V_{\mathrm{particle}} ) is the volume of each particle,
( V_{\text{unit cell}} ) is the volume occupied by the unit cell.

Mathematically, it has been proven that the most dense arrangement of identical spheres in a one-component structure yields an APF of approximately 0.74, a result derived from the Kepler conjecture . This maximum packing efficiency is achieved in close-packed structures, such as hexagonal close-packed (HCP) and face-centered cubic (FCC) lattices. However, in multiple-component structures, such as interstitial alloys, the APF can exceed 0.74 due to the presence of smaller atoms occupying interstitial sites between larger atoms.

The APF is of paramount importance in materials science , as it influences numerous physical properties of materials. For instance, metals with a high atomic packing factor tend to exhibit greater workability, encompassing enhanced malleability and ductility . This phenomenon can be analogized to a road surface: when stones are closely packed, the surface is smoother, allowing metal atoms to slide past one another with greater ease under mechanical stress.

Single-Component Crystal Structures

Various common sphere packings adopted by atomic systems are listed below, along with their corresponding packing fractions:

Hexagonal close-packed (HCP) : 0.74 [1]
Face-centered cubic (FCC) : 0.74 [1] (also referred to as cubic close-packed (CCP))
Body-centered cubic (BCC) : 0.68 [1]
Simple cubic : 0.52 [1]
Diamond cubic : 0.34

The majority of metallic elements adopt either the HCP, FCC, or BCC structure, as these configurations optimize spatial efficiency and mechanical properties. [2]

Simple Cubic Unit Cell

In a simple cubic packing arrangement, the unit cell contains one atom per unit cell. The side length of the unit cell is 2r, where r denotes the radius of the atom.

The APF for a simple cubic structure is calculated as follows:

[ \begin{aligned} \mathrm{APF} &= \frac{N_{\mathrm{atoms}} V_{\mathrm{atom}}}{V_{\text{unit cell}}} = \frac{1 \cdot \frac{4}{3} \pi r^{3}}{(2r)^{3}} \ &= \frac{\pi}{6} \approx 0.5236 \end{aligned} ]

This relatively low packing efficiency explains why the simple cubic structure is rare in metallic systems, as it leaves significant void space between atoms.

Face-Centered Cubic (FCC) Structure

The face-centered cubic (FCC) unit cell contains four atoms. A diagonal line drawn from the top corner of the cube to the bottom corner on the same face measures 4r. Using geometric relationships, the side length a of the unit cell can be expressed in terms of the atomic radius r as:

[ a = 2r \sqrt{2} ]

Given the volume of a sphere , the APF for the FCC structure is derived as:

[ \begin{aligned} \mathrm{APF} &= \frac{N_{\mathrm{atoms}} V_{\mathrm{atom}}}{V_{\text{unit cell}}} = \frac{4 \cdot \frac{4}{3} \pi r^{3}}{(2r \sqrt{2})^{3}} \ &= \frac{\pi}{3 \sqrt{2}} \approx 0.74048048 \end{aligned} ]

The FCC structure is highly efficient, which is why many metals, including copper, aluminum, and gold, crystallize in this arrangement.

Body-Centered Cubic (BCC) Structure

The body-centered cubic (BCC) primitive unit cell contains two atoms: one at each corner of the cube (shared among eight adjacent cells) and one at the center. The diagonal of the cube, passing through the center atom, measures 4r, where r is the atomic radius. Geometrically, the diagonal length is also a√3, allowing the side length a to be expressed as:

[ a = \frac{4r}{\sqrt{3}} ]

Using the volume of a sphere , the APF for the BCC structure is calculated as:

[ \begin{aligned} \mathrm{APF} &= \frac{N_{\mathrm{atoms}} V_{\mathrm{atom}}}{V_{\text{unit cell}}} = \frac{2 \cdot \frac{4}{3} \pi r^{3}}{\left( \frac{4r}{\sqrt{3}} \right)^{3}} \ &= \frac{\pi \sqrt{3}}{8} \approx 0.680174762 \end{aligned} ]

Metals such as iron (at room temperature) and tungsten adopt the BCC structure, balancing packing efficiency with mechanical strength.

Hexagonal Close-Packed (HCP) Structure

The hexagonal close-packed (HCP) structure consists of a hexagonal prism containing six atoms per unit cell. The unit cell comprises three atoms in the middle layer, with the top and bottom layers each contributing one central atom (shared with adjacent cells) and six corner atoms (each shared among six adjacent cells).

Let a represent the side length of the hexagonal base and c the height of the prism. The height c is twice the distance between adjacent layers, equivalent to twice the height of a regular tetrahedron formed by the central atom of the lower layer, two adjacent non-central atoms of the same layer, and one atom of the middle layer. If a = 2r, the height of this tetrahedron is:

[ \sqrt{\frac{8}{3}} a ]

Thus, the height c of the unit cell is:

[ c = 4 \sqrt{\frac{2}{3}} r ]

The volume of the HCP unit cell is then:

[ \frac{3 \sqrt{3}}{2} a^{2} c = 24 \sqrt{2} r^{3} ]

The APF for the HCP structure is calculated as:

[ \begin{aligned} \mathrm{APF} &= \frac{N_{\mathrm{atoms}} V_{\mathrm{atom}}}{V_{\text{unit cell}}} = \frac{6 \cdot \frac{4}{3} \pi r^{3}}{\frac{3 \sqrt{3}}{2} a^{2} c} \ &= \frac{6 \cdot \frac{4}{3} \pi r^{3}}{\frac{3 \sqrt{3}}{2} (2r)^{2} \sqrt{\frac{2}{3}} \cdot 4r} \ &= \frac{\pi}{\sqrt{18}} = \frac{\pi}{3 \sqrt{2}} \approx 0.74048048 \end{aligned} ]

This confirms that the HCP structure, like the FCC structure, achieves the theoretical maximum packing efficiency of 0.74.