APF Calculator

What is the Atomic Packing Factor (APF)?

In crystallography, the Atomic Packing Factor (APF), or packing efficiency, is the fraction of volume in a crystal structure that is occupied by constituent particles (atoms or ions). It is an indicator of how tightly atoms are packed together.

By definition, APF assumes that atoms are rigid spheres of a given radius.

The Equation

The formula for APF is:

$\text{APF} = \frac{N_{atoms} \times V_{atom}}{V_{unit\;cell}}$

Where:

• $N_{atoms}$ is the number of atoms per unit cell.
• $V_{atom}$ is the volume of a single atom, modeled as a sphere: $\frac{4}{3}\pi r^3$.
• $V_{unit\;cell}$ is the total volume of the unit cell.

APF of Common Structures

For pure metals with one type of atom, standard structures have fixed APFs:

• Simple Cubic (SC): 1 atom per cell, $a = 2r$. APF $\approx 0.52$
• Body-Centered Cubic (BCC): 2 atoms per cell, $a = \frac{4r}{\sqrt{3}}$. APF $\approx 0.68$
• Face-Centered Cubic (FCC): 4 atoms per cell, $a = 2\sqrt{2}r$. APF $\approx 0.74$
• Hexagonal Close-Packed (HCP): 6 atoms per cell. APF $\approx 0.74$

FCC and HCP represent the maximum possible packing efficiency for spheres of equal size, known as close-packed structures.