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Atomic Packing Factor (APF) — First Principles

Imagine you're packing oranges into a crate. No matter how carefully you arrange them, there will always be some empty space between the spheres. The Atomic Packing Factor (APF) is simply the fraction of that crate's volume that is actually filled by the oranges — except here, the "oranges" are atoms, and the "crate" is the unit cell of a crystal.

The Intuition

Atoms in a solid are not tiny cubes that stack perfectly. They're spheres. And spheres, when packed together, always leave gaps. The APF tells you how efficiently a given crystal structure uses space. A higher APF means atoms are packed more tightly together — less wasted volume.

For example, if you stack cannonballs in a pyramid, they pack more densely than if you just pile them randomly. Different crystal structures (like simple cubic, body-centered cubic, face-centered cubic) have different packing efficiencies. The APF is the number that quantifies this.

The Precise Definition

That's it. A ratio between 0 and 1 (or 0% to 100%). For most metals, APF values range from about 0.52 to 0.74.

How to Calculate It — Step by Step

You need three things:

1. Number of atoms per unit cell — Count carefully. Atoms at corners are shared by 8 cells, atoms on faces by 2 cells, atoms at edges by 4 cells, and atoms fully inside belong entirely to that cell.
2. Radius of the atom — Usually given as $R$.
3. Volume of the unit cell — Depends on the crystal structure. For a cube of side $a$, it's $a^3$. But $a$ itself depends on $R$ through the geometry of how atoms touch.

Worked Example: Simple Cubic (SC)

In a simple cubic cell:

• 8 corner atoms, each shared by 8 cells → $8 \times \frac{1}{8} = 1$ atom per cell
• Atoms touch along the cube edge: $a = 2R$
• Volume of cell: $a^3 = (2R)^3 = 8R^3$
• Volume of one atom: $\frac{4}{3}\pi R^3$

So only about 52.4% of the space is filled. Nearly half is empty — that's why simple cubic is rare in real metals.

Common APF Values