BCC, FCC and HCP crystal structures

Atomic packing factor (APF) is the fraction of the unit-cell volume occupied by atoms:

$$\text{APF} = \frac{n \times \frac{4}{3}\pi R^3}{a^3}$$

BCC (e.g., $\alpha$-Fe, Cr, W)

Atoms at the 8 corners and one at the body centre. Atoms touch along the body diagonal.

• Atoms per cell: $8 \times \tfrac{1}{8} + 1 = 2$
• Body diagonal $= \sqrt{3}\,a = 4R$, so $R = \dfrac{\sqrt{3}\,a}{4}$
• Coordination number = 8

$$\text{APF} = \frac{2 \times \frac{4}{3}\pi\left(\frac{\sqrt{3}a}{4}\right)^3}{a^3} = \frac{\sqrt{3}\,\pi}{8} = 0.68$$

FCC (e.g., $\gamma$-Fe, Cu, Al, Ni)

Atoms at the 8 corners and the centre of each of the 6 faces. Atoms touch along the face diagonal.

• Atoms per cell: $8 \times \tfrac{1}{8} + 6 \times \tfrac{1}{2} = 4$
• Face diagonal $= \sqrt{2}\,a = 4R$, so $R = \dfrac{\sqrt{2}\,a}{4}$
• Coordination number = 12

$$\text{APF} = \frac{4 \times \frac{4}{3}\pi\left(\frac{\sqrt{2}a}{4}\right)^3}{a^3} = \frac{\pi}{3\sqrt{2}} = 0.74$$

HCP (e.g., Zn, Mg, Ti, Co)

Two hexagonal layers (top and bottom) with a central atom each, plus a triangular layer of 3 atoms midway between them. Atoms per cell = $12 \times \tfrac{1}{6} + 2 \times \tfrac{1}{2} + 3 = 6$. Ideal axial ratio $c/a = 1.633$, coordination number 12, APF = 0.74.

FCC and HCP are both close-packed, differing in stacking sequence (ABCABC for FCC, ABABAB for HCP).