Z (Atoms per Unit Cell)

$Z_{SC} = 8 \times \frac{1}{8} = 1$

$Z_{BCC} = 8 \times \frac{1}{8} + 1 \times 1 = 2$

$Z_{FCC} = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4$

Coordination Number

• SC: CN = 6 (4 in same plane + 1 above + 1 below)
• BCC: CN = 8 (body centre touches all 8 corners)
• FCC: CN = 12 (4 in same plane + 4 above + 4 below)

Packing Efficiency

$PE_{SC} = \frac{\pi}{6} \approx 52.4\%$

$PE_{BCC} = \frac{\pi\sqrt{3}}{8} \approx 68.0\%$

$PE_{FCC} = \frac{\pi}{3\sqrt{2}} = \frac{\pi\sqrt{2}}{6} \approx 74.0\%$

Edge–Radius Relations

$SC: a = 2r \implies r = \frac{a}{2}$

$BCC: a\sqrt{3} = 4r \implies r = \frac{a\sqrt{3}}{4}$

$FCC: a\sqrt{2} = 4r \implies r = \frac{a\sqrt{2}}{4} = \frac{a}{2\sqrt{2}}$

Mnemonic: "S(1), B(2), F(4)" and "SC-edge, BCC-body, FCC-face" for the direction of atom contact.