Solid State: Step-by-Step Worked Examples (2)

Question

A metal crystallizes in a face-centered cubic (FCC) lattice with edge length $a = 400$ pm. Find (a) the radius of the metal atom, (b) the number of atoms per unit cell, and (c) the packing efficiency.

Solution — Step by Step

Final answers: $r \approx 141.4$ pm, atoms per cell = 4, packing efficiency $\approx 74\%$.

Why This Works

FCC packs atoms tighter than any other cubic lattice — touching along face diagonals leaves only $26\%$ empty space. That’s why most metals (Al, Cu, Au, Ag) crystallise FCC.

The relation $4r = a\sqrt{2}$ comes from Pythagoras on the face: face diagonal $= a\sqrt{2}$, and that diagonal contains 4 atomic radii (corner atom radius + atom-atom contact at face centre + corner atom radius on the other side).

Alternative Method

For BCC (body diagonal $= 4r = a\sqrt{3}$, packing $\approx 68\%$) or simple cubic (edge $= 2r$, packing $\approx 52\%$), the same logic. Different lattice → different geometric relation.

Common Mistake

Counting corner atoms as full atoms. Each corner is shared between 8 unit cells, so contributes $\frac{1}{8}$. Same for faces ($\frac{1}{2}$ each, shared between 2 cells) and edges ($\frac{1}{4}$ each, shared between 4 cells).