Solid State: Exam-Pattern Drill (4)

Question

In a face-centered cubic (FCC) lattice with edge length $a = 400$ pm, find (a) the radius of the atom, (b) the packing efficiency, (c) the number of atoms per unit cell.

Solution — Step by Step

Final answers: (a) $r \approx 141.4$ pm, (b) $74\%$, (c) $4$ atoms per unit cell.

Why This Works

FCC achieves the densest packing of identical spheres ($74\%$), tied with HCP. The face-diagonal relation comes from sphere-touching geometry — the four atoms along the face diagonal all touch.

The packing efficiency $\pi\sqrt{2}/6$ is a fixed number that depends only on the lattice type, not on $a$ or $r$ individually. That’s why FCC of any element has the same $74\%$ density.

Alternative Method

For Z, use the formula $\rho = ZM/(N_A a^3)$ if density and molar mass are given. Then back out Z. Useful when the problem gives density data instead of geometry.

Common Mistake

Using $a = 4r$ for FCC. That’s the body-diagonal relation for BCC. For FCC, atoms touch along the face diagonal, giving $a\sqrt{2} = 4r$. Mixing these costs full marks.