Which lattice has an APF of approximately 0.74?

• A. Face-Centered Cubic
• B. Body-Centered Cubic
• C. Simple Cubic
• D. Hexagonal Close-Packed

Answer: (A)

Atomic packing factor measures how much of a unit cell’s volume is actually filled by atoms. To get an APF around 0.74, you need a close-packed arrangement where spheres touch along the packing directions.

In a face-centered cubic lattice, there are 4 atoms per unit cell (one from each corner contributes 1 total, plus six face-centered atoms contribute 3, giving 4). The geometry of the corner and face atoms leads to a relation between the atomic radius r and the cell edge a: the atoms touch along the face diagonal, so √2 a = 4r, or r = a√2/4. The volume of the four atoms is 4*(4/3)πr^3. Dividing by the cell volume a^3 gives APF = (4*(4/3)πr^3)/a^3. Substituting r yields APF = π√2/6 ≈ 0.7405.

So this lattice indeed fills about 74% of its volume with atoms, matching the target value. The simple cubic and body-centered cubic lattices have lower APFs (0.52 and about 0.68, respectively). Hexagonal close-packed is another close-packed arrangement with a similar APF around 0.74, but the classic example often cited for this value is the face-centered cubic structure.