Which crystal structure has a packing factor around 0.74 and is commonly found in aluminum, copper, and nickel?

• A. BCC
• B. HCP
• C. FCC
• D. Simple Cubic

Answer: (C)

The main idea here is atomic packing efficiency in crystal structures, specifically how tightly atoms can be packed in a given arrangement. The face-centered cubic (FCC) structure achieves the highest packing density among common metal structures, with a packing factor of about 0.74.

In an FCC unit cell, atoms are at all corners and at the centers of all faces. There are four atoms effectively inside each unit cell. The atoms touch along the face diagonal, so the relation between the lattice parameter a and the atomic radius r is a√2 = 4r, giving a = 2√2 r. The cell volume is a^3, and the total volume occupied by the atoms is 4 × (4/3)πr^3 = 16/3 π r^3. Therefore, the packing factor is (16/3 π r^3) / (a^3) = (16/3 π r^3) / (16√2 r^3) = π/(3√2) ≈ 0.74.

Aluminum, copper, and nickel crystallize in the FCC structure, which is why this structure is associated with that packing density. Other common structures like simple cubic (packing ~0.52) or BCC (packing ~0.68) have lower packing factors, while hexagonal close-packed (HCP) also reaches about 0.74 but the metals listed are known to be FCC, making FCC the best match for this context.