Step 1: Grasping the fcc arrangement.
A face-centered cubic lattice hosts atoms at all eight corners and the centers of all six faces.
Step 2: Atoms per unit cell.
Corner atoms each supply 1/8: 8 × 1/8 = 1 atom. Face atoms each supply 1/2: 6 × 1/2 = 3 atoms. Total = 4 atoms per unit cell.
Step 3: Linking radius to edge length.
In fcc, the face diagonal equals 4r. Edge length a = 4r / √2 = 2√2 r.
Step 4: Volume of atoms inside the cell.
Four atomic spheres occupy 4 × (4/3)πr³ = (16/3)πr³.
Step 5: Total unit cell volume.
a³ = (2√2 r)³ = 16√2 r³.
Step 6: Packing efficiency calculation.
Efficiency = (atom volume / cell volume) × 100% = [(16/3)πr³ / (16√2 r³)] × 100% = 74%.
Step 7: Final statement.
The fcc lattice achieves 74% packing efficiency, the maximum among cubic structures.