How many atoms are contained in a unit cell of a face-centered cubic lattice?

• A. 1
• B. 2
• C. 4
• D. 8

Answer: (C)

Counting atoms in a unit cell by looking at how atoms are shared on the cell boundaries is the key idea. In a face-centered cubic lattice, atoms sit at the eight corners and at the centers of the six faces. An atom at a corner is shared by eight neighboring cells, so it contributes 1/8 of an atom to this unit cell. With eight corners, that gives 8 × 1/8 = 1 atom. An atom at a face center is shared by two cells, contributing 1/2 to this unit cell. With six faces, that gives 6 × 1/2 = 3 atoms. Total: 1 + 3 = 4 atoms per unit cell. The other numbers come from miscounting how boundary atoms are shared (for example, counting whole corner atoms or ignoring face-centered atoms).