Question

The number of atoms per unit cell in FCC structure is

• A. 8
• B. 6
• C. 4
• D. 2

Hint:

Effective number of atoms per unit cell for standard cubic lattices:
• Simple Cubic (SC) = \(8 \times \frac{1}{8} = 1\) atom
• Body-Centered Cubic (BCC) = \(\left(8 \times \frac{1}{8}\right) + 1 = 2\) atoms
• Face-Centered Cubic (FCC) = \(\left(8 \times \frac{1}{8}\right) + \left(6 \times \frac{1}{2}\right) = 4\) atoms

Answer 1

The Correct Option is C

Concept: When atoms are arranged at specific sites within a conventional unit cell, many of those atoms sit on shared corners or faces, meaning they are shared with neighboring cells. To find the effective number of atoms belonging exclusively to a single unit cell (\(N_{\text{eff}}\)), we must scale each atom based on how many cells share it.

Step 1: Counting contributions in a Face-Centered Cubic (FCC) lattice.
An FCC unit cell contains atoms at two types of geometric sites:
• Corner Atoms: There are 8 corner positions. Each corner point is shared equally among 8 adjacent surrounding cubic unit cells. Therefore, each corner atom contributes only an \(1/8\) fraction of its volume to a single cell: \[ N_{\text{corner}} = 8 \times \frac{1}{8} = 1 \text{ effective atom} \]
• Face Atoms: There are 6 faces on a cube. Each face-centered atom is shared directly between exactly 2 adjacent touching unit cells. Thus, each face atom contributes a \(1/2\) fraction of its volume to a single cell: \[ N_{\text{face}} = 6 \times \frac{1}{2} = 3 \text{ effective atoms} \]

Step 2: Summing up total effective atoms.
Adding the contributions from both types of lattice sites gives: \[ N_{\text{eff}} = N_{\text{corner}} + N_{\text{face}} = 1 + 3 = 4 \text{ atoms} \] Thus, a conventional Face-Centered Cubic unit cell contains exactly 4 effective atoms, matching option (C).