Step 1: Understanding the Question:
We need to determine the effective number of atoms that belong to a single unit cell in a Face-Centered Cubic (FCC) lattice structure. This involves summing the contributions of atoms at different positions within the cell.
Step 2: Key Formula or Approach:
The total number of atoms per unit cell is the sum of the contributions from atoms at different lattice positions. The key contribution factors are:
- Contribution of a corner atom = \(1/8\) (shared by 8 cells).
- Contribution of a face-centered atom = \(1/2\) (shared by 2 cells).
Step 3: Detailed Explanation:
An FCC unit cell has atoms at two types of locations:
(i) At the corners:
A cube has 8 corners. Each atom at a corner is shared by 8 adjacent unit cells.
Contribution from corner atoms = \(8 \, \text{corners} \times \frac{1}{8} \, \frac{\text{atom}}{\text{corner}} = 1 \, \text{atom}\).
(ii) At the center of the faces:
A cube has 6 faces. Each atom at the center of a face is shared by 2 adjacent unit cells.
Contribution from face-centered atoms = \(6 \, \text{faces} \times \frac{1}{2} \, \frac{\text{atom}}{\text{face}} = 3 \, \text{atoms}\).
(iii) Total number of atoms:
The total effective number of atoms per FCC unit cell is the sum of these contributions:
Total atoms = (Contribution from corners) + (Contribution from faces) = \(1 + 3 = 4\).
Step 4: Final Answer:
There are 4 atoms per unit cell in an FCC crystal structure.