What are the number of atoms per unit cell and the number of nearest neighbours in a face centered cubic structure?

Options

A. (a) 4,8
B. (b) 2,8
C. (c) 2,6
D. (d) 4,12

Answer

To solve the question regarding the number of atoms per unit cell and the number of nearest neighbors in a face-centered cubic (FCC) structure, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the FCC Structure**:
- In a face-centered cubic (FCC) unit cell, atoms are located at the corners and the centers of each face of the cube.

2. **Counting Corner Atoms**:
- There are 8 corners in a cube. Each corner atom is shared by 8 adjacent unit cells.
- Therefore, the contribution of each corner atom to the unit cell is \( \frac{1}{8} \).
- Total contribution from corner atoms = \( 8 \times \frac{1}{8} = 1 \).

3. **Counting Face-Centered Atoms**:
- There are 6 faces in a cube, and each face has 1 atom at its center.
- Each face-centered atom is shared by 2 adjacent unit cells.
- Therefore, the contribution of each face-centered atom to the unit cell is \( \frac{1}{2} \).
- Total contribution from face-centered atoms = \( 6 \times \frac{1}{2} = 3 \).

4. **Calculating Total Number of Atoms per Unit Cell (Z)**:
- Total number of atoms per unit cell \( Z \) = Contribution from corner atoms + Contribution from face-centered atoms
- \( Z = 1 + 3 = 4 \).

5. **Finding the Coordination Number**:
- The coordination number is defined as the number of nearest neighbors surrounding a given atom.
- In the FCC structure, each atom has 12 nearest neighbors.
- This is because each atom is surrounded by 4 atoms in the same plane and 4 atoms in the planes above and below it.

6. **Final Answer**:
- The number of atoms per unit cell in an FCC structure is **4**, and the number of nearest neighbors (coordination number) is **12**.

### Summary:
- Number of atoms per unit cell (Z) = 4
- Number of nearest neighbors (Coordination number) = 12