What would be the effective number of atoms per unit cell in end centred cubic uit cel, if this type of unit cell exist in nautre?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the effective number of atoms per unit cell in an end-centered cubic unit cell, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the End-Centered Cubic Unit Cell**:
- An end-centered cubic unit cell has atoms located at the corners of the cube and at the centers of two opposite faces.

2. **Counting Corner Atoms**:
- In a cubic unit cell, there are 8 corners.
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell because it is shared among 8 adjacent unit cells.
- Therefore, the total contribution from the corner atoms is:
\[ \text{Contribution from corner atoms} = 8 \times \frac{1}{8} = 1 \]

3. **Counting Face-Centered Atoms**:
- In the end-centered cubic unit cell, there are 2 face-centered atoms (one on each of the two opposite faces).
- Each face-centered atom contributes \( \frac{1}{2} \) of an atom to the unit cell because it is shared between two adjacent unit cells.
- Therefore, the total contribution from the face-centered atoms is:
\[ \text{Contribution from face-centered atoms} = 2 \times \frac{1}{2} = 1 \]

4. **Calculating the Total Effective Number of Atoms**:
- Now, we can sum the contributions from the corner atoms and the face-centered atoms to find the total effective number of atoms per unit cell:
\[ \text{Total effective number of atoms} = 1 \, (\text{from corners}) + 1 \, (\text{from faces}) = 2 \]

5. **Conclusion**:
- The effective number of atoms per unit cell in an end-centered cubic unit cell is **2**.

### Final Answer:
The effective number of atoms per unit cell in an end-centered cubic unit cell is **2**.

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