In a unit cell, atoms A, B, C and D are present at comers, face centres, body - centre and edge - centre respectively in a cubic unit cell. The total number of atoms present per unit cell is

Options

A. 4
B. 8
C. 15
D. 27

Answer

To find the total number of atoms present per unit cell in a cubic unit cell with atoms A, B, C, and D located at the corners, face centers, body center, and edge centers respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the contributions of atoms at different positions**:
- **Corner atoms (A)**: Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell. There are 8 corners in a cube.
- **Face-centered atoms (B)**: Each face-centered atom contributes \( \frac{1}{2} \) of an atom to the unit cell. There are 6 faces in a cube.
- **Body-centered atom (C)**: The body-centered atom contributes 1 whole atom to the unit cell.
- **Edge-centered atoms (D)**: Each edge-centered atom contributes \( \frac{1}{4} \) of an atom to the unit cell. There are 12 edges in a cube.

2. **Calculate the total contribution from each type of atom**:
- Contribution from corner atoms:
\[ \text{Total from corners} = 8 \times \frac{1}{8} = 1 \]
- Contribution from face-centered atoms:
\[ \text{Total from faces} = 6 \times \frac{1}{2} = 3 \]
- Contribution from the body-centered atom:
\[ \text{Total from body center} = 1 \times 1 = 1 \]
- Contribution from edge-centered atoms:
\[ \text{Total from edges} = 12 \times \frac{1}{4} = 3 \]

3. **Sum all contributions**:
\[ \text{Total number of atoms} = 1 \, (\text{from corners}) + 3 \, (\text{from faces}) + 1 \, (\text{from body}) + 3 \, (\text{from edges}) = 8 \]

4. **Conclusion**: The total number of atoms present per unit cell is 8.

### Final Answer:
The total number of atoms present per unit cell is **8**.