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 in a unit cell with atoms A, B, C, and D located at the corners, face centers, body center, and edge centers respectively, we will calculate the contribution of each type of atom based on their positions in the cubic unit cell.

### Step-by-Step Solution:

1. **Identify the Positions of Atoms:**
- Atom A is located at the corners of the cube.
- Atom B is located at the face centers of the cube.
- Atom C is located at the body center of the cube.
- Atom D is located at the edge centers of the cube.

2. **Calculate the Contribution of Atom A:**
- There are 8 corners in a cubic unit cell.
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell.
- Total contribution from A:
\[ \text{Contribution from A} = 8 \times \frac{1}{8} = 1 \text{ atom} \]

3. **Calculate the Contribution of Atom B:**
- There are 6 face centers in a cubic unit cell.
- Each face-centered atom contributes \( \frac{1}{2} \) of an atom to the unit cell.
- Total contribution from B:
\[ \text{Contribution from B} = 6 \times \frac{1}{2} = 3 \text{ atoms} \]

4. **Calculate the Contribution of Atom C:**
- There is 1 body center in a cubic unit cell.
- The body-centered atom contributes 1 whole atom to the unit cell.
- Total contribution from C:
\[ \text{Contribution from C} = 1 \text{ atom} \]

5. **Calculate the Contribution of Atom D:**
- There are 12 edges in a cubic unit cell.
- Each edge-centered atom contributes \( \frac{1}{4} \) of an atom to the unit cell.
- Total contribution from D:
\[ \text{Contribution from D} = 12 \times \frac{1}{4} = 3 \text{ atoms} \]

6. **Sum the Contributions:**
- Total number of atoms in the unit cell:
\[ \text{Total atoms} = \text{Contribution from A} + \text{Contribution from B} + \text{Contribution from C} + \text{Contribution from D} \]
\[ \text{Total atoms} = 1 + 3 + 1 + 3 = 8 \text{ atoms} \]

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