In a unit cell, atoms A, B, C and D are present at half of total corners, all face - centres, body - centre and one third of all edge - centres respectively. Then formula of unit cell is

Options

A. `AB_(3)CD_(3)`
B. ABCD
C. `AB_(6)C_(2)D_(4)`
D. `AB_(6)C_(2)D_(2)`

Answer

To determine the formula of the unit cell containing atoms A, B, C, and D, we will analyze the contributions of each type of atom based on their positions in the unit cell.

### Step-by-Step Solution:

1. **Identify the Positions of Atoms:**
- Atom A is present at half of the total corners.
- Atom B is present at all face centers.
- Atom C is present at the body center.
- Atom D is present at one-third of all edge centers.

2. **Count the Number of Each Position:**
- There are 8 corners in a cube.
- There are 6 face centers in a cube.
- There is 1 body center in a cube.
- There are 12 edges in a cube.

3. **Calculate Contributions:**
- **Atom A:** Since A is present at half of the corners, we have:
\[ \text{Contribution of A} = \frac{1}{2} \times \text{(number of corners)} = \frac{1}{2} \times 8 = 4 \]
- **Atom B:** Since B is present at all face centers, we have:
\[ \text{Contribution of B} = 6 \times \frac{1}{2} = 3 \]
- **Atom C:** Since C is present at the body center:
\[ \text{Contribution of C} = 1 \]
- **Atom D:** Since D is present at one-third of all edge centers, we have:
\[ \text{Contribution of D} = \frac{1}{3} \times 12 = 4 \]

4. **Summarize Contributions:**
- Total contributions:
- A: 4
- B: 3
- C: 1
- D: 4

5. **Write the Empirical Formula:**
- The formula based on the contributions is:
\[ A_4B_3C_1D_4 \]

6. **Convert to Whole Numbers:**
- To express the formula in whole numbers, we can divide each coefficient by the greatest common divisor if necessary. In this case, we can represent it as:
\[ A_4B_3C_1D_4 \]

### Final Formula:
The final formula of the unit cell is:
\[ A_4B_3C_1D_4 \]