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 at specified positions, we can follow these steps:

### Step 1: Identify the positions of the atoms

- **Atom A**: Present at half of the total corners.
- **Atom B**: Present at all face centers.
- **Atom C**: Present at the body center.
- **Atom D**: Present at one third of all edge centers.

### Step 2: Calculate the number of atoms for each type

1. **Atom A**:
- Total corners in a cube = 8.
- Half of the corners = 8 / 2 = 4 corners.
- Contribution per corner atom = 1/8.
- Total contribution from A = 4 corners × (1/8) = 4/8 = 1/2.

2. **Atom B**:
- Total face centers = 6 (one for each face of the cube).
- Contribution per face center atom = 1/2.
- Total contribution from B = 6 faces × (1/2) = 6/2 = 3.

3. **Atom C**:
- Body center = 1 (always one in a unit cell).
- Contribution from C = 1.

4. **Atom D**:
- Total edges in a cube = 12.
- One third of the edges = 12 / 3 = 4 edges.
- Contribution per edge atom = 1/4.
- Total contribution from D = 4 edges × (1/4) = 4/4 = 1.

### Step 3: Summarize the contributions

- Total number of atoms:
- A = 1/2
- B = 3
- C = 1
- D = 1

### Step 4: Convert to integer form

To express the formula in whole numbers, we can multiply all contributions by 2 (the least common multiple of the denominators):

- A = (1/2) × 2 = 1
- B = 3 × 2 = 6
- C = 1 × 2 = 2
- D = 1 × 2 = 2

### Step 5: Write the final formula

The final formula of the unit cell is:
\[ \text{A}_1\text{B}_6\text{C}_2\text{D}_2 \]
This can be simplified to:
\[ \text{AB}_6\text{C}_2\text{D}_2 \]

### Conclusion

Thus, the formula of the unit cell is **AB6C2D2**.

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