In a cubic unit cell atoms 'A' occupies the corners and centre, atoms 'B' occupies the face centres, and atoms 'C' occupies the edge centres. If all the atoms on one of the planes which is bisecting the unit cell is to two equal halves, are removed. Calculate the total number of atoms (X), present in the unit cell. What is the value of (X/4) ? (Given that, the plane is not a diagonal plane)

Explanation

To solve the problem, we need to calculate the total number of atoms present in the unit cell after removing the atoms on a bisecting plane. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the positions of the atoms in the unit cell
- **Atoms A** occupy the corners and the center of the unit cell.
- **Atoms B** occupy the face centers.
- **Atoms C** occupy the edge centers.

### Step 2: Calculate the contribution of each type of atom

1. **Atoms A:**
- There are 8 corners in a cubic unit cell. Each corner atom contributes \( \frac{1}{8} \) to the unit cell.
- Contribution from corner atoms = \( 8 \times \frac{1}{8} = 1 \)
- There is 1 atom at the center contributing fully (1).
- Total contribution from A = \( 1 + 1 = 2 \)

2. **Atoms B:**
- There are 6 faces in a cubic unit cell. Each face-centered atom contributes \( \frac{1}{2} \) to the unit cell.
- Contribution from face-centered atoms = \( 6 \times \frac{1}{2} = 3 \)

3. **Atoms C:**
- There are 12 edges in a cubic unit cell. Each edge-centered atom contributes \( \frac{1}{4} \) to the unit cell.
- Contribution from edge-centered atoms = \( 12 \times \frac{1}{4} = 3 \)

### Step 3: Total number of atoms before removal
- Total atoms \( X \) before removal = Contribution from A + Contribution from B + Contribution from C
- \( X = 2 + 3 + 3 = 8 \)

### Step 4: Removing atoms on the bisecting plane
- The bisecting plane divides the unit cell into two equal halves.
- Atoms A: 4 corner atoms are cut in half, contributing \( 4 \times \frac{1}{8} = \frac{1}{2} \) from each half, so total contribution from A after removal = \( 1 \) (the center atom remains).
- Atoms B: 4 face-centered atoms are cut in half, contributing \( 4 \times \frac{1}{2} = 2 \).
- Atoms C: 4 edge-centered atoms are cut in half, contributing \( 4 \times \frac{1}{4} = 1 \).

### Step 5: Calculate the total number of atoms after removal
- New total \( X \) after removal = Contribution from A + Contribution from B + Contribution from C
- \( X = 1 + 2 + 1 = 4 \)

### Step 6: Calculate \( \frac{X}{4} \)
- \( \frac{X}{4} = \frac{4}{4} = 1 \)

### Final Answer
- The total number of atoms present in the unit cell after removal is \( X = 4 \).
- The value of \( \frac{X}{4} = 1 \).