A face centred cubic is made up of two types of atoms A and B in which A occupies the corner positions and B occupies the face centres. If atoms along one of the body diagonal are removed, empirical formula of remainning solid will be

Options

A. `AB_(2)`
B. `A_(3)B`
C. `A_(7)B_(3)`
D. `AB_(4)`

Explanation

To solve the problem step by step, we will analyze the structure of the face-centered cubic (FCC) lattice and determine the empirical formula after removing specific atoms.

### Step 1: Determine the number of atoms in the FCC structure
In a face-centered cubic (FCC) lattice:
- **Atom A** occupies the corners.
- **Atom B** occupies the face centers.

**Atoms of A:**
- There are 8 corners in a cube.
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell.
- Therefore, the total contribution from corner atoms is:
\[ \text{Total A} = 8 \times \frac{1}{8} = 1 \text{ atom of A} \]

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

### Step 2: Total atoms in the unit cell
After calculating, we have:
- Total atoms of A = 1
- Total atoms of B = 3

### Step 3: Identify the atoms along the body diagonal
In a cube, the body diagonal connects two opposite corners and passes through the center atom:
- Along the body diagonal, we have:
- 2 corner atoms (A)
- 1 body-centered atom (which is B)

### Step 4: Remove the atoms along the body diagonal
When we remove the atoms along one body diagonal:
- We remove 2 corner atoms of A.
- The body-centered atom (B) does not contribute to the removal of A.

### Step 5: Calculate the remaining atoms
After removing the 2 corner atoms of A:
- Remaining atoms of A:
\[ \text{Remaining A} = 1 - 2 = -1 \text{ (not possible, so we take it as 0)} \]
Since we cannot have negative atoms, we consider that the contribution of A is effectively 0 after the removal.

- Remaining atoms of B:
\[ \text{Remaining B} = 3 \text{ (no B atoms were removed)} \]

### Step 6: Determine the empirical formula
Now we have:
- Remaining A = 0
- Remaining B = 3

The empirical formula can be expressed as:
\[ \text{Empirical formula} = A_0B_3 = B_3 \]

However, since we are looking for a ratio, we can express it as:
\[ \text{Empirical formula} = A_0B_3 \Rightarrow \text{We can represent it as } A_0B_3 \text{ or simply } B_3 \]

### Final Answer
The empirical formula of the remaining solid after the removal of atoms along the body diagonal is:
\[ \text{Empirical Formula} = A_0B_3 \text{ or simply } B_3 \]