A substance `A_(x)B_(y)` crystallizes in a face-centred cubic lattice in which atoms `A` occupy the centres of each face of the cube. Identify the correct composition of the substance `A_(x)B_(y)`.

Options

A. `AB_(3)`
B. `A_(4)B_(3)`
C. `A_(3)B`
D. Cannot be specified

Explanation

To determine the composition of the substance \( A_xB_y \) that crystallizes in a face-centered cubic (FCC) lattice with atoms \( A \) occupying the centers of each face of the cube, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the FCC Lattice Structure**:
- In a face-centered cubic lattice, there are atoms located at the corners of the cube and at the center of each face.

2. **Counting the Atoms**:
- **Atoms at the Corners**: There are 8 corners in a cube, and each corner atom is shared by 8 adjacent cubes. Therefore, the contribution of each corner atom to one cube is \( \frac{1}{8} \).
- Total contribution from corner atoms:
\[ 8 \times \frac{1}{8} = 1 \text{ atom of } B \]
- **Atoms at the Face Centers**: There are 6 faces in a cube, and each face atom is shared by 2 adjacent cubes. Therefore, the contribution of each face atom to one cube is \( \frac{1}{2} \).
- Total contribution from face-centered atoms:
\[ 6 \times \frac{1}{2} = 3 \text{ atoms of } A \]

3. **Determining the Composition**:
- From the above calculations, we find that there are 3 atoms of \( A \) and 1 atom of \( B \) in the unit cell.
- Thus, the composition of the substance can be expressed as:
\[ A_3B_1 \]

### Final Answer:
The correct composition of the substance \( A_xB_y \) is \( A_3B_1 \).

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