A mineral having the formula `AB_(2)`, crystallises in the cubic close -packed lattice, with the A atom occupying the lattice points. The co-ordination number of the A atoms is x, that of B atoms is y. Report your answer `( x- y)`

Options

A. 4, 8
B. 4, 4
C. 8, 8
D. 8, 4

Answer

To solve the problem, we need to analyze the crystal structure of the mineral with the formula \( AB_2 \) that crystallizes in a cubic close-packed (CCP) lattice. Here are the steps to find the coordination numbers \( x \) and \( y \) for the A and B atoms, respectively, and then calculate \( x - y \).

### Step-by-Step Solution:

1. **Understanding the Crystal Structure**:
- The mineral has the formula \( AB_2 \) and crystallizes in a cubic close-packed (CCP) lattice.
- In a CCP lattice, the A atoms occupy the lattice points (corners and face centers).

2. **Determine the Coordination Number of A**:
- In a face-centered cubic (FCC) structure (which is the same as CCP), each atom is surrounded by 12 other atoms.
- However, since we have \( AB_2 \), we need to consider the arrangement of B atoms around A.
- In this case, A atoms will have a coordination number of 8, as they are surrounded by B atoms in the tetrahedral voids.

3. **Determine the Coordination Number of B**:
- The B atoms occupy the tetrahedral voids in the FCC lattice.
- Each tetrahedral void is surrounded by 4 A atoms.
- Therefore, the coordination number of B is 4.

4. **Calculate \( x - y \)**:
- From the previous steps, we have:
- Coordination number of A, \( x = 8 \)
- Coordination number of B, \( y = 4 \)
- Now, calculate \( x - y \):
\[ x - y = 8 - 4 = 4 \]

### Final Answer:
The value of \( x - y \) is **4**.