In `BeO` (zinc blende structure), `Mg^(2+)` is introduced in available `TV` is. The coordination numbers of `Be^(2+)` and `Mg^(2+)` are, respectively,

Options

A. `8,8`
B. `6,6`
C. `4,4`
D. `8,6`

Explanation

To solve the problem, we need to determine the coordination numbers of `Be^(2+)` and `Mg^(2+)` in the zinc blende structure of `BeO` when `Mg^(2+)` is introduced into the available tetrahedral voids.

### Step-by-Step Solution:

1. **Understanding the Structure**:
- The zinc blende structure is characterized by a face-centered cubic (FCC) arrangement of anions (O²⁻ in this case) with cations (Be²⁺) occupying half of the tetrahedral voids.

2. **Determine the Number of O²⁻ Ions**:
- In an FCC unit cell, there are:
- 8 corner atoms, each contributing \( \frac{1}{8} \) (total contribution from corners = \( 8 \times \frac{1}{8} = 1 \)).
- 6 face-centered atoms, each contributing \( \frac{1}{2} \) (total contribution from faces = \( 6 \times \frac{1}{2} = 3 \)).
- Total number of O²⁻ ions in the unit cell = \( 1 + 3 = 4 \).

3. **Determine the Number of Tetrahedral Voids**:
- In a face-centered cubic structure, there are 8 tetrahedral voids.
- Since Be²⁺ occupies half of these tetrahedral voids, the number of Be²⁺ ions = \( \frac{8}{2} = 4 \).

4. **Determine the Number of Available Tetrahedral Voids for Mg²⁺**:
- The total number of tetrahedral voids = 8.
- The number of tetrahedral voids occupied by Be²⁺ = 4.
- Therefore, the number of available tetrahedral voids for Mg²⁺ = \( 8 - 4 = 4 \).

5. **Coordination Numbers**:
- The coordination number is defined as the number of nearest neighbors surrounding a central atom.
- In this structure, both Be²⁺ and Mg²⁺ occupy tetrahedral voids.
- Each cation (Be²⁺ and Mg²⁺) in a tetrahedral void has a coordination number of 4, as they are surrounded by 4 O²⁻ ions.

### Final Answer:
- The coordination numbers of `Be^(2+)` and `Mg^(2+)` are both 4.