The interionic distance for cesium chloride crystal will be

Options

A. `a`
B. `a//2`
C. `sqrt3a//2`
D. `2a//sqrt3`

Explanation

To find the interionic distance for a cesium chloride (CsCl) crystal, we need to analyze the structure of the crystal and the positions of the ions within it.

### Step-by-Step Solution:

1. **Identify the Structure of CsCl**:
- CsCl crystallizes in a face-centered cubic (FCC) lattice structure.
- In this structure, Cl⁻ ions are located at the corners of the cube, while Cs⁺ ions are located at the body center of the cube.

2. **Determine the Positions of Ions**:
- The corner positions of the cube can be represented as (0,0,0), (0,0,1), (0,1,0), (1,0,0), etc.
- The body center position is at (0.5, 0.5, 0.5) when considering a unit cell of length 'a'.

3. **Calculate the Distance Between Cs⁺ and Cl⁻**:
- The interionic distance is the distance between the Cs⁺ ion at the body center and a Cl⁻ ion at one of the corners.
- The distance can be calculated using the formula for the distance between two points in 3D space:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
- For the corner ion at (0, 0, 0) and the body center ion at (0.5, 0.5, 0.5), the distance 'd' becomes:
\[ d = \sqrt{(0.5 - 0)^2 + (0.5 - 0)^2 + (0.5 - 0)^2} \]
\[ d = \sqrt{(0.5)^2 + (0.5)^2 + (0.5)^2} = \sqrt{0.25 + 0.25 + 0.25} = \sqrt{0.75} = \frac{\sqrt{3}}{2} \]

4. **Relate the Distance to the Unit Cell Length**:
- The calculated distance is in terms of the unit cell length 'a'. To express it in terms of 'a', we note that the distance from the body center to a corner is:
\[ d = \frac{\sqrt{3}}{2} a \]
- Therefore, the interionic distance between Cs⁺ and Cl⁻ in the CsCl crystal is:
\[ \text{Interionic Distance} = \frac{\sqrt{3}}{2} a \]

5. **Select the Correct Option**:
- From the options given, the correct answer is:
- Option C: \(\frac{\sqrt{3}}{2} a\)

### Final Answer:
The interionic distance for cesium chloride crystal is \(\frac{\sqrt{3}}{2} a\) (Option C).