In an idaeal closet rock salt structure (edge length a) which of the following expression is correct?

Options

A. `r_(a)=sqrt2a`
B. `r_(a)=a//sqrt2`
C. `r_(a)=a//2sqrt2`
D. `r_(a)=a//4`

Answer

To solve the problem regarding the ideal rock salt structure (NaCl) and its edge length \( a \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Structure**:
The rock salt structure is a face-centered cubic (FCC) lattice where chloride ions (\( Cl^- \)) occupy the face-centered positions and sodium ions (\( Na^+ \)) occupy the octahedral voids.

2. **Identifying the Unit Cell**:
In the FCC structure, the unit cell has atoms at the corners and at the centers of each face. The edge length of the unit cell is given as \( a \).

3. **Finding the Face Diagonal**:
In an FCC lattice, the face diagonal can be calculated using the Pythagorean theorem. For a face of the cube, the face diagonal \( d \) can be expressed as:
\[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]

4. **Relating the Face Diagonal to Atomic Radii**:
In the rock salt structure, the face diagonal is equal to the sum of the diameters of the two corner atoms and the face-centered atom. Since the radius of each ion is \( r \), the face diagonal can be expressed as:
\[ d = 4r \]
Thus, we have:
\[ a\sqrt{2} = 4r \]

5. **Solving for the Radius \( r \)**:
Rearranging the equation gives:
\[ r = \frac{a\sqrt{2}}{4} \]
This can be simplified further:
\[ r = \frac{a}{2\sqrt{2}} \]

6. **Conclusion**:
The expression for the radius \( r \) of the ions in the ideal rock salt structure is:
\[ r = \frac{a}{2\sqrt{2}} \]
Therefore, the correct option that represents this expression is \( \frac{a}{2\sqrt{2}} \).