Which of the following expression is correct in case of a CsCl unit cell (edge length, a)?

Options

A. `r_(c)+r_(a)=a`
B. `r_(c)+r_(a)=a//sqrt2`
C. `r_(c)+r_(a)=sqrt3a//2`
D. `r_(c)+r_(a)=a//2`

Answer

To solve the question regarding the correct expression for the CsCl unit cell in terms of edge length \( a \), we can follow these steps:

### Step 1: Identify the Structure
The CsCl structure is a body-centered cubic (BCC) lattice. In this arrangement, chloride ions (Cl⁻) are located at the corners of the cube, while cesium ions (Cs⁺) are located at the body center.

### Step 2: Visualize the Unit Cell
Draw a cube to represent the unit cell. Mark the corners with chloride ions and the center with a cesium ion. There are 8 corners in a cube, each occupied by a chloride ion.

### Step 3: Calculate the Body Diagonal
In a cubic unit cell, the body diagonal can be calculated using the formula:
\[ \text{Body Diagonal} = \sqrt{3}a \]
This is derived from the Pythagorean theorem applied to the three dimensions of the cube.

### Step 4: Relate the Radii to the Body Diagonal
The body diagonal connects the two corner chloride ions to the body-centered cesium ion. The relationship between the radii of the ions and the body diagonal can be expressed as:
\[ \text{Body Diagonal} = r_A + 2r_C \]
Where:
- \( r_A \) is the radius of the chloride ion (anion),
- \( r_C \) is the radius of the cesium ion (cation).

### Step 5: Set Up the Equation
From the body diagonal relationship, we can equate:
\[ \sqrt{3}a = r_A + 2r_C \]

### Step 6: Rearranging the Equation
Rearranging the equation gives us:
\[ r_A + 2r_C = \sqrt{3}a \]
To express the sum of the radii in terms of the edge length \( a \), we can divide the entire equation by 2:
\[ \frac{r_A + 2r_C}{2} = \frac{\sqrt{3}}{2}a \]

### Step 7: Final Expression
We can express the relationship between the radii of the cation and anion as:
\[ r_A + r_C = \frac{\sqrt{3}}{2}a \]

### Conclusion
The correct expression for the CsCl unit cell in terms of edge length \( a \) is:
\[ r_A + r_C = \frac{\sqrt{3}}{2}a \]