In a CsCl structure, if edge length is a, then distance between one Cs atom and one Cl atoms is

Options

A. `(a sqrt(3))/(2)`
B. `(a sqrt(3))/(4)`
C. `a sqrt(2)`
D. `(a)/(sqrt(2))`

Answer

To find the distance between one Cs atom and one Cl atom in a CsCl structure, we can follow these steps:

### Step 1: Understand the CsCl Structure
In the CsCl crystal structure, cesium (Cs) atoms are located at the body center of the cube, while chloride (Cl) ions are located at the corners of the cube.

### Step 2: Identify the Edge Length
Let the edge length of the cube be denoted as \( a \).

### Step 3: Determine the Body Diagonal
The body diagonal of a cube can be calculated using the formula:
\[ \text{Body Diagonal} = a \sqrt{3} \]

### Step 4: Relate the Body Diagonal to Atomic Radii
In the CsCl structure, the body diagonal connects two corner Cl ions and one Cs ion at the center. The body diagonal can be expressed in terms of the radii of the ions:
\[ \text{Body Diagonal} = 2R_{Cl} + 2R_{Cs} \]
where \( R_{Cl} \) is the radius of the Cl ion and \( R_{Cs} \) is the radius of the Cs ion.

### Step 5: Set Up the Equation
From the above relationships, we can equate the two expressions for the body diagonal:
\[ a \sqrt{3} = 2R_{Cl} + 2R_{Cs} \]

### Step 6: Find the Distance Between Cs and Cl
The distance \( D \) between the Cs atom and the Cl atom is simply the sum of their radii:
\[ D = R_{Cs} + R_{Cl} \]

### Step 7: Solve for the Distance
From the equation derived in Step 5, we can express \( D \):
\[ D = \frac{a \sqrt{3}}{2} \]

### Final Answer
Thus, the distance between one Cs atom and one Cl atom in the CsCl structure is:
\[ D = \frac{a \sqrt{3}}{2} \]

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