The correct relation for radius of atom and edge - length in case of fcc arrangement is

Options

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

Answer

To find the correct relation between the radius of an atom (r) and the edge length (a) in a face-centered cubic (FCC) arrangement, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding FCC Structure**:
In a face-centered cubic (FCC) arrangement, atoms are located at each of the corners of the cube and at the center of each face.

2. **Visualizing the Face Diagonal**:
To find the relationship between the radius and edge length, we will consider the face diagonal of the cube. The face diagonal connects two corners of a face and passes through the center atom.

3. **Setting Up the Geometry**:
Let's denote the edge length of the cube as \( a \). Each side of the face forms a right triangle with the face diagonal. The face diagonal can be calculated using the Pythagorean theorem.

4. **Calculating the Face Diagonal**:
The length of the face diagonal (d) can be calculated as:
\[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]

5. **Identifying the Atoms on the Face Diagonal**:
On the face diagonal, there are three atoms contributing to the total length: one at each corner and one at the face center. Each corner atom contributes a radius \( r \), and the face-centered atom contributes another radius \( r \).

6. **Setting Up the Equation**:
The total length of the face diagonal can be expressed in terms of the radius:
\[ d = 4r \]
Therefore, we can equate the two expressions for the face diagonal:
\[ a\sqrt{2} = 4r \]

7. **Solving for the Radius**:
Rearranging the equation to solve for \( r \):
\[ r = \frac{a\sqrt{2}}{4} \]

8. **Simplifying the Expression**:
To simplify further, we can express \( r \) as:
\[ r = \frac{a}{2\sqrt{2}} \]

### Conclusion:
The correct relation for the radius of an atom and the edge length in the case of an FCC arrangement is:
\[ r = \frac{a}{2\sqrt{2}} \]