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 the FCC Structure**: In an FCC unit cell, atoms are located at each of the eight corners of the cube and at the centers of each of the six faces.

2. **Visualizing the Arrangement**: Consider the face of the cube. The atoms at the corners and the face center touch each other along the diagonal of the face.

3. **Diagonal of the Face**: The diagonal of the face of the cube can be expressed in terms of the edge length (a). For a square face, the diagonal (d) can be calculated using the Pythagorean theorem:
\[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]

4. **Atoms Along the Diagonal**: Along this diagonal, there are three atomic radii (r) since there is one atom at each corner and one atom at the center of the face. Therefore, we can express this relationship as:
\[ d = r + 2r + r = 4r \]

5. **Setting the Equations Equal**: Now, we can set the two expressions for the diagonal equal to each other:
\[ 4r = a\sqrt{2} \]

6. **Solving for r**: To find the radius (r), we can rearrange the equation:
\[ r = \frac{a\sqrt{2}}{4} = \frac{a}{2\sqrt{2}} \]

7. **Final Relation**: Thus, the correct relation for the radius of the atom and the edge length in case of FCC arrangement is:
\[ r = \frac{a}{2\sqrt{2}} \]

### Conclusion:
The correct option is option 3: \( r = \frac{a}{2\sqrt{2}} \).