For face centered cubic structure edge length 'a' can be related with radius 'r' as

Options

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

Answer

To find the relationship between the edge length 'a' and the radius 'r' in a face-centered cubic (FCC) structure, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the FCC Structure**:
- In a face-centered cubic (FCC) unit cell, atoms are located at each of the corners and at the center of each face of the cube.
- Each corner atom is shared among eight adjacent unit cells, and each face-centered atom is shared between two unit cells.

2. **Identifying Atom Positions**:
- The atoms at the corners contribute a total of 1 atom (8 corners × 1/8 atom per corner).
- The atoms at the faces contribute a total of 1 atom (6 faces × 1/2 atom per face).
- Therefore, the total number of atoms per FCC unit cell (Z) is 4.

3. **Relating Edge Length and Atomic Radius**:
- In an FCC structure, the face diagonal of the cube can be expressed in terms of the atomic radius.
- The face diagonal can be calculated using the Pythagorean theorem:
\[ \text{Face diagonal} = \sqrt{a^2 + a^2} = \sqrt{2}a \]
- The face diagonal also equals the sum of the diameters of the two face-centered atoms and the two corner atoms:
\[ \text{Face diagonal} = 4r \]

4. **Setting the Equations Equal**:
- From the two expressions for the face diagonal, we have:
\[ \sqrt{2}a = 4r \]

5. **Solving for Edge Length 'a'**:
- To find 'a' in terms of 'r', we rearrange the equation:
\[ a = \frac{4r}{\sqrt{2}} = 2\sqrt{2}r \]

6. **Conclusion**:
- The relationship between the edge length 'a' and the atomic radius 'r' in a face-centered cubic structure is:
\[ a = 2\sqrt{2}r \]

### Final Answer:
The correct relation is \( a = 2\sqrt{2}r \).