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

### Step-by-Step Solution:

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

2. **Identifying the Face Diagonal**:
- In an FCC unit cell, the atoms touch along the face diagonal.
- The length of the face diagonal can be calculated using the Pythagorean theorem.

3. **Calculating the Length of the Face Diagonal**:
- The face diagonal (d) of a cube with edge length 'a' can be expressed as:
\[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]

4. **Determining the Number of Atoms Along the Face Diagonal**:
- Along the face diagonal, there are 4 atomic radii (since there are two atoms at the ends and one atom at the center).
- Therefore, the relationship can be expressed as:
\[ d = 4r \]

5. **Setting Up the Equation**:
- Equating the two expressions for the face diagonal gives:
\[ a\sqrt{2} = 4r \]

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

### Final Relationship:
Thus, the relationship between the edge length 'a' and the atomic radius 'r' in a face-centered cubic structure is:
\[ a = 2\sqrt{2}r \]