The relation between atomic radius and edge length 'a' of a body centred cubic unit cell :

Options

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

Answer

To find the relationship between the atomic radius (r) and the edge length (a) of a body-centered cubic (BCC) unit cell, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Structure of BCC:**
In a body-centered cubic unit cell, there are atoms located at each of the eight corners of the cube and one atom at the center of the cube.

2. **Identify the Atomic Arrangement:**
The atoms at the corners are in contact with the atom at the center along the body diagonal of the cube.

3. **Determine the Length of the Body Diagonal:**
The body diagonal (d) of a cube can be calculated using the formula:
\[ d = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2} = \sqrt{3}a \]

4. **Relate the Body Diagonal to Atomic Radii:**
The body diagonal consists of the radius of the corner atom (r), the diameter of the center atom (2r), and the radius of the opposite corner atom (r). Therefore, the total length of the body diagonal can be expressed as:
\[ d = r + 2r + r = 4r \]

5. **Set the Two Expressions for the Body Diagonal Equal:**
From the previous steps, we have:
\[ \sqrt{3}a = 4r \]

6. **Solve for the Atomic Radius (r):**
Rearranging the equation gives:
\[ r = \frac{\sqrt{3}a}{4} \]

7. **Final Result:**
Thus, the relationship between the atomic radius and the edge length of a body-centered cubic unit cell is:
\[ r = \frac{\sqrt{3}}{4}a \]

### Conclusion:
The correct answer is option C: \( r = \frac{\sqrt{3}}{4}a \).