If a is the length of the side of a cube, the distance between the body centred atom and one corner atom in the cube will be:

Options

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

Answer

To find the distance between the body-centered atom and one corner atom in a body-centered cubic (BCC) structure, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the BCC Structure**: In a body-centered cubic (BCC) lattice, there is one atom at each corner of the cube and one atom at the center of the cube.

2. **Identify the Length of the Side of the Cube**: Let the length of the side of the cube be denoted as \( a \).

3. **Determine the Position of the Atoms**:
- The corner atom can be positioned at the coordinates (0, 0, 0).
- The body-centered atom is positioned at the coordinates (a/2, a/2, a/2).

4. **Calculate the Distance**:
- To find the distance \( d \) between the corner atom and the body-centered atom, we can use the distance formula in three-dimensional space:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
- Substituting the coordinates:
\[ d = \sqrt{(a/2 - 0)^2 + (a/2 - 0)^2 + (a/2 - 0)^2} \]
- This simplifies to:
\[ d = \sqrt{(a/2)^2 + (a/2)^2 + (a/2)^2} = \sqrt{3 \cdot (a/2)^2} = \sqrt{3} \cdot \frac{a}{2} \]
- Therefore, the distance \( d \) is:
\[ d = \frac{\sqrt{3}}{2} a \]

5. **Final Expression**: The distance between the body-centered atom and one corner atom in the cube is:
\[ d = \frac{\sqrt{3}}{2} a \]

### Conclusion:
The correct answer is \( \frac{\sqrt{3}}{2} a \), which corresponds to option 4.