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

Explanation

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

### Step-by-Step Solution:

1. **Understand the Structure**: In a BCC unit cell, there is one atom at each of the eight corners and one atom at the center of the cube. The length of the side of the cube is denoted as \( a \).

2. **Identify the Atoms**: We need to find the distance between the body-centered atom (located at the center of the cube) and one of the corner atoms (located at one of the corners of the cube).

3. **Determine the Coordinates**:
- The coordinates of the corner atom can be taken as \( (0, 0, 0) \).
- The coordinates of the body-centered atom are \( \left(\frac{a}{2}, \frac{a}{2}, \frac{a}{2}\right) \).

4. **Use the Distance Formula**: The distance \( d \) between two points in three-dimensional space can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the coordinates of the corner atom and the body-centered atom:
\[ d = \sqrt{\left(\frac{a}{2} - 0\right)^2 + \left(\frac{a}{2} - 0\right)^2 + \left(\frac{a}{2} - 0\right)^2} \]

5. **Simplify the Expression**:
\[ d = \sqrt{\left(\frac{a}{2}\right)^2 + \left(\frac{a}{2}\right)^2 + \left(\frac{a}{2}\right)^2} \]
\[ = \sqrt{3 \left(\frac{a}{2}\right)^2} \]
\[ = \sqrt{3} \cdot \frac{a}{2} \]
\[ = \frac{\sqrt{3} a}{2} \]

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