First three nearestneighbour distances for body centered cubic lattice are respectively :

Options

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

Answer

To find the first three nearest neighbor distances for a body-centered cubic (BCC) lattice, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the BCC Structure**:
In a body-centered cubic lattice, there are atoms located at each corner of the cube and one atom at the center of the cube.

2. **Identifying Nearest Neighbors**:
The nearest neighbors to the body-centered atom (let's call it atom A) are the corner atoms. In a BCC lattice, there are 8 corner atoms.

3. **Calculating the Distance to the First Nearest Neighbors**:
The distance from the body-centered atom (A) to each of the corner atoms can be calculated. The distance between atom A and any corner atom is equal to the edge length (a) of the cube. Therefore, the first three nearest neighbor distances are:
- Distance to corner atom 1 = a
- Distance to corner atom 2 = a
- Distance to corner atom 3 = a

4. **Calculating the Distance to the Second Nearest Neighbors**:
The second nearest neighbors are also located at the corners but at a different distance. The distance to the second nearest neighbors can be calculated using the face diagonal of the cube. The face diagonal can be calculated using the Pythagorean theorem:
\[ \text{Distance} = \sqrt{a^2 + a^2} = \sqrt{2a^2} = \sqrt{2}a \]
Thus, the second nearest neighbor distance is:
- Distance to second nearest neighbor = \(\sqrt{2}a\)

5. **Calculating the Distance to the Third Nearest Neighbors**:
The third nearest neighbors are located at the body diagonal of the cube. The body diagonal can also be calculated using the Pythagorean theorem:
\[ \text{Body diagonal} = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2} = \sqrt{3}a \]
Since the body-centered atom is at the midpoint of the body diagonal, the distance to the third nearest neighbor is half of this distance:
\[ \text{Distance} = \frac{\sqrt{3}a}{2} \]

6. **Final Summary of Distances**:
- First nearest neighbor distance = \( a \)
- Second nearest neighbor distance = \( \sqrt{2}a \)
- Third nearest neighbor distance = \( \frac{\sqrt{3}a}{2} \)