First three nearest neighboure distance for primitive cubic lattice are respectively (edge length of unit cell = a):

Options

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

Answer

To find the first three nearest neighbor distances for a primitive cubic lattice with edge length \( a \), we can follow these steps:

### Step 1: Identify the structure of the primitive cubic lattice
In a primitive cubic lattice, atoms are located at the corners of the cube. Each corner atom is shared by eight adjacent unit cells.

### Step 2: Determine the first nearest neighbor
The first nearest neighbor is the atom that is directly adjacent to the atom in question. For a primitive cubic lattice, this neighbor is located at the edge of the cube.

- **Distance to the first nearest neighbor (A)**:
\[ d_1 = a \]

### Step 3: Determine the second nearest neighbor
The second nearest neighbors are located at the face diagonals of the cube. To find the distance to this neighbor, we can use the Pythagorean theorem.

- **Distance to the second nearest neighbor (B)**:
\[ d_2 = \sqrt{a^2 + a^2} = \sqrt{2a^2} = \sqrt{2}a \]

### Step 4: Determine the third nearest neighbor
The third nearest neighbors are located at the body diagonal of the cube. To find this distance, we again apply the Pythagorean theorem.

- **Distance to the third nearest neighbor (C)**:
\[ d_3 = \sqrt{a^2 + (\sqrt{2}a)^2} = \sqrt{a^2 + 2a^2} = \sqrt{3a^2} = \sqrt{3}a \]

### Final Result
Thus, the first three nearest neighbor distances for a primitive cubic lattice are:
1. First nearest neighbor: \( a \)
2. Second nearest neighbor: \( \sqrt{2}a \)
3. Third nearest neighbor: \( \sqrt{3}a \)

### Summary
The distances to the first three nearest neighbors in a primitive cubic lattice with edge length \( a \) are:
- \( a \)
- \( \sqrt{2}a \)
- \( \sqrt{3}a \)