Consider a Body Centred Cube (BCC) arrangement, let `d_(e), d_(fd), d_(bd)` be the distances between successive atoms located along the edge, the face-diagonal, the body diagonal respectively in a unit cell. Their order is given as:

Options

A. `d_(e) lt d_(fd) lt d_(bd)`
B. `d_(fd) gt d_(bd) gt d_(e)`
C. `d_(fd) gt d_(e) gt d_(bd)`
D. `d_(bd) gt d_(e) gt d_(fd)`

Answer

To solve the problem regarding the distances between successive atoms in a Body-Centered Cubic (BCC) arrangement, we will follow these steps:

### Step 1: Understand the BCC Structure
In a Body-Centered Cubic (BCC) arrangement, there are 8 atoms located at the corners of the cube and 1 atom located at the center of the cube.

### Step 2: Identify the Distances
We need to find the distances between successive atoms along three different paths:
1. **Edge Distance (D_E)**: This is the distance between two atoms located at the corners of the cube along one edge.
2. **Face Diagonal Distance (D_FD)**: This is the distance between two atoms located at the corners of the cube along the face diagonal.
3. **Body Diagonal Distance (D_BD)**: This is the distance between two atoms located at the corners of the cube along the body diagonal.

### Step 3: Calculate the Distances
1. **Distance along the Edge (D_E)**:
- The distance between two atoms along the edge of the cube is equal to the edge length of the cube, denoted as \( A \).
- Therefore, \( D_E = A \).

2. **Distance along the Face Diagonal (D_FD)**:
- The face diagonal of a cube can be calculated using the Pythagorean theorem. The length of the face diagonal is given by \( \sqrt{2}A \).
- The distance between two atoms along the face diagonal is half of this length, so \( D_FD = \frac{\sqrt{2}A}{2} = \frac{A\sqrt{2}}{2} \).

3. **Distance along the Body Diagonal (D_BD)**:
- The body diagonal of a cube can also be calculated using the Pythagorean theorem. The length of the body diagonal is given by \( \sqrt{3}A \).
- The distance between two atoms along the body diagonal is half of this length, so \( D_BD = \frac{\sqrt{3}A}{2} \).

### Step 4: Compare the Distances
Now we have:
- \( D_E = A \)
- \( D_FD = \frac{A\sqrt{2}}{2} \)
- \( D_BD = \frac{\sqrt{3}A}{2} \)

To compare these distances, we can express them in terms of \( A \):
- \( D_E = A \)
- \( D_FD = 0.707A \) (approximately)
- \( D_BD = 0.866A \) (approximately)

### Step 5: Determine the Order of Distances
From the values calculated:
- \( D_E > D_BD > D_FD \)

Thus, the order of distances is:
1. \( D_E \) (Edge Distance)
2. \( D_BD \) (Body Diagonal Distance)
3. \( D_FD \) (Face Diagonal Distance)

### Final Answer
The correct order of distances is:
\[ D_E > D_BD > D_FD \]