Question

The BCC lattice has systematic extinction rules where reflections only occur if the sum of the indices is

• A. h+k+l=odd
• B. h+k+l=even
• C. h+k=l
• D. h+l=k

Hint:

Remember the systematic reflection rules for common cubic systems:
- BCC: \( h+k+l = \text{even} \)
- FCC: \( h, k, l \) must be all even or all odd (mixed indices are extinct)
- Simple Cubic: All planes can diffract (no systematic extinctions)

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the systematic extinction (or reflection) rule that determines which crystallographic planes \( (hkl) \) can produce X-ray diffraction peaks in a body-centered cubic (BCC) crystal structure.

Step 2: Key Formula or Approach:
The structure factor \( F_{hkl} \) for a unit cell determines the intensity of a diffracted beam:
\[ F_{hkl} = \sum_{j} f_j \exp\left[ 2\pi i (h x_j + k y_j + l z_j) \right] \]
For a BCC unit cell with two identical atoms located at fractional coordinates \( (0,0,0) \) and \( (1/2, 1/2, 1/2) \), this equation simplifies to:
\[ F_{hkl} = f \left( 1 + e^{\pi i (h+k+l)} \right) \]

Step 3: Detailed Explanation:

• Mathematical Conditions:
- If the sum \( h+k+l \) is an

even integer, then \( e^{\pi i (h+k+l)} = e^{\text{even}\cdot\pi i} = 1 \). This gives a non-zero structure factor:
\[ F_{hkl} = f(1 + 1) = 2f \quad \rightarrow \quad I \propto |F_{hkl}|^2 = 4f^2 \]
Therefore, diffraction can occur, and a reflection peak will be observed.
- If the sum \( h+k+l \) is an

odd integer, then \( e^{\pi i (h+k+l)} = e^{\text{odd}\cdot\pi i} = -1 \). This results in a structure factor of zero:
\[ F_{hkl} = f(1 - 1) = 0 \quad \rightarrow \quad I \propto |F_{hkl}|^2 = 0 \]
This leads to complete destructive interference, and no diffraction peak is observed (systematic extinction).

• First Allowed Planes: The first few planes that produce reflections in a BCC crystal are (110), (200), (211), and (220). Planes like (100) or (111) are extinct because their sum of indices is odd.


Step 4: Final Answer:
Hence, for a BCC lattice, reflections only occur if \( h+k+l = \text{even} \), which corresponds to Option (B).