Question

The magnitude of the reciprocal lattice vector is related to interplaner spacing \(d_{hkl}\):

• A. proportional to \(d_{hkl}\)
• B. inversely proportional to \(d_{hkl}\)
• C. proportional to \((d_{hkl})^2\)
• D. inversely proportional to \((d_{hkl})^2\)

Hint:

The very name "reciprocal" lattice should remind you of this inverse relationship. Distances in real space become inverse distances in reciprocal space. This is fundamental to understanding diffraction patterns, where larger spacings in the crystal lead to smaller spacings between diffraction spots.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The reciprocal lattice is a mathematical construct used to analyze periodic structures, especially in the context of diffraction. Each point in the reciprocal lattice corresponds to a set of parallel planes in the direct (real space) lattice. The reciprocal lattice vector, \(\vec{G}_{hkl}\), is associated with the set of planes having Miller indices (hkl).
Step 2: Key Formula or Approach:
The definition of the reciprocal lattice vector \(\vec{G}_{hkl}\) provides the relationship. Its direction is normal to the (hkl) planes, and its magnitude is given by:
\[ |\vec{G}_{hkl}| = \frac{2\pi}{d_{hkl}} \]
where \(d_{hkl}\) is the interplanar spacing for the (hkl) planes.
Step 3: Detailed Explanation:
From the formula \( |\vec{G}_{hkl}| = \frac{2\pi}{d_{hkl}} \), it is clear that the magnitude of the reciprocal lattice vector, \(|\vec{G}_{hkl}|\), is inversely proportional to the interplanar spacing, \(d_{hkl}\). As the spacing between planes in the real lattice decreases, the length of the corresponding vector in the reciprocal lattice increases.
Step 4: Final Answer:
The magnitude of the reciprocal lattice vector is inversely proportional to the interplanar spacing \(d_{hkl}\).