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: Concept Overview:
The reciprocal lattice is a mathematical tool used to analyze periodic structures, particularly in diffraction studies. Each point in this lattice relates to a set of parallel planes within the real-space lattice. The reciprocal lattice vector, \(\vec{G}_{hkl}\), corresponds to planes defined by the Miller indices (hkl).
Step 2: Core Formula:
The reciprocal lattice vector \(\vec{G}_{hkl}\) is defined such that its direction is perpendicular to the (hkl) planes. Its magnitude is determined by:
\[ |\vec{G}_{hkl}| = \frac{2\pi}{d_{hkl}} \]
where \(d_{hkl}\) represents the spacing between the (hkl) planes.
Step 3: Detailed Explanation:
The formula \( |\vec{G}_{hkl}| = \frac{2\pi}{d_{hkl}} \) demonstrates the inverse relationship between the magnitude of the reciprocal lattice vector, \(|\vec{G}_{hkl}|\), and the interplanar spacing, \(d_{hkl}\). Smaller real-space plane spacings result in larger corresponding reciprocal lattice vector lengths.
Step 4: Conclusion:
The magnitude of the reciprocal lattice vector is inversely proportional to the interplanar spacing \(d_{hkl}\).