Question

The primitive translation vectors of a two-dimensional lattice are \(\bm{a} = 2\bm{i} + \bm{j}\) and \(\bm{b} = 2\bm{j}\). The primitive translation vectors of its reciprocal lattice \(\bm{a}^*\) and \(\bm{b}^*\) are:

• A. \(\bm{a}^* = \frac{2\pi}{|\bm{a} \times \bm{b}|} (2 \times \bm{k}), \, \bm{b}^* = \frac{2\pi}{|\bm{a} \times \bm{b}|} (-\bm{i} + 2\bm{j})\)
• B. \(\bm{a}^* = \frac{-2\pi}{|\bm{a} \times \bm{b}|} (4 \times \bm{k}), \, \bm{b}^* = \frac{-2\pi}{|\bm{a} \times \bm{b}|} (-4\bm{i} + 4\bm{j})\)
• C. \(\bm{a}^* = \frac{-2\pi}{|\bm{a} \times \bm{b}|} (5 \times \bm{k}), \, \bm{b}^* = \frac{-2\pi}{|\bm{a} \times \bm{b}|} (-5\bm{i} + 2\bm{j})\)
• D. \(\bm{a}^* = \frac{-2\pi}{|\bm{a} \times \bm{b}|} (6 \times \bm{k}), \, \bm{b}^* = \frac{-2\pi}{|\bm{a} \times \bm{b}|} (-6\bm{i} + 2\bm{j})\)

Hint:

The reciprocal lattice vectors are critical in crystallography as they define the Fourier transform of the real lattice. For 2D lattices, ensure the cross-product a ×b is calculated correctly to determine the magnitude and orientation of a and b. These vectors are foundational in understanding diffraction patterns and Brillouin zones.

Answer 1

The Correct Option is A

The primitive translation vectors of the reciprocal lattice are calculated using:
\[\bm{a}^* = \frac{2\pi}{|\bm{a} \times \bm{b}|} (\bm{b} \times \bm{k}), \quad \bm{b}^* = \frac{2\pi}{|\bm{a} \times \bm{b}|} (\bm{k} \times \bm{a})\]
Using the given \(\bm{a}\) and \(\bm{b}\):
\[\bm{a}^* = \frac{2\pi}{|\bm{a} \times \bm{b}|} (2 \times \bm{k}), \quad \bm{b}^* = \frac{2\pi}{|\bm{a} \times \bm{b}|} (-\bm{i} + 2\bm{j})\]