GATE 2023 PH – Question 46
In the first Brillouin zone of a rectangular lattice (lattice constants a= 6 Å and b= 4 Å), three incoming phonons with the same wave vector 〈1.2 Å−1, 0.6 Å−1〉 interact to give one phonon. Which one of the following is the CORRECT wave vector of the resulting phonon?
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Correct answer: (C) $\langle0.48,0.23\rangle\ \mathrm{\AA^{-1}}$
Explanation
Phonons in a crystal obey **crystal momentum conservation** modulo a reciprocal lattice vector $\mathbf G$. The wave vector of the combined phonon is the sum of the three, reduced to the first Brillouin zone.
**Sum:**
$$3\times(1.2,\ 0.6)=(3.6,\ 1.8)\ \text{\AA}^{-1}.$$
**First Brillouin zone.** For lattice constants $a=6$ Å and $b=4$ Å the zone is $|k_x|\leq\dfrac\pi a=0.524$ Å⁻¹ and $|k_y|\leq\dfrac\pi b=0.785$ Å⁻¹. The reciprocal lattice vectors are $\left(\dfrac{2\pi m}{a},\dfrac{2\pi n}{b}\right)=(1.047\,m,\ 1.571\,n)$.
**Subtract a reciprocal lattice vector:** with $m=3$ and $n=1$,
$$\mathbf k=(3.6-3.142,\ 1.8-1.571)=(0.458,\ 0.229)\ \text{\AA}^{-1}\approx\langle0.48,\ 0.23\rangle\ \text{\AA}^{-1}.$$
This lies inside the first Brillouin zone (option C).