Discussion
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?
Answer: C
The resulting phonon has wave vector ⟨0.48 Å-1, 0.23 Å-1⟩, the only option inside the first Brillouin zone.
Three phonons combine to \(3 \times \langle 1.2, 0.6 \rangle = \langle 3.6, 1.8 \rangle\) Å-1. This lies outside the first zone, which runs over \(|k_{x}| \le \pi/a = 0.52\) Å-1 and \(|k_{y}| \le \pi/b = 0.79\) Å-1, so the process is an umklapp process and a reciprocal lattice vector is subtracted to bring the result back in. The reciprocal lattice spacings are \(2\pi/a \approx 1.05\) and \(2\pi/b \approx 1.57\) Å-1.
\[ \begin{aligned} k_{x} &= 3.6 - 3(1.05) \approx 0.46 \\ k_{y} &= 1.8 - 1.57 \approx 0.23 \end{aligned} \]
With \(2\pi/6\) rounded to 1.04 the \(x\) component comes to 0.48, the value printed in the keyed option. The other options lie outside the zone: ⟨3.60, 1.80⟩ is the unreduced sum, and ⟨2.56, 0.23⟩ subtracts only one \(2\pi/a\).
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