Lattice Vibrations and Phonons Questions and Answers

Q: In a two-dimensional square lattice, frequency \(\omega\) of phonons in the long wavelength limit changes linearly with the wave vector \(k\). Then the density of states of phonons is proportional to

The density of states is proportional to \(\omega\).In two dimensions the number of modes with wave vector less than \(k\) is proportional to the area of a circle in \(k\)-space, \(N \propto \pi k^{2}\). With the linear (acoustic) dispersion \(\omega = v k\), \(N \propto \omega^{2}\), and\[ g(\omega) = \frac{dN}{d\omega} \propto \omega \]\(\omega^{2}\) is the Debye result for three dimensions. \(\sqrt{\omega}\) is the 3D density of states for free particles with quadratic dispersion, and \(1/\sqrt{\omega}\) belongs to one dimension with quadratic dispersion.

Q: 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 &lang;1.2 Å-1, 0.6 Å-1&rang; interact to give one phonon. Which one of the following is the CORRECT wave vector of the resulting phonon?

The resulting phonon has wave vector &lang;0.48 Å-1, 0.23 Å-1&rang;, 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: &lang;3.60, 1.80&rang; is the unreduced sum, and &lang;2.56, 0.23&rang; subtracts only one \(2\pi/a\).

In a two-dimensional square lattice, frequency \(\omega\) of phonons in the long wavelength limit changes linearly with the wave vector \(k\). Then the density of states of phonons is proportional to

A.\(\omega\)
B.\(\omega^{2}\)
C.\(\sqrt{\omega}\)
D.\(\frac{1}{\sqrt{\omega}}\)

Exam Year(s)

GATE Physics - 2022,

Answer: A

The density of states is proportional to \(\omega\).

In two dimensions the number of modes with wave vector less than \(k\) is proportional to the area of a circle in \(k\)-space, \(N \propto \pi k^{2}\). With the linear (acoustic) dispersion \(\omega = v k\), \(N \propto \omega^{2}\), and

\[ g(\omega) = \frac{dN}{d\omega} \propto \omega \]

\(\omega^{2}\) is the Debye result for three dimensions. \(\sqrt{\omega}\) is the 3D density of states for free particles with quadratic dispersion, and \(1/\sqrt{\omega}\) belongs to one dimension with quadratic dispersion.

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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?

A.⟨2.56 Å\(^{-1}\), 0.23 Å\(^{-1}\)⟩
B.⟨3.60 Å\(^{-1}\), 1.80 Å\(^{-1}\)⟩
C.⟨0.48 Å\(^{-1}\), 0.23 Å\(^{-1}\)⟩
D.⟨3.60 Å\(^{-1}\), -0.80 Å\(^{-1}\)⟩

Exam Year(s)

GATE Physics - 2023,

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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