The GATE Grind

GATE 2022 PH – Question 43

Solid State Physics · lattice vibrations and thermal properties of solids · 2 marks · Multiple choice

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

  1. $\omega$
  2. $\omega^2$
  3. $\sqrt\omega$
  4. $1/\sqrt\omega$

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Correct answer: (A) $\omega$

Explanation

In a two-dimensional crystal the number of vibrational modes with wave vector below $k$ is proportional to the area in $k$-space:
$$N(k)\propto\pi k^2\;\Rightarrow\;g(k)\,dk\propto k\,dk .$$

**Linear dispersion** $\omega=vk$ at long wavelengths, so $k=\omega/v$ and $dk=d\omega/v$:
$$g(\omega)\,d\omega\propto\frac{\omega}{v}\frac{d\omega}{v}\;\Rightarrow\;g(\omega)\propto\omega\quad(\text{option A}).$$

(In 3D the same argument gives $g(\omega)\propto\omega^2$, and in 1D it gives a constant.)