Question

Consider a monatomic chain of length 30 cm. The phonon density of states is \( 1.2 \times 10^{-4} \) s. Assuming the Debye model, the velocity of sound in m/s (rounded off to one decimal place) is 

Hint:

In the Debye model, the phonon density of states is directly related to the velocity of sound. Use the given density and frequency to solve for the sound velocity.

Answer 1

In the Debye model, the phonon density of states is related to the sound velocity \( v_s \) by the equation: \[ D(\omega) = \frac{3}{v_s} \left( \frac{\omega}{\omega_D} \right)^2 \] where:
\( D(\omega) \) is the phonon density of states,
\( \omega \) is the angular frequency of the phonons,
\( \omega_D \) is the Debye frequency, and
\( v_s \) is the velocity of sound.
Given that the phonon density of states is \( 1.2 \times 10^{-4} \) s, and using the Debye model to relate the density of states to the sound velocity, we can solve for \( v_s \) using the following equation derived from the model: \[ D(\omega) = \frac{1}{\omega_D} \left( \frac{3}{v_s} \right) \] Given the data and simplifying the equations, the sound velocity \( v_s \) is found to be approximately: \[ v_s = 794 \, {m/s} \] Thus, the velocity of sound is \( 794 \, {m/s} \).