Question:
Which of the following is the Debye temperature?
Which of the following is the Debye temperature?
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Set the maximum phonon energy equal to a thermal energy: \(\hbar\omega_D = k_B\theta_D\).
Updated On: Jul 2, 2026
- \(\theta_D = \dfrac{\hbar\omega_D}{2k_B}\)
- \(\dfrac{\hbar\omega_D}{k_B}\)
- \(\dfrac{\hbar^2\omega_D}{k_B}\)
- \(\dfrac{\hbar^2\omega_D^2}{k_B}\)
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The Correct Option is B
Solution and Explanation
Step 1: In the Debye model of lattice vibrations there is a maximum (cut-off) angular frequency \(\omega_D\), the Debye frequency. The energy of this highest phonon mode is \(\hbar\omega_D\).
Step 2: The Debye temperature \(\theta_D\) is defined by equating this maximum phonon energy to a thermal energy scale \(k_B\theta_D\): \[k_B\theta_D = \hbar\omega_D.\]
Step 3: Solve for \(\theta_D\): \[\theta_D = \frac{\hbar\omega_D}{k_B}.\]
Step 4: Dimensional check: \(\hbar\omega_D\) has units of energy and \(k_B\) has units of energy per kelvin, so the ratio has units of kelvin, as required for a temperature. Options with \(\hbar^2\) or a factor of 2 are dimensionally or definitionally wrong.\[\boxed{\theta_D = \frac{\hbar\omega_D}{k_B}}\]
Step 2: The Debye temperature \(\theta_D\) is defined by equating this maximum phonon energy to a thermal energy scale \(k_B\theta_D\): \[k_B\theta_D = \hbar\omega_D.\]
Step 3: Solve for \(\theta_D\): \[\theta_D = \frac{\hbar\omega_D}{k_B}.\]
Step 4: Dimensional check: \(\hbar\omega_D\) has units of energy and \(k_B\) has units of energy per kelvin, so the ratio has units of kelvin, as required for a temperature. Options with \(\hbar^2\) or a factor of 2 are dimensionally or definitionally wrong.\[\boxed{\theta_D = \frac{\hbar\omega_D}{k_B}}\]
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