The Debye Model of Solids

16.1 From Einstein to Debye

The Einstein model treats all atoms as independent quantum oscillators with the same frequency $\omega_E$:

$C_V = 3Nk_B\left(\frac{\theta_E}{T}\right)^2 \frac{e^{\theta_E/T}}{(e^{\theta_E/T} - 1)^2}$

Where $\theta_E = \hbar\omega_E/k_B$. This correctly predicts $C_V \to 0$ as $T \to 0$But gives $C_V \propto e^{-\theta_E/T}$ at low $T$Whereas experiments show $C_V \propto T^3$.

The Debye model treats the lattice vibrations as a continuum of phonon modes with a cutoff frequency $\omega_D$:

$g(\omega) = \frac{3V\omega^2}{2\pi^2 v_s^3} \quad \text{for} 0 \leq \omega \leq \omega_D$

Where $v_s$ is the average sound speed. The cutoff is determined by the total number of modes:

$\int_0^{\omega_D} g(\omega)\,d\omega = 3N \implies \omega_D = v_s(6\pi^2 N/V)^{1/3}$

16.2 Debye Specific Heat

The internal energy:

$E = \int_0^{\omega_D} \frac{\hbar\omega}{e^{\beta\hbar\omega} - 1}\, g(\omega)\, d\omega = \frac{3V\hbar}{2\pi^2 v_s^3}\int_0^{\omega_D} \frac{\omega^3}{e^{\beta\hbar\omega} - 1}\, d\omega$

With $x = \hbar\omega/k_BT$ and $\theta_D = \hbar\omega_D/k_B$ (Debye temperature):

$E = 9Nk_BT\left(\frac{T}{\theta_D}\right)^3 \int_0^{\theta_D/T} \frac{x^3}{e^x - 1}\, dx$

The specific heat:

$C_V = 9Nk_B\left(\frac{T}{\theta_D}\right)^3 \int_0^{\theta_D/T} \frac{x^4 e^x}{(e^x - 1)^2}\, dx$

Low-temperature limit ($T \ll \theta_D$):

$C_V = \frac{12\pi^4}{5}Nk_B\left(\frac{T}{\theta_D}\right)^3 \propto T^3$

High-temperature limit ($T \gg \theta_D$): $C_V \to 3Nk_B$ (Dulong—Petit).