The Debye Model of Solids

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:

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

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).

In real solids, longitudinal and transverse waves have different speeds. The Debye model uses an average sound speed $v_s$ defined by:

$\frac{3}{v_s^3} = \frac{1}{v_L^3} + \frac{2}{v_T^3}$

where $v_L$ is the longitudinal speed and $v_T$ is the transverse speed. This accounts for one longitudinal mode and two transverse modes per wavevector.

The Einstein model fails at low temperatures because it assumes all oscillators have the same frequency, so only the exponentially small high-energy tail contributes. The Debye model correctly captures the $T^3$ law because the density of states $g(\omega) \propto \omega^2$ means that low- frequency (acoustic) modes have vanishing excitation energy.

At high temperatures, both models converge to the Dulong—Petit value $3Nk_B$.

The Debye wavevector $k_D$ is related to the Debye frequency by $\omega_D = v_s k_D$. The corresponding Debye wavelength $\lambda_D = 2\pi/k_D$ is comparable to the interatomic spacing.

The Debye temperature $\theta_D$ is a material property that correlates with the melting point and elastic constants. Materials with stiff bonds and light atoms (like diamond) have high $\theta_D$. Soft materials (like lead) have low $\theta_D$.

Debye’s model also describes thermal transport. The lattice thermal conductivity is:

$\kappa = \frac{1}{3} C_V v_s \ell$

where $\ell$ is the phonon mean free path. At low temperatures, $\ell$ is limited by boundary scattering; at high temperatures, by umklapp processes.

Problem 1. Estimate the Debye temperature of copper given: density $\rho = 8.96$ g/cm$^3$, molar mass $M = 63.55$ g/mol, and $v_s \approx 4700$ m/s.

Problem 2. Show that in the high-temperature limit, the Debye specific heat reduces to the Dulong-Petit law $C_V = 3Nk_B$.

Solution. For $T \gg \theta_D$, $x \ll 1$ in the integral, so $e^x \approx 1 + x$ and $x^4 e^x/(e^x - 1)^2 \approx x^2$. The integral $\int_0^{\theta_D/T} x^2\, dx \approx \frac{1}{3}(\theta_D/T)^3$. Then $C_V = 9Nk_B (T/\theta_D)^3 \cdot \frac{1}{3}(\theta_D/T)^3 = 3Nk_B$. $\blacksquare$

Problem 3. At what temperature does the Debye specific heat of aluminum reach 90% of its classical value? (Hint: use the Debye temperature $\theta_D = 428$ K.)

Problem 4. Derive the exact $T^3$ coefficient $\frac{12\pi^4}{5} Nk_B / \theta_D^3$ by evaluating $\int_0^\infty \frac{x^4 e^x}{(e^x - 1)^2}\, dx$ using the known value $\int_0^\infty \frac{x^3}{e^x - 1}\, dx = \pi^4/15$.

Graphite has highly anisotropic sound speeds due to its layered structure. The in-plane speed is $v_{\parallel} \approx 23,000$ m/s, while the out-of-plane speed is $v_{\perp} \approx 1,600$ m/s. This leads to a modified density of states and a different low-temperature behavior. The Debye temperature of graphite along different crystallographic directions can differ by a factor of 10.

The Debye model assumes a linear dispersion relation $\omega = v_s k$, which holds only for acoustic phonons at long wavelengths. Real phonon dispersion curves have optical branches and flatten near the Brillouin zone boundary. More accurate models include:

• Born-von Kármán model: Treats atoms as coupled oscillators with nearest-neighbor forces, giving realistic dispersion curves.
• First-principles DFT calculations: Compute phonon spectra directly from the electronic structure, giving the most accurate heat capacities.

• The Einstein model predicts $C_V \propto e^{-\theta_E/T}$ at low $T$, failing experimentally.
• The Debye model introduces a cutoff frequency $\omega_D$ and density of states $g(\omega) \propto \omega^2$.
• Low $T$: $C_V \propto T^3$ (Debye $T^3$ law). High $T$: $C_V \to 3Nk_B$ (Dulong-Petit).
• The Debye temperature $\theta_D$ is a material constant determined by sound speed and atomic density.
• The model is accurate for monatomic crystals but has limitations for anisotropic and polyatomic materials.

Problem 5. Diamond has $\theta_D \approx 2230$ K (very high due to strong bonds and light carbon atoms). Compute the specific heat of diamond at 100 K, 300 K, and 500 K using the Debye model.

Problem 6. Show that in the low-temperature limit, the Debye model gives $U = \frac{3\pi^4}{5} Nk_B T (T/\theta_D)^3$ by evaluating the integral $\int_0^\infty x^3/(e^x - 1)\, dx = \pi^4/15$.