Debye specific-heat theory — UPSC Physics Optional CSE 2026 Paper II

Debye Model of Lattice Specific Heat

Einstein's single-frequency model correctly predicts that the specific heat of a solid vanishes as T \to 0. However, it gives an exponential fall, whereas experiments show C_V \propto T^3. Debye (1912) resolved this by treating the solid as a continuous isotropic elastic medium whose vibrations are sound waves (phonons), rather than as independent oscillators of one frequency. He assumed a linear dispersion relation \omega = vk with the same velocity v for all modes and three polarizations (one longitudinal, two transverse) for each wavevector. Since a crystal of N atoms has only 3N degrees of freedom, he cut the spectrum off at a maximum frequency \omega_D, the Debye frequency.

Counting modes in k-space and including the three polarizations, the density of states is

g(\omega)\,d\omega = \frac{3V\omega^2}{2\pi^2 v^3}\,d\omega

The normalization condition \int_0^{\omega_D} g(\omega)\,d\omega = 3N fixes the cut-off and the Debye temperature:

\omega_D = v\left(\frac{6\pi^2 N}{V}\right)^{1/3}, \qquad g(\omega) = \frac{9N\omega^2}{\omega_D^3}, \qquad \theta_D = \frac{\hbar\omega_D}{k_B}

Each mode is a quantum oscillator obeying Bose–Einstein statistics, so the total vibrational energy is

U = \int_0^{\omega_D} \frac{\hbar\omega}{e^{\hbar\omega/k_BT}-1}\, g(\omega)\,d\omega

With the substitution x = \hbar\omega/k_BT this becomes

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

Differentiating with respect to temperature gives the Debye specific heat formula:

\boxed{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}

Because C_V depends only on T/\theta_D, it is a universal function for all solids.

High-temperature limit (T \gg \theta_D). Here x \ll 1, so the integrand reduces to x^2 and the integral equals \tfrac{1}{3}(\theta_D/T)^3. The result is C_V = 3Nk_B = 3R per mole, which recovers the classical Dulong–Petit law.

Low-temperature limit (T \ll \theta_D). The upper limit tends to infinity, and the integral takes the standard value \int_0^\infty x^4 e^x/(e^x-1)^2\,dx = 4\pi^4/15. This gives the Debye T^3 law:

\boxed{C_V = \frac{12\pi^4}{5}Nk_B\left(\frac{T}{\theta_D}\right)^3 \approx 1944\left(\frac{T}{\theta_D}\right)^3\ \text{J mol}^{-1}\text{K}^{-1}}

This agrees well with experiment for insulators. Physically, at low temperatures only long-wavelength acoustic phonons are excited, and for these modes the continuum approximation is essentially exact.

Limitations. The success of the model rests on approximations that limit its accuracy:

• Continuum picture. Treating the crystal as a continuum ignores its discrete atomic structure. As a result, the linear dispersion \omega = vk fails near the Brillouin zone boundary, where the real dispersion curve flattens.
• Single sound velocity. Longitudinal and transverse velocities actually differ, so the model must use an averaged velocity defined by 3/v^3 = 1/v_L^3 + 2/v_T^3.
• Idealized zone and density of states. The true polyhedral Brillouin zone is replaced by a sphere, and the parabolic g(\omega) misses the van Hove singularities of the real spectrum.
• Missing contributions. Optical phonons in crystals with more than one atom per primitive cell are neglected. So is the electronic contribution in metals, which gives C_V = \gamma T + AT^3 at low temperatures.
• Harmonic approximation. Anharmonic effects are ignored, although they make C_V slightly exceed 3R at high temperatures.
• Temperature-dependent \theta_D. Values of \theta_D extracted from experimental data vary with temperature, which shows that the model is not exact.
• Limited range of the T^3 law. The law holds accurately only for T \lesssim \theta_D/50, and it fails for strongly anisotropic or layered solids such as graphite, where C_V \propto T^2.