Debye Model of Phonon Heat Capacity

Statement

Treating the normal modes of a crystal as a gas of independent quantized harmonic oscillators (phonons) with a linear dispersion \( \omega = v_s k \) and a single cutoff frequency \( \omega_D \) fixed by requiring exactly \( 3N \) modes, the Debye model gives the lattice internal energy \( U(T) \) and hence the molar heat capacity \( C_V = 9 N k_B \left(\dfrac{T}{\Theta_D}\right)^{3} \displaystyle\int_0^{\Theta_D/T} \dfrac{x^4 e^x}{(e^x-1)^2}\,dx \), which reduces to the Dulong–Petit value \( 3 N k_B \) for \( T \gg \Theta_D \) and to the \( T^3 \) law \( C_V = \dfrac{12\pi^4}{5} N k_B \left(\dfrac{T}{\Theta_D}\right)^{3} \) for \( T \ll \Theta_D \).

Why it matters

The Dulong–Petit rule \( C_V = 3 N k_B \) is a triumph of classical equipartition, but experiment shows the heat capacity of every insulating solid collapses to zero as \( T \to 0 \). The Einstein model captured the fall but decayed exponentially, too fast. Debye's insight — that the low-energy modes are long-wavelength sound waves with arbitrarily small frequency — produces the correct universal \( T^3 \) approach to zero, one of the cleanest confirmations of the quantization of lattice vibrations.

The same phonon-gas machinery, with a Debye cutoff replacing the infinite photon spectrum, underlies thermal conductivity, the third law of thermodynamics for solids, and the entropy budget that sets equations of state at low temperature. Debye's \( \Theta_D \) is a single material parameter that collapses the heat-capacity curves of chemically unrelated solids onto one universal shape.

Assumptions

Derivation

Result

Limiting cases

• \( T\to 0 \): \( C_V\to \tfrac{12\pi^4}{5}Nk_B (T/\Theta_D)^3 \to 0 \) — the correct third-law behaviour; entropy \( S=\int C_V\,dT/T \) also vanishes as \( T^3 \).
• \( T\to\infty \): \( C_V\to 3Nk_B \), the Dulong–Petit plateau, independent of \( \Theta_D \) and of quantum mechanics.
• \( T=\Theta_D \): the crossover region; \( C_V\approx 0.95\times 3Nk_B \), already \( 95\% \) of classical, so \( \Theta_D \) roughly marks "nearly classical".
• \( \Theta_D\to 0 \) (very soft solid): classical everywhere above absolute zero; the \( T^3 \) window shrinks toward \( T=0 \).
• Two dimensions: the same argument with \( g(\omega)\propto\omega \) gives a \( T^2 \) low-temperature law; in \( d \) dimensions \( C_V\propto T^d \).
• Metals: add the electronic linear term \( \gamma T \); at low \( T \), \( C_V = \gamma T + \beta T^3 \), and a plot of \( C_V/T \) versus \( T^2 \) is a straight line whose slope gives \( \Theta_D \).

Breaks when

Failure modes

• Using two polarizations instead of three. Copying the photon result (transverse only) drops the longitudinal branch; the factor becomes \( 2 \) not \( 3 \) and Dulong–Petit comes out wrong. Phonons have one longitudinal plus two transverse modes.
• Forgetting the finite upper limit at high \( T \). Extending the integral to \( \infty \) is only legal at low \( T \). At high \( T \) the cutoff is essential — it is what caps \( C_V \) at \( 3Nk_B \) instead of diverging.
• Including zero-point energy in \( \langle n\rangle \). The occupation is \( 1/(e^{\hbar\omega/k_BT}-1) \), not \( \tfrac12 + 1/(e^{...}-1) \), for heat capacity — the constant \( \tfrac12\hbar\omega \) is \( T \)-independent and drops out of \( \partial U/\partial T \).
• Mislabelling the low-\( T \) constant. Writing \( \int_0^\infty x^4 e^x/(e^x-1)^2 dx \) as \( \pi^4/15 \) (the \( x^3/(e^x-1) \) value) instead of \( 4\pi^4/15 \). They differ by a factor \( 4 \); the correct coefficient is \( 12\pi^4/5 \), not \( 3\pi^4/5 \).
• Confusing \( C_V \) and \( C_p \). The derivation gives \( C_V \); comparing directly to measured \( C_p \) without the \( C_p-C_V=TV\alpha^2/\kappa_T \) correction misfits \( \Theta_D \) at higher \( T \).
• Treating \( \Theta_D \) as fundamental. \( \Theta_D \) is a fitted effective parameter, not a fixed material constant; the "best" value depends on which temperature window is fitted.

Discussion

The heart of the Debye result is that heat capacity counts thermally accessible modes. A mode contributes its full classical \( k_B \) only when \( k_BT \gtrsim \hbar\omega \); colder than that it is "frozen", exponentially unlikely to hold even one quantum. What makes solids different from the Einstein model is that there is no lowest nonzero frequency: acoustic modes run continuously down to \( \omega\to 0 \). No matter how cold, some modes are always active, so \( C_V \) never falls exponentially — it falls as a power law set by how the mode count grows from zero, namely \( g(\omega)\propto\omega^2 \), giving \( T^3 \).

The kinship with blackbody radiation is exact and illuminating. A photon gas and a phonon gas are both massless Bose gases with linear dispersion and \( \mu=0 \); the energy density integrand is identical. The only structural differences are that phonons have three polarizations (photons two) and, crucially, a finite spectrum cut off at \( \omega_D \) because a lattice has a shortest meaningful wavelength (\( \sim \) interatomic spacing) whereas the electromagnetic vacuum has none. Remove the cutoff and the phonon energy would grow as \( T^4 \) forever, the Stefan–Boltzmann law — indeed at low \( T \), before the cutoff is felt, the phonon energy does scale as \( T^4 \), which is exactly why its derivative \( C_V\propto T^3 \).

The Debye temperature is a compact fingerprint of a solid's stiffness and mass: \( \Theta_D\propto v_s (N/V)^{1/3} \), and since \( v_s\sim\sqrt{\text{stiffness}/\text{density}} \), hard light solids have high \( \Theta_D \) and soft heavy ones low. It sets not only the heat-capacity crossover but roughly the temperature above which classical transport and equipartition become good approximations, and it appears in the Lindemann melting criterion and the Bloch–Grüneisen resistivity of metals.

At a deeper level the \( T^3 \) law is a statement about the low-energy fixed point of the elastic continuum: at long wavelengths every crystal, whatever its microscopic detail, looks like an isotropic elastic medium with three linearly dispersing Goldstone modes of broken translational symmetry. The universality of the \( T^3 \) coefficient (fixed entirely by the sound speeds through \( \Theta_D \)) is a hydrodynamic, symmetry-protected result — the sharp cutoff and the density of states above it are model artefacts that never enter the \( T\to 0 \) limit. This is why the low-temperature Debye law is quantitatively reliable even though the model's mid-spectrum \( g(\omega) \) is crude.

Common misconceptions. Debye did not assume all atoms vibrate at one frequency (that is Einstein); he assumed a continuum of frequencies up to a cutoff. The \( T^3 \) law is not caused by the cutoff — it comes from the \( \omega\to 0 \) modes, and the cutoff is irrelevant at low \( T \). And \( \Theta_D \) is not a phase-transition temperature: nothing special happens to the solid at \( T=\Theta_D \); it is merely where the heat capacity is about \( 95\% \) of its classical value.

Worked examples

Problems

1. A solid has \( \Theta_D=200\,\mathrm{K} \). By what factor does its lattice heat capacity change when the temperature is doubled from \( 10\,\mathrm{K} \) to \( 20\,\mathrm{K} \)?
   Solution

   Both temperatures satisfy \( T\ll\Theta_D \), so \( C_V\propto T^3 \). Doubling \( T \) multiplies \( C_V \) by \( 2^3=8 \). Numerically, at \( 10\,\mathrm{K} \): \( C_V=233.78\times 8.314\times(10/200)^3=233.78\times8.314\times1.25\times10^{-4}=0.243\,\mathrm{J\,mol^{-1}K^{-1}} \); at \( 20\,\mathrm{K} \): \( 0.243\times 8=1.94\,\mathrm{J\,mol^{-1}K^{-1}} \). Factor \( =8 \).
2. Show that the phonon internal energy at low temperature scales as \( T^4 \), and relate its coefficient to the \( C_V \) coefficient.
   Solution

   At \( T\ll\Theta_D \), \( U=9Nk_BT(T/\Theta_D)^3\int_0^\infty x^3/(e^x-1)\,dx \). The integral is \( \pi^4/15 \), so \( U=9Nk_B\,\dfrac{\pi^4}{15}\dfrac{T^4}{\Theta_D^3}=\dfrac{3\pi^4}{5}\dfrac{Nk_B T^4}{\Theta_D^3} \). Then \( C_V=\partial U/\partial T=\dfrac{12\pi^4}{5}\dfrac{Nk_B T^3}{\Theta_D^3} \), consistent with the boxed low-\( T \) law. The factor \( 4 \) from differentiating \( T^4 \) turns the \( 3 \) into \( 12 \).
3. For a metal, low-temperature data give \( C_V/T = \gamma + \beta T^2 \) with \( \gamma=0.70\,\mathrm{mJ\,mol^{-1}K^{-2}} \) and \( \beta=0.048\,\mathrm{mJ\,mol^{-1}K^{-4}} \). Find \( \Theta_D \).
   Solution

   The lattice term is \( \beta T^3 \) with \( \beta=\dfrac{12\pi^4}{5}\dfrac{R}{\Theta_D^3} \) per mole. Solve \( \Theta_D=\left(\dfrac{12\pi^4 R}{5\beta}\right)^{1/3} \). With \( R=8.314\,\mathrm{J\,mol^{-1}K^{-1}} \) and \( \beta=4.8\times10^{-5}\,\mathrm{J\,mol^{-1}K^{-4}} \): \( \dfrac{12\pi^4\times8.314}{5\times4.8\times10^{-5}}=\dfrac{233.78\times8.314}{4.8\times10^{-5}}=4.05\times10^{7} \). Cube root: \( \Theta_D=(4.05\times10^{7})^{1/3}\approx 344\,\mathrm{K} \) (this is essentially copper).
4. At what temperature, as a fraction of \( \Theta_D \), does the exact Debye \( C_V \) reach \( 90\% \) of the Dulong–Petit value? Estimate using the high-\( T \) expansion \( C_V\approx 3Nk_B\left[1-\tfrac{1}{20}(\Theta_D/T)^2\right] \).
   Solution

   Set \( 1-\tfrac{1}{20}(\Theta_D/T)^2=0.90 \Rightarrow (\Theta_D/T)^2=0.10\times20=2.0 \Rightarrow \Theta_D/T=1.414 \Rightarrow T=\Theta_D/1.414=0.707\,\Theta_D \). So \( C_V \) is at \( 90\% \) of Dulong–Petit near \( T\approx 0.71\,\Theta_D \). (The exact numerical answer is close, \( T\approx 0.70\,\Theta_D \), confirming the leading correction is adequate here.)
5. Diamond has \( \Theta_D\approx 2230\,\mathrm{K} \) and lead \( \Theta_D\approx 105\,\mathrm{K} \). Compare their molar lattice heat capacities at \( T=300\,\mathrm{K} \).
   Solution

   Lead: \( T/\Theta_D=300/105=2.86\gg1 \), so \( C_V\approx 3R=24.9\,\mathrm{J\,mol^{-1}K^{-1}} \) (fully classical). Diamond: \( T/\Theta_D=300/2230=0.135\ll1 \), so use \( T^3 \): \( C_V=233.78\times8.314\times(0.1345)^3=233.78\times8.314\times2.43\times10^{-3}=4.72\,\mathrm{J\,mol^{-1}K^{-1}} \) — the \( T^3 \) formula overestimates here since \( 0.135 \) is not deeply in the low-\( T \) regime; the true value is \( \approx 6\,\mathrm{J\,mol^{-1}K^{-1}} \). Either way diamond sits far below Dulong–Petit at room temperature while lead is already there, because diamond is stiff and light (huge \( \Theta_D \)) whereas lead is soft and heavy.