---
title: Phonons and the Debye Model
module: Bosonic Systems
moduleNumber: 8
lessonNumber: 4
order: 804
summary: >
  The vibrations of a crystal lattice are quantized into phonons — bosons of zero
  chemical potential, counted exactly like cavity photons but with three
  polarizations, a finite sound speed, and a total of 3N modes. The Debye model
  replaces the true dispersion by a linear one cut off at a frequency that
  enforces that count. It gives the correct low-temperature T-cubed heat capacity
  the Einstein model missed and recovers the Dulong-Petit value at high
  temperature.
topics: [Bosonic Systems]
sources:
  - book: Schroeder
    ref: "Ch. 7 — Quantum Statistics; §7.5 Debye Theory of Solids"
  - book: Reif
    ref: "Ch. 10 — Systems of Interacting Particles; §10.1–10.2"
  - book: Pathria & Beale
    ref: "Ch. 7 — Ideal Bose Systems; §7.4"
  - book: Ashcroft & Mermin
    ref: "Ch. 23 — Classical Theory of the Harmonic Crystal (depth reference)"
draft: false
---

A crystal of $N$ atoms has $3N$ vibrational degrees of freedom. Classically each
contributes $k_B T$ to the energy by equipartition, giving the constant
heat capacity $3Nk_B$ that Dulong and Petit measured at room temperature, but
the heat capacity of every solid falls to zero at low temperature, and it falls
as $T^3$. The [Einstein solid](/statistical-mechanics/canonical/the-einstein-solid-and-harmonic-systems)
gets the freeze-out qualitatively but predicts an exponential falloff, too fast.
The defect is the assumption that every atom vibrates at one frequency. The
lattice actually supports a continuum of collective modes — sound waves of every
wavelength down to the atomic spacing — and quantizing those modes as bosons
gives the observed $T^3$ law.

## Phonons as a boson gas

The normal modes of a harmonic lattice are collective oscillations, each an
independent harmonic oscillator of some frequency $\omega$. Quantizing a mode
gives evenly spaced levels $(\,n + \tfrac{1}{2})\hbar\omega$; the excitation
quantum is a **phonon**, a boson carrying energy $\hbar\omega$ and crystal
momentum $\hbar\vec k$. Phonon number is not conserved — raising a mode's
excitation creates phonons, thermal contact destroys them — so, as for
[photons](/statistical-mechanics/bose-systems/the-photon-gas-and-plancks-radiation-law),
the chemical potential is zero and the mean occupation of a mode is

$$
\langle n(\omega)\rangle = \frac{1}{e^{\hbar\omega/k_B T} - 1}.
$$

The construction parallels the photon gas exactly, with three differences: the
wave speed is the speed of sound $v_s$ rather than $c$; there are three
polarizations per wavevector (one longitudinal, two transverse) rather than two;
and the number of modes is not infinite but exactly $3N$, because a lattice of
$N$ atoms cannot vibrate at a wavelength shorter than roughly the interatomic
spacing.

> **Definition (Phonon).** The quantum of a lattice vibrational mode: a boson of
> energy $\hbar\omega$ and crystal momentum $\hbar\vec k$, with $\mu = 0$ and
> mean occupation $1/(e^{\hbar\omega/k_B T} - 1)$. A solid in equilibrium is a
> gas of phonons filling its $3N$ normal modes.

## Dispersion and the Debye approximation

The true dispersion $\omega(\vec k)$ of a monatomic lattice is linear at long
wavelength, $\omega \approx v_s k$, and bends over to a maximum at the edge of
the first Brillouin zone, where the wavelength reaches twice the lattice spacing.
The exact relation is complicated and material-specific. The Debye model makes
two simplifications: it takes the dispersion to be linear, $\omega = v_s k$, for
every mode, and it replaces the polyhedral Brillouin zone by a sphere in
$\vec k$-space chosen to enclose exactly $3N$ modes. The sphere radius sets a
maximum frequency $\omega_D$, the **Debye frequency**.

$$
% caption: The true acoustic branch rises linearly then flattens at the Brillouin-zone edge; the Debye model keeps the linear slope and truncates it at a cutoff frequency chosen to preserve the total mode count.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (6.0,0) node[right] {wavevector k};
\draw[->, black] (0,0) -- (0,3.4) node[above] {frequency};
% true dispersion: sine-like to zone edge
\draw[acc, thick, smooth] plot[domain=0:4.2, samples=80] (\x,{2.7*sin(90*\x/4.2)});
\node[text=acc, anchor=west] at (4.3,2.5) {true branch};
% Debye linear
\draw[black, thick, dashed] (0,0) -- (3.3,3.15);
\node[text=black, anchor=north west] at (2.85,1.7) {Debye linear};
% cutoff
\draw[black, dashed] (0,2.7) -- (4.2,2.7);
\node[black, anchor=east] at (-0.05,2.7) {limit};
\draw[black, dashed] (4.2,0) -- (4.2,2.72);
\node[black, anchor=north] at (4.2,-0.05) {zone edge};
\end{tikzpicture}
$$

The density of modes follows the photon counting with the three replacements.
For one polarization the number of modes with wavevector up to $k$ is
$Vk^3/6\pi^2$, so with three polarizations and $\omega = v_s k$,

$$
g(\omega) = \frac{3V}{2\pi^2 v_s^3}\,\omega^2, \qquad 0 \le \omega \le \omega_D,
$$

and $g(\omega) = 0$ above the cutoff. The cutoff is fixed by requiring the total
mode count to equal $3N$:

$$
\int_0^{\omega_D} g(\omega)\,\d\omega = \frac{V\omega_D^3}{2\pi^2 v_s^3} = 3N
\quad\Longrightarrow\quad
\omega_D = v_s\left(6\pi^2\,\frac{N}{V}\right)^{1/3}.
$$

The cutoff wavelength $2\pi v_s/\omega_D$ is of order the interatomic spacing,
the physical shortest wave the lattice supports. The Debye temperature packages
the cutoff as an energy scale.

> **Definition (Debye temperature).** The temperature corresponding to the Debye
> cutoff frequency,
> $$
> \Theta_D = \frac{\hbar\omega_D}{k_B}
> = \frac{\hbar v_s}{k_B}\left(6\pi^2\,\frac{N}{V}\right)^{1/3}.
> $$
> It separates the low-temperature regime $T \ll \Theta_D$, where most modes are
> frozen, from the classical regime $T \gg \Theta_D$, where all $3N$ modes are
> thermally active. Typical values: $\Theta_D \approx 340\ \mathrm{K}$ for copper,
> $645\ \mathrm{K}$ for diamond's stiff lattice, $88\ \mathrm{K}$ for soft lead.

$$
% caption: The Debye density of states rises as frequency squared and stops abruptly at the cutoff; the Einstein model concentrates all modes at a single frequency instead, a spike.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (6.0,0) node[right] {frequency};
\draw[->, black] (0,0) -- (0,3.2) node[above] {g};
% Debye omega^2 up to cutoff
\draw[acc, thick, smooth] plot[domain=0:4.0, samples=60] (\x,{0.17*\x*\x});
\draw[acc, thick] (4.0,2.72) -- (4.0,0);
\fill[acc!12] plot[domain=0:4.0, samples=60] (\x,{0.17*\x*\x}) -- (4.0,0) -- (0,0) -- cycle;
\node[text=acc, anchor=south east] at (3.6,2.1) {Debye};
\draw[black, dashed] (4.0,0) -- (4.0,2.75);
\node[black, anchor=north] at (4.0,-0.05) {limit};
% Einstein spike
\draw[black, very thick, ->] (2.6,0) -- (2.6,2.9);
\node[text=black, anchor=west] at (2.65,2.7) {Einstein};
\end{tikzpicture}
$$

## Energy and heat capacity

Dropping the temperature-independent zero-point term, the thermal energy sums the
mode energy over the density of states,

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

Substituting $x = \hbar\omega/k_B T$ and $x_D = \Theta_D/T$, and using
$\omega_D^3 = 6\pi^2 N v_s^3/V$ to eliminate the constants,

$$
U = 9Nk_B T\left(\frac{T}{\Theta_D}\right)^3
\int_0^{x_D} \frac{x^3\,\d x}{e^x - 1}.
$$

Differentiating gives the Debye heat capacity,

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

a universal function of the single ratio $T/\Theta_D$. This scaling — that every
insulating solid has the same $C_V/3Nk_B$ once temperature is measured in units
of its own Debye temperature — is the model's sharpest prediction, and it holds
across materials whose $\Theta_D$ span an order of magnitude.[^debye]

### The two limits

- **High temperature ($T \gg \Theta_D$).** Then $x_D \to 0$ and the integrand
  $x^4 e^x/(e^x-1)^2 \to x^2$ over the whole range, so $\int_0^{x_D} x^2\,\d x =
  x_D^3/3$ and
  $$
  C_V \to 9Nk_B\left(\frac{T}{\Theta_D}\right)^3\frac{1}{3}\left(\frac{\Theta_D}{T}\right)^3
  = 3Nk_B.
  $$
  The Dulong-Petit value is recovered: all $3N$ modes carry $k_B T$ each.
- **Low temperature ($T \ll \Theta_D$).** Then $x_D \to \infty$ and the integral
  becomes the constant $\int_0^\infty x^3/(e^x-1)\,\d x = \pi^4/15$, so
  $$
  U \to \frac{3\pi^4}{5}\,Nk_B\,\frac{T^4}{\Theta_D^3},
  \qquad
  C_V \to \frac{12\pi^4}{5}\,Nk_B\left(\frac{T}{\Theta_D}\right)^3.
  $$

> **Theorem (Debye $T^3$ law).** At temperatures well below the Debye
> temperature, the lattice heat capacity of a solid is
> $$
> C_V = \frac{12\pi^4}{5}\,Nk_B\left(\frac{T}{\Theta_D}\right)^3 \propto T^3.
> $$

The cubic law has the same origin as the Stefan-Boltzmann $T^4$ energy: at low
temperature only the long-wavelength modes with $\hbar\omega \lesssim k_B T$ are
excited, and their number grows as $T^3$ from the $\omega^2$ density of states.
The finite cutoff is irrelevant at low $T$ because those high modes are frozen
anyway, which is why the Debye and exact dispersions agree there.

> **Worked example.** Copper has $\Theta_D = 343\ \mathrm{K}$. At $T = 20\
> \mathrm{K}$, deep in the $T^3$ regime, the molar lattice heat capacity is
> $$
> C = \frac{12\pi^4}{5}\,R\left(\frac{T}{\Theta_D}\right)^3
> = 234\times 8.314\times\left(\frac{20}{343}\right)^3
> \approx 0.39\ \mathrm{J\,mol^{-1}\,K^{-1}},
> $$
> about $1.6\%$ of the Dulong-Petit value $3R = 24.9\ \mathrm{J\,mol^{-1}\,K^{-1}}$.
> Almost every vibrational mode is frozen; the small residual comes from the thin
> shell of acoustic modes with $\hbar\omega \lesssim k_B T$.

$$
% caption: The Debye heat capacity rises from zero as temperature cubed at low temperature and saturates at the classical Dulong-Petit value once all modes are active above the Debye temperature.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (6.4,0) node[right] {T};
\draw[->, black] (0,0) -- (0,3.4) node[above] {$C_V$};
\draw[black, dashed] (0,2.9) -- (6.0,2.9);
\node[black, anchor=east] at (-0.05,2.9) {$3Nk_B$};
\draw[acc, very thick, smooth] plot coordinates
  {(0,0)(0.6,0.12)(1.2,0.45)(1.8,1.0)(2.4,1.65)(3.0,2.2)(3.7,2.6)(4.5,2.8)(5.5,2.88)(6.0,2.9)};
\node[text=acc, anchor=west] at (2.4,1.3) {Debye};
\draw[black, dashed] (2.2,0) -- (2.2,1.45);
\node[black, anchor=north] at (2.2,-0.05) {$\Theta_D$};
\end{tikzpicture}
$$

## Debye against Einstein and experiment

The two models share the high-temperature limit $3Nk_B$ but split at low
temperature. Einstein's single frequency $\omega_E$ gives every mode the same
Boltzmann suppression $e^{-\hbar\omega_E/k_B T}$, so $C_V \sim e^{-\Theta_E/T}$
falls exponentially — faster than any measured solid. Debye's spectrum of
arbitrarily low frequencies keeps a population of easily excited modes down to
$T = 0$, giving the gentler $T^3$ falloff that experiment shows.

| | Einstein model | Debye model |
| --- | --- | --- |
| Spectrum | single frequency $\omega_E$ | continuum up to $\omega_D$ |
| Density of states | delta spike | $g \propto \omega^2$ |
| Low-$T$ heat capacity | $\propto e^{-\Theta_E/T}$ | $\propto T^3$ |
| High-$T$ limit | $3Nk_B$ | $3Nk_B$ |
| Agreement with data | qualitative | quantitative below $\Theta_D$ |

Neither model is exact — the true density of states has van Hove peaks the Debye
$\omega^2$ curve smooths over — but the Debye interpolation is accurate to a few
percent across the whole temperature range with one fitted parameter, $\Theta_D$.
The Einstein model survives as the description of the optical phonon branches of
a polyatomic crystal, whose flat dispersion really does concentrate modes near a
single frequency.

$$
% caption: Measured heat capacity against temperature with both models fitted to the same high-temperature value. The Debye curve tracks the data down to low temperature; the Einstein curve drops below it, falling off exponentially.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (6.4,0) node[right] {T};
\draw[->, black] (0,0) -- (0,3.4) node[above] {$C_V$};
\draw[black, dashed] (0,2.9) -- (6.0,2.9);
\node[black, anchor=east] at (-0.05,2.9) {$3Nk_B$};
% Debye
\draw[acc, very thick, smooth] plot coordinates
  {(0,0)(0.7,0.16)(1.4,0.6)(2.1,1.35)(2.8,2.05)(3.6,2.55)(4.6,2.82)(6.0,2.9)};
\node[text=acc, anchor=west] at (3.0,2.15) {Debye};
% Einstein: lower at low T
\draw[black, thick, dashed, smooth] plot coordinates
  {(0,0)(1.0,0.05)(1.7,0.28)(2.4,0.95)(3.1,1.8)(3.9,2.45)(4.8,2.78)(6.0,2.9)};
\node[text=black, anchor=north west] at (2.4,0.75) {Einstein};
% data points near Debye
\foreach \pt in {(1.0,0.28),(1.7,0.85),(2.4,1.62),(3.1,2.25),(3.9,2.65),(4.8,2.83)}
  \fill[black] \pt circle (1.5pt);
\node[text=black, anchor=west] at (4.9,2.55) {data};
\end{tikzpicture}
$$

## Summary

- Lattice vibrations quantize into phonons: bosons of energy $\hbar\omega$,
  crystal momentum $\hbar\vec k$, and $\mu = 0$, with occupation
  $1/(e^{\hbar\omega/k_B T} - 1)$.
- The Debye model linearizes the dispersion $\omega = v_s k$ and cuts it at
  $\omega_D = v_s(6\pi^2 N/V)^{1/3}$, chosen so the $g(\omega) = 3V\omega^2/2\pi^2
  v_s^3$ spectrum holds exactly $3N$ modes. The Debye temperature is
  $\Theta_D = \hbar\omega_D/k_B$.
- The heat capacity is a universal function of $T/\Theta_D$: it reaches
  $3Nk_B$ (Dulong-Petit) for $T \gg \Theta_D$ and falls as
  $C_V = \tfrac{12\pi^4}{5}Nk_B(T/\Theta_D)^3$ for $T \ll \Theta_D$.
- The $T^3$ law matches experiment; the Einstein model's single frequency gives
  an exponential falloff that is too fast, though it describes flat optical
  branches well.

[^debye]: Schroeder, §7.5, derives the $9Nk_B(T/\Theta_D)^3$ prefactor and the
$T^3$ law; Pathria & Beale, §7.4, tabulate Debye temperatures and the
universal-curve collapse. Ashcroft & Mermin, Ch. 23, treat the true lattice
dynamics the Debye sphere approximates.