---
title: Phonons, Density of States, and Crystal Momentum
module: Lattice Dynamics
moduleNumber: 4
lessonNumber: 2
order: 402
summary: >
  Quantizing the normal modes of a harmonic crystal turns each vibrational mode
  into a quantum oscillator whose excitations are phonons. This lesson counts
  phonons with Bose-Einstein statistics, defines crystal momentum and the normal
  versus Umklapp distinction in momentum conservation, builds the density of
  states with its van Hove singularities, and shows how inelastic neutron
  scattering measures a dispersion curve point by point.
topics: [Lattice Dynamics]
draft: false
sources:
  - book: Ashcroft & Mermin
    ref: "Ch. 23 — Quantum Theory of the Harmonic Crystal; Ch. 24 — Measuring Phonon Dispersion"
  - book: Kittel
    ref: "Ch. 5 — Phonons II: Thermal Properties"
  - book: Hook & Hall
    ref: "Ch. 2 — Phonons"
---

The classical [harmonic crystal](/condensed-matter/lattice-dynamics/phonon-dispersion)
decomposes into independent normal modes, each a plane wave labelled by a
wavevector $\vec k$ and a branch $s$ oscillating at frequency $\omega_s(\vec k)$.
Quantum mechanics replaces each mode's continuous amplitude with a ladder of
discrete energy levels. A single quantum of one mode is a **phonon**: a particle-
like excitation carrying energy $\hbar\omega$ and crystal momentum $\hbar\vec k$.
This lesson quantizes the modes, counts the phonons at temperature $T$, defines
their momentum bookkeeping, and constructs the density of states that converts a
dispersion relation into a thermal or spectroscopic prediction.

## Normal modes as independent oscillators

A normal-mode transformation writes the crystal Hamiltonian as a sum over
independent oscillators. Each mode $(\vec k, s)$ contributes a term of the form

$$
H_{\vec k,s} = \frac{1}{2}P_{\vec k,s}^\ast P_{\vec k,s}
+ \frac{1}{2}\omega_s^2(\vec k)\,Q_{\vec k,s}^\ast Q_{\vec k,s},
$$

a harmonic oscillator in the normal coordinate $Q_{\vec k,s}$ with frequency
$\omega_s(\vec k)$. The full Hamiltonian is $H = \sum_{\vec k, s} H_{\vec k, s}$
with no coupling between modes. Reducing the crystal to independent oscillators
is what the harmonic approximation buys. Each oscillator quantizes in the usual way, with energy
levels

$$
E_{\vec k,s}(n) = \left(n + \tfrac{1}{2}\right)\hbar\,\omega_s(\vec k),
\qquad n = 0, 1, 2, \dots
$$

The integer $n$ is the number of phonons in the mode. Adding one phonon raises
the mode's energy by exactly $\hbar\omega_s(\vec k)$; the half-quantum
$\tfrac12\hbar\omega$ is the zero-point energy, present even at $T=0$.

> **Definition (Phonon).** A **phonon** is a quantum of a normal mode of lattice
> vibration. A mode with wavevector $\vec k$ on branch $s$ in its $n$-th excited
> state contains $n$ phonons, each carrying energy $\hbar\omega_s(\vec k)$ and
> crystal momentum $\hbar\vec k$. Phonons are the collective, quantized
> excitations of the whole crystal, not vibrations of a single atom; they are
> bosons, so any number may occupy the same mode.

Because the modes are independent oscillators and any number of phonons can share
a mode, the total lattice energy is a sum over modes of their occupation times
their quantum:

$$
U = \sum_{\vec k, s}\left(\langle n_{\vec k,s}\rangle + \tfrac{1}{2}\right)
\hbar\,\omega_s(\vec k).
$$

Everything thermal in the lattice reduces to knowing the mean occupation
$\langle n_{\vec k,s}\rangle$ and the spectrum $\omega_s(\vec k)$.

## Phonon statistics

Phonon number is not conserved: raising the temperature simply excites each
oscillator to a higher rung, creating phonons. A gas of particles whose number is
unconstrained and which obey Bose statistics has chemical potential zero, so the
mean occupation of a mode of frequency $\omega$ is the **Bose–Einstein**
(Planck) distribution

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

Two limits organize its behavior. When $\hbar\omega \ll k_B T$ the mode is
classically excited, $\langle n\rangle \approx k_B T/\hbar\omega \gg 1$, and its
mean energy $\hbar\omega\langle n\rangle \approx k_B T$ recovers equipartition.
When $\hbar\omega \gg k_B T$ the occupation is exponentially small, $\langle
n\rangle \approx e^{-\hbar\omega/k_B T}$: the mode is frozen out, its
contribution to the energy suppressed. The crossover at $\hbar\omega \sim k_B T$
is what makes heat capacity fall below the classical value at low temperature.

$$
% caption: A single vibrational mode as a quantum oscillator. Its equally spaced
% levels are separated by hbar omega; the mean occupation is the Bose-Einstein
% number, large at high temperature (many phonons) and exponentially small at low
% temperature (frozen out).
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % oscillator ladder on the left
  \foreach \i in {0,1,2,3,4}{
    \pgfmathsetmacro{\y}{0.6*\i}
    \draw[thick] (0,\y) -- (2.0,\y);
  }
  \draw[black, <->] (2.25,0) -- (2.25,0.6) node[midway, right] {spacing};
  \node[anchor=south] at (1.0,3.0) {mode levels};
  \node[anchor=north] at (1.0,-0.15) {$n=0$};
  % Bose factor curve on the right: n vs T (schematic, rising)
  \begin{scope}[xshift=4.6cm]
    \draw[->, black!70] (0,0) -- (4.0,0) node[below right] {$T$};
    \draw[->, black!70] (0,0) -- (0,3.1) node[left] {occupation};
    % 1/(exp(1/t)-1) using a stand-in smooth rise; avoid overflow
    \draw[acc, very thick, domain=0.35:3.7, samples=120, variable=\t]
      plot ({\t},{2.6*\t/(\t+0.6) - 0.35});
    \node[acc, anchor=west] at (1.6,1.55) {Bose factor};
    \node[black, anchor=west] at (0.15,2.55) {linear in $T$ at high $T$};
    \node[black, anchor=south west] at (0.2,0.05) {frozen at low $T$};
  \end{scope}
\end{tikzpicture}
$$

## Crystal momentum

A phonon carries $\hbar\vec k$, but $\vec k$ is defined only within the first
Brillouin zone, because shifting $\vec k$ by a reciprocal-lattice vector $\vec G$
leaves the displacement pattern unchanged. The quantity $\hbar\vec k$ is
therefore not true mechanical momentum — a single phonon actually carries no net
mass transport — but a bookkeeping label called **crystal momentum**, conserved
only modulo $\hbar\vec G$.

> **Definition (Crystal momentum).** The **crystal momentum** of a phonon is
> $\hbar\vec k$ with $\vec k$ restricted to the first Brillouin zone. In any
> process involving phonons and other excitations, the total crystal momentum is
> conserved up to an additive reciprocal-lattice vector: $\sum \hbar\vec k_{\text{in}}
> = \sum \hbar\vec k_{\text{out}} + \hbar\vec G$. Because $\vec k$ is defined only
> modulo $\vec G$, this is the strongest conservation law the lattice permits.

The two ways the ledger can balance are physically distinct. When the wavevectors
on both sides already add up within the zone, $\vec G = 0$, and the process is a
**normal** process; the total crystal momentum is literally unchanged. When the
sum of incoming wavevectors lands outside the first zone and must be folded back
by a nonzero $\vec G$, the process is an **Umklapp** process (from the German for
"flipping over"). Umklapp processes reverse the direction of momentum flow and
are the reason lattice heat conduction is finite, as the [thermal-transport
lesson](/condensed-matter/lattice-dynamics/anharmonicity-and-thermal-transport)
develops.

$$
% caption: Two three-phonon collisions in the Brillouin zone (dashed square). In
% the normal process the summed wavevector stays inside the zone. In the Umklapp
% process the raw sum leaves the zone and a reciprocal-lattice vector G folds the
% outgoing phonon back inside, reversing its direction.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % --- Normal process ---
  \begin{scope}
    \draw[black, dashed] (-1.4,-1.4) rectangle (1.4,1.4);
    \draw[acc, very thick, ->] (0,0) -- (0.8,0.5);
    \draw[acc, very thick, ->] (0.8,0.5) -- (1.2,-0.1);
    \draw[black, thick, ->] (0,0) -- (1.2,-0.1);
    \node[anchor=south] at (0,-2.0) {normal};
    \node[acc, anchor=south east] at (0.7,0.55) {$k_1$};
    \node[acc, anchor=south west] at (1.0,0.25) {$k_2$};
    \node[black, anchor=north] at (1.15,-0.2) {$k_3$};
  \end{scope}
  % --- Umklapp process ---
  \begin{scope}[xshift=6.0cm]
    \draw[black, dashed] (-1.4,-1.4) rectangle (1.4,1.4);
    \draw[acc, very thick, ->] (0,0) -- (1.0,0.7);
    \draw[acc, very thick, ->] (1.0,0.7) -- (1.9,1.35);
    % G vector folds it back
    \draw[black, thick, ->] (1.9,1.35) -- (0.1,0.55);
    \node[black, anchor=south] at (1.1,1.15) {$G$};
    \draw[acc, thick, ->] (0,0) -- (0.1,0.55);
    \node[anchor=south] at (0.3,-2.0) {Umklapp};
    \node[acc, anchor=west] at (0.55,0.35) {$k_1$};
    \node[acc, anchor=south] at (1.5,1.05) {$k_2$};
  \end{scope}
\end{tikzpicture}
$$

## The density of states

Thermal averages and spectra are sums over the discrete allowed $\vec k$. With
$N$ cells the $\vec k$ points are spaced by $(2\pi)^3/V$ and become dense, so
sums convert to integrals over the zone. It is efficient to first collect all
modes of a given frequency into the **density of states** $g(\omega)$, defined so
that $g(\omega)\,\d\omega$ counts the number of modes with frequency between
$\omega$ and $\omega + \d\omega$:

$$
g(\omega) = \sum_s \frac{V}{(2\pi)^3}\int_{\text{BZ}}
\delta\bigl(\omega - \omega_s(\vec k)\bigr)\,\d^3 k
= \sum_s \frac{V}{(2\pi)^3}\int_{S_\omega}
\frac{\d S}{\lvert\nabla_{\vec k}\,\omega_s(\vec k)\rvert}.
$$

The last form rewrites the delta function as a surface integral over the constant-
frequency surface $S_\omega$, weighted by the inverse of the group velocity
$\lvert\nabla_{\vec k}\omega\rvert$. Any thermodynamic average of a mode quantity
$f(\omega)$ becomes a single integral $\int f(\omega)\,g(\omega)\,\d\omega$.

The group-velocity weighting has a sharp consequence. Wherever the dispersion is
flat — at band extrema and saddle points, where $\nabla_{\vec k}\omega = 0$ — the
integrand diverges and $g(\omega)$ develops a **van Hove singularity**. In three
dimensions $g(\omega)$ stays finite but its slope jumps (a square-root kink); in
lower dimensions the singularity is stronger. Every maximum and saddle of the
dispersion prints a kink onto the density of states.

> **Definition (Van Hove singularity).** A **van Hove singularity** is a
> nonanalytic feature (a kink or divergence) in the phonon density of states
> $g(\omega)$ arising at a frequency where the dispersion $\omega_s(\vec k)$ has a
> critical point, $\nabla_{\vec k}\omega_s = 0$. In three dimensions these appear
> as square-root cusps at band edges and saddle points; they mark the maximum and
> minimum frequencies of each branch and any internal saddles.

$$
% caption: A phonon branch (left) and the density of states it produces (right).
% Flat regions of the dispersion at the zone center and boundary pile up many
% modes at nearly one frequency, printing van Hove kinks on g(omega); the sharp
% peak sits at the zone-boundary maximum where the group velocity vanishes.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % dispersion on the left: omega(k) rising then flattening
  \begin{scope}
    \draw[->, black!70] (0,0) -- (3.2,0) node[below right] {$k$};
    \draw[->, black!70] (0,0) -- (0,3.0) node[left] {frequency};
    \draw[acc, very thick, domain=0:3, samples=120, variable=\x]
      plot ({\x},{2.5*abs(sin(deg(pi*\x/6)))});
    \draw[black, dashed] (0,2.5) -- (3,2.5);
  \end{scope}
  % density of states on the right (rotated): g(omega) with divergence at top
  \begin{scope}[xshift=5.0cm]
    \draw[->, black!70] (0,0) -- (3.0,0) node[below right] {$g$};
    \draw[->, black!70] (0,0) -- (0,3.0) node[left] {frequency};
    % Debye-like rise then van Hove peak near omega_max=2.5
    \draw[acc, very thick, domain=0:2.42, samples=120, variable=\y]
      plot ({0.5 + 1.6*\y*\y/(6.25 - \y*\y + 0.35)},{\y});
    \draw[black, dashed] (0,2.5) -- (2.9,2.5);
    \node[black, anchor=west] at (1.4,2.62) {van Hove kink};
  \end{scope}
\end{tikzpicture}
$$

## Measuring dispersion by neutron scattering

The dispersion $\omega_s(\vec k)$ is measured directly by scattering a probe that
can exchange both energy and momentum with a single phonon. Thermal neutrons are
ideal: their de Broglie wavelength matches interatomic spacings, so their
momentum $\hbar\vec Q$ is comparable to a reciprocal-lattice vector, and their
kinetic energy ($\sim$ tens of meV) is comparable to $\hbar\omega$ for phonons.
A neutron that creates one phonon obeys

$$
\hbar\vec k_{\text{i}} - \hbar\vec k_{\text{f}} = \hbar\vec k + \hbar\vec G,
\qquad
\frac{\hbar^2 k_{\text{i}}^2}{2m_n} - \frac{\hbar^2 k_{\text{f}}^2}{2m_n}
= \hbar\,\omega_s(\vec k),
$$

energy conservation and crystal-momentum conservation together. The neutron
loses energy $\hbar\omega$ (phonon creation, Stokes) or gains it (phonon
absorption, anti-Stokes). Fixing the incident beam and measuring the outgoing
energy and angle determines both $\vec k$ and $\omega_s(\vec k)$ for one point on
one branch. Sweeping the geometry maps the entire dispersion.

$$
% caption: Inelastic neutron scattering. An incident neutron of wavevector k_i
% scatters to k_f, creating one phonon of wavevector k. The momentum triangle
% closes up to a reciprocal-lattice vector G, and the neutron's energy loss
% equals hbar omega of the phonon.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % sample
  \draw[thick] (0,0) circle (0.4);
  \node[anchor=north] at (0,-0.5) {sample};
  % incident
  \draw[acc, very thick, ->] (-3.0,0.9) -- (-0.35,0.12);
  \node[acc, anchor=south] at (-1.9,0.7) {$k_i$};
  % scattered
  \draw[acc, very thick, ->] (0.35,0.05) -- (2.7,1.4);
  \node[acc, anchor=south east] at (2.4,1.2) {$k_f$};
  % momentum triangle inset
  \begin{scope}[xshift=3.8cm, yshift=-0.6cm]
    \draw[acc, thick, ->] (0,0) -- (1.6,0.0);
    \node[acc, anchor=north] at (0.8,0) {$k_i$};
    \draw[acc, thick, ->] (1.6,0) -- (1.1,1.1);
    \node[acc, anchor=west] at (1.4,0.6) {$k_f$};
    \draw[black, thick, ->] (0,0) -- (1.1,1.1);
    \node[black, anchor=east] at (0.55,0.6) {$k+G$};
  \end{scope}
\end{tikzpicture}
$$

X-rays and electrons scatter from phonons on the same conservation laws, but
neutrons have two experimental advantages: their energies natively match phonon
energies, so the small energy loss is easy to resolve, and they scatter from
nuclei rather than the electron cloud, giving strong signal from light atoms that
X-rays barely see. The dispersion curves so measured feed directly into the
[thermal properties](/condensed-matter/lattice-dynamics/debye-einstein-heat-capacity)
through the density of states $g(\omega)$ built in this lesson.