---
title: Spin Waves and Magnons
module: Magnetism in Solids
moduleNumber: 9
lessonNumber: 4
order: 904
summary: >
  The lowest excitations of a ferromagnet are not single flipped spins but
  collective precessions in which every moment tips slightly and its phase
  advances along the crystal. These spin waves have a quadratic dispersion at long
  wavelength, quantize into magnons obeying Bose statistics, and their thermal
  population removes magnetization as the Bloch T-to-the-three-halves law.
  Antiferromagnetic magnons disperse linearly, and inelastic neutron scattering
  measures both.
topics: [Magnetism in Solids]
draft: false
sources:
  - book: Kittel
    ref: "Ch. 12 — Ferromagnetism and Antiferromagnetism; Spin Waves"
  - book: Ashcroft & Mermin
    ref: "Ch. 33 — Magnetic Ordering; Spin Waves"
  - book: Hook & Hall
    ref: "Ch. 8 — Magnetic Order"
---

At zero temperature a Heisenberg ferromagnet has all its spins aligned. The
cheapest way to excite it is not to reverse one spin completely — that would cost a
full exchange energy $\sim 2JS z$ against all $z$ neighbors — but to tip every spin
by a tiny angle and let the tilt precess with a phase that advances from site to
site. This collective mode is a **spin wave**; its quantum is a **magnon**. Spin
waves are to magnetic order what phonons are to the crystal lattice: the
low-energy, long-wavelength excitations that carry away order as the temperature
rises.

## The classical spin wave

Take a one-dimensional chain of spins $\vec S_n$ with nearest-neighbor
ferromagnetic coupling, described by the Heisenberg Hamiltonian
$\mathcal{H} = -2J\sum_n \vec S_n\cdot\vec S_{n+1}$ with $J > 0$. Treating each
spin as a classical vector, its motion is the torque from the exchange field of its
neighbors,

$$
\hbar\frac{\d\vec S_n}{\d t} = -2J\,\vec S_n\times(\vec S_{n-1} + \vec S_{n+1}).
$$

In the ground state every $\vec S_n = S\hat z$ and the right side vanishes. For a
small-amplitude wave, write $S_n^z \approx S$ and let the transverse components
$S_n^x, S_n^y$ be small. Linearizing and looking for a travelling-wave solution
$S_n^{x,y}\propto e^{i(kna - \omega t)}$ gives the **spin-wave dispersion**

$$
\hbar\omega_k = 4JS\bigl(1 - \cos ka\bigr).
$$

Each spin precesses on a cone about $\hat z$; the phase of the precession advances
by $ka$ from one site to the next, so the tips trace a helix frozen in a snapshot
and rotating in time.

$$
% caption: A spin wave on a chain. Side view: every spin tips by the same small
% angle. Top view (looking down the axis): the precession phase advances steadily
% along the chain, tracing a helix.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % side view: tilted arrows, tilt direction rotates along chain
  \node[black, anchor=east] at (-0.3,1.9) {side view};
  \foreach \i in {0,1,2,3,4,5,6,7,8}{
    \pgfmathsetmacro{\ph}{\i*45}
    \draw[acc, very thick, ->] (\i*0.72,1.5) -- ++({0.28*cos(\ph)},{0.8});
    \fill[black] (\i*0.72,1.5) circle (1.2pt);
  }
  \draw[black] (-0.1,1.5) -- (6.0,1.5);
  % top view: phase arrows around small circles
  \node[black, anchor=east] at (-0.3,-0.4) {top view};
  \foreach \i in {0,1,2,3,4,5,6,7,8}{
    \pgfmathsetmacro{\ph}{\i*45}
    \draw[black] (\i*0.72,-0.4) circle (0.28);
    \draw[acc, very thick, ->] (\i*0.72,-0.4) -- ++({0.28*cos(\ph)},{0.28*sin(\ph)});
  }
\end{tikzpicture}
$$

At long wavelength ($ka \ll 1$) the dispersion becomes quadratic,

$$
\hbar\omega_k \approx 2JS a^2 k^2 \equiv D k^2,
$$

with $D = 2JSa^2$ the **spin-wave stiffness**. The quadratic law $\omega\propto
k^2$ is the signature of a ferromagnetic spin wave and contrasts sharply with the
linear $\omega\propto k$ of an acoustic phonon. Its origin is the conservation of
total spin: a uniform tilt ($k = 0$) is a rotation of the whole magnetization and
costs no energy, so $\omega_k \to 0$ as $k\to 0$, and because the ground state
already extremizes the energy the leading correction is second order in $k$.

## Magnons as quantized excitations

Quantizing the transverse precession turns the classical spin wave into a set of
independent harmonic oscillators, one per wavevector $k$, exactly as for phonons.
Their quanta are magnons. A single magnon of wavevector $k$:

- **carries energy** $\hbar\omega_k$, given by the dispersion above;
- **lowers the total spin** $S^z_{\text{tot}}$ of the crystal by exactly one unit
  of $\hbar$ — the tilt is shared over all $N$ spins, each reduced by $1/N$, but
  the total reduction is one full quantum;
- **is a boson**: any number of magnons can occupy the same mode, and their
  thermal occupation follows the Bose–Einstein distribution

$$
\langle n_k\rangle = \frac{1}{e^{\hbar\omega_k/k_B T} - 1}.
$$

That one magnon reduces $S^z_{\text{tot}}$ by a single unit — rather than by the
$2S$ of a fully reversed spin — is why the spin wave is the cheap excitation. It
spreads the cost of one unit of demagnetization over the whole crystal.

$$
% caption: Magnon occupation grows with temperature through the Bose factor. Low
% frequency modes are populated first; the number of magnons rises without bound
% as their energy hbar omega falls below k_B T.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.4,0) node[below right] {temperature};
  \draw[->, black] (0,0) -- (0,3.6) node[left] {magnon number};
  % rising as T^{3/2}
  \draw[acc, very thick, domain=0:5.8, samples=120, variable=\x]
    plot ({\x},{0.28*\x*sqrt(\x)});
  \node[acc, anchor=south east] at (5.7,3.2) {grows with $T$};
\end{tikzpicture}
$$

## The Bloch law

The number of magnons excited at temperature $T$ fixes how much the magnetization
has fallen from its zero-temperature value. Each magnon removes one unit of spin,
so the fractional magnetization deficit is

$$
\frac{M(0) - M(T)}{M(0)} = \frac{1}{NS}\sum_k \langle n_k\rangle.
$$

Converting the sum to an integral over the Brillouin zone and using the
long-wavelength dispersion $\hbar\omega_k = Dk^2$ (only the low-energy modes are
populated at low $T$),

$$
\sum_k \langle n_k\rangle
= \frac{V}{(2\pi)^3}\int \frac{\d^3k}{e^{Dk^2/k_B T} - 1}
\propto \left(\frac{k_B T}{D}\right)^{3/2}.
$$

The three-dimensional phase space $\d^3 k \propto k^2\,\d k$ combined with the
quadratic dispersion produces the exponent $\tfrac{3}{2}$. This is the **Bloch
$T^{3/2}$ law**:

$$
M(T) = M(0)\left[1 - \left(\frac{T}{T_0}\right)^{3/2}\right],
$$

with $T_0$ set by $D$. It describes the low-temperature magnetization of iron,
cobalt, and nickel far more accurately than the mean-field Brillouin curve, which
predicts an exponentially small deficit and misses the data badly. Mean-field
theory treats each spin in an average field and knows nothing of the soft
collective modes; the Bloch law counts exactly those modes.

$$
% caption: Low-temperature magnetization. The Bloch T-to-the-three-halves law
% (from thermally excited magnons) falls faster than the exponential approach the
% mean-field Brillouin theory predicts, and matches the measured curve.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.4,0) node[below right] {temperature};
  \draw[->, black] (0,0) -- (0,3.6) node[left] {magnetization};
  \draw[black, dashed] (0,3.0) -- (0.4,3.0);
  \node[black, anchor=south west] at (0,3.0) {$M_s$};
  % Bloch: drops as 3.0*(1 - c T^{3/2})
  \draw[acc, very thick, domain=0:4.6, samples=120, variable=\x]
    plot ({\x},{3.0 - 0.28*\x*sqrt(\x)});
  \node[acc, anchor=north east] at (4.5,1.15) {Bloch law};
  % mean-field: flat then drops (exponential-ish) later
  \draw[black, thick, densely dotted, domain=0:4.6, samples=120, variable=\x]
    plot ({\x},{3.0 - 0.02*\x*\x*\x});
  \node[black, anchor=west] at (3.5,2.55) {single spin};
\end{tikzpicture}
$$

The same magnon gas carries a **heat capacity**. The internal energy is $U =
\sum_k\hbar\omega_k\langle n_k\rangle$, and the identical integral gives $U\propto
T^{5/2}$, so the magnon contribution to the specific heat is

$$
C_{\text{mag}} \propto T^{3/2}.
$$

In a magnetic insulator at low temperature this $T^{3/2}$ term adds to the phonon
$T^3$ term (from the [Debye model](/condensed-matter/lattice-dynamics/debye-einstein-heat-capacity)),
and their different exponents let a plot of $C$ versus $T$ separate the two.

## Antiferromagnetic magnons

An antiferromagnet also has spin-wave excitations, but their dispersion differs
fundamentally. The two-sublattice ground state is not an eigenstate of the
Heisenberg Hamiltonian (unlike the ferromagnet), and the linearized equations
couple the precessions of the two sublattices. The result is a **linear**
long-wavelength dispersion,

$$
\hbar\omega_k \approx \hbar c\,\lvert k\rvert,
$$

with $c$ a spin-wave velocity — the same form as an acoustic phonon or a photon,
not the quadratic ferromagnetic law. The linear dispersion changes the
thermodynamics: the low-temperature magnon heat capacity of an antiferromagnet
goes as $T^3$, matching the phonon exponent rather than the ferromagnetic
$T^{3/2}$.

$$
% caption: Magnon dispersion near the zone center. The ferromagnetic branch is
% quadratic in k; the antiferromagnetic branch is linear, resembling an acoustic
% phonon.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (5.4,0) node[below right] {wavevector $k$};
  \draw[->, black] (0,0) -- (0,3.6) node[left] {frequency};
  % ferromagnetic: quadratic
  \draw[acc, very thick, domain=0:4.8, samples=100, variable=\x]
    plot ({\x},{0.14*\x*\x});
  \node[acc, anchor=west] at (3.15,1.6) {ferromagnet (quadratic)};
  % antiferromagnetic: linear
  \draw[black, very thick, domain=0:4.8, samples=2]
    plot ({\x},{0.62*\x});
  \node[black, anchor=west] at (3.3,2.45) {antiferromagnet (linear)};
\end{tikzpicture}
$$

## Measuring magnons

Magnon dispersion curves are measured by **inelastic neutron scattering**, the
same technique used for phonons. A neutron carries both a magnetic moment, which
couples to the electron spins, and a wavelength comparable to the lattice spacing.
When it creates or absorbs a magnon of wavevector $\vec k$ and energy
$\hbar\omega_k$, its own energy and momentum change to conserve both,

$$
\vec k_i - \vec k_f = \vec k + \vec G,
\qquad E_i - E_f = \hbar\omega_k,
$$

where $\vec G$ is a reciprocal-lattice vector. Scanning the scattered neutron
energy at fixed momentum transfer maps out $\omega_k$ directly and confirms the
$k^2$ ferromagnetic and $k$ antiferromagnetic laws, together with the stiffness $D$
that the [Bloch law](/condensed-matter/magnetism/spin-waves-and-magnons) needs.
This closes the magnetism module: the exchange interaction sets the ordered ground
state, mean-field theory gives the transition and the sublattice structure,
domains and hysteresis govern the macroscopic magnet, and magnons are the
excitations that erode the order as temperature rises.