---
title: Exchange and Ferromagnetism
module: Magnetism in Solids
moduleNumber: 9
lessonNumber: 2
order: 902
summary: >
  Magnetic ordering at hundreds of kelvin cannot be dipolar; it is an exchange
  effect, the Coulomb repulsion sorted by the Pauli principle into a spin-dependent
  energy captured by the Heisenberg Hamiltonian. Weiss molecular-field theory
  replaces the exchange field by an average proportional to the magnetization,
  giving a self-consistent equation whose solution is spontaneous magnetization
  below a Curie temperature and a Curie–Weiss susceptibility above it. Itinerant
  ferromagnetism follows from the Stoner criterion on the band density of states.
topics: [Magnetism in Solids]
draft: false
sources:
  - book: Kittel
    ref: "Ch. 12 — Ferromagnetism and Antiferromagnetism"
  - book: Ashcroft & Mermin
    ref: "Ch. 32 — Electron Interactions and Magnetic Structure"
  - book: Hook & Hall
    ref: "Ch. 8 — Magnetic Order"
---

Iron orders ferromagnetically below $1043\ \text{K}$. The moments that align are
of order one Bohr magneton and sit about $0.25\ \text{nm}$ apart, so the magnetic
dipole–dipole energy between neighbors is

$$
E_{\text{dip}} \sim \frac{\mu_0 \mu_B^2}{4\pi r^3}
\approx 3\times10^{-24}\ \text{J} \approx 0.2\ \text{K}.
$$

That is four thousand times too small to hold the order together against thermal
agitation at $1000\ \text{K}$. The interaction responsible is not magnetic at all.
It is the electrostatic Coulomb repulsion between electrons, made spin-dependent
by the Pauli principle: the **exchange interaction**.

## Exchange as an electrostatic effect

The origin is the same two-electron antisymmetry seen in the
[hydrogen molecule](/condensed-matter/molecules-and-bonding/hydrogen-molecule-and-exchange).
A pair of electrons has a total wavefunction antisymmetric under exchange. A
symmetric spatial part pairs with the antisymmetric spin singlet ($S = 0$); an
antisymmetric spatial part pairs with the symmetric spin triplet ($S = 1$). The
two spatial states have different Coulomb energies because the antisymmetric one
keeps the electrons apart — its wavefunction vanishes when they coincide — and so
lowers their repulsion. The energy difference between singlet and triplet is the
**exchange energy**

$$
E_S - E_T = 2J,
$$

where $J$ is the exchange integral, an overlap of the two orbital densities
weighted by the Coulomb interaction. When $J > 0$ the triplet lies lower and
parallel spins are favored — the seed of ferromagnetism. The point is that a
purely electrostatic quantity, the Coulomb repulsion, carries an energy scale of
electron-volts, and even a small spin-dependent piece of it dwarfs the dipolar
term.

$$
% caption: Two overlapping atomic orbitals: the spin-triplet (parallel) state
% keeps the electrons in an antisymmetric spatial function that lowers their
% Coulomb repulsion, so a positive exchange integral favors parallel alignment.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % two orbitals overlapping
  \draw[acc, very thick] (0.9,0) circle (1.05);
  \draw[acc, very thick] (2.7,0) circle (1.05);
  \fill[black] (0.9,0) circle (1.6pt);
  \fill[black] (2.7,0) circle (1.6pt);
  \fill[acc!12] (1.8,0) ellipse (0.22 and 0.54);
  \node[black, anchor=south] at (1.8,0.95) {overlap};
  % spins parallel
  \draw[acc, very thick, ->] (0.9,-1.7) -- (0.9,-0.9);
  \draw[acc, very thick, ->] (2.7,-1.7) -- (2.7,-0.9);
  \node[black, anchor=north] at (1.8,-1.75) {parallel spins favored when $J>0$};
\end{tikzpicture}
$$

Summing the pairwise exchange over a lattice of localized spins $\vec S_i$ gives
the **Heisenberg Hamiltonian**

$$
\mathcal{H} = -2\sum_{\langle i,j\rangle} J_{ij}\,\vec S_i\cdot\vec S_j,
$$

the sum running over distinct pairs. A positive $J_{ij}$ between neighbors makes
the aligned configuration the ground state (ferromagnet); a negative $J_{ij}$
makes antiparallel neighbors favorable (antiferromagnet, the next lesson). The
sign of $J$ is a delicate matter — direct exchange, superexchange through an
intervening anion, and the RKKY interaction mediated by conduction electrons can
each dominate — but the Heisenberg form organizes all of them.

## The Weiss molecular field

Solving the Heisenberg Hamiltonian exactly is intractable. Pierre Weiss's 1907
approximation, predating the exchange concept, replaces the interaction of one
spin with its neighbors by an average internal field proportional to the
magnetization,

$$
\vec B_E = \lambda \mu_0 \vec M,
$$

the **molecular field**, with $\lambda$ a dimensionless coupling that the exchange
integral fixes. Each moment then behaves as an independent paramagnet in the total
field $B + B_E$, so the Brillouin result of the
[previous lesson](/condensed-matter/magnetism/diamagnetism-and-paramagnetism)
applies with that replacement:

$$
M = n g\mu_B J\, B_J\!\left(\frac{g\mu_B J\,(B + \lambda\mu_0 M)}{k_B T}\right).
$$

This is a self-consistent equation: $M$ appears on both sides, once as the
observable and once inside the field it creates. In zero applied field $B = 0$ it
can have a nonzero solution — a magnetization that sustains its own aligning field.

The graphical solution makes the structure plain. Write $M = M_s\,B_J(x)$ with $x
= g\mu_B J\lambda\mu_0 M/k_B T$, which is one relation between $M$ and $x$; the
definition of $x$ is a second, linear relation $M = (k_B T/g\mu_B J\lambda\mu_0)\,x$.
Their intersection is the solution. The Brillouin curve starts with slope
$(J+1)/3J$ at the origin and saturates; the straight line has slope proportional
to $T$. For high $T$ the line is steep and meets the curve only at the origin: no
spontaneous magnetization. Below a critical temperature the line is shallow enough
to cross the curve at a second point, and a nonzero $M$ appears.

$$
% caption: Graphical solution of the Weiss self-consistency in zero field. The
% saturating Brillouin curve and the straight line M proportional to x/T intersect
% away from the origin only when T is low enough; T_c is where the line is tangent
% at the origin.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (5.8,0) node[below right] {reduced variable $x$};
  \draw[->, black] (0,0) -- (0,3.6) node[left] {magnetization};
  \draw[black, dashed] (0,3.0) -- (5.6,3.0);
  \node[black, anchor=south east] at (5.6,3.0) {saturation};
  % Brillouin curve: asymptote 3.0, initial slope 2.1
  \draw[acc, very thick, domain=0:5.5, samples=120, variable=\x]
    plot ({\x},{3.0*(0.7*\x)/sqrt(1 + 0.49*\x*\x)});
  \node[acc, anchor=west] at (3.8,2.15) {Brillouin curve};
  % low T line: slope 0.9 < 2.1, crosses curve at (3.0, 2.71)
  \draw[black, thick] (0,0) -- (3.6,3.24);
  \node[black, anchor=north west] at (2.55,1.5) {$T<T_c$};
  \fill[black] (3.0,2.71) circle (2pt);
  \node[black, anchor=west] at (3.12,2.9) {solution};
  % high T line: slope 2.7 > 2.1, meets curve only at the origin
  \draw[black, thick, densely dotted] (0,0) -- (1.2,3.24);
  \node[black, anchor=south] at (1.15,3.28) {$T>T_c$};
\end{tikzpicture}
$$

Expanding $B_J$ to first order at small $x$ and demanding a nontrivial solution
locates the **Curie temperature**, the highest $T$ at which spontaneous
magnetization survives:

$$
T_c = \frac{n g^2\mu_B^2 J(J+1)\,\lambda\mu_0}{3 k_B}.
$$

At and below $T_c$ the moment grows from zero continuously — a second-order phase
transition. Near $T_c$ the spontaneous magnetization vanishes as $M \propto (T_c -
T)^{1/2}$ in the mean-field theory; the measured exponent is closer to $0.35$
because mean-field theory ignores the correlated fluctuations that dominate near a
critical point. Well below $T_c$ the magnetization approaches saturation, its
approach controlled at the lowest temperatures by spin waves rather than by the
single-spin Brillouin factor (the [spin-wave lesson](/condensed-matter/magnetism/spin-waves-and-magnons)
corrects it there).

$$
% caption: The spontaneous magnetization M(T) rises from zero at T_c and
% saturates at low temperature; the mean-field curve overshoots the data near T_c
% because it neglects critical fluctuations.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.2,0) node[below right] {temperature};
  \draw[->, black] (0,0) -- (0,3.4) node[left] {magnetization};
  \draw[black, dashed] (0,3.0) -- (5.6,3.0);
  \draw[black, dashed] (5.2,0) -- (5.2,3.2);
  \node[black, anchor=north] at (5.2,-0.05) {$T_c$};
  \node[black, anchor=south west] at (0,3.0) {$M_s$};
  % M(T): flat then drops to zero at T_c with vertical tangent
  \draw[acc, very thick, domain=0:5.19, samples=120, variable=\x]
    plot ({\x},{3.0*sqrt(1-\x/5.2)});
\end{tikzpicture}
$$

## The Curie–Weiss law above the transition

Above $T_c$ the spontaneous magnetization is zero, but a small applied field still
induces a magnetization, now enhanced by the molecular field. Keeping the linear
term in $B_J$ with both $B$ and $\lambda\mu_0 M$ present and solving for $M/B$
gives the **Curie–Weiss law**

$$
\chi = \frac{C}{T - T_c}, \qquad C = \frac{\mu_0 n\, p^2\mu_B^2}{3 k_B},
$$

with the same Curie constant $C$ and effective moment $p = g\sqrt{J(J+1)}$ as the
free paramagnet. The susceptibility diverges as $T \to T_c^+$: the response to an
infinitesimal field becomes infinite exactly where spontaneous order sets in. A
plot of $1/\chi$ against $T$ is again a straight line, but now it intercepts the
axis at $T_c > 0$ rather than at the origin — the shift measures the strength of
the exchange coupling. (For an antiferromagnet the same analysis gives a negative
intercept; that case is the next lesson.)

## Itinerant ferromagnetism and the Stoner criterion

The Heisenberg picture assumes moments localized on ions, appropriate for
insulating magnets and the rare earths. In iron, cobalt, and nickel the magnetic
electrons are the itinerant $3d$ band electrons, and the saturation moment per
atom ($2.2\,\mu_B$ for iron) is non-integer — a signature of band, not local,
magnetism. The **Stoner** model applies the exchange idea to the band.

Splitting the spin-up and spin-down bands by an energy $\Delta$ moves electrons
from the minority to the majority band. This costs kinetic energy, because the
transferred electrons occupy states above $E_F$, but it gains exchange energy,
because the now-more-numerous majority spins interact attractively. With an
exchange energy per aligned pair parametrized by $U$, the balance is favorable —
spontaneous band splitting occurs — when

$$
U\, g(E_F) \geq 1,
$$

the **Stoner criterion**. Ferromagnetism requires both a strong intra-atomic
exchange $U$ and a large density of states at the Fermi level $g(E_F)$. The
transition metals at the end of the $3d$ series satisfy it because their narrow
$d$ bands give a high $g(E_F)$; the broad $s$-$p$ metals do not, however large
their $U$.

$$
% caption: Stoner band splitting: the spin-up density of states shifts down and
% spin-down up by the exchange energy, giving unequal band fillings and a net
% moment when U g(E_F) exceeds one.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, ->] (0,0) -- (0,3.6) node[left] {energy};
  % spin up DOS (left), shifted down, filled higher
  \draw[acc, very thick, domain=0:3.0, samples=60, variable=\y]
    plot ({-1.15*sin(\y*60)},{\y});
  \fill[acc!12] plot[domain=0:2.5, samples=60, variable=\y] ({-1.15*sin(\y*60)},{\y}) -- (0,2.5) -- (0,0) -- cycle;
  \node[acc, anchor=north] at (-1.15,-0.1) {spin up};
  % spin down DOS (right), shifted up, filled lower
  \draw[black, very thick, domain=0:3.0, samples=60, variable=\y]
    plot ({1.15*sin(\y*60)},{\y});
  \fill[black] plot[domain=0:1.7, samples=60, variable=\y] ({1.15*sin(\y*60)},{\y}) -- (0,1.7) -- (0,0) -- cycle;
  \node[black, anchor=north] at (1.15,-0.1) {spin down};
  \draw[black, dashed] (-1.6,2.5) -- (1.6,2.5);
  \node[black, anchor=west] at (1.4,2.7) {$E_F$};
  \node[black, anchor=west] at (0.7,3.3) {net moment};
\end{tikzpicture}
$$

The two pictures are limits of one problem: strongly localized moments obey
Heisenberg exchange and give integer moments per ion; strongly itinerant electrons
obey the Stoner criterion and give band moments. Real ferromagnets sit between.
The molecular-field method carries over unchanged to antiferromagnets and
ferrimagnets once the lattice is split into sublattices, the subject of the
following lesson.