Exchange and Ferromagnetism

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

E_{dip} \sim \frac{μ _{0} μ _{B}^{2}}{4 π r ^{3}} \approx 3 \times 1 0^{- 24} J \approx 0.2 K .

That is four thousand times too small to hold the order together against thermal agitation at $1000 K$ . 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. 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} = 2 J,

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.

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.

Summing the pairwise exchange over a lattice of localized spins $S_{i}$ gives the Heisenberg Hamiltonian

H = - 2 ⟨ i, j ⟩ \sum J_{ij} S_{i} \cdot 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,

B_{E} = λ μ_{0} M,

the molecular field, with $λ$ 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 applies with that replacement:

M = n g μ_{B} J B_{J} (\frac{g μ _{B} J ( B + λ μ _{0} M )}{k _{B} T}) .

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 $x = g μ_{B} J λ μ_{0} M / k_{B} T$ , which is one relation between $M$ and $x$ ; the definition of $x$ is a second, linear relation $M = (k_{B} T / g μ_{B} J λ μ_{0}) x$ . Their intersection is the solution. The Brillouin curve starts with slope $(J + 1) /3 J$ 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.

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.

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} μ _{B}^{2} J ( J + 1 ) λ μ _{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 corrects it there).

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.

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 $λ μ_{0} M$ present and solving for $M / B$ gives the Curie–Weiss law

χ = \frac{C}{T - T _{c}}, C = \frac{μ _{0} n p ^{2} μ _{B}^{2}}{3 k _{B}},

with the same Curie constant $C$ and effective moment $p = g J (J + 1)$ 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 $1/ χ$ 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 $3 d$ band electrons, and the saturation moment per atom ( $2.2 μ_{B}$ 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 $Δ$ 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 $3 d$ series satisfy it because their narrow $d$ bands give a high $g (E_{F})$ ; the broad $s$ - $p$ metals do not, however large their $U$ .

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.

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.