Stoner Model

Intuition

The Heisenberg picture of ferromagnetism — fixed atomic moments coupled by exchange — works beautifully for insulating magnets like the rare earths or magnetic oxides. But it fails in an embarrassing way for the 3d transition metals that everyone actually uses:

• Iron carries ${\mu }_{\text{atom}}\approx 2.22{\mu }_B$ per atom — not an integer.
• Nickel carries ${\mu }_{\text{atom}}\approx 0.61{\mu }_B$ per atom — much less than one Bohr magneton.

Localized spins on an atom can only give integer (or half-integer) moments, so something else is going on. The resolution, due to Stoner (1938), is that the magnetism of the 3d metals lives in the band electrons themselves: the same electrons that carry the current also carry the magnetization. They are itinerant, not localized.

The mechanism is then geometric: the exchange interaction pushes up one spin band and pulls down the other; the bands fill up to a common Fermi level, and the difference in occupation is the magnetization. Whether this self-consistent splitting is energetically favourable is decided by a single number — the Stoner criterion.

Formal definition

Take a paramagnetic metal with a single density of states $g(\epsilon )$ per spin in zero field, so that $g_{\uparrow }(\epsilon )=g_{\downarrow }(\epsilon )=g(\epsilon )/2$. The Stoner ansatz assumes the two spin sub-bands remain rigid but are shifted in energy by an internal molecular field proportional to the magnetization itself:

$${\epsilon }_{\uparrow }(\vec{k})=\epsilon (\vec{k})-Im,{\epsilon }_{\downarrow }(\vec{k})=\epsilon (\vec{k})+Im,$$

where $m=(n_{\uparrow }-n_{\downarrow })/n$ is the dimensionless reduced magnetization and $I$ is the Stoner parameter — the exchange energy per pair of electrons of the same spin, a material constant of order $0.5$–$1$ eV in the 3d metals.

Key results

1. Band picture of a ferromagnet

Imagine the paramagnetic DOS $g(\epsilon )$ filled up to a common Fermi level ${\epsilon }_F$. Turn on the exchange splitting $2Im$:

1. The spin-↑ band slides down by $Im$ — it gains electrons.
2. The spin-↓ band slides up by $Im$ — it loses electrons.
3. The net imbalance $n_{\uparrow }-n_{\downarrow }$ is the magnetization.

Because the bands are continuous, the resulting moment per atom is in general non-integer — exactly what is observed in Fe, Co, Ni. Nickel is the textbook case: the majority d-band is completely filled, the minority d-band has $\approx 0.6$ holes per atom, and the magnetic moment is $\mu \approx 0.6{\mu }_B$.

2. The Stoner criterion

Whether the spin-split solution is energetically favourable comes from a competition at the Fermi level. Move a small slice $\delta n$ of electrons from spin-↓ to spin-↑:

• Kinetic cost (Pauli-like): promoting electrons across the Fermi level costs $$\Delta E_{\text{kin}}\approx \frac{(\delta n{)}^2}{g({\epsilon }_F)}.$$
• Exchange gain: the same spin imbalance lowers the exchange energy by $$\Delta E_{\text{ex}}\approx -I(\delta n{)}^2.$$

A spontaneous magnetization appears precisely when the gain beats the cost, i.e. when

$$\boxed{I\cdot g({\epsilon }_F)>1}\text{(Stoner criterion)}.$$

The criterion favours metals with a large density of states at the Fermi level, which is exactly what narrow, partially filled d-bands deliver. This is why ferromagnetism at room temperature is the privilege of a handful of 3d transition metals (Fe, Co, Ni) and not the noble metals (Cu, Ag, Au), whose Fermi level sits in a broad, dilute s-band.

3. Pd: the textbook near-miss

Palladium has $I\cdot g({\epsilon }_F)\approx 0.9$ — just below the threshold. It is not ferromagnetic, but its Pauli susceptibility is enormously enhanced ($\chi \sim 10^{-3}$, two orders of magnitude above an ordinary metal). A small impurity of Fe or Co can push it over the edge — Pd is on the brink of ferromagnetism.

4. Non-integer moments, explained

The non-integer atomic moments of Fe, Co, Ni follow from the rigid-band picture above:

Element	${\mu }_{\text{atom}}({\mu }_B)$	Band picture
Fe (bcc)	$2.22$	both d-bands partially filled
Co (hcp)	$1.72$	majority filled, minority partially filled
Ni (fcc)	$0.61$	majority filled, $\approx 0.6$ minority holes

There are no whole spins to count — only a continuous Fermi-level imbalance between two spin populations.

Limits and refinements

• The Stoner model overestimates $T_C$ because it ignores spin-wave excitations (magnons), which are the cheap low-temperature excitations that actually destroy long-range order. Modern theory combines Stoner physics with Heisenberg-like spin fluctuations.
• It also predicts only a smooth, mean-field transition; the true ferromagnetic transition is second-order with critical fluctuations.
• For rare-earth ferromagnets (Gd, Dy, …), the 4f electrons are localized and the Heisenberg model is the right starting point; the Stoner picture is reserved for the 3d itinerant magnets.

Summary

Stoner replaced the picture of localized atomic moments by a picture of spin-polarized bands: the same electrons that carry the current carry the magnetization. The transition to ferromagnetism is set by a single dimensionless number,

$$I\cdot g({\epsilon }_F)>1,$$

which is satisfied only by the 3d transition metals Fe, Co, Ni. The non-integer atomic moments measured in those metals are a direct fingerprint of itinerant magnetism.

Connections

• magnetic-materials — Pauli paramagnetism (the $I=0$ limit of Stoner)
• magnetic-moment — the Bohr magneton and the meaning of ${\mu }_{\text{atom}}$
• micromagnetic-energy — Heisenberg exchange, the localized counterpart
• magnetocrystalline-anisotropy — spin-orbit coupling on top of the band picture
• hysteresis — what a ferromagnet does once it exists

References

• E. C. Stoner, Proc. Roy. Soc. A 165, 372 (1938).
• S. Blundell, Magnetism in Condensed Matter (Oxford, 2001), Ch. 7.
• J. Kübler, Theory of Itinerant Electron Magnetism (Oxford, 2009).