Stuck to your fridge, right now, is a small piece of metal holding up a shopping list against the full pull of gravity — and it has been doing that for years, drawing no power, connected to nothing. Where does that endless force come from? It feels like something for nothing. It is not: it is one of the most quietly astonishing places in everyday life where quantum mechanics reaches all the way up to the human scale. The permanence of a fridge magnet is not explained by classical physics at all. It is the visible fingerprint of the Pauli exclusion principle and the spin of the electron.

This page is about where magnetism in solids comes from. We will start with the tiny magnet carried by every electron, meet the three great families of magnetic behaviour — diamagnetism, paramagnetism and ferromagnetism — and then confront the central surprise: the force that makes iron a permanent magnet is not magnetic at all. It is electrostatic, dressed up by quantum statistics into something called the exchange interaction. That is the twist worth waiting for.

The electron is a tiny magnet

A magnetic moment is what you get whenever electric charge circulates. A current loop is a little magnet; so is a charge with angular momentum. An electron in an atom contributes on two counts: its orbital motion around the nucleus, and its intrinsic spin — an angular momentum it carries even when it is sitting still, with no classical picture behind it. Both give the electron a magnetic moment measured in units of the Bohr magneton:

The moment tied to spin is \boldsymbol{\mu} = -g_s\,\mu_B\,\mathbf{S}/\hbar, with the spin g-factor g_s \approx 2. So each electron is a bar magnet about one Bohr magneton strong. In a filled shell the moments pair off and cancel — for every spin-up there is a spin-down — which is why the magnetism of a solid is decided almost entirely by its unpaired electrons. An atom with a half-filled shell (iron, cobalt, nickel, the rare earths) carries a big net moment; a closed-shell atom carries essentially none.

The magnetic susceptibility \chi is the number that ties a material's magnetisation M to the applied field H through M = \chi H. Its sign and its temperature dependence are the fingerprints that tell the three families apart.

Diamagnetism: the response hiding in everything

Switch on a magnetic field near any material — even one with no permanent moments at all, like water, copper, or wood — and the electron orbits readjust to oppose the change, exactly as an induced current in a loop opposes the flux that made it (Lenz's law, made microscopic). The induced moments point against the field, so the material is very slightly repelled from a strong magnet, and the susceptibility is small and negative, of order \chi \sim -10^{-5}. It barely depends on temperature, because it is a property of the orbits themselves, not of any thermal jostling.

Diamagnetism is feeble, but it is universal, and with a strong enough field its effects become spectacular — as the vignette below explains. In most everyday magnetic materials it is simply drowned out by the far stronger paramagnetic or ferromagnetic response of the unpaired spins.

Paramagnetism and the Curie law

Give a material unpaired moments and something new competes with the field: heat. The applied field B wants to line every moment up (lowest energy is -\boldsymbol{\mu}\cdot\mathbf{B}, moment parallel to field); random thermal kicks of energy \sim k_B T want to scramble them. The tug-of-war between the two sets how much net alignment survives. When the field is weak compared with temperature (\mu B \ll k_B T, the ordinary case), the alignment is small and proportional to B/T, giving the Curie law:

where C is the Curie constant, n the number of moments per unit volume, and \mu_{\text{eff}} the effective moment per atom. The message is simple: colder means more magnetic, because the thermal scrambling that fights the field weakens as T drops. Warm a paramagnet and its susceptibility falls in exact inverse proportion.

Worked example — a Curie-law ratio. A paramagnetic salt has susceptibility \chi_1 at T_1 = 300\ \text{K}. Cool it to T_2 = 100\ \text{K}. Because \chi = C/T with the same C,

The susceptibility triples. Notice the ratio flips the temperatures — a bigger \chi goes with a smaller T.

Ferromagnetism: alignment without a field

In iron, cobalt and nickel something far stronger happens. Below a material-specific Curie temperature T_C (for iron, T_C \approx 1043\ \text{K}) the moments line up spontaneously, all pointing the same way, with no applied field at all. The material becomes its own magnet. Heat it above T_C and the order melts: the spins scramble and the material turns back into an ordinary paramagnet.

What holds the spins in lockstep? Not the magnetic field one atom's moment makes at its neighbour — we will see in a moment that this is a thousand times too weak. The real agent is the quantum exchange interaction. It is born of two utterly non-magnetic facts working together: the Pauli exclusion principle (two electrons cannot share a quantum state, so electrons with parallel spins must keep out of each other's way) and the ordinary Coulomb repulsion between electrons. When two electrons align their spins, Pauli forces them into a spatially antisymmetric state that keeps them apart, which lowers their Coulomb energy. In the right materials, aligning spins is electrostatically cheaper — and so the spins align.

Why the Curie temperature is huge: exchange vs. dipole

Here is the number that settles the argument. Two magnetic moments of about one Bohr magneton, sitting a typical atomic spacing r \approx 0.25\ \text{nm} apart, interact through the classical dipole–dipole field with an energy of order

Set that equal to k_B T and you get an ordering temperature of roughly 1\ \text{K}. If magnetism came from the dipole field between atoms, iron would lose its magnetism the moment you took it out of liquid helium. But iron stays magnetic in a blazing oven up to 1043\ \text{K} — a thousand times higher. The exchange energy J, being an electrostatic Coulomb energy, is of order 0.1\ \text{eV}, and k_B T_C \sim J lands exactly in the right range. The factor of a thousand is the whole story. Permanent magnetism is a quantum-electrostatic effect wearing a magnetic disguise.

Above T_C: the Curie–Weiss law

The interactive graph shows how a ferromagnet's susceptibility behaves as it is cooled towards its ordering temperature. Above T_C a ferromagnet is a paramagnet, but a willing one: the exchange interaction is already trying to align the spins, giving the field a helping hand. The Curie law picks up a shift and becomes the Curie–Weiss law:

where \theta (the Weiss temperature) is close to T_C for a ferromagnet. Drag the slider for \theta and watch the curve. As T falls towards \theta the susceptibility diverges — the material becomes infinitely eager to magnetise, which is precisely the moment spontaneous order sets in. For a plain paramagnet \theta = 0 and the curve is the ordinary Curie law with its pole at absolute zero.

Worked example — Curie–Weiss. A ferromagnet has C = 2\ \text{K} and Curie temperature \theta = 300\ \text{K}. Just above ordering, at T = 320\ \text{K},

a large value. At T = 800\ \text{K}, far above ordering, \chi = 2/500 = 0.004 — small, ordinary paramagnetism. The closer you sit to \theta, the more violently the susceptibility grows.

Domains and the hysteresis loop

If exchange aligns every spin in a lump of iron, why doesn't every iron nail leap to your fridge? The answer is domains. A ferromagnet lowers its overall magnetic energy by breaking into many microscopic regions — domains — each fully magnetised, but pointing in different directions so their external fields largely cancel. An unmagnetised iron bar is not a bar of disordered spins; it is a patchwork of perfectly ordered domains whose arrows point every which way. Applying a field lets the favourably-aligned domains grow at the expense of the others, and the bar becomes magnetised as a whole.

Because domain walls stick and resist moving, the magnetisation M lags behind the applied field H: the material "remembers" its history. Plot M against H as you sweep the field up, down and back, and you trace a hysteresis loop:

The graph below draws an idealised loop so you can pick out the remanence on the vertical axis and the coercivity on the horizontal axis.