FERROMAGNETISM, DOMAINS, AND HYSTERESIS

Most everyday materials respond to a magnetic field by about one part in a hundred thousand. Their atomic moments either tilt slightly toward the field (paramagnets) or slightly against it (diamagnets), and the effect is tiny and linear. That is the whole previous topic.

Iron responds a million times more strongly than any paramagnet has any right to. And the response is not linear: iron has a memory. Magnetise a bar, remove the field, the bar stays magnetic. Its atoms are not independently tilting — they are locked onto each other, each one feeling its neighbours more strongly than it feels the field you apply.

Pierre Curie, in his 1895 doctoral thesis, mapped the temperature dependence of magnetic susceptibility and identified the critical temperature above which cooperative behaviour vanishes — the Curie temperature, T_c. Twelve years later his student Pierre Weiss published the picture that made cooperation work: inside a piece of iron, large regions are already fully magnetised, each pointing its own way, separated by thin walls. The bulk looks unmagnetised only because the regions cancel each other. Apply a field and the regions reorganise.

This is the surprise of ferromagnetism. Iron is not a mildly-biased paramagnet; it is a phase. Below T_c, cooperative. Above T_c, ordinary paramagnetic. Cross T_c and the behaviour changes qualitatively.

Weiss called the internally-aligned regions domains. A typical domain is a few micrometres across — millions of atoms, every single moment pointing the same way. A domain is not a defect. It is the ground state: exchange energy (neighbours want to align) fights magnetostatic energy (a uniformly-magnetised chunk would carry a huge field out into surrounding space), and the compromise is to break the piece into many regions whose directions cancel far away. A demagnetised piece of iron looks unmagnetised because its domains happen to point every which way.

Between two neighbouring domains lies a thin region — a domain wall — where the magnetisation swings over about 100 nanometres. Walls cost energy to exist, but less than the alternative of sending the field out into space.

Drag the slider. For small fields the wall bulges elastically — the magenta domain grows slightly into the cyan one, and when you release the field the bulge relaxes back. No permanent change. That is the reversible regime. Keep pushing. At some threshold the wall snaps free of the current pinning site — an impurity, a grain boundary, a dislocation — and jumps to the next pin. That jump is irreversible: releasing the field will not bring the wall back. The snap is a Barkhausen jump, after the 1919 experiment of winding a coil around an iron bar, connecting it to a speaker, and hearing the jumps as audible clicks while sweeping the field.

Every hysteresis loop is the vector sum of billions of these jumps.

Here is the live record of what the domains do as an applied field is cycled.

Start at the origin. The mosaic is randomly oriented — macroscopic magnetisation zero. Apply positive H. Domains whose switching threshold is low flip first; as H climbs, harder domains follow. The mosaic turns magenta; the B–H trace climbs, rounds over, saturates at +M_sat when every last domain has flipped. This initial magnetisation curve is travelled exactly once in the lifetime of the sample.

Reduce H back toward zero. Domains don't unflip just because you stopped pushing — each flipped domain is sitting in its own energy minimum, no restoring force. The trace does not retrace the initial curve; it stays high, curving gently down. At H = 0 the magnetisation is still substantial — the remanence B_r. Iron doesn't forget.

To bring B back to zero you have to reverse the applied field. A specific magnitude of reversed H, the coercive field H_c, is where enough domains flip back that the contributions cancel. Keep going and the piece saturates at −M_sat. Cycle once more and you retrace the same loop forever.

The symbol ∮ is a closed-loop integral — adding up B·dH as H makes one full round trip. Geometrically it is the area enclosed by the loop. In a soft iron transformer core that area becomes heat with every cycle of the AC — the hum and warmth of a plugged-in transformer.

The loop we draw is deliberately soft — realistic iron, Pr/Psat ≈ 0.38, Hc ≈ 15% of the saturation field. Textbooks sometimes draw a sharp rectangle, but that shape exists only in the limit of a hard magnet driven far past its coercive field, and it hides the whole physics of gradual domain reconfiguration.

Heat iron past 1043 kelvin — red-orange hot — and the ferromagnetism evaporates. Drop a bar magnet into that furnace and it comes out non-magnetic. Cool it and the ferromagnetism returns. This is a genuine phase transition, the first second-order phase transition this branch has met, and it is Weiss's other big idea.

Below T_c, exchange coupling wins against thermal agitation and domains exist. The spontaneous magnetisation of a single domain is the order parameter of the transition — zero in the disordered phase, non-zero in the ordered phase. Mean-field theory gives

the square-root shape visible on the magenta curve. Real iron deviates slightly (measured β ≈ 0.36), but √ is the clean textbook shape and carries the intuition.

Above T_c the domains dissolve. Iron becomes an ordinary paramagnet — but with a susceptibility that diverges as T drops toward T_c from above,

the divergence being the fingerprint of the order about to form. The Curie law's plain 1/T picks up the −T_c correction because atoms near the transition already feel their neighbours' influence, even though cooperation hasn't frozen in yet. The cyan dotted tail is this divergence.

You can watch the transition by eye: hang an iron paperclip from a thread, hold a magnet near it so the clip swings to attach, then heat the clip with a blowtorch. As it passes red heat the clip suddenly swings back, released. Cool it; it re-attaches. A phase transition, witnessed.

The difference between a transformer core and a permanent magnet is the shape of the loop.

Soft magnets have low H_c and low remanence. They respond eagerly, saturate quickly, release easily. Narrow loops — small area, small per-cycle loss. Silicon steel, soft iron, μ-metal. You want them inside transformers and motor stators and magnetic shields, where the field must follow an externally driven current without wasting heat.

Hard magnets have high H_c and high remanence. Once pushed to saturation they stay there. Wide boxy loops with huge area (fine — you are not cycling them; you drive them to saturation once and use the stored remanence forever). Neodymium-iron-boron, samarium-cobalt, Alnico, ferrites. These live inside speakers, hard drives, refrigerator magnets, EV rotors.

The two regimes trade off by microstructure: more grain boundaries and pinning sites mean higher H_c. Slow-cooled iron with large crystallites is soft; rapidly-quenched fine-grain alloy is hard. Designers pick the ratio for the application.

Every speaker and every motor rotor relies on a hard ferromagnet holding its polarisation against the reversing field from the voice coil or the drive windings. Every hard-disk platter stores bits as the local direction of the ferromagnetic film's magnetisation; the read-head senses the remanence; the write-head cycles H past H_c only in the regions that need flipping.

Every transformer core in every wall-wart, every pole transformer on every power line, every inductor labelled "μ_r = 2000" exploits the soft regime to concentrate the H from a coil into an enormous B inside the core. The price paid for the amplification is the loop area: core loss, why big transformers hum and need oil cooling.

Every MRI machine wraps its superconducting main coil in a huge soft-iron return yoke to channel the flux and contain the fringe field.

Every compass needle, every fridge-door latch, every fridge magnet is a hard ferromagnet frozen in its remanent state. Leave them on a shelf for decades and the domain configuration barely drifts — room-temperature thermal energy is a tiny fraction of the pinning energy. Iron is patient.

Iron remembers. That memory is what this topic is about. In the next one we will meet a class of materials that do exactly the opposite — they expel every field that tries to enter them, a reversal so sharp it looks like magic and is, like ferromagnetism, a genuine phase transition.