Magnetic Domains: Why Iron Isn't a Magnet Until You Ask It To Be

Why a magnet isn't magnetised everywhere at once

A chunk of iron is made of atoms whose electron spins want to align with their neighbours — that alignment is what makes it ferromagnetic. Yet an unmagnetised nail has no net field. The resolution, worked out by Pierre Weiss in 1907 before anyone knew about spin, is that the material splits into small regions called domains, each fully magnetised internally but pointing in a different direction from its neighbours, so the sample's net magnetisation averages to roughly zero.

Domains exist because of a competition between energies. Full alignment across the whole sample minimises exchange energy (the quantum-mechanical preference of neighbouring spins to line up) but maximises the magnetostatic energy stored in the stray field outside the sample — like the field around a bar magnet's poles. Splitting into oppositely-oriented domains keeps neighbouring spins mostly aligned (low exchange cost near the domain wall) while letting the field loop back inside the material instead of spilling outside, sharply cutting the magnetostatic energy. The material settles into whatever domain pattern minimises the total.

The Ising model: the simplest theory that grows domains

The workhorse toy model is the Ising model: a lattice of spins sⁱ = ±1, interacting only with their nearest neighbours plus any external field H.

E = −J ∑ sⁱs⁵ − H ∑ sⁱ
             ⟨i,j⟩         i

J > 0  ferromagnetic coupling (neighbours prefer to align)
below the Curie temperature T_c, the lattice spontaneously
breaks into domains of aligned spins even with H = 0

Run this on a lattice with a Monte Carlo update rule (flip a spin with a probability set by the energy change and temperature) and, below the Curie temperature, random noise is not enough to keep the spins disordered — they clump into patches, exactly the qualitative behaviour of real domains, even though the model has thrown away almost all of the real material's chemistry.

Domain walls and the width they choose

Between two domains pointing in different directions sits a domain wall — a thin transition region where spins rotate gradually from one orientation to the other rather than flipping abruptly. The wall's thickness is again a competition: exchange energy wants the rotation spread over many atomic spacings (a gentle change costs less exchange energy per step), while magnetic anisotropy — the material's preference for spins to point along specific crystal axes — wants the wall as thin as possible, since every spin outside the easy axis costs anisotropy energy. Balancing the two gives a characteristic wall width of hundreds of atomic spacings in typical ferromagnets, thin on a laboratory scale but enormous compared to a single atom.

Hysteresis: why the B-H loop doesn't retrace itself

Apply an external field H and domains aligned with it grow at the expense of others, by domain walls sweeping through the crystal; at high enough field every domain has flipped into a single one and the material is saturated. Reduce H back to zero and the walls do not fully retreat — lattice defects, impurities and grain boundaries pin them in place, so some magnetisation remains. This remnant magnetisation, and the reversed field needed to drive it back to zero (the coercivity, H_c), trace out the characteristic B-H hysteresis loop instead of a single reversible curve.

H increases 0 → H_sat : domain walls sweep, magnetisation M rises to saturation
H decreases to 0      : M stays above 0 (remnant magnetisation M_r) — walls are pinned
H reversed to −H_c     : M finally returns to 0 (coercive field H_c)

A large H_c means the walls are strongly pinned — a hard magnet, good for permanent magnets that must resist demagnetising fields. A small H_c means walls move easily — a soft magnet, good for transformer cores that must be magnetised and demagnetised many times per second with as little energy loss as possible, since the area enclosed by the hysteresis loop is exactly the energy dissipated as heat per cycle.

Frequently asked questions

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Everything above runs in your browser — open Magnetic Domains and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.