Magnetism of materials · susceptibility · MR safety

Magnetic Domains & Hysteresis

Why a paperclip is a projectile and a titanium screw is not. An Ising lattice of coupled moments, an applied field, and the memory that separates ferromagnets from everything else in the scanner room.

Below the Curie temperature: neighbouring moments pull each other into domains. Press Randomise at any time to start the growth over.

moment up moment down

Each cell is one magnetic moment. Regions of one colour are domains: the walls between them are where the material stores its magnetic history.

Magnetisation against applied field. A closed loop with area is a ferromagnet; a straight line through the origin is a paramagnet.

M applied field H

Material and field

Temperature T / T_c Below 1.0 the exchange coupling wins and domains survive. Above it, thermal motion wins.

Applied field H In units of the exchange energy. B₀ in the bore is enormous by this measure.

Lattice size Bigger lattices show cleaner domain walls.

Update rate Monte Carlo sweeps of the whole lattice per frame.

Field sweep speed How fast the automatic sweep drives H up and down.

Draw moments as arrows (small lattices)

M per moment0.000

H0.00

T / T_c0.85

remanence--

coercivity--

What this figure is for

Nuclear spins are not the only magnetism in the scanner room. The electrons in the surrounding material matter just as much, and how they respond to B₀ decides whether an object is harmless, an artifact, or a projectile. This is the lattice version of that story.

Ferromagnetic (T < T_c, the default). Neighbouring moments align with each other even without a field, forming domains. Sweep the field and you get a loop with area: the material remembers. Bring the field back to zero and magnetisation remains, which is remanence. Iron, nickel, cobalt, most steels. This is the projectile category, and the force does not switch off when the scan ends, because the magnet is always on.
Paramagnetic (raise T above 1.0). The domains dissolve and the loop collapses onto a single line through the origin: M follows H and forgets it immediately. Susceptibility is small and positive. Titanium, aluminium, gadolinium chelates, deoxyhaemoglobin. Safe to bring in, but still enough to distort the local field and cause susceptibility artifact.
Coercivity is the width of the loop. It is the reverse field needed to force the magnetisation back to zero. A wide loop is a hard magnet, a narrow one is soft. Lower the temperature and watch the loop widen.
Why this matters at the console. The field gradient at the bore entrance is what grabs a ferromagnetic object, so the danger peak is at the door, not at isocentre. And an implant labelled MR Conditional is one whose material sits in the second category, or whose geometry keeps the force and heating within limits.

Watch the domains coarsen. Straight after Randomise the lattice is a patchwork. Left alone, the domain walls cost energy, so the patches merge and the largest domain eats the rest until a single domain fills the lattice. Lower the temperature to freeze that process part-way; raise it toward 1.0 and the walls churn continuously.

Try this in class: set T/T_c to 0.6, press Sweep the field, and let one full loop draw. Then raise the temperature past 1.0 without clearing the plot. The fat loop and the thin line appear on the same axes, which is the clearest single picture of the difference between a ferromagnet and a paramagnet.

A note on the model: this is a two-dimensional Ising model with nearest-neighbour coupling, solved by Metropolis Monte Carlo. It is a cartoon of a real ferromagnet, but it produces genuine domains, a genuine Curie transition and a genuine hysteresis loop, all from one rule about neighbours.