Magnetic hysteresis

Why does a magnet's state depend on where the field has been rather than where it is now?

Most of physics is written as functions. Tell me the pressure and temperature of a gas and I can tell you its density; I do not need to know how it got there. That habit is so ingrained that we barely notice it as an assumption.

Now take a piece of iron and set the applied magnetic field to exactly zero. What is its magnetisation? The honest answer is that the question is unanswerable as asked. It could be strongly magnetised north, strongly magnetised south, or not magnetised at all — and the only way to know is to ask what field it experienced before. The present state of the field does not determine the present state of the iron. Why not?

Let us build the iron up from what is inside it. Each iron atom carries a magnetic moment, and neighbouring moments interact strongly enough that they align with each other over regions containing perhaps billions of atoms. Those regions are domains. Within a domain the material is fully magnetised; between domains the direction changes across a wall a few dozen atoms thick.

Why does a lump of iron divide into domains at all rather than sitting as one giant magnet? Because a single uniformly magnetised block would carry an enormous field in the space outside it, and that costs energy. Splitting into oppositely-directed domains lets the field close on itself inside the material. So the domain pattern is an energy compromise — and this is the first hint of the answer, because compromises can be struck in many different ways.

Now apply a field and watch. The domains already pointing roughly the right way grow at the expense of the others: the walls between them move. And here is the crucial observation — what happens when the wall meets a defect, a dislocation, a grain boundary, an impurity inclusion? The wall sticks. It costs energy to tear it loose. Push the field harder and the wall breaks free in a sudden jump to the next pinning site.

Those jumps are real and audible. Wind a coil around a piece of iron, amplify it, and slowly sweep a magnet past: you hear a hiss of discrete clicks, discovered by Barkhausen in 1919. The magnetisation does not change smoothly. It changes in avalanches.

Now reverse the field and ask the question that decides everything. Do the walls retrace their path?

They do not. Coming back, the walls are pinned at whatever sites they happened to land on, and freeing them requires a field of the opposite sign. So the material arrives back at zero applied field still magnetised — that leftover is the remanence — and you must apply a definite reverse field, the coercivity, before the magnetisation returns to zero. Trace the whole cycle and you get a loop, not a line.

So the underlying reason is this. The material's energy landscape is not a single valley with one bottom; it is rugged, with many local minima separated by barriers the field must pay to cross. The state occupies whichever minimum it was most recently walked into. A system like that cannot be described by a function of the present field, because the present field does not identify the minimum. Its history does. That is what path dependence is, stated generally — and magnetism is simply the cleanest laboratory example.

A few refinements worth having.

The loop's area is not decorative. It is the energy dissipated per unit volume per cycle, lost to heat as walls tear free of pinning sites. This is why the distinction between soft and hard magnetic materials is an engineering one. A transformer core is driven around its loop fifty or sixty times a second, so it wants the thinnest possible loop — low coercivity, small area — which is what silicon steel and ferrite alloys are for. A permanent magnet wants the opposite: a fat loop with high coercivity, so that stray fields cannot demagnetise it. Neodymium-iron-boron magnets are extreme on that axis.

Pinning is therefore a design variable, not a flaw: adding defects and grain structure raises coercivity, annealing them out lowers it.

The memory can be erased, in two ways. Heat the material above its Curie temperature — about 1043 K, or 770 degrees Celsius, for iron — and thermal agitation destroys the alignment altogether; on cooling it comes back with no memory of what went before. Or apply an alternating field of steadily decreasing amplitude, which walks the material around ever-smaller loops until it settles near zero. That second trick is what a tape degausser does.

One caveat on the tidy story above. Domain-wall pinning is the dominant mechanism in ordinary bulk ferromagnets, but not the only route to hysteresis. In particles small enough to hold only a single domain there are no walls to pin, and the loop instead comes from the moment having to rotate over an anisotropy barrier. And below a critical size, thermal energy hops that barrier unaided, the memory vanishes on its own, and the particle becomes superparamagnetic — which is precisely the limit that constrains how small a magnetic storage bit can be made.

• Preisach models, which reproduce a measured loop by summing many idealised two-state elements with different switching thresholds.
• The same shape elsewhere: sorption hysteresis in wood and canvas, elastic hysteresis in rubber, hysteresis deliberately designed into control loops as a dead band.
• Superparamagnetism and the areal density limit of magnetic recording.

Every term the collection defines is gathered in the glossary.