Ferromagnetism, Domains, and the Hysteresis Loop

The ferromagnet's puzzle: eager spins, yet a dead nail

Guide 1 left us with a striking picture: in iron, cobalt, and nickel the quantum exchange interaction makes each atom's magnetic dipole moment want to point the same way as its neighbours — not weakly, like the fickle atoms of a paramagnet, but so forcefully that whole neighbourhoods of atoms lock into alignment on their own, with no field applied at all. That is what ferromagnetism means. In iron each atom carries about 2.2 Bohr magnetons, and if you could line up every one of them the material would reach its saturation magnetization — the absolute ceiling, all dipoles pointing one way, worth about 2.1 tesla of magnetic flux density in iron. That is a colossal internal magnet.

This is why ferromagnets amplify a field so enormously. Recall the bookkeeping: the flux density inside a material is B = mu times H, where H is the field you apply and mu is the material's permeability. We write mu = mu_r times mu_0, where mu_0 = 4 times pi times 10^-7 is the permeability of empty space and mu_r is the dimensionless relative permeability — the amplification factor. For air mu_r is essentially 1; for a good soft iron it can be several thousand, and for special alloys like permalloy it reaches 10^5 or more. Wrap a coil around an iron core and the iron multiplies your field by thousands: that single fact is why every transformer, motor, and electromagnet has iron in it.

But now the paradox. If every iron atom is straining to align with its neighbours, an iron nail should be a permanent magnet the moment it solidifies — and yet the nail in your drawer pulls nothing until you stroke it against a magnet. All that internal alignment is real, but it produces no field outside the nail. The whole of this guide hangs on resolving that contradiction, and the answer is one of the prettiest ideas in materials science: the magnetic domain.

One honest caveat before we go on: do not treat permeability like the dielectric constant — a fixed number for the material. It is not. In a ferromagnet mu changes drastically with the applied field: it is small at low field, rises to a peak, then collapses toward mu_0 as the material saturates and has nothing left to give. So a quoted mu_r (say '5000 for soft iron') is really the initial or the maximum value, tagged to a particular point on the curve you are about to meet. The whole loop, not one number, is the honest description.

Domains: the material's quiet compromise

Here is the resolution. A lump of iron does not magnetize as one solid block. Instead it splits itself into many small regions called magnetic domains, and within each domain every atomic dipole is fully lined up — each domain is locally saturated. But different domains point in different directions, and they are arranged so that their fields cancel out on the outside. Picture a stadium crowd where each block of seats does a perfect little wave, but the blocks face different ways, so from a blimp overhead the motion averages to nothing. The exchange interaction still gets exactly what it wants locally; the material just hides the result by pointing its domains every which way.

Why go to this trouble? Because a single uniformly magnetized block would fling a strong field out into the surrounding space, and storing that external field costs energy (call it magnetostatic energy). By breaking into oppositely pointing domains, the material lets its field loop back internally from one domain into the next, keeping the flux inside and paying almost nothing to the outside world. The domains are not a defect or a disorder — they are the material minimizing its total energy, the same principle that governs everything from why a soap film pulls flat to why a solidifying alloy picks its phases.

Where two domains meet, the dipoles cannot flip direction in a single atomic step — the exchange interaction forbids such an abrupt clash. Instead the spins swivel gradually across a thin transition zone called a domain wall, twisting from one orientation to the other over roughly 100 nm, some hundreds of atoms wide. That wall itself costs a little energy, so the material settles on a domain size that balances the wall cost against the magnetostatic saving — typically domains a few micrometres to tens of micrometres across. How easily those walls can move, as we will see, decides almost everything about the magnet.

Turning on the magnet: how domains answer a field

Now apply an external field H to our unmagnetized iron and watch the domains respond. Start from zero, where the domains point every which way and cancel out (magnetization M = 0). As H grows, the material does not rotate all its dipoles at once — that would cost too much. Instead the domains already pointing near the field direction grow at the expense of their neighbours, by sliding their walls sideways to swallow the badly-aligned domains. Only once the easy wall motion is used up do the last stubborn domains rotate bodily to line up with the field, and the material reaches saturation. Trace M against H from zero and you get the initial, or virgin, magnetization curve.

1. Reversible wall bulging. At low H the domain walls bow out slightly, like a tent wall pushed by a breeze. Remove the field and they spring back — nothing is remembered yet.
2. Irreversible wall jumps. Push harder and each wall tears free of the defect pinning it, then leaps to the next obstacle. These sudden jumps are the Barkhausen effect — amplify a coil around the sample and you can literally hear them click.
3. Domain rotation. With the walls mostly gone, the remaining whole domains swing their direction toward the field. This is harder work, so the curve flattens.
4. Saturation. Every dipole now points along the field; the whole sample is one domain. Cranking H higher barely raises B — you have hit the saturation ceiling set by the atoms themselves.

Two things in that sequence are worth holding onto. First, saturation is a hard limit set by the atoms — you cannot squeeze more magnetization out of iron than 2.2 Bohr magnetons per atom, no matter how you process it. Second, and in sharp contrast, whether the walls jump easily or grudgingly depends entirely on the microstructure: on how many defects, precipitates, and grain boundaries lie in the walls' path to snag them. That split — an atomic ceiling on one hand, a structure-sensitive stickiness on the other — is the key to reading the loop we draw next.

The hysteresis loop: a magnet that remembers

Take the saturated sample and now turn the field back down. Here is the twist that names the whole subject: B does not retrace the curve it climbed. It lags behind. When H returns all the way to zero, the material is still magnetized — because the walls that jumped past pinning defects on the way up cannot all jump back, so the domains stay largely aligned. That leftover magnetization at H = 0 is the remanence, B_r: the field the magnet holds with nothing driving it, the reason a magnet is a magnet at all. To actually erase it you must push a reverse field, and the reverse H needed to force B back to zero is the coercivity, H_c. Keep going and the sample saturates the other way; reverse again and you close a loop. That lagging, memory-keeping loop is the hysteresis loop, and it is the single most useful diagram in magnetic materials.

  MAGNETIZATION  B   vs   APPLIED FIELD  H   -- the hysteresis loop

            B
            ^
     +Bs ...|.......########====   saturation: every domain aligned
            |        #      /
     +Br ---+-------#------o         Br = REMANENCE  (H=0, magnet still ON)
            |      #    /  |
   ---------+-----#----o---+---------> H
           /|    o    /   +Hc
     -Hc  o |   /    #                Hc = COERCIVITY (reverse H that zeroes B)
          | | /     #
     -Br  o-+------#
            |     /
     -Bs   ====########.....

   virgin curve (....) climbs once from 0; thereafter the loop is traced round

   SOFT magnet:  tall & THIN loop, tiny Hc  -> flips easily, wastes little heat
   HARD magnet:  wide & FAT  loop, huge Hc  -> hard to flip, keeps its field
   LOOP AREA  =  energy lost as heat per cycle, per unit volume  (J/m^3)

Read that loop and you can read the whole application space. The area enclosed by the loop is energy — the heat dissipated in the material every single time you take it once around the cycle. A transformer core is dragged around its loop fifty or sixty times a second, so you want that area as small as possible: a tall, whisker-thin loop with a coercivity of just a few amperes per metre. That is a soft magnetic material, the stuff of transformer and motor cores. A fridge magnet is the opposite: you want it to sit at its remanence forever and resist any stray field trying to wipe it, so you want a fat loop with a huge coercivity — a thousand kiloamperes per metre for a neodymium magnet. That is a hard magnetic material, a permanent magnet. Same physics, same axes; the difference is entirely the width of the loop, and guide 3 is devoted to engineering it in both directions.

Ferrimagnets, ferrites, and remembering a bit

Iron, cobalt, and nickel are not the only magnets. A whole family of oxides is magnetic through a subtler mechanism called ferrimagnetism. Here the crystal has two kinds of sites whose dipoles point in opposite directions — but the two sublattices carry unequal moments, so they do not fully cancel and a net magnetization survives. Magnetite, Fe3O4, works this way, and it is the original lodestone that swung the first compasses. Be honest about the trade: because the two sublattices partly fight each other, a ferrimagnet's saturation is noticeably weaker than a true ferromagnet's — but it still traces a full hysteresis loop and behaves, for engineering purposes, like a magnet.

Why bother with a weaker magnet? Because ferrites — the ceramic ferrimagnets like MnZn and NiZn oxides — carry one decisive advantage over metals: they are electrical insulators. Recall from the electrical rung that a changing magnetic field induces swirling eddy currents in any conductor, and in a solid metal core those currents waste energy as heat and get worse at high frequency. A ferrite hardly conducts, so eddy currents are throttled and the material stays efficient up into the megahertz — which is why the little core inside a radio antenna, a switching power supply, or a cable's noise-suppression bead is a ferrite, not iron. The same oxide family also gives us the cheap hard magnets: hexagonal barium and strontium ferrites are the black ceramic magnets in loudspeakers and on refrigerator doors.

Finally, the loop's memory is not just a curiosity — it is how the world stores data. A material with a squarish loop and a moderate coercivity can be pushed to one remanent state or the flipped opposite state, and it holds whichever one you left it in with no power at all: that stable choice is one bit. Writing flips a tiny region past its coercivity with a stronger local field; reading senses the direction of its remanent field. That is the physics inside every hard drive, magnetic-stripe card, and recording tape ever made. So the hysteresis loop you just learned to read is doing double duty right now — softly, in the transformer humming in your wall, and hard, in the drive remembering these very words. Guide 3 picks up the thread by pushing the soft-versus-hard split to its engineering limits, and by asking what happens when you heat a magnet past its Curie temperature and the exchange interaction finally lets go.