The magnet that reverses where its crystal is weakest

Nothing keeps a magnetisation for ever ended with a puzzle it could not compute. A commercial neodymium magnet is made of grains several micrometres across, thousands of times larger than the size below which a particle stays magnetised as a single domain, and yet each grain holds its magnetisation against reverse fields of a tesla or more. It does so, the essay said, because nothing has started the reversal: the grain is uniform not because it cannot divide but because a reversed region has nowhere to begin. Where that beginning happens, and why it happens at the field it does, were left as questions for metallurgy.

They have a sharper form than that, and part of the answer is a piece of quantum mechanics borrowed whole.

The field a perfect crystal could hold #

The magnet that has to fight its own field worked out the reversal of a particle so small that its magnetisation turns as one. The crystal prefers its magnetisation along an easy axis, with an anisotropy energy $K\sin^2\theta$ per unit volume for a tilt $\theta$ away from it; a reverse field $H$ along the axis pulls the other way. The uniform state stays stable until the field reaches the anisotropy field,

$$H_K = \frac{2K}{\mu_0 M_s},$$

where the curvature that held it in its minimum vanishes. For neodymium-iron-boron, with $K = 4.9$ megajoules per cubic metre and $\mu_0 M_s = 1.61$ tesla, $\mu_0 H_K$ is 7.7 tesla. A grain of perfect crystal, however large, would reverse at no lower field, because William Fuller Brown proved in 1945 that a perfect ellipsoid cannot begin to reverse by any non-uniform mode until the field exceeds $H_K$ less the demagnetising field of its own shape. Large grains can contain walls, and walls can move easily, but a grain with no wall in it has no way to make one below that field.

Real magnets come nowhere near it. Sintered neodymium-iron-boron reaches one to three tesla; samarium-cobalt, whose crystal could hold forty, manages a few; ferrites a fifth to a quarter of their anisotropy field. Brown’s theorem and the measured coercivities disagree by factors of three to twenty, in every hard magnetic material ever made. Brown called it a paradox. Its resolution is that Brown’s theorem is about perfect crystals, and the reversal begins precisely where a crystal is imperfect.

A tilt that has to start somewhere #

Consider a grain magnetised along $+z$ with a reverse field along $-z$, and allow the magnetisation to tilt by a small angle $\theta(x)$ that varies across the grain. Three energies compete. The anisotropy costs $K\theta^2$ per unit volume for a small tilt. The reverse field rewards it by $\tfrac12\mu_0 M_s H\theta^2$, because the magnetisation is pointing the wrong way and tilting moves it towards the field. And exchange — the coupling that what holds a magnet together traced to the Pauli principle — costs $A(d\theta/dx)^2$, because neighbouring moments resist pointing in different directions. To second order in the tilt, the energy per unit area of a planar region is

$$E = \int \left[A\left(\frac{d\theta}{dx}\right)^2 + \left(K(x) - \tfrac12\mu_0M_sH\right)\theta^2\right]dx.$$

The second-order expansion is the same step every minimum is a parabola takes for any stable state: near equilibrium the energy is a quadratic form, and the state is stable as long as the form is positive for every possible tilt. It stops being positive when the field reaches the smallest value of

$$\frac{\int \left[A\,\theta'^2 + K(x)\,\theta^2\right]dx}{\tfrac12\mu_0 M_s\int\theta^2\,dx}$$

over all tilt profiles — the lowest eigenvalue $\lambda$ of the operator $-A\,d^2/dx^2 + K(x)$, divided by $\tfrac12\mu_0M_s$.

That operator is Schrödinger’s. Put $\hbar^2/2m$ where $A$ is and a potential energy where $K(x)$ is, and the condition for the magnet to reverse becomes the energy of the ground state of a particle in the potential $K(x)$. The box that allows only some energies found that confining a wave raises its lowest energy above the bottom of the well, by an amount that grows as the well narrows; exactly the same arithmetic now decides at what field a magnet gives way.

The ground state of a soft layer #

In a perfect crystal the potential is flat at $K$, the ground state sits at $K$, and the nucleation field is $H_K$: Brown’s theorem. A defect where the anisotropy is lower — a grain boundary where the crystal structure is disordered, a thin layer of a different phase, a region of strained or misaligned lattice — is a well in the potential.

The ground state in the well lies above the well’s floor, as any confined wave’s must, and below the crystal’s anisotropy outside it. Its energy, as a fraction of $K$, is the factor $\alpha$ by which the nucleation field falls short of the anisotropy field: $H_N = \alpha H_K$. The mode itself is the shape of the first tilt — largest in the defect, leaking into the good crystal on either side over a distance set by the ratio of exchange to anisotropy, $\sqrt{A/K}$. That length is the wall parameter, the same one the first length that belongs to the substance found setting the thickness of a domain wall, and it is 1.25 nanometres in neodymium-iron-boron. Once the tilt in the defect grows past the linear regime it becomes a reversed nucleus, the nucleus becomes a domain wall, and in a hard magnet the wall then sweeps through the rest of the grain at once, because the field that nucleated it is far above the field needed to move it.

How wide a weak spot must be #

The quantum analogy immediately says which defects matter. A narrow well holds its ground state weakly: a particle in a well much narrower than its own wavelength spends most of its time outside, and its energy is nearly the energy outside. A wide well holds the ground state near its floor.

So a single atomic layer of disordered material at a grain boundary costs almost nothing, because exchange will not let the magnetisation turn inside it alone; the good crystal on either side holds it. A soft layer a couple of wall parameters thick — two or three nanometres in neodymium-iron-boron — already costs most of the anisotropy field. That is the scale of the disordered regions at the surfaces of real grains, of the thin intergranular phase that sintered magnets contain, and of the damaged layer left on any grain by grinding or by contact with an incompatible phase. The coercivity of the whole magnet, a property of grains micrometres across, is set by a few nanometres at their surfaces.

The arithmetic also explains why the composition of the grain-boundary phase matters so much. The intergranular layer in a good neodymium magnet is rich in neodymium and nearly non-magnetic, and its presence lowers the coercivity less than a ferromagnetic boundary would, because a non-magnetic layer decouples the grains instead of offering the reversal an easy start. Metallurgists found empirically that an annealing step at about five hundred degrees, which changes that layer’s structure and composition, triples the coercivity. In the language of the well, the anneal makes the defect shallower and narrower, raising the ground state.

A defect solved exactly #

A real defect does not have sharp walls; its anisotropy recovers smoothly into the crystal over some distance. Helmut Kronmüller chose a profile that makes the problem exactly solvable, $K(x) = K[1 - \Delta\,\mathrm{sech}^2(x/r_0)]$, which in quantum mechanics is the Pöschl–Teller well. The solitons a hump already contains met the same $\mathrm{sech}^2$ shape as the potential through which a wave passes without reflecting, and it is one of the few wells whose ground state has a closed form.

The ground-state energy gives $\alpha = 1 - (l/r_0)^2(s-1)^2$, with $l = \sqrt{A/K}$ and $s = \tfrac12 + \sqrt{\tfrac14 + \Delta(r_0/l)^2}$. For a defect whose anisotropy reaches zero at its centre and recovers over a wall parameter, $\alpha$ is exactly $(\sqrt5 - 1)/2 = 0.618$ — the golden ratio’s reciprocal turning up, as it occasionally does, in the ground state of a well with these proportions. Five times wider and $\alpha$ is 0.18. The exact solution and a numerical one on a fine grid agree to three decimal places, which is the check that the numerical method gives the right answer for the step-shaped wells where no exact solution exists. Kronmüller’s estimate, made in the late 1980s, that real neodymium magnets have $\alpha$ near a third and defects a few nanometres wide, is the standard resolution of Brown’s paradox — not a full calculation, since the defect profiles are not known independently, but a demonstration that defects of the observed size are enough.

Two ways to be hard #

Nucleation is not the only way a magnet can resist reversal, and the difference shows up before any reverse field is applied. A sintered neodymium magnet, heated above its Curie temperature and cooled again with no field, falls into domains; each grain then contains walls. Magnetising it from that state is easy — a modest field sweeps the walls out of every grain, in the small jumps the curve that is really a staircase followed — and the magnet reaches saturation at a field far below its eventual coercivity. Once the walls have gone, they have to be made again, and making them is what costs the large reverse field. A nucleation-controlled magnet is soft on the way up and hard on the way down.

The other kind is hard both ways. In samarium-cobalt of the composition Sm₂Co₁₇, a heat treatment grows a cellular structure: cells of the main phase tens of nanometres across, separated by thin walls of a second phase whose domain-wall energy is different. A domain wall moving through the material has to cross these cell boundaries, and at each one its energy changes abruptly, so it is held — pinned — until the field is strong enough to push it over. Here walls exist all the time, and the coercivity is the field needed to move them rather than to create them; magnetising the virgin magnet takes as much field as reversing it. The two mechanisms answer Brown’s paradox in opposite ways. In the first, the defect is where reversal starts; in the second, the defect is what stops it from spreading. Both put the coercivity in the microstructure rather than in the crystal, and the same eigenvalue arithmetic, applied to a wall crossing a layer of different anisotropy rather than to a tilt starting in one, gives the pinning field too.

The resemblance to an older problem is close. The barrier a new phase has to climb found supersaturated vapour sitting indefinitely in a clean vessel and condensing at once on the first speck of dust, because a surface lowers the barrier a droplet must climb. A reversed domain is a new phase in the same sense, a perfect crystal is the clean vessel, and a soft grain boundary is the speck of dust. In both cases the threshold for the transition is a property of the worst place in the sample, not of the substance, and the way to raise it is to remove the worst places rather than to improve the average.

The grain’s own field #

A defect is not the only thing that lowers the nucleation field. Each grain is a magnet, and its own magnetisation produces a field that opposes it — weakest at the centre of a smooth ellipsoid, much stronger at the edges and corners of a real, faceted grain, where the field from the surface charge concentrates. That demagnetising field adds to the applied field wherever reversal begins, and since reversal begins at the weakest place, it begins where the demagnetising field is strongest as well as where the anisotropy is lowest. Kronmüller wrote the coercivity as

$$H_c = \alpha H_K - N_{\text{eff}}M_s,$$

with an effective demagnetising factor $N_{\text{eff}}$ that can exceed one at a sharp corner.

Both terms are measurable together. The anisotropy field and the magnetisation change with temperature, and $\alpha$ and $N_{\text{eff}}$, which are set by the defects’ geometry, should not; plotting the measured coercivity against $H_K$ at a series of temperatures should give a straight line with slope $\alpha$ and intercept $-N_{\text{eff}}M_s$. For sintered neodymium magnets the lines come out straight, with $\alpha$ between about a third and a half and $N_{\text{eff}}$ near one. For a material used in the motors of electric cars, which run at a hundred and fifty degrees, the temperature dependence is the main engineering problem: $H_K$ falls steeply on heating, the term $\alpha H_K$ falls with it, and a magnet that is comfortably hard at room temperature can be demagnetised by the motor’s own field when hot.

Hardening only where it is needed #

The analysis turns into a recipe. Adding dysprosium or terbium to neodymium-iron-boron raises the anisotropy — dysprosium’s own compound has an anisotropy field about twice neodymium’s — but these heavy rare earths are scarce and costly, and they couple their moments antiparallel to iron’s, lowering the magnetisation and with it the energy the magnet can store. Alloying the whole grain buys coercivity at the price of strength.

The well says that only the anisotropy where the first tilt lives matters, and the first tilt lives within a few wall parameters of the grain’s surface. So the heavy rare earth is needed only there. In grain-boundary diffusion, developed in the 2000s, dysprosium or terbium is applied to the surface of a finished magnet and diffused inwards along the grain boundaries at high temperature, leaving each grain with a shell a few nanometres to a micrometre thick of high anisotropy around an unaltered core. The ground state of the reversal mode is raised almost as much as if the whole grain had been alloyed, with a fraction of the heavy rare earth, and the magnetisation of the cores is untouched. It is the quantum mechanics of the well, applied with a furnace.

What the well cannot show #

The drawings reduce reversal to one dimension: a planar defect, a tilt varying only across it, and a linear stability analysis that says where the uniform state fails and nothing about what follows. Real nucleation happens at edges and corners in three dimensions, where the demagnetising field is not uniform, and the mode that goes unstable first is a three-dimensional object found only by micromagnetic simulation. Those simulations, run on grains with realistic surface layers, now reproduce measured coercivities within tens of per cent, but each needs an assumed defect profile, and those are not known in detail.

The linear analysis also ignores temperature. Nothing keeps a magnetisation for ever found that a finite barrier is crossed by thermal activation, and near the nucleation field the barrier is small; a magnet at room temperature reverses at a field somewhat below the linear $\alpha H_K$, by an amount that depends on how long the field is applied. What the figures show is the field at which the barrier vanishes. Nor does a single defect in a single grain describe a whole magnet. In a sintered magnet the grains are mostly decoupled by their non-magnetic boundary layer, and each reverses at its own nucleation field, so the magnet’s loop is the sum of a distribution of grain coercivities. In nanocrystalline and hot-deformed magnets the grains are small and touch directly, exchange couples them, and a reversal nucleated in one grain can cascade into its neighbours — the cooperative behaviour that makes the coercivity of these magnets depend on grain size in ways a one-grain calculation cannot capture. The domain of the drawings is a uniaxial crystal with a planar defect, exchange stiffness and anisotropy as the only material constants, and reversal by a mode small enough to linearise.

Still open: what the weakest place actually is #

Electron microscopy can now resolve the structure and composition of a grain boundary in a neodymium magnet atom by atom, and finds layers of disordered crystal, a few nanometres of intergranular phase whose magnetism depends on its iron content, and misaligned surface grains. Which of these sets the coercivity of a given magnet — which is the deepest, widest well — is still argued, and the answer differs between processing routes. Measurements of the anisotropy of a single atomic layer at an interface do not exist, so the profiles fed into the calculations are inferred from the coercivity they are meant to explain.

The structure of the argument does not depend on those details. Reversal starts in the ground state of the operator $-A\,d^2/dx^2 + K(x)$, so a perfect crystal holds to $H_K$ but a soft layer is a well whose ground state sets the coercivity: harmless when much thinner than $\sqrt{A/K}$, 1.25 nm in neodymium-iron-boron, and decisive at a few times that — a fully softened defect a wall parameter in half-width gives α = 0.618 exactly. The strongest magnets are limited not by the crystals they are made of but by the few nanometres where those crystals end.