Illustrated Physics
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Series

Magnetisation — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Where a magnet actually sits on its own curve. The second quadrant of a magnet's B–H curve, for a material with a remanence of 1.28 T, and the load lines four shapes of it impose. A magnet's own poles put it in a reverse field of N·M, so the working point is where the curve meets the line B = −μ₀(1−N)/N·H. A long thin magnet with N = 0.02 keeps 98 per cent of its remanence and a squat one with N = 0.7 keeps 30 per cent — the same material, cut differently.

    The magnet that has to fight its own field

    A bar magnet's own poles put it in a reverse field, so the same material cut short and fat is weak and cut long and thin is strong. And a magnetised material does not have a magnetisation — it has a magnetisation and a history, which is why the word for what it does is the Greek for coming late.

    part 1 · electromagnetism
  2. The field a classical partition function cannot see. The momentum plane of one classical charge, in units of the root-mean-square thermal momentum. With no field the Boltzmann weight is a set of circles about the origin; with a field the same circles sit about p = qA, displaced and otherwise unaltered, because the energy depends on p only through p − qA. Integrating over the whole plane therefore cannot notice the displacement, and the numbers beside the drawing are that integral evaluated at 6 displacements: the largest departure from the zero-field value is 3.3e-16, which is the precision of the arithmetic and not a physical effect. The classical free energy has no B in it, so the classical magnetisation is exactly zero at every field and every temperature — no paramagnetism, no diamagnetism, no ferromagnetism. Every magnetic material is therefore evidence of something classical mechanics does not contain.

    The magnetism classical physics forbids

    Write down the partition function of any collection of classical charges in a magnetic field, and the field cancels out. Not approximately, not to leading order — the integral is over all of momentum space and the field only shifts where the middle of it is. So classical statistical mechanics predicts no paramagnetism, no diamagnetism and no ferromagnetism, and a compass needle is a quantum instrument.

    part 2 · electromagnetism
  3. A magnetisation curve, and the same curve magnified. On the left, the position of a domain wall against the applied field, over the whole of a crystal. It looks like a curve. On the right, a window 2.0% of its width, taken 42% of the way across, at the magnification a sensitive measurement reaches: the wall does not move at all while the field rises, then jumps, then stops again. The largest jump in the window covers 0.67 units of wall position in no field change at all. There is nothing smooth underneath this — the smooth curve on the left is a staircase with 1400 steps in it, drawn small.

    The curve that is really a staircase

    A magnetisation curve is drawn as a smooth line and is nothing of the sort. Measured finely enough it is a sequence of jumps of every size, audible as a crackle in a coil, with no typical jump and no smooth motion underneath.

    part 3 · electromagnetism
  4. The interaction that orders a magnet is not the magnetic one. For six ordered magnets, two temperatures on a logarithmic scale: the energy of the magnetic interaction between two neighbouring moments, expressed as a temperature, and the temperature at which the material actually orders. iron orders at 1043 K against a dipolar scale of 0.201 K, a factor of 5182; cobalt orders at 1394 K against a dipolar scale of 0.117 K, a factor of 11961; nickel orders at 627 K against a dipolar scale of 0.015 K, a factor of 42312; gadolinium orders at 293 K against a dipolar scale of 0.790 K, a factor of 371; europium oxide orders at 69 K against a dipolar scale of 0.615 K, a factor of 112; lithium holmium fluoride orders at 1.53 K against a dipolar scale of 1.230 K, a factor of 1.24. The five ferromagnets order between a hundred and forty thousand times above the only interaction their moments have with each other, so whatever aligns them is not magnetism. The sixth is the control: lithium holmium fluoride is a magnet whose ordering really is dipolar, and its two temperatures agree.

    What holds a magnet together is not magnetism

    Two neighbouring moments in iron interact magnetically with an energy worth a fifth of a kelvin, and iron keeps its order to 1,043 kelvin. Whatever aligns them is five thousand times stronger than the only force they exert on one another — and it is electrostatic, with the exclusion principle deciding which of two spatial arrangements two electrons may use.

    part 4 · electromagnetism
  5. Two energies with opposite slopes, and the width between them. The energy per unit area of a domain wall in iron, against the width the wall is assumed to have, split into the two terms that decide it. The exchange term falls as one over the width, because a reversal spread over more atoms turns through a smaller angle between each neighbouring pair. The anisotropy term rises in proportion to the width, because every atom inside the wall points away from an easy direction and pays for it. The sum has a least value, found here by scanning four hundred thousand widths: 92.9 nanometres and 4.46 millijoules per square metre. Neither constant is a dimension of the sample, so neither is the width: it is the first length in magnetism that belongs to the material. The assumed profile turns at a uniform rate, which is not the cheapest way to turn, so this width is 41 per cent above the conventional πδ of the exact profile, and this energy 11 per cent above the exact 4√(AK). A trial function can only ever overestimate, and eleven per cent is how much this one costs.

    The first length that belongs to the substance

    Every length in magnetism so far has been a length of the sample — a demagnetising factor is a shape, an avalanche cutoff is a sample's own restoring field. A domain wall's width is not. It is √(A/K), made of two material constants and nothing else, and across seven ordinary magnets it runs from two and a half nanometres to nine hundred.

    part 5 · electromagnetism
  6. The barrier a reverse field takes away. The energy of a single-domain particle against the direction its moment points, in units of its anisotropy energy, for four strengths of reverse field along the easy axis. With no field the two directions are equally good and the barrier between them is exactly the anisotropy energy. A reverse field tilts the landscape and lowers the barrier out of the forward well as the square of the field: at 0.0 of the anisotropy field the barrier is 1.000 KV, at 0.1 of the anisotropy field the barrier is 0.810 KV, at 0.3 of the anisotropy field the barrier is 0.490 KV, at 0.6 of the anisotropy field the barrier is 0.160 KV. Each barrier is located by scanning two hundred thousand directions for the stationary points rather than by substituting the formula. The whole of the argument follows from the barrier being finite: a particle does not need the field that removes the barrier, only time enough to be shaken over what is left of it.

    Nothing keeps a magnetisation for ever

    A magnetised particle sits in a well with a barrier between it and the other direction, and a barrier of finite height is crossed eventually. So remanence has a lifetime, coercivity is a different number depending on how fast it is measured, and a grain below about twenty nanometres of iron forgets within a second at room temperature.

    part 6 · electromagnetism
  7. The tilt that starts the reversal, in a well of soft crystal. A planar defect three wall parameters wide (three times √(A/K)) in which the crystal's anisotropy falls to zero, drawn as the anisotropy K(x) across it; the curve is the shape of the first tilt of the magnetisation to become unstable as a reverse field is raised, the lowest mode of −A d²/dx² + K(x), and the dashed line is its eigenvalue. The equation is Schrödinger's, with the exchange stiffness A in place of ħ²/2m and the anisotropy as the potential, and the reversal begins in its ground state. Here the ground state lies at 0.373 K, so the magnet reverses at 37 per cent of the anisotropy field. The mode reaches out of the well a distance of about √(A/K) on each side, because exchange will not let the tilt change abruptly.

    The magnet that reverses where its crystal is weakest

    A perfect crystal of neodymium-iron-boron should hold its magnetisation against a reverse field of 7.7 tesla. The best magnets made of it give way at one to three. The shortfall, known for eighty years as Brown's paradox, is not a failure of the crystal but of its edges: reversal begins wherever the crystal is weakest, in a layer a few nanometres thick. Working out how weak that layer can afford to be turns out to be a problem already solved in quantum mechanics — the nucleation field is the ground-state energy of a particle in a well shaped like the defect.

    part 7 · electromagnetism
  8. A loop, and the butterfly it makes. Left: magnetisation against applied field for a model ferromagnet, as a fraction of saturation, with the field in units of the coercive field — the rising branch and the falling one, crossing zero magnetisation at plus and minus one. Right: the strain the same material shows along the field, as a fraction of its saturation magnetostriction, on the same two branches. Because the strain goes as the square of the magnetisation it does not care which way the magnetisation points, so it is the same at full positive and full negative saturation, and it falls to nothing at the two points where the magnetisation passes through zero — the coercive fields. A loop traced once produces a butterfly traced once: the strain goes through two full stretches for every one cycle of the field.

    The hum at twice the mains

    A transformer hums, and the note is not the frequency of the mains but twice it: a hundred hertz in Europe, a hundred and twenty in America. The reason is that iron changes shape when it is magnetised, by a few parts in a million, and the change does not care which way the magnetisation points. A core driven at fifty cycles a second therefore stretches a hundred times a second. The same evenness makes the strain trace a butterfly against the field, puts harmonics at every even multiple of the mains and none at the odd ones, and makes the hum climb steeply as the steel nears saturation.

    part 8 · electromagnetism

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