Illustrated Physics
physics, drawn
Concept

Magnetisation — where it appears

The magnetic moment per unit volume of a material, and the quantity that makes a magnet a magnet. Classical statistical mechanics predicts it to be exactly zero at every field and every temperature, so a compass needle is direct evidence of angular momentum coming in units.

Named by 14 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

electromagnetism · Magnetisation 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.

electromagnetism · Magnetisation Each stage takes 25.0 per cent of what is left. Entropy against temperature for a spin-½ paramagnet at 0.25 T and 1 T, with the cooling cycle drawn between them: a vertical drop is isothermal magnetisation, a horizontal move is adiabatic demagnetisation. Starting from 1 K the treads are at 1.000 K, 0.250 K, 0.062 K, 0.016 K, 3.91 mK. Each is 0.2500 of the one before — a ratio read back off the drawn treads rather than written into them, and equal to the field ratio 0.25/1 because this refrigerant's entropy depends on the field and the temperature only through their quotient. The steps therefore shrink in proportion to what is left, and no finite number of them arrives.

The staircase that never reaches the floor

Absolute zero is unreachable, and the reason is not that the apparatus is not good enough. Every stage of cooling removes a fixed fraction of what is left rather than a fixed amount, so the steps shrink in proportion to the distance remaining — and the fixed fraction cannot be made one, because the entropy curves at two field strengths are required to meet where the axis is.

thermodynamics · Third law 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.

electromagnetism · Magnetisation The entropy each degree of freedom has not yet given up. The entropy carried by each kind of degree of freedom, drawn against the temperature at which it orders and hands that entropy over. Lattice vibrations freeze out around room temperature; electron spins in a paramagnetic salt order in the millikelvin range, which is what makes adiabatic demagnetisation work; and nuclear spins hold R ln(2I+1) — 11.5 joules per kelvin per mole for copper — down to some tens of nanokelvin, where their own dipolar interactions finally sort them out. A copper sample at a microkelvin therefore has a large entropy and violates nothing: its nuclear spin system has not reached its ground state, and the third law is a statement about ground states rather than about thermometers.

A law about spectra, not about heat

The third law is usually met as a statement about cooling. Its statistical form is a statement about a spectrum: the entropy of a system in its ground state is k ln g, and it vanishes only when the ground state is unique. Every apparent exception is a degeneracy or a system that never reached its ground state — and copper nuclei carry eleven joules per kelvin per mole down to a hundred nanokelvin without violating anything.

thermodynamics · Third law 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.

electromagnetism · Magnetisation 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.

electromagnetism · Magnetisation 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.

electromagnetism · Magnetisation Two fields of the same magnet, and inside they point opposite ways. A uniformly magnetised sphere, with the field lines of B on the left and of H on the right, both computed from the exact solution — uniform inside, a dipole outside. Outside the sphere the two pictures are identical up to a constant, because there B is μ₀ times H and nothing else. Inside they are opposite: B is 0.67 tesla pointing along the magnetisation and H is 267 kiloamps a metre pointing against it. The B lines close on themselves and never end; the H lines begin on the top face and end on the bottom, which is what a field with sources looks like. Nothing about the magnet changed between the two panels — only which currents the circulation is allowed to count.

The field that points against the magnet it is in

There are two magnetic fields in use and the difference between them is which currents a loop is allowed to count. The consequence nobody expects on being told the definitions: inside a permanent magnet H points the other way from B. It has to — a loop inside the magnet threads no wire, so its H circulation is zero, and the only arrangement left has H running backwards.

electromagnetism · Ampere law Circles that go nowhere, and a current across the line. Gyrating ions in a uniform magnetic field pointing out of the page, with 60 guiding centres drawn from a density that falls by a factor of e every 3 gyroradii to the right, and Maxwellian speeds. Every ion goes round clockwise and none of the circles moves. Of the circles that cross the dashed vertical line, those centred to its left cross it moving down and those centred to its right cross it moving up; in this sample 11 cross moving down and 7 moving up, a count a sample this small could turn either way. Because there are more circles on the left, the ions at the line move downwards on average over every speed and phase, at exactly the thermal speed squared over the gyrofrequency times L, 0.33 thermal speeds, computed by averaging over speeds and phases and checked against that value. It is a current, carried by circles whose centres are still.

The current no particle carries

A magnetised plasma holds its own pressure against the field only if a current flows across the pressure gradient, and the fluid equations say exactly how much. Follow the particles in a uniform field and none of them is going anywhere; every guiding centre is still. The current is real all the same. It is made of circles that are more crowded on one side of a line than the other, and when the field is not uniform, the drifts that do move the guiding centres flow the wrong way.

electromagnetism · Magnetism How a field enters a superconductor that pins it. The magnetic field inside a slab of type-II superconductor, against position across its thickness, as an applied field is raised from zero, in units of the full-penetration field H = Jc a. Each line is the profile at an applied field of 0.5 H, 1.0 H, 1.5 H, 2.0 H. The field enters from both faces as a wedge whose slope is the most current the pinned vortices can carry, Jc; at 0.5 H it reaches halfway to the middle, at H* it just reaches the middle, and beyond that the whole slab carries Jc and the profile rides up with the applied field. A type-I superconductor would show no field inside at all, and a normal metal the applied field everywhere.

The field a superconductor keeps after all

A superconductor is famous for throwing magnetic field out, and the experiment that showed it is the one that made superconductivity a state of matter. Most superconductors that are actually used do something closer to the opposite: they let the field in as threads, grip the threads where they are, and keep them when the field is taken away. The result is a slope of field inside the material that is as steep as the grip allows and no steeper, which is all that is needed to explain why such a superconductor has a hysteresis loop, heats up in an alternating field, can hang a magnet underneath itself, and can be made into a magnet stronger than any of iron.

electromagnetism · Superconductivity A row of magnets with its field on one side. Two wavelengths of a Halbach row: blocks of permanent magnet, 4 to a wavelength, each magnetised a 90° turn from the last (block arrows), with the magnetic field drawn as arrows above and below, their lengths proportional to its strength. Just above the row the field is 5 times stronger than the same distance below it. The magnetisation turning one way along the row puts the field on that side; turned the other way, it would put it underneath. With finer blocks the weak side empties further, and a continuously turning magnetisation leaves it with no field at all.

The magnets that put their whole field on one side

Lay bar magnets in a row, each turned a quarter turn from the last, and the row's field appears almost entirely on one side, with almost nothing on the other — although every magnet in it, alone, is perfectly symmetric. Turn the magnetisation smoothly instead of in steps and the weak side has no field at all, exactly. Nothing is shielding it. The two halves of the pattern each make a symmetric field, and their fields line up on one side and cancel on the other, because a magnetisation that turns as it goes can only make the field that decays in one direction.

electromagnetism · Magnetism Where the two theories part: the magnetic permeability. The voltage across the spinning wall as a fraction of a metal's, against the material's relative permeability μ, for a permittivity ε = 6. Lorentz's electron theory (flat) polarises the material only through the force on its charges, so μ does not enter: (ε − 1)/ε = 0.833 at every μ. Relativity gives (εμ − 1)/εμ, which agrees exactly at μ = 1 — so no ordinary insulator, such as the ebonite H. A. Wilson spun in 1904, can tell them apart — and climbs towards a metal's value as μ grows. At μ = 3 (dot) the relativistic voltage is 13.3 per cent larger. That is why Wilson and Wilson needed an insulator that was also magnetic, and made one from steel spheres set in wax.

The voltage a moving magnet adds to an insulator

Spin a hollow cylinder of insulator in a magnetic field along its axis and a voltage appears between its inner and outer surfaces, because every piece of it is moving through the field. How large that voltage is turns out to depend on whether the insulator is magnetic, and the dependence was predicted by one theory and missed by the other. Maxwell's equations hold in a moving material only if the material's response is written in its own frame and carried into the laboratory by relativity — and then a moving magnet is also an electric dipole.

electromagnetism · Maxwell equations Reversing the magnetism in step with the swing. The rod hung from a fibre as a torsion pendulum with a period of 10 s and a quality factor of 200, its magnetisation reversed 40 times, once every half swing, in step with its motion, then left alone; the angle it turns through, integrated, in degrees. Each reversal adds the same small kick of angular momentum and the kicks add up, as pushes on a swing do: the swing grows to 15.46° if the magnetism is spin, 30.92° if it were orbital currents. A single reversal would turn the rod by about 0.45° — measurable in principle, buried in practice by the much larger twist that the coil's own field puts on any slight misalignment, which is why Einstein and de Haas drove the rod at its resonance and why their first result, g′ = 1, came out wrong.

The rod that turns when its magnetism flips

Hang an iron rod on a thin fibre inside a coil, reverse the current, and the rod turns a little. Nothing has pushed it sideways: reversing the magnetisation reversed the angular momentum the electrons inside it carry, and since nothing outside can twist the rod about its axis, the iron lattice has to take up the difference. Einstein and de Haas did it in 1915 to prove that magnetism is electric charge going round. They got the right effect and the wrong number, and the right number — twice what they expected — was evidence, years before anyone proposed it, that the electron's magnetism comes from something that is not a rotation at all.

quantum · Spin

Named alongside it

The objects these essays reach for when they reach for this one.

HysteresisCoercivityDemagnetising fieldAngular momentumConstitutive lawDegeneracyDiamagnetismDomain wallEntropyEquilibriumExchange interactionFerromagnetism

All concepts