Illustrated Physics
physics, drawn
Quantum

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.
16 min read 5 figures What stays the sameMade into an instrument

Assumes: The angular momentum that is not a rotation · The loop that behaves like a needle

The angular momentum that is not a rotation found that the electron carries an intrinsic angular momentum, ℏ/2\hbar/2 along any axis it is measured on, and a magnetic moment to go with it, and that the ratio of the two is twice what any spinning ball of charge would have. That essay met the factor of two in atomic spectra and in the Stern–Gerlach experiment, where single atoms are deflected by a field. It can also be met with a piece of iron hanging on a thread, and that is how it was first met — in 1915, a decade before anyone proposed that the electron spins, by Albert Einstein and Wander de Haas in Berlin, who were trying to prove something else.

The question they were trying to answer was Ampère’s. A hundred years earlier André-Marie Ampère had suggested that a magnet is magnetic because each of its molecules carries a tiny circulating electric current, and by 1915 the electron had been discovered and the obvious candidate for Ampère’s currents was electrons going round in atoms. If so, every atomic magnet in a piece of iron is a little current loop, and every current loop made of moving mass carries angular momentum as well as a magnetic moment. Magnetise the iron, and the atomic loops line up; their angular momenta line up too. The rod as a whole then carries a hidden spin, and if the magnetisation is suddenly reversed, so is the hidden spin — and something must turn the other way.

The angular momentum a reversal hands to the rod. An iron rod 2 mm thick and 10 cm long, magnetised to saturation along its axis (magnetic moment 0.54 A m²), in units of 10⁻¹² kg m²/s about its axis. If the magnetism is the electrons' spin, the moment comes with 3.05 of angular momentum; reversing the magnetisation reverses it, a change of 6.11, and the rod — the only other thing that can turn — must take up the difference. If the magnetism were orbiting electrons, as Ampère's molecular currents were imagined, the same moment would carry twice the angular momentum, because an orbit's magnetic moment per unit angular momentum is half a spin's, and the rod would take 12.22. Measuring the rod's kick measures which.
Fig. 1 An iron rod 2 mm thick and 10 cm long, magnetised to saturation along its axis (moment 0.54 A m20.54\ \mathrm{A\,m^2}), in units of 10−12 kg m2/s10^{-12}\ \mathrm{kg\,m^2/s}. If its magnetism is the electrons’ spin, the moment carries 3.05 of angular momentum; reversal changes it by 6.11, which the rod must take up. If the magnetism were orbiting electrons, the same moment would carry twice as much and the rod would take 12.22.

The bookkeeping

The first figure keeps the accounts. An iron rod two millimetres thick and ten centimetres long, magnetised along its length to saturation, has a magnetic moment of about half an ampere square metre. The angular momentum that goes with it depends on the ratio between magnetic moment and angular momentum, the gyromagnetic ratio, and for any charged body going round on an orbit that ratio is fixed: a charge ee of mass mm on any orbit has a magnetic moment of e/2me/2m times its angular momentum, whatever the orbit’s size or speed. The loop that behaves like a needle found the current-loop half of that statement; the other half is that the same current is moving mass.

With orbital magnetism the rod’s moment would come with about 1.2×10−111.2\times10^{-11} kilogram square metres per second of angular momentum in its electrons. Reverse the magnetisation, as a coil round the rod can do in a fraction of a second, and the electrons’ angular momentum reverses: a change of twice that amount. No external torque acts about the rod’s axis — the coil’s field points along the axis and twists nothing — so the total must be conserved, and the iron lattice, carrying the electrons, must take up the change. The rod turns.

It turns very slowly. Its moment of inertia about its axis is about a billionth of a kilogram square metre, and an angular momentum of 10−1110^{-11} gives it a turning rate of about a hundredth of a radian per second. Einstein and de Haas expected exactly that, with the ratio of the orbit. The number they would actually have found, had the experiment been clean, was half of it.

Why classical physics could not have settled it

Ampère’s picture had a deeper problem than the factor of two, and it was not known in 1915. The magnetism classical physics forbids found that in classical statistical mechanics a collection of charges in thermal equilibrium can have no net magnetisation at all — the Bohr–van Leeuwen theorem, proved by Niels Bohr in 1911 and Hendrika van Leeuwen in 1919 — because a magnetic field does no work on moving charges and so cannot change the thermal distribution of their energies. Permanent atomic current loops of fixed size, which Ampère’s theory needed, are impossible classically; an orbiting electron radiates and spirals in. So the magnetism of iron had to be quantum from the start, and the question the rod could answer was which kind: quantised orbits, with the orbital ratio between moment and angular momentum, or something else.

The rod answered without needing any theory of what the something else was. The ratio of magnetic moment to angular momentum is a measurable number, independent of how many electrons contribute or how they are arranged, and a ratio of e/me/m rather than e/2me/2m cannot be produced by any arrangement of orbits. The experiment measures that ratio for the carriers of the magnetism, whatever they are, by weighing their angular momentum mechanically.

A kick too small to see, made large by resonance

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.
Fig. 2 The rod hung as a torsion pendulum (period 10 s, quality factor 200), its magnetisation reversed 40 times, once every half swing, in step with its motion; the angle it turns, integrated. The kicks add as pushes on a swing do: the swing grows to 15.5° for spin, 31° for orbital magnetism. A single reversal would turn the rod by only 0.45°.

A single reversal turns the rod by under half a degree if it hangs as a torsion pendulum with a ten-second period, and that is a difficult thing to measure cleanly. The coil’s field is enormously stronger than anything the experiment is trying to detect, and any slight misalignment between the rod’s axis and the field, any small permanent magnetism across the rod, or the Earth’s field acting on the rod’s moment, gives a torque far larger than the one sought. Einstein and de Haas got round part of the problem by resonance. They reversed the current periodically, at the natural frequency of the rod hanging on its fibre, so that each reversal pushed in step with the swing the previous ones had started, and the second figure integrates what that does. Each kick adds the same small amount, and with a high enough quality factor the swing grows over dozens of reversals to many degrees — as the frequency that gets an answer found for any oscillator driven at its own frequency, the response grows until the losses per cycle match the drive.

The resonance amplifies the spurious torques just as much as the real one if they too reverse with the current, and the experiment’s difficulty was in separating them. Einstein and de Haas reported a result consistent with the orbital ratio, g′=1g' = 1 in the notation that would come later. It was not. The repeats over the next few years — by John Stewart in 1918 and Emil Beck in 1919, and with growing precision afterwards — found the swing half as large as orbital magnetism predicted: g′g' close to 2. Einstein, who was otherwise not wrong about much in those years, had found the effect and believed the number his theory predicted.

The inverse experiment, and the same factor of two

The field a spinning body behaves as if it were in. The inverse effect, found by Samuel Barnett in 1915: a body spun at a rate f magnetises itself as if it sat in a field of 2mₑ(2πf)/g′e, on logarithmic axes, for spin (g′ = 2, solid) and orbital currents (g′ = 1, dashed). The electrons' spins, carried round with the body, try to line up with the rotation, and the field that would line them up equally is tiny: 3.6·10⁻⁹ T at a hundred turns a second, and still only 1.1 μT, against the Earth's 50, at the speed of an ultracentrifuge. To match the Earth's field a body would have to turn 1.4·10⁶ times a second. Barnett measured it anyway, in iron rods, with the Earth's field cancelled around them, and found g′ close to 2 — the experiment that, with Emil Beck's repeat of Einstein and de Haas, showed the magnetism of iron was not orbital before anyone knew the electron spun.
Fig. 3 Magnetisation by rotation (Barnett, 1915): a body spun at ff turns a second magnetises itself as if in a field 2me(2πf)/g′e2m_e(2\pi f)/g'e, on logarithmic axes, for spin (solid) and orbital currents (dashed): 3.6×10−93.6\times10^{-9} T at a hundred turns a second and still only 1.1 μT, against the Earth’s 50, at the speed of an ultracentrifuge. To match the Earth’s field a body would have to turn 1.4×1061.4\times10^6 times a second.

In the same year, in Ohio, Samuel Barnett did the experiment backwards. If magnetising a rod makes it turn, then turning a rod should magnetise it: the electrons’ angular momenta, carried round with the rotating body, are biased towards the rotation axis exactly as they would be by a magnetic field along it. The equivalent field is the rotation rate divided by the gyromagnetic ratio, and the third figure shows how small it is. A rod turning a hundred times a second is magnetised as if it sat in four billionths of a tesla, a ten-thousandth of the Earth’s field; to equal the Earth’s field it would have to turn more than a million times a second.

Barnett measured it anyway, by spinning iron rods inside a coil with the Earth’s field carefully cancelled and detecting the tiny change of magnetisation, and his result, like Stewart’s and Beck’s, came out near g′=2g' = 2. Two different experiments — one turning a rod by magnetising it, the other magnetising a rod by turning it — agreed that the magnetism of iron carries half the angular momentum that orbiting electrons would give it. In 1915 nobody could say why. In 1925 George Uhlenbeck and Samuel Goudsmit proposed that the electron has an intrinsic angular momentum ℏ/2\hbar/2 with a magnetic moment of one Bohr magneton, a ratio twice the orbital one, and in 1928 Dirac’s equation produced the factor of two from relativity. The magnetism of iron is spin.

There is a superconducting relative of Barnett’s effect. The field a spinning superconductor makes of itself found that a rotating superconductor generates a magnetic field 2mω/e2m\omega/e throughout its interior, the London moment, because its electrons, unable to lag behind the lattice, must carry a current that makes exactly that field. It is the same ratio of mass to charge, appearing because a body’s charge carriers have angular momentum that the body’s rotation drags.

Two g-factors that are not quite 2

Two g-factors for one magnet, and the relation between them. For the three ferromagnetic metals, the magnetomechanical ratio g′ — measured by twisting rods, as Einstein and de Haas did, or by spinning them — against the spectroscopic g measured by ferromagnetic resonance, with the curve 1/g + 1/g′ = 1. Pure spin would put both at 2 (cross). A small orbital moment, left over because the crystal's field does not quite quench the electrons' orbits, adds magnetic moment and angular momentum in a different ratio and pushes the two the opposite ways: iron at g′ = 1.919 and g = 2.09, cobalt 1.838 and 2.18, nickel 1.835 and 2.19. The two numbers come from a torque on a rod and a microwave absorption line, and they agree with each other to a few parts in a thousand through Kittel's relation: magnetism in these metals is about 95 per cent spin.
Fig. 4 For iron, cobalt and nickel, the magnetomechanical ratio g′g' (from rotation experiments) against the spectroscopic gg (from ferromagnetic resonance), with Kittel’s relation 1/g+1/g′=11/g + 1/g' = 1. Pure spin would put both at 2 (cross). A small unquenched orbital moment pushes them opposite ways: iron at g′=1.919g' = 1.919, g=2.09g = 2.09; cobalt 1.838 and 2.18; nickel 1.835 and 2.19.

Careful measurements in the decades after found g′g' close to 2 but not equal to it: 1.92 for iron, 1.84 for cobalt and nickel. The difference is a small orbital contribution. In a crystal, the electric fields of neighbouring atoms lock the electrons’ orbits to the lattice and largely cancel their orbital magnetism — the orbital moment is said to be quenched — but not entirely, and the few per cent that survive add magnetic moment and angular momentum in the orbital ratio, pulling the mixture away from 2.

The fourth figure compares that with a second measurement of the same physics. Ferromagnetic resonance, the microwave absorption of a magnetised sample in a steady field, measures a different g-factor, the spectroscopic gg, and for the same metals it comes out slightly above 2: 2.09 for iron, 2.18 for cobalt and nickel. Charles Kittel showed in 1949 that when the orbital contribution is small, the two are related by 1/g+1/g′=11/g + 1/g' = 1, so that a gg above 2 must come with a g′g' below it. The measured pairs sit on that curve to a few parts in a thousand. A torque on a rod hanging on a fibre and the frequency of a microwave absorption line, measured in different laboratories by different methods, agree through a relation that follows only if the magnetism is almost entirely spin with a small orbital admixture.

The gyromagnetic ratio, read as a precession

The gyromagnetic ratio that the rod measures as a torque can also be measured as a frequency, and that is the more familiar way. A magnetic moment that carries angular momentum does not simply line up with a magnetic field; the field’s torque turns the angular momentum, so the moment precesses around the field at a rate equal to the field times the gyromagnetic ratio. For electron spins that is 28 gigahertz per tesla, the basis of electron spin resonance and of the ferromagnetic resonance of the previous figure. For protons it is 42.6 megahertz per tesla, and the image made of frequencies found that a hospital scanner builds its pictures out of exactly that precession, with the field graded across the body so that each place has its own frequency. A proton’s g-factor is 5.59, not 2, because a proton is not an elementary particle; its moment comes from its quarks and their motion, and the number is a measurement of its structure.

The precession and the rod’s turning are two faces of one relation. A moment with no angular momentum would align with a field directly, like a compass needle with no inertia; one that carries angular momentum precesses, and when its direction is reversed something else must take up the change. In the rod the something is the lattice; in a resonance experiment it is the microwave field that drives the transition. Either way the angular momentum is real and is conserved, and the angular momentum that is in nothing at all found the extreme case, in which a charged disc in a magnetic field starts turning when the field is switched off because the angular momentum had been stored in the electromagnetic field itself. In the iron rod the field’s share is negligible and the ledger closes between the spins and the lattice.

Where the angular momentum goes in a fraction of a picosecond

The thinner the rod, the faster a reversal spins it. The rate at which a reversal sets an iron rod turning, in radians per second, against its radius, on logarithmic axes; the length cancels. The angular momentum handed over grows with the rod's volume, the radius squared, but its moment of inertia grows with the fourth power, so the rate grows as one over the radius squared: 0.0049 rad/s for a rod 2 mm thick, 4939 for a micrometre wire, 5.5·10⁸ for a crystal a few nanometres across. Reversing a nanocrystal's magnetism would set it spinning at hundreds of millions of radians per second if nothing held it; held in a lattice, the same angular momentum goes into the lattice as a twisting wave, which was seen in 2019 within a few hundred femtoseconds of a laser pulse demagnetising a ferromagnetic film.
Fig. 5 The rate at which a reversal sets an iron rod turning, against its radius, on logarithmic axes; the length cancels. The angular momentum handed over grows as the volume and the moment of inertia as the radius to the fourth, so the rate grows as one over the radius squared: 0.0049 rad/s for a rod 2 mm thick, 4,939 for a micrometre wire, 5.5×1085.5\times10^8 for a crystal a few nanometres across.

The effect becomes dramatic as the rod becomes small. The angular momentum handed over grows only as the rod’s volume, but its moment of inertia grows as its mass times its radius squared, so the turning rate after a reversal grows as one over the radius squared; the fifth figure follows it from Einstein and de Haas’s rod to a nanometre-sized crystal, which would be spun at hundreds of millions of radians per second if nothing held it. Mechanical versions of the experiment on micrometre-scale cantilevers have been used to measure the magnetism of thin films directly as a torque.

In a solid nothing is free to spin; the lattice is held by its neighbours. So where does the angular momentum go when magnetism vanishes suddenly? The question became sharp in 1996, when it was found that a short laser pulse can demagnetise a nickel film in less than a picosecond, far faster than anyone had thought the spins could shed their angular momentum. In 2019 an experiment using ultrashort X-ray pulses to watch the lattice found the answer: within a few hundred femtoseconds of the demagnetisation, the lattice develops a transverse strain, a twist, carrying away the spins’ angular momentum as a sound wave propagating from the surface. The angular-momentum ledger of the first figure is balanced at the femtosecond scale exactly as on the fibre — and a rod hanging in Berlin in 1915 and a film watched with X-rays a century later measure the same conservation law.

Why the measurement was so hard to get right

The size of the effect is set by a ratio, 2me/g′e2m_e/g'e, about 5.7×10−125.7\times10^{-12} kilograms per coulomb for spin, that is fixed by the electron’s charge and mass. Everything else in the experiment is far larger: the coil’s field acting on the rod’s moment, the Earth’s field, the torque on any eddy currents induced by the changing field, the twist of the fibre under its own magnetisation. Einstein and de Haas’s result of g′≈1g' \approx 1 is now generally put down to those, and to a measurement that, with the resonance technique, could not easily separate a real angular-momentum kick from a torque that happened to reverse with the current. Their result was consistent with what they expected, and it was the expectation that was at fault. It is a recurring lesson in precision measurement that the systematic errors of an experiment are most easily overlooked when they push the answer towards the predicted one.

The later experiments separated the effects by changing the rod’s shape and material, by reversing the magnetisation in different ways, and by comparing the rotation with the inverse effect. By the 1960s the magnetomechanical ratios of the ferromagnetic metals were known to a part in a thousand, and they were among the first precise evidence that electron spin, not orbital motion, is the source of ferromagnetism — complementing what holds a magnet together is not magnetism, which found that the force aligning the spins is not magnetic at all but a consequence of the exclusion principle.

The ledger, not the spins

The figures follow the angular momentum as numbers on a ledger and the rod’s swing as an integrated curve. They do not show the spins themselves, or the process by which their angular momentum reaches the lattice — through the spin–orbit coupling that ties each spin to the crystal and through the magnetoelastic coupling of the lattice’s strain to the magnetisation — which is the part of the effect that took a century to watch. The pendulum is driven by ideal impulses, with no spurious torques, which is the version of the experiment that is easy to draw and was never easy to do. And the g-factors are the measured ones; the figures do not derive them, and nothing in the figures explains why the electron’s spin has a gyromagnetic ratio of exactly twice the orbital one, which is a statement of relativistic quantum mechanics.

Still open: how fast spin angular momentum can be handed to a lattice

The ultrafast experiments show that a demagnetising film passes its spin angular momentum to the lattice within a few hundred femtoseconds, but how — which microscopic channels carry it, how much goes into lattice vibrations directly, how much into electrons’ orbital motion first, how much leaves the film altogether as spin currents into neighbouring layers — is still being worked out, and the answers matter for how fast magnetic memory can be written. The magnetic state of a bit can now be switched with a single laser pulse in some materials, and the speed limit on that switching is set in part by how quickly angular momentum can be moved out of the spins. The Einstein–de Haas effect has become a question about femtoseconds.

The habit worth carrying away is to ask what a magnet’s magnetism weighs in angular momentum. Reversing the magnetisation of an iron rod reverses the angular momentum its electrons carry, 2me/g′e2m_e/g'e per unit of magnetic moment, and the lattice takes up the difference: a 2 mm rod turns at 0.005 radians per second, a swing pumped to fifteen degrees by reversals in step with it. The measured g′=1.92g' = 1.92, close to 2 rather than the orbital 1, showed a decade before spin was named that the magnetism of iron is not electrons going round.

Part 8 of 8

This essay is one argument about Spin. The others:

A signal that dies, and comes back when the spins are turned over. The size of the signal from an ensemble of proton spins whose precession rates are spread by 20 Hz (root mean square) about the mean, with a true relaxation time T₂ of 80 ms, against time after they are tipped. The free signal falls to 1/e in about 10 ms and to 0.03 by 20 ms: not because any spin has relaxed but because they have fanned out. A π pulse at 30 ms (marked) turns every spin over, and at 60 ms the signal returns to 0.472 — exactly e^(−2τ/T₂), the dashed envelope, since every static difference in rate has cancelled. Without the pulse the signal at 60 ms is below one part in a million. The echo that gives back everything but the wandering Part 7 — Tip a sample of protons in a slightly uneven magnetic field and their signal dies within milliseconds. None of them has relaxed: they have fanned out, each precessing at the rate its own patch of field sets. Turn every spin over halfway through and the fan closes again, and the signal comes back as an echo, smaller only by the decay that was real. What the echo cannot return is a phase gained while a spin moved, and that one loss, which grows as the cube of the waiting time, is how a scanner finds a stroke within minutes.

Where this goes next

How fast a spinning electron would have to turn. The equatorial speed of a uniform sphere with the electron's mass and an angular momentum of ħ/2, against the radius it is given, both logarithmically. The expression is 5ħ/4mr and it passes the speed of light at 4.83e-13 m — half a picometre, which is four hundred times larger than a hydrogen nucleus. Giving it a smaller radius only makes the answer worse: at the experimental upper bound, 10⁻¹⁸ m the equator would move at 4.8e+5 times the speed of light; at the classical electron radius the equator would move at 1.7e+2 times the speed of light; at the reduced Compton wavelength the equator would move at 1.3e+0 times the speed of light. No radius the electron is permitted to have gets anywhere near a legal answer, and the experimental bound is off the scale by eight orders of magnitude. So the angular momentum is not the angular momentum of anything going round. It is a property the particle has, in the same way a charge is, and the only thing it shares with a spinning top is the algebra it obeys — which is, admittedly, the whole of what angular momentum means in physics. The angular momentum that is not a rotation An electron has angular momentum, and it is not going round anything. A sphere of its mass carrying that much angular momentum would need its equator moving at half a million times the speed of light at any size the electron is allowed to have. What survives of the analogy is the algebra — and the algebra turns out to require that turning the thing all the way round leaves it changed. Rotation is a magnetic field, and a superconductor cancels it. Magnetic fields along the spin axis inside a spinning metal, in units of 2mω/e, as seen by its electrons in the frame turning with it. In that frame the Coriolis force on an electron is exactly the force a magnetic field of 2mω/e would exert, pointing against the spin for a negative charge (Larmor's theorem). A normal metal spun in no field has no real field inside, and its electrons feel the rotation's equivalent field in full. A superconductor will not let its carriers feel any field in their interior — that is the Meissner effect — so it generates a real field of exactly the same size pointing the other way, and the sum vanishes. The real field is the London moment. The field a spinning superconductor makes of itself Spin a lump of superconductor with no magnet anywhere near it, and a magnetic field appears inside it, pointing along the axis of spin: about eleven trillionths of a tesla for every radian per second, the same in lead, niobium or a copper-oxide ceramic. The field is not left over from anything. In a turning frame the Coriolis force on a moving charge is indistinguishable from a magnetic force, a superconductor will not let its carriers feel any field inside it, and so it makes a real field to cancel the one rotation imitates. The size depends on nothing but the carriers' mass and charge, which makes the field a balance for weighing them — and the one time it was read to a part in a hundred thousand, it gave an answer theory has not matched. 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.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Angular momentumConservation lawsFerromagnetismG factorGyromagnetic ratioMagnetisationSpinTorsion pendulum