The field a spinning superconductor makes of itself

The field that is pushed out found that a superconductor does more than conduct without resistance: it expels a magnetic field from its interior, whatever the field was doing when it became superconducting, and lets it in only to a depth $\lambda$ of a few tens of nanometres. The two in the flux quantum found that the carriers are pairs of electrons, with charge $2e$, and that a ring traps flux in units of $h/2e$. Two lengths, and which one is longer found how the expulsion fails in a strong field, and the circuit that forgets its charge followed the pairs through a junction.

All of those put a superconductor in a field and asked what it does. This essay asks what it does in no field at all, when it is simply set spinning. The answer is that it magnetises itself. A field appears inside it, uniform and pointing along the axis, with a strength fixed by the rate of spin and by two constants of nature, and by nothing about the metal. Fritz London predicted it in 1950 from an argument already implicit in 1933, and it was measured in 1964. It turned out to be useful in two unexpected ways: as a balance for weighing the superconducting carriers, and as the only way to tell which way a perfectly smooth sphere is spinning.

Rotation is a magnetic field #

The starting point is a fact about rotating frames that has nothing to do with superconductivity. The forces that are not there found that an observer turning with a platform at angular velocity $\omega$ sees every moving object pushed sideways by the Coriolis force, $-2m\,\omega \times v$. A magnetic field pushes a moving charge sideways too, with the Lorentz force $q\,v \times B$. The two have the same form — both perpendicular to the velocity, both proportional to it — and they are equal when

$$B_L = \frac{2m}{q}\,\omega.$$

That is Larmor’s theorem: for a charge in the rotating frame, rotation at $\omega$ acts exactly like a magnetic field $B_L$, up to a centrifugal term that is second order in $\omega$ and irrelevant for slow rotation. For an electron, with its negative charge, the equivalent field points against the rotation. Larmor used the theorem in 1897 to show that a magnetic field applied to an atom makes its electrons precess as a whole — the effect behind the diamagnetism every material has. Here it is used in reverse: a rotation applied to a metal looks, to its electrons, like a field.

In a normal metal spun steadily nothing much follows. The electrons are dragged round with the lattice by collisions, and the rotation’s equivalent field bends their thermal motions a little, as a real field would; there is no current and no magnetism worth measuring. In a superconductor the rule is different. The Meissner effect is a statement about the field the superconducting carriers feel: deep inside the material it must vanish. For a stationary superconductor that field is the real magnetic field. For a spinning one, viewed from the frame in which the metal is at rest, it is the real field plus the rotation’s equivalent. The sum must vanish, so the superconductor must contain a real field

$$B = \frac{2m_e}{e}\,\omega,$$

pointing along the spin axis. That is the London moment, and it is 1.137 × 10⁻¹¹ tesla for every radian per second of rotation.

The same field in every superconductor #

The field is small. A superconducting rotor turning eighty times a second contains 5.7 nanotesla, and one turning a thousand times a second 71 nanotesla — about a thousandth of the Earth’s field. Measuring it needs a magnetic shield made of another superconductor and a detector sensitive enough to see a fraction of a flux quantum, which is why the prediction waited fourteen years for a measurement.

The striking thing about the formula is what is missing from it. It contains the electron’s mass and charge and nothing else: not the metal, not its transition temperature, not its penetration depth or its density of carriers. The reason is that the argument above never used any of them. It needed only that the carriers must feel no field, and the field rotation imitates depends only on their ratio of mass to charge. The carriers are pairs, with mass $2m$ and charge $2e$, and the ratio is the same as for one electron. Conventional superconductors of several kinds have been spun, and in 1990 so was one of the copper-oxide ceramics discovered four years earlier, whose pairing is of a quite different character; within the precision of each measurement all gave the same field.

A superconductor therefore turns rotation into a magnetic field with a conversion factor that is a fundamental constant, and that is a strange thing for a lump of metal to do. It is the magnetic counterpart of the voltage that is a frequency, where a junction converts a voltage into an oscillation at $2e/h$ per volt whatever it is made of. In both, the superconducting state is so rigid a quantum object that the only numbers left in its response are the ones that describe its carriers.

Where the current flows #

A real field needs a current to make it, and in a superconductor that current can flow only within the penetration depth of the surface.

The field is zero at the surface of a long spinning cylinder, since there is nothing outside to make one, and rises to its full value within a few penetration depths, as the figure shows. The current that makes it flows in the same thin layer, and it is a current of electrons that do not keep up with the rotating metal. Deep inside, the superconducting electrons turn exactly with the lattice; at the surface they fall behind it.

By how much is a clean result. The current needed is the field divided by $\mu_0\lambda$, and the density of superconducting electrons is fixed by the penetration depth itself, since $\lambda^2 = m/\mu_0 n e^2$. Putting the two together, the speed at which the surface electrons trail the metal is exactly $2\omega\lambda$. For niobium, spinning at a hundred radians per second, that is 7.8 micrometres per second, while the surface of a centimetre-sized body moves at a metre per second. The electrons slip behind the metal by one part in a hundred thousand, in a layer a few hundred atoms deep, and that is the whole of the current.

This is the other half of the reason the answer is universal. A superconductor with a longer penetration depth has fewer superconducting electrons, which must lag further to carry the current, and the field they make is the same.

Why ordinary metals barely answer #

If rotation looks like a field to every electron, it is fair to ask why a spinning copper bar is not magnetised in the same way. The answer is one of the more surprising theorems in physics, and it is what makes the superconductor’s response special rather than typical.

The magnetism classical physics forbids found that a collection of classical charges in thermal equilibrium develops no magnetisation at all in an applied field: the field bends every orbit, but the currents of orbits bent one way are cancelled exactly by those skipping along the boundary the other way. By Larmor’s theorem the same cancellation holds for the equivalent field of rotation. A normal metal, spun, has electrons whose paths are all curved by the Coriolis force, and the net orbital current is zero — or rather, it is only the small residue that quantum mechanics leaves, which is Landau’s diamagnetism, a susceptibility of about one part in a hundred thousand.

Something does survive in ordinary matter, and it is not orbital. The electron’s spin is an angular momentum with a magnetic moment attached, and in a rotating body the spins tend to line up with the rotation, just as they tend to line up with a field. Samuel Barnett measured this in 1915 by spinning iron rods and detecting the magnetisation they acquired, and the size of the effect told him the ratio of magnetic moment to angular momentum of whatever was doing the aligning. It came out twice what orbiting charges would give — one of the first signs that the magnetism of iron is a property of the electron’s spin rather than of its motion. For spins the equivalent field of rotation is $m\omega/e$, half the orbital value, and it magnetises a body only through that body’s own susceptibility, which is small in most metals and large in iron.

The superconductor does neither. Its field does not depend on a susceptibility, because it is not a response to an equivalent field but a cancellation of it, carried out by a condensate that refuses to feel any field at all. And it is orbital, the very kind of magnetism the classical theorem forbids in equilibrium — permitted here because a superconductor’s carriers are not a classical gas but a single coherent state. The London moment is the orbital magnetism that ordinary metals are not allowed to have, turned on by making the electrons one object — a whole body behaving, as the loop that behaves like a needle found a current loop does, like a single magnet.

A superfluid rotates by making whirlpools; a superconductor by making a field #

The London moment has a neutral twin, and setting them side by side shows what the charge buys.

A superconductor and a superfluid are both described by a single quantum phase, and in both the velocity of the condensate is fixed by how fast the phase changes from place to place. For a neutral superfluid that forces the flow to be irrotational: the whirlpool that comes in one size found that liquid helium in a rotating bucket cannot turn as a solid body, because solid-body rotation has a curl of $2\omega$ everywhere and the superfluid can have none. It turns instead by threading itself with an array of quantised vortices, each carrying one quantum of circulation, packed at a density of $2\omega/\kappa$ so that on average the fluid keeps up with the bucket.

A charged condensate has another option. Its velocity is the change of phase plus a term from the magnetic vector potential, and the curl of that term is the magnetic field. So a superconducting condensate can have a curl of $2\omega$ — can rotate as a solid body, with no vortices at all — provided a magnetic field of exactly $2m\omega/e$ runs through it. The London moment is that field. The superconductor makes it, from a surface current, because making it costs far less energy than making vortices would.

Both answer the question the rotating bucket has posed since Newton: how does a body that cannot rotate in the ordinary way keep up with the vessel it is in? The neutral fluid punctures itself; the charged one magnetises itself. The difference is the single term in the velocity that a charge carries.

Weighing a pair #

Since the field depends only on the carriers’ mass-to-charge ratio, measuring it precisely measures that ratio, and the charge is known exactly from the flux quantum. So a precise measurement of the London moment is a measurement of the mass of a Cooper pair.

That is less trivial than it sounds, because the pair’s mass is not quite twice the mass of a free electron. The electrons in a metal are bound, and their binding energy, divided by $c^2$, subtracts from their mass; they move at speeds where relativistic corrections are measurable at a part in a million; and the relevant mass is the one in the equation for the condensate’s motion, which is the bare mass of the electron, corrected for those effects, rather than the effective mass that describes an electron’s motion through the crystal lattice. Theory predicted that for niobium the pair’s mass should be about eight parts per million less than twice the free-electron mass.

In 1989 Janet Tate, Blas Cabrera, Stephen Felch and John Anderson measured it. They spun a niobium ring and compared the flux its London field threaded through it with the flux quantum, reading both with a superconducting quantum interference device, so that the result came out as a ratio of the pair’s mass to Planck’s constant. With Planck’s constant and the electron mass from other experiments, they found the pair’s mass to be 84 parts per million more than twice the free electron’s, with an uncertainty of 21. The difference from theory, 92 parts per million, is more than four times the uncertainty.

No accepted explanation has emerged. Corrections to the theory have been proposed and found too small; systematic errors in the experiment have been looked for and not found; and the measurement, which took years to make, has not been repeated at that precision in any laboratory. In 2006 claims were made of a much larger anomalous effect in spinning superconductors — a gravitational analogue of the London moment, many orders of magnitude above what general relativity allows — and they were not reproduced by later experiments. The 1989 discrepancy is modest by comparison and has never been refuted. It sits in the literature as an unresolved measurement of a quantity that the theory of superconductivity treats as known.

The only mark on a perfect sphere #

The London moment’s most demanding use was in space. Gravity Probe B, launched in 2004, carried four gyroscopes whose rotors were spheres of fused quartz 38 millimetres across, polished round to within forty atomic layers and coated with a thin film of niobium. They spun at up to eighty turns a second in a polar orbit 640 kilometres up, and the experiment’s purpose was to measure how their spin axes drifted relative to a distant guide star: general relativity predicted 6.6 arcseconds a year from the curvature of space around the Earth, and 39 milliarcseconds a year from the dragging of space by the Earth’s rotation.

A perfect sphere has a problem that a gyroscope with a rim does not. Nothing on it marks the axis. Any mark or asymmetry that could be read optically would also be a lever by which stray forces could turn it. The rotors were made as featureless as possible precisely so that nothing could torque them, and that left nothing to read.

The London moment was the mark. Once the niobium film was superconducting, the spinning rotor carried a field of a few nanotesla along its spin axis, invisible and exerting no torque, and a superconducting loop on the housing around it caught a flux proportional to the tilt of the axis. The figure computes the ideal case: a few thousand flux quanta threaded at full tilt, and fifteen millionths of a flux quantum per milliarcsecond of tilt. Resolving the frame-dragging drift meant reading changes of a few ten-thousandths of a flux quantum and averaging over months. The measurement succeeded, with larger uncertainties than planned, and found both effects in agreement with general relativity. The electrons that carried the information were the ones trailing their rotor’s surface by a few micrometres a second.

Where the model stops #

The argument is exact within London’s description of a superconductor, and three things lie outside it. The London field is the field inside a body that is superconducting throughout its bulk and larger than the penetration depth; a film thinner than $\lambda$, or a body threaded by trapped vortices from a field present when it cooled, adds fields of its own, and in practice the trapped flux in a real rotor had to be reduced below the London field itself before the moment could be read cleanly. The Earth’s field, or any residual field, adds to the London field in a way that does not rotate with the body, and must be screened out by an outer superconducting shield. And the centrifugal term in Larmor’s theorem, second order in the rotation rate, produces a tiny electric field and redistributes charge within the rotor; it is negligible at any speed a rotor survives.

At the level of the 1989 measurement the question is not what London’s theory leaves out but which of the many small corrections to the pair’s mass has been computed incorrectly or omitted — the contribution of the lattice ions’ own motion, the work function at the surface, thermoelectric and contact potentials between the ring and its leads. Each has been estimated and none accounts for ninety parts per million.

What the pictures cannot show #

The figures draw a field that is uniform inside and a current in a planar surface layer, which is the limit of a body much larger than the penetration depth. They do not draw the field outside a spinning sphere, which is a dipole, nor its ends, nor the way it varies near corners and edges. They show the Coriolis force and the Lorentz force as the same arrow, which is true of the force on a moving charge and false of the energy: a magnetic field stores energy in space, and the rotation’s equivalent field stores none. And the figure of the pair’s mass shows a single measurement and a single prediction, which is all that exists at that precision; it cannot show what a second experiment would find.

Still open: the four parts in a hundred thousand #

The discrepancy between the measured and predicted Cooper-pair mass in niobium is small, precise and unexplained, and it concerns a quantity that the theory of superconductivity would ordinarily treat as settled. Several possibilities remain: an unrecognised systematic effect in a difficult experiment; a correction to the pair’s mass in a real metal that has been left out of the theory; or something about the relation between the flux quantum, Planck’s constant and the pair’s inertia that is less simple than supposed. A modern repetition, with the improvements in superconducting detectors made since 1989, could decide between them, and it has been proposed more than once. It has not been done.

The habit worth carrying away is to ask what a rotating observer’s physics looks like before deciding that a rotating body has nothing to do. In a turning frame the Coriolis force on a moving charge is a magnetic field of 2mω/q, and a superconductor, which insists that its carriers feel no field, answers rotation by generating a real one of the same size — 1.137 × 10⁻¹¹ tesla per radian per second, the same in every metal. A neutral superfluid, with no such option, has to riddle itself with vortices to do what a charged one does with a field.