Spin
Two Answers, and Two Turns to Come Back
A Magnet With No Spinning Ball
Every electron is a tiny magnet. Every electron also carries a fixed amount of turning, the angular momentum a spinning wheel has. The amount never changes: nothing can speed it up or slow it down. This built-in turning is called spin. Protons and neutrons have it, quarks have it, and so does light.
The turning is real. Hang an iron rod on a thread inside a coil, and flip its magnetism with the coil’s current. The coil’s field runs along the thread, so it cannot twist the rod by itself, yet the rod twists, to keep its total turning the same. Iron’s magnetism, like a fridge magnet’s, comes mostly from the spins of its electrons, lined up, and the turning they carry has flipped with them.
The size of the twist holds a clue. The first measurement found the twist expected if iron’s magnetism came from the turning of its electrons’ clouds round the nuclei. More careful ones found only half of it: each bit of the magnetism carries half the turning that such clouds would give it. The half was a puzzle until spin explained it: for its turning, a spin is twice as strong a magnet as a turning cloud.
An X-ray laser has caught the same handover in a thin film of iron. A flash of laser light knocked out about a tenth of its magnetism, and within a fifth of a trillionth of a second about four fifths of the turning the spins had lost passed to the iron’s atoms. The atoms at its surface moved sideways, much as the hanging rod turns.
The name comes from the obvious picture, a tiny ball of charge spinning on its axis. It was put forward to explain why the colors that atoms give off come in close pairs. Its trouble is that the ball’s surface would have to move faster than light.
The pairs of colors are spin’s doing. In many of an atom’s patterns, the electron’s cloud carries turning round the nucleus. An electric field is partly magnetic for anything moving through it, so to an electron in such a cloud, part of the nucleus’s electric field acts as a magnetic one. So its energy is a little higher with its spin turning the same way as its cloud than the other way, and many colors split in two, as sodium’s yellow does. Worked out this way, the split comes out twice as large as measured; a second effect of relativity, which comes from the nucleus’s pull itself, takes back half.
The ball does not survive. Collisions in particle accelerators have found no size for the electron at all, down to less than a thousandth of a proton’s width. Even a ball as wide as a proton would need its surface to race round hundreds of times faster than light to carry the electron’s spin. The turning is real and the magnet is real, but no ball of charge is spinning.
Two Spots, Never Between
Spin can be seen directly. Physicists sent a beam of silver atoms between the poles of a magnet. One pole was shaped to a sharp edge and the other hollowed into a groove, so the field was far stronger near the edge. A small magnet in such a field is pulled toward the stronger part or away from it, harder the more it points along the field. A silver atom’s magnet is almost all the spin of its one outer electron, as its other 46 electrons pair off and cancel.
The atoms leave a hot oven with their magnets pointing every which way, like compass needles tossed into a box. Each should be pulled by an amount set by its tilt, and the beam should smear into a line on the glass plate where the atoms land. Instead the silver gathered in two patches, one above the other, with nothing between. Every atom landed in one patch or the other, as if its magnet pointed straight along the field or straight against it.
Turn the magnet, and the atoms split in two along the new direction instead. An electron’s spin answers any question of the form “which way along this line?” with one of two answers, up or down. Its turning along the line is always the same size: half the smallest step of turning that an electron’s cloud can carry round a nucleus.
Hydrogen atoms, with one electron each, show that the split is spin’s doing. At the atom’s lowest energy, that electron’s cloud is round and carries no turning round the nucleus at all, yet a beam of hydrogen atoms splits in two as well. Turning in a cloud comes in whole steps, and it would split a beam into an odd number of spots, never two.
The Answer Depends on the Question
Put a second magnet after the first, the same way up, and block the atoms that went down. Every atom that reaches the second magnet goes up again. Asked the same question twice, an atom gives the same answer.
Now turn the second magnet on its side. The atoms that all went up split evenly, half to one side and half to the other. Block one side, and send the other on to a third magnet, upright again. Half of them now go down. Asking sideways has wiped out the answer to the first question.
Polarizing filters do the same to light, where a third filter set between two crossed ones lets light through. An atom that went up is, for a sideways magnet, a superposition of left and right. For an upright magnet it is simply up. The two questions clash, as turnings about two different lines always do in the uncertainty principle: no atom can carry a sharp answer to both, and asking one wipes out the answer to the other.
The answer is a direction. An electron found up along one line will be found up along that line every time. Asked along another line, it is up by chance: three times in four for a line 60 degrees away, half the time at 90 degrees, and never at 180. So a spin can point any way, and each way is a sure answer to one question and a chance at the others. For light the sure “no” lies 90 degrees away, where a crossed filter blocks everything. For spin it lies 180 degrees away, so its chances change with half the angle. That half is the key to the strangest thing about spin.
Two Turns to Come Back
A magnetic field swings a spin’s direction round the field, much as gravity swings the axis of a tilted top round. Turn an electron’s spin through one full turn like this, and it points just as it did before. Every measurement on it gives what it gave before. Yet its wave is not back where it started. Its chances are the square of the height of its wave, and it is the wave that follows half the angle: after one turn the wave has gone only halfway round, and it comes back upside down, every crest where a trough was. Squared, upside down looks the same, so nothing measured on the electron alone can tell. Only a second full turn brings the wave itself back.
An upside-down wave shows only where it meets a wave that was not turned. Physicists split a beam of neutrons, which have spin as electrons do, into two paths through a crystal of silicon. A magnetic field turned the spins on one path, and then the paths met again. One full turn swapped the exit at which the neutrons were found, as making one path half a wave longer would. A second full turn swapped it back. Measured closely, the period came out at 716.8 degrees, within four degrees of two full turns.
Things tied to their surroundings can show the same arithmetic. Hang a cube between two posts by two straps, and turn it one full turn. The cube looks as it did, but each strap now has a full twist in it, and no moving of the straps will take the twists out while the cube is held still. Turn it a second full turn the same way, and now the straps can swing round the cube and come back flat, without the cube ever turning back. A belt shows the same thing. After one turn of the buckle, passing a loop of the belt round it only turns the twist the other way; after two turns, it takes the twist out. One turn leaves a trace in how a thing is tied to the world, and two turns clear it. A spin of one half is a wave that carries that trace.
Halves and Wholes
What spin is comes down to fields. A field has a value at every point, and the kind of value it has decides what happens when you turn it. The Higgs field’s value points no way at all, so turning does not change it, and its particle has spin zero. The electromagnetic field’s value is a pair of arrows, its electric and its magnetic part, which come back after one turn, and its particle, the photon, has spin one. The electron field’s value is a set of numbers that turn by half the angle and come back only after two turns, and the electron has spin one half. A particle carries spin because the field it is a ripple in has that kind of value.
The photon’s spin can be felt. In circularly polarized light, the field’s arrow turns round as the wave passes. Hang a thin crystal plate on a fine quartz fiber and shine such light through it. The plate reverses the light’s turning, and it twists on its fiber by about the amount that the light’s turning, one unit of spin for each photon, says it should.
Today the same push is a tool. Laser tweezers hold a speck of crystal or glass in the focus of a beam. In light whose field swings along one line, a speck of crystal, or of glass that is not quite round, lines up with it. When the field turns, the speck spins, and it spins the other way when the field turns the other way. Held in a vacuum, where no air slows them, specks of glass some five hundred times thinner than a hair have been spun faster than a billion turns a second, and one has reached about six billion.
Every particle’s spin is a whole number or an odd number of halves: zero, a half, one, three halves, two. In our three dimensions of space, nothing between is possible, because a thing turned once round either comes back as it was or comes back upside down. The particles of matter, the electrons and the quarks, have spin one half. The particles that carry forces, the photon, the gluons and the W and Z, have spin one, and the Higgs has spin zero. Gravity’s particle, if it exists, would have spin two. A particle built of others gets its spin from its parts’ spins and their motion inside it together: a proton’s is a half, and a common helium atom’s is zero. An odd number of parts whose spins are halves always makes a whole whose spin is an odd number of halves, such as a half or three halves.
The size of a spin sets how many answers a magnet gets: two for a half, three for a spin of one, four for three halves. The photon has spin one but only two answers, its two circular polarizations. A particle with no mass can never be at rest, and its spin can lie only along or against the way it moves.
Why Matter Takes Up Room
The halves and the wholes behave in opposite ways in a crowd. Particles whose spin is an odd number of halves, called fermions, never share a state. Particles with whole spin, called bosons, crowd into one readily. For electrons the rule shows in the pattern of the colors atoms give off: no two electrons in an atom can have both the same wave round the nucleus and the same spin. In a world that obeys both relativity and quantum physics, it can be proved that spin must decide it: the spin-statistics theorem. There is a picture of why, though no simple proof that covers every case: swapping two things tied to the world by straps twists the straps as one full turn of one of them does.
The rule builds atoms. Each wave round a nucleus can hold two electrons, one with spin up and one with spin down. The pair is not up and down in any settled way: asked along any line, the two always answer opposite ways, yet neither has an answer of its own until asked. A third has to settle in a wave farther out. So lithium’s third electron spreads about four times farther out than helium’s two, and each new row of the periodic table begins with a large atom. Which electrons sit outermost decides how an atom bonds, so chemistry comes from this rule.
The rule also makes magnets. In iron, several waves have nearly the same energy, and its electrons spread over them with their spins the same way. By the rule, electrons with matching spins keep apart, and the electric energy of the whole arrangement falls. That is why iron’s spins line up against the jostling of warmth, which their magnets’ own pull on each other could never beat.
Without the rule, every electron would crowd into the innermost wave, and atoms heavier than helium would shrink. Matter itself would then fall in on itself. The same rule holds up white dwarf stars against their own weight, and, for neutrons, helps hold up neutron stars.
Bosons do the opposite. A photon is more likely to be sent into a state that photons already fill, which is how a laser builds its beam. The two kinds of helium show the divide most plainly. An atom of helium-4 is made of six fermions, an even number, and so is a boson. Below 2.17 degrees above absolute zero, part of the liquid, and nearly all of it in the deeper cold, flows without friction and creeps as a film up the walls of its cup. Helium-3, one neutron short and so a fermion, stays an ordinary liquid until a thousandth or two of a degree, when its atoms pair up and the pairs share one wave, as electrons do in a superconductor.
The split into just two kinds needs three dimensions of space. Electrons held to a flat layer in a strong magnetic field move together in ripples that act as particles of another kind, neither fermions nor bosons. Experiments have caught these anyons, both by colliding them and by sending them round loops.
The Best-Measured Magnet
The electron’s equation, written to obey both quantum physics and relativity, describes a particle of spin one half. With the simplest tie to the electric and magnetic field, it also fixes the strength of its magnet. For its turning, the electron is twice as strong a magnet as a spinning ball would be with its charge spread like its mass. The doubling comes from that simple tie, not from relativity: a version of the equation that leaves relativity out gives it too. That factor of two is why the hanging iron rod twists only half as far as turning clouds would make it.
Measured closely, the magnet is about a tenth of a percent stronger still. The cause is the electron acting on itself through its own field. The popular picture of an electron jostled by a jittery vacuum gets even the direction of the shift wrong. A single electron held in a small trap has given the strength of its magnet to 0.13 parts in a trillion. Given the strength of electric forces as measured with atoms, the theory agrees with it to about one part in a trillion, one of the closest tests of any prediction. The two best measurements of that strength disagree with each other, and until they agree, the test can go no closer.
The muon, about two hundred times heavier than the electron, is a magnet of the same kind, and heavy particles no one has found would show in it more. In a ring of magnets, its spin turns a little faster than its path does, and the difference measures its magnet’s extra tenth of a percent beyond the factor of two. For two decades the measured extra disagreed with the prediction, a hint, many hoped, of new particles. The final measurement pinned it down to 127 parts per billion, and a prediction reworked with the strong force computed on grids now agrees with it. The older way of working out the strong force’s share, from collisions of electrons and positrons, is stuck: those measurements disagree with each other, and why is not settled.
Spin at Work
Almost every hydrogen nucleus is a single proton with spin, and so a magnet. Inside a scanner whose field is 1.5 tesla, thirty thousand times Earth’s, the protons in your body are not lined up. The jostling of warmth knocks them about far harder than the field can turn them. Their spins point every which way, with only a slight lean toward the field: together they make the magnet of about five in a million lined up. Every MRI image is made from that small surplus. The protons’ spins wobble round the field, 64 million times a second at 1.5 tesla. A pulse of magnetic field from a coil, turning at that rate, tips the surplus over. As it wobbles, its turning magnetism drives a tiny current in a coil, as a turning magnet does in a generator. Extra fields that vary across the body tell where each signal came from, and how fast it fades tells one tissue from another.
An electron’s two answers make a natural quantum bit, a qubit. Up and down stand for 0 and 1, and a spin pointing any other way is a superposition of both, drawn as an arrow to a point on a sphere. Spins are good qubits because they keep their direction for a long time, and pulses of microwaves or radio waves at the rate they wobble can turn them precisely. Recently a silicon chip made by the methods of ordinary chip making, with room for 18 qubits, ran seven of them together in a code that catches errors. Each is kept in the spins of three electrons, so that pulses of voltage alone can turn it, without microwaves. A grid of laser beams has held 6,100 cesium atoms, each storing its bit in the joint spin of its outer electron and its nucleus for about 13 seconds. Chips can be made the way phone chips are; atoms are all exactly alike, but harder to hold and slower to work.
The two answers that split a beam of silver atoms in two are now being built into computers.
