Spin: an electron’s intrinsic angular momentum

Think of a bicycle wheel turning on its axle. If we want to stop it, two things matter. One is how fast it turns. The other is how much mass it has, and how far that mass is from the axle. A heavy wheel turning slowly can be as hard to stop as a light wheel turning fast. Physicists combine these into one quantity, the angular momentum, written $L$. Take one small piece of mass $m$ going round a circle of radius $r$ at speed $v$. Its angular momentum is $L = mvr$. For a whole wheel, we add up the $mvr$ of all its pieces. The unit is $\text{kg}\,\text{m}^2/\text{s}$.

Rotation also has a direction: a wheel can turn one way or the other, about an axle pointing in any direction. We describe all of this with a single arrow drawn along the axle. Curl the fingers of your right hand the way the wheel turns; your thumb then points the way the arrow points. This is the right-hand rule. The arrow’s length shows how much angular momentum there is, and its direction shows the axle and the sense of turning. Try it below: turn the wheel the other way and the arrow flips; turn it faster and the arrow grows.

Angular momentum has one more property we will need. In a closed system, the total never changes. This is called the conservation of angular momentum. A skater who pulls in her arms spins faster. Her mass moves closer to the axis ($r$ smaller), so her speed must rise to keep $mvr$ the same. And if one part of a system starts turning one way, another part must turn the other way. That way the arrows still add up to the same total.

Now for magnetism. In 1820 Hans Christian Ørsted put a compass needle beside a wire. When an electric current flowed through the wire, the needle turned. A current is moving charge, so moving charge makes a magnetic field. Bend the wire into a circle, and the loop of current behaves exactly like a small bar magnet. One face of the loop is a north pole and the other a south pole. The right-hand rule tells us which is which. Curl your fingers the way positive charge goes round the loop, and your thumb points out of the north face.

Now spin a ball that carries electric charge spread over its surface. Every bit of charge moves round in a circle about the axis. So the turning ball is a stack of current loops, all circulating the same way. Together they make the ball a magnet. Its north pole lies along the axis, on the side the right-hand-rule arrow points to. In the model below, the orange dots are the charges and the gold rings show the loops of current. The red-and-blue arrow is the magnet (red end north). Turn the ball faster, and the magnet becomes stronger. Reverse the spin, and the magnet flips.

There is a simple link between how strong this magnet is and how much the ball rotates. The strength of a small magnet is measured by its magnetic moment, $\mu$. For a flat loop it is the current times the area of the loop, $\mu = IA$. Take one charge $q$ going round a circle of radius $r$ at speed $v$. It takes a time $T = 2\pi r / v$ to go round once, so the current, the charge passing a point per second, is $I = q/T = qv/(2\pi r)$. The area is $A = \pi r^2$. Multiplying,

The same charge, if it has mass $m$, has angular momentum $L = mvr$. Compare the two: $\mu = qvr/2$ and $L = mvr$ contain the same $vr$, so dividing one by the other gives $\mu / L = q/(2m)$. The speed and the radius cancel. So for a spinning charged object, the magnet is proportional to the rotation: twice the angular momentum, twice the magnet. Remember this link; it is the key to spin.

A magnetic field is the region around a magnet where another magnet feels a force. We draw it with field lines. A compass needle placed anywhere lines up along the line through that point. Where the lines crowd together, the field is stronger. Between the poles of a magnet, the lines run from the north pole to the south pole.

What happens to a small magnet placed in a field? First suppose the field is uniform: the lines are parallel and evenly spaced, the same strength everywhere. The small magnet’s north end is pulled along the lines and its south end is pulled the other way, equally hard. The two pulls twist the magnet, but they cancel as a push: a small magnet in a uniform field turns but does not move.

Now make the field uneven. In the picture, the field lines run upwards from a north pole below to a pointed south pole above. They crowd together near the point, so the field is stronger at the top. Suppose the small magnet points up, along the field. Its north end is pulled up (along the lines), and its south end is pulled down. The north end sits higher, in the stronger field, so its pull wins. The whole magnet is pushed upwards. Point it down, against the field, and the same reasoning pushes it downwards. Tilt it at an angle $\theta$ to the vertical, and the push is in between. Only the upright part of the magnet, $\mu\cos\theta$, counts. Use the controls below to switch the field and turn the magnet.

One more point will matter later. A spinning magnet behaves like a spinning top. The field tries to twist it, but it does not swing round to line up. Instead it slowly circles round the field direction, keeping the same tilt $\theta$. So a spinning magnet flying through an uneven field keeps its angle. Its push, $\mu\cos\theta$, stays steady all the way through.

Now the electron. Many measurements show that every electron is a tiny magnet, and that all electrons are magnets of exactly the same strength. If an electron is a magnet, our spinning-ball reasoning suggests it should also carry angular momentum. Does it?

Is the electron then a tiny spinning ball? Three facts say no. First, its spin quantum number is fixed: changing its spin state is not speeding up or slowing down a material surface. Second, experiments have found no size for the electron at all; it is smaller than $10^{-18}\,\text{m}$. A ball that small would need its surface to move impossibly fast even to account for the observed component magnitude $\hbar/2$. Using $L = mvr$ with all the mass on the rim (the most favourable case),

about 200 000 times the speed of light, which a material surface cannot reach. Third, the electron’s magnet is approximately twice as strong as our rule $\mu/L = q/(2m)$ predicts for a spinning charged ball with the same angular momentum. Thus spin shares important properties with rotational angular momentum: it is angular momentum, it adds into the conservation law, and it makes a magnet. But nothing is actually turning. It is a property, not a motion.

Other particles have spin too. Electrons, protons and neutrons have spin quantum number $s=\tfrac12$; photons have $s=1$. These numbers classify their quantum angular momentum, not the speed of a turning surface. We now concentrate on ideal two-outcome measurements of spin-$\tfrac12$ systems. The same probability rule applies when such a state is prepared and measured along a chosen axis.

How can we measure the spin of something as small as an electron? In 1922 Otto Stern and Walther Gerlach used its magnet. A free electron is hard to use, because its charge makes it swerve in any magnetic field. So they used silver atoms. A silver atom has 47 electrons. Forty-six of them sit in pairs whose spins point in opposite directions, so their magnets cancel. One outer electron is left over. The 47 electrons balance the 47 protons in the nucleus, so the atom has no overall charge. The field cannot deflect it by pulling on a charge. Yet the atom carries the magnet of one unpaired electron.

Follow the apparatus in the model from left to right. An oven heated to about $1000\,^{\circ}\text{C}$ lets silver atoms escape through a small hole. Two narrow slits cut them down to a thin, flat ribbon. The beam then passes through the gap of a magnet. Its lower pole, the north pole (red band), has a groove. Its upper pole, the south pole (blue band), is shaped into a sharp edge. The field lines run up from the north pole to the south pole and crowd together under the sharp edge. So the field is stronger near the top. This uneven field is exactly what is needed to push a small magnet. Finally the atoms land on a glass plate and build up a visible silver deposit.

Before switching it on, what should we expect? The atoms leave a hot oven, so their magnets point in every direction at random. In chapter 3 we saw that each atom is pushed by an amount proportional to $\cos\theta$. Here $\theta$ is the angle between its magnet and the vertical, and the atom keeps that angle through the magnet. An atom pointing straight up is pushed up the most. One pointing down is pushed down the most. One lying sideways is not pushed at all.

How are the pushes shared out? Picture each atom’s magnet as an arrow of the same length, starting at the centre of a ball. Random directions spread the arrow tips evenly over the ball’s surface. The push depends on the height of the tip, $\cos\theta$, which lies between $-1$ and $+1$. Now slice the ball into horizontal bands of equal thickness. Archimedes first proved a remarkable fact: every band has the same area. A band near the top has a smaller radius, but its surface is more steeply tilted. The two effects exactly balance. So every height between $-1$ and $+1$ is equally common. The atoms should spread out evenly into one continuous band on the glass. Choose “If electrons were ordinary magnets” to see that prediction.

Now choose “What really happens”. The silver lands in two separate strips, one above the centre and one below it, with nothing in between. (The strips curve like a pair of lips. The field is strongest under the middle of the sharp edge, so atoms near the middle of the ribbon are pushed farthest.) Gerlach sent the result to Niels Bohr on a postcard. No atom lands near the middle, where atoms lying sideways should have landed. To make the two groups easy to follow, the model colours the upper deposit orange and the lower one blue. The real silver is all the same colour.

What do the two strips mean? Each atom was pushed either fully up or fully down, never partly. The push measures the upright part of the atom’s magnet, and the magnet is proportional to the spin. So the upright part of the spin can have only two values: $+\hbar/2$ or $-\hbar/2$. We call these two results spin up and spin down. A quantity that can only take certain separate values is called quantised. The energy levels of an atom are another example of this idea.

Count the atoms as they arrive. About half land in each strip, and the more atoms you count, the closer the two fractions come to $\tfrac12$. For atoms straight from the oven, the chance of each result is $\tfrac12$. We cannot say in advance which strip a particular atom will land in, only how likely each one is.

Could the oven be sending out only atoms with spins pointing straight up or straight down? We can check by turning the magnet and its slits around the beam. Use the angle control and watch the screen. At $90^{\circ}$ the magnet measures sideways, and the two strips appear on the left and right. At every chosen angle there are still two strips, along the direction measured by the magnet.

This is already strange for an ordinary arrow. An arrow pointing straight up has no sideways part at all, so measured sideways it should give $0$. A spin never gives $0$ along any direction: each measurement gives $+\hbar/2$ or $-\hbar/2$ along whichever direction we choose to ask about. The answer exists only as an answer to the question we put to it.

To go further, we need a handier tool. An analyser is a small Stern–Gerlach magnet with two exits instead of a glass plate. Atoms with result $+$ leave through the upper port (orange ring). Atoms with result $-$ leave through the lower port (blue ring). We can put a cap on either port to stop those atoms, and send the others on into a second analyser. At the end, two cups count the atoms that arrive.

Put two vertical analysers in a row. Cap the lower port of the first, so only atoms that gave “up” go on, and measure them again vertically. Every one of them gives “up” again; the blue cup stays empty. Once an atom has given “up”, it keeps giving “up” when asked the same question. The first analyser has prepared it: we say its spin is now in the state up along the vertical.

Now turn the second analyser on its side, to $90^{\circ}$, while the first still keeps the vertical “up” atoms. These atoms are definitely up along the vertical, yet about half of them give “+” sideways and half give “−”. Knowing the vertical answer tells us nothing about the sideways one.

Try other angles. At $0^{\circ}$ every atom gives “+”. At $180^{\circ}$ the analyser is upside down. Its “+” port now collects atoms that are up along the downward direction. Our atoms are up along the vertical, so every atom gives “−”. In between, the fraction giving “+” falls smoothly. Many careful experiments agree on one rule. Prepare a spin up along one direction, then measure it along a direction turned by $\theta$. Then

Check it with the counts below for a few angles. At $\theta = 60^{\circ}$, $\cos 30^{\circ} \approx 0.866$ and $0.866^2 = 0.75$: three atoms in four give “+”. At $\theta = 90^{\circ}$, $\cos 45^{\circ} \approx 0.707$ and $0.707^2 = 0.5$: half. At $\theta = 120^{\circ}$, $\cos 60^{\circ} = 0.5$ and $0.5^2 = 0.25$: one in four. Each single atom still gives only “+” or “−”. The rule tells us how often each happens.

This rule has a neat link to the ordinary arrow picture. Score each “+” as $+1$ and each “−” as $-1$, and take the average over many atoms. A fraction $P_+$ score $+1$ and the rest, $1 - P_+$, score $-1$, so the average is $P_+ - (1 - P_+) = 2P_+ - 1$. Put in $P_+ = \cos^2(\theta/2)$:

The last step uses the double-angle formula, $\cos\theta = 2\cos^2(\theta/2) - 1$. So the average of many results is $\cos\theta$, exactly what an ordinary arrow of length 1 tilted by $\theta$ would give. Every single result is all-or-nothing, but on average spin behaves like the old arrow. That is why the arrow picture is useful, and also why it can never be the whole story.

We can use three analysers to test whether spin behaves like an ordinary arrow. The first is vertical and keeps “up”, so every atom leaving it is up along the vertical. The last is vertical too. With nothing between them, every arriving atom gives “up”, leaving the blue cup empty. Now put a sideways analyser in the middle and keep only its “+” atoms. Does the last measurement still give the same result?

Run it. Half the atoms are stopped at the first analyser. Half of the rest are stopped at the sideways one. That leaves a quarter to reach the last analyser, and half of these now give “down”. We added a filter that only removed atoms, yet “down” atoms appeared where there were none. Suppose each atom carried a little arrow with a definite vertical part and a definite sideways part at the same time. Then the middle analyser could only throw atoms away. It could never turn an “up” atom into a “down” one.

So measuring sideways changed the atoms. After the middle analyser, each atom that passed is “+” along the sideways direction, and that is all it is. Its earlier “up” is gone. Then the last analyser asks the vertical question again. By the rule from chapter 9 with $\theta = 90^{\circ}$, the answer is “up” or “down”, with probability $\tfrac12$ each. A spin can be definite along only one direction at a time. Each measurement leaves it definite along the direction just measured.

We can now put the whole answer together. Each part is backed by an experiment we have seen.

So spin is not a ball turning. Nor is it an arrow with definite parts along every direction at once. A useful picture to keep is an arrow that can answer only one question at a time. Ask it along any direction, and it answers “with me” or “against me”. And the act of asking turns it to the direction you asked about.

Chemistry. Two electrons can share the same orbital in an atom only if their spins are opposite, one up and one down. No third electron can join them. This rule is called the Pauli exclusion principle. Because of it, electrons fill atoms shell by shell instead of all crowding into the lowest level. So it shapes the whole periodic table. A covalent bond is a pair of electrons with opposite spins, shared between two atoms.

Magnets. In most materials electron spins pair up and their magnets cancel. In iron, cobalt and nickel, neighbouring spins prefer to line up, so large regions become magnetised: that is where the fridge magnet’s magnetism comes from.

Medicine. The protons in the hydrogen atoms of water also have spin $\tfrac12$. An MRI scanner places the body in a strong magnet and uses radio waves to flip these spins. As the spins flip back, they give out signals. The signals show where the water is, and so the shape of tissues inside the body.

Particles. Spin sorts every kind of particle into two families. Particles with half-integer spin quantum number ($\tfrac12$, $\tfrac32$, …), such as electrons and quarks, are called fermions. They obey the exclusion principle, and they make up matter. Particles with integer spin quantum number ($0$, $1$, …), such as photons, are called bosons. The force carriers are bosons, but not every boson is a force carrier; the Higgs boson is another example. The Standard Model builds on this.