### Spinning Electron Got to Go Round

Does one rotation always mean the same thing? It might seem so: if you turn around (bright eyes), or spin a desktop globe, or take your car all the way around the block, you’re moving through 360°. Even something very smooth like a cue ball in pool has some slight distinguishing features that allow you to see how much you’ve rotated it.

However, things on the microscopic scale get more interesting. For one thing, particles are identical: for example, you can’t tell two electrons apart, so the details of color, imperfections of shape, and so forth are nonexistent. Rotations in the quantum world also have different character than in macroscopic systems. While an ice skater has angular momentum that depends on her size, mass, and rate of revolution, electrons and other subatomic particles have a type of angular momentum called spin that is intrinsic: they don’t speed up or slow down their spin, you can’t make them stop, and the rate of spin depends only on the type of particle. All electrons, quarks, protons, neutrons, and neutrinos have the same spin: everything that constitutes ordinary matter. Photons, gravitons (the carriers of the gravitational force), and particles called mesons have different spins.

Spin is both simpler and more challenging to grasp than rotation in the macroscopic world, but we can use everyday analogies and a bit of imagination to help us understand it. A normal macroscopic object like you (at least pretend to be normal, you freak) is analogous to a spin-1 particle: one full rotation of a spin-1 particle is 360°, just like you’d expect from experience. Examples of spin-1 particles are photons, the particles of light that carry the electromagnetic force, and some kinds of mesons, which are particles made up of two quarks. If you want to know the technical term, these are vector particles: take the fingers of your right hand and curl them in the direction of rotation. Make a “thumbs-up” gesture, and your thumb will point along the direction of the spin vector. (For photons, the direction of spin is the polarization of light, which is always perpendicular to the direction the photon is traveling.) My post on Emmy Noether has more information about rotational symmetries.

Spin-0 is the simplest spin system. Imagine a perfectly smooth featureless sphere. Rotating this sphere by any angle around any axis doesn’t change how it looks, so in a sense any rotation is equivalent to a full rotation, which is equivalent to no rotation at all! No matter how you spin it, you don’t change a spin-0 particle. Examples of spin-0 particles are the scalar mesons (also made up of quarks) and that funky hypothetical particle that keeps making the news over and over again, the Higgs boson.

In fact, we can make the featureless sphere into a kind of spin-1 particle by adding a dot on one side: to get the dot back into its original position requires rotating the sphere by 360°. (This isn’t precisely true: if you rotated the sphere by an axis running through the dot, it wouldn’t make a difference. Please ignore that fact for the sake of argument, and remember we aren’t literally talking about spheres and dots, but quantum-mechanical particles.) Now let’s add a second completely identical dot on the exact opposite side of the sphere as the first: if you rotate the sphere by 180°, it looks the same as when you started. This represents a spin-2 particle: a half-rotation is sufficient to bring it back to its original configuration. Gravitons are the only spin-2 particles you’ll see discussed normally, though like the Higgs boson, they haven’t been detected experimentally yet. Spin-2 particles are known as tensor particles for reasons a little beyond the scope of this post, but you can also think of them like an oval: the long axis of the oval is the direction of the polarization. Unlike a vector particle, there isn’t a “thumb” that tells you which way the graviton is spinning. There’s a lot more we can say about gravitational waves and polarization, but I’ll save that for another day.

Note that the particles that make up atoms in our bodies aren’t in any of those lists above: electrons, quarks, protons, neutrons, and related particles fall into a separate category known as spin-1/2 particles. Despite their fundamental nature and importance, understanding them is more challenging than spin-0, spin-1, or spin-2: they require 720° (twice 360°) to return to their original configuration. You can’t represent that easily with a sphere, or everyday examples. I’ve seen a number of physical analogies, and different teachers may show you different ones, but here’s my favorite (borrowed from The Big Black Book: Gravitation, by Charles Misner, Kip Thorne, and John Archibald Wheeler).

My quick-and-dirty electron model, made from scrap wood, elastic, and a styrofoam ball. Note the repair in the string where my cats chewed through it. Stern and Gerlach didn't have to put up with that sort of thing!

We’ll start again with a sphere, but we’ll suspend it using elastic bands from a frame. My quick-and-dirty electron model uses scrap pieces of 1×4, 3/8” doweling, 1/8” elastic, and a styrofoam ball (the latter items from a craft store). I used two elastic bands running through holes in the styrofoam ball, but you could get away with one thick band if you design it right. (Three or more bands might be more stable, but harder to manipulate.) For convenience, I put a dot on one side of the ball; turning the ball 360° around the axis brings the dot back to its original configuration, but the elastic is twisted, so the entire system is not restored.

Rotating the ball by 360 degrees (one twist) can't be untangled. Even though the ball is facing the same direction as it started, the entire system is not in an equivalent state.

Turning the ball around an additional 360° degrees (to make a total of 720°) may seem like we’ve made the situation worse, but here’s the trick. Take the upper twist of elastic and loop it all the way around the ball in the opposite direction from the rotation you made, all the while holding the ball as steady as you can. Then, take the lower twist of elastic and pull it over the top of the ball…and suddenly the elastic is untangled. The whole system is back to its original configuration, indistinguishable from where it started! This trick can’t be done with a rotation of 360°, 1080° (three full rotations), half-rotations, etc.: it requires 720° (or other multiples of two full rotations).

However, two full rotations of the ball can be untangled: loop the top elastic twist around the ball, in the opposite direction from the way you rotated the ball. Then stretch the lower elastic up over the ball so that it escapes from the loop created by the upper elastic and voila! everything is back in its initial state!

This type of object is known as a spinor particle, since it first was discovered in the context of spin. My earlier post on quaternions even hides some information about spinors in the mathematical sense: they show up in interesting places, including the physics of spacetime. Math aside, however, you should be wondering two things right now: how does this spin-1/2 character show up in the laboratory? and if matter is spinor-like microscopically, why don’t we see it behave like that macroscopically?

In the lab, the spin-1/2 nature of electrons and other particles shows up when they are subjected to magnetic fields. In the early 1920s, physicists Otto Stern and Walther Gerlach built a magnet where one pole had a sharp point, so that the magnetic field was concentrated more toward that pole. They then sent a stream of silver atoms through the magnetic field. Neutral silver has an unpaired electron, which actually is the important thing for determining the total magnetic character of the entire atom, so Stern and Gerlach could measure the properties of that electron without worrying about the electrical effects they would get by experimenting on electrons directly. What they found was surprising: the beam split neatly into two pieces, demonstrating that electrons rotate in a special way when influenced by a magnetic field. The details are a bit much for this blog post, but suffice to say that the special rotational property is precisely spin-1/2. Otto Stern was awared the Nobel Prize in physics in 1943 for his work; Gerlach was overlooked for reasons I haven’t been able to determine (perhaps owing partly to his remaining in Nazi Germany while so many of his colleagues—including Stern—became refugees).

All matter is made up of spin-1/2 particles. So why is nothing in the macroscopic world spin-1/2? The answer, as with so many things in quantum mechanics, is in the combinations of a large number of particles, which averages out a lot of the microscopic effects. If two spin-1/2 particles interact, they make a composite system that (depending on the relative orientations of the spins) either is spin-0 or spin-1. We see that on the quantum level with mesons: one type consists either of an up quark paired with an down anti-quark or a down quark paired with an up anti-quark, and is a spin-0 system called the pion (or π meson). However, if the spins of the same quarks are aligned differently, the result is a ρ (rho) meson, which is spin-1.

Mesons aren’t constituents of atoms, but they help us see how atoms work. Even if an atom contains particles that make it act like a spin-1/2 particle, if you get enough atoms of that type together, they’ll tend to behave in a more “normal” way. Any macroscopic object contains a vast number of atoms, so it’s frankly statistically impossible for the spinor character to show up in our daily lives. (Physicists have created special quantum systems that are pretty close to being macroscopic, which exhibit microscopic spin characteristics. However, these currently all require very cold temperatures and other specialized conditions, so as much as I’d like to build and demonstrate one for you, I just can’t. My apologies.)

Even though spin-1/2 seems exotic, it’s part of our lives in a literal sense. We can understand it, and someone as unhandy as myself can even build models to illustrate how it works.