What is "spin"?

Hello Everybody! This is the first of what I hope will be a number of short little entries explaining some interesting things in the world of physics, biology and chemistry. One thing I'd like to discuss is the quantum mechanical property known as spin. Spin is an interesting little property that is not commonly understood outside the world of physics and chemistry, but is responsible for an enormous number of the things we observe in the world around us. Spin is responsible for how chemical bonds can and cannot form between atoms (the Pauli principle and Hund's rules) and is also one of the major quantities that gives rise to the phenomenon of magnetism (meaning it is responsible for why medical imaging techniques like MRI work). But what is spin really?

Spin is kind of a poor name, and the main purpose of this article is to explain why. When we think of spin, what comes to mind is an object that is rotating about some axis (like a bicycle wheel or a planet). But does this have anything to do with the physical quantity we call spin? First of all, let's start out by saying that spin is a property of subatomic particles. These are things like protons, neutrons, and electrons that make up atoms. Let's talk specifically about electrons. These are very small and very light (they are about 2000 times lighter than protons) and they have a negative charge. This charge bit is important. It turns out that this is actually one way to create a magnetic field. An electric current (i.e. a moving charge) gives rise to a magnetic field. This is known as the Biot-Savart Law. Now, let's imagine that our electron is like a little planet spinning around its axis. This little ball of spinning charge would create a magnetic field. And it is this little magnetic field that causes magnets to work (and allows a doctor to give you an MRI).

Figure 1: if an electron is a spinning ball of charge it would create a magnetic field.

The reason this property is called spin is because it is a type of angular momentum. We know from quantum mechanics that spin (angular momentum) can only take on specific values called quanta. As a matter of fact, this is where quantum mechanics gets its name: when objects are really small their physical properties (such as energy, momentum, or position) can only have certain allowed values. In the case of the spin angular momentum of an electron this can be $\pm\hbar/2$ ($\hbar$ is a special number in quantum mechanics and its value is $1.05457\times10^{-34}m^{2}kg/s$; it has an interesting history and a profound meaning but we won't talk about that now). For arguments sake we can say that $+\hbar/2$ refers to an electron spinning counter-clockwise and $-\hbar/2$ to one spinning clockwise.

Now let me pose a question: if you were somehow sitting on the surface of the electron (which has an angular momentum of $\hbar/2$) how fast would you be moving? To figure this out we need to know what the radius of the electron is. Modern particle physics does not give an electron a radius or a shape (it is essentially a point particle), but we can use classical physics and relativity to set an upper bound on how big an electron could be.

If we assume that an electron is a little ball of uniform charge, and we know both its mass and charge, how big would it have to be? Well we can calculate the electric field associated with a spherical charged object using Gauss' Law and from that we can calculate the potential energy associated with that electric field. If we set this equal to the energy arising from the rest mass of an electron (you know Einstein's good old $E=mc^2$) we can figure out the approximate size of an electron.
$$U=\int{\mathbf{F}\cdot d\mathbf{r}}=\int{q\mathbf{E}\cdot d\mathbf{r}}=\frac{q^2}{4\pi\epsilon r}=mc^2$$

Don't worry if you don't understand the particulars, all this is saying is that we will assume that the energy associated with an electron's mass is equal to the energy associated with its charge. Solving this little equation for r gives us:
$$r_{c}=\frac{q^2}{4\pi\epsilon mc^2}=2.819\times10^{-15}m$$

This is known as the Compton radius for the electron (AKA the classical electron radius). This is unimaginably small (ten billionths of a millimeter!). As we said previously the spin of an electron is quantized in amounts of $\hbar/2$. If we assume that corresponds to the angular momentum of a spinning object, we can use classical physics to figure out the speed on the surface of that object. We would use the classical formula for the angular momentum of a sphere:
$$S=I\omega=\frac{2}{5}mr^2\omega=\frac{2}{5}mrv$$

Now if we plug in the values for S (spin angular momentum), m (mass of an electron), and r (Compton radius of an electron) we can solve for v:
$$v=\frac{5\hbar}{4m_{e}r}=5.135\times10^{10}m/s$$

How fast is this? It is 171 times faster than the speed of light. As you may already know, objects with a non-zero rest mass must move slower than the speed of light (more on this in a future blog post). So the "spin" of an electron cannot really correspond to it spinning around its axis!

So how should we think of spin? A better term than spin is "intrinsic angular momentum" or better yet, a particle's "intrinsic magnetic moment". We can convert spin into a magnetic moment using the simple formula:
$$\mu=g\mu_{b}S/\hbar$$

This way, we can think of particles like electrons as little bar magnets and the spin of these particles is really just their intrinsic magnetic moment. So rather than thinking about spin as an electron spinning clockwise or counter-clockwise, you can think of it as a little bar magnet pointed up or down (and in quantum mechanics we do in fact call $+\hbar/2$ as "spin up" and $-\hbar/2$ as "spin down").

Figure 2: electrons are more like little bar magnets than little spinning balls!

I hope you liked my first little blog post. There are lots of other cool things to talk about related to spin: superfluidity and quantum computing to name just a few. But we will leave that for another time. If you have any comments or corrections please feel free to leave them below. Thanks and stay tuned for more fun science stuff!