First the basics: spin is an intrinsic property of matter, like charge or mass. It is measurable in the real world by observing interactions with magnetism, and is the basis of technologies like MRI and hard disk drives!
We of course recognize the verb ‘to spin’, which means to rotate around a fixed axis the way that wheels, figure skaters, and the Earth do. But the word spin is also used to describe a fundamental property of particles. We have already talked a little about a fundamental property, charge, which was useful because a lot of the important forces at the atomic scale are electromagnetic and thus related to charge. And we remember that mass, another fundamental property, determines how matter interacts via the gravitational force. Spin is a bit different.
The idea of particles having an intrinsic spin first arose during the development of quantum mechanics, when Wolfgang Pauli and others noticed that part of the mathematical solution for particle states resembled angular motion, as if the particles were physically spinning around an axis. But unlike spinning at the macroscopic scale, quantum spin can only occur at a few discrete values: integer and half-integer multiples of ħ, the reduced Planck constant. The allowed values of spin are clustered around zero, and the ħ factor is dropped by convention because particle physicists like to make things look simple. So a photon, the quantum of light, has spin 0, whereas electrons and quarks, which make up protons and neutrons, have spin 1/2. There are also particles with spin 1, 3/2, and 2. As with charge, spin is reminiscent of a behavior we see in the macroscopic world, but its values are quantized into a few allowed values.
Spin can have one of two polarities, meaning we can have an electron with spin +1/2 and one with spin -1/2. And charged particles like the electron actually respond to magnetic fields differently if they have positive or negative spin! This is because the motion of a charged particle creates a small magnetic moment, which will be aligned in one direction for positive spin and the opposite direction for negative spin. This is the basis of the famous Stern-Gerlach experiment, in which atoms with one free electron are sorted by their spin under the influence of a magnetic field. But it’s also the basis of nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI), two related techniques for determining the composition and structure of either chemical substances or human patients! Strong magnetic fields can be used to align spins within any object, and how quickly the spins decay back to their original orientation gives information about what is inside the object. Currently, researchers are trying to build circuits that use spin instead of charge to carry information, which is called ‘spintronics’.
But at a more basic level, when we talked about chemical bonds we skipped over the importance of spin. The reason spin matters for bonding is due to the Pauli exclusion principle, the idea that no two electrons can share the same quantum state. In the development of quantum mechanics, it became clear from the data that even if all the available energy states were mathematically accounted for, there still seemed to be a degeneracy in which two electrons shared what was thought to be the same quantum state. This can be explained with a new quantum number, which we call spin. So spin is another factor of the electron cloud shape and is critical in the understanding of chemical bonding.
But there are actually even more strange things about spin than I can fit in this post, including the fact that the Pauli exclusion principle only applies to particles with half-integer spin! Half-integer and whole-integer spin particles are fundamentally different from each other, in some pretty interesting ways, but why is a story for another time!