What is the evidence (experimental observation) that elementary particles have spin angular momentum?

Question

From what I have read here, the concept of spin is attributed to a calculation based on a mathematical model using quantum mechanics. For example,

How can a particle with no size have angular momentum?

The experiment by Einstein and deHaas

https://www.dwc.knaw.nl/DL/publications/PU00012546.pdf

https://en.wikipedia.org/wiki/Einstein%E2%80%93de_Haas_effect

relied on the conservation of angular momentum. The simple youtube video

https://www.youtube.com/watch?v=4UK10VAVzXk

implies that you can get an iron cylinder dangling from a string to spin by imposing a magnetic field in the direction of the supporting string. The actual paper tho describes that a magnetic field has to be reversed ("Then, on reversing a current in $K$ a rotation of C ought to be observed.") This makes more sense as you first need to have all the angular momenta lined up. Then the reversal of the field induces angular momenta to 'flip'. I assume that the 'flipping' is caused by the Lorentz force of the new magnetic field on the intrinsic magnetic field of all the lined up atoms. I dont get why that should cause the cylinder to spin. For the Lorentz force to make the cylinder spin, the magnetic field of the coil should be transverse to the magnetic field of the cylinder.

However the authors then say " In reality, however, this simple method cannot be thought of. As the field of the coil will not be uniform the cylinder would probably show highly irregular motions completely masking the effect that is sought for. "

Obviously, they saw variations that obscured their data and 'non uniformity' is how they explained it. Personally, I cannot understand what 'non-uniformity' they are talking about.

They then go on to talk about using 'resonance', basically using an AC current to cause the field to oscillate and thence to cause the cylinder to spin back and forth.

The complexities introduced by pursuing the analysis of resonance are numerous, including the effect of the Earth's magnetic field. Their use of mathematics borders on being a demonstration of their virtuoso skills. Perhaps I am too dense as I get lost in the weeds.

In addition remarks like "Unfortunately, when our experiments had been brought to a conclusion and one of us had left Berlin it came out that a mistake had been made in the application of the method, so that we must consider as a failure this part of our investigation." make me less confident about the whole experiment and its results.

Later experiments by Barnett

https://en.wikipedia.org/wiki/Barnett_effect

talk about the formation of a magnetic field simply by spinning a ferromagnetic material. I totally do not understand why that should happen if all the angular momenta in a lump of iron are randomly distributed. Anyway, I don't get how that tells you that electrons in an atom have angular momentum.

Further phenonmena such as electron spin resonance have been attributed to the intrinsic 'spin' of electrons.

https://en.wikipedia.org/wiki/Electron_paramagnetic_resonance#:~:text=Electron%20paramagnetic%20resonance%20(EPR)%20or,the%20spins%20of%20atomic%20nuclei.

Certainly, I understand that "increasing an external magnetic field, the gap between the ${\displaystyle m_{\mathrm {s} }=+{\tfrac {1}{2}}} m_\mathrm{s} = + \tfrac{1}{2} and {\displaystyle m_{\mathrm {s} }=-{\tfrac {1}{2}}} m_\mathrm{s} = - \tfrac{1}{2} $ energy states is widened until it matches the energy of the microwaves,..."

but how do you get from a widening energy gap to saying that it reflects 'spin'?

The wiki article says 'At this point the unpaired electrons can move between their two spin states. '

It's an energy gap, like the energy gap between orbitals of electrons in an atom. What is the evidence that tells us it is angular momentum and therefore prompts the confusing use of the word 'spin'.

I would appreciate referral to an honest, simple experiment that shows electrons in an atom have angular momentum.

Answer 1

It is perhaps best to start this out from the classical perspective. In classical electromagnetism, a spinning electrically-charged object generates a magnetic field due to the fact that the spinning of the charged object is charge in motion and thus technically is an electric current (even though it may not be what one thinks of an electric current which is where charges within the material are flowing with respect to other charges that are remaining stationary, while here the whole thing is in motion). This is Ampere's law.

Indeed, the guy for which that law is named was one of those to first indirectly observe evidence for what we'd now call as electron spin. You see, if you take an object carrying an electric current - and that would include a spinning static charge - and you put it in a magnetic field, a force is developed on the current thanks to the magnetic force law, $\mathbf{F}_\text{mag} = q\mathbf{v} \times \mathbf{B}$, and this is obvious in the case of electromagnets, where you have an electric circuit and run current through it. And what was a going hypothesis at the time was that electric currents must somehow account for all magnetic fields - but if that's the case, then we have a seeming problem: there exist so-called permanent magnets which were historically by far the first observations of what we now call magnetism, all the way back to ancient Greece (the terms "magnet" and "magnetism" themselves come from the name of a place in Greece, Magnesia, where lots of naturally magnetized ores [magnetite] could be found), and yet seem to possess no detectable internal current!

And that leaves the question of how to account for them, and what Ampere suggested was that it was due to so-called microcurrents (sadly with yet more eponymism also called Amperian currents) within the material, extraordinarily small, ever-flowing electric currents of some sort, each of which would have to be something akin to a tiny loop because otherwise you'd have a large-scale current, and which would each produce a small dipole, but by virtue of their phenomenal minuteness, would be unamenable to detection by an ordinary instrument. In some materials, those dipoles would line up, and you get a large-scale magnetic field; in others, they don't, and instead they contribute randomly and the fields average out to approximately nothing.

So from that alone, there's a strong hint that something in the material must be undergoing some sort of continuous motion that is resulting in the generation of these magnetic fields; but it was not clear what that was until better understandings of atomic structure and the nature of electric currents were probed more closely, and the electron was discovered and more importantly, was discovered by finding it to be separable from the rest of matter (this is typically done using a thermionic valve, i.e. a vacuum tube: heat up a filament like a lightbulb until it's glowing super hot - yellow hot, white hot - and it will be roiling with electrons), and thus allowing it to be moved about on its own independently of a material, and with that available, it was possible to probe its properties more closely by suitable manipulations of the now-liberated electrons with electromagnetic fields, and that revealed it to contain, in addition to its negative electric charge, a small but not zero dipole moment that is what you might expect were it a spinning object of some kind - at last, Ampere's famous micro-current.

Of course then we know with further work that quantum mechanics is a thing, and the behavior of these spins - and all other motions on the atomic scale - is very much different from Newtonian mechanics: from a very modern perspective, we'd say this results because the spin axis of the rotating electron is ill-defined as to which way it points in space, in turn because, as an "elementary system" (as far as we know), the electron can only hold a single bit of information, and with a single bit, you have far too little to write down a complete $(\theta, \phi)$ pair-of-real-numbers spatial orientation for the axis of rotation of an object!

So basically, an earlier stage of systematically splitting up matter into smaller chunks, as has been going on in research all the way up until now.

Answer 2

I am posting this comment in the answer box because it does not fit in the comment box. So I am not answering but rather trying to refine the question so that people dont think I am 'moving the goalposts'. Asking a precise question is difficult sometimes - it's like trying to get directions to a destination if I dont know the street names.

Thank you so much Jon Custer for that reference. The account by Goudsmit begins exactly with the puzzle that spin seeks to solve: the splitting of the Lyman alpha line (the 2P to 1 S transition resulting in emission of 121.6 nm light). And Goudsmit's account is full of 'humanity' which makes the abstract notion of theoretical physics warmer and more palatable. But I wish there were more stories that recounted the 'other ideas that failed' to account for the splitting.

For example, the whole idea that a 'jumping electron' gives off a photon is still mysterious. Something happens within the size of a hydrogen atom (120 picometers) that generates a wavelength a thousand times longer. That splitting happens indicates two different kinds of jump. There has to be 'another degree of freedom' (in Goudsmit's words) to account for the fine splitting of that line. So sometimes the jump is from a slightly higher energy and sometimes the jump is from a slightly lower energy. I wish I could hear the discussion between Pauli, Goudsmit, Ehrenfest and others so that I could understand where they got the idea of 'spin'.

It could have been anything. Why didn't they say the electron had two isomers or isotopes - large and small. The larger isomer would give a slightly larger energy change and hence a shorter wavelength. If you start talking about spin, then I would expect electrons with different spins to be deflected slightly differently by a magnet - but I have not read about beta decay resulting in two different electron paths when a magnet is brought near. So evidently, it is 'not really spin' but rather a label for a property - similar to those imagined for quarks - charm, direction,etc

So that is the reason I asked for experiments that show angular momentum. If you really want to talk about something spinning, you have to measure its angular momentum compared to something else you really know is spinning.

And if 'spin' is just being used as a label for another degree of freedom, why did the 'inventors of spin' come to use units of angular momentum to describe it?