What is Spin?

by Markus Ehrenfried

Early models of the atom used to be strongly influenced by our image of the solar system: a massive nucleus in the center surrounded by electrons which orbit the nucleus like planets revolving around the sun. But apart from circling on their orbits the planets (electrons) have another possible motion: they can rotate around their own axis -- exactely like the earth does, travelling around the sun within one year but in addition rotating around its own axis in 24 hours.

Speaking in this analogy our planet has an orbital angular momentum (around the sun / the nucleus) and in addition a spin angular momentum (around its own axis).

Electrons are particles which have an electric charge. Whenever an electric charge moves a magnetic field is created. There is nothing you can do about it: if you move a charge you'll induce a magnetic field. An electron moving on an orbit around the nucleus is basically a tiny loop of electrical current and it sets up a magnetic field. The spin of the electron sets up another magnetic field. Therefore, atoms behave like small magnets. Magnets can interact with other magnets which means that atoms can be influenced by external magnetic fields.

The Discovery of Spin

In 1921 two physicists named Otto Stern and Walther Gerlach made an interesting experiment. They took a beam of electrically-neutral silver atoms and let it pass through a non-uniform magnetic field. This magnetic field deflected the silver atoms like it would deflect little dipole magnets if you threw them through the field. After passing through the field the deflected atoms hit a photoplate and made little visible dots. (You may ask why they took silver atoms; silver atoms behave like hydrogen atoms with only one electron but they are much easier to handle in the lab and easier to detect with a photoplate.)

The result of this experiment was totaly unexpected and very surprising. Keep in mind: those atoms were just coming out of an oven where silver was evaporated, they had no special orientation in space, therefore the spins of the outer electrons in these atoms should point into all possible directions in space. Depending on their orientation the magnetic force our little dipole magnets 'feel' is different and therefore the deflection is different. Some of them would be oriented in a way that the deflection is very strong, others would have an orientation which results in almost no deflection at all, so Stern and Gerlach expected a pattern like the one depicted as 'Classical Expectation' in the picture above: a spot on their photographic plate produced by many many tiny dots caused by silver atoms hitting the plate all over the place. What they got instead was a pattern like the one shown below: only a contour was hit by the atoms and in the middle was nothing!!!

There was only one possible explanation for that behaviour: The magnetic moments -- and therefore the spins -- can only have two certain orientations in space. (To be precise, this picture is only true if the orbital momentum of the electron is zero (l=0 ground state)).

Let's summarise what we've found out up to now:

1. Electrons behave like little tops.
2. The rotation around their own axis is called 'spin'.
3. An electrical charge which moves (spins) sets up a magnetic field. This is how we can measure spin: we measure the interaction between an external magnetic field and the magnetic field caused by the spinning electron.
4. The spin of the electron behaves very strange in a certain respect: it can have only two orientations in space. This is something which cannot be explained by the model of a spinning top.
5. Perhaps the electron is not at all like a little top....????

The quantum mechanical picture of spin

The problem with analogies like 'the electron is like a little top' is that they work only up to a certain point and in quantum theory this point is reached rather soon. Reasoning by analogy will usually lead to conclusions which have nothing to do with reality. The model of the little spinning top gives us the comfortable feeling that we understand what's going on (because we compare it to something which is easy to understand) but in fact we're deceiving ourselfs. It's not that the analogy is plainly wrong (because an analogy is just an analogy and in this sense will never be 'right') and indeed the picture of the electron as a little top was historically extremely helpful: it's hard to imagine how one could get an understanding of the world of atoms and subatomic particles without starting with pictures like little solar systems and spinning tops, but progress in atomic and particle physics pretty soon made clear that there are a lot of properties which are impossible to explain if we try to stick to these simple ideas.

In fact, spin is something very very weird. In textbooks about quantum theory you'll usually find the formulation that spin is a non-classical degree of freedom. That's the way physicists say that it is unlike anything you've ever seen with your eyes.

Classical physics is the physics of Galileo, Newton and Maxwell. It's about billiard balls, spinning tops, electromagnetic waves. It's the sum of the theories at the end of the 19th century: everything apart from quantum mechanics and Einsteins theories of special and general relativity. It was so complete that physicists of that era thought physics would be finalized soon and only a few, small problems had to be sorted out (like for example the problem of black body radiation). Classical physics describes the macroscopic world with amazing accuracy.

In the beginning of what is often called the Quantum Revolution people tried to fix their pictures based on classical physics by introducing a few additional rules. A famous example is Bohr's model of the hydrogen atom which is very close to the picture of a miniature solar system with a few additional rules which are in contradiction to classical physics. (For example in classical physics accelerated charges will always emit radiation and thus lose energy. The electrons in Bohr's atoms should lose energy with every turn around the nucleus which would lower their orbit according to classical mechanics: they would pretty soon spiral down and crash into the nucleus. Bohr's way to prevent this was to postulate something like: 'okay, let's assume that this just doesn't happen'.)

To make a long story short, to rescue the naive pictures you'll have to introduce so many exceptions that you'll sooner or later give up and accept that the world at the quantum level is a totaly different place with weird and bizarre rules. That's what is called 'non-classical': it has nothing to do anymore with the laws of classical physics which describe so nicely our everyday world.

That's about the point to abandon the picture of the tiny spinning tops.

Spin and symmetry

There is another way to look at spin: it tells us something about the symmetry a particle has. Stephen W. Hawking explains this aspect of spin in chapter 5 of his book A Brief History of Time with a nice example:

A particle with spin zero behaves like a point: it looks the same way regardless of the direction from which you look at it.
 

Something which has a symmetry like the playing card on the left needs a full rotation of 360° until it looks again the same. That is the sort of symmetry a spin-1 particle has: after a full rotation it is again in the same state. (Obviously it makes no sense to say something like 'a particle looks the same' as we are not able to see it, but particles have well defined states which can be detected, that's why we can say that a particle is again in the same state).

A spin-2 particle behaves under rotation like the playing card on the right hand side. It already looks the same (is again in the same state) after half a rotation (180°).

The electron is a spin-1/2 particle, and now things become strange: a spin-1/2 particle needs two full rotations (2x360°=720°) until it is again in the same state. There is nothing in our macroscopic world which has a symmetry like that. Common sense tells us that something like that cannot exist, that it simply is impossible. Yet that's how it is. Actually, it is even relatively easy to set up an experiment in a lab which demonstrates that electrons behave exactely in this weird way: if you 'turn' them around once they are not in the same state but in minus that state and only after another full rotation they are again in the state they had initially. There is no way to explain this if we imagine spin as a little arrow in the three-dimensional space of our everyday life! (For fellow physicists: please note that we're talking about spin itself, not the expectancy value of some projected z-component.)

Even if we cannot imagine a behaviour like that it fortunately doesn't mean that we are unable to calculate it. If you look into a basic textbook about quantum mechanics you'll see that from a mathematical point of view spin is no problem at all. There are just some factors of 1/2 which appear in the formulas and everything works fine. It only sounds bizarre if we try to apply the same rules to things we know in the macroscopic world. The quantum world is indeed very different and far from our experience.

This special, non-classical symmetry under rotations is formulated by matrices named after the Austrian/Swiss physicist Wolfgang Pauli and they look like this:

-- but really, this is something you'll find in textbooks about quantum theory and the point is: even though it sounds bizarre and contra-intuitive we can perfectly understand it on a mathematical level and work with it. The results of these calculations will describe our experiments with very high precision. Quantum theory works.

Let's summarise once again: ;-)

1. Electrons are not like little tops. Only very few aspects of spin can be understood by this picture. The name 'Spin' has historical reasons, there is nothing which 'spins'.
2. Spin is an intrinsic property of particles. There is no picture in the framework of classical physics which can explain what spin is.
3. Quantum mechanics can describe spin, but can it explain it? Spin is a property which we discover if we study nature and we can rig up mathematical structures which perfectly reflect how it behaves -- but in the first place we have to accept that there is something which is very different from the macroscopic things we experience with our senses.
4. Even if spin is beyond our imagination it is no problem at all from a theoretical point of view.

Is Spin important?

The Standard Model of particle physics distinguishes between two basic types of elementary particles: particles which are matter and particles which transmit forces (interactions). If you hear that for the first time you'll probably find the idea that forces like e.g. gravitation or the electromagnetic force are transmitted by particles very surprising but please simply accept it for the moment without any further explanation.

Let's concentrate on the building blocks of matter first. These are the elementary matter particles we know today:
 

You can find our old friend the electron in the left table, it has an electrical charge of minus one unit charge. In case you wonder where in this table the protons and the neutrons are: protons and neutrons, the building blocks of the atomic nucleus (that's why they are also called 'nucleons') are not elementary particles, they are 'molecules' in the language of chemistry; their building blocks are called 'quarks'. Quarks are as elementary as electrons as far as we know today. The proton is a 'molecule' of two up quarks and one down quark (if you sum up the electrical charges of this combination you'll see that you get +1, the charge of the proton: 2/3 + 2/3 - 1/3 = 3/3 = 1). The neutron is a combination of two down quarks and one up quark (and again, if you combine the electrical chages, you'll see that they sum up to zero: it's electrically neutral). 99,9% of you consists of quark matter: up and down quarks combined to protons and neutrons which are again combined to nuclei in the atoms inside your body. The tiny remaining rest is the little mass which is contributed by the electrons in the shells of the atoms.

All the particles in the table above have one in common: they all have a spin of 1/2:

ALL MATTER IN THE UNIVERSE IS BUILT FROM A SMALL NUMBER OF FUNDAMENTAL SPIN 1/2 PARTICLES.

Now let's have a look at the second category of particles: particles which transmit forces. The standard model knows four fundamental forces (strong force, weak force electromagnetic force and gravitation) and three of the forces are mediated by these particles here:

As you can see all the particles listed in the table have a spin of 1. Nobody has yet seen the particle which mediates the gravitational force, the graviton, so it is not listed here. There are theories which predict that the graviton will have a spin of 2.

ALL FORCES ARE MEDIATED BY PARTICLES WHICH HAVE INTEGRAL SPIN.

So it looks like spin is the main difference between 'force-particles' and 'matter-particles'. Spin certainly seems to be a very important property in quantum mechanics!

Fermions and Bosons

The trick is, that particles with half-integral spin (like 1/2, 3/2, ...) behave in a totaly different way when you put a bunch of them together than particles with integral spin (like 0, 1, 2, ...) do. This is something very theoretical and it is not possible to explain here with words why this is the case. If you would like to understand it you'll have to learn quantum mechanics, there is no easy way.

But remember that spin is basically something about symmetry:

• spin 1/2 particles have this weird symmetry that one full turn brings them not back into the same state but into minus that state and only a second full turn brings them back to the state they had initially. That is something which is called in mathematics 'antisymmetric' behaviour. The same is true for any other half-integral value of the spin. (For physicists: The multiparticle wavefunction is antisymmetric under exchange of identical fermions.)
• Particles with integral spin behave in a way which is called 'symmetric'. (For physicists: The multiparticle wavefunction is symmetric under exchange of identical bosons.)

Particles with half-integral spin are called 'Fermions' because they obey the Fermi-Dirac statistics while particles with integral spin obey Bose-Einstein statistics and are called 'Bosons'.

Of course it is impossible to understand the meaning of the sentences above without already knowing all about this stuff. ;-)

The point is: There is a rule in nature which is called the Pauli principle. The Pauli principle forbids particles to be in the same state: if you take a bunch of them and put them together they cannot be all in the same condition (although they would very much like to be as they always try to minimize their energy). This is something which can be derived with mathematics but it is not possible to explain it with some analogy. It's a rule like in chess implicitly the rule exists that two chessmen are not allowed to be on the same field at the same time. You probably know the Pauli principle from chemistry: the electrons inside an atom are not allowed to be all on the lowest shell, they have to occupy higher and higher shells. The rule is that they have to be different in at least one quantum number: that is exactely the Pauli principle. Without the Pauli principle all the electrons would sit on the lowest shell because this is the minimum energy nature always tries to reach. Matter would not build up higher and higher structures if it were not for the Pauli principle. But the Pauli principle applies only to particles with half-integral spin. The electrons of the atomic shell have half-integral spin and also the protons and neutrons in the nucleus and one level deeper the quarks inside the protons and neutrons.

The way how a bunch of particles which you put together arranges to obey the Pauli principle is expressed by statistics, and this statistics was found by and named after Enrico Fermi and Paul Dirac. That's the reason why particles with half-integral spin are called 'Fermions'.

But what's about those other particles which have not half-integral but integral spin? For them the Pauli principle is just not valid. They underlie other rules. They are allowed to go all into the same state. Their statistical behaviour was first calculated by the Indian physicist Satyendra Nath Bose for photons (spin 1) in 1920 and later generalized by Albert Einstein in 1924. Particles with integral spin are named after Bose. It is (quite literally) a cool effect if billions and billions of particles go into the same state: this is called Bose-Einstein condensation.

A very nice description of the Stern-Gerlach Experiment can be found here. There is also a java applet which shows what happens under certain conditions.

Have also a look at HyperPhysics, especially on their spin page.
 

The 'real' discovery of the electron spin was made by the two Dutch physicists G.E. Uhlenbeck and S.A. Goudsmit in 1925. They demonstrated that atomic spectra can be understood by the concept that the electron has a spin.

You might want to read this lecture given by Samuel Goudsmit about his discovery (for which he and Uhlenbeck never got the Nobel prize).
 

A not-so-obvious fact is that a Stern-Gerlach Experiment with free electrons wouldn't work. (With 'free electrons' we mean electrons which are not part of an atom.) You might think that you could sort electrons according to their spin orientation by shooting them through the inhomogeneous field of a Stern-Gerlach-type magnet and that the beam would split into two beams, each of them spin-polarized.

Unfortunately it isn't that easy to get polarized electrons! Free electrons are not electrically neutral (like the silver atoms) and therefore they are deflected in addition by the Lorentz force. This, combined with the uncertainty principle, prevents the separation of spin-up and spin-down electrons. You can find a detailed explanation of this effect in Joachim Kessler's book Polarized Electrons (Springer, Heidelberg 1976)

Spin is an intrinsic property of particles which cannot be interpreted in a classical way as a rotation of some structure inside the particle. There is nothing which 'spins'.

Let's imagine the electron as a 'ball' or a 'cloud' of electrical charge which rotates around some axis and let's assume that this rotating charge produces the magnetic moment. From high-energy scattering experiments we know that the upper limit on the size of this sphere is less than 10^-19 m. The upper limit for the rotation speed is that no point is allowed to travel faster than the speed of light. Even if you assume that the whole charge is located in a thin ring around the 'equator', the angular momentum would be far too low to explain the experimentally observed spin of the electron. Any other distribution would give an even lower value.

It doesn't help: we are forced to give up the picture that the electron is a rotating object, we just have to accept that the spin is an intrinsic property of the electron with no classical explanation for it whatsoever.

(Ref.: The Charm of Strange Quarks - Mysteries and Revolutions of Particle Physics by R.M. Barnett, H. Mühry and H.R. Quinn, Springer, New York 2000.)
 

A virtual exhibition about Wolfgang Pauli and his ideas at the ETH Zürich.

An article about Paul Dirac on physicsweb.org.

The Gluon, the particle which transmits the so-called strong force, was directly observed for the first time in 1979 at DESY's storage ring PETRA. Gluons 'glue' the quarks together to form particles like the proton and the neutron. Gluons were discovered in three-jet events like the one shown in the picture above. The angular distribution of the jets proved that the gluon is a spin 1 particle.

Ref.: Donald H. Perkins, Introduction to High Energy Physics, 4th Edition, p180-183.

There is a short biography of Isaac Newton on the science pages of Wolfram Research.

The Newton Project allows you to have a look at the original publications of Isaac Newton -- online.

It's a subtle irony of history that gravitation was the first fundamental force which was mathematically described (by Isaac Newton in 1665) but till today is the one which is least understood. The standard model includes the 'graviton', the particle which is blamed for transmitting the gravitational force, but up to now nobody could prove its existence in an experiment. One reason for this is that gravitation is by far the faintest force among the four fundamental ones, much much weaker than the so-called weak force.

Enrico Fermi won the Nobel Prize for Physics in 1938.

Richard Rhodes, author of the two fascinating books The Making of the Atomic Bomb and Dark Sun, wrote an article about Enrico Fermi in Time Magazine -- worth reading!

Satyendra Nath Bose (1894-1974) was professor of physics at the University of Calcutta 1945-58. There is a long biography of Bose on calcuttaweb.com.

(c) Andrew Truscott & Randall Hulet

Today we're actually able to 'see' how the Pauli principle works at ultra-cold temperatures. Please click on the picture above to read an explanation on the NASA webpages.

An easily understandable explanation of Bose-Einstein condensation can be found on colorado.edu.

The 2001 Nobel Prize for Physics was awarded to Eric Cornell, Wolfgang Ketterle and Carl Wieman for their "achievement of Bose-Einstein condensation in dilute gases of alkali atoms and for early fundamental studies of the properties of the condensates". A nice article about their work can be found here.

To have a look on the formulas which describe the symmetric and antisymmetric wave functions and the Pauli principle click here.

The Spin of the Proton

Admittedly it took a bit to get to this point, but unfortunately it is impossible to discuss what the problem with the proton's spin is without having a rough idea what spin is. (If we talk about protons, in fact we mean both protons and neutrons, merely it is less confusing to explain it with just one particle.)

Basically we should be able to explain the spin of the proton with what we know up to now:

• We've already mentioned that the proton is a spin 1/2 particle. This is the reason why protons have to obey the Pauli principle and build up complicated atomic nuclei. If the proton had an integral spin (like zero or 1), atoms wouldn't look like they actually do, matter would collapse to something which doesn't form higher structures and certainly we wouldn't be here to wonder about that.
• The proton consists of three quarks: two up quarks and one down quark.
• These quarks also have spin 1/2 so they have to obey the Pauli principle, too. This means that two particles are not allowed to be in exactely the same state: the two up quarks have to align their spins into opposite directions. (If you remember your chemistry lessons that is what you already know from the electrons in the atomic shell.) The spins of the two up quarks would cancel due to the antiparallel alignement -- which leaves one quark over, the remaining down quark -- et voila: the down quark has also spin 1/2, so the protons spin is in fact the down quarks spin!

Indeed this was assumed for many years but then a measurement at CERN's EMC experiment in the mid-eighties showed the surprising result that the three quarks are only responsible for a small part of the proton spin. In fact, the proton is a lot more complicated than only three quarks plus Pauli principle. This is the problem HERMES is investigating.

The HERMES Experiment at DESY tries to understand how the proton spin is built up by the proton's constituents, quarks and gluons.