Can anyone give a simple explanation for what the fractions and integers mean in particle physics when describing spin? I've seen on another forum (the naked scientist) that it should not be thought of as angular momentum as described for spinning objects like the earth, so I don't understand what it actually is.
I also don't understand how it was measured from collisions i.e. if it was calculated from the momentum and energy of the collisions of particles?
Spin in quantum mechanics is an intrinsic component of the angular momentum of a quantum system, as might be measured via the magnetic moment of that system (e.g. a Stern-Gerlach experiment). Since angular momentum has things to say about the rotational symmetry of a quantum system, spin is also related to the probability distribution and magnetic moment of that system. Viewing spin in this not-so-intuitive way avoids the awkwardness of trying to build a misleading picture of little spinning tops! The "spinning material" view falls over when it hits issues like how spin angular momentum generates twice as much magnetic moment as "normal" orbital angular momentum...
Quantised orbital angular momenta (which is what spin is one contributor to) can be viewed as related to rotational symmetry in an inverse way: spin 0 systems look the same under any amount of rotation, spin 1 systems look the same when rotated by a full turn, and spin 1/2 weirdly requires two full turns before they "look" the same as they did at the start. There's no classical analogue for that, but the maths is consistent.
For your last question about how it's measured -- angular momentum is conserved, so the spins of particles entering and leaving a collision have to be correlated. The spin thus influences the angular distributions of the outgoing particles, which can be measured.
In classical mechanics, a charge going round a circle produces a magnetic field and has a magnetic dipole moment. We can thus relate angular momentum of the charge with the magnetic dipole moment.
Moving to quantum mechanics, some particles have an intrinsic magnetic moment, the way they could be having an electric charge or a rest mass. For such particles, it helps to interpret the intrinsic magnetic moment in terms of an intrinsic angular momentum. Spin is another name to the intrinsic angular momentum.
Instead of trying to visualize an electron spinning, you may just accept the experimental fact that it has an intrinsic magnetic moment and therefore spin.
The relation between intrinsic magnetic moment and spin angular momentum is not just a mathematical construct. The spin part of angular momentum (S) is essential to ensure conservation of angular momentum. By that, I mean, orbital angular momentum (L) alone is not conserved but the sum of it (J = L + S) and spin, called total angular momentum is conserved.
