What is used to measure the spin of a particle? I was wondering what is the specific system or method that is used to measure spin of a particle? e.g. In a lab what would they use to tell what a particles spin is?
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 A: First, it is worth mentioning that while the word measurement is used, the experimental process involved actually changes the state rather than revealing some preexisting property.
This can be known by considering an interaction for the z component followed by the z component again followed by the x component as compared to the interaction for z component followed by the x component again followed by the z component. In the first case the two results for the z component always agree, so the interaction for the z component apparently n leaves the particle on a state that gives 199% determined results for z component interactions but the x component interaction changes it from that state to a new state.
So the spin interactions literally change the state.
That said, the main idea is that you first pick a direction. Then you arrange an interaction that interacts with the different states in different ways such that when repeated, they don't get split further. Then you split the state into those states, then eigenstates, and do it in a way that allows the different states to become so separated that each is capable of acting like it is the one state in the universe. At this point it is acceptable to say a measurement has occurred.
A Stern-Gerlach device is a common device that can do this interaction.
A: Once the stable particles , electron and proton have had their spin determined by the stern gerlach method as discussed in the other answer, one can start building up the spins of the elementary particles and the resonances.
The spins of the particles have been determined by the  angular distributions of decay products.
Example: the Higgs has been declared the Higgs because it is a scalar

Examining decay patterns. Spin-1 had been ruled out at the time of initial discovery by the observed decay to two photons (γ γ), leaving spin-0 and spin-2 as remaining candidates.



Distribution of |cosθ⁎| for events in the signal region defined by 122GeV(lt)mγγ(lt)130GeV. The data (dots) are overlaid with the projection of the signal (blue/dark band) and background (yellow/light histogram) components obtained from the inclusive fit of the data under the spin-0 hypothesis.

The various analysis use extensively also  conservation of angular momentum, i.e. knowing the spins of all the other particles except the one under study. 
