What gives an atom its atomic spin?

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What gives an atom its atomic spin?
 

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How deeply do you want this question answered? An atom is given its spin by the spin of its constituent particles. Protons and Neutrons are fermions, as are the electrons, all of which have spin 1/2. These constituent particles' spins combine quantum mechanically to give you the atom's total spin.
 
In addition to the intrinsic spin of the constituents, there is also the orbital angular momentum which also adds to the total spin of an atom.