Nuclear shell model, spin and parity

The shell model treats the nucleus the way the Bohr model treats an atom: protons and neutrons each fill their own set of discrete energy levels, from the bottom up, obeying the Pauli exclusion principle just as electrons do. A nucleus is unusually stable whenever a shell closes completely — at the "magic numbers" 2, 8, 20, 28, 50, 82 and 126 protons or neutrons — because closing a shell leaves no loosely bound nucleon sitting exposed at a higher, less-bound level.

Pairing is the second half of the picture. Within a sub-shell, nucleons pair up with opposite orientations, and each such pair contributes zero net spin and even parity. That is why the ground-state spin of almost every even-proton, even-neutron nucleus is exactly zero: every nucleon is paired off, and nothing is left over to point in any particular direction. Odd-mass nuclei, by contrast, have one unpaired nucleon whose own orbital state fixes the whole nucleus's spin and parity — which is how physicists read off shell occupancy from measured spins.

This is also the reason light nuclei like helium-4, oxygen-16 and calcium-40 are conspicuously more stable than their neighbours a few nucleons either side: each sits at a doubly magic number, with both its proton shell and its neutron shell closed at once.