I keep reading that
Nuclei with even number of protons and of neutrons have a net spin of zero.
This I understand, as to have the lowest energy state, by the Pauli exclusion principle, all protons will pair wth one of each pair being spin up and th other spin down; and all neutrons will pair as well.
The next 'rule' is
Nuclei with odd number of protons and neutrons will have integer spin of 1,2,3….
In all of the sources I have read, I have not seen spin of 0 mentioned here. This is my first point of confusion. Could it not be that the net proton and net neutron aline with opposite spins to give 0? Perhaps the answer lies in considering energy- maybe the nucleus is in a lower energy state if the spins of the net proton an neutron are aligned? But then surely a spin-zero excited nucleus CAN exist.
And when considering energy, I do not see how we can achieve nuclear spin states of 2, 3 etc. This would imply that there are some protons/neutrons which could pair with another of the same nucleon, but are not doing so? Why would this be energetically favourable, and a spin 0 nucleus is not?
Final 'rule':
A nucleus with odd number of protons or neutrons but the other nucleon has an even number has integer spin: 1/2, 3/2, 5/2…
Again, my confusion with this rule is that for spin magnitudes greater than 1/2, there must be nucleons which could be pairing but are not?
Best Answer
Nuclear angular momentum includes not only nucleon spin but also nucleon orbital angular momentum. The rules developed for building a nucleus in the nuclear shell model therefore bear a lot of resemblance to Hund's rules for filling electron shells, although the pairing interaction is stronger among nucleons (for which there are two species) than among electrons.
Here's a hand-waving way to think of it:
As you say, even-even nuclei like to relax to spin and parity $J^P = 0^+$.
An even-odd or odd-even nucleus must have half-integer $J$, because of the spin of the unpaired nucleon. If the extra nucleon also carries orbital angular momentum — for instance, if the extra nucleon is in a $p$-shell or a $d$-shell — the orbital angular momentum must be added to the total spin of the nucleus. Here are a couple of examples where you can think of a single extra nucleon orbiting a $0^+$ core.
An odd-odd nucleus must have integer $J$, but not necessarily $J=0$. One model is an even-even core orbited by a deuteron.