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I have recently been learning about nuclear magnetic resonance spectroscopy. It is mentioned that the spin of the $^{1}$H nucleus is $\frac{1}{2}\ $and the spin of the $^{2}$H nucleus is $1$. This made me think that this likely has something to do with the number of protons and neutrons and this is indeed the case. This site reveals that:

  1. If the number of neutrons and the number of protons are both even, then the nucleus has no spin.
  2. If the number of neutrons plus the number of protons is odd, then the nucleus has a half-integer spin (i.e. $\frac{1}{2}\ $, $\frac{3}{2}\ $, $\frac{5}{2}\ $).
  3. If the number of neutrons and the number of protons are both odd, then the nucleus has an integer spin (i.e. $1$, $2$, $3$).

However, I was unable to find a site which stated the exact formula needed to determine the spins of atomic nuclei. Thus, I would like to ask if such a formula exists or is the determination of nuclear spin only done through experiment?

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    $\begingroup$ There is no formula. $\endgroup$ – Ivan Neretin Dec 19 '17 at 10:36
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    $\begingroup$ @IvanNeretin So the nuclear spin of an atomic nucleus can only be known through experimental means? $\endgroup$ – Tan Yong Boon Dec 19 '17 at 11:27
  • $\begingroup$ @Ivan what happens if we try to use nuclear shells? $\endgroup$ – Oscar Lanzi Dec 19 '17 at 11:56
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    $\begingroup$ @OscarLanzi It works poorly. The system is extremely computationally complicated and involves relativistic effects and strong spin-orbit coupling. Add that nuclon-nuclon interactions are a shadow of quark-quark interactions and things become... sketchy. $\endgroup$ – permeakra Dec 19 '17 at 12:12
  • $\begingroup$ I'm afraid you could as well ask how to get spin of arbitrary molecule and it's too broad. Also are you interested in spins of ground states? Nuclei can be excited just as molecules can be. $\endgroup$ – Mithoron Dec 19 '17 at 21:11

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