Why does the alpha particle have zero spin?

Question

Wouldn't it have multiple total spin quantum number possibilities by spin addition of the 4 nucleons it is composed of? Why is it specifically zero? I get that they would behave as bosons due to being composed of four spin-1/2 fermions, thus resulting in 0, 1 or 2 spin. However, I don't understand how angular momentum addition comes into this.

Answer 1

To a first approximation, the alpha particle has zero spin because the only bound $ppnn$ state has spins $p\uparrow p\downarrow n\uparrow n\downarrow$ with zero orbital angular momentum (S-state).

Any other S-state arrangement of spins requires either the two protons or the two neutrons to have parallel spins, so the Pauli Exclusion Principle forces one of the nucleons into a higher orbital momentum state. Putting any of the nucleons into such a higher orbital state requires more energy than the alpha particle binding energy, so the excited state would immediately decay with the nucleon escaping.

More precisely, according to the Table of Nuclides, the neutron binding energy in a $^4$He nucleus (alpha particle) is $20578$ keV and the proton binding energy is $19814$ keV. The lowest energy $^4$He$^*$ spin-1 state has an excitation energy of $23640$ keV with a half-life of $7.4\times 10^{-23}$ seconds, and the lowest energy spin-2 state has an excitation energy of $21840$ keV with a half-life of $22.7\times 10^{-23}$seconds.

This is related to "Why don't neutrons and protons have variable half-integer spin?".

Note added: As several comments have noted, there is likely some S-D mixing in the $^4$He wave function (e.g. up to $10$% according to Doma & El-Gendy 2018). This means there is a small contribution from D-states where all four nucleons have parallel spins but this is cancelled out by opposite $l=2$ orbital angular momentum. This does not affect the general point that all $^4$He$^*$ states with spin greater than zero are unbound, but the full theoretical understanding is a bit messier than my simple S-state explanation above.

Answer 2

I would say it is related to Pauli's exclusion principle. An alpha particle is formed by two protons and two neutrons (both fermions). Since equal fermions cannot occupy equal quantum states, you need one of the protons and one of the neutrons to have spin 1/2, while the other proton and neutron have spin -1/2. This results in a net-spin value of zero. You could still have orbital angular momentum, but in processes that result in the emission of alpha-particles, those are generally ejected radially.