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Consider the pairing term in Weizsäcker formula. Here it is claimed that:

Due to the Pauli exclusion principle the nucleus would have a lower energy if the number of protons with spin up were equal to the number of protons with spin down.

I don't understand how Pauli exclusion principle should be the cause of this. This term comes from spin-spin interaction (or "coupling"), but I do not see the link with the fact that protons (or neutrons) with the same quantum numbers cannot occupy the same quantum state within a quantum system simultaneously

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Two protons with the same quantum numbers (other than spin) will have completely overlapping wavefunctions, and thus will have greater strong force interaction between them and stronger binding energy.

This makes it energetically favorable (having lower energy) for protons to pair in pairs of opposite spins.

In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. These particles obey the Pauli exclusion principle. Fermions include all quarks and leptons, as well as all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics.

A fermion can be an elementary particle, such as the electron, or it can be a composite particle, such as the proton. According to the spin-statistics theorem in any reasonable relativistic quantum field theory, particles with integer spin are bosons, while particles with half-integer spin are fermions.

In addition to the spin characteristic, fermions have another specific property: they possess conserved baryon or lepton quantum numbers. Therefore, what is usually referred to as the spin statistics relation is in fact a spin statistics-quantum number relation.[1]

As a consequence of the Pauli exclusion principle, only one fermion can occupy a particular quantum state at any given time. If multiple fermions have the same spatial probability distribution, then at least one property of each fermion, such as its spin, must be different.

Please see here:

https://en.wikipedia.org/wiki/Fermion

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It is nothing to do with any spin-spin interaction.

Consider protons (or neutrons) filling their shell model levels. An empty level can take a spin up (or down) proton. The next proton can be down (or up) in the same level. Adding a third proton cannot be done with this level but requires the next level up which has a higher energy. So as you add protons 1,2,3,4,5..... the energies reuired for 1,3,5... are bigger than those for 2,4,8...

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