Question

I have a silly confusion. The Pairing term of the semi-empirical mass formula tells that nucleons tend to pair up. I believe that pairing up means forming a state in which two nucleons combine to give a total S=0 (or total J=0?). Pairing increases the binding energy.

On the other hand, we know that nuclear forces favour the alignment of spin i.e. S=1 is favoured over S=0.

Don't these two statements contradict each other and say completely opposite things?

Answer 1

In atomic orbitals, electrons follow the Pauli exclusion principle and fill orbitals with opposite spins. However, in the nucleus, nucleons such as protons and neutrons pair in a spin-0 state for increased stability despite nuclear forces favoring aligned spins when nucleons are apart. This pairing increases the binding energy of the nucleus.

Step-by-step explanation:

The confusion arises from conflating the behavior of nucleons in nuclei with the behavior of electrons in atomic and molecular orbitals. In atomic orbitals, the Pauli exclusion principle mandates that no two electrons may occupy the same quantum state, thus requiring that electrons fill orbitals in such a way that they have opposite spins when they occupy the same orbital.

In contrast, the nuclear forces in nucleons (protons and neutrons within an atomic nucleus) favor aligned spins (spin 1) when apart, but in a bound state, nucleons in a nucleus pair up in a spin-0 state for increased stability due to the strong force. This pairing results in a lower energy state and thus a more tightly bound nucleus, reflected by an increased binding energy.

The semi-empirical mass formula includes a pairing term that represents this stability gained from nucleon pairing.

This can appear contradictory because, in isolation, nucleons prefer aligned spins due to the nuclear force, but within the nucleus, paired nucleons (in a spin-0 state) lead to greater stability, a factor that's particularly relevant for nuclei with even numbers of protons and/or neutrons.

Thus, in the context of the nucleus, the pairing term reflects the balance of forces resulting in a stable, paired configuration with a lower energy state, even though individual nuclear forces might favor spin alignment when nucleons are not paired.