system
Given the following post, respond as if you're commenting on Reddit:
human
How can adding a neutron to atomic nuclei make them unstable?

The traditional model for atomic nuclei is that they're a balance of the strong force between nucleons and the electrical repulsion of the protons. This model cannot explain why adding in an additional neutron would make it fall apart since this only increases the effect of the strong force while letting the protons move further away from eachother to decrease electrical repulsion. In fact, the prediction is that the more neutrons, the more stable the nucleus. What is going on? Why is this not the case?


EDIT: Conclusion (that I think is correct):


Neutrons cannot be in the same state because of Pauli exclusion princible. Neutrons differ from one another via spin and energy. You can only have two neutrons of the same energy (spin up and spin down), after that an added neutron must be at a higher energy state than the last two. So therefore, every other neutron added goes into a higher energy level than the ones before. At some point this energy is so high, that a neutron decaying into a proton releases energy (neutron's energy is higher than the energy of the resulting proton (electrical repulsion) and other decay products). Please someone who actually knows refute or confirm.
gpt
You can't consider a nucleus in isolation, you have to think about what other nuclei it can break apart into. If its possible that there is an arrangement of the same neutrons and protons into two nuclei that are more stable, then the original nucleus will eventually decay to that.

For small numbers of protons and neutrons a model call the shell-model works pretty well for thinking about nuclei. In the shell model, protons and neutrons fill up energy levels just like electrons in an atom. However, protons and neutrons have separate "towers" of levels; so, no matter how empty the proton tower is, if you want to add more neutrons they have to go in the neutron tower. The neutrons higher in the neutron tower have higher energy, and thus the nuclei as a whole is less tightly bound. If it's possible for the system to break apart into two nuclei where the protons and neutrons are more tightly bound, that system will be more stable and so the original nucleus will eventually decay.