Why is there a limit to the number of neutrons in a nucleus?

[BillKet]

Hello! Why can't we have as many neutrons as we want inside a nucleus? I understand that for protons you have the Coulomb repulsion, but what leads to an increase of energy when adding more neutrons (which in turns lead to beta decay or fission)?

 

[Gaussian97]

Nuclear physics is not an easy subject and you need some advanced quantum mechanics to really understand some things, but you can think this way: Is not the fact that what you can have or not, you want stability, and that is, if there is some possible decay that will lead to a less energy state, you probably won't find this in nature. Then, neutrons have a bigger mass than protons, and with beta decay, you can transform a neutron to a proton (emitting other particles). Then seams reasonable to search some equilibrium between having sufficient neutrons to keep the nucleus hold together but having the minimum mass.

You can read about the semi-empirical mass formula that predicts $$\frac{N}{Z}\approx1+0.02(N+Z)^{2/3}$$ where $N,Z$ are the number of neutrons and protons respectively.

 

[BillKet]

Nuclear physics is not an easy subject and you need some advanced quantum mechanics to really understand some things, but you can think this way: Is not the fact that what you can have or not, you want stability, and that is, if there is some possible decay that will lead to a less energy state, you probably won't find this in nature. Then, neutrons have a bigger mass than protons, and with beta decay, you can transform a neutron to a proton (emitting other particles). Then seams reasonable to search some equilibrium between having sufficient neutrons to keep the nucleus hold together but having the minimum mass.

You can read about the semi-empirical mass formula that predicts $$\frac{N}{Z}\approx1+0.02(N+Z)^{2/3}$$ where $N,Z$ are the number of neutrons and protons respectively.

Oh I see. So basically the extra mass of the neutron must add more energy than the mass of the proton plus the coulomb repulsion (roughly), in order for beta decay to happen, right?

 

[Gaussian97]

Oh I see. So basically the extra mass of the neutron must add more energy than the mass of the proton plus the coulomb repulsion (roughly), in order for beta decay to happen, right?

Is one argument, another argument, slightly more accurate, is using the Pauli exclusion principle, that says that two identical fermions (the proton and the neutron are fermions) cannot be in the same quantum state.

Therefore since the energies are quantized, if you add a lot of neutrons they will go to higher energy states, while the protons will go to states with lower energies (since if there are few protons, they will not be occupied), so adding a neutron will be less efficient than adding a proton.

 

[snorkack]

Hello! Why can't we have as many neutrons as we want inside a nucleus? I understand that for protons you have the Coulomb repulsion, but what leads to an increase of energy when adding more neutrons (which in turns lead to beta decay or fission)?

You have Fermi repulsion. Since the strong force potential hole is small, with only very shallow tails, you can only have a finite number of states in it - unlike electrostatic monopole-monopole attraction whose long tail of attraction allows an infinite number of states. With fermions, you can fill all of them. Example He-5. Neither an extra proton nor an extra neutron can be bound to an alpha.