Undergrad Shell Model: Filling Up Nuclei

[pierce15]

I'm a bit confused about how things fill up in the shell model. Here is my understanding: the potential we are working with is the net potential due to the other nucleons that a proton or neutron feels throughout the nucleus; it is a combination of a square and harmonic oscillator potential. When this is solved (not accounting for spin-orbit coupling, yet), we get energy levels that depend on the quantum numbers $n_r$ and $l$.

What I don't understand is how this is used to "build up" the nucleus. I have a hard time believing that when accounting for spin-orbit coupling, the levels always fill up in the order spdsfpgdshfpig for every nucleus, since the potential changes (radius of well $\propto A^{1/3}$). I recall that while studying the related Hartree-Fock method for atoms, the order of filling of inner subshells changed dramatically while increasing in atomic number -- i.e. the Aufbau principle is actually not a good rule, and the order of filling can't be accurately determined without a graph/table on hand.

Of course, this is an important question since if the levels aren't the same for everything, then the "magic numbers" for that potential won't be the same magic numbers for everything -- i.e. a different nucleus will have different states which are unexpectedly stable.

So is it really the case that everything has the same energy levels/magic numbers? As much as I hate exceptions to the rules, I'm a bit suspicious.

 

[drvrm]

I'm a bit confused about how things fill up in the shell model.

one must consider the a sizable spin-orbit interaction which
splits the levels by an energy difference which depends on orbital quantum number.
as l increases the split-energy also increases.

This leads to the overlapping levels .say e.g. 1d3/2 going above 2s , 1f1/2 going above 2p 3/2 so the filling up order changes
so if a fresh energy level diagram with spin orbit coupling is drawn then the magic number nuclei can emerge as closed shell structures.

 

[e.bar.goum]

I'm a bit confused about how things fill up in the shell model. Here is my understanding: the potential we are working with is the net potential due to the other nucleons that a proton or neutron feels throughout the nucleus; it is a combination of a square and harmonic oscillator potential. When this is solved (not accounting for spin-orbit coupling, yet), we get energy levels that depend on the quantum numbers $n_r$ and $l$.

What I don't understand is how this is used to "build up" the nucleus. I have a hard time believing that when accounting for spin-orbit coupling, the levels always fill up in the order spdsfpgdshfpig for every nucleus, since the potential changes (radius of well $\propto A^{1/3}$). I recall that while studying the related Hartree-Fock method for atoms, the order of filling of inner subshells changed dramatically while increasing in atomic number -- i.e. the Aufbau principle is actually not a good rule, and the order of filling can't be accurately determined without a graph/table on hand.

Of course, this is an important question since if the levels aren't the same for everything, then the "magic numbers" for that potential won't be the same magic numbers for everything -- i.e. a different nucleus will have different states which are unexpectedly stable.

So is it really the case that everything has the same energy levels/magic numbers? As much as I hate exceptions to the rules, I'm a bit suspicious.

No, it's not really the case that all nuclei have the same energy levels or magic numbers! For instance, the N=20 shell closure disappears for neutron rich isotones. ( http://www.sciencedirect.com/science/article/pii/0370269392913209 and http://www.sciencedirect.com/science/article/pii/037026939500012A ). Have you come across the Nilsson model? You'll see very quickly that energy levels (and the order of filling) changes rapidly away from spherical nuclei. Here's rather a good set of slides.