The Nuclear Shell Model

We see that the atomic shell model works even though the hydrogen states are not very good approximations due to the coulomb repulsion between electrons. It works because of the tight binding and simplicity of closed shells. This is based on angular momentum and the Pauli principle.

Even with the strong nuclear force, a shell model describes important features of nuclei. Nuclei have tightly bound closed shells for both protons and neutrons. Tightly bound nuclei correspond to the most abundant elements. What elements exist is governed by nuclear physics and we can get a good idea from a simple shell model. Nuclear magic numbers occur for neutron or proton number of 2, 8, 20, 28, 50, 82, and 126, as indicated in the figure below. Nuclei where the number of protons or neutrons is magic are more tightly bound and often more abundant. Heavier nuclei tend to have more neutrons than protons because of the coulomb repulsion of the protons (and the otherwise symmetric strong interactions). Nuclei which are doubly magic are very tightly bound compared to neighboring nuclei. $_{82}{\mathrm{Pb}}^{208}$ is a good example of a doubly magic nucleus with many more neutrons than protons.

Remember, its only hydrogen states which are labeled with a principle quantum number $n=n_r+\ell+1$. In the nuclear shell model, $n$ refers only to the radial excitation so states like the $1h_{9\over 2}$ show up in real nuclei and on the following chart. The other feature of note in the nuclear shell model is that the nuclear spin orbit interaction is strong and of the opposite sign to that in atoms. The splitting between states of different $j$ is smaller than that but of the same order as splitting between radial or angular excitations. It is this effect and the shell model for which Maria Mayer got her Nobel prize.

Another feature of nuclei not shown in the table is that the spin-spin force very much favors nucleons which are paired. So nuclear isotopes with odd numbers of protons or odd numbers of neutrons have less binding energy and nuclei with odd numbers of both protons and neutrons are unstable (with one exception).