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I have read that, in the nuclear shell model, nucleons are distributed filling energy levels following the Pauli exclusion principle. These energy levels can be designed with the $nl_j$ notation: $1s_{1/2},1p_{3/2},1p_{1/2}, 1d_{5/2},...$

I have two questions in relation to this matter:

  • Do protons and neutrons fill the same levels or do they fill separate levels? For example, would level $1s_{1/2}$ be filled by a neutron and a proton or would there be two different levels $1s_{1/2}$, one for each particle type?
  • Is there a mnemonic rule or diagram with which to remember the order of filling of the energy levels, in the style of that for the electrons?

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Protons and neutrons fill different levels. On the one hand they are distinct fermions and on the other they evolve in a slightly different average potential because of the residual electromagnetic interaction.

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There is no obvious mnemonic device. In fact it depends a bit on how one models the nuclear potential. Phenomenologically at least, the ordering of shell model levels is given by

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which predicts the correct magic numbers, and which are the same for protons and neutrons. The ordering is based on a harmonic oscillator core potential, with a small $\ell^2$ and spin-orbit interactions added, with interaction strengths chosen to produce the correct magic numbers.

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