Nuclear shell model

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The nuclear shell model describes atomic nuclei as groups of protons and neutrons that fill discrete energy levels, like electrons in an atom. The Pauli exclusion principle ensures each level can hold only a certain number of nucleons. When a shell is full, the nucleus is unusually stable.

Key ideas:
- Magic numbers: 2, 8, 20, 28, 50, 82, 126 (and possibly 184) are the numbers of nucleons that complete shells and give extra stability. Protons and neutrons have their own shells, so a nucleus can be “magic” for protons, neutrons, or both (doubly magic).
- Independent shells: The proton shells and neutron shells fill separately, which leads to different possible combinations and kinds of stable nuclei.

How the shells are built:
- A starting average potential: Nucleons move in a mean field that is between a square well and a harmonic oscillator. A spin-orbit term is added, and this strong interaction changes the ordering of levels.
- Spin-orbit splitting: This interaction shifts energies so that high-angular-momentum states can move down in energy, producing the observed magic numbers. Realistic descriptions use potentials like Woods–Saxon to better match experiments.
- Filling rule: Protons fill the lowest available states, then neutrons fill theirs. A nucleus with a full outer shell for protons or neutrons is more tightly bound.

Intruders and islands:
- The simple ordering changes when spin-orbit coupling and more realistic potentials are included, creating “intruder” levels from higher shells that mix into lower ones. This reshapes shell sizes and explains new magic numbers.
- Predictions include the so-called island of stability around very heavy nuclei, where certain numbers of protons and neutrons may produce extra stability (e.g., a possible neutron magic number near 184).

Beyond the basic picture:
- Residual interactions: For nuclei with several valence nucleons outside a closed shell, additional two-body interactions are important. These interactions mix different configurations and lift degeneracies. Calculations are done in a reduced model space with an effective Hamiltonian tuned to the problems being studied.
- No-core shell model: A more fundamental approach that treats all nucleons as active, requiring three-body forces to match experiments.
- Ground states and excited states: The shell model does a good job predicting ground-state spins and parities and often gives insight into some excited states, but the exact ordering of excited levels can be more complex.
- Deformed nuclei and rotational bands: Some nuclei are not spherical. The Nilsson model modifies the shell picture by allowing ellipsoidal (deformed) shapes. A cranking term helps describe rotation and explains rotational bands seen in nuclei.

Related ideas:
- Magnetic moments: Often estimated from the last unpaired nucleon, with real nuclei showing mixtures of possibilities.
- Connections to other models: The shell model underpins ideas that lead to the interacting boson model and other approaches; modern methods include the no-core shell model and other ab initio techniques.

Example:
- Oxygen-17 has eight protons and eight neutrons filling the first three shells, plus one extra neutron. That lone neutron determines the nucleus’s spin and parity, typically giving a specific positive-parity, nonzero total angular momentum.

In short, the nuclear shell model explains why nuclei have certain stable numbers of nucleons, how spins and parities arise, and how more complex behavior like deformation and collective rotation fits into a unified framework.