Nuclear shell model and magic numbers

In a spherical field, label an orbital by a radial index, orbital angular momentum l, and total single-particle angular momentum j. For a spin-one-half nucleon, j is l plus or minus one half, except that l zero permits only one half. Each j orbital has two j plus one magnetic substates. Pauli exclusion applies separately to protons and neutrons: the same spatial-spin orbital can contain one proton and one neutron because their internal species states differ.

A completely filled j multiplet is rotationally invariant and has zero total angular momentum. For an even-even closed core with positive parity, one additional valence nucleon has the spin and parity of its orbital in the extreme independent-particle approximation. A single hole in a filled orbital carries the same j and parity labels. Residual coupling to core excitations can change the content of the actual state, so these assignments are conditional predictions rather than an automatic rule for every odd nucleus.

The isotropic three-dimensional oscillator provides a useful solvable reference. Its major shell number K counts total Cartesian oscillator quanta. The number of triples of nonnegative integers summing to K is half of K plus one times K plus two. Including two spin states doubles that count. Filling major shells gives cumulative populations two, eight, twenty, forty, and so on for one species, before adding spin-orbit splitting or changing the central potential.

The early agreement with familiar magic populations does not establish the oscillator as an exact nuclear Hamiltonian. Near ordinary stability, prominent closures include two, eight, twenty, twenty-eight, fifty, eighty-two, and one hundred twenty-six. The higher sequence requires substantial changes in level ordering, especially the nuclear spin-orbit interaction. Far from stability even familiar gaps can weaken or new ones can appear. A magic number summarizes observed or predicted shell behavior in a region, not a topological invariant of the nucleon count.

A practical valence-space calculation selects orbitals outside a core and diagonalizes an effective Hamiltonian among many allowed occupation configurations. Even if each basis configuration resembles independent filling, the eigenstate may be a superposition with no unique last-particle orbital. Good total angular momentum and parity remain useful when the Hamiltonian preserves those symmetries. Two or more valence nucleons require angular-momentum coupling and exchange constraints, not simple addition of their individual spin labels.

The core and truncated space also influence transition operators. Polarization of excluded orbitals can enter effective interactions and effective electromagnetic charges. Parameters fitted in one region should not be transported unchanged without checking their domain. Comparing spectra alone can conceal an inadequate transition operator or missing configuration mixing. Mass differences, transfer strengths, moments, and transition probabilities provide complementary tests of whether the selected space captures the relevant structure.