Although nucleons interact strongly, many discrete nuclear properties can be understood by treating each proton or neutron as moving approximately in an average potential created by all the others. This is the nuclear shell model.

The allowed single-particle states are organized by orbital angular momentum and spin. A strong spin-orbit interaction splits states according to total angular momentum $$\mathbf j=\mathbf \ell+\mathbf s.$$ Filling proton and neutron states subject to the Pauli principle produces especially stable closed shells at the observed magic numbers $$2,\ 8,\ 20,\ 28,\ 50,\ 82,\ 126.$$ Nuclei with magic proton or neutron numbers often have unusually high binding, characteristic excitation spectra and reduced tendency to deform.

For even-even nuclei whose proton and neutron states are both paired, the ground-state angular momentum is often $0$. In nuclei with one unpaired nucleon outside a closed shell, the ground-state spin and parity are frequently dominated by that nucleon's orbital quantum numbers.

The shell model emphasizes discrete single-particle quantum structure. Other nuclear models emphasize smooth collective properties such as overall binding, deformation and surface behavior. Real nuclei display both kinds of physics, so no single simplified model captures every observable equally well.

Magic numbers show that the nucleus is not a featureless drop of matter. Quantized orbital structure, spin, exclusion and interactions leave experimentally visible signatures in nuclear stability and spectra.