Where the smooth model fails
Nuclei with certain proton or neutron counts — 2, 8, 20, 28, 50, 82, and 126 — are conspicuously special. They are more abundant in nature, have unusually high energy needed to knock out a nucleon, resist capturing extra neutrons, and cluster more stable isotopes around them. These are the magic numbers, and the smooth liquid-drop formula cannot produce a single one of them.
Nucleons in a self-made well
The nuclear shell model makes a bold move: even though every nucleon interacts strongly with every other, treat each one as moving independently in the average potential produced by all the others — a mean field. Solve the quantum problem of one particle in that well (a rounded box, the Woods–Saxon potential) and its energy levels come in bunches, with gaps between them. Fill the levels with protons and neutrons separately, obeying the Pauli principle, and a closed bunch is a magic number.
There is a catch. A plain harmonic-oscillator or Woods–Saxon well reproduces the first three magic numbers — 2, 8, 20 — and then predicts 40 and 70, which are not magic. The model was tantalisingly close and stubbornly wrong above 20. Something was missing.
The spin–orbit fix
In 1949 Maria Goeppert-Mayer and independently Hans Jensen added a strong spin–orbit coupling: the energy of a nucleon depends on whether its intrinsic spin is aligned or anti-aligned with its orbital motion. Unlike the tiny spin–orbit effect in atoms, in nuclei it is large and of the opposite sign, so it splits each level with orbital angular momentum l into two, with the higher-j member pushed sharply down.
The spin–orbit energy splits j = l + 1/2 from j = l − 1/2; in nuclei the high-j level drops far enough to jump the next gap.
Those depressed high-j 'intruder' states drop across the old gaps and open new ones exactly at 28, 50, 82 and 126. With one physically motivated term, all seven magic numbers fall out. Goeppert-Mayer and Jensen shared the 1963 Nobel Prize.
What the shell model explains
The payoff is broad. The model predicts ground-state spins and parities (usually set by the last unpaired nucleon), rough magnetic moments, and which nuclei are exceptionally stable. Doubly magic nuclei — magic in both Z and N — are the standouts: helium-4 (2,2), oxygen-16 (8,8), calcium-40 (20,20), the neutron-rich calcium-48 (20,28), and lead-208 (82,126), the heaviest stable nucleus.
Two models, one nucleus
Do not let the two pictures fight. The liquid drop is a collective view — the nucleus as a whole, good for bulk energy, fission and giant vibrations. The shell model is an independent-particle view — good for spins, magic numbers and low-lying states. Real nuclei do both at once: mid-shell nuclei deform, rotate and vibrate collectively (the Nilsson and collective models), while their individual nucleons still occupy shells.