Two protons, side by side, are a study in contradiction. Each carries a positive charge, and like charges repel — pushed together to within a femtometre (1\ \text{fm} = 10^{-15}\ \text{m}, roughly the width of a proton) they shove each other apart with a Coulomb energy of a few hundred keV. And yet the nucleus of every atom heavier than hydrogen is a crowd of protons packed exactly that closely, and it does not fly apart. Something is holding it together — something far stronger than electromagnetism, but with a reach so short it never leaks out to your everyday world.

That something is the nuclear force (more precisely the residual strong force between nucleons). This page is about that force and, because no single mental picture captures a nucleus, about the two great models we use to tame it: the liquid-drop model, which treats the nucleus as a wobbling charged raindrop, and the shell model, which treats each nucleon as a wave in an orbit. They sound like rivals. They are in fact partners, each seeing a different face of the same object.

How strong, and how a Coulomb estimate sets the scale

Let us pin down the numbers that make a nucleus paradoxical. The electrostatic (Coulomb) potential energy of two point charges z_1 e and z_2 e a distance r apart is, in the handy nuclear units of MeV and femtometres,

For two protons (z_1 = z_2 = 1) touching at r \approx 1\ \text{fm} this gives about 1.44/1 \approx 1.44\ \text{MeV} of repulsion. (People often quote a smaller figure, \sim 0.23\ \text{MeV}, for protons spread over the whole nucleus rather than in contact — either way it is a fraction of an MeV.) Compare that to the binding energy: each nucleon in a typical nucleus is bound by about 8\ \text{MeV}. The attractive nuclear force is winning by roughly an order of magnitude at these separations — and it must, or matter as we know it would not exist.

But push the two nucleons even closer, below about 0.5\ \text{fm}, and the story flips again: the force turns fiercely repulsive. This "hard core" is why a nucleus is nearly incompressible, with a density that barely changes from the lightest to the heaviest — every nucleus is packed to the same \sim 0.17\ \text{nucleons/fm}^3.

The shape of the force: a well with a wall

Plotting the potential energy of two nucleons against their separation r makes every property visible at once. Far apart, the curve is flat at zero — the force has no reach. Coming inward it plunges into a deep attractive well around 1\ \text{fm}, roughly 50\text{–}100\ \text{MeV} deep. Closer still it rears up into a repulsive core. Laid on the same axes, the Coulomb repulsion between two protons is almost invisible: a gentle 1/r rise dwarfed by the nuclear well.

Read off the four headline properties from the picture: the force is short-ranged (dies by 2\ \text{fm}), strongly attractive at nucleon spacing, has a repulsive core at very short range, and is far bigger than the Coulomb force it overwhelms.

Charge independence and saturation

Two more properties do not show on the graph but matter enormously. First, charge independence: strip away the electrostatic part and the nuclear force between two protons, two neutrons, or a proton and a neutron is very nearly the same. The nuclear force does not care about electric charge at all — to it a proton and a neutron are two states of one particle, the nucleon. (It does care about spin: two nucleons attract more strongly when their spins are aligned, which is why the deuteron, a proton and neutron with parallel spins, is bound, while two nucleons with opposite spins fail to bind.)

Second, saturation. Because the force is so short-ranged, a nucleon deep inside a nucleus feels only its immediate neighbours — not the whole crowd. Add more nucleons far away and they contribute nothing to its binding. Each nucleon therefore binds roughly the same fixed number of neighbours, so the total binding grows in step with the number of nucleons:

Contrast this with a force of unlimited range, like gravity or Coulomb, where every particle attracts every other: there the count of interacting pairs grows like A^2/2, and binding per particle would keep climbing with size. The fact that B/A is instead flat across the whole chart of nuclides is direct fingerprint evidence that the nuclear force saturates. It behaves, in this respect, exactly like the short-range attractions between molecules in a drop of water — which is the clue that launches our first model.

Model 1: the liquid drop

Saturation and incompressibility are precisely the traits of a liquid: molecules in water also attract only their nearest neighbours, and water too resists being squeezed. So Bohr, Gamow and Weizsäcker pictured the nucleus as a tiny, incompressible, positively charged drop of nuclear fluid. From that single analogy the semi-empirical mass formula for the total binding energy falls out term by term:

• The volume term a_V A is saturation itself: bulk binding proportional to the number of nucleons (a_V \approx 15.8\ \text{MeV}).
• The surface term is surface tension: nucleons at the drop's skin have fewer neighbours, so they are under-bound — a correction scaling with surface area A^{2/3}.
• The Coulomb term is the proton–proton repulsion spread through the drop, the one long-range force that grows like Z^2 and eventually limits how big a nucleus can be.

This drop picture explains the bulk facts beautifully: the average binding per nucleon, the overall shape of the binding-energy curve, and — most dramatically — fission. A heavy drop, straining under Coulomb repulsion, can be nudged into an oscillation; stretch it into a dumbbell and surface tension can no longer pull it back, so it splits into two smaller drops, releasing energy. The liquid-drop model is what let Bohr and Wheeler explain nuclear fission in 1939. What it cannot explain is why certain specific nuclei are freakishly more stable than their neighbours. For that, you must abandon the featureless drop and look at individual orbits.

Model 2: shells and magic numbers

Electrons in an atom fill quantized orbitals, and closing a shell (the noble gases) buys extraordinary chemical stability. Nucleons, it turns out, do the same thing. In the shell model each nucleon moves not in the jagged detail of every pairwise force but in the smooth average (mean-field) potential made by all the others — roughly a rounded well. Solve for the allowed energy levels and they bunch into shells separated by large gaps. Fill a nucleus exactly up to a gap and you get a closed shell that is unusually tightly bound and hard to disturb.

A naïve well gives the wrong gaps. The breakthrough, from Maria Goeppert Mayer and Hans Jensen in 1949, was a strong spin–orbit coupling: a nucleon's energy depends on whether its spin is aligned with or against its orbital motion, and this splitting is large enough to reshuffle the levels. With spin–orbit coupling in place, the shell closures land at exactly the numbers experiment had been shouting about — the magic numbers.

How do we know these numbers are special? Three independent lines of evidence, all pointing the same way:

• Extra binding & abundance. Magic nuclei sit above the smooth binding-energy trend and are unusually common in the cosmos (tin, with a magic Z=50, has more stable isotopes than any other element).
• Separation-energy jumps. The energy to remove one neutron drops sharply after a magic number — just as ionization energy plunges after a noble gas. You have to pay extra to break open a closed shell.
• Reluctance to react. Magic nuclei have small neutron-capture cross-sections and often no low-lying excited states — a closed shell is hard to excite.