The strong force and the push-apart

Look at a nucleus such as helium-4: two protons squeezed into a space about 3 fm across. Protons are positive, so they repel each other.

Let us put a number on it. Two protons 1 fm apart push on each other with an electric force of

F = k e² / r² ≈ (8.99 × 10⁹) × (1.60 × 10⁻¹⁹)² / (10⁻¹⁵)² ≈ 230 N.

That is about the weight of a 23 kg mass, acting on one proton. What about gravity between them? It comes to about 2 × 10⁻³⁴ N, which is around 10³⁶ times weaker. Gravity is irrelevant in a nucleus.

Something else must bind the nucleus together, and it must be stronger than 230 N at that distance. That something is the strong nuclear force.

What the strong force is like.

• It is attractive between nucleons when they are about 1 to 2 fm apart, and it is much stronger than the electric repulsion there.
• Its range is very short: by about 2 to 3 fm it has died away. Beyond that distance the electric repulsion wins.
• At less than about 0.5 to 0.7 fm it turns repulsive, which stops the nucleons from collapsing into each other.
• It acts between protons and neutrons alike. It is nearly charge-blind: the force between two protons, two neutrons or a proton and a neutron is about the same. (Spin details mean that a bound pair of two protons or two neutrons does not exist, while a proton-neutron pair does. That is the deuteron.)

Deeper down, nucleons are made of quarks held together by gluons. The force between nucleons is a leftover, or "residual", of that quark force, a little like the way neutral molecules still attract each other.

Why heavy nuclei need extra neutrons. The strong force has such a short range that a nucleon only feels the few neighbours touching it. Adding more nucleons does not make each one bind harder. The electric repulsion is different: every proton pushes on every other proton, however far across the nucleus, and the total grows roughly with the square of the number of protons.

Neutrons help because they add strong-force attraction without adding any repulsion. They spread the protons out and act like glue. That is why light nuclei such as calcium-40 (20 protons, 20 neutrons) have about equal numbers, while lead-208 has 82 protons and 126 neutrons: N/Z ≈ 1.5. Past lead, no amount of neutron glue is enough, and every nucleus is unstable.