Two forces. Two protons repel each other with the electrostatic (Coulomb) force, F = 1.44 MeV fm / r². At 1 fm that is about 230 N, which is enormous for two particles that weigh 10⁻²⁷ kg. Yet nuclei hold together, because between nucleons (protons and neutrons) there is another force that is much stronger.

The strong nuclear force is about 50 to 100 times stronger than the Coulomb force at 1 fm (about 10⁴ N). It has three important features. It is short-ranged: it is negligible beyond about 3 fm, while the Coulomb force decreases only as 1/r². It is attractive from about 0.8 fm out to about 3 fm, which holds the nucleus together. And it is repulsive at shorter distances, below about 0.8 fm, which keeps nucleons from collapsing into each other (the “hard core”). It acts equally between protons and neutrons, since the strong force does not care about charge.

V(r) = [ −626.9 e^{−1.55 r} + 1438.7 e^{−3.11 r} ] / r MeV (r in fm), F = −dV/dr

Where does the shape come from? Hideki Yukawa proposed in 1935 that nucleons exchange particles. A particle of mass m can exist for a time Δt ≈ ℏ/(mc²) (the uncertainty principle), so it can travel no further than about ℏc/mc². For the pion (139.6 MeV) this is 1.41 fm. The potential of an exchanged particle has the form e^−r/range/r, a Yukawa potential. The model on this page combines an attractive Yukawa term of longer range with a repulsive one of shorter range (the Malfliet–Tjon potential, which describes the average of the nucleon–nucleon force and reproduces the binding of the deuteron). The real force also depends on spin and on the direction between the nucleons, and underneath it all, it is the strong force between the quarks, carried by gluons.

Why nuclei need neutrons. The strong force acts only between close neighbours, but every proton repels every other proton in the nucleus. As nuclei get larger the electric repulsion adds up and the strong force does not, so large nuclei need more neutrons than protons, which add attraction without any repulsion. For two protons alone the attraction is not enough: the “diproton” does not exist, but the proton–neutron pair (the deuteron) is bound.

The forces are drawn as arrows on a logarithmic scale, because the strong force is hundreds of times larger than the electric force. The nucleons are drawn with the proton charge radius of 0.84 fm. They overlap when they are closer than 1.7 fm.