The Nuclear Force and the Shell Model

The hydrogen atom yields to the Schrödinger equation because the electron-proton potential $V(r)=-k{e}^2/r$ is known exactly. The deuteron, a single proton bound to a single neutron, does not: the nucleon-nucleon potential has no known closed form. What holds the nucleus together must nonetheless be strong. Two protons $1\mathrm{fm}$ apart repel with Coulomb energy

yet removing a nucleon from ${}^4\mathrm{He}$ costs about $20\mathrm{MeV}$. The binding force is attractive and far stronger than electromagnetism — the strong or nuclear force.

Properties of the nuclear force

Scattering experiments (proton-proton, neutron-proton) cannot invert to give the force law uniquely, but they constrain its features sharply enough to characterize it.¹

• Strong: about $10^2$ times the Coulomb force at nuclear separations. A square-well estimate of the well depth from the ground-state energy of a nucleon confined to $a\approx 2\mathrm{fm}$ gives $V\approx 50\mathrm{MeV}$, against $0.72\mathrm{MeV}$ of Coulomb repulsion at the same distance.
• Short-range: attractive within about $2.5\mathrm{fm}$, dropping to essentially zero beyond $3\mathrm{fm}$.
• Charge-independent: the $n$-$p$, $n$-$n$, and (Coulomb-subtracted) $p$-$p$ potentials are the same. Proton and neutron are two charge states of one particle, the nucleon.
• Saturated: each nucleon interacts with only a fixed number of nearest neighbors, which is why $B/A$ is nearly constant and the density is uniform.
• Spin-dependent with a hard core: strongly repulsive within about $0.5\mathrm{fm}$, preventing the nucleons from collapsing and keeping the central density constant as $A$ grows.

The exchange force

Yukawa (1935) asked what mechanism produces such a force and answered by analogy with electromagnetism. In quantum theory a charge continually emits and absorbs virtual photons, and the exchange of a virtual photon between two charges is the Coulomb force. A charge may emit a photon of energy $\Delta E=hf$ without violating energy conservation provided it lives no longer than the uncertainty-principle time $\Delta t=\hslash /\Delta E$. In that time the photon reaches at most

For a massless photon $\Delta E$ can be arbitrarily small, so $R$ is infinite — the electromagnetic force has unlimited range.

Yukawa proposed the nuclear force is the exchange of a massive virtual particle, the meson. Giving the exchanged particle rest energy $m{c}^2$ makes $\Delta E\ge m{c}^2$, so the range is finite:

This is the meson's reduced Compton wavelength. Inverting, the known $\approx 1\mathrm{fm}$ range implies

A particle of this mass, the pion, was found in cosmic rays in 1947 with $m{c}^2=140\mathrm{MeV}$, in three charge states ($+e$, $0$, $-e$) as charge-independence requires. The exchange of a charged pion swaps the nucleons' identities; a neutral pion leaves them unchanged.¹

A static exchange field obeys the relativistic Klein-Gordon equation, whose time-independent solution is the Yukawa potential

an exponentially screened Coulomb form. As $m\to 0$, $R\to \infty$ and $\Phi \to A/r$, recovering the Coulomb potential. The screening length $R$ sets the force range.

In the modern picture the pion is a quark-antiquark pair and the underlying force between quarks is carried by the gluon, developed in fundamental interactions.

The shell model

The smooth liquid-drop binding energy misses sharp local structure. The binding energy of the last neutron, $M[(A-1),Z]+m_n-M(A,Z)$ measured against the mass-formula prediction, jumps at $N=20,28,50,82,126$: these neutrons are much more tightly bound than the next one added. The same numbers appear as drops in the neutron-capture cross section and as extra stable isotopes.

These are the magic numbers. The success of an independent-particle model — each nucleon moving in an average potential produced by all the others — is surprising given how strongly the nucleons interact. The exclusion principle rescues it: in the ground state the low levels are filled, so a collision that would merely swap two nucleons into occupied states is forbidden, and only nucleons near the top filled level (the nuclear Fermi level) can scatter. Most nucleons therefore orbit almost freely.

A plain square well gives the wrong magic numbers. Mayer and Jensen (1949) fixed this with a strong spin-orbit coupling: the spin-dependence of the nuclear force lowers a level when a nucleon's spin and orbital angular momentum are parallel and raises it when antiparallel, following $j$-$j$ coupling rather than the $L$-$S$ coupling of atomic spin-orbit. The splitting rearranges the level ordering so that the large gaps fall exactly at $2,8,20,28,50,82,126$.

The two models divide the labor cleanly.

The shell model predicts nuclear spins and magnetic moments well, especially near closed shells, and complements the liquid drop: the drop captures the smooth bulk trends of binding energy, while the shell model captures the quantized structure riding on top. The evidence for the magic numbers is multiple and independent.

• Extra stable species: elements with magic $Z$ have anomalously many stable isotopes (tin, $Z=50$, has ten); magic $N$ gives extra isotones.
• Binding-energy jumps: the last nucleon of a magic nucleus is bound several $\mathrm{MeV}$ more tightly than the next one added.
• Capture cross sections: the neutron-capture cross section drops by nearly two orders of magnitude at $N=50,82,126$, since a closed shell resists binding another neutron.
• Vanishing quadrupole moments: magic nuclei are spherical, so their electric quadrupole moments pass through zero.

The predicted stability of $Z,N=126$ drives the ongoing search for a superheavy island of stability; the heaviest nuclei synthesized so far reach $Z=118$.