---
title: The Nuclear Force and the Shell Model
module: The Nuclear Force
moduleNumber: 2
lessonNumber: 1
order: 201
summary: >
  The strong force between nucleons is short-range, charge-independent,
  saturated, and repulsive at its core, about a hundred times stronger than
  Coulomb. Yukawa explained it as an exchange of massive mesons, tying the
  force's range to the meson mass through the uncertainty principle. Layered on
  top, an independent-particle shell model with strong spin-orbit coupling
  reproduces the magic numbers 2, 8, 20, 28, 50, 82, 126.
topics: [The Nuclear Force]
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 11 — Nuclear Physics; §11-5 The Nuclear Force, Exchange Mesons"
  - book: Tipler & Llewellyn
    ref: "§11-6 The Shell Model"
draft: false
---

The hydrogen atom yields to the Schrödinger equation because the electron-proton
potential $V(r) = -ke^2/r$ 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

$$
V = \frac{ke^2}{r} = \frac{1.44\ \mathrm{MeV{\cdot}fm}}{1\ \mathrm{fm}} = 1.44\,\mathrm{MeV},
$$

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.[^tl-force]

- **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.

$$
% caption: The nucleon-nucleon potential: a deep attractive well of a few fm
% width with a repulsive hard core inside 0.5 fm; the p-p curve adds a Coulomb
% tail that dominates beyond 3 fm.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.4,0) node[right, black!70] {$r$ (fm)};
  \draw[->, black] (0,-2.4) -- (0,2.2) node[above, black!70] {$V(r)$ (MeV)};
  \foreach \x/\lab in {1.2/1,2.4/2,3.6/3,4.8/4,6.0/5} \node[anchor=north, black, font=\scriptsize] at (\x,-0.02) {\lab};
  % n-p potential: hard core rise then deep well then back to 0
  \draw[acc, very thick]
    (0.5,2.0) -- (0.62,2.0)
    .. controls (0.75,1.2) and (0.9,-2.1) .. (1.5,-2.1)
    .. controls (2.2,-2.1) and (2.6,-0.9) .. (3.2,-0.35)
    .. controls (3.9,-0.05) and (4.8,-0.02) .. (6.6,0.0);
  \node[acc, anchor=west, font=\scriptsize] at (1.6,-1.7) {$n$-$p$, $n$-$n$};
  % p-p: same shape but positive Coulomb tail
  \draw[black, thick, dashed]
    (3.2,0.15)
    .. controls (4.0,0.55) and (4.8,0.85) .. (6.6,1.2);
  \node[black, anchor=west, font=\scriptsize] at (5.0,0.95) {$p$-$p$ tail};
  % hard-core label
  \node[black, anchor=south, font=\scriptsize] at (0.56,2.0) {core};
\end{tikzpicture}
$$

## 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 $\Delta t = \hbar/\Delta E$. In that time the photon
reaches at most

$$
R = c\,\Delta t = \frac{c\hbar}{\Delta E}.
$$

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

$$
% caption: Two nucleons interact by exchanging a virtual meson: emission by the
% left nucleon and absorption by the right, drawn on a spacetime (Feynman)
% diagram with time upward.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % axes
  \draw[->, black] (0,0) -- (0,4.4) node[above, black] {time};
  \draw[->, black] (0,0) -- (6.4,0) node[right, black] {space};
  % left nucleon worldline
  \draw[very thick] (1.4,0.3) -- (1.4,1.9) -- (1.1,2.4) -- (1.1,4.1);
  % right nucleon worldline
  \draw[very thick] (4.8,0.3) -- (4.8,1.9) -- (5.1,2.4) -- (5.1,4.1);
  % meson exchange (wavy-ish drawn as dashed diagonal)
  \draw[acc, thick, dashed] (1.4,1.9) -- (4.8,2.4);
  % vertices
  \fill[black] (1.4,1.9) circle (2pt);
  \fill[black] (4.8,2.4) circle (2pt);
  % labels
  \node[black, anchor=east, font=\scriptsize] at (1.35,0.6) {nucleon};
  \node[black, anchor=west, font=\scriptsize] at (4.85,0.6) {nucleon};
  \node[acc, anchor=south, font=\scriptsize] at (3.1,2.15) {meson};
\end{tikzpicture}
$$

Yukawa proposed the nuclear force is the exchange of a massive virtual particle,
the **meson**. Giving the exchanged particle rest energy $mc^2$ makes
$\Delta E \geq mc^2$, so the range is finite:

$$
R = \frac{\hbar}{mc}.
$$

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

$$
mc^2 = \frac{\hbar c}{R} \approx 200\ \mathrm{MeV},
\qquad m \approx 380\,m_e.
$$

A particle of this mass, the **pion**, was found in cosmic rays in 1947 with
$mc^2 = 140\,\mathrm{MeV}$, 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.[^tl-force]

> **Example (Range of the nuclear force).** Using the measured pion mass,
> $$
> R = \frac{\hbar c}{mc^2} = \frac{(1.06\times10^{-34}\,\mathrm{J{\cdot}s})(3.0\times10^{8}\,\mathrm{m/s})}{(140\,\mathrm{MeV})(1.6\times10^{-13}\,\mathrm{J/MeV})} = 1.4\times10^{-15}\,\mathrm{m},
> $$
> matching the observed short range.

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

$$
\Phi(r) = \frac{A\,e^{-r/R}}{r},
\qquad
|\Phi(r)|^2 = \frac{|A|^2\,e^{-2r/R}}{r^2},
$$

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.

$$
% caption: The Yukawa potential tracks the Coulomb potential near the origin but
% is cut off by the exponential factor beyond a few times the range R, giving the
% nuclear force its short reach.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.4,0) node[right, black!70] {$r$};
  \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {$V(r)$};
  % Coulomb 1/r (slow decay)
  \draw[black, thick, dashed] (0.45,4.1)
    .. controls (0.9,2.1) and (1.8,1.2) .. (3.0,0.85)
    .. controls (4.6,0.55) and (5.8,0.45) .. (7.0,0.38);
  \node[black, anchor=west, font=\scriptsize] at (4.2,0.75) {Coulomb};
  % Yukawa (fast exponential cutoff)
  \draw[acc, very thick] (0.45,4.1)
    .. controls (0.85,2.0) and (1.3,1.0) .. (2.0,0.55)
    .. controls (2.8,0.25) and (3.4,0.12) .. (4.2,0.06)
    -- (5.6,0.03);
  \node[acc, anchor=west, font=\scriptsize] at (2.1,0.9) {Yukawa};
  % range marker
  \draw[black, dashed] (1.4,0) -- (1.4,1.2);
  \node[anchor=north, black!70] at (1.4,0) {$R$};
\end{tikzpicture}
$$

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](/particle-physics/foundations/fundamental-interactions-force-carriers).

| Force | Carrier | Carrier mass | Range | Relative strength |
| --- | --- | --- | --- | --- |
| Electromagnetic | photon | $0$ | infinite | $10^{-2}$ |
| Strong (nuclear) | pion (meson) | $140\,\mathrm{MeV}/c^2$ | $\sim 1\,\mathrm{fm}$ | $1$ |
| Weak | (heavy boson) | large | $\ll 1\,\mathrm{fm}$ | $10^{-13}$ |

## The shell model

The smooth liquid-drop
[binding energy](/nuclear-physics/nuclear-properties/nuclear-constituents-nuclide-chart)
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.

$$
% caption: The binding-energy anomaly: the last nucleon is bound far more tightly
% just at the magic numbers, then a nucleon added past a closed shell is loosely
% bound, exactly as ionization energy jumps at closed atomic shells.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.4,0) node[right, black!70] {$N$};
  \draw[->, black] (0,-1.8) -- (0,1.8) node[above, black!70] {binding anomaly};
  \draw[black] (0,0) -- (7.2,0);
  % sawtooth-ish: peaks just below magic numbers then sharp drops after
  \draw[acc, very thick]
    (0.3,0.4) .. controls (0.9,1.3) and (1.3,1.3) .. (1.5,1.3)
    -- (1.6,-1.2)
    .. controls (2.2,0.2) and (2.8,1.2) .. (3.1,1.2)
    -- (3.2,-1.1)
    .. controls (3.9,0.2) and (4.6,1.15) .. (4.9,1.15)
    -- (5.0,-1.0)
    .. controls (5.8,0.3) and (6.4,1.0) .. (6.7,1.0);
  \foreach \x/\lab in {1.55/28,3.15/50,4.95/82} {
    \draw[black, dashed] (\x,0) -- (\x,1.4);
    \node[anchor=north, black, font=\scriptsize] at (\x,-0.05) {\lab};
  }
\end{tikzpicture}
$$

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.

> **Definition (Nuclear shell model).** An independent-particle model in which
> each nucleon occupies quantized levels of an average central potential, filling
> proton and neutron ladders separately by the exclusion principle. Large energy
> gaps between groups of levels produce the magic numbers, the nuclear analog of
> closed electron shells.

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](/atomic-physics/fine-structure-and-the-dirac-atom/spin-orbit-thomas-precession).
The splitting rearranges the level ordering so that the large gaps fall exactly at
$2, 8, 20, 28, 50, 82, 126$.

$$
% caption: Single-particle levels with spin-orbit splitting; the number of
% nucleons through each level (running total in brackets) shows the large gaps
% that land at the magic numbers.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % energy axis
  \draw[->, black] (0,0) -- (0,7.2) node[above, black!70] {energy};
  % levels: draw as short bars at increasing energy; magic-gap levels in accent
  \foreach \y/\tot/\magic in {0.6/2/1, 1.4/8/1, 2.3/20/1, 3.0/28/1, 4.0/50/1, 5.1/82/1, 6.3/126/1} {
    \draw[very thick] (1.2,\y) -- (3.0,\y);
    \node[black, anchor=west, font=\scriptsize] at (3.15,\y) {[\tot]};
  }
  % a few intermediate (non-magic) levels between the gaps, thinner black
  \foreach \y in {1.0,1.9,2.65,3.5,3.7,4.5,4.7,5.6,5.9} \draw[black] (1.4,\y) -- (2.8,\y);
  % magic-number callouts on the right
  \foreach \y/\m in {0.6/2, 1.4/8, 2.3/20, 3.0/28, 4.0/50, 5.1/82, 6.3/126}
    \node[black, anchor=west, font=\scriptsize] at (4.0,\y) {magic \m};
\end{tikzpicture}
$$

The two models divide the labor cleanly.

| | Liquid-drop model | Shell model |
| --- | --- | --- |
| Picture | classical charged droplet | independent nucleons in a mean field |
| Captures | smooth bulk binding, fission | magic numbers, spins, moments |
| Key input | volume/surface/Coulomb terms | strong spin-orbit coupling |
| Misses | shell structure, magic numbers | collective deformation, fission |

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 $Z = 118$.

[^tl-force]: Tipler & Llewellyn, "Modern Physics," 5th ed., §11-5.