---
title: Meson Exchange, the Yukawa Potential, and Isospin
module: The Nuclear Force
moduleNumber: 2
lessonNumber: 4
order: 204
summary: >
  Yukawa's massive-field propagator turns the range of the nuclear force into a
  meson mass: the exchanged quantum's Compton wavelength is the range. One-pion
  exchange fixes the long-range tail, complete with the tensor structure the
  deuteron demanded, while heavier mesons build the intermediate attraction and
  the repulsive core. Charge independence becomes an isospin symmetry, the force
  is diagonalized by the total isospin through a tau-dot-tau interaction, and the
  whole picture sits inside QCD as a residual color force between color-neutral
  nucleons.
topics: [The Nuclear Force]
sources:
  - book: Krane
    ref: "Ch. 4 — The Force Between Nucleons; §4.4 The Exchange Force Model"
  - book: Wong
    ref: "Ch. 3 — Nuclear Force and Two-Nucleon Systems; §3-7; Ch. 2 §2-6 Isospin"
  - book: Povh, Rith, Scholz, Zetsche
    ref: "Particles and Nuclei; Ch. 17 The Nucleon-Nucleon Force"
draft: false
---

The scattering data give the shape of the nuclear force channel by channel but not
its origin. Yukawa's proposal supplies the mechanism: nucleons interact by
exchanging massive quanta, and the mass of the quantum sets the range. This lesson
derives the Yukawa potential from the field equation of a massive particle, builds
the long-range nuclear force from pion exchange (recovering the tensor term the
deuteron required), and recasts charge independence as invariance under
isospin.[^kr-ex]

## The Yukawa potential from a massive field

A massless field obeys $\nabla^2\phi = 0$ outside sources, with the Coulomb
solution $\phi \propto 1/r$. Yukawa asked what field would fall off faster, cut off
at a finite range. The relativistic energy-momentum relation
$E^2 = p^2c^2 + m^2c^4$, promoted to an operator with $E \to i\hbar\,\partial_t$ and
$\vec p \to -i\hbar\nabla$, gives the **Klein-Gordon equation**

$$
\frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2} - \nabla^2\phi + \mu^2\phi = 0,
\qquad
\mu \equiv \frac{mc}{\hbar}.
$$

For a static point source of strength $g$ the time derivative drops and the
equation becomes

$$
-\nabla^2\phi + \mu^2\phi = g\,\delta^3(\vec r).
$$

Away from the origin the spherically symmetric solution of
$-\nabla^2\phi + \mu^2\phi = 0$ is $\phi = C\,e^{-\mu r}/r$. Matching to the source
as $r \to 0$, where the $\mu^2$ term is negligible and the equation reduces to the
Poisson form $-\nabla^2\phi = g\,\delta^3(\vec r)$ with solution $g/4\pi r$, fixes
$C = g/4\pi$. The result is the **Yukawa potential**

$$
\phi(r) = \frac{g}{4\pi}\,\frac{e^{-\mu r}}{r},
\qquad
R = \frac{1}{\mu} = \frac{\hbar}{mc}.
$$

The range is the exchanged particle's reduced Compton wavelength. Equivalently, in
momentum space the potential is the Fourier transform of the massive propagator,

$$
\tilde V(\vec q) \propto \frac{g^2}{q^2 + \mu^2},
\qquad
V(r) \propto \frac{g^2}{4\pi}\,\frac{e^{-\mu r}}{r},
$$

so a heavier exchanged quantum means a larger $\mu$, a shorter range, and a more
strongly damped force.

> **Worked example (From range to pion mass).** The nuclear force reaches about
> $R \approx 1.4\,\mathrm{fm}$. Inverting the range-mass relation,
> $$
> mc^2 = \frac{\hbar c}{R} = \frac{197.3\ \mathrm{MeV{\cdot}fm}}{1.4\ \mathrm{fm}}
> = 141\,\mathrm{MeV}.
> $$
> Yukawa predicted a quantum of this mass in 1935; the pion, $m_\pi c^2 = 138\,\mathrm{MeV}$,
> was found in cosmic rays in 1947.[^pdg]

## One-pion exchange and the meson hierarchy

The pion is pseudoscalar ($J^\pi = 0^-$), so its coupling to nucleons carries the
nucleon spins and isospins. The one-pion-exchange potential (OPEP) that results has
the schematic structure

$$
V_{\pi}(r) = \frac{f^2}{4\pi}\,\frac{m_\pi c^2}{3}\,
(\vec\tau_1\cdot\vec\tau_2)\!
\left[\vec\sigma_1\cdot\vec\sigma_2
+ S_{12}\!\left(1 + \frac{3}{\mu r} + \frac{3}{\mu^2 r^2}\right)\right]
\frac{e^{-\mu r}}{\mu r}.
$$

Two features follow without further input.

- **A spin-spin central term** $\vec\sigma_1\cdot\vec\sigma_2$, which distinguishes
  singlet from triplet — the spin dependence the deuteron and the np cross section
  both revealed.
- **A tensor term** $S_{12}$, with exactly the operator that mixes $^3S_1$ and
  $^3D_1$. The deuteron's quadrupole moment is the long-range tail of pion
  exchange.

The full interaction needs more than the pion. In the **one-boson-exchange** model
the force decomposes by range according to the mass of the lightest meson that can
reach:

- **Long range** ($r \gtrsim 2\,\mathrm{fm}$): single-pion exchange
  ($m_\pi = 138\,\mathrm{MeV}$), giving the tensor tail.
- **Intermediate range** ($1$–$2\,\mathrm{fm}$): correlated two-pion exchange,
  often modeled as a scalar $\sigma$ meson ($\sim 500\,\mathrm{MeV}$), the main
  source of attraction.
- **Short range** ($r \lesssim 0.8\,\mathrm{fm}$): the vector mesons $\rho$
  ($770\,\mathrm{MeV}$) and $\omega$ ($782\,\mathrm{MeV}$); $\omega$ exchange
  produces the repulsive hard core and a short-range spin-orbit term.

$$
% caption: Yukawa potentials for three exchanged mesons. Heavier mesons (rho and
% omega near 775 MeV) fall off within a few tenths of a fm, so they govern only the
% short-range core; the light pion (138 MeV) reaches out past 1 fm and carries the
% long-range tail. Curves are distinguished by weight and dash, not colour.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.6,0) node[right, black!70] {$r$ (fm)};
  \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {$V(r)$};
  \foreach \x/\lab in {1.4/0.5,2.8/1.0,4.2/1.5,5.6/2.0,7.0/2.5} \node[anchor=north, black, font=\scriptsize] at (\x,-0.02) {\lab};
  % pion (long range): slow Yukawa, mu=0.7/fm -> use gentle decay; focal acc solid
  \draw[acc, very thick, domain=0.35:7.2, samples=140, smooth]
    plot (\x, {0.95*exp(-0.70*\x/1.4*1.4/1.0)/(\x/1.4)*0.55});
  \node[acc, anchor=west, font=\scriptsize] at (3.0,1.05) {pion};
  % rho/omega (short range): fast Yukawa mu~3.9/fm, neutral dashed
  \draw[black, thick, dashed, domain=0.28:2.4, samples=120, smooth]
    plot (\x, {1.1*exp(-3.9*\x/1.4)/(\x/1.4)*0.9});
  \node[black, anchor=west, font=\scriptsize] at (1.3,2.3) {rho, omega};
\end{tikzpicture}
$$

$$
% caption: One-pion-exchange vertex. Each nucleon emits or absorbs a pion at a
% pseudoscalar vertex; the exchanged pion carries momentum q, and the propagator
% 1/(q^2+mu^2) Fourier-transforms to the Yukawa tail. A charged pion swaps the
% nucleon identities, a neutral pion leaves them fixed.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % two incoming/outgoing nucleon lines (left and right verticals)
  \draw[very thick, ->] (0.4,0.4) -- (1.6,1.6);
  \draw[very thick, ->] (1.6,1.6) -- (0.4,2.8);
  \draw[very thick, ->] (5.2,0.4) -- (4.0,1.6);
  \draw[very thick, ->] (4.0,1.6) -- (5.2,2.8);
  % vertices
  \fill[black] (1.6,1.6) circle (2pt);
  \fill[black] (4.0,1.6) circle (2pt);
  % pion exchange (dashed horizontal)
  \draw[acc, thick, dashed] (1.6,1.6) -- (4.0,1.6);
  \node[acc, anchor=south, font=\scriptsize] at (2.8,1.65) {pion, momentum $q$};
  % labels
  \node[black, anchor=east, font=\scriptsize] at (0.35,0.4) {nucleon};
  \node[black, anchor=west, font=\scriptsize] at (5.25,0.4) {nucleon};
\end{tikzpicture}
$$

## Charge independence as isospin symmetry

The near-equality of the $pp$, $nn$, and $np$ forces has a group-theoretic
statement. Treat the proton and neutron as two states of one particle, the
**nucleon**, forming a doublet of an internal $SU(2)$ symmetry — **isospin** — in
formal analogy with ordinary spin-$\tfrac12$. Assign the third component

$$
T_3 = +\tfrac12\ (\text{neutron}),
\qquad
T_3 = -\tfrac12\ (\text{proton}),
$$

so that a nucleus has $T_3 = (N - Z)/2$. Charge independence of the nuclear force
is the statement that the strong Hamiltonian commutes with the total isospin,
$[H_{\text{strong}}, \vec T] = 0$: it is invariant under rotations in isospin space
and cannot depend on $T_3$, only on the total $T$.

$$
% caption: The isospin assignments. The nucleon is a doublet (T=1/2): neutron and
% proton at the two values of T3. The pion is a triplet (T=1): three charge states
% along the T3 axis. With the neutron-positive convention T3=+1/2 for the neutron,
% nuclear charge falls as T3 rises, while the pion charge rises with T3.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % nucleon doublet
  \draw[black, ->] (-2.3,1.4) -- (2.3,1.4) node[right, black!70] {$T_3$};
  \fill[black] (-0.9,1.4) circle (2.6pt);
  \fill[black] (0.9,1.4) circle (2.6pt);
  \node[black, anchor=south, font=\scriptsize] at (-0.9,1.55) {proton};
  \node[black, anchor=south, font=\scriptsize] at (0.9,1.55) {neutron};
  \node[black, anchor=south, font=\scriptsize] at (0.0,2.15) {nucleon doublet};
  % pion triplet
  \draw[black, ->] (-2.7,-0.6) -- (2.7,-0.6) node[right, black!70] {$T_3$};
  \fill[black] (-1.7,-0.6) circle (2.6pt);
  \fill[black] (0.0,-0.6) circle (2.6pt);
  \fill[black] (1.7,-0.6) circle (2.6pt);
  \node[black, anchor=south, font=\scriptsize] at (-1.7,-0.5) {pi minus};
  \node[black, anchor=south, font=\scriptsize] at (0.0,-0.5) {pi zero};
  \node[black, anchor=south, font=\scriptsize] at (1.7,-0.5) {pi plus};
  \node[black, anchor=north, font=\scriptsize] at (0.0,-0.8) {pion triplet};
\end{tikzpicture}
$$

Two nucleons combine to total isospin $T = 0$ (antisymmetric) or $T = 1$
(symmetric), just as two spins combine to singlet and triplet. The isospin
dependence of the force is carried by the scalar $\vec\tau_1\cdot\vec\tau_2$, whose
eigenvalues follow from $\vec T^2 = \tfrac14(\vec\tau_1+\vec\tau_2)^2$:

$$
\vec\tau_1\cdot\vec\tau_2 = 2T(T+1) - 3 =
\begin{cases}
+1 & T = 1, \\
-3 & T = 0.
\end{cases}
$$

The force is therefore the same for every member of an isospin multiplet — the
three $T = 1$ pairs ($pp$, $nn$, and the singlet $np$) share one interaction — but
differs between $T = 0$ and $T = 1$. The deuteron lives in the $T = 0$ channel,
where the attraction is stronger, which is why it binds while the $T = 1$ states do
not. Charge symmetry ($pp \leftrightarrow nn$) is the weaker statement that $H$ is
invariant under the single isospin rotation exchanging $p$ and $n$; charge
independence is full isospin invariance. Both are broken at the percent level by
electromagnetism and by the up-down quark mass difference.

> **Definition (Isospin).** The internal $SU(2)$ quantum number under which the
> proton and neutron form a doublet of total isospin $T = \tfrac12$. Strong
> interactions conserve total $\vec T$; the third component is fixed by the charge
> through $Q = \tfrac12 A - T_3$ for a nucleus of mass number $A$ (with the
> neutron-positive convention $T_3 = +\tfrac12$ for the neutron), and
> electromagnetic and mass effects break the symmetry weakly.

## The residual color force

Meson exchange is an effective description of a deeper interaction. Each nucleon is
a color-singlet bound state of three quarks held together by gluon exchange, the
[fundamental strong interaction](/particle-physics/foundations/fundamental-interactions-force-carriers).
The force between two nucleons is what remains when two color-neutral objects
approach — a **residual** strong force, analogous to the van der Waals force
between two neutral atoms, whose electromagnetic interaction cancels internally but
leaves a short-range remnant.

$$
% caption: The nucleon-nucleon force as a residual color interaction. Each nucleon
% is three quarks in a color-neutral bound state; when two approach, the exchanged
% quark-antiquark pair is a meson (the pion for the longest range). The strong
% force between nucleons is the leftover of the color force, as the van der Waals
% force is the leftover of the atomic Coulomb force.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % left nucleon: three quarks in a blob
  \draw[very thick] (1.4,0) ellipse (0.95 and 0.7);
  \fill[black] (0.95,0.15) circle (2pt);
  \fill[black] (1.55,0.28) circle (2pt);
  \fill[black] (1.35,-0.28) circle (2pt);
  \node[black, anchor=north, font=\scriptsize] at (1.4,-0.85) {nucleon (3 quarks)};
  % right nucleon
  \draw[very thick] (6.0,0) ellipse (0.95 and 0.7);
  \fill[black] (5.6,0.2) circle (2pt);
  \fill[black] (6.25,0.25) circle (2pt);
  \fill[black] (6.0,-0.3) circle (2pt);
  \node[black, anchor=north, font=\scriptsize] at (6.0,-0.85) {nucleon (3 quarks)};
  % exchanged meson (quark-antiquark pair) in the middle
  \draw[acc, thick, dashed] (2.35,0.1) -- (5.05,0.1);
  \draw[acc, thick, dashed] (2.35,-0.1) -- (5.05,-0.1);
  \node[acc, anchor=south, font=\scriptsize] at (3.7,0.15) {meson exchange};
  \node[black, anchor=north, font=\scriptsize] at (3.7,-0.15) {(quark pair)};
\end{tikzpicture}
$$

The pion is special in this picture: it is the lightest meson because it is the
near-Goldstone boson of spontaneously broken chiral symmetry, so its exchange
dominates the long-range force. The complete two-nucleon interaction, from the
one-pion tail through the intermediate attraction to the repulsive core, is the
low-energy face of quantum chromodynamics, and modern **chiral effective field
theory** derives it as a controlled expansion in $m_\pi$ and momentum rather than
by fitting one boson at a time.

| Range | Dominant exchange | Meson mass | Effect |
| --- | --- | --- | --- |
| $\gtrsim 2\,\mathrm{fm}$ | one pion | $138\,\mathrm{MeV}$ | tensor tail, spin-spin |
| $1$–$2\,\mathrm{fm}$ | two pions ($\sigma$) | $\sim 500\,\mathrm{MeV}$ | central attraction |
| $\lesssim 0.8\,\mathrm{fm}$ | $\rho$, $\omega$ | $\sim 775\,\mathrm{MeV}$ | repulsive core, spin-orbit |

The two-nucleon system is now complete: the deuteron fixes the bound-state
properties, scattering measures the force across energy and spin, and meson
exchange with isospin symmetry supplies the mechanism and the organizing
principle. The models of the many-nucleon nucleus — Fermi gas, liquid drop, shell,
and collective — build on this interaction.

[^kr-ex]: Krane, "Introductory Nuclear Physics," §4.4; isospin formalism in Wong, "Introductory Nuclear Physics," 2nd ed., §2-6.
[^pdg]: Meson masses from the Particle Data Group, "Review of Particle Physics," [pdg.lbl.gov](https://pdg.lbl.gov/).