---
title: Nucleon-Nucleon Scattering and the Interaction's Structure
module: The Nuclear Force
moduleNumber: 2
lessonNumber: 3
order: 203
summary: >
  Scattering probes the nuclear force above threshold. Partial-wave analysis
  reduces low-energy data to a single s-wave phase shift, and the effective-range
  expansion packages that into a scattering length and an effective range. The
  triplet channel binds (the deuteron) while the singlet is only virtual, which
  together explain the anomalously large free neutron-proton cross section.
  Comparing pp, nn, and np results establishes charge symmetry and charge
  independence, and polarization experiments expose the spin-orbit and tensor
  pieces.
topics: [The Nuclear Force]
sources:
  - book: Krane
    ref: "Ch. 4 — The Force Between Nucleons; §4.2 Nucleon-Nucleon Scattering"
  - book: Wong
    ref: "Ch. 3 — Nuclear Force and Two-Nucleon Systems; §3-3, §3-4"
  - book: Povh, Rith, Scholz, Zetsche
    ref: "Particles and Nuclei; Ch. 16"
draft: false
---

The deuteron gives one bound state and a handful of static moments. Scattering
supplies the rest: a continuous energy range over which the two-nucleon
interaction can be measured directly. Because the force has no closed form, the
strategy is not to solve for it but to parametrize the observable — the phase the
force imprints on the scattered wave — and to read the interaction's structure off
the energy and spin dependence of that phase.[^kr-scat]

## Partial waves and phase shifts

A beam of relative momentum $\hbar k$ incident on a short-range potential produces
the asymptotic stationary state

$$
\psi(\vec r) \xrightarrow{r\to\infty} e^{ikz} + f(\theta)\,\frac{e^{ikr}}{r},
$$

with $\d\sigma/\d\Omega = |f(\theta)|^2$. Expanding in Legendre polynomials, each
angular momentum $\ell$ evolves independently under a central force. Outside the
range of the potential the radial function $u_\ell(r) = rR_\ell(r)$ is a free
solution, and the only trace of the interaction is a shift of its asymptotic
phase relative to the force-free case:

$$
u_\ell(r) \xrightarrow{r\to\infty} \sin\!\left(kr - \frac{\ell\pi}{2} + \delta_\ell\right).
$$

The **phase shift** $\delta_\ell(k)$ carries everything measurable about the force
in the $\ell$-th wave. The scattering amplitude and cross section follow:

$$
f(\theta) = \frac{1}{k}\sum_{\ell}(2\ell+1)\,e^{i\delta_\ell}\sin\delta_\ell\,P_\ell(\cos\theta),
\qquad
\sigma = \frac{4\pi}{k^2}\sum_\ell (2\ell+1)\sin^2\delta_\ell .
$$

$$
% caption: Scattering as a phase shift. The dashed reference wave is the
% force-free radial function; the solid wave has been pulled inward by an
% attractive potential, emerging with the same wavelength but advanced in phase by
% delta. A positive shift signals attraction, a negative shift repulsion.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.0,0) node[right, black!70] {$r$};
  \draw[black] (0,0) -- (7.8,0);
  % free reference wave (dashed) sin(kr)
  \draw[black, thick, dashed, domain=0:7.6, samples=120, smooth]
    plot (\x, {1.1*sin(180*\x/1.3)});
  % shifted wave (solid), pulled inward: phase advance
  \draw[acc, very thick, domain=0:7.6, samples=140, smooth]
    plot (\x, {1.1*sin(180*(\x+0.55)/1.3)});
  \node[black, anchor=south, font=\scriptsize] at (6.7,0.9) {free wave};
  \node[acc, anchor=north, font=\scriptsize] at (6.7,-0.9) {shifted wave};
  % mark the phase advance near the first zero
  \draw[black, ->] (1.3,-1.35) -- (0.75,-1.35);
  \node[black, anchor=west, font=\scriptsize] at (1.35,-1.35) {pulled inward};
\end{tikzpicture}
$$

## The low-energy limit is pure s-wave

A partial wave $\ell$ probes impact parameters near $b \approx \ell/k$. It feels a
force of range $R$ only when $b \lesssim R$, i.e. $\ell \lesssim kR$. For the
nuclear range $R \approx 2\,\mathrm{fm}$, the boundary $kR = 1$ falls at

$$
E_{\text{lab}} \approx \frac{(\hbar c)^2 k^2}{\mu c^2}
\approx 20\,\mathrm{MeV}.
$$

Below roughly $10\,\mathrm{MeV}$ only $\ell = 0$ contributes, and the cross section
collapses to a single term,

$$
\sigma = \frac{4\pi}{k^2}\sin^2\delta_0 .
$$

This is the regime where the two-nucleon force is cleanest: one number,
$\delta_0(k)$, against energy. The scattering is isotropic (no $\theta$ dependence
from $P_0$), so the whole content is in the total cross section.

## Scattering length and effective range

Near zero energy $\delta_0 \to 0$, and the useful expansion is not of $\delta_0$
but of $k\cot\delta_0$, which is analytic in $k^2$. The **effective-range
expansion** truncates it after two terms:

$$
k\cot\delta_0 = -\frac{1}{a} + \frac{1}{2}r_0 k^2 + \cdots
$$

- **Scattering length** $a$: the zero-energy intercept, defined by
  $\displaystyle\lim_{k\to0} k\cot\delta_0 = -1/a$. At threshold the cross section
  is $\sigma_0 = 4\pi a^2$.
- **Effective range** $r_0$: the leading energy correction, a measure of the
  width over which the force acts.

Two parameters therefore summarize all low-energy scattering, independent of the
detailed shape of the potential. The scattering length has a clean geometric
meaning: the zero-energy exterior wavefunction is a straight line
$u_0(r)\propto (r - a)$, and $a$ is where its extrapolation crosses the axis.

$$
% caption: Geometric meaning of the scattering length. Outside the range the
% zero-energy wavefunction is a straight line; the point where its extrapolation
% meets the r-axis is a. A bound state bends the interior down so the intercept
% sits at positive a (left); a not-quite-bound virtual state gives a large
% negative intercept (right).
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % LEFT: bound, positive a
  \draw[->, black] (0,0) -- (3.7,0) node[right, black!70] {$r$};
  \draw[->, black] (0,-0.4) -- (0,2.3) node[above, black!70] {$u_0$};
  % interior curving up
  \draw[very thick] (0,0) .. controls (0.4,1.2) and (0.9,1.55) .. (1.3,1.5);
  % exterior straight line
  \draw[very thick] (1.3,1.5) -- (3.3,0.55);
  % extrapolation to axis (positive a to the right of origin? intercept at r=a)
  \draw[black, densely dotted] (3.3,0.55) -- (4.4,0.03);
  \node[black!70, anchor=north, font=\scriptsize] at (1.85,-0.15) {positive $a$};
  \node[black, anchor=south, font=\scriptsize] at (1.9,1.55) {bound (triplet)};
  % RIGHT: virtual, negative a
  \begin{scope}[xshift=5.2cm]
  \draw[->, black] (0,0) -- (3.7,0) node[right, black!70] {$r$};
  \draw[->, black] (0,-0.4) -- (0,2.3) node[above, black!70] {$u_0$};
  \draw[very thick] (0,0) .. controls (0.5,0.9) and (1.0,1.15) .. (1.4,1.25);
  \draw[very thick] (1.4,1.25) -- (3.3,2.0);
  % extrapolation backward to negative axis
  \draw[black, densely dotted] (1.4,1.25) -- (0.0,0.7);
  \node[black!70, anchor=north, font=\scriptsize] at (1.4,-0.15) {negative $a$ (behind origin)};
  \node[black, anchor=south, font=\scriptsize] at (1.9,1.9) {virtual (singlet)};
  \end{scope}
\end{tikzpicture}
$$

## Triplet binds, singlet is virtual

Neutron-proton scattering has two spin channels, and they behave oppositely.

- **Triplet** $^3S_1$: $a_t = +5.42\,\mathrm{fm}$, $r_{0t} = 1.75\,\mathrm{fm}$.
  The positive scattering length signals a bound state near threshold — the
  deuteron. At the bound-state pole $k = i\kappa$ the expansion gives
  $$
  \frac{1}{a_t} = \kappa - \frac{1}{2}r_{0t}\,\kappa^2 ,
  $$
  which with $\kappa = 0.232\,\mathrm{fm}^{-1}$ returns $a_t = 5.4\,\mathrm{fm}$,
  larger than $1/\kappa = 4.3\,\mathrm{fm}$ by the effective-range correction. The
  scattering length and the deuteron binding energy are the same physics seen from
  two sides of threshold.
- **Singlet** $^1S_0$: $a_s = -23.7\,\mathrm{fm}$, $r_{0s} = 2.73\,\mathrm{fm}$.
  The large negative scattering length signals a **virtual state** — a level that
  just fails to bind, a pole on the unphysical sheet just above threshold. The
  singlet force is attractive but slightly too weak to hold a bound state.

$$
% caption: Zero-energy wavefunctions in the two np spin channels. The triplet
% (solid) turns over and its asymptote crosses at a small positive scattering
% length, marking the bound deuteron; the singlet (dashed) barely bends, and its
% asymptote crosses far out on the negative side, the signature of a virtual state.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-1.2,0) -- (7.4,0) node[right, black!70] {$r$};
  \draw[->, black] (0,-0.3) -- (0,2.6) node[above, black!70] {$u_0(r)$};
  % triplet: turns over, asymptote crosses positive a small
  \draw[acc, very thick] (0,0) .. controls (0.4,1.4) and (0.9,1.85) .. (1.3,1.8) -- (3.6,0.9);
  \draw[acc, densely dotted] (3.6,0.9) -- (5.4,0.1);
  \node[acc, anchor=south west, font=\scriptsize] at (1.3,1.85) {triplet (bound)};
  \node[black!70, anchor=north, font=\scriptsize] at (0.55,-0.06) {small};
  % singlet: barely bends, rising, asymptote to negative side
  \draw[black, thick, dashed] (0,0) .. controls (0.6,0.55) and (1.1,0.75) .. (1.5,0.82) -- (6.6,2.4);
  \draw[black, densely dotted] (1.5,0.82) -- (-1.1,-0.35);
  \node[black, anchor=west, font=\scriptsize] at (4.4,1.7) {singlet (virtual)};
  \node[black, anchor=east, font=\scriptsize] at (-0.8,-0.2) {far negative};
\end{tikzpicture}
$$

## The neutron-proton cross section

An unpolarized beam populates the triplet with weight $3/4$ (three $m_S$ states)
and the singlet with weight $1/4$. The zero-energy cross section is the weighted
sum,

$$
\sigma_0 = 4\pi\!\left(\tfrac34 a_t^2 + \tfrac14 a_s^2\right)
= \pi\!\left(3a_t^2 + a_s^2\right).
$$

Inserting the two scattering lengths,

$$
\sigma_0 = \pi\!\left[3(5.42)^2 + (23.7)^2\right]\mathrm{fm}^2
= \pi(88.1 + 561.7)\,\mathrm{fm}^2
= 2041\,\mathrm{fm}^2 = 20.4\,\mathrm{b},
$$

in agreement with the measured thermal-neutron value. This number was a historic
puzzle: using only the triplet (deuteron) channel predicts about $4\,\mathrm{b}$,
a factor of five too small. The resolution is that the force is **spin-dependent**
and the singlet channel, with its huge $|a_s|$, dominates the cross section even
though it binds nothing. The np cross section thus measures the singlet
interaction that the deuteron cannot show.

$$
% caption: Total neutron-proton cross section against laboratory energy. It rises
% to about 20 barn at thermal energies, set almost entirely by the large singlet
% scattering length, then falls smoothly as 1/E once k exceeds the inverse range;
% partial waves beyond s add structure only above roughly 10 MeV.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.6,0) node[right, black!70] {$\log E$};
  \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {cross section};
  % plateau near 20 b then fall
  \draw[acc, very thick]
    (0.4,3.7) -- (1.8,3.65)
    .. controls (2.6,3.5) and (3.2,2.9) .. (3.9,2.1)
    .. controls (4.7,1.3) and (5.6,0.7) .. (7.0,0.35);
  \draw[black, dashed] (0,3.7) -- (1.8,3.7);
  \node[black, anchor=east, font=\scriptsize] at (-0.05,3.7) {$20$};
  \node[black, anchor=west, font=\scriptsize] at (0.5,3.95) {thermal plateau};
  \node[black, anchor=west, font=\scriptsize] at (4.6,1.35) {decreasing};
\end{tikzpicture}
$$

## The phase shift changes sign

Following $\delta_0$ up in energy exposes the hard core. In the $^3S_1$ channel it
starts near $180^\circ$ at threshold (Levinson's theorem: one bound state
contributes $\pi$), falls through the energy range, and in the $^1S_0$ channel the
phase shift rises from zero, peaks, and passes through zero near
$E_{\text{lab}} \approx 250\,\mathrm{MeV}$. A phase shift that turns from positive
to negative marks the transition from a net attractive to a net repulsive
interaction: at short distance the wave is pushed **out**, the signature of the
repulsive core inferred from saturation and constant nuclear density.

$$
% caption: The singlet s-wave phase shift versus energy. It rises from zero at
% threshold under the attractive well, reaches a maximum, then descends and crosses
% zero near 250 MeV, where the short-range repulsive core begins to dominate the
% net interaction.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.6,0) node[right, black!70] {$E_{\text{lab}}$};
  \draw[->, black] (0,-1.6) -- (0,2.4) node[above, black!70] {phase shift};
  \draw[black] (0,0) -- (7.4,0);
  % rise, peak, fall through zero, go negative
  \draw[acc, very thick]
    (0.2,0.15) .. controls (0.9,1.6) and (1.6,2.1) .. (2.3,2.0)
    .. controls (3.4,1.75) and (4.2,0.9) .. (5.0,0.0)
    .. controls (5.8,-0.75) and (6.6,-1.15) .. (7.2,-1.35);
  \draw[black, dashed] (5.0,0) -- (5.0,-0.6);
  \node[black, anchor=north, font=\scriptsize] at (5.0,-0.05) {sign change};
  \node[black, anchor=south, font=\scriptsize] at (2.3,2.05) {attractive};
  \node[black, anchor=west, font=\scriptsize] at (6.0,-1.05) {repulsive core};
\end{tikzpicture}
$$

## Charge symmetry and charge independence

Comparing the singlet scattering lengths across the three nucleon pairs tests two
distinct symmetries. After subtracting the Coulomb interaction from the
proton-proton data, the nuclear singlet scattering lengths are

$$
a_{pp} \approx -17.3\,\mathrm{fm},
\qquad
a_{nn} \approx -18.9\,\mathrm{fm},
\qquad
a_{np} \approx -23.7\,\mathrm{fm}.
$$

- **Charge symmetry** ($pp \leftrightarrow nn$): $a_{pp} \approx a_{nn}$ holds to
  the precision of the extraction. The $pp$ and $nn$ nuclear forces are equal.
- **Charge independence** ($pp,nn \leftrightarrow np$): $a_{np}$ differs from
  $a_{pp}, a_{nn}$ by more than the charge-symmetry gap, a few-percent violation
  traced mainly to the pion mass difference $m_{\pi^\pm} \neq m_{\pi^0}$.

The apparent size of these differences is exaggerated because the singlet system
sits at a virtual-state pole: a $1\%$ change in the potential swings the
scattering length by several $\mathrm{fm}$. The underlying interactions agree to
about $1\%$, which is why proton and neutron are treated as one particle carrying
an internal [isospin](/nuclear-physics/nuclear-force-deuteron/meson-theory-isospin)
label.

## Spin-orbit and tensor terms from polarization

Above the s-wave region the higher partial waves resolve the spin structure.
Scattering a polarized beam, or measuring the polarization induced in an
unpolarized one, gives a left-right asymmetry that no central force can produce:

- A **spin-orbit** term $V_{LS}(r)\,\vec L\cdot\vec S$ splits the $^3P_0$, $^3P_1$,
  $^3P_2$ phase shifts, which are degenerate under a central force. The measured
  splitting, and the analyzing power in $p$-wave scattering, fix its strength and
  sign.
- A **tensor** term $V_T(r)\,S_{12}$ mixes coupled partial waves of the same $J$
  and parity — the $^3S_1$-$^3D_1$ pair, parametrized by a mixing angle
  $\varepsilon_1$ — exactly the coupling that gives the deuteron its quadrupole
  moment.

$$
% caption: Double-scattering polarization test. The first target polarizes the
% beam (a spin-dependent force deflects spin-up and spin-down differently); the
% second target then shows a left-right asymmetry. A purely central force gives
% none, so any measured asymmetry isolates the spin-orbit and tensor pieces.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % beam in
  \draw[black, very thick, ->] (0,0) -- (2.0,0);
  \node[black, anchor=south, font=\scriptsize] at (1.0,0.05) {beam};
  % first target
  \draw[very thick] (2.35,0) circle (0.32);
  \node[black, anchor=north, font=\scriptsize] at (2.35,-0.42) {target 1};
  % polarized beam out (two spin components split)
  \draw[black, very thick, ->] (2.7,0.1) -- (4.6,0.5);
  \draw[black, very thick, ->] (2.7,-0.1) -- (4.6,-0.5);
  \node[black, anchor=west, font=\scriptsize] at (3.4,0.75) {polarized};
  % second target
  \draw[very thick] (4.95,0) circle (0.32);
  \node[black, anchor=north, font=\scriptsize] at (4.95,-0.42) {target 2};
  % asymmetric outgoing
  \draw[acc, very thick, ->] (5.3,0.15) -- (7.2,0.95);
  \draw[black, thick, ->] (5.3,-0.15) -- (6.7,-0.7);
  \node[black!70, anchor=west, font=\scriptsize] at (6.6,0.9) {more left};
  \node[black, anchor=west, font=\scriptsize] at (6.3,-0.7) {less right};
\end{tikzpicture}
$$

Assembling the channels, the nucleon-nucleon interaction has a central part, a
spin-spin part (splitting singlet from triplet), a tensor part (the
$S$-$D$ coupling), and a spin-orbit part, each with its own radial shape and each
extracted from a distinct feature of the data. The next lesson derives the
long-range piece of this force from meson exchange and recasts charge independence
as an isospin symmetry.

| Channel | $a$ (fm) | $r_0$ (fm) | State |
| --- | --- | --- | --- |
| $np$ triplet $^3S_1$ | $+5.42$ | $1.75$ | bound (deuteron) |
| $np$ singlet $^1S_0$ | $-23.7$ | $2.73$ | virtual |
| $pp$ singlet (nuclear) | $-17.3$ | $2.8$ | virtual |
| $nn$ singlet | $-18.9$ | $2.8$ | virtual |

[^kr-scat]: Krane, "Introductory Nuclear Physics," §4.2; low-energy scattering parameters from Krane Table 4.1 and the current values compiled by the NNDC, [nndc.bnl.gov](https://www.nndc.bnl.gov/).