Nucleon-Nucleon Scattering and the Interaction's Structure

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.¹

Partial waves and phase shifts

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

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

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

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

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

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:

• Scattering length $a$: the zero-energy intercept, defined by $\underset{k\to 0}{\lim }k\cot {\delta }_0=-1/a$. 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.

Triplet binds, singlet is virtual

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

• Triplet ${}^3{S}_1$: $a_t=+5.42\mathrm{fm}$, $r_{0t}=1.75\mathrm{fm}$. 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 ${}^1{S}_0$: $a_s=-23.7\mathrm{fm}$, $r_{0s}=2.73\mathrm{fm}$. 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.

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,

Inserting the two scattering lengths,

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.

The phase shift changes sign

Following ${\delta }_0$ up in energy exposes the hard core. In the ${}^3{S}_1$ channel it starts near $180^{\circ }$ at threshold (Levinson's theorem: one bound state contributes $\pi$), falls through the energy range, and in the ${}^1{S}_0$ 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.

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

• 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 $m_{{\pi }^{\pm }}\ne m_{{\pi }^0}$.

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 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 ${}^3{P}_0$, ${}^3{P}_1$, ${}^3{P}_2$ 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 ${}^3{S}_1$-${}^3{D}_1$ pair, parametrized by a mixing angle ${\epsilon }_1$ — exactly the coupling that gives the deuteron its quadrupole moment.

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.