advanced30 min studyLesson 573 in Physics

Nucleon–nucleon scattering and the deuteron

Connect low-energy phase shifts and shallow bound-state tails while retaining spin and tensor-channel distinctions.

Before you begin

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The question for today

What you’ll understand

Calculate a low-energy s-wave cross section in a specified spin channel.
Relate a shallow bound-state pole to its exponential tail scale.
Explain why the deuteron requires more than a central spin-independent potential.

Two-nucleon measurements constrain the forces used in larger nuclei, but the word force does not identify a single universal central potential. Scattering depends on energy, total spin, orbital motion, and isospin, while noncentral terms couple channels. The deuteron provides a complementary bound-state constraint. Its existence and structure must agree with the same interaction that describes continuum data; an attractive well tuned to one binding energy alone leaves much of the physics undetermined.

Read a low-energy scattering amplitude

For elastic scattering in one uncoupled s-wave channel, describe the outgoing wave by a phase shift. A useful low-momentum expansion expresses k times the cotangent of that phase through a scattering length and effective range. The scattering length is not the geometric radius of the interaction. It can be large and either positive or negative, particularly near a bound or virtual state. Its sign must follow the phase-shift convention used in the amplitude.

Neutron and proton are distinguishable particles for this cross-section calculation, so the single-channel s-wave total is four pi times the squared amplitude. An unpolarized beam samples spin-singlet and spin-triplet channels with statistical weights one quarter and three quarters when no spin selection is made. Their scattering lengths differ. Adding amplitudes from those incoherent initial spin populations would be wrong; the corresponding cross sections are averaged with their population weights.

k\cot\delta_0=-\frac1a+\frac12r_e k^2+O(k^4),\qquad f_0=\frac{1}{k\cot\delta_0-ik},\qquad \sigma_0=4\pi|f_0|^2

Wave number k is inverse length, scattering length a and effective range r-e are lengths, and sigma has area units. The expansion assumes low energy in an elastic short-range channel.↔ Scroll wide equations horizontally.

Continue the amplitude to a bound pole

A bound state has an exponentially decreasing exterior radial wave function. Its inverse length kappa follows from binding energy and reduced mass. Analytically continuing the scattering amplitude to imaginary k identifies a bound-state pole when its denominator vanishes. If the effective range and higher terms are negligible, a large positive scattering length gives kappa approximately its reciprocal. This is a shallow-state approximation, not a proof that every positive scattering length implies a bound state in every potential.

The exterior s-wave solution is a reduced radial function proportional to an exponential. The full radial wave function also includes division by radius, and the interior must satisfy regularity and matching conditions. A weakly bound system can therefore have a tail extending well beyond the interaction range. The binding energy fixes that tail exponent but not the complete interior density, normalization partition, tensor mixing, or all electromagnetic observables.

u(r)\propto e^{-\kappa r},\qquad B=\frac{\hbar^2\kappa^2}{2\mu},\qquad \kappa=\frac1a+\frac12r_e\kappa^2+\cdots

B is positive binding energy and mu is the neutron-proton reduced mass. The pole relation uses the displayed effective-range convention and is meaningful only within its low-energy domain.↔ Scroll wide equations horizontally.

Use the deuteron to test operator structure

The physical deuteron has total angular momentum one and positive parity. Its dominant spin-triplet S component is coupled to a D component by the tensor force, which depends on the direction joining the nucleons as well as their spins. Both orbital angular momenta can combine with spin one to total angular momentum one. A nonzero quadrupole response signals physics beyond a pure spherical S-wave charge configuration, though its precise calculation also requires appropriate electromagnetic operators.

A numerical D-component probability depends on the chosen interaction representation and is not an independently measured observable by itself. Observables such as scattering amplitudes, binding, form factors, and moments must be compared consistently after any unitary change of representation. This is why a two-component drawing cannot extract the deuteron wave function from first principles. It can illustrate amplitude mixing while leaving radial dynamics, tensor coupling, and currents outside the picture.

S_{12}=3(\boldsymbol\sigma_1\cdot\widehat{\mathbf r})(\boldsymbol\sigma_2\cdot\widehat{\mathbf r})-\boldsymbol\sigma_1\cdot\boldsymbol\sigma_2,\qquad {}^3S_1\leftrightarrow{}^3D_1

The tensor operator is dimensionless; its radial coupling coefficient has energy units. Spectroscopic notation labels spin multiplicity, orbital angular momentum, and total angular momentum.↔ Scroll wide equations horizontally.

Work it through

A toy shallow s-wave channel

In one distinguishable-particle spin channel take toy values $a=5\,\mathrm{fm}$ , $r_e=0$ , $k=0.10\,\mathrm{fm}^{-1}$ , and $\hbar^2/(2\mu)=40\,\mathrm{MeV\,fm}^2$ . Calculate the s-wave cross section and shallow-pole binding estimate.

The amplitude denominator has real part $-1/a=-0.2\,\mathrm{fm}^{-1}$ and imaginary part $-0.1\,\mathrm{fm}^{-1}$ . Its squared magnitude is $0.04+0.01=0.05\,\mathrm{fm}^{-2}$ , so $|f_0|^2=20\,\mathrm{fm}^2$ .
Multiplying by four pi gives $\sigma_0=80\pi\,\mathrm{fm}^2\approx251.3\,\mathrm{fm}^2=2.513\,\mathrm{barn}$ . One barn is one hundred square femtometers.
With effective range neglected, the pole has $\kappa=1/a=0.2\,\mathrm{fm}^{-1}$ . Its binding estimate is $40(0.2)^2=1.6\,\mathrm{MeV}$ . The toy channel is not a fit to the physical deuteron.

The result

The stipulated single-channel model gives cross section 2.513 barns and shallow-pole binding 1.6 MeV under its zero-range assumptions.

The living notebook

Explore the model

Interactive lab

Interactive lab · quantum

An amplitude-mixing analogy

Use the two basis states only as an abstract two-component analogy. Reset and set mixing angle $60^\circ$ , phase $0^\circ$ ; predict component probabilities three quarters and one quarter. Compare a fresh Reset run at $0^\circ,0^\circ$ . This pure-qubit lab does not calculate deuteron radial functions, S-D mixing, spin coupling, or scattering cross sections; the one-quarter value is deliberately a toy parameter.

Bloch vector / P(0)P(1)Phase latitude

Probability |0⟩: 71.1%
Probability |1⟩: 28.9%
Bloch (x, y, z): 0.78, 0.45, 0.42

Mixing angle θ65°Relative phase φ30°

t = 0 s

A normalized two-level pure state, not an interacting field simulation. Playback adds one radian per second to the relative phase; basis probabilities stay fixed. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA normalized two-component amplitude can illustrate mixing, while physical partial waves require angular and radial dynamics.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

There is one singlet and three triplet spin states for two spin-one-half particles. Incoherent initial populations require a weighted sum of probabilities or cross sections, not interfering amplitudes.

Your turn

Try it yourself

In the same zero-effective-range toy model compare $a=+10\,\mathrm{fm}$ with $a=-10\,\mathrm{fm}$ . Find each zero-energy cross section. With $\hbar^2/(2\mu)=40\,\mathrm{MeV\,fm}^2$ , which sign supports the shallow bound-pole estimate and what is its binding?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

The zero-energy cross section contains a squared length, but a decaying bound tail requires positive kappa.

Compare with the solution

Both signs give $\sigma_0(0)=4\pi a^2=400\pi\,\mathrm{fm}^2\approx12.566\,\mathrm{barn}$ . The positive scattering length gives $\kappa=0.1\,\mathrm{fm}^{-1}$ and $B=40(0.1)^2=0.4\,\mathrm{MeV}$ . The negative sign gives a virtual-state pole in this zero-range description, not a normalizable decaying bound state with positive kappa. Equal zero-energy cross sections therefore do not determine the pole character.

Keep the idea

Take it with you

Scattering length is a channel-dependent amplitude parameter rather than a nuclear radius.
A shallow binding energy constrains a tail scale without fixing the full wave function.
Tensor mixing and consistent electromagnetic operators matter for deuteron structure.

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