Two-Body Problems: Nucleon-Nucleon Scattering & Low-Energy Dynamics

1. Kinematics of Elastic Neutron-Proton Scattering

Consider a beam of monoenergetic neutrons with laboratory kinetic energy $E_{\text{lab}}$ incident on a stationary hydrogen target ($m_p \approx m_n \equiv M$). In the center-of-mass (CM) frame, the relative wavevector $k$ and center-of-mass energy $E_{\text{cm}}$ are related to $E_{\text{lab}}$ by:

$$E_{\text{cm}} = \frac{1}{2} E_{\text{lab}}, \quad k = \frac{\sqrt{M E_{\text{cm}}}}{\hbar} = \frac{\sqrt{M E_{\text{lab}}/2}}{\hbar}$$

For low incident energies ($E_{\text{lab}} < 10\text{ MeV}$), the de Broglie wavelength of the relative motion is large compared to the range of the nuclear potential ($R \approx 1.5\text{ to }2.0\text{ fm}$):

$$\lambdabar = \frac{1}{k} = \frac{\hbar}{\sqrt{M E_{\text{lab}}/2}} = \frac{197.3\text{ MeV}\cdot\text{fm}}{\sqrt{938.9 \times E_{\text{lab}}/2}}$$

For example, at $E_{\text{lab}} = 1\text{ MeV}$, $k \approx 0.11\text{ fm}^{-1}$, yielding $k R \approx (0.11)(2.0) \approx 0.22 \ll 1$. According to the semiclassical impact parameter argument ($l_{\max} \approx k R$), particles with orbital angular momentum $l \ge 1$ cannot penetrate the centrifugal barrier. Therefore, low-energy $n$-$p$ scattering is governed exclusively by $s$-wave ($l = 0$) partial waves.

2. Partial Wave Analysis for S-Wave Scattering

In the asymptotic region ($r > R$) where the nuclear potential vanishes, the radial Schrödinger equation for $l = 0$ is:

$$\frac{d^2 u_0(r)}{dr^2} + k^2 u_0(r) = 0$$

The general asymptotic solution with boundary condition $u_0(0) = 0$ is phase-shifted relative to the unperturbed free wave $\sin(kr)$:

$$u_0(r) = C \sin(k r + \delta_0)$$

where $\delta_0$ is the $s$-wave phase shift.

• If the potential is attractive ($V(r) < 0$), the wavefunction is pulled inward toward the origin, and $\delta_0 > 0$.
• If the potential is repulsive ($V(r) > 0$), the wavefunction is pushed outward, and $\delta_0 < 0$.

The total elastic scattering cross section for pure $s$-wave scattering is:

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

1. Spin Combinations of the Two-Nucleon System

Both the proton and neutron are spin-$1/2$ fermions. Coupling their spins $\vec{s}_1$ and $\vec{s}_2$ yields total spin $\vec{S} = \vec{s}_1 + \vec{s}_2$:

• Spin-Triplet State ($S = 1$): Three symmetric magnetic sub-states ($M_S = +1, 0, -1$): $$|1, 1\rangle = |\!\uparrow\uparrow\rangle, \quad |1, 0\rangle = \frac{1}{\sqrt{2}}(|\!\uparrow\downarrow\rangle + |\!\downarrow\uparrow\rangle), \quad |1, -1\rangle = |\!\downarrow\downarrow\rangle$$ Statistical weight $g_t = \frac{2S+1}{(2s_1+1)(2s_2+1)} = \frac{3}{4}$.
• Spin-Singlet State ($S = 0$): One antisymmetric sub-state ($M_S = 0$): $$|0, 0\rangle = \frac{1}{\sqrt{2}}(|\!\uparrow\downarrow\rangle - |\!\downarrow\uparrow\rangle)$$ Statistical weight $g_s = \frac{1}{4}$.

2. The Spin-Dependent Total Cross Section

Because the nuclear force depends strongly on the relative spin orientation, the phase shifts for the triplet state ($\delta_{0t}$) and singlet state ($\delta_{0s}$) are completely different.

For an unpolarized incident neutron beam hitting an unpolarized proton target, the total unpolarized elastic scattering cross section is the statistically weighted sum of triplet and singlet cross sections:

$$\sigma = \frac{3}{4}\sigma_t + \frac{1}{4}\sigma_s = \frac{3}{4}\left( \frac{4\pi}{k^2}\sin^2\delta_{0t} \right) + \frac{1}{4}\left( \frac{4\pi}{k^2}\sin^2\delta_{0s} \right)$$

If the nuclear force were spin-independent, then $\delta_{0t} = \delta_{0s}$, and the scattering cross section at zero energy would be determined entirely by the deuteron bound-state parameters:

$$\sigma_{\text{spin-indep}} = \frac{4\pi}{\gamma^2} = \frac{4\pi \hbar^2}{M B} \approx 4.3\text{ barns}$$

However, experimental measurements of the low-energy thermal neutron-proton scattering cross section (Wigner, 1935) yielded:

$$\sigma_{\text{exp}} \approx 20.4\text{ barns}$$

The enormous factor-of-five discrepancy ($20.4\text{ b} \gg 4.3\text{ b}$) provided direct historical proof that the nuclear force is strongly spin-dependent ($\sigma_s \gg \sigma_t$).

1. Definition of the Scattering Length $a$

In the limit of zero incident kinetic energy ($k \to 0$), the phase shift $\delta_0$ approaches zero proportional to $k$. The scattering length $a$ is defined as the negative limit of the ratio:

$$a \equiv -\lim_{k\to 0} \frac{\tan\delta_0}{k} = -\lim_{k\to 0} \frac{\delta_0}{k}$$

Geometrically, if the asymptotic zero-energy radial wavefunction $u_0(r) \propto 1 - r/a$ is extrapolated linearly toward the origin, $a$ represents the intercept of the asymptotic wavefunction on the $r$-axis:

$$\lim_{k\to 0} u_0(r) \propto (r - a)$$

2. Physical Interpretation of Positive vs Negative Scattering Lengths

• Positive Scattering Length ($a > 0$): The potential is sufficiently attractive to produce a real bound state. The interior wavefunction bends over and has a negative slope at the potential boundary, causing the linear extrapolation to intercept the positive $r$-axis ($a > 0$). This is the case for the triplet $n$-$p$ state: $$a_t = +5.424 \pm 0.004\text{ fm}$$ In the zero-range approximation, $a_t \approx \frac{1}{\gamma} = \frac{\hbar}{\sqrt{M B}} \approx 4.32\text{ fm}$.
• Negative Scattering Length ($a < 0$): The potential is attractive but not deep enough to form a bound state. The interior wavefunction does not reach a negative slope at $r = R$; it curves toward the origin, causing the linear extrapolation to intercept the negative $r$-axis ($a < 0$). This indicates an unbound virtual state. For the singlet $n$-$p$ state: $$a_s = -23.740 \pm 0.020\text{ fm}$$ The colossal negative value of $a_s$ indicates that the singlet state is extraordinarily close to binding (the virtual pole lies at $E_s = -\frac{\hbar^2}{M a_s^2} \approx -66\text{ keV}$).

3. Zero-Energy Cross Section Evaluation

In the zero-energy limit $k \to 0$:

$$\sigma_t = 4\pi a_t^2 = 4\pi (5.424\text{ fm})^2 = 4\pi (29.42\text{ fm}^2) \approx 3.698\times 10^{-24}\text{ cm}^2 = 3.70\text{ b}$$ $$\sigma_s = 4\pi a_s^2 = 4\pi (-23.74\text{ fm})^2 = 4\pi (563.59\text{ fm}^2) \approx 7.082\times 10^{-23}\text{ cm}^2 = 70.82\text{ b}$$

Weighting the triplet and singlet contributions:

$$\sigma_0 = \frac{3}{4}\sigma_t + \frac{1}{4}\sigma_s = \frac{3}{4}(3.70\text{ b}) + \frac{1}{4}(70.82\text{ b}) = 2.775\text{ b} + 17.705\text{ b} = 20.48\text{ b}$$

This matches the experimental value of $20.4\text{ b}$ to within $0.5\%$.

1. The Bethe-Schwinger Effective Range Expansion

As the neutron energy increases away from zero, the cross section begins to vary with $k$. Hans Bethe and Julian Schwinger developed effective range theory, which parameterizes low-energy scattering independently of the specific shape of the nuclear potential.

By comparing the true radial wavefunction $u(k, r)$ inside the potential to the asymptotic wavefunction $v(k, r) = \frac{\sin(kr + \delta_0)}{\sin\delta_0}$ extrapolated to all $r$, one derives the exact shape-independent expansion:

$$k \cot\delta_0 = -\frac{1}{a} + \frac{1}{2} r_0 k^2 - P r_0^3 k^4 + \mathcal{O}(k^6)$$

where:

• $a$ is the scattering length defined at $k = 0$.
• $r_0$ is the effective range, defined by the integral: $$r_0 \equiv 2 \int_0^\infty \left[ v_0^2(r) - u_0^2(r) \right] dr$$ where $u_0(r)$ is the true zero-energy wavefunction and $v_0(r) = 1 - r/a$ is its linear asymptotic continuation.
• $P$ is a dimensionless shape parameter (typically $|P| \le 0.05$).

2. Empirical Values for Triplet and Singlet Channels

By measuring $n$-$p$ total scattering cross sections over the energy range $0.1\text{ MeV} \le E_{\text{lab}} \le 10\text{ MeV}$, the parameters are experimentally determined:

$$\text{Triplet Channel: } a_t = +5.424 \pm 0.004\text{ fm}, \quad r_{0t} = 1.759 \pm 0.005\text{ fm}$$ $$\text{Singlet Channel: } a_s = -23.740 \pm 0.020\text{ fm}, \quad r_{0s} = 2.77 \pm 0.05\text{ fm}$$

3. Relation to Deuteron Binding Energy

In the triplet channel, the effective range expansion can be evaluated at the negative bound-state energy $E = -B$, corresponding to $k = i\gamma$:

$$i\gamma \cot\delta_t(i\gamma) = -\gamma = -\frac{1}{a_t} + \frac{1}{2} r_{0t} (i\gamma)^2 = -\frac{1}{a_t} - \frac{1}{2} r_{0t} \gamma^2$$ $$\gamma = \frac{1}{a_t} + \frac{1}{2} r_{0t} \gamma^2 \implies \frac{1}{a_t} = \gamma\left( 1 - \frac{1}{2}\gamma r_{0t} \right)$$

Using $\gamma = 0.2316\text{ fm}^{-1}$ and $r_{0t} = 1.759\text{ fm}$:

$$\frac{1}{a_t} = 0.2316 \left( 1 - \frac{1}{2}(0.2316)(1.759) \right) = 0.2316(1 - 0.2037) = 0.2316(0.7963) \approx 0.1844\text{ fm}^{-1}$$ $$a_t = \frac{1}{0.1844} \approx 5.42\text{ fm}$$

This remarkable agreement demonstrates that effective range theory unifies bound-state properties ($B$) and low-energy scattering data ($a_t, r_{0t}$) in a single coherent framework.

1. Onset of Higher Partial Waves ($P$, $D$, and $F$ Waves)

As the laboratory neutron energy increases above $10\text{ MeV}$, the condition $k R \ll 1$ breaks down:

• At $E_{\text{lab}} = 20\text{ MeV}$: $k \approx 0.49\text{ fm}^{-1} \implies k R \approx 1.0$. $P$-waves ($l = 1$) become significant.
• At $E_{\text{lab}} = 100\text{ MeV}$: $k \approx 1.10\text{ fm}^{-1} \implies k R \approx 2.2$. $D$-waves ($l = 2$) and $F$-waves ($l = 3$) contribute substantially.

The differential cross section becomes highly anisotropic and non-spherical:

$$\frac{d\sigma}{d\Omega} = |f(\theta)|^2 = \left| \frac{1}{k}\sum_{l=0}^\infty (2l+1) e^{i\delta_l} \sin\delta_l P_l(\cos\theta) \right|^2$$

2. The Repulsive Hard Core

Phase-shift analysis of high-energy $p$-$p$ and $n$-$p$ scattering data (Robert Jastrow, 1951) revealed that the $s$-wave phase shift $\delta_0(E)$ decreases with energy, crosses zero at $E_{\text{lab}} \approx 250\text{ to }300\text{ MeV}$, and becomes increasingly negative at higher energies.

A negative phase shift at high energy is the unmistakable signature of a short-range repulsive core. At high incident energies, the de Broglie wavelength is small enough for nucleons to probe the innermost region of the potential:

$$V(r) = +\infty \quad (\text{or } +1\text{ to }2\text{ GeV}) \quad \text{for } r \le r_c \approx 0.4\text{ to }0.5\text{ fm}$$

Physical consequences of the hard core:

• Nuclear Saturation: It prevents nuclei from collapsing under the attractive nuclear forces. Without a repulsive core, the binding energy would scale as $A^2$ rather than $A$.
• Incompressibility of Nuclear Matter: It establishes the nearly constant interior density of atomic nuclei ($\rho_0 \approx 0.16\text{ nucleons/fm}^3$).
• Origin in QCD: The hard core originates from the Pauli exclusion principle acting on the constituent quarks and vector meson ($\omega$) exchange.

1. The Anomalous U-Shaped Differential Cross Section

In classical scattering or ordinary attractive central potential scattering, high-energy collisions produce a strong forward scattering peak ($\theta_{\text{cm}} \approx 0^\circ$) due to diffraction, with the differential cross section dropping monotonically toward backward angles ($\theta_{\text{cm}} \to 180^\circ$).

However, high-energy neutron-proton scattering experiments at $E_{\text{lab}} = 90\text{ to }400\text{ MeV}$ (conducted at Berkeley, Rochester, and Harwell) revealed a startling, highly symmetric U-shaped differential cross section:

$$\frac{d\sigma}{d\Omega}(\theta_{\text{cm}}) \text{ has prominent peaks at both } \theta_{\text{cm}} \approx 0^\circ \text{ and } \theta_{\text{cm}} \approx 180^\circ$$

In fact, the backward cross section at $180^\circ$ is comparable in magnitude to the forward cross section at $0^\circ$!

2. Physical Mechanism: Virtual Charged Pion Exchange

A neutron scattered backward at $180^\circ$ in the center-of-mass frame appears in the laboratory frame as a high-energy proton continuing forward in the beam direction!

This phenomenon is known as charge-exchange scattering:

$$n + p \to p + n$$

In the language of quantum field theory and Feynman diagrams, this reaction proceeds via the exchange of a virtual charged pion ($\pi^+$ or $\pi^-$):

$$n \to p + \pi^-, \quad \text{followed by} \quad \pi^- + p \to n$$

The fast incident neutron emits a virtual $\pi^-$ and transforms into a proton, which continues forward with nearly all of the incident momentum. The target proton absorbs the $\pi^-$ and becomes a slow neutron.

In potential scattering, this corresponds to a Majorana space-exchange potential $V_M(r) P_r$:

$$V(r) = V_W(r) + V_M(r) P_r$$

where $P_r \psi(\vec{r}) = \psi(-\vec{r})$. In partial-wave expansion, $P_r P_l(\cos\theta) = (-1)^l P_l(\cos\theta)$, meaning even-$l$ waves experience $V_W + V_M$ while odd-$l$ waves experience $V_W - V_M$. This alternating sign naturally produces the backward peaking at $\cos\theta = -1$.

1. Molecular Hydrogen States: Ortho vs Para

Molecular hydrogen ($H_2$) exists in two nuclear spin isomers depending on the coupling of the two proton spins:

• Ortho-Hydrogen: Symmetric nuclear spin triplet ($I_{\text{mol}} = 1$). By the Pauli exclusion principle for identical proton fermions, the rotational wavefunction must be antisymmetric: odd rotational quantum numbers $J_{\text{rot}} = 1, 3, 5, \dots$
• Para-Hydrogen: Antisymmetric nuclear spin singlet ($I_{\text{mol}} = 0$). Symmetrical rotational wavefunction: even rotational quantum numbers $J_{\text{rot}} = 0, 2, 4, \dots$ At low temperatures ($T < 20\text{ K}$), liquid hydrogen converts almost entirely to the para-ground state ($J_{\text{rot}} = 0$).

2. Coherent and Incoherent Scattering Amplitudes

When very slow, sub-thermal neutrons ($\lambda \gg \text{molecular bond length } d \approx 0.74\text{ \AA}$) scatter from an $H_2$ molecule, the scattered neutron waves from the two proton centers interfere coherently.

Let $a_t$ and $a_s$ be the triplet and singlet $n$-$p$ scattering lengths. The neutron-proton scattering amplitude operator is:

$$\hat{f} = -\left( \frac{3a_t + a_s}{4} + \frac{a_t - a_s}{4} \vec{\sigma}_n\cdot\vec{\sigma}_p \right) \equiv -(a_{\text{coh}} + 2 a_{\text{inc}} \vec{\sigma}_n\cdot\vec{s}_p)$$

The coherent scattering length and incoherent scattering amplitude are:

$$a_{\text{coh}} = \frac{3}{4}a_t + \frac{1}{4}a_s = \frac{3(5.42) + (-23.74)}{4} = \frac{16.26 - 23.74}{4} = \frac{-7.48}{4} = -1.87\text{ fm}$$ $$a_{\text{inc}} = \frac{a_t - a_s}{4} = \frac{5.42 - (-23.74)}{4} = \frac{29.16}{4} = +7.29\text{ fm}$$

3. Extreme Cross-Section Contrast

For para-hydrogen ($I_{\text{mol}} = 0$), the total nuclear spin of the molecule is zero. The spin-dependent terms cancel identically, leaving only the coherent amplitude:

$$\sigma_{\text{para}} = 4\pi |2 a_{\text{coh}}|^2 \left( \frac{M_{\text{red}}}{M} \right)^2 \propto (3a_t + a_s)^2$$

For ortho-hydrogen ($I_{\text{mol}} = 1$), both coherent and incoherent amplitudes contribute:

$$\sigma_{\text{ortho}} \propto (3a_t + a_s)^2 + 2(a_t - a_s)^2$$

Evaluating the ratio: because $3 a_t \approx 16.3\text{ fm}$ is nearly equal in magnitude and opposite in sign to $a_s \approx -23.7\text{ fm}$, the coherent amplitude $3a_t + a_s$ undergoes strong destructive interference:

$$\sigma_{\text{para}} \approx 3.2\text{ to }4.0\text{ barns}$$ $$\sigma_{\text{ortho}} \approx 125\text{ to }140\text{ barns}$$

The experimental cross-section ratio is colossal:

$$\frac{\sigma_{\text{ortho}}}{\sigma_{\text{para}}} \approx 35 \text{ to } 40$$

This dramatic ratio provided historic verification of the sign and magnitude of the singlet scattering length ($a_s < 0$), establishing that the virtual singlet state is unbound.