Chapter 6

Nucleon-Nucleon Scattering

Probing the Nuclear Force

Nucleon-nucleon (NN) scattering experiments are the primary source of information about the nuclear force. By measuring differential cross sections and polarization observables at various energies, one extracts the NN phase shifts, which encode the strength and range of the interaction in each partial wave channel.

The NN system has four scattering channels: $pp$, $nn$,$np$ (with $T=1$), and $np$ (with $T=0$). The $np$ system accesses both isospin channels, making it especially rich. High-precision NN potentials (Argonne $v_{18}$, CD-Bonn, Nijmegen) fit the world database of over 4000 $pp$ and $np$ scattering data points with$\chi^2/\text{datum} \approx 1$.

Partial Wave Analysis

In the center-of-mass frame, the scattering amplitude for spinless particles is expanded in partial waves:

$$f(\theta) = \sum_{\ell=0}^{\infty} (2\ell + 1)\,\frac{e^{2i\delta_\ell} - 1}{2ik}\,P_\ell(\cos\theta)$$

For nucleons with spin, we use the spectroscopic notation $^{2S+1}L_J$ where$S$ is the total spin, $L$ is the orbital angular momentum, and $J$ is the total angular momentum. The generalized Pauli principle (antisymmetry under exchange of identical fermions) constrains the allowed states:

$$(-1)^{L+S+T} = -1$$

This means:

Singlet ($S=0$), even $L$: $T=1$

States: $^1S_0$, $^1D_2$, $^1G_4$, ... These exist in $pp$, $nn$, and $np$ ($T=1$).

Singlet ($S=0$), odd $L$: $T=0$

States: $^1P_1$, $^1F_3$, ... These exist only in $np$ ($T=0$).

Triplet ($S=1$), even $L$: $T=0$

States: $^3S_1\text{--}^3D_1$, $^3D_2$, ... The tensor force couples $L$ and $L+2$ when $S=1$. The deuteron lives in the coupled $^3S_1\text{--}^3D_1$ channel.

Triplet ($S=1$), odd $L$: $T=1$

States: $^3P_0$, $^3P_1$, $^3P_2\text{--}^3F_2$, ... These exist in all NN channels.

Effective Range Expansion

At low energies, the S-wave ($\ell = 0$) phase shift dominates and can be parameterized model-independently using the effective range expansion (ERE):

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

where $a$ is the scattering length, $r_0$ is the effective range, and$P$ is the shape parameter. The scattering length has a geometric interpretation: it is the intercept of the asymptotic wave function with the $r$-axis.

Scattering Length Sign Convention

$a > 0$: The potential supports a bound state near threshold. For the $^3S_1$ channel, $a_t = +5.424$ fm (deuteron exists).

$a < 0$: No bound state, but the potential is nearly strong enough to bind. For the $^1S_0$ channel, $a_s = -23.714$ fm (virtual state very close to threshold).

$|a| \to \infty$: Unitarity limit. A new bound state is forming at zero energy. The $^1S_0$ np system is remarkably close to this with $|a_s| \gg r_0$.

The low-energy np total cross section is given by:

$$\sigma_{\text{total}} = \frac{\pi}{k^2}\left[3\sin^2\delta_t + \sin^2\delta_s\right] \xrightarrow{k\to 0} \pi\left(3a_t^2 + a_s^2\right) \approx 20.4\;\text{barn}$$

The factor of 3 for the triplet reflects the spin statistical weight ($2S+1 = 3$ for $S=1$ vs 1 for $S=0$).

Charge Independence and Charge Symmetry

Two related symmetries constrain the nuclear force:

Charge Symmetry (CS)

The nuclear force is invariant under interchange of protons and neutrons (rotation by$\pi$ about the 2-axis in isospin space). This implies $V_{pp} = V_{nn}$ in corresponding states. Evidence: the $^1S_0$ scattering lengths are:

$$a_{pp} = -7.806\;\text{fm}, \qquad a_{nn} = -18.95\;\text{fm}$$

After Coulomb correction, the nuclear part of $a_{pp}$ becomes $a_{pp}^N \approx -17.3$ fm, in reasonable agreement with $a_{nn}$. Charge symmetry is broken at the$\sim 1\text{--}3\%$ level by electromagnetic effects and quark mass differences ($m_d - m_u$).

Charge Independence (CI)

The nuclear force is the same in all $T=1$ states regardless of $T_z$:$V_{pp} = V_{nn} = V_{np}^{(T=1)}$. This is a stronger condition than CS. The $^1S_0$ scattering lengths provide a test:

$$a_s^{np} = -23.714\;\text{fm} \quad \text{vs} \quad a_{nn} \approx -18.95\;\text{fm}$$

Charge independence is broken at a larger level ($\sim 5\%$ in the potential), primarily by pion mass differences ($m_{\pi^\pm} \neq m_{\pi^0}$) which affect one-pion exchange.

Yukawa Potential and One-Pion Exchange

Hideki Yukawa (1935) proposed that the nuclear force is mediated by the exchange of massive bosons, leading to a potential of the form:

$$V(r) = -g^2\,\frac{e^{-\mu r}}{r}, \qquad \mu = \frac{m_\text{meson} c}{\hbar}$$

The range of the force is set by the Compton wavelength of the exchanged meson:$\lambda = \hbar/(m_\text{meson}c)$. For the pion ($m_\pi \approx 140$ MeV),$\lambda_\pi = \hbar c/m_\pi c^2 \approx 1.4$ fm, consistent with the observed range of the nuclear force.

The one-pion exchange potential (OPEP) in coordinate space is:

$$V_{\text{OPEP}}(\mathbf{r}) = \frac{f_{\pi NN}^2}{4\pi}\,\frac{m_\pi c^2}{3}\,(\boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2)\left[(\boldsymbol{\sigma}_1 \cdot \boldsymbol{\sigma}_2)\,\frac{e^{-x}}{x} + S_{12}\left(1 + \frac{3}{x} + \frac{3}{x^2}\right)\frac{e^{-x}}{x}\right]$$

where $x = m_\pi r / \hbar$ and $f_{\pi NN}^2/(4\pi) \approx 0.08$ is the pion-nucleon coupling constant. The OPEP contains both a central spin-spin term and a tensor term $S_{12}$. The tensor component is responsible for the D-state admixture in the deuteron.

Multi-Meson Exchange: Building the Full Potential

The NN potential at different distance scales is dominated by different meson exchanges:

Long range ($r > 2$ fm): One-pion exchange (OPEP), well established
Medium range ($1 < r < 2$ fm): Two-pion exchange and scalar $\sigma$ meson, provides main attraction
Short range ($r < 1$ fm): Vector meson ($\omega, \rho$) exchange, provides repulsive core
Very short range ($r < 0.5$ fm): Quark-gluon degrees of freedom become relevant

Properties of the Nuclear Force

Decades of scattering experiments have established the following properties of the nuclear force:

1. Short Range

The nuclear force is negligible beyond $r \approx 2\text{--}3$ fm. This is reflected in the exponential falloff of the Yukawa potential and the saturation of nuclear binding energy. Each nucleon interacts only with its nearest neighbors.

2. Repulsive Core

At distances $r \lesssim 0.5$ fm, the nuclear force becomes strongly repulsive. Evidence comes from the high-energy S-wave phase shift passing through zero and becoming negative. The repulsive core prevents nuclear collapse and is essential for nuclear saturation. In modern NN potentials, the core height is $\sim 1$--2 GeV.

3. Spin Dependence

The force depends on the total spin $S$: the $^3S_1$ (triplet) potential is stronger than $^1S_0$ (singlet), as evidenced by the deuteron existing only in the triplet channel. The spin-orbit force $V_{LS}(r)\,\mathbf{L}\cdot\mathbf{S}$ is also significant, as shown by P-wave splitting ($^3P_0$, $^3P_1$, $^3P_2$ have different phase shifts).

4. Tensor Component

The tensor force $V_T(r)\,S_{12}$ is a non-central force that couples orbital and spin angular momenta. It arises naturally from one-pion exchange and is essential for understanding the deuteron quadrupole moment and the D-state admixture. The tensor force mixes partial waves with $\Delta L = 2$.

5. Approximate Charge Independence

The strong interaction is approximately the same for $pp$, $nn$, and $np$in the same isospin state. This isospin symmetry follows from the approximate$u$-$d$ quark mass degeneracy and is broken at the few-percent level.

Python Simulation: Phase Shifts and Cross Sections

Computes S-wave phase shifts from square well and effective range expansion models, meson exchange potentials, and total np cross sections showing triplet and singlet contributions.

NN Scattering Phase Shifts

Python

Phase shifts, meson exchange potentials, and np cross sections

script.py169 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ──────────────────────────────────────────────
# Nucleon-Nucleon Scattering Phase Shifts
# Partial wave analysis and Yukawa potential
# ──────────────────────────────────────────────

hbar2_2mu = 41.47  # hbar^2/(2*mu) in MeV*fm^2, mu = m_N/2
hbar_c = 197.327   # MeV*fm

# Yukawa (one-pion exchange) potential
m_pi = 139.6  # MeV, pion mass
mu_pi = m_pi / hbar_c  # fm^-1
g2_4pi = 14.0  # pion-nucleon coupling constant squared / 4pi

def yukawa_potential(r, V0, mu_range):
    """Yukawa potential V(r) = -V0 * exp(-mu*r) / (mu*r)"""
    return -V0 * np.exp(-mu_range * r) / (mu_range * r)

def square_well_phase(k, V0, R):
    """Analytic S-wave phase shift for square well"""
    if V0 + hbar2_2mu * k**2 < 0:
        return 0.0
    K = np.sqrt((V0 + hbar2_2mu * k**2) / hbar2_2mu)
    delta = np.arctan(k * np.tan(K * R) / K) - k * R
    # Unwrap to keep continuity
    return delta

# Energy range
E_lab = np.linspace(0.1, 350, 500)  # MeV lab energy
k_cm = np.sqrt(E_lab / (2 * hbar2_2mu))  # CM momentum in fm^-1

# Phase shifts for different channels using square well approximation
# Triplet S (3S1) - attractive, has bound state
V0_3S1 = 36.0  # MeV
R_3S1 = 2.1    # fm

# Singlet S (1S0) - attractive but no bound state
V0_1S0 = 25.0  # MeV (weaker than triplet)
R_1S0 = 2.5    # fm

delta_3S1 = np.array([square_well_phase(k, V0_3S1, R_3S1) for k in k_cm])
delta_1S0 = np.array([square_well_phase(k, V0_1S0, R_1S0) for k in k_cm])

# Convert to degrees
delta_3S1_deg = np.degrees(delta_3S1)
delta_1S0_deg = np.degrees(delta_1S0)

# Effective range expansion
a_t = 5.424   # fm, triplet scattering length
r_t = 1.759   # fm, triplet effective range
a_s = -23.714 # fm, singlet scattering length (negative = no bound state)
r_s = 2.73    # fm, singlet effective range

# ERE phase shifts
k_ere = np.linspace(0.01, 1.5, 200)
E_ere = hbar2_2mu * k_ere**2  # MeV

# k*cot(delta) = -1/a + r*k^2/2
kcotd_t = -1.0/a_t + 0.5*r_t*k_ere**2
delta_ere_t = np.arctan(k_ere / kcotd_t)
# Fix branch for positive scattering length (bound state)
for i in range(len(delta_ere_t)):
    if kcotd_t[i] < 0 and delta_ere_t[i] < 0:
        delta_ere_t[i] += np.pi

kcotd_s = -1.0/a_s + 0.5*r_s*k_ere**2
delta_ere_s = np.arctan(k_ere / kcotd_s)
for i in range(len(delta_ere_s)):
    if kcotd_s[i] < 0 and delta_ere_s[i] < 0:
        delta_ere_s[i] += np.pi

# Cross sections
# Total np cross section from S-waves
sigma_t = np.pi / k_ere**2 * 3 * np.sin(delta_ere_t)**2  # triplet (weight 3)
sigma_s = np.pi / k_ere**2 * 1 * np.sin(delta_ere_s)**2  # singlet (weight 1)
sigma_total = sigma_t + sigma_s
sigma_total_barn = sigma_total * 10.0  # fm^2 to barn (1 barn = 100 fm^2... actually 1 barn = 100 fm^2)
# Convert fm^2 to barns: 1 barn = 100 fm^2, so divide by 100
sigma_total_barn = sigma_total / 100.0

fig, axes = plt.subplots(2, 2, figsize=(14, 11))
fig.patch.set_facecolor('#0a0a1a')

for row in axes:
    for ax in row:
        ax.set_facecolor('#0a0a1a')
        for spine in ax.spines.values():
            spine.set_edgecolor('#334155')

# Top left: Phase shifts vs energy (square well model)
axes[0,0].plot(E_lab, delta_3S1_deg, color='#ef4444', linewidth=2.5, label='$^3S_1$ (triplet)')
axes[0,0].plot(E_lab, delta_1S0_deg, color='#3b82f6', linewidth=2.5, label='$^1S_0$ (singlet)')
axes[0,0].axhline(y=0, color='#64748b', linewidth=0.5, linestyle='--')
axes[0,0].axhline(y=90, color='#64748b', linewidth=0.5, linestyle=':', alpha=0.5)
axes[0,0].set_xlabel('$E_{lab}$ [MeV]', color='#94a3b8', fontsize=11)
axes[0,0].set_ylabel('$\\delta_0$ [degrees]', color='#94a3b8', fontsize=11)
axes[0,0].set_title('S-wave Phase Shifts (Square Well)', color='white', fontsize=12)
axes[0,0].legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')
axes[0,0].tick_params(colors='#94a3b8')

# Top right: ERE phase shifts
axes[0,1].plot(E_ere, np.degrees(delta_ere_t), color='#ef4444', linewidth=2.5,
               label=f'$^3S_1$: a={a_t:.1f} fm, r={r_t:.1f} fm')
axes[0,1].plot(E_ere, np.degrees(delta_ere_s), color='#3b82f6', linewidth=2.5,
               label=f'$^1S_0$: a={a_s:.1f} fm, r={r_s:.1f} fm')
axes[0,1].axhline(y=0, color='#64748b', linewidth=0.5, linestyle='--')
axes[0,1].set_xlabel('$E_{cm}$ [MeV]', color='#94a3b8', fontsize=11)
axes[0,1].set_ylabel('$\\delta_0$ [degrees]', color='#94a3b8', fontsize=11)
axes[0,1].set_title('ERE Phase Shifts', color='white', fontsize=12)
axes[0,1].legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')
axes[0,1].tick_params(colors='#94a3b8')

# Bottom left: Yukawa potential and variations
r_pot = np.linspace(0.3, 6.0, 500)
V_opep = yukawa_potential(r_pot, g2_4pi * mu_pi * hbar2_2mu * 2, mu_pi)
# Central part of OPEP (simplified)
V_central_opep = -(g2_4pi / 3.0) * (mu_pi**2 * hbar_c) * np.exp(-mu_pi * r_pot) / (mu_pi * r_pot)
# Heavier meson exchange (sigma, omega, rho)
V_sigma = -300.0 * np.exp(-2.5 * r_pot) / (2.5 * r_pot)  # scalar sigma
V_omega = 200.0 * np.exp(-3.97 * r_pot) / (3.97 * r_pot)  # vector omega (repulsive)
V_total = V_central_opep + V_sigma + V_omega

axes[1,0].plot(r_pot, V_central_opep, color='#22c55e', linewidth=2, label='OPEP (pion)')
axes[1,0].plot(r_pot, V_sigma, color='#3b82f6', linewidth=2, linestyle='--', label='Sigma exchange')
axes[1,0].plot(r_pot, V_omega, color='#f59e0b', linewidth=2, linestyle='--', label='Omega exchange')
axes[1,0].plot(r_pot, V_total, color='#ef4444', linewidth=2.5, label='Total (schematic)')
axes[1,0].axhline(y=0, color='#64748b', linewidth=0.5, linestyle='--')
axes[1,0].set_xlabel('r [fm]', color='#94a3b8', fontsize=11)
axes[1,0].set_ylabel('V(r) [MeV]', color='#94a3b8', fontsize=11)
axes[1,0].set_title('Meson Exchange Potentials', color='white', fontsize=12)
axes[1,0].set_xlim(0.3, 5)
axes[1,0].set_ylim(-120, 100)
axes[1,0].legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')
axes[1,0].tick_params(colors='#94a3b8')

# Bottom right: Total np cross section
axes[1,1].semilogy(E_ere, sigma_total_barn * 1000, color='#ef4444', linewidth=2.5,
                    label='S-wave total (ERE)')
axes[1,1].semilogy(E_ere, sigma_t / 100.0 * 1000, color='#22c55e', linewidth=1.5,
                    linestyle='--', label='Triplet $^3S_1$')
axes[1,1].semilogy(E_ere, sigma_s / 100.0 * 1000, color='#3b82f6', linewidth=1.5,
                    linestyle='--', label='Singlet $^1S_0$')
axes[1,1].set_xlabel('$E_{cm}$ [MeV]', color='#94a3b8', fontsize=11)
axes[1,1].set_ylabel('$\\sigma$ [mb]', color='#94a3b8', fontsize=11)
axes[1,1].set_title('np Total Cross Section', color='white', fontsize=12)
axes[1,1].legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')
axes[1,1].tick_params(colors='#94a3b8')

plt.tight_layout()
plt.savefig('nn_scattering.png', dpi=130, bbox_inches='tight',
            facecolor=fig.get_facecolor())

print("Nucleon-Nucleon Scattering Analysis")
print("=" * 50)
print(f"\nEffective Range Parameters:")
print(f"  Triplet (3S1): a_t = {a_t:+.3f} fm,  r_t = {r_t:.3f} fm")
print(f"  Singlet (1S0): a_s = {a_s:+.3f} fm,  r_s = {r_s:.3f} fm")
print(f"\nNote: a_t > 0  => bound state exists (deuteron)")
print(f"      a_s < 0  => no bound state in singlet channel")
print(f"      |a_s| >> range => near-threshold virtual state")
print(f"\nLow-energy np cross section (thermal):")
sigma_0 = np.pi * (3*a_t**2 + a_s**2)  # fm^2
print(f"  sigma_0 = pi*(3*a_t^2 + a_s^2) = {sigma_0:.0f} fm^2 = {sigma_0/100:.1f} barn")
print(f"  Experimental: ~20.4 barn")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran Implementation: Scattering Analysis

Tabulates effective range parameters, tests charge independence, connects to the deuteron bound state, and computes phase shifts at multiple energies.

NN Scattering Calculator

Fortran

Effective range analysis, charge independence tests, and phase shift tables

nn_scattering.f90113 lines

program nn_scattering
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  real(dp), parameter :: pi = 3.14159265358979323846_dp
  real(dp), parameter :: hbar2_2mu = 41.47_dp  ! MeV*fm^2
  real(dp), parameter :: hbar_c = 197.327_dp   ! MeV*fm

! Effective range parameters
  real(dp), parameter :: a_t = 5.424_dp      ! triplet scattering length (fm)
  real(dp), parameter :: r_t = 1.759_dp      ! triplet effective range (fm)
  real(dp), parameter :: a_s = -23.714_dp    ! singlet scattering length (fm)
  real(dp), parameter :: r_s = 2.73_dp       ! singlet effective range (fm)

! pp scattering (Coulomb corrected)
  real(dp), parameter :: a_pp = -7.8063_dp   ! pp scattering length (fm)
  real(dp), parameter :: r_pp = 2.794_dp     ! pp effective range (fm)

! nn scattering
  real(dp), parameter :: a_nn = -18.95_dp    ! nn scattering length (fm)

real(dp) :: k, E_cm, kcot_t, kcot_s, delta_t, delta_s
  real(dp) :: sigma_t, sigma_s, sigma_total
  real(dp) :: kappa_d
  integer :: i

! Pion exchange parameters
  real(dp), parameter :: m_pi = 139.6_dp     ! MeV
  real(dp), parameter :: mu_pi = 0.7073_dp   ! fm^-1 (m_pi/hbar_c)
  real(dp), parameter :: g2_4pi = 14.0_dp    ! coupling

write(*,'(A)') '=== Nucleon-Nucleon Scattering Analysis ==='
  write(*,'(A)') ''

! Effective range parameters summary
  write(*,'(A)') '--- Effective Range Parameters ---'
  write(*,'(A)') ''
  write(*,'(A,A12,A12,A12)') '  Channel   ', 'a [fm]', 'r [fm]', '|a|/r'
  write(*,'(A)') '  ' // repeat('-', 40)
  write(*,'(A,F12.3,F12.3,F12.1)') '  np 3S1    ', a_t, r_t, abs(a_t)/r_t
  write(*,'(A,F12.3,F12.3,F12.1)') '  np 1S0    ', a_s, r_s, abs(a_s)/r_s
  write(*,'(A,F12.3,F12.3,F12.1)') '  pp 1S0    ', a_pp, r_pp, abs(a_pp)/r_pp
  write(*,'(A,F12.3,A,A)') '  nn 1S0    ', a_nn, '      ---', '     ---'

! Charge symmetry test
  write(*,'(A)') ''
  write(*,'(A)') '--- Charge Independence Test ---'
  write(*,'(A,F10.3,A)') '  np singlet:  a_s  = ', a_s, ' fm'
  write(*,'(A,F10.3,A)') '  nn singlet:  a_nn = ', a_nn, ' fm'
  write(*,'(A,F10.3,A)') '  pp singlet:  a_pp = ', a_pp, ' fm (Coulomb corrected)'
  write(*,'(A)') '  Charge symmetry: a_nn ~ a_pp? Approximately yes'
  write(*,'(A)') '  Charge independence: a_nn ~ a_pp ~ a_s(np)? Broken at ~25%'
  write(*,'(A)') '  => Nuclear force is approximately charge-independent'

! Deuteron connection
  kappa_d = sqrt(2.2246_dp / hbar2_2mu)
  write(*,'(A)') ''
  write(*,'(A)') '--- Deuteron-Scattering Connection ---'
  write(*,'(A,F10.4,A)') '  kappa_d = ', kappa_d, ' fm^-1'
  write(*,'(A,F10.4,A)') '  1/a_t   = ', 1.0_dp/a_t, ' fm^-1'
  write(*,'(A)') '  At leading order: kappa ~ 1/a_t (within 20%)'
  write(*,'(A)') '  Effective range correction improves agreement'

! Phase shift table
  write(*,'(A)') ''
  write(*,'(A)') '--- S-wave Phase Shifts (ERE) ---'
  write(*,'(A)') ''
  write(*,'(A,A10,A12,A12,A14)') '  ', 'E_cm(MeV)', 'delta_t', 'delta_s', 'sigma(fm^2)'
  write(*,'(A)') '  ' // repeat('-', 50)

do i = 1, 10
    E_cm = real(i, dp) * 5.0_dp
    k = sqrt(E_cm / hbar2_2mu)

! Triplet
    kcot_t = -1.0_dp/a_t + 0.5_dp*r_t*k**2
    if (abs(kcot_t) > 1.0d-10) then
      delta_t = atan(k / kcot_t)
      if (kcot_t < 0.0_dp .and. delta_t < 0.0_dp) delta_t = delta_t + pi
    else
      delta_t = pi / 2.0_dp
    end if

! Singlet
    kcot_s = -1.0_dp/a_s + 0.5_dp*r_s*k**2
    if (abs(kcot_s) > 1.0d-10) then
      delta_s = atan(k / kcot_s)
      if (kcot_s < 0.0_dp .and. delta_s < 0.0_dp) delta_s = delta_s + pi
    else
      delta_s = pi / 2.0_dp
    end if

! Cross sections
    sigma_t = pi/k**2 * 3.0_dp * sin(delta_t)**2
    sigma_s = pi/k**2 * sin(delta_s)**2
    sigma_total = sigma_t + sigma_s

write(*,'(A,F10.1,F12.2,F12.2,F14.1)') '  ', E_cm, &
         delta_t * 180.0_dp / pi, delta_s * 180.0_dp / pi, sigma_total
  end do

! Nuclear force properties summary
  write(*,'(A)') ''
  write(*,'(A)') '--- Nuclear Force Properties ---'
  write(*,'(A)') '  1. Short range: R ~ 1-2 fm (pion Compton wavelength)'
  write(*,'(A,F6.2,A)') '     Pion range: 1/mu_pi = ', 1.0_dp/mu_pi, ' fm'
  write(*,'(A)') '  2. Strongly attractive at r ~ 1 fm (V ~ -50 to -100 MeV)'
  write(*,'(A)') '  3. Repulsive core at r < 0.5 fm (omega exchange)'
  write(*,'(A)') '  4. Spin dependent: V(3S1) /= V(1S0)'
  write(*,'(A)') '  5. Tensor component from pion exchange: S_12 operator'
  write(*,'(A)') '  6. Approximately charge independent'
  write(*,'(A,F8.3,A)') '  7. OPEP coupling: g^2/4pi = ', g2_4pi, ''

end program nn_scattering

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