Chapter 7

Cross Sections & Resonances

Reaction Cross Sections

The cross section $\sigma$ is the fundamental measure of the probability for a nuclear reaction. It has dimensions of area and is measured in barns (1 b = $10^{-24}$ cm$^2$ = 100 fm$^2$):

$$R = n_{\text{beam}}\,n_{\text{target}}\,\sigma\,v$$

where R is the reaction rate, $n$ are number densities, and $v$ is the relative velocity. The geometric cross section of a nucleus is:

$$\sigma_{\text{geom}} = \pi R^2 = \pi r_0^2 A^{2/3} \approx 45 A^{2/3} \text{ mb}$$

Actual cross sections can be much larger (resonances) or much smaller than geometric, depending on the energy and reaction mechanism.

Q-Values and Kinematics

For a reaction $a + A \to b + B$, the Q-value is the net energy released:

$$Q = (M_a + M_A - M_b - M_B)c^2 = T_b + T_B - T_a$$

For endothermic reactions ($Q < 0$), there is a threshold energy below which the reaction cannot proceed:

$$T_{\text{thresh}} = -Q\left(1 + \frac{M_a}{M_A} + \frac{|Q|}{2M_A c^2}\right) \approx -Q\frac{M_a + M_A}{M_A}$$

Compound Nucleus Model

Niels Bohr proposed that many nuclear reactions proceed through a compound nucleus: the projectile is absorbed, its energy is shared among all nucleons, and the compound nucleus decays independently of how it was formed. Near an isolated resonance at energy $E_0$, the cross section is given by the Breit-Wigner formula:

$$\sigma_{ab}(E) = \pi\lambdabar^2 \frac{(2J+1)}{(2s_a+1)(2I_A+1)} \frac{\Gamma_a\Gamma_b}{(E-E_0)^2 + (\Gamma/2)^2}$$

where:

- $\lambdabar = \hbar/p$ is the reduced de Broglie wavelength
- $J$ is the spin of the compound nucleus state
- $\Gamma_a, \Gamma_b$ are partial widths for entrance and exit channels
- $\Gamma = \sum_i \Gamma_i$ is the total width (related to the lifetime: $\tau = \hbar/\Gamma$)

The resonance has a Lorentzian shape with FWHM = $\Gamma$. For neutron capture on heavy nuclei, the level density is high and resonances overlap at higher energies, leading to the statistical (Hauser-Feshbach) treatment.

Python Simulation: Breit-Wigner Resonance

Visualization of the Breit-Wigner resonance cross section for U-238 neutron capture, showing the famous 6.67 eV resonance and the first few resonance structures.

Breit-Wigner Resonance Cross Section

Python

U-238 neutron capture resonance structure using Breit-Wigner formula

script.py86 lines

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

# ──────────────────────────────────────────────
# Breit-Wigner Resonance Cross Section
# ──────────────────────────────────────────────

# Breit-Wigner formula for isolated resonance
# sigma(E) = pi * lambdabar^2 * g * Gamma_a*Gamma_b / ((E-E0)^2 + (Gamma/2)^2)

def breit_wigner(E, E0, Gamma_a, Gamma_b, Gamma_tot, l=0, m_proj=938.0, m_targ=11000.0):
    """Breit-Wigner resonance cross section (barns)"""
    # de Broglie wavelength squared in fm^2
    hbar_c = 197.3  # MeV fm
    mu = m_proj * m_targ / (m_proj + m_targ)  # reduced mass MeV
    k2 = 2 * mu * E / hbar_c**2  # fm^-2
    lambdabar2 = 1.0 / k2  # fm^2

# Statistical spin factor (simplified)
    g = 1.0

sigma = np.pi * lambdabar2 * g * Gamma_a * Gamma_b / ((E - E0)**2 + (Gamma_tot/2)**2)
    return sigma * 10.0  # convert fm^2 to millibarns (*10) then to barns

# Resonance parameters (typical neutron capture)
E0 = 6.67  # eV (famous U-238 resonance)
Gamma_n = 1.52e-3  # eV (neutron width)
Gamma_gamma = 0.026  # eV (capture width)
Gamma_tot = Gamma_n + Gamma_gamma

E = np.linspace(5.0, 8.5, 2000)  # eV

# Cross section (simplified, showing shape)
sigma = np.pi * (0.4516 / E) * Gamma_n * Gamma_gamma / ((E - E0)**2 + (Gamma_tot/2)**2)
# 0.4516 eV*barns is lambdabar^2 in appropriate units at ~6.67 eV for U-238

fig, axes = plt.subplots(1, 2, figsize=(14, 6))
fig.patch.set_facecolor('#0a0a1a')

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

# Left: Linear scale
axes[0].plot(E, sigma * 1e24, color='#ef4444', linewidth=2)
axes[0].set_xlabel('Neutron Energy [eV]', color='#94a3b8', fontsize=12)
axes[0].set_ylabel('Cross Section [arb. units]', color='#94a3b8', fontsize=12)
axes[0].set_title('Breit-Wigner: U-238 (n,gamma) at 6.67 eV', color='white', fontsize=13)
axes[0].tick_params(colors='#94a3b8')
axes[0].axvline(x=E0, color='#fbbf24', linestyle='--', alpha=0.5, label=f'E0 = {E0} eV')
axes[0].axvline(x=E0 - Gamma_tot/2, color='#64748b', linestyle=':', alpha=0.5)
axes[0].axvline(x=E0 + Gamma_tot/2, color='#64748b', linestyle=':', alpha=0.5)
axes[0].legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')

# Right: Multiple resonances
E_wide = np.linspace(0.5, 50, 5000)
E_resonances = [6.67, 20.87, 36.68]
Gamma_ns = [1.52e-3, 10.1e-3, 34.1e-3]
Gamma_gs = [0.026, 0.023, 0.024]

sigma_total = np.zeros_like(E_wide)
colors = ['#ef4444', '#f59e0b', '#22c55e']
for j, (E0_j, Gn_j, Gg_j) in enumerate(zip(E_resonances, Gamma_ns, Gamma_gs)):
    Gt_j = Gn_j + Gg_j
    sig_j = np.pi * (0.4516 / E_wide) * Gn_j * Gg_j / ((E_wide - E0_j)**2 + (Gt_j/2)**2)
    sigma_total += sig_j
    axes[1].axvline(x=E0_j, color=colors[j], linestyle='--', alpha=0.4)

axes[1].semilogy(E_wide, sigma_total * 1e24, color='#ef4444', linewidth=1.5)
axes[1].set_xlabel('Neutron Energy [eV]', color='#94a3b8', fontsize=12)
axes[1].set_ylabel('Cross Section [arb. units]', color='#94a3b8', fontsize=12)
axes[1].set_title('U-238 Resonance Structure (first 3 resonances)', color='white', fontsize=13)
axes[1].tick_params(colors='#94a3b8')
axes[1].grid(True, alpha=0.15, color='#334155')

plt.tight_layout()
plt.savefig('breit_wigner.png', dpi=130, bbox_inches='tight',
            facecolor=fig.get_facecolor())
print(f"U-238 first resonance: E0 = {E_resonances[0]} eV")
print(f"Total width: Gamma = {Gamma_tot*1000:.1f} meV")
print(f"Neutron width: {Gamma_ns[0]*1000:.2f} meV")
print(f"Capture width: {Gamma_gs[0]*1000:.1f} meV")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran Implementation

Calculates Q-values and threshold energies for various nuclear reactions.

Q-Value and Threshold Energy Calculator

Fortran

Computes reaction Q-values and threshold energies for various nuclear reactions

qvalue_calculator.f9061 lines

program qvalue_calculator
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  real(dp), parameter :: amu = 931.494_dp  ! MeV/c^2

integer, parameter :: n_reactions = 7
  character(len=30) :: reaction(n_reactions)
  real(dp) :: M_a(n_reactions), M_A(n_reactions)
  real(dp) :: M_b(n_reactions), M_B(n_reactions)
  real(dp) :: Q, T_thresh
  integer :: i

! Reaction data (masses in amu)
  reaction(1) = 'D(d,n)He-3              '
  M_a(1) = 2.01410_dp; M_A(1) = 2.01410_dp
  M_b(1) = 1.00866_dp; M_B(1) = 3.01603_dp

reaction(2) = 'D(d,p)T                 '
  M_a(2) = 2.01410_dp; M_A(2) = 2.01410_dp
  M_b(2) = 1.00783_dp; M_B(2) = 3.01605_dp

reaction(3) = 'T(d,n)He-4              '
  M_a(3) = 2.01410_dp; M_A(3) = 3.01605_dp
  M_b(3) = 1.00866_dp; M_B(3) = 4.00260_dp

reaction(4) = 'Li-7(p,alpha)He-4       '
  M_a(4) = 1.00783_dp; M_A(4) = 7.01600_dp
  M_b(4) = 4.00260_dp; M_B(4) = 4.00260_dp

reaction(5) = 'N-14(n,p)C-14           '
  M_a(5) = 1.00866_dp; M_A(5) = 14.00307_dp
  M_b(5) = 1.00783_dp; M_B(5) = 14.00324_dp

reaction(6) = 'O-16(p,alpha)N-13       '
  M_a(6) = 1.00783_dp; M_A(6) = 15.99491_dp
  M_b(6) = 4.00260_dp; M_B(6) = 13.00574_dp

reaction(7) = 'U-235(n,fission)        '
  M_a(7) = 1.00866_dp; M_A(7) = 235.04393_dp
  M_b(7) = 94.91360_dp; M_B(7) = 138.91780_dp  ! approximate

write(*,'(A)') '=== Nuclear Reaction Q-Value Calculator ==='
  write(*,'(A)') ''
  write(*,'(A)') 'Reaction                    Q (MeV)    Threshold (MeV)'
  write(*,'(A)') '--------                    -------    ---------------'

do i = 1, n_reactions
    Q = (M_a(i) + M_A(i) - M_b(i) - M_B(i)) * amu

if (Q < 0.0_dp) then
      T_thresh = -Q * (1.0_dp + M_a(i)/M_A(i))
      write(*,'(A30, F10.3, F14.3)') reaction(i), Q, T_thresh
    else
      write(*,'(A30, F10.3, A)') reaction(i), Q, '     (exothermic)'
    end if
  end do

write(*,'(A)') ''
  write(*,'(A)') 'D-T fusion has the largest cross section at low energies'
  write(*,'(A)') 'due to the large Q-value and favorable resonance structure.'
end program qvalue_calculator

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

← Gamma Decay Fission →