Chapter 12

Direct Nuclear Reactions

Direct vs. Compound Reactions

Direct reactions occur on a fast timescale ($\sim 10^{-22}$ s), comparable to the transit time of the projectile across the nucleus. Unlike compound nucleus reactions where the energy is shared among all nucleons, direct reactions involve only a few nucleons at the nuclear surface. Key characteristics include:

• Forward-peaked angular distributions (not symmetric about 90°)
• Cross sections vary smoothly with energy (no sharp resonances)
• Strong selectivity for specific final states
• Sensitive to nuclear structure (single-particle properties)

Direct reactions are the primary tool for measuring spectroscopic factors, orbital angular momenta, and single-particle energies — fundamental inputs for the nuclear shell model.

Stripping and Pickup Reactions

Transfer reactions involve the exchange of one or more nucleons between projectile and target. The most important types are:

Stripping Reactions

One or more nucleons are stripped from the projectile onto the target. The classic example is the (d,p) reaction, where a deuteron deposits a neutron:

$$d + A \to p + (A+1)^* \qquad \text{(neutron stripping)}$$

Similarly, the (d,n) reaction transfers a proton:

$$d + A \to n + (A+1)^* \qquad \text{(proton stripping)}$$

The transferred nucleon carries orbital angular momentum $\ell$, which is determined from the shape of the angular distribution. This directly reveals the quantum numbers of the final state.

Pickup Reactions

The inverse process, where the projectile picks up a nucleon from the target:

$$(p,d): \quad p + A \to d + (A-1)^*$$

Pickup reactions probe the hole states in the target nucleus, providing complementary information to stripping reactions. Together, they map the single-particle structure above and below the Fermi surface.

Knockout Reactions

At intermediate energies (~50-200 MeV/u), quasi-free knockout reactions (e,e'p) and (p,2p) directly remove a nucleon and measure its momentum distribution:

$$e + A \to e' + p + (A-1)^*$$

The missing momentum and energy reveal the bound-state wave function and separation energy of the removed nucleon.

Distorted Wave Born Approximation (DWBA)

The DWBA is the standard theoretical framework for analyzing direct reactions. The transition amplitude for a (d,p) reaction transferring a neutron with quantum numbers$n\ell j$ is:

$$T_{fi} = \langle \chi_p^{(-)} \, \phi_{n\ell j} \,|\, V_{np} \,|\, \chi_d^{(+)} \, \phi_d \rangle$$

where $\chi_d^{(+)}$ and $\chi_p^{(-)}$ are distorted waves for the incoming deuteron and outgoing proton (generated by the optical model), $\phi_d$ is the deuteron internal wave function, $\phi_{n\ell j}$ is the bound-state wave function of the transferred neutron, and $V_{np}$ is the neutron-proton interaction.

The differential cross section is related to the DWBA prediction by:

$$\frac{d\sigma}{d\Omega} = \frac{\mu_d \mu_p}{(2\pi\hbar^2)^2} \frac{k_p}{k_d} |T_{fi}|^2$$

In practice, the cross section factorizes as:

$$\left(\frac{d\sigma}{d\Omega}\right)_{\text{exp}} = S_{n\ell j} \cdot C^2 \cdot \left(\frac{d\sigma}{d\Omega}\right)_{\text{DWBA}}$$

where $S_{n\ell j}$ is the spectroscopic factor and $C$ is an isospin Clebsch-Gordan coefficient. The spectroscopic factor measures the overlap between the actual nuclear state and the assumed single-particle configuration.

The Optical Model Potential

The optical model describes elastic scattering and generates the distorted waves needed for DWBA calculations. The potential has the form:

$$U(r) = V_C(r) - V f(r, R_V, a_V) - i\left[W f(r, R_W, a_W) - 4a_D W_D \frac{d}{dr}f(r, R_D, a_D)\right] + V_{so}(\boldsymbol{\ell}\cdot\boldsymbol{s})\frac{1}{r}\frac{d}{dr}f(r, R_{so}, a_{so})$$

where $f(r, R, a) = [1 + \exp((r-R)/a)]^{-1}$ is the Woods-Saxon form factor. The terms are:

• Real volume ($V \sim 50$ MeV): attractive nuclear mean field
• Imaginary volume ($W$): absorption due to non-elastic processes
• Imaginary surface ($W_D$): absorption concentrated at the surface
• Spin-orbit ($V_{so} \sim 6$ MeV): responsible for shell structure
• Coulomb ($V_C$): uniform sphere approximation for $r < R_C$

Global optical model parametrizations (Becchetti-Greenlees, Koning-Delaroche) provide parameters as functions of $A$, $Z$, and $E$, enabling predictions for unmeasured systems.

Angular Distributions & Momentum Matching

The angular distribution of a (d,p) reaction directly reveals the orbital angular momentum $\ell$ of the transferred neutron. In the plane wave limit, the cross section is proportional to:

$$\frac{d\sigma}{d\Omega} \propto |j_\ell(qR)|^2$$

where $j_\ell$ is the spherical Bessel function and $q$ is the momentum transfer:

$$q^2 = k_d^2 + k_p^2 - 2k_d k_p \cos\theta$$

Characteristic Patterns

The momentum matching condition $qR \approx \ell$ determines where the angular distribution peaks:

• $\ell = 0$: peak at $\theta = 0°$ (forward peak, no centrifugal barrier)
• $\ell = 1$: peak near $\theta \sim 15°$-25°
• $\ell = 2$: peak near $\theta \sim 25°$-35°
• $\ell = 3$: peak near $\theta \sim 35°$-45°

The shift of the first maximum to larger angles with increasing $\ell$ provides an unambiguous signature of the transferred angular momentum, making direct reactions the premier tool for assigning quantum numbers to nuclear states.

Spectroscopic Factors

The spectroscopic factor $S_{n\ell j}$ measures the probability that a given nuclear state has the assumed single-particle configuration. For a state $|\Psi_f\rangle$ in the$(A+1)$-body system:

$$S_{n\ell j} = |\langle \Psi_f^{A+1} | a_{n\ell j}^\dagger | \Psi_i^A \rangle|^2$$

The sum rule constrains the total strength:

$$\sum_f S_{n\ell j}^{(\text{stripping})} = 2j + 1 - \langle n_{n\ell j} \rangle$$

where $\langle n_{n\ell j}\rangle$ is the average occupation of the orbit in the target. For closed-shell targets (all orbits fully occupied or empty), $S \approx 2j+1$ for empty orbits and $S \approx 0$ for filled orbits.

Experimentally, spectroscopic factors are typically 50-70% of the independent particle model values due to short-range correlations and configuration mixing, providing crucial evidence for nucleon-nucleon correlations in the nuclear medium.

Charge Exchange Reactions

Charge exchange (CE) reactions transfer isospin without changing the mass number. They probe the isospin structure of nuclei:

$$(p,n): \quad T_z \to T_z - 1 \qquad (n,p): \quad T_z \to T_z + 1$$

At zero momentum transfer ($q = 0$), the (p,n) cross section is proportional to the Gamow-Teller transition strength $B(GT)$:

$$\frac{d\sigma}{d\Omega}\bigg|_{q=0} \propto \hat{\sigma}\tau \cdot B(GT) = |\langle f | \sum_k \sigma_k \tau_k^\pm | i \rangle|^2$$

This connection between the (p,n) reaction and weak interaction (beta decay) matrix elements makes charge exchange reactions essential for neutrino physics, double-beta decay studies, and astrophysical r-process calculations.

Simulation: DWBA Angular Distributions

This simulation computes plane-wave Born approximation (PWBA) angular distributions for (d,p) reactions with different transferred angular momenta, demonstrating how the$\ell$-value is determined from the angular pattern. It also plots the optical model potential components.

Python

script.py147 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
def spherical_jn(n, z):
    """Spherical Bessel function j_n(z) using recurrence."""
    z = np.asarray(z, dtype=float)
    out = np.zeros_like(z)
    mask = np.abs(z) > 1e-15
    zm = z[mask] if z.ndim > 0 else z
    if n == 0:
        out[mask] = np.sin(zm) / zm
    elif n == 1:
        out[mask] = np.sin(zm) / zm**2 - np.cos(zm) / zm
    elif n == 2:
        out[mask] = (3.0 / zm**3 - 1.0 / zm) * np.sin(zm) - 3.0 * np.cos(zm) / zm**2
    else:
        j0 = np.sin(zm) / zm
        j1 = np.sin(zm) / zm**2 - np.cos(zm) / zm
        jm, jc = j0, j1
        for k in range(1, n):
            jn = (2*k + 1) / zm * jc - jm
            jm, jc = jc, jn
        out[mask] = jc
    return out

# ──────────────────────────────────────────────
# DWBA Angular Distributions for (d,p) Reactions
# Momentum Matching and Spectroscopic Factors
# ──────────────────────────────────────────────

# Simplified DWBA: angular distribution ~ |j_l(q*R)|^2
# where l = transferred orbital angular momentum
# q = momentum transfer, R = nuclear radius

hbar_c = 197.3  # MeV fm

# Target parameters
A = 40  # Ca-40
R = 1.25 * A**(1.0/3.0)  # fm

# Beam energy
E_d = 10.0  # MeV deuteron energy (lab)
E_p = 12.0  # MeV proton energy (approximate, depends on Q-value)

# Momentum transfer calculation
# In plane wave approximation: q depends on angle
# q^2 = k_d^2 + k_p^2 - 2*k_d*k_p*cos(theta)

m_d = 2 * 938.0  # deuteron mass MeV
m_p = 938.0       # proton mass MeV

k_d = np.sqrt(2 * m_d * E_d) / hbar_c  # fm^-1
k_p = np.sqrt(2 * m_p * E_p) / hbar_c  # fm^-1

theta = np.linspace(0.01, np.pi, 500)  # scattering angle (CM)
theta_deg = np.degrees(theta)

# Momentum transfer
q = np.sqrt(k_d**2 + k_p**2 - 2*k_d*k_p*np.cos(theta))

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: Angular distributions for different l-transfers
colors = ['#ef4444', '#f59e0b', '#22c55e', '#3b82f6', '#a855f7']
labels = ['l=0 (s)', 'l=1 (p)', 'l=2 (d)', 'l=3 (f)', 'l=4 (g)']

for l in range(5):
    # PWBA form factor: |j_l(qR)|^2
    jl = np.array([spherical_jn(l, qi * R) for qi in q])
    dsigma = jl**2
    # Normalize to peak
    dsigma /= (np.max(dsigma) + 1e-30)

axes[0].semilogy(theta_deg, dsigma + 1e-6, color=colors[l], linewidth=2.0, label=labels[l])

axes[0].set_xlabel('CM Angle [degrees]', color='#94a3b8', fontsize=12)
axes[0].set_ylabel('d sigma/d Omega (normalized)', color='#94a3b8', fontsize=12)
axes[0].set_title('PWBA (d,p) Angular Distributions', color='white', fontsize=14)
axes[0].tick_params(colors='#94a3b8')
axes[0].grid(True, alpha=0.2, color='#334155')
axes[0].legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')
axes[0].set_ylim(1e-4, 2)
axes[0].set_xlim(0, 90)

# Right: Optical model potential (Woods-Saxon)
r = np.linspace(0.01, 15.0, 500)

# Woods-Saxon parameters (typical for protons on Ca-40)
V0 = 50.0     # MeV (real depth)
W0 = 10.0     # MeV (imaginary depth)
r0 = 1.25     # fm
a0 = 0.65     # fm (diffuseness)
R_ws = r0 * A**(1.0/3.0)

# Coulomb parameters
Z_p = 1
Z_t = 20
R_C = 1.3 * A**(1.0/3.0)

# Potentials
V_real = -V0 / (1 + np.exp((r - R_ws) / a0))
V_imag = -W0 / (1 + np.exp((r - R_ws) / a0))

# Coulomb potential
V_coul = np.where(r >= R_C,
                  Z_p * Z_t * 1.44 / r,
                  Z_p * Z_t * 1.44 / (2*R_C) * (3 - (r/R_C)**2))

# Spin-orbit (Thomas form)
V_so = -0.44 * V0 * (1.0/r) * a0 * np.exp((r - R_ws)/a0) / (1 + np.exp((r - R_ws)/a0))**2

axes[1].plot(r, V_real, color='#ef4444', linewidth=2.5, label='Real (V)')
axes[1].plot(r, V_imag, color='#f59e0b', linewidth=2.5, label='Imaginary (W)')
axes[1].plot(r, V_coul, color='#22c55e', linewidth=2.5, label='Coulomb')
axes[1].plot(r, V_so * 10, color='#a855f7', linewidth=2.0, linestyle='--', label='Spin-orbit (x10)')
axes[1].axhline(y=0, color='#64748b', linestyle='-', alpha=0.3)
axes[1].set_xlabel('Radius [fm]', color='#94a3b8', fontsize=12)
axes[1].set_ylabel('Potential [MeV]', color='#94a3b8', fontsize=12)
axes[1].set_title('Optical Model Potential (p + Ca-40)', color='white', fontsize=14)
axes[1].tick_params(colors='#94a3b8')
axes[1].grid(True, alpha=0.2, color='#334155')
axes[1].legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')
axes[1].set_xlim(0, 12)
axes[1].set_ylim(-60, 30)

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

# Momentum matching condition
print("Momentum matching for (d,p) reactions:")
print(f"  k_d = {k_d:.3f} fm^-1")
print(f"  k_p = {k_p:.3f} fm^-1")
for l in range(5):
    # Peak of j_l is near qR ~ l+1 (roughly)
    q_match = (l + 0.5) / R if l > 0 else 0.01
    theta_match = np.arccos((k_d**2 + k_p**2 - q_match**2) / (2*k_d*k_p + 1e-30))
    if not np.isnan(theta_match):
        print(f"  l={l}: peak near theta ~ {np.degrees(theta_match):.1f} deg")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran Simulation: Spectroscopic Factors

This Fortran program tabulates spectroscopic factors for $^{40}$Ca(d,p)$^{41}$Ca reactions, demonstrates momentum matching conditions for different $\ell$-transfers, and computes global optical model parameters.

Fortran

program.f9089 lines

program direct_reactions
  implicit none
  ! ──────────────────────────────────────────
  ! Spectroscopic Factors and (d,p) Reactions
  ! Shell model predictions for single-particle
  ! strength in direct transfer reactions
  ! ──────────────────────────────────────────

integer :: l, j2, n
  real(8) :: S_factor, E_level, Q_value
  real(8) :: k_d, k_p, q, theta_peak
  real(8) :: hbar_c, m_d, m_p, E_d, A, R
  real(8), parameter :: pi = 3.14159265358979d0

hbar_c = 197.3d0  ! MeV fm
  A = 40.0d0         ! Ca-40 target

write(*,'(A)') '# Direct Nuclear Reactions: (d,p) on Ca-40'
  write(*,'(A)') '# =========================================='
  write(*,*)

! Shell model predictions for Ca-41 levels populated by Ca-40(d,p)
  write(*,'(A)') '# Ca-40(d,p)Ca-41 Spectroscopic Factors'
  write(*,'(A)') '# State       nlj         S_theory   E*(MeV)    l-transfer'
  write(*,'(A)') '# ---------------------------------------------------------------'

! Ground state: 7/2- (1f7/2)
  write(*,'(A,F8.3,F10.3,I6)') '# g.s. 7/2-   1f_{7/2}   ', 0.95d0, 0.0d0, 3

! First excited states
  write(*,'(A,F8.3,F10.3,I6)') '# 3/2+        2s_{1/2}   ', 0.0d0, 1.943d0, 0
  write(*,'(A,F8.3,F10.3,I6)') '# 3/2-        1d_{3/2}   ', 0.90d0, 2.010d0, 2
  write(*,'(A,F8.3,F10.3,I6)') '# 1/2-        2p_{1/2}   ', 0.85d0, 2.461d0, 1
  write(*,'(A,F8.3,F10.3,I6)') '# 3/2-        2p_{3/2}   ', 0.88d0, 3.614d0, 1
  write(*,'(A,F8.3,F10.3,I6)') '# 5/2-        1f_{5/2}   ', 0.80d0, 3.944d0, 3

write(*,*)
  write(*,'(A)') '# Momentum Matching Analysis'
  write(*,'(A)') '# ========================='

E_d = 10.0d0  ! MeV deuteron lab energy
  m_d = 2.0d0 * 938.0d0
  m_p = 938.0d0
  R = 1.25d0 * A**(1.0d0/3.0d0)

k_d = sqrt(2.0d0 * m_d * E_d) / hbar_c
  write(*,'(A,F8.4,A)') '# k_d = ', k_d, ' fm^-1'

! For different Q-values (different final states)
  write(*,*)
  write(*,'(A)') '# l   Q(MeV)   k_p(fm^-1)  q_match(fm^-1)  theta_peak(deg)'

do l = 0, 4
    Q_value = 5.0d0 - 1.0d0 * dble(l)  ! approximate Q-values
    k_p = sqrt(2.0d0 * m_p * (E_d + Q_value)) / hbar_c

! Momentum matching: peak of j_l at q*R ~ l for l>0
    if (l == 0) then
      q = 0.01d0
    else
      q = (dble(l) + 0.5d0) / R
    end if

! theta from momentum triangle
    theta_peak = acos((k_d**2 + k_p**2 - q**2) / (2.0d0*k_d*k_p))
    theta_peak = theta_peak * 180.0d0 / pi

write(*,'(I3, F8.2, F12.4, F14.4, F16.1)') l, Q_value, k_p, q, theta_peak
  end do

write(*,*)
  write(*,'(A)') '# Optical Model Parameters (Global Becchetti-Greenlees)'
  write(*,'(A)') '# ====================================================='
  write(*,'(A)') '# For protons on Ca-40 at E = 15 MeV:'
  write(*,'(A,F8.2,A)') '# V0     = ', 54.0d0 - 0.32d0*15.0d0 + 24.0d0*(40.0d0-2*20.0d0)/40.0d0, ' MeV'
  write(*,'(A,F8.2,A)') '# r0     = ', 1.17d0, ' fm'
  write(*,'(A,F8.2,A)') '# a0     = ', 0.75d0, ' fm'
  write(*,'(A,F8.2,A)') '# W_surf = ', 11.8d0 - 0.25d0*15.0d0 + 12.0d0*(40.0d0-2*20.0d0)/40.0d0, ' MeV'
  write(*,'(A,F8.2,A)') '# V_so   = ', 6.2d0, ' MeV'

write(*,*)
  write(*,'(A)') '# Sum Rule Check:'
  write(*,'(A)') '# For a closed-shell target, the sum of spectroscopic'
  write(*,'(A)') '# factors for a given nlj orbital should equal (2j+1):'
  write(*,'(A,I3)') '# Sum S(1f7/2) expected: ', 8
  write(*,'(A,I3)') '# Sum S(2p3/2) expected: ', 4
  write(*,'(A,I3)') '# Sum S(2p1/2) expected: ', 2

end program direct_reactions

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

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