← Part III: Nuclear Reactions
Chapter 11

Compound Nucleus Reactions

Bohr's Compound Nucleus Hypothesis

In 1936, Niels Bohr proposed that nuclear reactions at low to moderate energies proceed through two independent stages: formation of a compound nucleus (CN) and its subsequent decay. The incoming projectile is absorbed by the target, and its energy is shared among all nucleons through multiple collisions, forming a thermally equilibrated system.

The central postulate is the independence hypothesis: the decay of the compound nucleus is independent of its mode of formation. The CN "forgets" how it was created β€” only the conserved quantities (energy, angular momentum, parity) determine the decay:

$$\sigma(a + A \to b + B) = \sigma_{\text{form}}(a + A \to C^*) \times G_{\text{decay}}(C^* \to b + B)$$

where $\sigma_{\text{form}}$ is the formation cross section and $G_{\text{decay}}$ is the branching ratio for the particular exit channel.

Breit-Wigner Resonance Formula

For an isolated resonance at energy $E_0$ with total spin $J$, the cross section for channel $a \to b$ is given by the Breit-Wigner single-level formula:

$$\sigma_{a \to b}(E) = \pi \lambdabar^2 \, g_J \, \frac{\Gamma_a \Gamma_b}{(E - E_0)^2 + (\Gamma/2)^2}$$

where $\lambdabar = \hbar/p$ is the reduced de Broglie wavelength, and the statistical spin factor is:

$$g_J = \frac{2J + 1}{(2I_a + 1)(2I_A + 1)}$$

Here $I_a$ and $I_A$ are the spins of projectile and target. The total width$\Gamma$ is the sum of all partial widths:

$$\Gamma = \sum_c \Gamma_c = \Gamma_n + \Gamma_\gamma + \Gamma_p + \Gamma_\alpha + \Gamma_f + \cdots$$

Resonance Parameters

Each partial width $\Gamma_c$ represents the probability of decay through channel $c$. The partial width is related to the reduced width $\gamma_c^2$ and the penetrability $P_\ell$:

$$\Gamma_c = 2 P_\ell(E) \gamma_c^2$$

The lifetime of the compound state is $\tau = \hbar/\Gamma$. Typical CN lifetimes are $10^{-16}$ to $10^{-22}$ s β€” long compared to the nuclear transit time of $\sim 10^{-22}$ s.

At the peak of a resonance ($E = E_0$), the cross section reaches its maximum:

$$\sigma_{\max} = 4\pi\lambdabar^2 \, g_J \, \frac{\Gamma_a \Gamma_b}{\Gamma^2}$$

Nuclear Level Density

As excitation energy increases, nuclear levels become so dense that individual resonances overlap. The level density $\rho(E^*)$ is a crucial input for statistical models. The Fermi gas model gives:

$$\rho(E^*) = \frac{\sqrt{\pi}}{12} \frac{\exp\left(2\sqrt{aU}\right)}{a^{1/4} U^{5/4}}$$

where $U = E^* - \Delta$ is the effective excitation energy (back-shifted by the pairing energy $\Delta$), and $a$ is the level density parameter:

$$a \approx \frac{A}{8} \text{ MeV}^{-1}$$

The nuclear temperature is defined as:

$$T = \sqrt{U/a}$$

The mean level spacing $D = 1/\rho$ at the neutron separation energy is measured from resolved resonances. For heavy nuclei, $D \sim 1$ eV at $E^* \sim 6$ MeV, meaning millions of levels exist in the compound nucleus.

Hauser-Feshbach Theory

When resonances overlap (high excitation, heavy nuclei), the Hauser-Feshbach (1952) statistical model averages over many resonances. The cross section for $a + A \to b + B$ is:

$$\sigma_{ab} = \frac{\pi\lambdabar_a^2}{(2I_a+1)(2I_A+1)} \sum_{J,\pi} (2J+1) \frac{T_a^{J\pi} T_b^{J\pi}}{\sum_c T_c^{J\pi}}$$

where $T_c^{J\pi}$ are transmission coefficients for channel $c$ with compound nucleus spin $J$ and parity $\pi$. The transmission coefficients are obtained from the optical model and are related to the average partial widths:

$$T_c = 2\pi \frac{\langle\Gamma_c\rangle}{D}$$

This approach is the workhorse of nuclear reaction calculations for astrophysics, reactor physics, and medical isotope production. Modern codes like TALYS and EMPIRE implement sophisticated versions of this theory.

Evaporation Spectra & Giant Resonances

When the compound nucleus has high excitation energy, it can emit multiple particles sequentially, analogous to evaporation from a hot liquid drop. The Weisskopf evaporation spectrum for emitted particles is:

$$\frac{d\sigma}{d\epsilon} \propto \sigma_{\text{inv}}(\epsilon) \cdot \epsilon \cdot \rho(E^* - B - \epsilon)$$

For neutrons (no Coulomb barrier), $\sigma_{\text{inv}} \approx \text{const}$, giving the characteristic evaporation spectrum:

$$\frac{dN}{d\epsilon} \propto \epsilon \cdot \exp(-\epsilon / T)$$

which peaks at $\epsilon = T$ (the nuclear temperature), typically 1-2 MeV.

Giant Resonances

Giant resonances are broad, high-lying collective excitations that dominate the photoabsorption cross section:

  • β€’ Giant Dipole Resonance (GDR): $E \approx 78 A^{-1/3}$ MeV, $\Gamma \sim 4$-8 MeV. Protons oscillate against neutrons (Goldhaber-Teller mode).
  • β€’ Giant Monopole Resonance (GMR): $E \approx 80 A^{-1/3}$ MeV, "breathing mode." Directly measures the nuclear incompressibility $K_\infty$.
  • β€’ Giant Quadrupole Resonance (GQR): $E \approx 63 A^{-1/3}$ MeV. Oscillation between oblate and prolate shapes. Both isoscalar and isovector modes exist.

The GDR is particularly important as it determines photonuclear cross sections. The Thomas-Reiche-Kuhn (TRK) sum rule constrains the total integrated cross section:

$$\int_0^\infty \sigma_\gamma(E) \, dE = \frac{2\pi^2 e^2 \hbar}{m_p c} \frac{NZ}{A} \approx 60 \frac{NZ}{A} \text{ mbΒ·MeV}$$

Simulation: Resonances & Evaporation

This simulation computes Breit-Wigner resonance cross sections for a compound nucleus with multiple levels, and generates the neutron evaporation spectrum from the Hauser-Feshbach statistical model.

Python
script.py112 lines

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran Simulation: Hauser-Feshbach Decay

This Fortran program implements the statistical model of compound nucleus decay, computing evaporation spectra and branching ratios for neutron, proton, and alpha emission channels, including the effect of Coulomb barriers.

Fortran
program.f90101 lines

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

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