Key Takeaways — Chapter 18
Core Concepts
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The compound nucleus model (Bohr, 1936): When a projectile is absorbed by a target nucleus, the kinetic energy is shared among all nucleons via nucleon-nucleon collisions. The resulting compound nucleus is a thermalized system that has lost all memory of how it was formed.
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The Bohr independence hypothesis (threshold concept): The cross section for $a + A \to C^* \to b + B$ factorizes into a formation cross section (depending only on the entrance channel) and a decay probability (depending only on $E^*$, $J$, $\pi$ of the compound nucleus). Formation and decay are independent.
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The Breit-Wigner resonance formula: $$\sigma_{a \to b}(E) = \pi\lambdabar^2 \, g_J \, \frac{\Gamma_a \Gamma_b}{(E - E_R)^2 + (\Gamma/2)^2}$$ - $E_R$ = resonance energy, $\Gamma$ = total width (sum of all partial widths) - $g_J = (2J+1)/[(2i+1)(2I+1)]$ = statistical spin factor - Lorentzian line shape with FWHM = $\Gamma$ - Derived from the S-matrix pole structure
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Resonance widths encode physics: - $\Gamma = \hbar / \tau$ (uncertainty relation) - $\Gamma_n$ (neutron width) varies strongly, follows Porter-Thomas distribution - $\Gamma_\gamma$ (radiation width) is nearly constant ($\sim 25$ meV for heavy nuclei) - $\Gamma_f$ (fission width) present only for fissile/fissionable nuclei
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Nuclear level densities (Bethe/Fermi gas model): $$\rho(E^*) = \frac{\sqrt{\pi}}{12} \frac{\exp(2\sqrt{aE^*})}{a^{1/4}(E^*)^{5/4}}$$ - Level density parameter: $a \approx A/8$ MeV$^{-1}$ - Exponential growth with excitation energy - Shell effects wash out at high excitation
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The Hauser-Feshbach statistical model extends the compound nucleus model to the regime of overlapping resonances by averaging over many levels using transmission coefficients from the optical model.
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The $1/v$ law for neutron capture: $\sigma_\gamma(E) \propto 1/\sqrt{E} \propto 1/v$ at energies below the first resonance. Arises from $\lambdabar^2 \propto 1/E$ and $\Gamma_n \propto \sqrt{E}$ in the Breit-Wigner formula.
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The resonance integral $I_\gamma = \int \sigma_\gamma(E) \, dE/E$ quantifies total neutron absorption in the epithermal (resonance) region, weighted by the $1/E$ slowing-down spectrum. Critical for reactor design.
Key Numbers to Remember
| Quantity | Typical value | Context |
|---|---|---|
| First resonance of n + ${}^{238}$U | $E_R = 6.67$ eV | Reactor physics benchmark |
| Typical $\Gamma_\gamma$ (heavy nuclei) | $\sim 25$ meV | Nearly constant |
| Level density parameter | $a \approx A/8$ MeV$^{-1}$ | Fermi gas model |
| $D_0$ for ${}^{238}$U + n | $\sim 20$ eV | s-wave spacing |
| $D_0$ for ${}^{56}$Fe + n | $\sim 25$ keV | Light nucleus, much larger |
| Resonance integral, ${}^{238}$U | 275 b | Reactor criticality |
| Thermal $\sigma_\gamma$, ${}^{238}$U | 2.68 b | $1/v$ regime |
Connections
| Concept | Connects to |
|---|---|
| Partial wave analysis, S-matrix | Chapter 17 (reaction fundamentals) |
| Shell model, magic numbers | Chapter 6 (shell model) |
| Fission width, fissile vs. fissionable | Chapter 20 (fission) |
| Resonance escape probability | Chapter 26 (reactor physics) |
| $s$-process, $r$-process abundances | Chapters 22–24 (nucleosynthesis) |
| Level densities, statistical model | Chapter 19 (contrast with direct reactions) |
| Neutron star mergers, GW170817 | Chapter 25 (neutron stars) |
Common Mistakes
- Confusing $\Gamma$ (total width) with $\Gamma_n$ (neutron partial width). The total width is the sum: $\Gamma = \Gamma_n + \Gamma_\gamma + \Gamma_f + \ldots$
- Forgetting the $\sqrt{E}$ energy dependence of the s-wave neutron width when evaluating the Breit-Wigner formula away from the resonance energy.
- Applying the compound nucleus model at too high an energy, where direct reactions dominate.
- Treating the level density parameter $a$ as exactly $A/8$ — this is only an approximation. Shell corrections, pairing, and deformation all modify $a$.
- Confusing the $1/v$ law (which describes the cross section's energy dependence below the first resonance) with the behavior at resonance energies.