Quiz — Chapter 18

Instructions: Select the best answer for each question. Each question has exactly one correct answer.

Q1. The Bohr independence hypothesis states that:

(A) The compound nucleus always decays by the same channel through which it was formed. (B) The formation and decay of the compound nucleus are independent — the decay probabilities do not depend on the entrance channel. (C) The compound nucleus is formed only in direct reactions. (D) The energy of the compound nucleus is independent of the projectile energy.

Q2. In the Breit-Wigner formula $\sigma(E) = \pi\lambdabar^2 g_J \Gamma_a \Gamma_b / [(E - E_R)^2 + (\Gamma/2)^2]$, what does $\Gamma$ represent?

(A) The partial width for the entrance channel only. (B) The radiation width $\Gamma_\gamma$. (C) The total width, equal to the sum of all partial widths. (D) The level spacing between adjacent resonances.

Q3. A nuclear resonance has total width $\Gamma = 0.10$ eV. What is the lifetime of the compound nuclear state?

(A) $6.6 \times 10^{-12}$ s (B) $6.6 \times 10^{-15}$ s (C) $6.6 \times 10^{-16}$ s (D) $6.6 \times 10^{-22}$ s

Q4. The statistical spin factor $g_J$ in the Breit-Wigner formula accounts for:

(A) The probability that the compound nucleus decays by gamma emission. (B) The fraction of entrance-channel spin states that can couple to the compound nuclear state of spin $J$. (C) The effect of Coulomb barriers on the cross section. (D) The density of states at the resonance energy.

Q5. The neutron capture cross section of most nuclei follows a $1/v$ dependence at low energies. This arises because:

(A) The nuclear force is inversely proportional to velocity. (B) The neutron de Broglie wavelength increases as $1/v$, and the s-wave neutron width $\Gamma_n \propto \sqrt{E}$. (C) All resonances are at zero energy. (D) The radiation width $\Gamma_\gamma$ is proportional to $1/v$.

Q6. Which of the following is approximately constant from resonance to resonance in heavy nuclei?

(A) The neutron width $\Gamma_n$. (B) The radiation width $\Gamma_\gamma$. (C) The fission width $\Gamma_f$. (D) The resonance energy $E_R$.

Q7. In the Fermi gas model for nuclear level densities, the level density parameter $a$ is approximately:

(A) $a \approx A/80$ MeV$^{-1}$ (B) $a \approx A/8$ MeV$^{-1}$ (C) $a \approx 8A$ MeV$^{-1}$ (D) $a \approx A^2$ MeV$^{-1}$

Q8. The level density in the Bethe formula grows as:

(A) $\rho \propto E^2$ (B) $\rho \propto \exp(E/kT)$ (C) $\rho \propto \exp(2\sqrt{aE})$ (D) $\rho \propto A^2$

Q9. The average level spacing $D_0$ for s-wave neutron resonances in ${}^{238}$U is about 20 eV, while for ${}^{56}$Fe it is about 25 keV. The main reason for this difference is:

(A) Iron has a larger neutron capture cross section. (B) Uranium has a much higher level density at the neutron separation energy due to its larger mass number and higher number of valence nucleons. (C) Iron is doubly magic. (D) The Coulomb barrier in uranium is much higher.

Q10. The Hauser-Feshbach model is most useful when:

(A) Individual resonances are well resolved and isolated. (B) The reaction proceeds by a direct mechanism. (C) Many resonances overlap and a statistical description is appropriate. (D) The projectile energy is below the Coulomb barrier.

Q11. The resonance integral $I_\gamma = \int \sigma_\gamma(E) \, dE/E$ is important in reactor physics because:

(A) It gives the total cross section at thermal energies. (B) It quantifies the total neutron capture probability for neutrons slowing down through the epithermal region in a $1/E$ flux spectrum. (C) It determines the critical mass of the reactor. (D) It measures the Doppler broadening of resonances.

Q12. Doppler broadening of nuclear resonances at higher temperatures:

(A) Decreases both the peak cross section and the effective resonance integral. (B) Increases the peak cross section but narrows the resonance. (C) Decreases the peak cross section, broadens the resonance, and increases the effective resonance integral (in a thick sample). (D) Has no effect on reactor reactivity.

Q13. In the $s$-process of stellar nucleosynthesis, the abundance peaks at $A \approx 88, 138, 208$ occur because:

(A) These nuclei are the most tightly bound. (B) Nuclei with magic neutron numbers ($N = 50, 82, 126$) have small neutron capture cross sections, causing material to accumulate at these points. (C) These nuclei have the shortest beta-decay half-lives. (D) The $s$-process path terminates at these mass numbers.

Q14. The Porter-Thomas distribution describes:

(A) The distribution of resonance energies. (B) The distribution of total widths. (C) The distribution of reduced neutron widths, which follow a chi-squared distribution with one degree of freedom. (D) The angular distribution of scattered particles.

Q15. In a time-of-flight (TOF) measurement, the neutron energy is determined from the measured flight time $t$ over a known distance $L$. To improve the energy resolution at fixed energy, one should:

(A) Increase the flight path length $L$. (B) Decrease the flight path length $L$. (C) Increase the sample thickness. (D) Decrease the neutron energy.

Answer Key

  1. (B) — The Bohr independence hypothesis: formation and decay are independent.
  2. (C) — $\Gamma$ is the total width, the sum of all partial widths.
  3. (B) — $\tau = \hbar/\Gamma = 6.582 \times 10^{-16}\,\text{eV}\cdot\text{s} / 0.10\,\text{eV} = 6.6 \times 10^{-15}$ s.
  4. (B) — $g_J = (2J+1)/[(2i+1)(2I+1)]$ is the spin coupling fraction.
  5. (B) — The $\lambdabar^2 \propto 1/E$ factor combined with $\Gamma_n \propto \sqrt{E}$ gives $\sigma \propto 1/\sqrt{E} \propto 1/v$.
  6. (B) — $\Gamma_\gamma$ is a sum over many gamma transitions and fluctuates little.
  7. (B) — $a \approx A/8$ MeV$^{-1}$ is the standard Fermi gas estimate.
  8. (C) — The Bethe formula: $\rho \propto \exp(2\sqrt{aE})/E^{5/4}$; the exponential dominates.
  9. (B) — Uranium at $A = 238$ has far more nucleons contributing to the level density at the neutron separation energy.
  10. (C) — Hauser-Feshbach is the statistical limit for overlapping resonances.
  11. (B) — The resonance integral weights $\sigma_\gamma$ by the $1/E$ slowing-down flux.
  12. (C) — Doppler broadening reduces the peak but widens the resonance; the net effect in a thick sample is increased absorption (negative temperature coefficient).
  13. (B) — Magic neutron numbers give small cross sections and high $s$-process abundances ($\sigma N \approx$ const).
  14. (C) — The Porter-Thomas distribution is a chi-squared distribution with $\nu = 1$.
  15. (A) — Longer flight path gives smaller $\Delta E/E$ because $\Delta t/t$ decreases for fixed $\Delta t$.