Self-Assessment Quiz — Chapter 20

Test your understanding of the core concepts before moving on. Try to answer each question before checking the solutions at the end.

Q1. (Multiple Choice) The fissility parameter $x$ is defined as:

(a) $E_C / E_S$ (b) $2E_C / E_S$ (c) $E_C / 2E_S$ (d) $E_S / 2E_C$

Q2. (True/False) A nucleus with fissility parameter $x > 1$ has no fission barrier and is unstable against even infinitesimal deformations.

Q3. (Short Answer) In the liquid drop model of fission, which energy term acts as the restoring force opposing deformation, and which acts as the driving force favoring deformation?

Q4. (Multiple Choice) The approximate value of $(Z^2/A)_{\text{crit}}$ above which nuclei have no fission barrier is:

(a) $\sim 25$ (b) $\sim 35$ (c) $\sim 50$ (d) $\sim 100$

Q5. (True/False) The fission barrier height for $^{236}$U is approximately 200 MeV — roughly equal to the total energy released in fission.

Q6. (Multiple Choice) $^{235}$U is fissile with thermal neutrons but $^{238}$U is not. The fundamental reason is:

(a) $^{235}$U has a lower fission barrier than $^{238}$U (b) The compound nucleus $^{236}$U is even-even, gaining pairing energy that makes the excitation energy exceed the barrier (c) $^{235}$U has a larger fission cross section because it is lighter (d) $^{238}$U absorbs thermal neutrons but emits them before fission can occur

Q7. (Short Answer) State the general rule: which type of target nuclei (odd-$N$ or even-$N$) are fissile with thermal neutrons, and why?

Q8. (Multiple Choice) The fission product mass distribution for thermal fission of $^{235}$U is:

(a) Symmetric, peaked at $A \approx 118$ (b) Asymmetric, with peaks near $A \approx 95$ and $A \approx 140$ (c) Flat (all fragment masses equally probable) (d) Symmetric with a secondary peak near $A \approx 140$

Q9. (True/False) The asymmetric fission product distribution is predicted by the liquid drop model.

Q10. (Short Answer) What nuclear shell closures in the heavy fission fragment explain the preference for asymmetric fission? Give the magic numbers involved.

Q11. (Multiple Choice) The average number of prompt neutrons per thermal fission of $^{235}$U is approximately:

(a) 1.0 (b) 2.4 (c) 5.0 (d) 10

Q12. (Short Answer) Define "delayed neutrons" and explain why they are crucial for reactor control. What is the typical timescale associated with delayed neutrons?

Q13. (True/False) The delayed neutron fraction $\beta$ for $^{235}$U is approximately 6.5%.

Q14. (Multiple Choice) The total energy released per fission of $^{235}$U is approximately:

(a) 2 MeV (b) 20 MeV (c) 200 MeV (d) 2,000 MeV

Q15. (Short Answer) Of the approximately 200 MeV released per fission, which component carries the most energy? Which component is unrecoverable?

Q16. (Multiple Choice) A nuclear reactor operating at steady power has an effective multiplication factor of:

(a) $k_{\text{eff}} < 1$ (b) $k_{\text{eff}} = 1$ (c) $k_{\text{eff}} > 1$ (d) $k_{\text{eff}} = \bar{\nu}$

Q17. (Short Answer) State the four-factor formula $k_\infty = \eta f p \varepsilon$ and briefly explain what each factor represents physically.

Q18. (Multiple Choice) A reactor is "prompt critical" when:

(a) $k_{\text{eff}} = 1$ (b) $\rho = 0$ (c) $\rho = \beta$ (reactivity equals the delayed neutron fraction) (d) All neutrons are delayed neutrons

Q19. (True/False) In the problem of nuclear waste, fission products require isolation for hundreds of years, while transuranic actinides require isolation for thousands to millions of years.

Q20. (Short Answer) Explain how the binding energy per nucleon curve (Chapter 4) predicts both the energy release in fission and the energy release in fusion. Why does fission work for heavy nuclei and fusion for light nuclei?

Solutions

Q1. (c) $x = E_C / 2E_S = a_C Z(Z-1) / (2 a_S A)$. The factor of 2 arises because the condition for instability is $E_C = 2E_S$ (the Coulomb deformation coefficient is $-1/5$ while the surface deformation coefficient is $+2/5$).

Q2. True. When $x > 1$, the quadratic coefficient in the deformation energy is negative, meaning any infinitesimal deformation lowers the energy. There is no barrier, and the nucleus disassembles immediately.

Q3. The surface energy acts as the restoring force (deformation increases surface area, increasing surface energy). The Coulomb energy acts as the driving force (deformation spreads the charge, decreasing Coulomb energy). Fission occurs when the Coulomb driving force overwhelms the surface restoring force.

Q4. (c) $(Z^2/A)_{\text{crit}} = 2a_S/a_C \approx 2 \times 17.8/0.714 \approx 50$.

Q5. False. The fission barrier for $^{236}$U is approximately 5.8 MeV — a modest activation energy. The 200 MeV is the total energy released after the barrier is overcome, mostly from the Coulomb repulsion between the separating fragments.

Q6. (b) When $^{235}$U (odd-$N$) captures a neutron, the compound nucleus $^{236}$U is even-even, gaining pairing energy. This makes $S_n = 6.55$ MeV, which exceeds the fission barrier of 5.8 MeV. For $^{238}$U (even-$N$), the compound nucleus $^{239}$U is odd-$N$, with a lower $S_n = 4.81$ MeV that falls below the barrier of 6.2 MeV.

Q7. Odd-$N$ targets are fissile because the captured neutron pairs with the previously unpaired neutron, forming an even-$N$ compound nucleus with extra pairing energy. This makes $S_n$ large enough to exceed the fission barrier. Even-$N$ targets form odd-$N$ compound nuclei without the pairing bonus, giving smaller $S_n$ that typically falls below the barrier.

Q8. (b) Asymmetric, with peaks near $A \approx 95$ (light fragment) and $A \approx 140$ (heavy fragment). Symmetric fission is suppressed by about two orders of magnitude.

Q9. False. The liquid drop model predicts symmetric fission. The asymmetry arises from nuclear shell effects — specifically, the doubly-magic shell closures near $^{132}$Sn in the heavy fragment.

Q10. The heavy fragment preferentially forms near $Z = 50$ (magic proton number) and $N = 82$ (magic neutron number), corresponding to the doubly-magic region near $^{132}$Sn. The extra binding energy from these shell closures favors the asymmetric split.

Q11. (b) $\bar{\nu}_p \approx 2.43$ for thermal fission of $^{235}$U.

Q12. Delayed neutrons are neutrons emitted following the beta decay of certain fission products (called precursors). The beta-decay parent has a half-life of seconds to minutes, so these neutrons appear on a much longer timescale than prompt neutrons. They constitute about 0.65% of all fission neutrons for $^{235}$U. They are crucial for reactor control because they stretch the effective neutron generation time from $\sim 10^{-4}$ s (prompt) to $\sim 0.1$ s (effective with delayed neutrons), making mechanical control feasible. Typical delayed neutron timescale: weighted mean half-life $\approx 13$ seconds.

Q13. False. The delayed neutron fraction $\beta \approx 0.0065$, which is 0.65%, not 6.5%.

Q14. (c) Approximately 200 MeV per fission (208 MeV total, of which 196 MeV is recoverable).

Q15. The kinetic energy of the fission fragments carries the most energy (~169 MeV out of ~208 MeV total). The antineutrino energy (~12 MeV) is unrecoverable because neutrinos interact too weakly to be absorbed in the reactor.

Q16. (b) A reactor operating at steady power has $k_{\text{eff}} = 1$ (critical). Each generation of neutrons produces exactly as many neutrons as the previous generation, maintaining constant power.

Q17. $k_\infty = \eta \cdot f \cdot p \cdot \varepsilon$, where: $\eta$ = reproduction factor (neutrons produced per thermal neutron absorbed in fuel); $f$ = thermal utilization factor (fraction of thermal neutrons absorbed in fuel vs. total absorptions); $p$ = resonance escape probability (fraction of neutrons that slow down without being captured in resonance absorptions, especially in $^{238}$U); $\varepsilon$ = fast fission factor (enhancement from fissions caused by fast neutrons before thermalization).

Q18. (c) A reactor is prompt critical when $\rho = \beta$, meaning it is critical on prompt neutrons alone without needing delayed neutrons. This is the threshold beyond which the reactor responds on the prompt neutron generation timescale ($\sim 10^{-4}$ s) rather than the delayed neutron timescale ($\sim$ seconds), making control extremely difficult.

Q19. True. Fission products (mainly $^{137}$Cs and $^{90}$Sr) have half-lives of $\sim 30$ years, so after $\sim 300$ years (10 half-lives), their activity drops by a factor of $\sim 1000$. Transuranic actinides ($^{239}$Pu at 24,110 yr, $^{237}$Np at $2.14 \times 10^6$ yr) require isolation for thousands to millions of years.

Q20. The binding energy per nucleon $B/A$ has a peak near $A \approx 56$–$62$. For heavy nuclei (actinides, $A \approx 235$), $B/A \approx 7.6$ MeV, which is less than the peak value. Splitting them into medium-mass fragments ($B/A \approx 8.5$ MeV) increases $B/A$, releasing the difference as energy — this is fission. For light nuclei (hydrogen, helium), $B/A$ is much less than the peak. Combining them into heavier nuclei increases $B/A$ — this is fusion. Both processes move nuclei toward the peak of the $B/A$ curve from opposite sides. Fission works for heavy nuclei because there is "room" to increase $B/A$ by splitting; fusion works for light nuclei because there is room to increase $B/A$ by combining.