Self-Assessment Quiz — Chapter 26

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 four factors in the four-factor formula $k_\infty = \eta f p \varepsilon$ are, in order:

(a) Enrichment factor, thermal utilization, resonance escape, fast fission factor (b) Reproduction factor, thermal utilization, resonance escape probability, fast fission factor (c) Neutron yield, fuel fraction, parasitic absorption, epithermal fission factor (d) Reproduction factor, fuel fraction, resonance capture, delayed neutron factor

Q2. (True/False) A reactor with $k_\infty > 1$ will always sustain a chain reaction, regardless of its size.

Q3. (Multiple Choice) The thermal neutron fission cross section of ${}^{235}\text{U}$ is approximately:

(a) 1.2 barns (b) 98 barns (c) 584 barns (d) 2,650,000 barns

Q4. (Short Answer) Why does a light-water-moderated reactor require enriched uranium, while a heavy-water-moderated reactor (CANDU) can use natural uranium?

Q5. (Multiple Choice) The moderating ratio (figure of merit for a moderator) is:

(a) $\xi / \Sigma_a$ (b) $\xi \Sigma_s / \Sigma_a$ (c) $\Sigma_s / \xi \Sigma_a$ (d) $\xi \Sigma_a / \Sigma_s$

Q6. (True/False) The delayed neutron fraction $\beta$ for ${}^{235}\text{U}$ thermal fission is approximately 6.5%.

Q7. (Short Answer) Explain in two sentences why delayed neutrons make reactor control possible. What would happen if all fission neutrons were prompt?

Q8. (Multiple Choice) A reactor is "prompt critical" when:

(a) $k_{\text{eff}} > 1$ (b) $k_{\text{eff}} = 1$ and all delayed neutron precursors have decayed (c) The reactivity $\rho$ equals or exceeds the delayed neutron fraction $\beta$ (d) The temperature coefficient of reactivity becomes positive

Q9. (True/False) A negative temperature coefficient of reactivity means the reactor power increases when the temperature increases.

Q10. (Multiple Choice) The RBMK reactor design had a dangerous positive void coefficient because:

(a) The graphite moderator expanded when heated, increasing moderation (b) The light-water coolant provided moderation, and losing it reduced $k_{\text{eff}}$ (c) The graphite provided moderation while the light water primarily absorbed neutrons; losing water (voiding) reduced parasitic absorption while moderation continued (d) The fuel enrichment was too high

Q11. (Short Answer) Why does the ${}^{135}\text{Xe}$ concentration increase after a reactor shuts down? What is the source of the xenon?

Q12. (Multiple Choice) The thermal neutron absorption cross section of ${}^{135}\text{Xe}$ is approximately:

(a) 584 barns (b) 20,600 barns (c) 277,000 barns (d) 2,650,000 barns

Q13. (True/False) In a PWR, the primary coolant (which passes through the reactor core) also drives the turbine directly.

Q14. (Short Answer) What is the purpose of the enrichment process? Why is it technically difficult?

Q15. (Multiple Choice) Approximately what fraction of a reactor's thermal power continues as decay heat immediately after shutdown?

(a) 0.1% (b) 1% (c) 6% (d) 20%

Q16. (Short Answer) In one sentence each, state the primary cause of each of the three major nuclear accidents: TMI, Chernobyl, and Fukushima.

Q17. (True/False) Finland's Onkalo facility is the world's first deep geological repository licensed for high-level nuclear waste disposal.

Q18. (Multiple Choice) According to mortality statistics (deaths per TWh), the safest major energy sources are:

(a) Natural gas and hydropower (b) Nuclear, wind, and solar (c) Coal and oil (because of extensive safety regulations) (d) Hydropower and nuclear

Q19. (Short Answer) What is MOX fuel? What is the proliferation concern associated with its production?

Q20. (Multiple Choice) A "Small Modular Reactor" (SMR) is generally defined as having an electrical output below:

(a) 50 MWe (b) 100 MWe (c) 300 MWe (d) 1,000 MWe

Solutions

Q1. (b) The four factors are: $\eta$ (reproduction factor — neutrons produced per neutron absorbed in fuel), $f$ (thermal utilization — fraction of thermal neutrons absorbed in fuel), $p$ (resonance escape probability — fraction of neutrons that slow down past ${}^{238}\text{U}$ resonances without capture), and $\varepsilon$ (fast fission factor — ratio of total fission neutrons to thermal fission neutrons).

Q2. False. A finite reactor has neutron leakage through the surface. If the reactor is too small, leakage makes $k_{\text{eff}} < 1$ even though $k_\infty > 1$. The reactor must exceed a minimum critical size.

Q3. (c) $\sigma_f({}^{235}\text{U}) = 584$ barns at thermal energies. Option (a) is the fast fission cross section; (b) is the capture cross section; (d) is the absorption cross section of ${}^{135}\text{Xe}$.

Q4. Light water (H$_2$O) absorbs more neutrons than heavy water (D$_2$O) — the macroscopic absorption cross section of H$_2$O is about 670 times larger than D$_2$O. This neutron loss must be compensated by enrichment to maintain criticality. D$_2$O absorbs so few neutrons that the natural abundance of ${}^{235}\text{U}$ (0.72%) is sufficient to sustain the chain reaction.

Q5. (b) The moderating ratio $\xi \Sigma_s / \Sigma_a$ balances slowing-down efficiency ($\xi \Sigma_s$) against parasitic absorption ($\Sigma_a$). Heavy water has the highest moderating ratio (~21,000) among practical moderators.

Q6. False. The delayed neutron fraction is $\beta = 0.0065$, which is 0.65%, not 6.5%. This small fraction is nonetheless essential for reactor control.

Q7. Delayed neutrons increase the effective neutron generation time from $\sim 10^{-4}$ s (prompt only) to $\sim 0.1$ s, slowing the reactor period by a factor of $\sim 1000$ and making mechanical control feasible. If all neutrons were prompt, the reactor period at small positive reactivity would be milliseconds — far too fast for any control rod or human operator.

Q8. (c) Prompt critical means the chain reaction is self-sustaining on prompt neutrons alone: $\rho \geq \beta$ (or $\rho \geq 1\$$). This is catastrophic because the reactor period drops to the prompt neutron lifetime ($\sim 10^{-4}$ s).

Q9. False. A negative temperature coefficient means that a temperature increase causes $k_{\text{eff}}$ to decrease, reducing power. This is stabilizing (negative feedback). A positive coefficient would cause power to increase with temperature — dangerous positive feedback.

Q10. (c) In the RBMK, the graphite provides the bulk of the moderation. The light-water coolant absorbs neutrons (parasitic absorption). When the water boils (voids), the absorption decreases, but the graphite moderation continues unaffected, so $k_{\text{eff}}$ increases — a positive void coefficient.

Q11. After shutdown, the neutron flux drops to zero, so ${}^{135}\text{Xe}$ is no longer being destroyed by neutron absorption ($\sigma_a \phi X \to 0$). But the ${}^{135}\text{I}$ precursor (half-life 6.57 hours) continues to beta-decay into ${}^{135}\text{Xe}$. Since production continues but the dominant removal mechanism has ceased, the Xe concentration rises until the I-135 inventory is depleted, peaking around 10–12 hours after shutdown.

Q12. (d) $\sigma_a({}^{135}\text{Xe}) = 2.65 \times 10^6$ barns — the largest thermal neutron absorption cross section of any known nuclide.

Q13. False. In a PWR, the primary coolant (which is radioactive) transfers heat to a secondary loop via a steam generator. The secondary-loop steam drives the turbine. This two-loop design keeps radioactivity out of the turbine building. (In a BWR, the primary coolant does drive the turbine directly.)

Q14. Enrichment increases the fraction of fissile ${}^{235}\text{U}$ from its natural abundance (0.72%) to the 3–5% required by most reactors. It is technically difficult because ${}^{235}\text{U}$ and ${}^{238}\text{U}$ are chemically identical — they differ only in mass by 1.26% ($\Delta m = 3$ u out of 238 u), so separation must exploit this tiny mass difference using physical processes like gas centrifugation.

Q15. (c) Approximately 6% of the reactor's thermal power appears as decay heat immediately after shutdown, from the radioactive decay of fission products. This decays over time but remains significant for hours to days.

Q16. TMI (1979): A stuck-open relief valve caused a loss-of-coolant accident that operators misdiagnosed, leading them to reduce emergency cooling. Chernobyl (1986): A positive void coefficient combined with severe operating procedure violations and the graphite-tipped control rod design caused a prompt-critical power excursion. Fukushima (2011): A beyond-design-basis tsunami destroyed all backup power, preventing decay heat removal from the shutdown reactors.

Q17. True. Onkalo, located on Olkiluoto island in Finland, was licensed in 2015 and is the world's first purpose-built deep geological repository for spent nuclear fuel.

Q18. (b) By deaths per TWh, nuclear (~0.03), wind (~0.04), and solar (~0.02) are the safest major energy sources. Coal and oil are the most dangerous, primarily due to air pollution.

Q19. MOX (Mixed OXide) fuel is a blend of plutonium dioxide (PuO$_2$) and uranium dioxide (UO$_2$), fabricated from plutonium recovered by reprocessing spent fuel. The proliferation concern is that the PUREX reprocessing process separates plutonium from the highly radioactive fission products, producing separated plutonium that could potentially be diverted for weapons use.

Q20. (c) The IAEA defines SMRs as reactors with electrical output below 300 MWe, designed for factory fabrication and modular deployment.