Exercises — Chapter 20
The Fission Barrier and Fissility Parameter
Problem 20.1 ⭐ Calculate the fissility parameter $x$ for the following nuclei, using $a_S = 17.8$ MeV and $a_C = 0.714$ MeV:
(a) $^{208}$Pb ($Z = 82$, $A = 208$)
(b) $^{235}$U ($Z = 92$, $A = 235$)
(c) $^{252}$Cf ($Z = 98$, $A = 252$)
(d) $^{298}$Fl ($Z = 114$, $A = 298$), the predicted center of the island of stability
For which of these nuclei does the liquid drop model predict a fission barrier? Which is closest to the instability limit $x = 1$?
Problem 20.2 ⭐ Show that the critical value of $Z^2/A$ for which the fission barrier vanishes is:
$$\left(\frac{Z^2}{A}\right)_{\text{crit}} = \frac{2a_S}{a_C}$$
Using $a_S = 17.8$ MeV and $a_C = 0.714$ MeV, calculate the numerical value. No known nucleus reaches this limit — explain why in terms of the competition between Coulomb instability and alpha/beta decay.
Problem 20.3 ⭐⭐ Starting from the deformation energy expression:
$$\Delta E(\varepsilon) = \frac{1}{5}(2E_S^{(0)} - E_C^{(0)})\varepsilon^2$$
(a) Derive the condition $2E_S^{(0)} = E_C^{(0)}$ for instability against quadrupole deformation.
(b) Show that this is equivalent to $x = 1$.
(c) For $x < 1$, the coefficient is positive. Sketch the deformation energy $\Delta E(\varepsilon)$ for $x = 0.5$, $x = 0.7$, and $x = 0.9$, noting that higher-order terms eventually cause $\Delta E$ to decrease at large $\varepsilon$ and become negative (the fragments separate). Qualitatively, how does the barrier height depend on $x$?
Problem 20.4 ⭐⭐ The liquid drop model barrier height for actinide nuclei can be approximated as:
$$B_f \approx 0.83 \, E_S^{(0)} (1 - x)^3$$
(a) Calculate $B_f$ for $^{240}$Pu ($Z = 94$, $A = 240$). Compare with the experimental value $B_f \approx 6.05$ MeV (outer barrier).
(b) Calculate $B_f$ for $^{252}$Cf ($Z = 98$, $A = 252$). Why is the spontaneous fission half-life of $^{252}$Cf so much shorter than that of $^{240}$Pu?
(c) Discuss why the liquid drop model systematically overestimates actinide fission barriers. What physics is missing?
Problem 20.5 ⭐⭐ The surface area of a prolate ellipsoid with semi-axes $a$ and $b = c$ (where $a > b$) and eccentricity $e = \sqrt{1 - b^2/a^2}$ is:
$$S = 2\pi b^2 \left(1 + \frac{a}{b \cdot e}\sin^{-1}(e)\right)$$
(a) For a volume-conserving deformation $a = R_0(1 + \varepsilon)$, $b = R_0(1 + \varepsilon)^{-1/2}$, expand $S$ to second order in $\varepsilon$ and verify that $S = 4\pi R_0^2(1 + \frac{2}{5}\varepsilon^2 + \cdots)$.
(b) The electrostatic self-energy of a uniformly charged prolate ellipsoid is:
$$U_C = \frac{3}{5}\frac{Z^2 e^2}{4\pi\epsilon_0 R_0}\frac{1}{e}\left(\frac{1}{2}\ln\frac{1+e}{1-e}\right) \cdot (1+\varepsilon)^{-1/2}$$
Expand to second order in $\varepsilon$ and verify the coefficient $-1/5$ in the Coulomb deformation energy.
Induced Fission and Pairing Energy
Problem 20.6 ⭐ For each of the following neutron-capture reactions, determine whether the compound nucleus has even or odd neutron number, and predict qualitatively whether the target is fissile (fissions with thermal neutrons) or only fissionable:
(a) $n + {}^{233}$U
(b) $n + {}^{237}$Np
(c) $n + {}^{240}$Pu
(d) $n + {}^{241}$Pu
(e) $n + {}^{242}$Am
Problem 20.7 ⭐⭐ The pairing energy in the SEMF is approximately $\delta = \pm a_P A^{-1/2}$ with $a_P \approx 12$ MeV ($+$ for even-even, $-$ for odd-odd, $0$ for odd-$A$).
(a) Estimate the difference in neutron separation energy $S_n$ between the reaction $n + {}^{235}$U $\to$ $^{236}$U$^*$ (compound nucleus is even-even) and $n + {}^{238}$U $\to$ $^{239}$U$^*$ (compound nucleus is odd-$A$). Use the pairing term only.
(b) The actual separation energies are $S_n(^{236}\text{U}) = 6.55$ MeV and $S_n(^{239}\text{U}) = 4.81$ MeV. The difference is 1.74 MeV. Compare with your estimate from part (a). How does the pairing energy account for the difference?
(c) Explain why this energy difference — less than 2 MeV — has such profound consequences for nuclear technology.
Problem 20.8 ⭐⭐ $^{232}$Th is not fissile with thermal neutrons, but it is "fertile" — it can be converted to the fissile isotope $^{233}$U via neutron capture:
$$^{232}\text{Th} + n \to {}^{233}\text{Th} \xrightarrow[\beta^-]{22\text{ min}} {}^{233}\text{Pa} \xrightarrow[\beta^-]{27\text{ d}} {}^{233}\text{U}$$
(a) Verify that $^{233}$U is fissile by checking that $S_n(^{234}\text{U}) > B_f(^{234}\text{U})$. Use $S_n(^{234}\text{U}) = 6.84$ MeV and $B_f(^{234}\text{U}) \approx 5.5$ MeV.
(b) Why is $^{233}$U fissile while $^{232}$Th is not? Frame your answer in terms of the neutron number (odd/even) of the respective compound nuclei.
(c) The thorium fuel cycle ($^{232}$Th $\to$ $^{233}$U) is an alternative to the uranium fuel cycle ($^{238}$U $\to$ $^{239}$Pu). List one advantage and one disadvantage from a physics perspective.
Problem 20.9 ⭐⭐⭐ The threshold energy for neutron-induced fission of $^{238}$U is approximately 1 MeV. Using $S_n(^{239}\text{U}) = 4.81$ MeV and $B_f(^{239}\text{U}) \approx 6.2$ MeV:
(a) Show that the threshold kinetic energy in the lab frame is approximately:
$$E_n^{\text{thresh}} \approx B_f - S_n + \frac{B_f}{A} \approx 1.4 \text{ MeV}$$
where the last term is the center-of-mass correction (recoil energy).
(b) The cross section for $^{238}$U(n,f) rises from zero at threshold to about 0.5 barn at $E_n = 2$ MeV, then increases to about 1 barn at 5 MeV. Sketch this cross section as a function of $E_n$ from 0 to 10 MeV. How does it compare to the $^{235}$U(n,f) cross section in the same energy range?
(c) In a fast reactor (no moderator, average neutron energy ~0.1–0.5 MeV), most neutrons are below the $^{238}$U fission threshold. Why are fast reactors nevertheless often designed to use $^{238}$U as part of the fuel?
Fission Products and Energy Release
Problem 20.10 ⭐ Using the unchanged charge distribution (UCD) approximation, calculate the expected proton number $Z_f$ of a fission fragment with mass number $A_f$ for the fission of $^{236}$U ($Z = 92$, $A = 236$):
(a) $A_f = 95$ (light fragment)
(b) $A_f = 140$ (heavy fragment)
(c) For $A_f = 95$, the most probable primary fragment is $^{95}$Sr ($Z = 38$). How many beta decays must it undergo to reach the most stable isobar at $A = 95$ (which is $^{95}$Mo, $Z = 42$)?
Problem 20.11 ⭐ Calculate the total energy released in the fission reaction:
$$^{236}\text{U}^* \to {}^{92}\text{Kr} + {}^{141}\text{Ba} + 3n$$
using the following atomic masses: $M(^{236}\text{U}) = 236.04557$ u, $M(^{92}\text{Kr}) = 91.92616$ u, $M(^{141}\text{Ba}) = 140.91440$ u, $m_n = 1.008665$ u.
Compare your result with the typical ~200 MeV total energy release.
Problem 20.12 ⭐⭐ The kinetic energy of the two fission fragments can be estimated from the Coulomb repulsion at the scission point. At scission, the two fragments (modeled as touching spheres with $R_i = r_0 A_i^{1/3}$) have a separation distance:
$$d = r_0(A_1^{1/3} + A_2^{1/3})$$
(a) For the fission $^{236}$U $\to$ $^{95}$Sr + $^{141}$Ba (ignoring the neutrons), calculate the Coulomb energy at scission:
$$E_C = \frac{Z_1 Z_2 e^2}{4\pi\epsilon_0 d}$$
Use $r_0 = 1.3$ fm and the UCD charges from Problem 20.10. Compare with the experimental total fragment kinetic energy of approximately 169 MeV.
(b) Why does this Coulomb calculation overestimate the fragment kinetic energy? Consider the deformed shapes at scission.
(c) Show that the total kinetic energy release in fission is approximately proportional to $Z^2/A^{1/3}$ by writing $Z_1 Z_2 \approx Z^2/4$ and $A_1^{1/3} + A_2^{1/3} \approx 2(A/2)^{1/3} = 2^{2/3}A^{1/3}$.
Problem 20.13 ⭐⭐ The fission product $^{137}$Cs decays to $^{137}$Ba with a half-life of 30.17 years, emitting a 0.662 MeV gamma ray (via the $^{137m}$Ba isomeric state).
(a) If a reactor has been operating for 1 year and is then shut down, how long must the spent fuel be stored before the $^{137}$Cs activity has decreased to 0.1% of its value at shutdown? (Assume saturation activity — the reactor operated for many half-lives.)
(b) After 300 years of storage, what fraction of the original $^{137}$Cs remains?
(c) Compare 300 years with the half-lives of the transuranic actinides $^{239}$Pu (24,110 yr) and $^{237}$Np ($2.14 \times 10^6$ yr). Why does this comparison motivate interest in transmutation of actinides?
Problem 20.14 ⭐⭐⭐ The average number of prompt neutrons $\bar{\nu}_p$ in fission increases approximately linearly with the excitation energy of the compound nucleus:
$$\bar{\nu}_p(E^*) \approx \bar{\nu}_p(S_n) + \frac{dv}{dE_n} E_n$$
where $dv/dE_n \approx 0.13$ neutrons/MeV for $^{235}$U.
(a) Calculate $\bar{\nu}_p$ for $^{235}$U fission induced by 2 MeV and 14 MeV neutrons.
(b) The 14 MeV neutron case is relevant for fusion-fission hybrid reactors. How does the increase in $\bar{\nu}_p$ affect $k_{\text{eff}}$?
(c) An important consequence: for a 14 MeV neutron inducing fission in $^{238}$U (which has a fission threshold of ~1 MeV), the compound nucleus $^{239}$U has $E^* = S_n + E_n = 4.81 + 14 = 18.81$ MeV. Estimate the number of prompt neutrons using $d\nu/dE_n \approx 0.11$ neutrons/MeV for $^{238}$U and $\bar{\nu}_p(S_n) \approx 2.5$.
The Chain Reaction and Reactor Physics
Problem 20.15 ⭐ For an infinite, homogeneous reactor with: - $\eta = 2.04$ - $f = 0.87$ - $p = 0.80$ - $\varepsilon = 1.04$
(a) Calculate $k_\infty$.
(b) If the non-leakage probability is $P_{\text{NL}} = 0.96$, calculate $k_{\text{eff}}$.
(c) Is this reactor critical, subcritical, or supercritical? What is the reactivity $\rho$?
Problem 20.16 ⭐ For thermal neutrons on $^{235}$U, the relevant cross sections are:
- Fission cross section: $\sigma_f = 584$ b
- Radiative capture cross section: $\sigma_\gamma = 99$ b
- Average neutrons per fission: $\bar{\nu} = 2.43$
(a) Calculate the absorption cross section $\sigma_a = \sigma_f + \sigma_\gamma$.
(b) Calculate $\eta = \bar{\nu} \cdot \sigma_f / \sigma_a$.
(c) For $^{239}$Pu, $\sigma_f = 748$ b, $\sigma_\gamma = 271$ b, and $\bar{\nu} = 2.87$. Calculate $\eta$ for $^{239}$Pu. Why is $\eta$ for Pu comparable to that of $^{235}$U despite the larger $\bar{\nu}$?
Problem 20.17 ⭐⭐ A power reactor operates at steady state ($k_{\text{eff}} = 1.000$) with a thermal power of 3,000 MW$_{\text{th}}$.
(a) How many fission events per second are required to sustain this power? Use 196 MeV of recoverable energy per fission.
(b) How many kilograms of $^{235}$U are consumed per day? (Each fission destroys one $^{235}$U nucleus.)
(c) A 1 GW coal-fired plant burns approximately 10,000 tonnes of coal per day. Compare the daily fuel consumption of the nuclear and coal plants.
Problem 20.18 ⭐⭐ The reactor period $T$ (the time for the neutron population to change by a factor of $e$) is given approximately by:
$$T \approx \frac{\ell}{\rho} \quad \text{(without delayed neutrons)}$$
$$T \approx \frac{\beta - \rho}{\lambda \rho} \quad \text{(with delayed neutrons, for } 0 < \rho \ll \beta\text{)}$$
where $\ell \approx 10^{-4}$ s is the prompt neutron generation time, $\beta = 0.0065$, and $\lambda \approx 0.08$ s$^{-1}$ is the effective delayed neutron decay constant.
(a) For a reactivity insertion of $\rho = 0.001$ (about 15 cents), calculate $T$ with and without delayed neutrons.
(b) For $\rho = 0.0065 = \beta$ (prompt critical), what happens to the delayed-neutron formula? What is the physical meaning?
(c) Calculate the time for the reactor power to increase by a factor of 10 for each case in part (a).
Problem 20.19 ⭐⭐⭐ Derivation of the four-factor formula. Trace one neutron through the complete life cycle in an infinite thermal reactor:
(a) A thermal neutron is absorbed in the fuel. The probability that it causes fission (rather than radiative capture) is $\sigma_f / \sigma_a$. Each fission produces $\bar{\nu}$ fast neutrons. Therefore, each thermal neutron absorbed in fuel produces $\eta = \bar{\nu} \sigma_f / \sigma_a$ fast neutrons. Derive this expression.
(b) The probability that a thermal neutron is absorbed in the fuel (rather than in the moderator or structural materials) is $f$. Therefore, of the $\eta$ fast neutrons, only $\eta f$ are produced per thermal neutron absorbed anywhere in the reactor. Explain why $f < 1$ in any practical reactor.
(c) Before reaching thermal energies, each neutron must slow down through the resonance region of $^{238}$U without being captured. The probability of escaping resonance capture is $p$. Why is the resonance escape probability less than 1? What happens to neutrons that are captured?
(d) A small fraction of the fast neutrons cause fission in $^{238}$U before they slow down. The fast fission factor $\varepsilon$ accounts for this bonus. Show that the total number of neutrons produced per initial thermal neutron absorbed in the reactor is $k_\infty = \eta f p \varepsilon$.
(e) For $\eta = 2.04$, $f = 0.87$, $p = 0.80$, $\varepsilon = 1.04$, how many neutrons are "lost" at each stage per fission neutron?
Problem 20.20 ⭐⭐⭐ The natural enrichment of uranium is 0.72% $^{235}$U and 99.28% $^{238}$U.
(a) For a thermal reactor using natural uranium fuel and graphite moderator, estimate $\eta$ using the weighted absorption cross sections: $\sigma_f^{25} = 584$ b, $\sigma_\gamma^{25} = 99$ b for $^{235}$U, and $\sigma_a^{28} = 2.68$ b for $^{238}$U. The absorption cross section of the fuel mixture is $\Sigma_a = N^{25}\sigma_a^{25} + N^{28}\sigma_a^{28}$, and $\eta = \bar{\nu} N^{25}\sigma_f^{25}/\Sigma_a$.
(b) With graphite as moderator ($\sigma_a = 0.0035$ b), and a moderator-to-fuel ratio $N_{\text{mod}}/N_{\text{fuel}} = 400$ (typical for a graphite-moderated reactor), estimate $f$.
(c) Can you achieve $k_\infty > 1$ with natural uranium and graphite? This is the design space of the Chicago Pile-1 and the Hanford production reactors. Discuss why Fermi chose graphite rather than water as the moderator.
Critical Mass and Weapons Physics
Problem 20.21 ⭐⭐ The critical mass of a bare sphere of fissile material can be estimated from the condition that the neutron leakage rate equals the neutron production rate. For a sphere of radius $R$:
$$k_{\text{eff}} \approx k_\infty \left(1 - \frac{\pi^2}{\Sigma_a \Sigma_{\text{tr}} R^2}\right)$$
where $\Sigma_a$ is the macroscopic absorption cross section and $\Sigma_{\text{tr}}$ is the macroscopic transport cross section.
(a) The critical condition is $k_{\text{eff}} = 1$. Show that the critical radius is:
$$R_{\text{crit}} = \frac{\pi}{\sqrt{\Sigma_a \Sigma_{\text{tr}} (k_\infty - 1)}}$$
(b) For metallic $^{235}$U ($\rho = 18.7$ g/cm$^3$), $\Sigma_a \approx 0.066$ cm$^{-1}$ and $\Sigma_{\text{tr}} \approx 0.31$ cm$^{-1}$ for fission-spectrum neutrons, and $k_\infty \approx 2.3$. Calculate $R_{\text{crit}}$ and the critical mass $M_{\text{crit}} = \frac{4}{3}\pi R_{\text{crit}}^3 \rho$.
(c) A reflector (a shell of natural uranium or beryllium surrounding the core) reduces the critical mass by reflecting escaping neutrons back into the core. If a good reflector reduces the critical mass by a factor of ~3, estimate the reflected critical mass.
Problem 20.22 ⭐⭐⭐ The spontaneous fission rate of $^{240}$Pu is $4.7 \times 10^4$ fissions/(g$\cdot$s). Reactor-grade plutonium contains approximately 6% $^{240}$Pu.
(a) For a plutonium weapon core of 6 kg (total), calculate the number of spontaneous fission events per second from the $^{240}$Pu content.
(b) In a gun-type assembly, the two subcritical pieces take about 1 ms to fully assemble. What is the probability that at least one spontaneous fission neutron appears during assembly? (Use Poisson statistics.)
(c) Explain why this probability is unacceptably high for a gun-type design and how the implosion design solves the problem by reducing the assembly time to $\sim 1$ $\mu$s.
Energy and Technology
Problem 20.23 ⭐ A typical 1 GW$_e$ nuclear power plant has a thermal efficiency of 33%.
(a) What is the thermal power output?
(b) How many fissions per second are required?
(c) If the fuel is UO$_2$ enriched to 4.5% $^{235}$U, how many kilograms of $^{235}$U are consumed per year (assuming 365 days at full power)?
(d) A typical fuel assembly remains in the reactor for 3 fuel cycles (each ~18 months), and only about 4% of the total uranium (including bred $^{239}$Pu) is fissioned. Estimate the annual uranium consumption (total, not just $^{235}$U).
Problem 20.24 ⭐⭐ Spent nuclear fuel from a PWR contains approximately (per tonne of heavy metal): - 955 kg $^{238}$U - 8 kg $^{235}$U (residual) - 10 kg plutonium isotopes - 3.5 kg minor actinides (Am, Cm, Np) - 35 kg fission products
(a) What fraction of the original energy content of the uranium has been extracted? (Natural uranium contains 0.72% $^{235}$U, and the fuel was enriched to 4.5%.)
(b) The plutonium contains approximately 70% fissile isotopes ($^{239}$Pu and $^{241}$Pu). If this plutonium were recycled as MOX (mixed oxide) fuel, how much additional energy could be extracted?
(c) In a closed fuel cycle with fast reactors, virtually all of the $^{238}$U can eventually be fissioned (via conversion to $^{239}$Pu). What factor of improvement in energy extraction does this represent compared to a once-through LWR fuel cycle?
Problem 20.25 ⭐⭐ The NuScale SMR module produces 77 MW$_e$ from a core approximately 2 m in diameter and 2 m tall.
(a) Estimate the average volumetric power density in MW/m$^3$ (thermal, assuming 33% efficiency).
(b) A conventional PWR core (e.g., AP1000) produces 3,400 MW$_{\text{th}}$ from a core approximately 3.4 m in diameter and 4.3 m tall. Calculate its volumetric power density.
(c) Compare the two power densities. What are the safety implications of a lower power density?
Advanced Problems
Problem 20.26 ⭐⭐⭐ The asymmetric fission of $^{236}$U produces fragments near the shell closures $Z = 50$, $N = 82$ in the heavy fragment. For a heavy fragment of $A_H = 132$:
(a) Using the UCD approximation, calculate $Z_H$ and $N_H$ for a heavy fragment of $A_H = 132$ from $^{236}$U fission.
(b) How close is this to the doubly magic $^{132}$Sn ($Z = 50$, $N = 82$)?
(c) The shell correction energy for $^{132}$Sn is approximately $-11$ MeV (strongly stabilizing). Argue qualitatively why this shell effect favors asymmetric fission over symmetric fission, even though the liquid drop model predicts the symmetric split should be preferred.
(d) For the spontaneous fission of $^{252}$Cf, the heavy fragment peak is near $A_H \approx 140$. Is this still consistent with the shell closure argument? What is the corresponding $N_H$ for $Z_H = 50$?
Problem 20.27 ⭐⭐⭐ Double-humped barrier and fission isomers. The nucleus $^{240}$Pu has a double-humped fission barrier with: - Inner barrier: $B_A = 6.05$ MeV - Outer barrier: $B_B = 5.15$ MeV - Second well depth: approximately 3 MeV above ground state
(a) Sketch the double-humped potential energy surface as a function of deformation, labeling the ground state, first barrier, second well, second barrier, and scission point.
(b) A "fission isomer" is a metastable state trapped in the second well. The isomer in $^{240}$Pu has a half-life of 3.7 ns. Using the WKB approximation, argue why the isomer preferentially decays by fission through the (lower) outer barrier rather than by gamma decay back through the (higher) inner barrier to the ground state.
(c) Fission isomers were first discovered via delayed fission following the reaction $^{238}$U(d,p)$^{239}$U. The deuteron transfers a neutron to $^{238}$U, populating excited states in $^{239}$U that decay into the second well. Explain why this reaction preferentially populates states near the second well.
Problem 20.28 ⭐⭐⭐ The Oklo natural reactor. In 1972, French scientists discovered that the isotopic composition of uranium in ore from Oklo, Gabon was anomalous: the $^{235}$U abundance was 0.440% instead of the usual 0.720%.
(a) Given that $T_{1/2}(^{235}\text{U}) = 7.04 \times 10^8$ yr and $T_{1/2}(^{238}\text{U}) = 4.47 \times 10^9$ yr, calculate the $^{235}$U enrichment 2 billion years ago.
(b) Was natural uranium 2 billion years ago enriched enough to sustain a chain reaction with water as moderator? (Modern light-water reactors require ~3% enrichment. Why was less needed at Oklo? Consider the moderator and the absence of fission product poisons at startup.)
(c) The Oklo reactor operated intermittently for several hundred thousand years, consuming approximately 6 tonnes of $^{235}$U. Estimate the total energy released.
Problem 20.29 ⭐⭐⭐ (Research) The r-process of nucleosynthesis (Chapter 22) produces nuclei so neutron-rich that they eventually reach the region where spontaneous fission terminates further neutron capture. Using the systematic trend of spontaneous fission half-lives with $Z^2/A$, estimate the maximum $Z$ that the r-process can produce before fission recycling returns material to lighter elements. Discuss how fission fragment distributions from the r-process contribute to the observed abundance peaks in the solar system.
Problem 20.30 ⭐⭐⭐ (Research) Investigate the physics of accelerator-driven subcritical systems (ADS) for nuclear waste transmutation. The basic concept: a high-energy proton beam ($\sim 1$ GeV) strikes a spallation target (lead-bismuth), producing $\sim 20$–$30$ neutrons per proton. These neutrons maintain a subcritical assembly ($k_{\text{eff}} \approx 0.95$–$0.97$) containing minor actinide waste.
(a) For $k_{\text{eff}} = 0.97$, the source multiplication $M = 1/(1-k_{\text{eff}})$ amplifies the external neutron source. Calculate $M$.
(b) If the spallation source produces $10^{18}$ neutrons per second, how many fissions per second occur in the subcritical assembly? What thermal power does this correspond to?
(c) What are the advantages of a subcritical system compared to a critical fast reactor for actinide transmutation? What are the disadvantages (consider the accelerator power requirements)?
Selected Answers and Hints
20.1: (a) $x \approx 0.65$; (b) $x \approx 0.72$; (c) $x \approx 0.76$; (d) $x \approx 0.88$. All have $x < 1$, so all have a fission barrier. $^{298}$Fl is closest to the instability limit — but shell effects near $Z = 114$ stabilize it far more than the liquid drop model alone would suggest.
20.2: $(Z^2/A)_{\text{crit}} \approx 49.9$. The heaviest naturally occurring nucleus, $^{238}$U, has $Z^2/A = 35.6$, well below the limit. Nuclei with $Z^2/A > 40$ decay rapidly by alpha emission or spontaneous fission before accumulating in nature.
20.6: (a) $^{234}$U, even-$N$ compound, fissile; (b) $^{238}$Np, even-$N$ compound, fissile; (c) $^{241}$Pu, odd-$N$ compound, not fissile (fissionable with fast neutrons); (d) $^{242}$Pu, even-$N$ compound, fissile; (e) $^{243}$Am, odd-$N$ compound, not fissile.
20.11: Using atomic masses: $Q = [236.04557 - 91.92616 - 140.91440 - 3 \times 1.008665] \times 931.494 \approx 173.2$ MeV. This is the kinetic energy of fragments + neutrons; the total energy release including subsequent decays is $\sim 200$ MeV.
20.15: (a) $k_\infty = 2.04 \times 0.87 \times 0.80 \times 1.04 = 1.476$; (b) $k_{\text{eff}} = 1.476 \times 0.96 = 1.417$; (c) Supercritical, $\rho = (1.417 - 1)/1.417 = 0.294$.
20.17: (a) $P = 3000 \times 10^6$ W, each fission releases $196 \times 1.602 \times 10^{-13}$ J $= 3.14 \times 10^{-11}$ J, so $N = 9.55 \times 10^{19}$ fissions/s; (b) 3.68 kg/day; (c) Nuclear: 3.68 kg $^{235}$U/day vs. Coal: ~10,000 tonnes/day — a factor of $\sim 2.7 \times 10^6$.
20.18: (a) Without delayed neutrons: $T = 10^{-4}/0.001 = 0.1$ s. With delayed neutrons: $T = (0.0065 - 0.001)/(0.08 \times 0.001) = 69$ s. The delayed neutrons increase the period by a factor of ~700.
20.21: (b) $R_{\text{crit}} \approx 8.7$ cm, $M_{\text{crit}} \approx 52$ kg for a bare sphere.
20.28: (a) $^{235}$U enrichment 2 Gyr ago: $\approx 3.7$%. This is comparable to modern low-enriched reactor fuel.
Full solutions and additional hints are available in the Instructor Guide. Problems marked with ⭐ are basic, ⭐⭐ intermediate, and ⭐⭐⭐ challenging or research-level.