Exercises — Chapter 28

Critical Mass and Weapons Physics

Problem 28.1 ⭐ Using one-speed diffusion theory, estimate the bare critical radius and critical mass for a sphere of metallic ${}^{235}\text{U}$ with the following fast-neutron parameters: $\sigma_f = 1.24\,\text{b}$, $\sigma_a = 1.69\,\text{b}$ (includes fission + capture), $\sigma_{\text{tr}} = 6.8\,\text{b}$, $\bar{\nu} = 2.52$, $\rho = 19.1\,\text{g/cm}^3$, $A = 235$.

(a) Calculate the number density $n$, the macroscopic cross sections $\Sigma_f$, $\Sigma_a$, and $\Sigma_{\text{tr}}$, the diffusion length $L$, and the infinite-medium multiplication factor $k_\infty$.

(b) Determine the critical radius $R_c = \pi L / \sqrt{k_\infty - 1}$ and the critical mass $M_c = (4/3)\pi R_c^3 \rho$.

(c) Compare your result to the accepted value of $\sim 52\,\text{kg}$ and discuss the sources of error in the one-speed diffusion approximation.

Problem 28.2 ⭐ Repeat Problem 28.1 for ${}^{239}\text{Pu}$ in its $\alpha$-phase ($\rho = 19.8\,\text{g/cm}^3$) using $\sigma_f = 1.73\,\text{b}$, $\sigma_a = 2.11\,\text{b}$, $\sigma_{\text{tr}} = 6.2\,\text{b}$, $\bar{\nu} = 2.88$, $A = 239$.

(a) Calculate $M_c$ for a bare sphere.

(b) Why is the critical mass of ${}^{239}\text{Pu}$ so much smaller than that of ${}^{235}\text{U}$? Identify which nuclear parameter contributes most to the difference.

Problem 28.3 ⭐⭐ A fissile sphere is surrounded by a thick reflector of natural uranium, which reduces the critical mass by a factor $f \approx 3$ relative to the bare sphere.

(a) Estimate the reflected critical mass of ${}^{235}\text{U}$ and ${}^{239}\text{Pu}$ using your results from Problems 28.1 and 28.2.

(b) If the implosion design compresses the ${}^{239}\text{Pu}$ core to twice its normal density, by what factor does the critical mass change? What is the new critical mass (with reflector)?

(c) Explain physically why compression is so effective at reducing critical mass. (Hint: think about what happens to the mean free path.)

Problem 28.4 ⭐⭐ The generation time for prompt fission neutrons in a supercritical assembly of ${}^{235}\text{U}$ metal is approximately $\tau \sim 10\,\text{ns}$. Assume a multiplication factor $k = 1.8$ per generation.

(a) How many generations are required for the number of fissions to reach $10^{24}$ (starting from a single initiating neutron)?

(b) How much time does this take?

(c) Calculate the total energy released if $10^{24}$ fissions occur, each releasing $200\,\text{MeV}$. Express your answer in joules and in kilotons of TNT ($1\,\text{kt} = 4.18 \times 10^{12}\,\text{J}$).

(d) What fraction of 60 kg of ${}^{235}\text{U}$ has fissioned? Comment on the efficiency of this idealized scenario.

Problem 28.5 ⭐⭐⭐ Pre-initiation probability. A gun-type weapon using plutonium with a ${}^{240}\text{Pu}$ mass fraction $f_{240}$ has total plutonium mass $m$ and assembly time $t_a$.

(a) The spontaneous fission rate of ${}^{240}\text{Pu}$ is $R_{\text{SF}} = 1.02 \times 10^3\,\text{s}^{-1}\,\text{g}^{-1}$. For $m = 6\,\text{kg}$ of plutonium with $f_{240} = 0.06$ (weapons-grade), calculate the total spontaneous fission rate.

(b) For an assembly time $t_a = 1\,\text{ms}$ (gun-type), compute the probability that no spontaneous fission occurs during assembly. Is a gun-type weapon feasible with this material?

(c) For $t_a = 5\,\mu\text{s}$ (implosion), recalculate the probability of no spontaneous fission. Does the implosion design solve the pre-initiation problem for weapons-grade plutonium?

(d) Repeat parts (b) and (c) for reactor-grade plutonium with $f_{240} = 0.24$. Discuss the implications for whether reactor-grade plutonium could be used in a weapon.

Enrichment Physics

Problem 28.6 ⭐ The separation factor for a gas centrifuge is $\alpha - 1 \approx \Delta M v^2 / (2RT)$, where $\Delta M$ is the molecular mass difference between ${}^{238}\text{UF}_6$ and ${}^{235}\text{UF}_6$.

(a) Calculate $\Delta M$ in kg/mol.

(b) For a centrifuge with peripheral speed $v = 500\,\text{m/s}$ operating at $T = 310\,\text{K}$, calculate $\alpha$.

(c) How does $\alpha$ change if $v$ is increased to $700\,\text{m/s}$? By what factor does the separative power increase? (The separative power of a single centrifuge scales approximately as $(\alpha - 1)^2$.)

Problem 28.7 ⭐⭐ A country wishes to produce 25 kg of weapons-grade uranium (90% ${}^{235}\text{U}$) from natural uranium feed (0.72% ${}^{235}\text{U}$) with a tails assay of 0.3% ${}^{235}\text{U}$.

(a) Using the formula for the number of enriching stages:

$$N_E \approx \frac{2}{\alpha - 1} \ln\left(\frac{x_P(1-x_F)}{x_F(1-x_P)}\right)$$

calculate $N_E$ for $\alpha = 1.20$.

(b) Calculate the number of stripping stages:

$$N_S \approx \frac{2}{\alpha - 1} \ln\left(\frac{x_F(1-x_W)}{x_W(1-x_F)}\right)$$

where $x_W = 0.003$ is the tails assay.

(c) The separative work required to produce $P$ kg of product at enrichment $x_P$ from feed at $x_F$ with tails at $x_W$ is:

$$\text{SWU} = P \cdot V(x_P) + W \cdot V(x_W) - F \cdot V(x_F)$$

where $V(x) = (2x - 1)\ln(x/(1-x))$ is the value function, and $F = P(x_P - x_W)/(x_F - x_W)$ and $W = F - P$ from mass balance. Calculate the total SWU required for 25 kg of product.

(d) If each IR-1 centrifuge produces 0.9 SWU/year, how many centrifuges are needed to produce 25 kg of WGU in one year?

Problem 28.8 ⭐⭐ Cascade reconfiguration (breakout scenario). A cascade of 5,000 IR-1 centrifuges is configured to produce 3.5%-enriched LEU from natural feed.

(a) Estimate the LEU production rate in SWU/year and kg of 3.5% product per year (use tails assay 0.3%).

(b) If the cascade is reconfigured to produce 90% HEU by batch recycling (feeding the product back through the cascade), estimate the time required to produce 25 kg of WGU. Assume the cascade SWU capacity remains 4,500 SWU/year.

(c) This is the "breakout time." Discuss what IAEA inspectors would observe during such a reconfiguration and how safeguards are designed to detect it.

Problem 28.9 ⭐⭐⭐ Gaseous diffusion vs. centrifuge. The gaseous diffusion process has a separation factor $\alpha_{\text{diff}} = \sqrt{M({}^{238}\text{UF}_6)/M({}^{235}\text{UF}_6)}$.

(a) Calculate $\alpha_{\text{diff}}$. ($M({}^{235}\text{UF}_6) = 349.03\,\text{g/mol}$, $M({}^{238}\text{UF}_6) = 352.04\,\text{g/mol}$.)

(b) How many diffusion stages are needed to enrich from 0.72% to 90%? Compare to the centrifuge cascade.

(c) Each gaseous diffusion stage requires compressing the $\text{UF}_6$ gas against a porous barrier. The energy consumption per SWU is approximately 2,400 kWh for diffusion vs. $\sim 50\,\text{kWh}$ for centrifuges. For 25 kg of WGU ($\sim 5,750\,\text{SWU}$), compare the total electricity consumption. Express in GWh and as a fraction of a year's output from a 1 GW power plant.

(d) Explain why the enormous energy consumption of gaseous diffusion plants made them historically detectable by intelligence agencies, while centrifuge facilities are harder to detect.

Nuclear Forensics

Problem 28.10 ⭐ A sample of uranium is analyzed by mass spectrometry and found to have the following isotopic composition: ${}^{234}\text{U}$: 1.02%, ${}^{235}\text{U}$: 93.15%, ${}^{236}\text{U}$: 0.42%, ${}^{238}\text{U}$: 5.41%.

(a) Is this LEU, HEU, or weapons-grade? What does the IAEA classify this as?

(b) What does the presence of ${}^{236}\text{U}$ indicate about the material's history?

(c) The ${}^{234}\text{U}$ abundance in natural uranium is 0.0055%. In this sample, the ${}^{234}\text{U}/{}^{235}\text{U}$ ratio is $1.02/93.15 = 0.01095$. Compare this to the natural ratio ($0.0055/0.72 = 0.00764$). What does the elevated ${}^{234}\text{U}$ suggest about the enrichment process? (Hint: ${}^{234}\text{U}$ is lighter than ${}^{235}\text{U}$.)

Problem 28.11 ⭐⭐ A seized sample of plutonium has the isotopic composition: ${}^{238}\text{Pu}$: 0.03%, ${}^{239}\text{Pu}$: 93.8%, ${}^{240}\text{Pu}$: 5.7%, ${}^{241}\text{Pu}$: 0.4%, ${}^{242}\text{Pu}$: 0.07%.

(a) Classify this material (weapons-grade, fuel-grade, or reactor-grade).

(b) Calculate the ${}^{240}\text{Pu}/{}^{239}\text{Pu}$ ratio. What does it tell you about the reactor burnup?

(c) The sample also contains ${}^{241}\text{Am}$ at a ratio of ${}^{241}\text{Am}/{}^{241}\text{Pu} = 0.15$ (atom ratio). Using the decay constant $\lambda_{241} = \ln 2 / (14.33\,\text{yr}) = 0.04836\,\text{yr}^{-1}$, calculate the time since the plutonium was last chemically separated.

(d) What additional forensic measurements would help identify the origin of this material?

Problem 28.12 ⭐⭐ The ${}^{241}\text{Am}$ chronometer. ${}^{241}\text{Pu}$ ($t_{1/2} = 14.33\,\text{yr}$) undergoes $\beta^-$ decay to ${}^{241}\text{Am}$. At the time of chemical separation ($t = 0$), all americium is removed.

(a) Write the expression for the number of ${}^{241}\text{Am}$ atoms as a function of time, starting from $N_0$ atoms of ${}^{241}\text{Pu}$ and assuming ${}^{241}\text{Am}$ is stable on this timescale ($t_{1/2}({}^{241}\text{Am}) = 432\,\text{yr}$).

(b) Show that the ${}^{241}\text{Am}/{}^{241}\text{Pu}$ atom ratio at time $t$ is:

$$\frac{N_{\text{Am}}}{N_{\text{Pu}}} = e^{\lambda t} - 1$$

(c) A sample is measured to have ${}^{241}\text{Am}/{}^{241}\text{Pu} = 0.34$ (atom ratio). Calculate the age of the sample.

(d) Discuss the precision requirements. If the ratio can be measured to $\pm 5\%$, what is the uncertainty in the age?

Problem 28.13 ⭐⭐⭐ Post-detonation forensics. After a nuclear detonation, analysts measure the ratio of fission products ${}^{140}\text{Ba}$ to ${}^{95}\text{Zr}$ in debris samples. The cumulative fission yields for these isotopes differ between ${}^{235}\text{U}$ and ${}^{239}\text{Pu}$:

(a) Calculate the ${}^{140}\text{Ba}/{}^{95}\text{Zr}$ ratio expected for pure ${}^{235}\text{U}$ fission and for pure ${}^{239}\text{Pu}$ fission.

(b) If the measured ratio is 0.98, which fissile material was the likely fuel?

(c) If the weapon used a combination of ${}^{235}\text{U}$ and ${}^{239}\text{Pu}$ (as in some designs), write an expression for the expected ratio as a function of the fraction $f$ of fissions from ${}^{239}\text{Pu}$.

(d) Solve for $f$ given the measured ratio of 0.98. Discuss the limitations of this simple two-component model.

Radiation Detection and Security

Problem 28.14 ⭐ A portal monitor uses a NaI:Tl detector with dimensions $10\,\text{cm} \times 10\,\text{cm} \times 40\,\text{cm}$. The natural background count rate in the energy window 100–300 keV is 250 counts/s. A cargo container passes through the portal in 15 seconds.

(a) Calculate the expected number of background counts during the passage.

(b) The alarm threshold is set at $N_{\text{bg}} + 3\sqrt{N_{\text{bg}}}$ (a "$3\sigma$ above background" criterion). Calculate the alarm threshold in counts.

(c) A 10 kg sphere of HEU (${}^{235}\text{U}$, 93% enriched) at a distance of 2 m from the detector produces an additional signal of 25 counts/s in the same energy window. Will the alarm trigger?

(d) If the HEU is shielded by 5 mm of lead, the 185.7 keV gamma-ray signal is attenuated by a factor of $\sim 50$. Will the alarm still trigger?

Problem 28.15 ⭐⭐ The energy resolution of a gamma-ray detector is typically expressed as the FWHM (full width at half maximum) of the photopeak, $\Delta E$, divided by the peak energy $E$:

$$R = \frac{\Delta E}{E} \times 100\%$$

(a) A NaI detector has $R = 7.0\%$ at $E = 662\,\text{keV}$ (${}^{137}\text{Cs}$). Calculate $\Delta E$. Can this detector resolve the 661.7 keV line of ${}^{137}\text{Cs}$ from the 609 keV line of ${}^{214}\text{Bi}$ (a NORM indicator)?

(b) An HPGe detector has $R = 0.20\%$ at $662\,\text{keV}$. Calculate $\Delta E$. Can it resolve the same two lines?

(c) A LaBr$_3$ detector has $R = 3.0\%$ at $662\,\text{keV}$. Can it resolve the two lines?

(d) Explain why energy resolution matters for distinguishing threat sources from NORM in a security context.

Problem 28.16 ⭐⭐ Neutron detection of plutonium. The spontaneous fission rate of ${}^{240}\text{Pu}$ is $R_{\text{SF}} = 1.02 \times 10^3\,\text{fissions}\,\text{s}^{-1}\,\text{g}^{-1}$, and each fission produces $\bar{\nu}_{\text{SF}} \approx 2.2$ neutrons.

(a) For 8 kg of weapons-grade plutonium ($6\%$ ${}^{240}\text{Pu}$), calculate the total neutron emission rate.

(b) At a distance of 3 m, assuming isotropic emission, calculate the neutron flux (neutrons per cm$^2$ per second).

(c) A ${}^{3}\text{He}$ neutron detector with an effective area of $200\,\text{cm}^2$ and an efficiency of $15\%$ for fast neutrons is placed at 3 m. How many counts per second does it register?

(d) The cosmic-ray neutron background at sea level is approximately 20 neutrons cm$^{-2}$ hr$^{-1}$. Compare the signal from part (c) to the background count rate and assess whether detection is feasible in a 30 s measurement.

Problem 28.17 ⭐ A ${}^{137}\text{Cs}$ source has activity $A = 100\,\text{GBq}$ ($100 \times 10^9$ disintegrations/s). The specific gamma-ray dose constant for ${}^{137}\text{Cs}$ is $\Gamma = 7.8 \times 10^{-5}\,\text{mSv}\cdot\text{m}^2/(\text{MBq}\cdot\text{h})$.

(a) Calculate the dose rate at 1 m from the unshielded source.

(b) Calculate the dose rate at 100 m (outdoor dispersal scenario, ignoring air attenuation).

(c) If this source were dispersed uniformly over a circular area of radius 100 m by an RDD, estimate the approximate dose rate at the center (this requires integration — approximate by treating the contamination as a uniform surface source with areal activity $A / (\pi r^2)$ and using the infinite-plane result $\dot{H} \approx \Gamma_{\text{surface}} \cdot S_A$, where $S_A$ is the surface activity in Bq/m$^2$ and $\Gamma_{\text{surface}} \approx 5 \times 10^{-6}\,\text{mSv}\cdot\text{m}^2/(\text{MBq}\cdot\text{h})$ for ${}^{137}\text{Cs}$ surface contamination).

(d) How does this dose rate compare to the natural background ($\sim 0.3\,\mu\text{Sv/h}$)? How long would a person need to remain in the contaminated area to accumulate a 1 mSv dose (the annual public dose limit)?

IAEA Safeguards and Nonproliferation

Problem 28.18 ⭐ The IAEA defines a "significant quantity" of different nuclear materials (Table in Section 28.4.2). A state has the following declared nuclear material inventory:

• 3,000 kg of LEU ($4.5\%$ ${}^{235}\text{U}$) as $\text{UO}_2$ fuel assemblies
• 150 kg of separated plutonium ($1\%$ ${}^{240}\text{Pu}$) in MOX fuel
• 2 kg of HEU ($93\%$ ${}^{235}\text{U}$) in a research reactor

(a) Calculate the ${}^{235}\text{U}$ content of the LEU. How many significant quantities does this represent?

(b) How many significant quantities of plutonium does the state possess?

(c) How many significant quantities of HEU does the state possess?

(d) Which material category poses the greatest proliferation risk, and why?

Problem 28.19 ⭐⭐ Environmental sampling sensitivity. An IAEA inspector collects a swipe sample from a centrifuge cascade hall. The sample is analyzed by LG-SIMS, which can measure the ${}^{235}\text{U}/{}^{238}\text{U}$ isotopic ratio of individual micrometer-sized uranium particles.

(a) The facility is declared to enrich uranium to 3.5%. What ${}^{235}\text{U}/{}^{238}\text{U}$ atom ratio corresponds to 3.5 wt% enrichment? (For practical purposes, the weight and atom ratios are nearly identical for uranium isotopes.)

(b) The inspector finds several particles with ${}^{235}\text{U}/{}^{238}\text{U} = 0.25$. What enrichment does this correspond to?

(c) Is this consistent with the declared activities? What follow-up actions would the IAEA take?

(d) Could the particles have been introduced by contamination from another facility (e.g., via shared equipment)? How might the IAEA distinguish between undeclared enrichment and cross-contamination?

Problem 28.20 ⭐⭐⭐ Breakout time calculation. A state has 1,000 IR-6 centrifuges (each producing 10 SWU/year) and a stockpile of 200 kg of uranium enriched to 5% ${}^{235}\text{U}$.

(a) Calculate the SWU required to enrich 1 kg of 5% uranium to 90% (with 0.7% tails from the re-enrichment step, using the formula from Problem 28.7(c)).

(b) How many kg of 90%-enriched uranium can be produced from 200 kg of 5% feed? (Mass balance: $F \cdot x_F = P \cdot x_P + W \cdot x_W$.)

(c) Calculate the total SWU required for the 90% product from part (b).

(d) With 10,000 SWU/year cascade capacity, how long would it take to accumulate one significant quantity (25 kg) of WGU? Express in days.

Synthesis and Analysis

Problem 28.21 ⭐⭐ Comparative proliferation pathways. Compare the uranium enrichment and plutonium production routes to weapons material. Create a table with rows for: technical difficulty, infrastructure scale, time to one SQ, detectability by IAEA, detectability by intelligence, dual-use ambiguity. Discuss which route is more concerning for nonproliferation, and why the answer depends on context (state vs. non-state actor).

Problem 28.22 ⭐⭐⭐ Energy budget of a thermonuclear weapon. The "Ivy Mike" test (1 November 1952) had a yield of approximately 10.4 Mt.

(a) Convert this to joules.

(b) Estimate the number of D-T fusions required if all the yield came from fusion ($Q = 17.6\,\text{MeV}$ per D-T reaction). What mass of tritium (or lithium-6 deuteride) would be needed?

(c) In practice, a significant fraction of the yield comes from fission of the ${}^{238}\text{U}$ tamper by 14.1 MeV fusion neutrons. If 60% of the yield is fission and 40% is fusion, recalculate the masses.

(d) The fission primary (trigger) had a yield of approximately 500 kt. What fraction of the total yield came from the primary?

Problem 28.23 ⭐⭐ The CTBT verification system. The seismic component of the International Monitoring System can detect underground nuclear explosions with yields as low as approximately 1 kt.

(a) A 1 kt underground explosion produces a seismic wave equivalent to a magnitude $\sim 4$ earthquake. Using the approximate relation between yield $Y$ (kt) and body-wave magnitude $m_b$:

$$m_b \approx 4.0 + 0.75 \log_{10}(Y/1\,\text{kt})$$

what magnitude would a 10 kt test produce? A 100 kt test?

(b) The radionuclide monitoring component detects noble gas isotopes (${}^{133}\text{Xe}$, ${}^{135}\text{Xe}$, ${}^{133m}\text{Xe}$, ${}^{131m}\text{Xe}$) that seep from an underground test site. The ratios between these isotopes help distinguish a nuclear explosion from a reactor (which also releases xenon). Why are noble gases particularly useful for this purpose? (Consider their chemistry.)

(c) In September 2017, the DPRK conducted a nuclear test that produced a seismic signal corresponding to $m_b \approx 6.1$. Estimate the yield.

Problem 28.24 ⭐⭐⭐ (Research) The "nuclear archaeology" concept. It has been proposed that graphite-moderated reactor cores retain a permanent record of their neutron exposure history in the form of isotopic changes in the graphite itself (e.g., production of ${}^{14}\text{C}$ from ${}^{14}\text{N}$ impurities, and ${}^{36}\text{Cl}$ from ${}^{35}\text{Cl}$). By measuring these activation products, one could determine the total neutron fluence the core experienced and hence infer the total plutonium production.

(a) Write the activation reactions for ${}^{14}\text{C}$ and ${}^{36}\text{Cl}$ production in graphite.

(b) If the thermal neutron capture cross section of ${}^{14}\text{N}$ is $\sigma = 1.82\,\text{b}$ and the concentration of nitrogen impurities in reactor-grade graphite is $\sim 10\,\text{ppm}$ by weight, estimate the ${}^{14}\text{C}$ production rate per gram of graphite in a thermal neutron flux of $10^{13}\,\text{cm}^{-2}\text{s}^{-1}$.

(c) Discuss how this technique could be applied to verify the plutonium production history of a reactor as part of a nuclear disarmament verification program.

Problem 28.25 ⭐⭐⭐ (Research) Reactor-grade plutonium and weapons. It is sometimes stated that reactor-grade plutonium (high ${}^{240}\text{Pu}$ content) cannot be used in a nuclear weapon. Evaluate this claim.

(a) What are the two main problems with reactor-grade plutonium for weapons? (Consider pre-initiation from ${}^{240}\text{Pu}$ spontaneous fission and the heat generation from ${}^{238}\text{Pu}$ decay.)

(b) Pre-initiation reduces the yield but does not prevent a nuclear explosion. Estimate the probability that an implosion weapon using reactor-grade plutonium ($24\%$ ${}^{240}\text{Pu}$, 8 kg total Pu, assembly time $5\,\mu\text{s}$) suffers pre-initiation.

(c) Even with pre-initiation, the weapon would produce a nuclear yield (a "fizzle yield"). The fizzle yield of an implosion weapon is estimated at $\sim 1$–$2\,\text{kt}$. Compare this to the Hiroshima bomb (15 kt). Is a fizzle yield still devastating?

(d) Based on your analysis, discuss whether the distinction between "weapons-grade" and "reactor-grade" plutonium is a reliable barrier to proliferation.