Exercises — Chapter 33
The Nuclear Landscape and Drip Lines
Problem 33.1 ⭐ As of 2025, approximately 3,400 nuclides have been experimentally observed out of an estimated 7,000 bound species.
(a) What fraction of the nuclear chart has been explored?
(b) If FRIB discovers new isotopes at an average rate of 50 per year, how many years would it take to observe all predicted bound nuclei? (This is a dramatic oversimplification — explain why.)
(c) The neutron drip line is experimentally known up to $Z = 10$. Given that the periodic table extends to $Z = 118$, what fraction of the neutron drip line has been mapped?
Problem 33.2 ⭐ The one-neutron separation energy of the most neutron-rich bound oxygen isotope, $^{24}$O ($Z = 8$, $N = 16$), is approximately $S_n \approx 4.1$ MeV. The nucleus $^{25}$O ($Z = 8$, $N = 17$) is unbound.
(a) What does "unbound" mean in this context? Is $^{25}$O's neutron separation energy positive or negative?
(b) The standard shell model with traditional magic numbers predicts $N = 20$ as the next neutron magic number after $N = 8$. What is the expected heaviest bound oxygen isotope in this picture?
(c) Explain, using concepts from Chapter 10, why the actual drip line at $N = 16$ differs from this prediction.
Problem 33.3 ⭐⭐ Different nuclear mass models predict the neutron drip line for calcium ($Z = 20$) at locations ranging from $^{60}$Ca to $^{70}$Ca.
(a) How many neutrons separate these two predictions? What is the fractional uncertainty in the drip line location, expressed as $\Delta N / N_{\text{avg}}$?
(b) The semi-empirical mass formula (Chapter 4) predicts the neutron drip line by setting $S_n = 0$. Using the SEMF parameters from Chapter 4 ($a_V = 15.56$ MeV, $a_S = 17.23$ MeV, $a_C = 0.697$ MeV, $a_A = 23.29$ MeV, $a_P = 12$ MeV), estimate the neutron drip line for calcium. Compare to the range quoted above and comment on the reliability of the SEMF for this extrapolation.
(c) What nuclear physics effects does the SEMF miss that are important at the drip line?
Equation of State and Neutron Stars
Problem 33.4 ⭐⭐ The NICER experiment measured the radius of PSR J0740+6620 ($M \approx 2.08 M_\odot$) to be $R \approx 12.4$ km.
(a) Calculate the average density of this neutron star in units of nuclear saturation density $\rho_0 = 0.16$ fm$^{-3}$.
(b) If the central density is approximately three times the average density, estimate the central density in fm$^{-3}$ and in kg/m$^3$.
(c) At what density would you expect hyperons ($\Lambda$ baryons, rest mass 1116 MeV/$c^2$) to begin appearing, if their threshold is set by the neutron chemical potential equaling the $\Lambda$ mass? Express your answer qualitatively — is this density above or below the estimated central density?
Problem 33.5 ⭐⭐ The symmetry energy $S(\rho)$ can be parameterized near saturation density as:
$$S(\rho) = S_0 + \frac{L}{3}\left(\frac{\rho - \rho_0}{\rho_0}\right) + \frac{K_{\text{sym}}}{18}\left(\frac{\rho - \rho_0}{\rho_0}\right)^2$$
where $S_0 \approx 32$ MeV, $L \approx 60$ MeV (uncertain), and $K_{\text{sym}}$ is poorly constrained.
(a) Calculate $S(\rho)$ at $\rho = 2\rho_0$ and $\rho = 3\rho_0$ using only the first two terms ($K_{\text{sym}} = 0$).
(b) How sensitive is your answer to the value of $L$? Recalculate for $L = 40$ MeV and $L = 80$ MeV. What is the fractional variation at $3\rho_0$?
(c) The PREX-II measurement of the $^{208}$Pb neutron skin ($R_n - R_p = 0.283 \pm 0.071$ fm) implies a relatively large $L$. Does this favor a stiff or soft EOS at high density? What are the implications for neutron star radii?
Problem 33.6 ⭐⭐⭐ The tidal deformability $\Lambda$ of a neutron star is related to its compactness $C = GM/(Rc^2)$ approximately as:
$$\Lambda \approx \frac{2}{3} k_2 C^{-5}$$
where $k_2 \approx 0.05$–$0.15$ is the tidal Love number (depending on the EOS).
(a) For a $1.4 M_\odot$ neutron star with $R = 12$ km, calculate the compactness $C$.
(b) Estimate $\Lambda$ for $k_2 = 0.08$ and compare to the GW170817 constraint $\Lambda_{1.4} = 190^{+390}_{-120}$.
(c) How does $\Lambda$ change if the radius is 11 km instead of 12 km (keeping $k_2$ the same)? This illustrates why tidal deformability is such a powerful EOS constraint.
Superheavy Elements
Problem 33.7 ⭐ The fissility parameter for a nucleus with $Z$ protons and mass number $A$ is $x \approx Z^2 / (50.88 A)$.
(a) Calculate $x$ for $^{298}$Fl ($Z = 114$, $A = 298$) and for the hypothetical $^{304}$120 ($Z = 120$, $A = 304$).
(b) For what value of $Z$ does $x = 1$ for a nucleus on the line of beta stability (approximately $A \approx 2.5Z$ for heavy nuclei)?
(c) The fact that elements beyond $Z \approx 104$ exist at all, despite $x$ approaching 1, is evidence for what phenomenon?
Problem 33.8 ⭐⭐⭐ To synthesize element 120, one proposed reaction is $^{54}$Cr + $^{248}$Cm $\to ^{302}$120$^*$.
(a) Calculate the Coulomb barrier for this reaction using $V_C = k Z_1 Z_2 e^2 / (r_0(A_1^{1/3} + A_2^{1/3}))$ with $r_0 = 1.44$ fm and $ke^2 = 1.44$ MeV fm.
(b) The compound nucleus is formed at an excitation energy $E^* = E_{\text{CM}} + Q$. If $Q \approx -200$ MeV and the beam energy is just above the Coulomb barrier, estimate $E^*$.
(c) At $E^* \approx 40$ MeV, the compound nucleus must evaporate approximately 3–4 neutrons. If the survival probability per neutron evaporation step is $\Gamma_n/(\Gamma_n + \Gamma_f) \approx 0.3$, estimate the total survival probability. This explains why superheavy element cross sections are in the femtobarn range.
r-Process and Nucleosynthesis
Problem 33.9 ⭐ The kilonova associated with GW170817 ejected approximately $0.05 M_\odot$ of r-process material. The estimated rate of neutron star mergers in the Milky Way is $\sim 30$ per million years.
(a) Calculate the total mass of r-process material produced per million years from mergers alone.
(b) The total mass of r-process elements in the Galaxy is estimated at $\sim 30{,}000 M_\odot$. If the Galaxy is $\sim 10{,}000$ million years old, what average production rate is needed?
(c) Compare your answers to (a) and (b). Can mergers alone account for all Galactic r-process material? What caveats apply to this simple estimate?
Problem 33.10 ⭐⭐ The r-process path passes through the neutron magic number $N = 82$, where nuclei are "waiting-point" nuclei because their high neutron separation energies slow neutron capture. The beta-decay half-lives of these waiting-point nuclei set the timescale for the r-process to proceed past $N = 82$.
(a) Why does a large $S_n$ at a magic number slow down neutron capture during the r-process? (Think about the $(n,\gamma) \leftrightarrow (\gamma,n)$ equilibrium.)
(b) The waiting-point nucleus $^{130}$Cd ($Z = 48$, $N = 82$) has a measured half-life of about 162 ms. If the r-process duration is ~1 s, approximately how many half-lives of $^{130}$Cd occur during the r-process?
(c) Why is measuring the half-lives of neutron-rich $N = 82$ nuclei at FRIB so important for r-process simulations?
Neutrinos and Fundamental Symmetries
Problem 33.11 ⭐ Two-neutrino double beta decay ($2\nu\beta\beta$) has been observed in $^{76}$Ge with a half-life of $T_{1/2}^{2\nu} \approx 1.9 \times 10^{21}$ years. The current experimental lower limit on the neutrinoless mode is $T_{1/2}^{0\nu} > 1.8 \times 10^{26}$ years.
(a) Express both half-lives in seconds.
(b) What is the ratio $T_{1/2}^{0\nu} / T_{1/2}^{2\nu}$? What does this ratio tell you about the relative difficulty of observing $0\nu\beta\beta$?
(c) If the LEGEND-1000 experiment observes zero events in 10 tonne-years of exposure, and the expected background is 0.5 events, what does this imply about the sensitivity? (Qualitative answer only — no detailed statistical analysis required.)
Problem 33.12 ⭐⭐ The effective Majorana mass $\langle m_{\beta\beta} \rangle$ is related to the $0\nu\beta\beta$ half-life by:
$$[T_{1/2}^{0\nu}]^{-1} = G^{0\nu} |M^{0\nu}|^2 \left(\frac{\langle m_{\beta\beta} \rangle}{m_e}\right)^2$$
For $^{76}$Ge, $G^{0\nu} \approx 2.36 \times 10^{-15}$ yr$^{-1}$ and $|M^{0\nu}|^2 \approx 4$–$16$ (model-dependent).
(a) If the experimental limit is $T_{1/2}^{0\nu} > 1.8 \times 10^{26}$ years, calculate the upper limit on $\langle m_{\beta\beta} \rangle$ for $|M^{0\nu}|^2 = 10$. Express in meV.
(b) Repeat for $|M^{0\nu}|^2 = 4$ and $|M^{0\nu}|^2 = 16$. What is the range of $\langle m_{\beta\beta} \rangle$ limits? This illustrates why nuclear matrix element uncertainty matters.
(c) The inverted hierarchy prediction is $\langle m_{\beta\beta} \rangle \approx 15$–$50$ meV. What half-life sensitivity is needed to probe this range?
Dark Matter and Nuclear Detectors
Problem 33.13 ⭐⭐ A WIMP with mass $m_\chi = 50$ GeV/$c^2$ scatters elastically off a xenon nucleus ($A = 131$).
(a) What is the maximum nuclear recoil energy $E_R^{\max}$ for a WIMP moving at the Galactic escape velocity $v \approx 550$ km/s? Use non-relativistic kinematics: $E_R^{\max} = 2\mu^2 v^2 / m_A$, where $\mu$ is the WIMP-nucleus reduced mass.
(b) Convert your answer to keV. This sets the energy scale that dark matter detectors must be sensitive to.
(c) For spin-independent scattering, the cross section scales as $A^2$. Calculate the ratio of the expected event rate in xenon ($A = 131$) to that in germanium ($A = 73$), assuming the same detector mass. (Ignore the reduced mass difference for this estimate.)
Problem 33.14 ⭐⭐⭐ The "neutrino fog" (or "neutrino floor") sets a practical limit on dark matter detection sensitivity because coherent elastic neutrino-nucleus scattering (CE$\nu$NS) produces nuclear recoils indistinguishable from WIMP scattering.
(a) For CE$\nu$NS, the cross section scales as $N^2$ (the square of the neutron number, approximately). For xenon ($N \approx 78$), estimate the ratio of CE$\nu$NS cross section per nucleus to that of a single nucleon.
(b) Solar $^8$B neutrinos have energies up to $\sim 15$ MeV. Estimate the maximum recoil energy of a xenon nucleus struck by a 10 MeV neutrino. Compare to your answer in Problem 33.13.
(c) Explain qualitatively why the neutrino fog is a more serious problem for dark matter searches targeting low WIMP masses than for high WIMP masses.
Fusion Energy
Problem 33.15 ⭐ The D-T fusion reaction produces a 3.5 MeV alpha particle and a 14.1 MeV neutron.
(a) How much energy is released per gram of D-T fuel? Compare to the energy released per gram of $^{235}$U fission ($\sim 8 \times 10^{10}$ J/g).
(b) ITER aims to produce 500 MW of fusion power. At what rate (in grams per second) must D-T fuel be consumed?
(c) The world's tritium inventory is approximately 25 kg. If a 1 GW fusion power plant consumes tritium at the rate you calculated in (b) (scaled appropriately), how long would 25 kg last?
Problem 33.16 ⭐⭐ The Lawson criterion for D-T fusion can be written as:
$$n \tau_E > 1.5 \times 10^{20} \text{ m}^{-3} \text{s}$$
at the optimal temperature of approximately 15 keV.
(a) ITER's design parameters are $n \approx 10^{20}$ m$^{-3}$ and $\tau_E \approx 3.7$ s. Verify that these satisfy the Lawson criterion.
(b) The NIF implosion achieves $n \approx 10^{32}$ m$^{-3}$ for a confinement time of $\tau \approx 10^{-10}$ s. Does this satisfy the Lawson criterion?
(c) Explain the fundamental difference in approach between magnetic (ITER) and inertial (NIF) confinement: how does each achieve the required $n\tau_E$?
Proton Spin and EIC
Problem 33.17 ⭐ The proton spin is $1/2$ (in units of $\hbar$). The naive quark model assigns all the spin to the three valence quarks.
(a) In the simplest $SU(6)$ quark model, what fraction of the proton spin is carried by up quarks and what fraction by the down quark?
(b) The EMC experiment found that quark spins contribute only $\sim 30\%$ of the proton spin. If gluons contribute $\sim 40\%$, what fraction must come from orbital angular momentum?
(c) Why is measuring orbital angular momentum experimentally much harder than measuring spin contributions?
Problem 33.18 ⭐⭐⭐ The EIC will operate at center-of-mass energies from 20 to 140 GeV.
(a) For an electron-proton collider with electron energy $E_e$ and proton energy $E_p$ (both highly relativistic), the center-of-mass energy is $\sqrt{s} = 2\sqrt{E_e E_p}$. If the proton energy is 275 GeV, what electron energy is needed for $\sqrt{s} = 140$ GeV?
(b) The minimum Bjorken-$x$ accessible at a given $\sqrt{s}$ is approximately $x_{\min} \sim Q^2_{\min}/s$, where $Q^2_{\min} \sim 1$ GeV$^2$. For $\sqrt{s} = 140$ GeV, estimate $x_{\min}$.
(c) Why is access to small $x$ important for resolving the proton spin puzzle? (Hint: think about where the gluon contribution is largest.)
Machine Learning in Nuclear Physics
Problem 33.19 ⭐⭐ Nuclear mass models trained on machine learning (neural networks, Gaussian processes) have achieved rms deviations of $\sim 200$ keV on known masses.
(a) The r-process abundance pattern for the rare-earth peak ($A \approx 160$) is sensitive to nuclear masses at the level of $\sim 500$ keV. Is the ML model accuracy sufficient for this application?
(b) The extrapolation uncertainty of ML models (predicting masses far from training data) is typically larger than the interpolation uncertainty. Why is this a concern for r-process calculations?
(c) How can experiments at FRIB reduce the extrapolation uncertainty of ML mass models?
Comprehensive and Research Problems
Problem 33.20 ⭐⭐⭐ (Synthesis problem) Create a table listing the ten open questions from this chapter. For each, identify: (a) the primary experimental facility that will address it, (b) the approximate timeline for a major advance, and (c) the nuclear physics concept (from earlier in this textbook) that is most directly relevant. This is an exercise in synthesis — use the entire textbook.
Problem 33.21 ⭐⭐⭐ (Critical analysis) The 2023 Long Range Plan for Nuclear Science identified four overarching questions. Choose one open question from this chapter and write a 500-word argument for why it should be the highest priority for nuclear physics funding. Your argument should address scientific importance, feasibility, broader impacts, and timeline.
Problem 33.22 ⭐⭐⭐⭐ (Research) The tetraneutron ($^4n$) claim from RIKEN (2022) reported a resonance at $\sim 2$ MeV above threshold with a width of $\sim 1.8$ MeV. Search the recent literature (2022–2025) for:
(a) The original experimental paper. What reaction was used? What was the measured observable?
(b) Subsequent theoretical analyses. Do they support or challenge the interpretation?
(c) Any follow-up experiments. Has the result been independently confirmed?
Write a brief critical assessment (300 words maximum) of the current status of the tetraneutron claim.
Problem 33.23 ⭐⭐⭐⭐ (Research) Choose one of the following upcoming experiments: LEGEND-1000, nEXO, DARWIN/XLZD, or the EIC. Using the experiment's technical design report or published white papers:
(a) Describe the experimental technique and the primary physics goal.
(b) Identify the three most significant technical challenges the experiment faces.
(c) Assess the projected sensitivity and compare it to the physics targets (e.g., for $0\nu\beta\beta$ experiments, compare the projected $\langle m_{\beta\beta} \rangle$ sensitivity to the inverted hierarchy prediction).
Problem 33.24 ⭐⭐⭐⭐ (Research) The role of machine learning in nuclear physics is rapidly evolving. Find a recent paper (2023 or later) that applies machine learning to one of the following: nuclear mass prediction, r-process simulation, or nuclear data evaluation.
(a) Summarize the ML method used and the physics problem addressed.
(b) What was the key result? How did the ML approach improve on previous methods?
(c) What are the limitations of the approach as described by the authors?
Problem 33.25 ⭐⭐ (Career exploration) Choose one of the career paths described in Section 33.13 (national laboratory, university, medical physics, nuclear engineering, national security, policy, or industry). Research:
(a) The typical educational pathway (degree level, postdoc requirements, certifications).
(b) A specific job opening or position description in that area (search laboratory or university websites).
(c) How the nuclear physics skills from this course apply to the position. Be specific — cite at least three specific topics from this textbook.