Exercises — Chapter 10: Exotic Nuclei

Section 10.1: The Expanding Nuclear Landscape

10.1. The 2020 Atomic Mass Evaluation contains measured or estimated masses for approximately 3,557 nuclides. The FRDM (Finite-Range Droplet Model) predicts roughly 9,800 nuclides between the drip lines.

(a) What fraction of all predicted bound nuclei have been experimentally observed?

(b) If FRIB discovers new isotopes at an average rate of 40 per year, how many years would it take to observe all predicted nuclei? Comment on why this is an unrealistic estimate.

(c) The neutron drip line has been experimentally determined only for elements with $Z \leq 10$. The proton drip line is known for elements up to $Z \approx 91$. Explain the physical reason for this asymmetry.

10.2. For a nucleus at the neutron drip line, the one-neutron separation energy is $S_n = 0$. Using the semi-empirical mass formula from Chapter 4 (with coefficients $a_V = 15.56$ MeV, $a_S = 17.23$ MeV, $a_C = 0.697$ MeV, $a_A = 23.29$ MeV, $a_P = 12.0$ MeV), calculate the predicted neutron drip line nucleus for oxygen ($Z = 8$) by finding the value of $N$ that makes $S_n \approx 0$. Compare your result to the experimentally determined drip line at $^{24}$O ($N = 16$). What does the discrepancy tell you about the role of shell effects?

10.3. The discovery rate of new isotopes has accelerated from approximately 15 per year in the 1970s to over 40 per year in the 2020s. Plot (or sketch) the cumulative number of known nuclides as a function of year, using the approximate data points: 1940 (300), 1960 (1,200), 1980 (2,000), 2000 (2,700), 2020 (3,300). Estimate when the total might reach 5,000 if the current pace of discovery continues.

Section 10.2: Radioactive Ion Beam Facilities

10.4. In the in-flight fragmentation method, a $^{238}$U beam at 200 MeV/nucleon strikes a beryllium target. Calculate:

(a) The total kinetic energy (in GeV) of a single $^{238}$U ion at 200 MeV/nucleon.

(b) The velocity $\beta = v/c$ of the uranium beam (use the relativistic relation $E_k = (\gamma - 1)mc^2$ with $mc^2 \approx 238 \times 931.5$ MeV).

(c) The magnetic rigidity $B\rho = \gamma m v / q$ (in T$\cdot$m) for a fully stripped $^{238}$U$^{92+}$ ion at this energy.

10.5. The ISOL method at ISOLDE uses a 1.4 GeV proton beam striking a thick uranium carbide target.

(a) Estimate the energy per nucleon of the proton beam. (This is trivial, but the point is to compare: why does ISOL use a proton beam while in-flight facilities use heavy-ion beams?)

(b) The diffusion time for reaction products to escape from a heated ISOL target is typically 10–100 ms. What is the shortest half-life that can be studied with the ISOL method? Express your answer in terms of the fraction of the produced isotopes that survive the diffusion delay, assuming $t_{\text{diff}} = 50$ ms.

(c) For a radioactive species with $t_{1/2} = 5$ ms, what fraction survives a 50 ms diffusion delay? Why does this make the in-flight method essential for short-lived species?

10.6. FRIB's primary beam power for $^{238}$U is 400 kW at 200 MeV/nucleon.

(a) Calculate the beam current in particle-$\mu$A (where $1$ p$\mu$A $= 6.24 \times 10^{12}$ ions/s).

(b) Calculate the number of uranium ions per second striking the production target.

(c) If the fragmentation cross section for producing $^{78}$Ni from $^{238}$U on beryllium is approximately $10^{-6}$ mb, and the target thickness is $10^{23}$ atoms/cm$^2$, estimate the production rate of $^{78}$Ni (in atoms per second). Comment on the feasibility of studying this nucleus.

Section 10.3: Halo Nuclei

10.7. For a particle bound in a square well potential with separation energy $E_b$, the asymptotic wavefunction is $\psi(r) \propto e^{-\kappa r}/r$ with $\kappa = \sqrt{2\mu E_b}/\hbar$.

(a) Calculate $\kappa$ and the decay length $1/\kappa$ for $^{11}$Be ($S_n = 0.502$ MeV, reduced mass $\mu \approx m_n \times m_{^{10}\text{Be}} / (m_n + m_{^{10}\text{Be}}) \approx 857$ MeV/$c^2$).

(b) Calculate $\kappa$ and $1/\kappa$ for a typical well-bound neutron with $S_n = 8$ MeV and $\mu \approx 900$ MeV/$c^2$.

(c) What is the ratio of the two decay lengths? Explain why this ratio is the key to understanding halo formation.

10.8. The centrifugal barrier for orbital angular momentum $\ell$ is $V_\ell(r) = \ell(\ell+1)\hbar^2/(2\mu r^2)$.

(a) Calculate the height of the centrifugal barrier at $r = 4$ fm for $\ell = 1$, $\ell = 2$, and $\ell = 3$, using $\mu = 900$ MeV/$c^2$. Express your answers in MeV.

(b) Compare these barrier heights to the typical halo separation energy of 0.5 MeV. For which values of $\ell$ can a halo form?

(c) Explain why halo nuclei are predominantly observed in light nuclei (say $A < 40$). Hint: Consider which single-particle orbits lie near the Fermi surface for light versus heavy neutron-rich nuclei.

10.9. The interaction cross section of a projectile on a target is approximately $\sigma_I \approx \pi (R_P + R_T)^2$, where $R_P$ and $R_T$ are the matter radii.

(a) For $^{9}$Li ($r_{\text{rms}} = 2.44$ fm) on $^{12}$C ($r_{\text{rms}} = 2.47$ fm), estimate $\sigma_I$ using $R \approx \sqrt{5/3} \, r_{\text{rms}}$. Express your answer in mb ($1$ fm$^2 = 10$ mb).

(b) Repeat for $^{11}$Li ($r_{\text{rms}} = 3.55$ fm) on $^{12}$C.

(c) What is the percentage increase in $\sigma_I$ going from $^{9}$Li to $^{11}$Li? Compare to the $A^{1/3}$ expectation (which predicts $R_{11}/R_9 = (11/9)^{1/3} \approx 1.07$, i.e., a ~14% increase in $\sigma_I$).

10.10. The momentum distribution of $^{9}$Li fragments after one-neutron knockout from $^{11}$Li has a narrow component with FWHM $\approx 45$ MeV/$c$.

(a) Use the uncertainty principle $\Delta x \sim \hbar / \Delta p$ to estimate the spatial extent of the halo wavefunction. Use $\Delta p \approx \text{FWHM}/2 \approx 22.5$ MeV/$c$.

(b) Compare this to the $^{9}$Li core radius. Is the result consistent with a halo extending well beyond the core?

(c) For a deeply bound neutron ($S_n \sim 8$ MeV), the fragment momentum FWHM is typically 150–200 MeV/$c$. Estimate the corresponding spatial extent and compare.

10.11. (Research-level) $^{22}$C ($Z = 6$, $N = 16$) has $S_{2n} \approx 110$ keV, making it one of the most weakly bound nuclei known.

(a) Calculate the decay length $1/\kappa$ for the two-neutron halo (use $\mu_{2n} \approx 2m_n \times m_{^{20}\text{C}} / (2m_n + m_{^{20}\text{C}})$).

(b) Compare the decay length to the $^{20}$C core radius ($r_{\text{rms}} \approx 2.8$ fm). What do you conclude about the extent of the halo?

(c) The measured reaction cross section of $^{22}$C is consistent with a matter radius of approximately 5.4 fm. How does this compare to the $r_0 A^{1/3}$ expectation?

Section 10.4–10.5: Shell Evolution and the Island of Inversion

10.12. The first $2^+$ excitation energy $E(2^+_1)$ is a key indicator of shell closure. Using the data below:

(a) For which nuclei does $N = 20$ or $N = 28$ appear to be a good magic number? For which has the magic number disappeared?

(b) Quantitatively, what is the ratio of $E(2^+_1)$ values for $^{48}$Ca and $^{42}$Si? What does this ratio tell you about the collectivity of $^{42}$Si?

(c) Does $^{52}$Ca support the existence of a new magic number at $N = 32$? Compare its $E(2^+_1)$ to the stable doubly-magic nuclei.

10.13. The tensor force produces a monopole shift when nucleons fill or empty specific orbits. Consider the oxygen isotopic chain ($Z = 8$):

(a) The proton configuration in oxygen fills the $0p_{1/2}$ and $0p_{3/2}$ orbits. The neutrons fill the $0d_{5/2}$, $1s_{1/2}$, and $0d_{3/2}$ orbits as $N$ increases from 8 to 20. Using the rule that the tensor force is attractive between $j_> = \ell + 1/2$ (proton) and $j_< = \ell' - 1/2$ (neutron), which proton-neutron orbital pairs experience attractive tensor interaction?

(b) In fluorine ($Z = 9$), one proton occupies the $0d_{5/2}$ orbit. How does adding this proton change the neutron $0d_{3/2}$ single-particle energy relative to oxygen? (Hint: is the tensor interaction between $\pi 0d_{5/2}$ and $\nu 0d_{3/2}$ attractive or repulsive?)

(c) Use this to explain qualitatively why the $N = 16$ gap is prominent in oxygen but not in fluorine.

10.14. $^{32}$Mg lies in the island of inversion. The standard shell model predicts a ground state in the "normal" $sd$-shell configuration, while the observed ground state is dominated by intruder $fp$-shell configurations.

(a) In the normal configuration, all neutrons occupy $sd$-shell orbits, and the nucleus is approximately spherical. The intruder configuration promotes two neutrons from the $0d_{3/2}$ orbit (below $N = 20$) to the $0f_{7/2}$–$1p_{3/2}$ orbits (above $N = 20$). If the $N = 20$ gap is $\Delta = 4$ MeV (reduced from its stable-nucleus value by the tensor force), what is the energy cost of promoting two neutrons?

(b) The intruder configuration gains correlation energy from deformation. If the total binding energy gain from quadrupole correlations is 10 MeV, what is the net energy difference between the intruder and normal configurations? Which is the ground state?

(c) Repeat for $\Delta = 6$ MeV (the stable-nucleus value). Is the intruder still the ground state? What does this tell you about the role of the weakened shell gap?

10.15. The quadrupole deformation parameter $\beta_2$ can be extracted from the $B(E2)$ value using:

$$B(E2; 0^+ \to 2^+) = \left(\frac{3}{4\pi}\right)^2 Z^2 e^2 R_0^4 \beta_2^2$$

where $R_0 = r_0 A^{1/3}$ with $r_0 = 1.2$ fm.

(a) For $^{32}$Mg ($Z = 12$, $A = 32$), calculate $\beta_2$ using $B(E2) = 454$ e$^2$fm$^4$.

(b) For $^{40}$Ca ($Z = 20$, $A = 40$), calculate $\beta_2$ using $B(E2) = 113$ e$^2$fm$^4$.

(c) Compare the two values. A nucleus with $\beta_2 > 0.2$ is considered "well-deformed." Does $^{32}$Mg qualify? Does $^{40}$Ca?

Section 10.6: Borromean Nuclei

10.16. $^{6}$He is a Borromean nucleus: $^{6}$He $= ^{4}$He $+ n + n$ is bound, but $^{5}$He $= ^{4}$He $+ n$ is unbound (the $p_{3/2}$ resonance in $^{5}$He has a width $\Gamma \approx 0.6$ MeV).

(a) The two-neutron separation energy of $^{6}$He is $S_{2n} = 0.975$ MeV. Calculate the decay length $1/\kappa$ for the two-neutron halo.

(b) The matter radius of $^{6}$He is 2.48 fm, while the core $^{4}$He radius is 1.46 fm. Calculate the rms distance of the halo neutrons from the center of mass, assuming a simple core + halo decomposition: $r_{\text{rms}}^2 = (4/6) r_{\text{core}}^2 + (2/6) r_{\text{halo}}^2$. Solve for $r_{\text{halo}}$.

(c) Compare $r_{\text{halo}}$ to the decay length $1/\kappa$. Are they consistent?

10.17. (Conceptual) Explain in your own words why a Borromean nucleus can be bound even though no two-body subsystem is bound. Use a potential energy diagram or a qualitative quantum mechanical argument.

10.18. In Jacobi coordinates for a three-body system (core $c$, neutrons $n_1$, $n_2$):

$$\vec{x} = \vec{r}_{n_1} - \vec{r}_{n_2}, \qquad \vec{y} = \vec{r}_c - \frac{1}{2}(\vec{r}_{n_1} + \vec{r}_{n_2})$$

(a) In the "dineutron" limit where both neutrons are at the same point, what are $|\vec{x}|$ and $|\vec{y}|$?

(b) In the "cigar" limit where the core is at the midpoint between two widely separated neutrons, what are $|\vec{x}|$ and $|\vec{y}|$ relative to the core-neutron distance $d$?

(c) Real three-body calculations for $^{11}$Li find the dineutron configuration has about 50–60% of the probability. Does this mean the two halo neutrons are "almost bound" as a dineutron? Why or why not?

Section 10.7: Proton-Rich Nuclei

10.19. The half-life of a proton emitter depends on the Coulomb barrier penetration. For proton emission from $^{151}$Lu ($Q_p = 1.233$ MeV, $\ell = 5$):

(a) Calculate the Coulomb barrier height $V_C = Z_d e^2 / (4\pi\epsilon_0 R)$ for the daughter $^{150}$Yb ($Z_d = 70$), using $R = r_0 (A_d^{1/3} + 1) \approx 7.5$ fm. Use $e^2/(4\pi\epsilon_0) = 1.44$ MeV$\cdot$fm.

(b) Compare the Coulomb barrier to $Q_p = 1.233$ MeV. By what factor does the barrier exceed the available energy?

(c) The measured half-life is $t_{1/2} = 85$ ms. If the orbital angular momentum were $\ell = 0$ instead of $\ell = 5$, would the half-life be longer or shorter? Explain using the concept of the centrifugal barrier.

10.20. Two-proton radioactivity has been observed in $^{45}$Fe.

(a) Why can't $^{45}$Fe simply emit one proton? (Hint: consider the one-proton separation energy $S_p$ versus the two-proton separation energy $S_{2p}$.)

(b) Two-proton emission can proceed either as simultaneous two-proton tunneling (a "diproton" mechanism) or as sequential emission via an intermediate state. What experimental observable would distinguish between these mechanisms?

(c) How is two-proton radioactivity analogous to two-neutron halo structure in Borromean nuclei?

Section 10.8: The Neutron-Rich Frontier

10.21. The r-process path at a given temperature and neutron density follows a contour of approximately constant neutron separation energy $S_n \approx 2$–$3$ MeV.

(a) On a sketch of the chart of nuclides, indicate the valley of stability, the neutron drip line, and the approximate location of the r-process path (between stability and the drip line).

(b) At the $N = 82$ shell closure, $S_n$ drops sharply. Explain why this creates an r-process "waiting point" where material accumulates.

(c) If the $N = 82$ shell gap is reduced far from stability (as suggested by some theoretical models), what would happen to the height and width of the $A \approx 130$ abundance peak? Would it sharpen, broaden, or shift?

10.22. (Research-level) The mass of $^{78}$Ni ($Z = 28$, $N = 50$) was first directly measured at RIKEN in 2022 using a multi-reflection time-of-flight mass spectrograph, with only 12 detected events.

(a) $^{78}$Ni is doubly magic. If the $N = 50$ shell closure weakens far from stability (as $N = 20$ and $N = 28$ do), how would this affect the binding energy of $^{78}$Ni relative to the shell model prediction?

(b) The measured mass was consistent with a robust $N = 50$ shell closure. Why is this significant for r-process calculations? (Hint: $^{78}$Ni is near the $N = 50$ r-process waiting point that produces the $A \approx 80$ abundance peak.)

(c) With only 12 events, the mass uncertainty was relatively large. How could FRIB, with its higher production rates, improve this measurement?

Synthesis and Review

10.23. Create a table listing five exotic nuclear phenomena discussed in this chapter (halo nuclei, shell evolution, island of inversion, Borromean nuclei, proton radioactivity). For each, list: (a) one specific nuclear example, (b) the key experimental observable, (c) the facility where the key measurement was performed, and (d) one connection to nuclear astrophysics or applications.

10.24. A recurring theme in this chapter is that nuclear properties change qualitatively far from stability. For each of the following "rules" that work near stability, give a specific example of where the rule breaks down:

(a) Nuclear radii scale as $R = r_0 A^{1/3}$. (b) Magic numbers are 2, 8, 20, 28, 50, 82, 126. (c) All nuclei near a magic number are spherical. (d) Nuclei with $S_n > 0$ are bound and those with $S_n < 0$ are unbound (no further subtlety needed).

10.25. (Estimation) The r-process requires a neutron flux sufficiently intense that neutron capture is faster than beta decay. If the average neutron capture cross section is $\langle \sigma v \rangle \approx 10^{-20}$ cm$^3$/s and the typical beta-decay half-life of a neutron-rich nucleus on the r-process path is $t_{1/2} \sim 100$ ms:

(a) Estimate the minimum neutron number density $n_n$ required for neutron capture to compete with beta decay.

(b) Convert this to a mass density (assume all the material is free neutrons). Is this reasonable for the interior of a neutron star merger or the outer layers of a core-collapse supernova?

(c) Why does the r-process path not simply follow the neutron drip line?

10.26. (Conceptual) The nuclear symmetry energy $a_{\text{sym}}$ characterizes the energy cost of converting protons to neutrons in nuclear matter. Explain how measurements of the neutron skin thickness $\Delta r_{np}$ in $^{208}$Pb constrain $a_{\text{sym}}$, and why this is relevant for the equation of state of neutron star matter. What role do exotic neutron-rich nuclei play in extending these constraints?

10.27. (Computational) Using the exotic_nuclei_chart.py code from this chapter:

(a) Modify the SEMF parameters by $\pm 10\%$ for the asymmetry coefficient $a_A$. How does this change the predicted neutron drip line location? Plot the original and modified drip lines on the same chart.

(b) For the oxygen isotopic chain ($Z = 8$), plot the SEMF neutron separation energy $S_n$ as a function of $N$ from $N = 6$ to $N = 22$. Mark the experimental drip line at $N = 16$ and the shell model prediction of $N = 16$ magicity. Where does the SEMF predict $S_n = 0$?

(c) Add the magic number $N = 16$ to the grid lines on the chart of nuclides. Discuss where $N = 16$ matters (oxygen, fluorine) and where it does not (heavier elements).

10.28. (Essay) In 500–800 words, compare the scientific case for FRIB with the scientific case for a next-generation particle physics facility (e.g., a future circular collider). Both are multi-billion-dollar investments in fundamental physics. What questions does each address? What is the discovery potential of each? How do they complement each other?

10.29. The Borromean nucleus $^{22}$C has $S_{2n} \approx 110$ keV. This is so weakly bound that the precise value of $S_{2n}$ is difficult to measure — different experiments have reported values ranging from 40 keV to 400 keV.

(a) Calculate the halo decay length $1/\kappa$ for $S_{2n} = 40$ keV, 110 keV, and 400 keV. Use a reduced mass $\mu_{2n} \approx 1670$ MeV/$c^2$.

(b) For the smallest separation energy (40 keV), compare the decay length to the $^{20}$C core radius (~2.8 fm). What fraction of the total nuclear size would be "halo"?

(c) Why is measuring $S_{2n}$ so precisely important for theoretical models of $^{22}$C? How would a Penning trap mass measurement at FRIB help?