Chapter 11 Exercises: Superheavy Elements

Section A: Conceptual Questions

Exercise 11.1 — Shell Corrections and Existence

(a) Explain in your own words why the liquid drop model predicts that no nucleus beyond Z $\approx$ 104 should exist with a measurable half-life. What physical quantity vanishes, and what does that imply?

(b) The Strutinsky shell correction method adds a correction $\delta E_{\text{shell}}$ to the liquid drop energy. Under what conditions is this correction negative (stabilizing)? Why is the correction typically small at the saddle-point deformation?

(c) For $^{208}$Pb, the shell correction energy is approximately $-13$ MeV. For a predicted doubly magic superheavy nucleus like $^{298}$Fl, it is approximately $-7$ MeV. Why is the superheavy shell correction smaller in magnitude? Give at least two physical reasons.

Exercise 11.2 — Cold vs. Hot Fusion Comparison

(a) Complete the following table comparing cold fusion and hot fusion approaches to superheavy element synthesis:

(b) Explain physically why $^{208}$Pb is an ideal target for cold fusion reactions. Connect your answer to the shell model.

(c) Explain why $^{48}$Ca is an ideal projectile for hot fusion reactions. Your answer should address at least three distinct advantages of $^{48}$Ca.

Exercise 11.3 — Relativistic Chemistry

(a) Calculate the velocity of a 1s electron in element 120 (Z = 120) as a fraction of the speed of light. What is the corresponding Lorentz factor $\gamma$?

(b) Explain the distinction between the direct relativistic effect (s and p$_{1/2}$ orbital contraction) and the indirect relativistic effect (d and f orbital expansion). Why do both effects originate from the same cause?

(c) Oganesson (Z = 118) is placed in Group 18 (noble gases) of the periodic table. List three predicted properties of oganesson that differ from those of a typical noble gas. For each, explain how relativistic effects cause the deviation.

Section B: Quantitative Problems

Exercise 11.4 — Fissility Parameter

The fissility parameter is $x = Z^2 / (50.88 \, A)$ (using $a_S = 17.8$ MeV, $a_C = 0.711$ MeV).

(a) Calculate $x$ for $^{208}$Pb (Z = 82), $^{238}$U (Z = 92), $^{252}$Cf (Z = 98), and $^{298}$Fl (Z = 114).

(b) The liquid drop fission barrier height can be approximated as:

$$B_f^{\text{LD}} \approx 0.38 \, E_{\text{surface}}^{(0)} (1 - x)^3$$

where $E_{\text{surface}}^{(0)} = a_S A^{2/3}$. Calculate $B_f^{\text{LD}}$ for each nucleus in part (a).

(c) For $^{298}$Fl, if the shell correction adds approximately 6 MeV to the fission barrier, what is the total barrier height? Compare this to the fission barrier of $^{240}$Pu ($B_f \approx 6$ MeV). Comment on the relative importance of the shell correction.

Exercise 11.5 — Compound Nucleus Excitation Energy

When a projectile with laboratory kinetic energy $T_{\text{lab}}$ strikes a stationary target, the center-of-mass kinetic energy is:

$$E_{\text{CM}} = T_{\text{lab}} \frac{A_{\text{target}}}{A_{\text{proj}} + A_{\text{target}}}$$

The compound nucleus excitation energy is $E^* = E_{\text{CM}} + Q$, where $Q$ is the fusion Q-value.

(a) For the cold fusion reaction $^{70}$Zn + $^{208}$Pb $\rightarrow$ $^{278}$Cn$^*$, the optimum beam energy is $T_{\text{lab}} \approx 344$ MeV. Calculate $E_{\text{CM}}$.

(b) The atomic masses (in u) are: $m(^{70}$Zn) = 69.92532, $m(^{208}$Pb) = 207.97664, $m(^{278}$Cn) = 278.0 (estimated). Calculate the Q-value in MeV (1 u = 931.494 MeV/c$^2$). Then find $E^*$.

(c) Repeat for the hot fusion reaction $^{48}$Ca + $^{244}$Pu $\rightarrow$ $^{292}$Fl$^*$ at the optimum beam energy $T_{\text{lab}} \approx 245$ MeV. Use $m(^{48}$Ca) = 47.95253 u, $m(^{244}$Pu) = 244.06420 u, $m(^{292}$Fl) = 292.0 (estimated).

(d) Compare the excitation energies from (b) and (c). Explain how the difference in $E^*$ affects the number of neutrons that must be evaporated and the survival probability.

Exercise 11.6 — Production Cross Sections

The production rate of superheavy atoms is:

$$R = \sigma \cdot \Phi \cdot N_T$$

where $\sigma$ is the cross section, $\Phi$ is the beam flux (ions per second), and $N_T$ is the number of target atoms per cm$^2$.

(a) For the discovery of element 118 via $^{48}$Ca + $^{249}$Cf, the cross section was $\sigma \approx 0.5$ pb, the beam intensity was $\Phi = 4 \times 10^{12}$ ions/s, and the target thickness was 0.35 mg/cm$^2$ of $^{249}$Cf. Calculate $N_T$ (in atoms/cm$^2$) and the production rate $R$ (in atoms/second). Convert to atoms/day and atoms/month.

(b) If the overall detection efficiency (geometric acceptance, transmission through separator, implantation and detection) is 40%, how many atoms would be detected per month? How many months of continuous operation would be needed to accumulate 5 detected events?

(c) For element 120 via $^{54}$Cr + $^{248}$Cm, the predicted cross section is $\sigma \approx 50$ fb. Assuming the SHE Factory delivers $\Phi = 5 \times 10^{13}$ ions/s onto a similar target, calculate the expected production rate. How long would it take to produce 3 detected events (assuming 40% efficiency)?

Exercise 11.7 — Alpha Decay Energetics

The alpha-decay chain of $^{294}$Og is:

$$^{294}\text{Og} \xrightarrow{\alpha} ^{290}\text{Lv} \xrightarrow{\alpha} ^{286}\text{Fl} \xrightarrow{\alpha} ^{282}\text{Cn} \xrightarrow{\text{SF}}$$

The measured alpha energies are approximately: - $^{294}$Og $\rightarrow$ $^{290}$Lv: $E_\alpha = 11.65$ MeV - $^{290}$Lv $\rightarrow$ $^{286}$Fl: $E_\alpha = 10.84$ MeV - $^{286}$Fl $\rightarrow$ $^{282}$Cn: $E_\alpha = 10.19$ MeV

(a) For each step, calculate the Q-value using $Q_\alpha = E_\alpha (1 + m_\alpha / m_{\text{daughter}})$, where $m_\alpha = 4$ u and $m_{\text{daughter}} = A - 4$ u (a good approximation).

(b) The Q-value is related to binding energies by $Q_\alpha = B(A-4, Z-2) + B(4,2) - B(A,Z)$, where $B(4,2) = 28.296$ MeV. Use the Q-values from part (a) and an estimated binding energy for $^{282}$Cn of $B \approx 2034$ MeV to reconstruct the binding energies of $^{286}$Fl, $^{290}$Lv, and $^{294}$Og.

(c) Compare the Q-values in the chain. Do they increase or decrease with increasing Z? What does this trend tell you about the proximity of these nuclei to the island of stability center?

Exercise 11.8 — Coulomb Barrier

The Coulomb barrier for the head-on collision of two nuclei with charges $Z_1$ and $Z_2$ and radii $R_1$ and $R_2$ is approximately:

$$V_C = \frac{Z_1 Z_2 e^2}{4\pi\epsilon_0 (R_1 + R_2)} = \frac{1.44 \, Z_1 Z_2}{R_1 + R_2} \text{ MeV}$$

where $R_i = 1.25 A_i^{1/3}$ fm.

(a) Calculate the Coulomb barrier for $^{48}$Ca + $^{244}$Pu and for $^{48}$Ca + $^{249}$Cf.

(b) Calculate the Coulomb barrier for $^{54}$Cr + $^{248}$Cm and for $^{70}$Zn + $^{208}$Pb.

(c) In practice, the beam energy in the center-of-mass frame must be slightly above the Coulomb barrier for fusion to occur (by about 3-5% for sub-barrier enhancement from quantum tunneling). For each reaction above, calculate the required laboratory beam energy using $T_{\text{lab}} = E_{\text{CM}} (A_1 + A_2)/A_2$.

(d) Compare the Coulomb barriers for reactions using $^{48}$Ca vs. $^{54}$Cr projectiles on similar targets. How does the higher barrier for $^{54}$Cr affect the excitation energy and survival probability?

Exercise 11.9 — Shell Correction Magnitude

The Strutinsky shell correction can be roughly estimated from the gap in the single-particle spectrum near the Fermi surface. If the gap between the last filled and first empty single-particle levels is $\Delta \varepsilon$ and there are approximately $n$ levels that contribute to the shell correction, a crude estimate is:

$$|\delta E_{\text{shell}}| \sim n \cdot \frac{\Delta \varepsilon}{2}$$

For $^{208}$Pb, the proton gap at Z = 82 is $\Delta \varepsilon_p \approx 3.5$ MeV and the neutron gap at N = 126 is $\Delta \varepsilon_n \approx 3.8$ MeV. Approximately 6-8 levels contribute on each side of the gap.

(a) Estimate $\delta E_{\text{shell}}$ for $^{208}$Pb using this crude formula with $n = 7$ for both protons and neutrons. Compare to the accepted value of approximately $-13$ MeV.

(b) For the predicted Z = 114 gap, $\Delta \varepsilon_p \approx 2.0$ MeV (from relativistic mean-field calculations). For the predicted N = 184 gap, $\Delta \varepsilon_n \approx 2.5$ MeV. Estimate $\delta E_{\text{shell}}$ for $^{298}$Fl with $n = 6$. Why is this correction smaller than for $^{208}$Pb?

(c) If the actual proton magic number is Z = 120 instead of Z = 114, and the Z = 120 gap is $\Delta \varepsilon_p \approx 2.5$ MeV, how would the shell correction for the corresponding doubly magic nucleus $^{304}120$ compare?

Section C: Synthesis and Analysis Problems

Exercise 11.10 — Decay Mode Competition

For a superheavy nucleus, the dominant decay modes are alpha decay ($\alpha$) and spontaneous fission (SF). The partial half-life for alpha decay can be estimated from the Viola-Seaborg formula:

$$\log_{10} t_{1/2}^{\alpha}(\text{s}) = \frac{aZ + b}{\sqrt{Q_\alpha}} + cZ + d$$

with $a = 1.66175$, $b = -8.5166$, $c = -0.20228$, $d = -33.9069$ for even-even nuclei.

(a) Using $Q_\alpha = 11.81$ MeV for $^{294}$Og, estimate $t_{1/2}^{\alpha}$. Compare to the measured value of $\sim 0.7$ ms.

(b) The measured half-life of $^{294}$Og is $\sim 0.7$ ms and the decay is by alpha emission. What does this tell you about the spontaneous fission half-life? (It must be longer than the alpha half-life.)

(c) For the predicted nucleus $^{298}$Fl with $Q_\alpha \approx 7.5$ MeV (estimated), calculate the alpha-decay half-life. Explain why the much lower Q-value leads to a dramatically longer half-life.

Exercise 11.11 — The RIKEN Discovery of Element 113

RIKEN's discovery of element 113 required 553 days of beam time over 9 years to observe 3 events.

(a) The beam intensity was approximately $2.4 \times 10^{12}$ ions/s of $^{70}$Zn and the $^{209}$Bi target was 0.5 mg/cm$^2$. The cross section was 22 fb. Verify the expected production rate and show that 3 events in 553 days is consistent with the stated cross section (assume 30% detection efficiency).

(b) Compare this to element 112 at GSI ($\sigma \approx 1$ pb, $^{70}$Zn + $^{208}$Pb). What is the ratio of cross sections? If GSI had attempted element 113 via cold fusion instead of RIKEN, how long might it have taken?

(c) Why did RIKEN pursue cold fusion for element 113 rather than the hot fusion approach used by Dubna? What was the scientific advantage of the cold fusion route for establishing the discovery claim?

Exercise 11.12 — Neutron Evaporation and Survival

In the hot fusion reaction $^{48}$Ca + $^{248}$Cm $\rightarrow$ $^{296}$Lv$^*$, the compound nucleus is formed with excitation energy $E^* \approx 35$ MeV. Each neutron evaporation removes approximately 8 MeV (binding energy of the neutron plus kinetic energy carried away).

(a) How many neutrons must be evaporated to cool the compound nucleus to below the neutron separation energy? (This gives the evaporation channel: 3n, 4n, or 5n.)

(b) At each evaporation step, the survival probability is approximately $P_{\text{surv}} = \Gamma_n / (\Gamma_n + \Gamma_f)$. If $P_{\text{surv}} \approx 0.3$ at each step (typical for actinide-region nuclei with shell-enhanced fission barriers), what is the total survival probability for 4 neutron evaporation steps?

(c) The capture cross section is approximately 30 mb and the fusion probability is $P_{\text{fusion}} \approx 5 \times 10^{-4}$. Using your answer to (b), estimate the evaporation residue cross section $\sigma_{\text{SHE}}$. Express your answer in picobarns.

(d) How does $P_{\text{surv}}$ per step change if the shell correction energy increases (i.e., the nucleus is closer to a magic number)? Explain physically.

Exercise 11.13 — Island of Stability Half-Life Trends

The measured half-lives of even-even flerovium isotopes are approximately:

(a) Plot $\log_{10} t_{1/2}$ versus N. Is the trend linear? Extrapolate the trend to N = 184 and estimate the half-life of $^{298}$Fl.

(b) Your extrapolation assumes a smooth trend. Explain why a shell closure at N = 184 would cause the actual half-life to be even longer than your extrapolation predicts.

(c) If the half-life of $^{298}$Fl were 10$^{6}$ years (as some models predict), how many atoms would you need to produce in order to reliably measure the half-life? (Hint: for a half-life of $10^6$ years, what fraction of a sample decays per year? How many decay events would you need to observe?)

Exercise 11.14 — Relativistic Electron Velocity

(a) Plot $v_{1s}/c = Z/137$ and the corresponding Lorentz factor $\gamma = 1/\sqrt{1-(Z/137)^2}$ as functions of Z for Z = 1 to Z = 140. At what Z does $\gamma$ exceed 2? At what Z does $\gamma$ exceed 5?

(b) The relativistic contraction of the 1s orbital radius relative to the Bohr radius $a_0/Z$ is approximately $r_{1s}^{\text{rel}} \approx (a_0/Z) \times (1/\gamma)$. Plot the ratio $r_{1s}^{\text{rel}}/r_{1s}^{\text{NR}}$ as a function of Z. For oganesson (Z = 118), by what percentage is the 1s orbital contracted?

(c) The point-nucleus model predicts that the 1s binding energy diverges at Z = 137. For an extended nucleus of radius $R = 1.2 A^{1/3}$ fm, the critical Z is pushed to approximately 170-173. Explain physically why the finite nuclear size raises the critical Z.

Section D: Research and Discussion Problems

Exercise 11.15 — Discovery Priority Controversies

The discovery of superheavy elements has been marked by priority disputes, particularly between GSI (Germany) and JINR (Russia/Soviet Union) for elements in the Z = 104-106 range, and later between JINR (Russia) and RIKEN (Japan) for element 113.

(a) Research the historical dispute over element 104. What were the competing names (rutherfordium vs. kurchatovium)? How was the dispute resolved by IUPAC?

(b) For element 113, the IUPAC/IUPAP Joint Working Party considered claims from both Dubna (via hot fusion, element 113 as daughter of element 115) and RIKEN (via cold fusion, direct production). What criteria did the JWP use to evaluate the claims? Why was priority awarded to RIKEN?

(c) Discuss the challenges of establishing "discovery" when only a few atoms are produced. What constitutes sufficient evidence that a new element has been synthesized? How many independent decay chains are typically required?

Exercise 11.16 — Designing an Experiment for Element 119

You are a nuclear physicist tasked with proposing an experiment to synthesize element 119. Address the following:

(a) Choose a projectile-target combination. Justify your choice based on the expected cross section, target availability, and compound nucleus excitation energy. Consider at least two options and explain which you prefer.

(b) Estimate the Coulomb barrier for your chosen reaction and the required beam energy.

(c) At the SHE Factory, beam intensities of $5 \times 10^{13}$ ions/s for $^{50}$Ti may be achievable. If the cross section is 30 fb, calculate the expected production rate. How many months of beam time would you request?

(d) What decay mode would you expect for element 119? If it alpha-decays, trace the expected decay chain to a known nucleus. How would you confirm the identification?

Exercise 11.17 — The Periodic Table at Its Limits

(a) The 8th period of the periodic table (Z = 119-168) is predicted to involve the filling of 8s, 5g, 6f, 7d, and 8p orbitals. In the non-relativistic Madelung rule, what would the filling order be? How do relativistic effects alter this predicted order?

(b) Pekka Pyykko has proposed an extended periodic table up to Z = 172. Find and examine Pyykko's extended table. How does it differ from a simple extrapolation of the current periodic table?

(c) Discuss: is the concept of "element" itself well-defined for nuclei with half-lives shorter than $10^{-14}$ seconds? What operational definition of "element" should we use in the superheavy region?

Exercise 11.18 — Multi-Nucleon Transfer Reactions

A possible alternative to complete fusion for reaching neutron-rich superheavy nuclei is multi-nucleon transfer (MNT) — collisions where many nucleons are exchanged between projectile and target without full fusion.

(a) In a collision of $^{238}$U + $^{248}$Cm at energies slightly above the Coulomb barrier, nucleons from $^{238}$U can transfer to $^{248}$Cm (or vice versa). Explain why this process can produce more neutron-rich products than complete fusion.

(b) The cross sections for producing specific superheavy isotopes via MNT are predicted to be on the order of nanobarns to microbarns (much larger than complete fusion). Why, then, has MNT not yet been used to discover new elements? What is the experimental challenge?

(c) The MNT approach is being explored at the SHE Factory and at FRIB. Design a conceptual experiment to produce neutron-rich isotopes of flerovium (Z = 114) via MNT. What would be your projectile and target? What detection technique would you use?

Exercise 11.19 — Computational: SHE Half-Life Visualization

Using the superheavy_stability.py code from this chapter (or your own implementation), complete the following:

(a) Plot the known half-lives of all isotopes with Z = 104-118 as a function of neutron number N. Use a logarithmic y-axis. Color-code the data points by dominant decay mode ($\alpha$, SF, $\beta^+$/EC).

(b) For each element from Z = 112 to Z = 118, fit a straight line to $\log_{10} t_{1/2}$ vs. N (using only $\alpha$-decaying isotopes). Extrapolate each fit to N = 184 and plot the predicted half-lives. Do the extrapolations converge to the expected island of stability region?

(c) Plot the predicted shell correction energies from the code alongside the half-life data. Is there a correlation between larger (more negative) shell corrections and longer half-lives?

Exercise 11.20 — The Complete Z = 118 Decay Chain

The decay chain of $^{294}$Og $\rightarrow$ $^{290}$Lv $\rightarrow$ $^{286}$Fl $\rightarrow$ $^{282}$Cn $\rightarrow$ SF contains measured alpha energies and half-lives:

(a) Construct a nuclear level scheme showing the alpha-decay chain and the SF endpoint. For each step, show the Q-value (calculated from $E_\alpha$), the half-life, and the decay mode.

(b) The large uncertainties in the half-lives reflect the very small number of events (3 chains for $^{294}$Og). If the production rate at the SHE Factory is 5 times higher than the original Dubna experiment, how many chains could be collected in 6 months? How would this improve the half-life uncertainties?

(c) $^{282}$Cn terminates the chain by spontaneous fission rather than alpha decay. What does this tell you about the relative magnitude of the fission barrier at N = 170 compared to N = 172-176 (where the heavier isotopes alpha-decay)? Connect your answer to the concept of shell stabilization.

Exercise 11.21 — Evaporation Residue Cross-Section Estimate

The production of element 116 via $^{48}$Ca + $^{248}$Cm proceeds through the compound nucleus $^{296}$Lv$^*$.

(a) The Coulomb barrier for this reaction is approximately $V_C = 210$ MeV in the center-of-mass frame. The optimal beam energy is $E_{\text{CM}} = 215$ MeV (slightly above the barrier). Using $Q_{\text{fusion}} \approx -175$ MeV (estimated from the mass excess difference), calculate the excitation energy $E^*$ of the compound nucleus.

(b) If each neutron evaporation step removes approximately 8 MeV and the ground-state neutron separation energy is about 7 MeV, how many neutrons can be evaporated? Identify the dominant evaporation channel (2n, 3n, 4n, or 5n).

(c) The capture cross section at this energy is $\sigma_{\text{cap}} \approx 25$ mb. The fusion probability (ratio of complete fusion to capture) has been estimated as $P_{\text{fus}} \approx 4 \times 10^{-4}$. If the survival probability (product over all evaporation steps) is $P_{\text{surv}} \approx 2 \times 10^{-4}$, calculate the predicted evaporation residue cross section. Compare to the measured value of approximately 3 pb.

(d) Discuss the sources of uncertainty in each factor of the cross-section estimate. Which factor is most uncertain, and why?

Exercise 11.22 — Mass Predictions in the Superheavy Region

Different nuclear mass models make distinct predictions for the binding energies of superheavy nuclei. Consider the nucleus $^{298}$Fl (Z = 114, N = 184):

(a) The SEMF (Chapter 4) with standard parameters ($a_V = 15.75$, $a_S = 17.8$, $a_C = 0.711$, $a_{\text{sym}} = 23.7$ MeV, $a_P = 12.0$ MeV) predicts a total binding energy. Calculate $B_{\text{SEMF}}$ for $^{298}$Fl. (This is the liquid drop prediction, without shell corrections.)

(b) If the shell correction at the ground state is $\delta E_{\text{shell}} = -7.2$ MeV (from macroscopic-microscopic models), what is the total binding energy including shell effects? Express the result as binding energy per nucleon $B/A$.

(c) Calculate the two-proton separation energy $S_{2p} = B(Z, N) - B(Z-2, N) - B(2, 0)$ using the SEMF for both $^{298}$Fl (Z = 114) and $^{296}$Cn (Z = 112), both at N = 184. A sudden increase in $S_{2p}$ at a particular Z would indicate a proton shell closure. Does the SEMF show such an increase at Z = 114? Why or why not?

(d) Explain why the SEMF cannot predict shell closures, and why self-consistent models (Skyrme-HFB, RMF) are necessary for this purpose. What additional physics do they include?

Exercise 11.23 — The Naming of Elements

The naming of elements 113-118 was announced by IUPAC on November 28, 2016 — a date that completed the seventh row of the periodic table and placed new names on science's most famous diagram.

(a) IUPAC rules for element naming allow names derived from: (i) a mythological concept, (ii) a mineral or similar substance, (iii) a place or geographic region, (iv) a property of the element, or (v) a scientist. Classify each of elements 113-118 according to which naming category it falls into. Note any elements named after more than one category.

(b) The "transfermium wars" of the Cold War era (1960s-1990s) saw competing names for elements 104-109, with American and Soviet/Russian laboratories proposing different names (e.g., rutherfordium vs. kurchatovium for Z = 104). What compromise was reached by IUPAC in 1997? How did the resolution of these disputes influence the naming process for elements 113-118?

(c) Element 118 was named oganesson after Yuri Oganessian, who was still alive at the time. Glenn Seaborg had similarly had seaborgium (Z = 106) named after him while living. Why is naming an element after a living person controversial? What arguments can be made for and against this practice?

(d) If elements 119 and 120 are discovered, what names might be appropriate? Consider the contributions of the discovering laboratory, the relevant scientists, and the IUPAC naming guidelines. Propose and justify one name for each element.