Self-Assessment Quiz — Am I Ready for the Capstone?

Before beginning the capstone project, assess your readiness across the six core competency areas. For each question, try to answer from memory before checking. If you cannot answer more than half the questions in a section, review the indicated chapters before starting the project.

Part I: Nuclear Structure (Chapters 1–11)

Q1. (Short Answer) Write the shell model configuration for the ground state of ${}^{40}\text{Ca}$ ($Z = 20$, $N = 20$). What is the predicted spin and parity? Why is this nucleus doubly magic?

Review if unsure: Chapters 1, 6

Q2. (Calculation) Using the SEMF with parameters $a_v = 15.56$, $a_s = 17.23$, $a_c = 0.697$, $a_a = 23.29$, $\delta_0 = 12$ MeV, calculate the binding energy of ${}^{208}\text{Pb}$ ($Z = 82$, $A = 208$). The experimental value is $B = 1636.43\,\text{MeV}$. What is the residual, and what does its sign tell you?

Review if unsure: Chapter 4

Q3. (Multiple Choice) The energy ratio $E(4^+_1)/E(2^+_1) = 3.30$ for a particular even-even nucleus. This nucleus is best described as:

(a) A good spherical vibrator (b) A transitional nucleus (c) A good rigid rotor (d) A superdeformed nucleus

Review if unsure: Chapter 8

Q4. (Short Answer) What is a Weisskopf unit? If the measured $B(E2; 0^+ \to 2^+_1) = 50\,\text{W.u.}$, what does this tell you about the transition?

Review if unsure: Chapter 9

Q5. (True/False) The Schmidt magnetic moment for a proton in a $j = \ell + 1/2$ state is always larger than the measured magnetic moment of the corresponding nucleus.

Review if unsure: Chapter 6

Part II: Radioactive Decay (Chapters 12–16)

Q6. (Calculation) The $Q$-value for alpha decay of ${}^{238}\text{U}$ is $Q_\alpha = 4.270\,\text{MeV}$. Using the Geiger-Nuttall relation $\log_{10} t_{1/2} = a/\sqrt{Q_\alpha} + b$ with $a = 1.61$ and $b = -28.9$ (for even-even nuclei with $Z = 92$), estimate the half-life. Compare with the measured value of $4.468 \times 10^9$ years.

Review if unsure: Chapter 13

Q7. (Short Answer) Explain the difference between secular equilibrium and transient equilibrium. Under what condition on the parent and daughter half-lives does secular equilibrium apply?

Review if unsure: Chapter 12

Q8. (Multiple Choice) A $\beta^-$ transition with $\log ft = 3.5$ is classified as:

(a) Superallowed (b) Allowed (c) First forbidden (d) Second forbidden

Review if unsure: Chapter 14

Q9. (Short Answer) Write the $Q$-value formula for $\beta^-$ decay in terms of atomic masses $M(A,Z)$ and $M(A, Z+1)$. Why do the electron masses cancel when using atomic masses?

Review if unsure: Chapter 14

Q10. (True/False) The Bethe-Bloch formula for stopping power is proportional to $Z_{\text{projectile}}^2 / v^2$ at high velocities, where $v$ is the projectile speed.

Review if unsure: Chapter 16

Part III: Nuclear Reactions (Chapters 17–21)

Q11. (Calculation) Calculate the $Q$-value for the reaction ${}^{12}\text{C}(\alpha,\gamma){}^{16}\text{O}$, given: - $M({}^{12}\text{C}) = 12.000000\,\text{u}$ - $M({}^{4}\text{He}) = 4.002603\,\text{u}$ - $M({}^{16}\text{O}) = 15.994915\,\text{u}$

Is this reaction exothermic or endothermic?

Review if unsure: Chapter 17

Q12. (Short Answer) Explain the compound nucleus hypothesis. Why does the compound nucleus "forget" how it was formed?

Review if unsure: Chapter 18

Q13. (Multiple Choice) The Breit-Wigner cross section for an isolated resonance at energy $E_r$ with total width $\Gamma$ is proportional to:

(a) $1/(E - E_r)^2$ (b) $\Gamma^2 / [(E - E_r)^2 + \Gamma^2/4]$ (c) $\exp[-(E - E_r)^2 / \Gamma^2]$ (d) $\Gamma / (E - E_r)$

Review if unsure: Chapter 18

Q14. (Short Answer) Define the astrophysical S-factor. Why is it more useful than the bare cross section for extrapolating charged-particle reaction rates to stellar energies?

Review if unsure: Chapter 21

Q15. (Calculation) For the D-T fusion reaction ($Q = 17.59\,\text{MeV}$), calculate the kinetic energy of the outgoing neutron and alpha particle in the center-of-mass frame.

Review if unsure: Chapter 17

Part IV: Nuclear Astrophysics (Chapters 22–25)

Q16. (Short Answer) What is the Gamow peak? Explain qualitatively why the peak exists as the product of two competing effects, and write the formula for the peak energy $E_0$.

Review if unsure: Chapter 22

Q17. (Multiple Choice) The rate-limiting step of the pp chain is:

(a) $p + p \to d + e^+ + \nu_e$ (b) $d + p \to {}^3\text{He} + \gamma$ (c) ${}^3\text{He} + {}^3\text{He} \to {}^4\text{He} + 2p$ (d) ${}^7\text{Be} + e^- \to {}^7\text{Li} + \nu_e$

Review if unsure: Chapter 22

Q18. (Short Answer) Explain why the r-process requires neutron densities of $n_n \gtrsim 10^{20}\,\text{cm}^{-3}$ and what happens at the "waiting points."

Review if unsure: Chapter 23

Q19. (True/False) The primordial helium abundance from Big Bang nucleosynthesis ($Y_p \approx 0.245$) is primarily determined by the neutron-to-proton ratio at the time of deuterium bottleneck breakout.

Review if unsure: Chapter 24

Q20. (Short Answer) What is the Tolman-Oppenheimer-Volkoff equation? What nuclear physics input does it require?

Review if unsure: Chapter 25

Part V: Applications and Tools (Chapters 26–30)

Q21. (Short Answer) Define the four-factor formula for the infinite multiplication factor $k_\infty = \eta f p \epsilon$. What does each factor represent physically?

Review if unsure: Chapter 26

Q22. (Multiple Choice) In PET imaging, the detected radiation consists of:

(a) A single 511 keV gamma ray (b) Two 511 keV gamma rays emitted back-to-back (c) A positron and an electron (d) A characteristic X-ray from the radioactive isotope

Review if unsure: Chapter 27

Q23. (Short Answer) Name three nuclear data resources and describe what type of data each provides.

Review if unsure: Section 34.5

Part VI: Integration and Synthesis

Q24. (Essay — 1 paragraph) Explain how the binding energy per nucleon curve connects nuclear structure (Part I), stellar nucleosynthesis (Part V), and nuclear energy applications (Part VI). Your answer should mention specific nuclei.

This tests your readiness for the synthesis component of the capstone.

Q25. (Essay — 1 paragraph) Choose any nucleus and explain how its shell model configuration affects at least two of the following: its decay properties, its role in nucleosynthesis, and its technological applications.

This tests your ability to make the cross-topic connections required for a strong capstone.

Scoring Guide

Interpretation:

• 24–25 correct: You are fully prepared. Proceed to the capstone with confidence.
• 20–23 correct: You are mostly prepared. Review the weak areas before beginning, but you can start data gathering immediately.
• 15–19 correct: You have gaps in key areas. Spend a week reviewing the indicated chapters before beginning the capstone analysis.
• Below 15: Significant preparation needed. Meet with your instructor to develop a review plan before starting the project.

Solutions

Q1. Configuration: $1s_{1/2}^2 \; 1p_{3/2}^4 \; 1p_{1/2}^2 \; 1d_{5/2}^6 \; 2s_{1/2}^2 \; 1d_{3/2}^4$ for both protons and neutrons. $I^\pi = 0^+$ (even-even). Doubly magic because $Z = 20$ and $N = 20$ are both magic numbers, corresponding to the closure of the $sd$ shell.

Q2. $B_{\text{SEMF}} = 15.56(208) - 17.23(208)^{2/3} - 0.697 \frac{82 \times 81}{208^{1/3}} - 23.29 \frac{(208-164)^2}{4 \times 208} + \frac{12}{208^{1/2}} = 3236.5 - 605.6 - 967.9 - 541.2 + 0.83 \approx 1622.6\,\text{MeV}$. Residual: $1636.4 - 1622.6 = +13.8\,\text{MeV}$. The positive residual indicates ${}^{208}\text{Pb}$ is more tightly bound than the SEMF predicts, a clear signature of the double shell closure at $Z = 82$, $N = 126$.

Q3. (c) A good rigid rotor. The rotational limit gives $E(4^+)/E(2^+) = 10/3 = 3.33$.

Q4. A Weisskopf unit is the single-particle estimate of the reduced transition probability. $B(E2) = 50\,\text{W.u.}$ means the transition rate is 50 times the single-particle estimate, indicating strong collective enhancement — many nucleons contribute coherently to the transition.

Q5. False. The Schmidt values provide upper and lower limits; measured moments typically fall between the $j = \ell + 1/2$ and $j = \ell - 1/2$ Schmidt lines due to configuration mixing, core polarization, and meson exchange currents.

Q6. $\log_{10} t_{1/2} = 1.61/\sqrt{4.270} + (-28.9) = 1.61/2.066 - 28.9 = 0.779 - 28.9 = -28.1$. This is in seconds: $t_{1/2} \approx 10^{-28.1}\,\text{s}$. This is clearly wrong for ${}^{238}\text{U}$, because the Geiger-Nuttall parameters given are approximate. The measured half-life is $4.468 \times 10^9\,\text{yr} \approx 1.41 \times 10^{17}\,\text{s}$. The point of this exercise is to recognize that the Geiger-Nuttall relation is a semiempirical fit and the parameters $a$ and $b$ must be carefully matched to the nuclear system. More careful parameterizations (using the correct coefficients for the $Z = 92$ region, e.g., $a \approx 128$, $b \approx -57$) give results within an order of magnitude.

Q7. Secular equilibrium: $t_{1/2,\text{parent}} \gg t_{1/2,\text{daughter}}$. After a time $t \gg t_{1/2,\text{daughter}}$, the daughter activity equals the parent activity: $A_d = A_p$. Transient equilibrium: $t_{1/2,\text{parent}} > t_{1/2,\text{daughter}}$ (but not $\gg$). The daughter activity exceeds the parent activity by a constant ratio: $A_d/A_p = t_{1/2,\text{parent}} / (t_{1/2,\text{parent}} - t_{1/2,\text{daughter}})$.

Q8. (a) Superallowed. $\log ft < 3.7$ corresponds to superallowed transitions (Fermi type, $0^+ \to 0^+$).

Q9. $Q_{\beta^-} = [M(A,Z) - M(A, Z+1)]c^2$. The atomic mass $M(A,Z)$ includes $Z$ electrons. In $\beta^-$ decay, the daughter has $Z+1$ protons, so its atomic mass includes $Z+1$ electrons. The emitted electron plus the daughter's extra electron accounts for all electrons, and the masses cancel cleanly.

Q10. True. The Bethe-Bloch formula gives $-dE/dx \propto z^2 Z_{\text{target}} / (A_{\text{target}} v^2)$ in the high-velocity limit (before relativistic corrections).

Q11. $Q = [M({}^{12}\text{C}) + M({}^{4}\text{He}) - M({}^{16}\text{O})]c^2 = [12.000000 + 4.002603 - 15.994915] \times 931.494 = 0.007688 \times 931.494 = 7.162\,\text{MeV}$. Exothermic ($Q > 0$).

Q12. The compound nucleus hypothesis (Bohr, 1936) states that a nuclear reaction proceeds in two independent steps: (1) formation of an excited compound nucleus that is fully equilibrated, and (2) statistical decay of the compound nucleus. The compound nucleus "forgets" its formation because the excitation energy is shared among all nucleons through many collisions before decay occurs — the decay probability depends only on the total energy, angular momentum, and parity, not on the specific entrance channel.

Q13. (b). The Breit-Wigner single-level formula is $\sigma \propto \Gamma_a \Gamma_b / [(E - E_r)^2 + \Gamma^2/4]$.

Q14. The astrophysical S-factor is defined as $S(E) = E\sigma(E)\exp(2\pi\eta)$, where $\eta = Z_1 Z_2 e^2 / (\hbar v)$ is the Sommerfeld parameter. It removes the exponential Coulomb barrier dependence and the geometric $1/E$ factor, leaving a slowly varying function of energy that is much easier to extrapolate to the very low energies (keV) relevant for stellar interiors.

Q15. In the CM frame, the total kinetic energy of the products equals $Q + E_{\text{CM}}$. For threshold (zero beam energy): $T_n + T_\alpha = Q = 17.59\,\text{MeV}$. By momentum conservation in the CM frame: $m_n T_n = m_\alpha T_\alpha$. So $T_n = Q \cdot m_\alpha/(m_n + m_\alpha) = 17.59 \times 4/5 = 14.07\,\text{MeV}$ and $T_\alpha = 17.59 \times 1/5 = 3.52\,\text{MeV}$.

Q16. The Gamow peak is the energy at which the product of the Maxwell-Boltzmann distribution (decreasing exponentially with $E$) and the barrier penetration factor (increasing exponentially with $E$, as $\exp(-2\pi\eta) \propto \exp(-bE^{-1/2})$) is maximized. $E_0 = (bkT/2)^{2/3}$ where $b = \pi Z_1 Z_2 e^2 \sqrt{2\mu}/\hbar$.

Q17. (a). The $p + p$ reaction is mediated by the weak interaction and has a cross section of order $10^{-47}\,\text{cm}^2$ at solar energies, making it by far the slowest step.

Q18. The r-process requires extreme neutron densities so that neutron capture is faster than $\beta$-decay. Nuclei capture neutrons rapidly along isotopic chains until reaching the neutron drip line or until the neutron separation energy drops low enough that photodisintegration $(\gamma,n)$ balances further capture. At these "waiting points," the nucleus must $\beta$-decay to increase $Z$ before the r-process path can continue to heavier elements. Waiting points occur at magic neutron numbers ($N = 50, 82, 126$) where the enhanced stability leads to longer $\beta$-decay lifetimes.

Q19. True. The helium mass fraction $Y_p$ is primarily set by the $n/p$ ratio at weak freeze-out ($T \sim 0.8\,\text{MeV}$), with a small correction from free neutron decay before the deuterium bottleneck breaks at $T \sim 0.07\,\text{MeV}$.

Q20. The TOV equation is the general-relativistic equation of hydrostatic equilibrium for a spherically symmetric, non-rotating star: $dP/dr = -G(\rho + P/c^2)(m + 4\pi r^3 P/c^2) / [r(r - 2Gm/c^2)]$. It requires the nuclear equation of state $P(\rho)$ — the pressure as a function of energy density — as input, which depends on the nuclear force at densities above nuclear saturation density.

Q21. $k_\infty = \eta f p \epsilon$. $\eta$ = reproduction factor (neutrons produced per neutron absorbed in fuel). $f$ = thermal utilization factor (fraction of thermal neutrons absorbed in fuel vs. other materials). $p$ = resonance escape probability (fraction of neutrons that avoid capture while slowing down). $\epsilon$ = fast fission factor (ratio of total fissions to thermal fissions).

Q22. (b). PET detects the two 511 keV annihilation gamma rays emitted back-to-back when the positron from $\beta^+$ decay annihilates with an electron.

Q23. (1) NNDC/NuDat: nuclear structure and decay data (level schemes, half-lives, decay modes). (2) ENSDF: evaluated nuclear structure data (adopted levels, gamma-ray transitions, reduced transition probabilities). (3) ENDF: evaluated nuclear reaction cross sections for applications (energy-dependent cross sections, angular distributions, fission yields).