Capstone Project — Detailed Requirements and Rubric

This document specifies the full requirements for each project option, provides detailed checklists, and gives the grading rubric. Use it alongside the analysis guides in Chapter 34.

Option A: Complete Analysis of a Single Nucleus

Deliverable Checklist

Complete all of the following for your chosen nucleus:

1. Ground-State Properties (15 points)

• [ ] Write the full proton and neutron shell model configuration, specifying each subshell occupation
• [ ] Predict the ground-state spin and parity $I^\pi$; compare with the NNDC experimental value
• [ ] Calculate the Schmidt magnetic moment; compare with the measured value
• [ ] For odd-$A$ nuclei: predict the electric quadrupole moment from the single-particle estimate
• [ ] Discuss the quality of the single-particle predictions: where does the extreme single-particle model succeed and where does it fail?

2. Binding Energy and Mass Systematics (15 points)

• [ ] Calculate the SEMF binding energy $B_{\text{SEMF}}(A,Z)$ using the five-term formula
• [ ] Look up the experimental binding energy from AME2020 and compute the residual $\delta B$
• [ ] Calculate $B/A$ and locate your nucleus on the binding energy per nucleon curve
• [ ] Calculate the one-neutron separation energy $S_n$ and one-proton separation energy $S_p$
• [ ] Calculate the two-neutron separation energy $S_{2n}$ and two-proton separation energy $S_{2p}$
• [ ] Plot $S_n$ or $S_{2n}$ for the isotopic chain of your element; identify any shell closures
• [ ] Interpret the SEMF residual: what does it tell you about shell effects?

3. Excited States and Electromagnetic Transitions (15 points)

• [ ] Retrieve the level scheme from ENSDF for the first 8–10 excited states
• [ ] Present the level scheme as a properly formatted energy-level diagram
• [ ] Calculate the energy ratio $E(4^+_1)/E(2^+_1)$ (for even-even nuclei) and classify the nucleus as spherical, vibrational, transitional, or rotational
• [ ] Calculate the Weisskopf single-particle estimate for at least two electromagnetic transitions
• [ ] Compare with the measured $B(E\lambda)$ or $B(M\lambda)$ values from ENSDF
• [ ] Determine the enhancement factor (ratio of measured to Weisskopf estimate) and interpret it in terms of collectivity

4. Stability and Decay Modes (15 points)

• [ ] Calculate $Q$-values for all relevant decay modes: $\alpha$, $\beta^-$, $\beta^+$/EC, $p$-emission, $n$-emission
• [ ] For each mode: state whether it is energetically allowed or forbidden
• [ ] If the nucleus is unstable: identify the dominant decay mode, calculate the expected half-life using the relevant model (Geiger-Nuttall for $\alpha$, $\log ft$ systematics for $\beta$), and compare with experiment
• [ ] If the nucleus is stable: explain why all decay modes are forbidden in terms of the energetics
• [ ] Discuss the nucleus in the context of the valley of $\beta$-stability

5. Nucleosynthesis (10 points)

• [ ] Identify the primary nucleosynthesis process(es) responsible for producing this nucleus
• [ ] Describe the astrophysical site and conditions (temperature, density, timescale)
• [ ] Explain the production mechanism in terms of the nuclear reactions involved
• [ ] Discuss the predicted abundance and compare with solar system observations

6. Applications (10 points)

• [ ] Identify at least one technological application involving your nucleus or its isotopes
• [ ] Explain the nuclear physics behind the application (e.g., which nuclear property is exploited)
• [ ] Discuss any connections between the nuclear properties analyzed in Steps 1–4 and the application

7. Synthesis and Integration (20 points)

• [ ] Write a 2–3 page synthesis section that explicitly connects findings across all six analysis steps
• [ ] Identify at least three specific connections where the same nuclear property appears in different contexts (e.g., shell closure affects both binding energy and decay stability)
• [ ] Discuss where different models agree and disagree for your nucleus
• [ ] Reflect on what this nucleus reveals about nuclear physics as a unified discipline

Option B: Complete Analysis of a Decay Chain

Deliverable Checklist

1. Chain Mapping (15 points)

• [ ] Present the complete decay chain in a table: parent, decay mode, daughter, $Q$-value (MeV), half-life, branching ratios at branch points
• [ ] Draw the chain on an $N$-$Z$ chart, color-coded by decay mode
• [ ] Calculate all $Q$-values from AME2020 masses and verify against NNDC
• [ ] For $\alpha$ decays: verify that $Q_\alpha$ values follow Geiger-Nuttall systematics (plot $\log t_{1/2}$ vs. $Q_\alpha^{-1/2}$)
• [ ] For $\beta$ decays: look up $\log ft$ values and identify transition types (allowed, first-forbidden, etc.)

2. Bateman Equation Solutions (20 points)

• [ ] Write the coupled differential equations $dN_i/dt = -\lambda_i N_i + \lambda_{i-1} N_{i-1}$ for all members of the chain
• [ ] Solve numerically using the toolkit's decay_chains.py module (or equivalent ODE solver)
• [ ] Plot the number of atoms $N_i(t)$ and activity $A_i(t) = \lambda_i N_i(t)$ for each member of the chain as a function of time
• [ ] Verify the numerical solution against the analytic Bateman solution for the first three members
• [ ] Discuss the relevant time scales: which members build up quickly? Which take the longest?

3. Equilibrium Analysis (15 points)

• [ ] Classify each parent-daughter pair as transient equilibrium, secular equilibrium, or no equilibrium
• [ ] For secular equilibrium pairs: verify that $A_{\text{daughter}}/A_{\text{parent}} \to 1$ at late times
• [ ] Calculate the total activity of the chain as a function of time
• [ ] Determine the time required for the chain to reach secular equilibrium (to within 1%)
• [ ] Plot the activity spectrum (activity vs. nuclide) at equilibrium

4. Radiometric Dating (15 points)

• [ ] Derive the isochron equation for the relevant dating system
• [ ] Apply the method to a realistic geological sample (you may use published data from a geology textbook or paper, with citation)
• [ ] Propagate uncertainties through the age calculation
• [ ] Discuss assumptions and potential sources of systematic error
• [ ] Compare with independent age determinations if available

5. Applications (15 points)

• [ ] Connect the chain to at least two of: reactor physics, environmental radiation, nuclear forensics, medical isotope production
• [ ] For reactor physics: trace the ${}^{238}\text{U}(n,\gamma){}^{239}\text{U} \to {}^{239}\text{Np} \to {}^{239}\text{Pu}$ production chain (if analyzing the uranium series)
• [ ] For environmental radiation: calculate the radon (${}^{222}\text{Rn}$ or ${}^{220}\text{Rn}$) emanation rate and indoor exposure estimate

6. Synthesis (20 points)

• [ ] Connect the decay chain analysis to nuclear structure (why certain nuclei in the chain are long-lived while others are short-lived)
• [ ] Explain how the chain traverses the chart of nuclides in terms of the interplay between alpha decay (reducing $Z$ and $N$) and beta decay (adjusting $N/Z$ ratio)
• [ ] Discuss the cosmic origin of the chain's parent nucleus

Option C: Complete Analysis of a Nuclear Reaction

Deliverable Checklist

1. Kinematics (15 points)

• [ ] Calculate the $Q$-value from AME2020 masses
• [ ] If endothermic: derive and evaluate the threshold energy in the lab frame
• [ ] Perform the CM/lab transformation: calculate CM energy, momentum, and velocity at three different beam energies
• [ ] Plot the energy-angle correlation of the ejectile in the lab frame
• [ ] For inverse kinematics (heavy beam on light target): discuss the kinematic focusing effect

2. Cross Section Analysis (20 points)

• [ ] Retrieve experimental cross section data from EXFOR or ENDF
• [ ] Plot $\sigma(E)$ over the relevant energy range
• [ ] Identify resonance features: extract center energy $E_r$, total width $\Gamma$, and partial widths $\Gamma_a$, $\Gamma_b$ for at least two prominent resonances
• [ ] Fit Breit-Wigner profiles and overlay on the data
• [ ] For charged-particle reactions: extract the astrophysical $S$-factor $S(E) = E\sigma(E)\exp(2\pi\eta)$ and plot $S(E)$
• [ ] For neutron reactions: identify the $1/v$ region, resolved resonance region, and unresolved resonance region

3. Reaction Mechanism (15 points)

• [ ] Determine whether the reaction proceeds via compound nucleus, direct, or both mechanisms
• [ ] For compound nucleus: estimate the formation cross section from the optical model
• [ ] For direct reactions: identify the transferred quantum numbers ($\Delta L$, $\Delta S$)
• [ ] Discuss the role of angular momentum conservation and selection rules

4. Astrophysical or Technological Context (20 points)

• [ ] For astrophysical reactions: calculate the Gamow window at the relevant stellar temperature
• [ ] Evaluate the Maxwellian-averaged reaction rate $\langle\sigma v\rangle$ as a function of temperature
• [ ] Discuss the reaction's role in the relevant nucleosynthesis process or burning stage
• [ ] For technological reactions: explain the role in energy production, isotope production, or detection
• [ ] Compare your calculated quantities with values in the JINA REACLIB database

5. Synthesis (30 points)

• [ ] Connect the reaction to nuclear structure: how do the properties of the entrance and exit channel nuclei affect the cross section?
• [ ] Discuss the current experimental status: how well is this cross section measured? What are the dominant uncertainties?
• [ ] Identify any open questions or ongoing experiments related to this reaction
• [ ] Discuss the impact of cross section uncertainties on the astrophysical or technological application

Option D: Complete Analysis of a Stellar Burning Stage

Deliverable Checklist

1. Reaction Network (15 points)

• [ ] List all reactions in the burning cycle/chain with their $Q$-values
• [ ] Draw the reaction flow diagram on a section of the chart of nuclides
• [ ] Identify the rate-limiting step and explain why it limits the cycle
• [ ] Classify each reaction by interaction type: strong, electromagnetic, or weak
• [ ] For cyclic processes (pp, CNO): verify that the net reaction produces the expected products

2. Reaction Rates (20 points)

• [ ] For each charged-particle reaction: calculate the Gamow peak energy $E_0$ and width $\Delta$ at the stellar temperature
• [ ] Extract or parameterize the astrophysical $S$-factor for the key reactions
• [ ] Compute $\langle\sigma v\rangle$ for each reaction at the relevant temperature
• [ ] For weak reactions: compute the rate from phase space and matrix element considerations
• [ ] Plot all rates as a function of temperature and identify crossover temperatures

3. Energy Generation (20 points)

• [ ] Calculate the total energy released per cycle, clearly distinguishing nuclear energy from neutrino losses
• [ ] Derive the energy generation rate $\epsilon(\rho, T, X_i)$ and express it in erg g$^{-1}$ s$^{-1}$
• [ ] Determine the temperature exponent $n$ where $\epsilon \propto T^n$ and discuss its implications for stellar stability (e.g., CNO cycle: $n \approx 16-17$ implies convective core)
• [ ] Compare your derived $\epsilon$ with the parameterizations in the literature

4. Neutrino Physics (15 points)

• [ ] Identify every neutrino-producing reaction in the network
• [ ] Determine the neutrino energy: monoenergetic for EC, continuous spectrum for $\beta^+$
• [ ] Calculate the average neutrino energy for each branch
• [ ] Compute the total neutrino luminosity as a fraction of the total nuclear energy generation
• [ ] Discuss detectability: are these neutrinos within the sensitivity range of current detectors (Super-Kamiokande, Borexino, SNO+)?

5. Stellar Evolution Context (10 points)

• [ ] Specify the temperature, density, and composition conditions for this burning stage
• [ ] Estimate the duration of this burning stage for a representative stellar mass
• [ ] Describe what happens when fuel is exhausted: which burning stage follows?
• [ ] Discuss observational signatures (surface abundances, luminosity, neutrino flux)

6. Synthesis (20 points)

• [ ] Connect the nuclear physics (reaction rates, $Q$-values, weak interaction rates) to the macroscopic stellar behavior
• [ ] Discuss the sensitivity of stellar evolution to nuclear physics uncertainties
• [ ] Identify the most uncertain reaction rate in the network and discuss ongoing efforts to measure it

General Grading Rubric (All Options)

Total: 100 points. Passing: 70 points.

Timeline

Warm-Up Exercises

Before beginning your capstone project, complete the following exercises to verify that your core skills are ready.

Exercise 34.1 ⭐ (Shell Model Warm-Up)

Write the full shell model configuration for ${}^{48}\text{Ca}$ ($Z = 20$, $N = 28$). Identify the last occupied subshell for both protons and neutrons. Predict $I^\pi$. Why is this nucleus doubly magic?

Exercise 34.2 ⭐ (SEMF Warm-Up)

Calculate the SEMF binding energy for ${}^{120}\text{Sn}$ ($Z = 50$, $A = 120$). The experimental value is $B_{\text{exp}} = 1020.54\,\text{MeV}$. Compute the residual $\delta B = B_{\text{exp}} - B_{\text{SEMF}}$. Is tin-120 more or less bound than the liquid-drop prediction? What does this tell you about the $Z = 50$ shell closure?

Exercise 34.3 ⭐ (Q-Value Warm-Up)

Using the atomic masses $M({}^{226}\text{Ra}) = 226.025410\,\text{u}$, $M({}^{222}\text{Rn}) = 222.017578\,\text{u}$, and $M({}^{4}\text{He}) = 4.002603\,\text{u}$, calculate the $Q$-value for the alpha decay ${}^{226}\text{Ra} \to {}^{222}\text{Rn} + \alpha$. Compare with the tabulated value of $Q_\alpha = 4.871\,\text{MeV}$.

Exercise 34.4 ⭐⭐ (Bateman Warm-Up)

Consider the three-member chain: ${}^{226}\text{Ra} \to {}^{222}\text{Rn} \to {}^{218}\text{Po}$ with $t_{1/2}(\text{Ra}) = 1600\,\text{yr}$, $t_{1/2}(\text{Rn}) = 3.823\,\text{d}$, and $t_{1/2}(\text{Po}) = 3.10\,\text{min}$.

(a) Show that both ${}^{222}\text{Rn}$ and ${}^{218}\text{Po}$ satisfy the secular equilibrium condition with ${}^{226}\text{Ra}$.

(b) If you start with 1 gram of pure ${}^{226}\text{Ra}$, calculate the activity (in Bq) of ${}^{226}\text{Ra}$ and the equilibrium activity of ${}^{222}\text{Rn}$.

(c) How many atoms of ${}^{222}\text{Rn}$ are present at secular equilibrium?

Exercise 34.5 ⭐⭐ (Kinematics Warm-Up)

For the reaction ${}^{14}\text{N}(p,\gamma){}^{15}\text{O}$:

(a) Calculate the $Q$-value using masses: $M({}^{14}\text{N}) = 14.003074\,\text{u}$, $M({}^{1}\text{H}) = 1.007825\,\text{u}$, $M({}^{15}\text{O}) = 15.003066\,\text{u}$.

(b) Calculate the Gamow peak energy at $T_9 = 0.025$ (25 million K, appropriate for the hydrogen-burning shell of an RGB star).

(c) Explain why this reaction is the rate-limiting step of the CNO cycle.

Exercise 34.6 ⭐⭐ (Weisskopf Warm-Up)

The first excited state of ${}^{208}\text{Pb}$ is the $3^-$ state at $2614.5\,\text{keV}$.

(a) Calculate the Weisskopf single-particle estimate for the $E3$ transition rate $T(E3)$ from the $3^-$ to the $0^+$ ground state. Use $R = r_0 A^{1/3}$ with $r_0 = 1.21\,\text{fm}$.

(b) The measured $B(E3; 0^+ \to 3^-_1) = 34\,\text{W.u.}$ What is the measured transition rate in s$^{-1}$?

(c) Calculate the half-life of the $3^-$ state from the measured $B(E3)$ and compare with the experimental half-life of $T_{1/2} = 16.7\,\text{ps}$.

(d) The enhancement of 34 W.u. indicates collective behavior. Explain in one paragraph why the $E3$ (octupole) mode is collective while the $E2$ (quadrupole) mode is not ($B(E2; 0^+ \to 2^+_1) \approx 0.6\,\text{W.u.}$). (Hint: consider the available particle-hole excitations near the Fermi surface.)

Exercise 34.7 ⭐⭐⭐ (NNDC Data Retrieval)

Go to the NNDC NuDat website (https://www.nndc.bnl.gov/nudat3/) and look up the nucleus you intend to analyze for your capstone project. Record:

(a) The ground-state spin, parity, and magnetic moment (compare with your shell model prediction)

(b) The energies, spins, and parities of the first five excited states

(c) The dominant decay mode and half-life (if unstable) or confirm stability

(d) The natural isotopic abundance (if stable)

This exercise serves as your initial data-gathering step for the capstone project.

Exercise 34.8 ⭐⭐⭐ (Integration Exercise)

Choose any one of the following nuclei: ${}^{16}\text{O}$, ${}^{40}\text{Ca}$, ${}^{90}\text{Zr}$, ${}^{197}\text{Au}$. In a single page, explain:

(a) Its shell model configuration and why the ground-state spin is what it is

(b) Whether it is magic (and if so, what observable signatures confirm this)

(c) How it was synthesized (which nucleosynthesis process, in what astrophysical site)

(d) One technological application

This mini-analysis is a rehearsal for the full capstone project.

Frequently Asked Questions

Q: Can I choose a nucleus/reaction not on the recommended list?

A: Yes, with instructor approval. Your choice must be complex enough to support a full analysis. A good rule of thumb: if you can find the nucleus in ENSDF with a level scheme of at least 10 states, it is rich enough.

Q: What if I haven't completed all the toolkit modules?

A: The capstone pipeline is modular. Use the modules you have, and for missing modules, perform the calculations by hand or write simplified versions. The key is demonstrating understanding, not code completeness.

Q: How much of the analysis should be computational vs. analytical?

A: Aim for a balance. Every calculation should have both an analytical setup (showing the physics and the equation) and a numerical result (from the code or by hand). Pure code output without physics explanation is not sufficient; pure theory without numbers is not sufficient.

Q: Can I work with a partner?

A: No. This is an individual project. You may discuss concepts, but all calculations, code, and writing must be your own.

Q: What format should the final document be in?

A: PDF is preferred. LaTeX is encouraged but not required. Markdown converted to PDF via Pandoc is also acceptable. The document should look professional — consistent formatting, properly typeset equations, numbered figures with captions.

Q: How long should the final document be?

A: Target 15--25 pages including figures and tables. Quality matters more than length, but a thorough analysis of any nuclear system, with proper figures and comparisons, will naturally fill this range. If your document is under 12 pages, you are probably missing analysis steps or figures.

Q: Should I include error analysis?

A: Yes. At minimum, discuss the sources of uncertainty in your calculations and how they propagate to your final results. For experimental data, report the published uncertainties. For SEMF predictions, discuss the systematic limitations of the model. A thoughtful uncertainty discussion is one of the hallmarks of a physicist's analysis (as opposed to an engineer's calculation).

Q: What if different models give contradictory predictions?

A: This is expected and is one of the most interesting aspects of the capstone. When models disagree, discuss why they disagree: what physics does each model include or neglect? Which model is more appropriate for this particular observable? The disagreement teaches you more than the agreement.

Extended Exercises

These exercises provide additional practice for students who want to deepen their preparation.

Exercise 34.9 ⭐⭐ (Geiger-Nuttall Systematics)

The following data are for alpha emitters in the thorium ($4n$) decay series:

(a) Plot $\log_{10}(t_{1/2}/\text{s})$ versus $Q_\alpha^{-1/2}$ for the even-even alpha emitters in this chain. Fit a straight line and extract the slope and intercept.

(b) Use your fitted Geiger-Nuttall parameters to predict the half-life of ${}^{232}\text{Th}$ from its $Q_\alpha$ value alone. How does this compare with the measured value?

(c) ${}^{212}\text{Bi}$ ($Z = 83$, odd-$Z$) has a much longer half-life than the even-even systematics would predict. Estimate the hindrance factor (ratio of actual to predicted half-life) and explain its origin in terms of the angular momentum barrier for the alpha particle.

Exercise 34.10 ⭐⭐⭐ (Cross Section Comparison)

Consider the two neutron capture reactions: - ${}^{55}\text{Fe}(n,\gamma){}^{56}\text{Fe}$ - ${}^{208}\text{Pb}(n,\gamma){}^{209}\text{Pb}$

(a) Without looking up the cross sections, predict which reaction has the larger thermal neutron capture cross section. Justify your prediction based on nuclear structure arguments (shell closures, level density at the neutron binding energy).

(b) The measured thermal capture cross sections are approximately $2.6\,\text{b}$ for ${}^{55}\text{Fe}$ and $0.00050\,\text{b}$ for ${}^{208}\text{Pb}$. Calculate the ratio. Does this confirm your prediction?

(c) Explain why the ${}^{208}\text{Pb}$ cross section is so extraordinarily small in terms of the $N = 126$ shell closure and its effect on the compound nucleus level density.

(d) Discuss the astrophysical consequence: what does this cross section ratio imply for the s-process abundances of iron-region versus lead-region elements?

Exercise 34.11 ⭐⭐⭐ (Nucleosynthesis Pathways)

The elements barium ($Z = 56$) and europium ($Z = 63$) are both heavy elements, but they are produced by different nucleosynthesis processes.

(a) Barium is primarily an s-process element. Explain why, in terms of the neutron capture cross sections and half-lives of isotopes along the s-process path near $Z = 56$.

(b) Europium (specifically ${}^{151}\text{Eu}$ and ${}^{153}\text{Eu}$) has a significant r-process contribution. Explain why certain elements receive r-process contributions while others are dominated by the s-process.

(c) The ratio [Eu/Ba] (the europium-to-barium abundance ratio, relative to solar) is used as an observational diagnostic of r-process versus s-process enrichment in old stars. Explain the physical reasoning: why does a high [Eu/Ba] ratio indicate r-process dominance?

(d) How does the observation of [Eu/Ba] in the kilonova AT2017gfo (associated with GW170817) connect to the nuclear physics you have studied in this book?