Exercises — Chapter 35

Reading a Paper

Problem 35.1 ⭐ Go to the arXiv (https://arxiv.org) and find the daily listing for nucl-ex. Read the titles and abstracts of all papers submitted in the most recent day. For each paper, classify it into one of the following categories: (a) nuclear structure, (b) nuclear reactions, (c) nuclear astrophysics, (d) fundamental symmetries, (e) nuclear applications, (f) nuclear data. How many papers were posted? What fraction fall into each category?

Problem 35.2 ⭐ Select any Physical Review C paper from the past year. Without reading the text, examine only the figures.

(a) List each figure and describe (in one sentence each) what is being plotted.

(b) From the figures alone, write a one-paragraph summary of what the paper measured or calculated.

(c) Now read the abstract. How accurate was your figure-based summary?

Problem 35.3 ⭐⭐ Read the following paper (or choose a recent PRC paper assigned by your instructor):

A. Gade et al., "In-beam gamma-ray spectroscopy of ${}^{32}\text{Mg}$," Phys. Rev. C 99, 011301(R) (2019).

(a) Identify the: (i) nucleus studied, (ii) experimental technique, (iii) facility where the experiment was performed, (iv) key result.

(b) What is the significance of ${}^{32}\text{Mg}$ in nuclear structure? (Hint: island of inversion.)

(c) The paper compares experimental results to shell model calculations. Which effective interaction was used? What model space?

(d) Does the paper report systematic uncertainties? If so, what are the dominant sources?

Problem 35.4 ⭐⭐ Find a Physical Review Letters paper on a nuclear physics topic published in the past two years.

(a) The PRL format limits papers to approximately 4 pages. What information is compressed or omitted compared to a typical PRC paper?

(b) Does the PRL paper reference a longer companion paper (sometimes called a "long paper") in PRC or NPA? If so, find it.

(c) In your judgment, does the result warrant publication in PRL (i.e., is it of broad significance to physics), or would PRC have been more appropriate? Justify your answer.

Problem 35.5 ⭐⭐⭐ Find two papers that measure the same nuclear quantity (e.g., an excitation energy, a half-life, a cross section, or a spectroscopic factor) and reach results that disagree by more than $2\sigma$.

(a) Describe the quantity measured and the two results with their uncertainties.

(b) Identify possible sources of the discrepancy: different techniques, different analysis methods, different systematic corrections.

(c) Does the more recent paper acknowledge and discuss the discrepancy? If ENSDF has evaluated this quantity, what adopted value did the evaluator choose, and how was the discrepancy handled?

Statistical Analysis

Problem 35.6 ⭐ A nuclear physics experiment measures the gamma-ray energy of a transition and reports:

$$E_\gamma = 1332.50 \pm 0.03\,(\text{stat}) \pm 0.10\,(\text{syst})\,\text{keV}$$

(a) What is the total uncertainty if the statistical and systematic components are combined in quadrature?

(b) The ENSDF adopted value for this transition (${}^{60}\text{Co}$ 4$^+$ → 2$^+$) is $1332.508 \pm 0.004\,\text{keV}$. Is the new measurement consistent with the adopted value?

(c) The experiment claims to have improved the precision. Is this claim justified?

Problem 35.7 ⭐ An experimentalist fits a theoretical curve to 25 data points using a model with 3 free parameters and obtains $\chi^2 = 44$.

(a) Calculate the reduced chi-squared $\chi^2_\nu$.

(b) Is this a good fit? What might be wrong?

(c) If the experimentalist doubles all error bars and re-fits, what happens to $\chi^2_\nu$? Is this an acceptable solution?

Problem 35.8 ⭐⭐ A search for a rare decay mode reports observing 7 events where the expected background is $3.2 \pm 0.5$ events.

(a) Calculate the statistical significance of the excess in standard deviations (assume Poisson statistics for the observed counts and treat the background as known).

(b) Does this constitute "evidence" (3$\sigma$) for the decay?

(c) What additional information would you want before concluding that the decay has been observed?

Problem 35.9 ⭐⭐ A cross section measurement reports 15 data points with the following $\chi^2$ values when compared to three different theoretical models:

(a) Calculate $\chi^2_\nu$ for each model.

(b) Which model provides the best fit? Is the best fit statistically acceptable?

(c) Model B has no free parameters. Why is this significant for the interpretation?

(d) A colleague argues that Model A fits better simply because it has more free parameters. How would you respond?

Problem 35.10 ⭐⭐⭐ A paper reports the half-life of a radioactive nucleus as $T_{1/2} = 125 \pm 15\,\text{ms}$ based on fitting an exponential decay curve to 200 detected events.

(a) Estimate the expected statistical uncertainty on $T_{1/2}$ given $N = 200$ events. (Hint: The relative uncertainty on $T_{1/2}$ from a maximum-likelihood fit to $N$ exponential decays is approximately $T_{1/2}/\sqrt{N}$.)

(b) Compare your estimate to the quoted uncertainty. Is the uncertainty dominated by statistics or systematics?

(c) What systematic effects might contribute to the uncertainty in a half-life measurement?

Nuclear Data Resources

Problem 35.11 ⭐ Use NuDat (https://www.nndc.bnl.gov/nudat3/) to look up the following information for ${}^{60}\text{Co}$:

(a) Ground-state spin-parity and half-life.

(b) The energies and intensities of the two most prominent gamma rays emitted in the decay.

(c) The $Q$-value for the $\beta^-$ decay.

(d) Draw a simplified decay scheme showing the ground state of ${}^{60}\text{Co}$, the two excited states of ${}^{60}\text{Ni}$ populated in the decay, and the gamma-ray transitions.

Problem 35.12 ⭐ Use the AME2020 data (available at https://www-nds.iaea.org/amdc/) or the interactive mass explorer at NNDC to calculate:

(a) The one-neutron separation energy $S_n$ for ${}^{208}\text{Pb}$ and ${}^{209}\text{Pb}$.

(b) The one-proton separation energy $S_p$ for ${}^{208}\text{Pb}$ and ${}^{209}\text{Bi}$.

(c) Explain why the dramatic difference between $S_n({}^{208}\text{Pb})$ and $S_n({}^{209}\text{Pb})$ provides evidence for the $N = 126$ shell closure.

Problem 35.13 ⭐⭐ Use the NNDC Sigma tool (https://www.nndc.bnl.gov/sigma/) to plot the neutron capture cross section $\sigma(n,\gamma)$ for ${}^{197}\text{Au}$ from 0.001 eV to 1 MeV.

(a) Identify the energy region where the cross section follows a $1/v$ law.

(b) Identify the resolved resonance region. What is the approximate energy of the lowest-energy neutron resonance?

(c) At thermal energy ($E = 0.0253\,\text{eV}$), what is the cross section? Compare to the value quoted in nuclear data tables.

(d) Download the ENDF/B-VIII.0 cross section data for this reaction and plot it using Python.

Problem 35.14 ⭐⭐ Use ENSDF (via NNDC) to find the adopted level scheme of ${}^{48}\text{Ca}$.

(a) List the first five excited states with their energies and spin-parity assignments.

(b) ${}^{48}\text{Ca}$ is doubly magic ($Z = 20$, $N = 28$). How does the excitation energy of the first $2^+$ state compare to that of neighboring even-even calcium isotopes (${}^{46}\text{Ca}$, ${}^{44}\text{Ca}$)? What does this tell you about the $N = 28$ shell gap?

(c) Find the ENSDF evaluation reference for $A = 48$. When was it last updated? Who performed the evaluation?

Problem 35.15 ⭐⭐⭐ Use the TENDL-2023 library (via the IAEA Nuclear Data Services) to find the cross section for the ${}^{68}\text{Zn}(p,n){}^{68}\text{Ga}$ reaction.

(a) Plot the excitation function (cross section vs. proton energy) from threshold to 30 MeV.

(b) This reaction is used to produce ${}^{68}\text{Ga}$ for PET imaging. At what proton energy is the cross section maximum? What is the peak cross section?

(c) Compare the TENDL prediction to any available experimental data (search EXFOR). How well does TENDL reproduce the measurements?

arXiv and Literature Searching

Problem 35.16 ⭐ Create a Google Scholar alert for a nuclear physics topic of your choice. After one week, report:

(a) How many alerts did you receive?

(b) How many of the papers were relevant to the topic?

(c) Were any of the papers from journals you had not previously encountered?

Problem 35.17 ⭐⭐ Use INSPIRE-HEP (https://inspirehep.net) to find the 10 most-cited nuclear physics papers published in Physical Review C in the past 5 years.

(a) List the papers by citation count.

(b) Classify each paper as: experimental measurement, theoretical calculation, data evaluation, or review.

(c) What subfields of nuclear physics are represented? Are there any surprises?

Problem 35.18 ⭐⭐ Choose a specific nuclear reaction (e.g., ${}^{12}\text{C}(\alpha,\gamma){}^{16}\text{O}$, ${}^{56}\text{Ni}(p,\gamma){}^{57}\text{Cu}$, or ${}^{130}\text{Sn}(d,p){}^{131}\text{Sn}$).

(a) Search arXiv and INSPIRE for all papers on this reaction published in the past 10 years.

(b) Construct a timeline showing when each measurement or calculation was published.

(c) Has the accepted cross section or reaction rate changed significantly over this period? What drove the changes?

Conferences and Career Paths

Problem 35.19 ⭐⭐ Find the program for the most recent APS Division of Nuclear Physics (DNP) meeting.

(a) How many parallel sessions were there? List the session topics.

(b) Identify three invited talks that relate to topics covered in this textbook. For each, identify the relevant chapter(s).

(c) How many contributed talks were given by graduate students? (The abstract index usually indicates student status.)

Problem 35.20 ⭐⭐ Interview a nuclear physicist (in person, by email, or by reading a published interview). This can be a professor, a national lab scientist, a medical physicist, or anyone working in a nuclear-physics-related career.

(a) What was their career path? (Undergraduate → graduate school → postdoc(s) → current position.)

(b) What fraction of their time is spent reading papers? How do they stay current?

(c) What advice would they give to a student starting in nuclear physics today?

Write a 500–800 word summary of the interview.

Problem 35.21 ⭐⭐⭐ Research one of the following career paths and write a 1000-word career guide for a graduating nuclear physics PhD:

(a) Clinical medical physicist (ABR certification path)

(b) Staff scientist at a DOE national laboratory

(c) Nuclear security analyst at a national laboratory or government agency

(d) Nuclear engineer at a company developing advanced reactors (SMRs, MSRs, etc.)

Your guide should include: educational requirements, typical timeline, salary range, day-to-day responsibilities, and the aspects of nuclear physics training that are most relevant.

Computational Exercises

Problem 35.22 ⭐⭐ Using the literature_tools.py script (or your own code), retrieve and plot the known energy levels of ${}^{208}\text{Pb}$ below 5 MeV excitation energy.

(a) Create a level scheme diagram showing energy levels as horizontal lines, labeled with spin-parity.

(b) Identify the first $3^-$ state. This is the lowest collective excitation (octupole vibration) of ${}^{208}\text{Pb}$. What is its energy?

(c) Compare the level density (number of levels per MeV) below 4 MeV and between 4 and 5 MeV. What does this tell you about the doubly-magic nature of ${}^{208}\text{Pb}$?

Problem 35.23 ⭐⭐⭐ Write a Python script that reads an AME-format mass table and produces the following plots for a user-specified element (all isotopes of that element):

(a) Binding energy per nucleon $B/A$ vs. neutron number $N$

(b) One-neutron separation energy $S_n$ vs. $N$

(c) Two-neutron separation energy $S_{2n}$ vs. $N$

Mark the magic neutron numbers (8, 20, 28, 50, 82, 126) with vertical dashed lines. Test your script on the tin ($Z = 50$) isotopic chain.

Problem 35.24 ⭐⭐⭐ (Research) Choose one of the open questions from Section 35.9.2 and write a 2000-word research brief that:

(a) States the question precisely and explains why it matters.

(b) Summarizes the current experimental and theoretical status (cite at least 5 papers from the past 3 years).

(c) Describes the key experiments or calculations planned for the next 5 years.

(d) Assesses the prospect for a definitive answer within the next decade.

This exercise requires genuine literature research. Use arXiv, INSPIRE, and the NSAC Long Range Plan as starting points.