Chapter 2 Exercises — Nuclear Properties: Size, Shape, Mass, Spin, and Moments

These problems use real nuclear data. Where experimental values are quoted, they are from the NNDC evaluated data or AME2020 unless otherwise noted. Constants: $e^2/(4\pi\epsilon_0) = 1.4400$ MeV$\cdot$fm, $\mu_N = 5.0508 \times 10^{-27}$ J/T, $u = 931.494$ MeV/$c^2$, $r_0 = 1.21$ fm (half-density radius).

Section A: Nuclear Sizes (Problems 1–7)

Problem 1 — Radius Systematics

Calculate the half-density radius $R_{1/2}$ and the rms charge radius $R_\text{rms}$ (assuming a Fermi distribution with $a = 0.54$ fm) for the following nuclei:

(a) ${}^{12}\text{C}$ (b) ${}^{56}\text{Fe}$ (c) ${}^{120}\text{Sn}$ (d) ${}^{238}\text{U}$

Compare your $R_\text{rms}$ values to the experimental charge radii: 2.470 fm, 3.738 fm, 4.654 fm, and 5.860 fm, respectively. Comment on the quality of the agreement.

Problem 2 — Nuclear Density

(a) Using the saturation density $\rho_0 = 0.17$ nucleons/fm$^3$, calculate the mass density of nuclear matter in kg/m$^3$.

(b) A neutron star with mass $1.4 M_\odot$ has a radius of approximately 12 km. Calculate its average density and compare to $\rho_0$. What does this comparison tell you about the interior of a neutron star?

(c) If the entire Earth ($M = 5.97 \times 10^{24}$ kg) were compressed to nuclear density, what would its radius be?

Problem 3 — Electron Scattering Form Factor

The first diffraction minimum in elastic electron scattering from a uniform sphere of radius $R$ occurs at $qR = 4.493$.

(a) If the minimum is observed at electron scattering angle $\theta = 50°$ with beam energy $E = 500$ MeV, find $q$ (assume the electron is ultrarelativistic so that $q = 2E\sin(\theta/2)/(\hbar c)$). Then determine the nuclear radius $R$.

(b) For which nucleus is this radius most consistent?

(c) Why does the uniform sphere model predict exact zeros (minima) while the experimental cross section shows only dips that don't reach zero?

Problem 4 — The Neutron Skin

The charge radius of ${}^{208}\text{Pb}$ is $R_\text{ch} = 5.501$ fm and the neutron radius from PREX-2 is $R_n = 5.727 \pm 0.071$ fm.

(a) Calculate the neutron skin thickness $\Delta r_{np} = R_n - R_p$, taking $R_p \approx R_\text{ch}$.

(b) Estimate the fraction of neutrons that reside in the skin region (the shell between $R_p$ and $R_n$), assuming uniform density in this region and using the neutron density $\rho_n \approx (N/A)\rho_0$.

(c) The neutron skin thickness is correlated with the slope of the symmetry energy $L$ in the nuclear equation of state. If $\Delta r_{np} = 0.283$ fm corresponds to $L \approx 106$ MeV, explain qualitatively why a larger symmetry energy slope produces a thicker neutron skin.

Problem 5 — Isotope Shift

The charge radii of the calcium isotopes show the following pattern:

(a) Plot (or sketch) $R_\text{ch}$ vs. $N$ and describe the trend.

(b) Why does the charge radius decrease from ${}^{44}\text{Ca}$ to ${}^{48}\text{Ca}$ despite adding four neutrons?

(c) The charge radius of ${}^{52}\text{Ca}$ ($N = 32$) was recently measured to be 3.57 fm. This is significantly larger than predicted by many models. What might this imply about shell structure near $N = 32$?

Problem 6 — Muonic Atom Energies

A muonic atom has a muon (mass $m_\mu = 105.66$ MeV/$c^2$) in the 1$s$ orbit around a nucleus with charge $Z$ and radius $R$.

(a) Calculate the Bohr radius of the muonic 1$s$ orbit for $Z = 82$ (lead). Compare this to the nuclear charge radius of ${}^{208}\text{Pb}$ (5.50 fm). Is the muon significantly inside the nucleus?

(b) For a muon inside a uniformly charged sphere, the potential is $V(r) = -(Ze^2/2R)(3 - r^2/R^2)$. Estimate the energy shift of the 1$s$ level due to the finite nuclear size compared to a point-nucleus calculation, for ${}^{208}\text{Pb}$.

(c) If the charge radius changes by $\delta R = 0.01$ fm, estimate the change in the muonic 2$p \to 1s$ transition energy. This is how muonic x-ray spectroscopy measures nuclear charge radii.

Problem 7 — Halo Nuclei

The nucleus ${}^{11}\text{Li}$ has a matter radius of $R_m \approx 3.55$ fm, while ${}^{9}\text{Li}$ has $R_m \approx 2.44$ fm.

(a) Calculate the expected radii using $R = r_0 A^{1/3}$ with $r_0 = 1.21$ fm for both nuclei. Compare to the measured values.

(b) The discrepancy for ${}^{11}\text{Li}$ is a signature of its neutron halo — two loosely bound neutrons extending far from the ${}^{9}\text{Li}$ core. Estimate the average distance of the halo neutrons from the center, modeling ${}^{11}\text{Li}$ as a ${}^{9}\text{Li}$ core plus two neutrons at distance $d$:

$$R_m^2({}^{11}\text{Li}) = \frac{9 R_m^2({}^{9}\text{Li}) + 2d^2}{11}$$

(c) The two-neutron separation energy of ${}^{11}\text{Li}$ is only $S_{2n} = 0.369$ MeV (compared to a typical value of $\sim 10$–15 MeV for nuclei near stability). Using the uncertainty principle estimate for the extent of a loosely bound state, $d \sim \hbar / \sqrt{2\mu S_{2n}}$ (where $\mu$ is the reduced mass of the neutron pair relative to the core), estimate $d$ and compare to your result from (b).

(d) ${}^{11}\text{Li}$ is a Borromean nucleus: neither ${}^{10}\text{Li}$ (core + 1 neutron) nor the dineutron (2 neutrons alone) is bound, yet the three-body system (core + $n$ + $n$) is bound. Why does the term "Borromean" (from the Borromean rings of topology) apply?

Section B: Nuclear Masses (Problems 8–13)

Problem 8 — Binding Energy Calculations

Using the AME2020 mass excesses given below, calculate the total binding energy and binding energy per nucleon for each nucleus.

Use $\Delta_n = 8071.3$ keV and $\Delta_H = 7288.97$ keV. Which of these nuclei has the highest $B/A$? (The answer may surprise students who think it is always ${}^{56}\text{Fe}$.)

Problem 9 — Separation Energies

(a) Using $\Delta({}^{208}\text{Pb}) = -21749.6$ keV, $\Delta({}^{207}\text{Pb}) = -22451.8$ keV, and $\Delta_n = 8071.3$ keV, calculate the one-neutron separation energy $S_n$ of ${}^{208}\text{Pb}$.

(b) Using $\Delta({}^{209}\text{Pb}) = -17613.0$ keV, calculate $S_n$ of ${}^{209}\text{Pb}$.

(c) The difference $S_n({}^{208}\text{Pb}) - S_n({}^{209}\text{Pb})$ quantifies the "shell gap" at $N = 126$. Calculate this gap and comment on its magnitude.

Problem 10 — Penning Trap Precision

A Penning trap at ISOLTRAP measures the cyclotron frequency ratio of ${}^{76}\text{Ge}^+$ to ${}^{76}\text{Se}^+$ as:

$$R = \frac{\nu_c({}^{76}\text{Se}^+)}{\nu_c({}^{76}\text{Ge}^+)} = 1.000\,000\,097\,3(8)$$

(a) Given $M({}^{76}\text{Ge}) = 75.921\,402\,7$ u, calculate $M({}^{76}\text{Se})$.

(b) From these masses, calculate the double beta decay Q-value: $Q_{\beta\beta} = [M({}^{76}\text{Ge}) - M({}^{76}\text{Se}) - 2m_e]c^2$ (using atomic masses, the electron masses cancel except for the two emitted electrons, but in atomic masses we need $Q_{\beta\beta} = [\Delta({}^{76}\text{Ge}) - \Delta({}^{76}\text{Se})]$).

(c) The GERDA and LEGEND experiments search for neutrinoless double beta decay of ${}^{76}\text{Ge}$. Why does a 1 keV uncertainty in the Q-value matter for these experiments?

Problem 11 — Proton Drip Line

The proton separation energy of ${}^{151}\text{Lu}$ ($Z = 71$, $N = 80$) is $S_p = -1.24$ MeV (negative, meaning it is proton-unbound).

(a) What does $S_p < 0$ mean physically?

(b) Despite being proton-unbound, ${}^{151}\text{Lu}$ has a measurable half-life of $80$ ms. This is because the emitted proton must tunnel through the Coulomb barrier. Estimate the height of the Coulomb barrier (take $R_C = r_0(A_\text{daughter}^{1/3} + 1)$ with $r_0 = 1.2$ fm).

(c) Compare the Q-value for proton emission ($|S_p| = 1.24$ MeV) to the barrier height. Why is the tunneling probability very sensitive to the Q-value?

Problem 12 — Two-Neutron Separation Energies in Tin

This problem is a pencil-and-paper version of the Chapter 2 code checkpoint. Use the following mass excesses for tin isotopes to compute $S_{2n}$ and plot the results:

Recall $S_{2n}(A) = \Delta(A-2) + 2\Delta_n - \Delta(A)$ (where all are the same element, and $\Delta_n = 8071.3$ keV).

(a) Compute $S_{2n}$ for each isotope from ${}^{116}\text{Sn}$ onward (you need the $A-2$ isotope for each).

(b) Plot $S_{2n}$ vs. $N$ and mark the $N = 82$ shell closure. What is the magnitude of the drop across $N = 82$?

(c) The smooth decline in $S_{2n}$ from $N = 66$ to $N = 80$ (before the shell closure) reflects the gradual filling of the $N = 50$–82 shell. The slope is approximately 1.5 MeV per pair of neutrons. Relate this slope qualitatively to the behavior expected from the asymmetry term in the semi-empirical mass formula (Chapter 4).

(d) After the $N = 82$ gap, $S_{2n}({}^{134}\text{Sn})$ is notably low. If you extrapolate the post-gap trend, at what neutron number would you predict $S_{2n} = 0$ (i.e., the two-neutron drip line for tin)? Compare to the theoretical prediction of $N \approx 100$–110.

Problem 13 — The Proton-Neutron Mass Difference

The neutron is heavier than the proton by $\Delta m = m_n - m_p = 1.293\,332\,36(46)$ MeV/$c^2$.

(a) If the neutron and proton masses were equal, would the hydrogen atom be stable? Would the neutron undergo beta decay?

(b) If $\Delta m$ were 2.0 MeV instead of 1.29 MeV, which common nucleus would become unstable against electron capture? (Hint: consider the ${}^{1}\text{H}$ atom.)

(c) If $\Delta m$ were 0.5 MeV, estimate the effect on Big Bang nucleosynthesis by considering the neutron-to-proton freeze-out ratio $n/p \sim \exp(-\Delta m c^2 / k_B T_f)$ at freeze-out temperature $T_f \approx 0.7$ MeV.

Section C: Spin and Parity (Problems 14–17)

Problem 14 — Ground-State Assignments

Using the shell model level ordering (given below for reference), predict the ground-state $J^\pi$ for the following nuclei and compare with the experimental values:

Level ordering: $1s_{1/2}$, $1p_{3/2}$, $1p_{1/2}$, $1d_{5/2}$, $2s_{1/2}$, $1d_{3/2}$, $1f_{7/2}$, $2p_{3/2}$, $1f_{5/2}$, $2p_{1/2}$, $1g_{9/2}$, ...

(a) ${}^{15}\text{N}$ ($Z = 7$, $N = 8$) — Experimental: $1/2^-$ (b) ${}^{39}\text{K}$ ($Z = 19$, $N = 20$) — Experimental: $3/2^+$ (c) ${}^{93}\text{Nb}$ ($Z = 41$, $N = 52$) — Experimental: $9/2^+$ (d) ${}^{13}\text{C}$ ($Z = 6$, $N = 7$) — Experimental: $1/2^-$

Problem 15 — Even-Even Nuclei

(a) Explain why all even-even nuclei have $J^\pi = 0^+$ ground states. What interaction is responsible?

(b) The first excited state of nearly all even-even nuclei is $2^+$. What does this tell you about the nature of the excitation — is it single-particle or collective?

(c) The energy of the first $2^+$ state in ${}^{208}\text{Pb}$ is 4.085 MeV, while in ${}^{166}\text{Er}$ it is 0.0808 MeV. What do these very different energies imply about the structure of these two nuclei?

Problem 16 — Odd-Odd Nuclei

The ground state of ${}^{6}\text{Li}$ ($Z = 3$, $N = 3$) is $J^\pi = 1^+$. Both the unpaired proton and unpaired neutron are in the $1p_{3/2}$ orbit.

(a) What values of $J$ are allowed when coupling two $j = 3/2$ particles? (Hint: the triangle rule gives $|j_1 - j_2| \leq J \leq j_1 + j_2$.)

(b) The parity of the coupled state is $(-1)^{\ell_p + \ell_n} = (-1)^{1+1} = +$. Verify that all the $J$ values in (a) are consistent with positive parity.

(c) Apply the Nordheim strong rule to predict $J$. Both nucleons have $j = \ell + 1/2$ (since $j = 3/2$ and $\ell = 1$), so the strong rule predicts $J = |j_p - j_n| = 0$. Does this give the correct answer? What does this failure tell you about the limitations of simple coupling rules for odd-odd nuclei?

(d) The ground state has $T = 0$. The first $T = 1$ state lies at 3.56 MeV excitation. This $T = 1$ state is the isobaric analog of the ground states of ${}^{6}\text{He}$ and ${}^{6}\text{Be}$. Explain why the $T = 1$ state lies higher in ${}^{6}\text{Li}$ than the $T = 0$ ground state, in terms of the isospin dependence of the nuclear force.

Problem 17 — Parity Conservation

(a) The reaction ${}^{16}\text{O}(\alpha, \gamma){}^{20}\text{Ne}$ proceeds through a $J^\pi = 1^-$ resonance in ${}^{20}\text{Ne}$. If the $\alpha$ particle is captured in an $\ell = 1$ partial wave and ${}^{16}\text{O}$ has $J^\pi = 0^+$ and $\alpha$ has $J^\pi = 0^+$, verify that this is consistent with parity conservation.

(b) Why can the $0^+$ ground state of ${}^{16}\text{O}$ not capture an $\alpha$ particle in an $s$-wave ($\ell = 0$) to form the $1^-$ state of ${}^{20}\text{Ne}$?

Section D: Electromagnetic Moments (Problems 18–24)

Problem 18 — Schmidt Moments

Calculate the Schmidt magnetic moment for the following nuclei and compare to the experimental value:

(a) ${}^{3}\text{He}$ ($1/2^+$, last neutron in $1s_{1/2}$). Experimental: $\mu = -2.128\,\mu_N$. (b) ${}^{7}\text{Li}$ ($3/2^-$, last proton in $1p_{3/2}$). Experimental: $\mu = +3.256\,\mu_N$. (c) ${}^{41}\text{Ca}$ ($7/2^-$, last neutron in $1f_{7/2}$). Experimental: $\mu = -1.595\,\mu_N$. (d) ${}^{15}\text{N}$ ($1/2^-$, last proton hole in $1p_{1/2}$). Experimental: $\mu = -0.283\,\mu_N$.

Hint for (d): A proton hole in the $1p_{1/2}$ orbit has the same magnetic moment as a proton particle in the $1p_{1/2}$ orbit (with a sign change for holes in filled orbits — work this out).

Problem 19 — Effective g-Factors

Using the data from Problem 18, calculate the effective $g_s$ that would be needed to reproduce each experimental magnetic moment exactly. Is the quenching factor $g_s^\text{eff}/g_s^\text{free}$ approximately constant?

Problem 20 — Quadrupole Moments: Single-Particle Estimates

Calculate the single-particle quadrupole moment for the following nuclei and compare to experiment:

(a) ${}^{17}\text{O}$ ($5/2^+$, neutron in $1d_{5/2}$). Experimental: $Q = -2.578$ fm$^2$. (b) ${}^{209}\text{Bi}$ ($9/2^-$, proton in $1h_{9/2}$). Experimental: $Q = -51.6$ fm$^2$.

Comment on the discrepancy in each case.

Problem 21 — Deformation Parameters

The spectroscopic quadrupole moment of ${}^{176}\text{Lu}$ ($J^\pi = 7^-$) is $Q = +8.0$ b.

(a) Assuming the ground-state rotational band has $K = J = 7$, extract the intrinsic quadrupole moment $Q_0$.

(b) From $Q_0$, calculate the deformation parameter $\beta_2$, using $R_0 = 1.2 \times 176^{1/3}$ fm and $Z = 71$.

(c) Compute the ratio of the semimajor to semiminor axis for this deformation: $\delta R / R_0 \approx \beta_2 \sqrt{5/(16\pi)}$. Is this a strongly deformed nucleus?

Problem 22 — Spherical vs. Deformed

The following table gives $Q$ values for select indium ($Z = 49$) isotopes. Indium has one proton hole below the $Z = 50$ shell closure.

(a) Calculate the single-particle estimate for a proton hole in the $1g_{9/2}$ orbit. (Note: a hole has the opposite sign to a particle.)

(b) Compare to the measured values. The measured $Q$ is positive while the proton-hole single-particle estimate would give negative $Q$. What does this reversal in sign tell you about core polarization?

Problem 23 — Quadrupole Moment and $J = 0$ or $1/2$

(a) Prove that $Q = 0$ for any state with $J = 0$ by evaluating $\langle J = 0, m = 0 | \hat{Q}_{20} | J = 0, m = 0 \rangle$ using the Wigner-Eckart theorem.

(b) Show that $Q = 0$ for $J = 1/2$ as well, using the fact that $Q_{20}$ is a rank-2 tensor operator and the triangle condition $|J - J| \leq 2 \leq J + J$ requires $J \geq 1$.

(c) The ground state of ${}^{208}\text{Pb}$ has $J^\pi = 0^+$ and therefore $Q = 0$. Does this mean ${}^{208}\text{Pb}$ is spherical? How would you determine its intrinsic shape experimentally?

Problem 24 — Magnetic Moments of Mirror Nuclei

The magnetic moments of the mirror pair ${}^{3}\text{H}$ and ${}^{3}\text{He}$ (both $J^\pi = 1/2^+$) are:

$$\mu({}^{3}\text{H}) = +2.979\,\mu_N, \quad \mu({}^{3}\text{He}) = -2.128\,\mu_N$$

(a) In the simplest shell model picture, the unpaired nucleon in ${}^{3}\text{H}$ is a proton in $1s_{1/2}$ and in ${}^{3}\text{He}$ is a neutron in $1s_{1/2}$. Calculate the Schmidt values for both.

(b) A better model treats the $A = 3$ system as having the odd nucleon carry all the angular momentum while the even pair couples to $J = 0$. The magnetic moment is then $\mu = g_j j\,\mu_N$ with the appropriate $g$-factors. Compute this for both nuclei.

(c) The sum $\mu({}^{3}\text{H}) + \mu({}^{3}\text{He}) = +0.851\,\mu_N$ is called the isoscalar moment, and the difference $\mu({}^{3}\text{H}) - \mu({}^{3}\text{He}) = +5.107\,\mu_N$ is the isovector moment. Compare these to the isoscalar and isovector nucleon moments $(\mu_p + \mu_n)/2$ and $(\mu_p - \mu_n)/2$. What do the ratios tell you about the quenching of spin in the nuclear medium?

Section E: Isospin (Problems 25–28)

Problem 25 — Isospin Assignments

Assign the total isospin $T$ and $T_3$ for the ground states of:

(a) ${}^{4}\text{He}$ (b) ${}^{14}\text{N}$ (c) ${}^{14}\text{C}$ (d) ${}^{14}\text{O}$

Identify which of (b), (c), (d) form an isospin triplet.

Problem 26 — Mirror Nuclei Coulomb Energies

For the mirror pair ${}^{17}\text{O}$ ($Z = 8$, $N = 9$) and ${}^{17}\text{F}$ ($Z = 9$, $N = 8$):

(a) Both have $J^\pi = 5/2^+$. Verify that both have $T = 1/2$ and give the correct $T_3$ values.

(b) The mass excess of ${}^{17}\text{O}$ is $\Delta = -808.8$ keV and ${}^{17}\text{F}$ is $\Delta = +1951.7$ keV. Calculate the Coulomb energy difference $\Delta E_C = M({}^{17}\text{F}) - M({}^{17}\text{O}) - (m_n - m_p)c^2$.

(c) Compare this to a uniform-sphere Coulomb estimate. The fact that the estimate works well demonstrates the dominance of Coulomb effects over genuine charge-symmetry-breaking nuclear forces.

Problem 27 — Isobaric Multiplet Mass Equation

The $A = 21$, $T = 3/2$ isobaric multiplet consists of the ground states of ${}^{21}\text{O}$ ($T_3 = -3/2$), ${}^{21}\text{F}$ ($T_3 = -1/2$), ${}^{21}\text{Ne}$ ($T_3 = +1/2$), and ${}^{21}\text{Na}$ ($T_3 = +3/2$). Their mass excesses (keV) are:

(a) Fit the IMME $M = a + bT_3 + cT_3^2$ to these four data points using a least-squares fit (or matrix equation — four points, three parameters means one residual).

(b) What is the residual? Is the quadratic IMME adequate for this multiplet?

(c) Interpret the coefficients $a$, $b$, and $c$ physically. ($a$ is the charge-independent average mass, $b$ reflects the Coulomb displacement, $c$ arises from charge-dependent nuclear forces and Coulomb correlations.)

Problem 28 — Isospin Selection Rules in Reactions

(a) The reaction $d + d \to {}^{4}\text{He} + \pi^0$ has never been observed, despite adequate beam energy. Explain why this reaction is forbidden by isospin conservation.

(b) The reaction $p + p \to d + \pi^+$ is allowed. Verify this using isospin conservation.

(c) In the decay of the $\Delta^{++}$ resonance ($T = 3/2$, $T_3 = 3/2$), what are the allowed final states? Why is $\Delta^{++} \to n + \pi^+$ forbidden?

Challenge Problems

Problem 29 — Nuclear Charge Form Factor Analysis

The elastic electron scattering cross section for ${}^{12}\text{C}$ shows diffraction minima at the following momentum transfers:

(a) A uniform sphere has minima at $qR = 4.493, 7.725, 10.904, \ldots$ Extract the radius from each minimum. Are the values consistent?

(b) Fit a Fermi distribution $\rho(r) = \rho_0/[1 + \exp((r - c)/a)]$ to the data by computing the form factor numerically (or look up the analytic result) and adjusting $c$ and $a$. What values do you obtain?

(c) Real data shows that the minima are not true zeros but have finite cross sections. This is primarily due to the surface diffuseness ($a \neq 0$). Explain qualitatively why a fuzzy surface fills in the diffraction minima.

Problem 30 — Binding Energy Differences and the Nolen-Schiffer Anomaly

After careful Coulomb corrections, the binding energy differences between mirror nuclei systematically exceed the predictions by about 5–10%, an effect known as the Nolen-Schiffer anomaly (first noted in 1969).

(a) For the mirror pair ${}^{41}\text{Sc}$/${}^{41}\text{Ca}$, the measured binding energy difference is 7.278 MeV. A Hartree-Fock calculation with a realistic charge distribution gives a Coulomb energy difference of 6.76 MeV. What is the magnitude of the anomaly?

(b) List three possible sources of this anomaly. (Hint: consider charge-symmetry breaking of the nuclear force, the neutron-proton mass difference inside the nucleus, and correlations not captured by Hartree-Fock.)

(c) Recent ab initio calculations using chiral effective field theory potentials with explicit charge-symmetry breaking have reduced but not fully resolved the anomaly. Why is resolving this discrepancy important for tests of the Standard Model using nuclear beta decay?

Problem 31 — The Nuclear Landscape

Using the data in the chapter, estimate the following:

(a) The total number of protons and neutrons in ${}^{238}\text{U}$. If each nucleon occupies a volume of approximately $(4/3)\pi r_0^3$ with $r_0 = 1.2$ fm, what fraction of the nuclear volume is "occupied" by nucleons? (This is the nuclear packing fraction — compare to the closest-packing fraction for hard spheres, $\pi/(3\sqrt{2}) \approx 0.74$.)

(b) The nuclear force has a range of approximately 1.4 fm (the pion Compton wavelength $\hbar/(m_\pi c) \approx 1.4$ fm). How many nearest neighbors does a nucleon in the nuclear interior have, given the internucleon spacing $d \approx (4\pi\rho_0/3)^{-1/3}$?

(c) Based on the number of nearest neighbors from (b), explain qualitatively why the binding energy per nucleon saturates at $B/A \approx 8$ MeV rather than growing with $A$. (Hint: if each nucleon interacted with all $A - 1$ others, $B$ would scale as $A^2$, giving $B/A \propto A$.)

Problem 32 — Compilation Exercise

Look up the following quantities in the NNDC chart of nuclides (https://www.nndc.bnl.gov/nudat3/) or the AME2020 tables and verify them against the values in this chapter:

(a) The mass excess of ${}^{132}\text{Sn}$. (b) The ground-state spin and parity of ${}^{209}\text{Bi}$. (c) The magnetic moment of ${}^{41}\text{Sc}$. (d) The quadrupole moment of ${}^{176}\text{Lu}$.

This exercise practices the essential skill of using nuclear data compilations — a skill you will need throughout this course and in any future nuclear physics research.