Chapter 2 Self-Assessment Quiz

Test your understanding of nuclear sizes, masses, spins, electromagnetic moments, and isospin. Answers are provided at the end. Attempt each question before checking.

Question 1

The nuclear radius formula $R = r_0 A^{1/3}$ implies that the nuclear volume is proportional to:

(a) $A^{1/3}$ (b) $A^{2/3}$ (c) $A$ (d) $A^2$

Question 2

The nuclear density saturation value is approximately $\rho_0 \approx 0.17$ nucleons/fm$^3$. This means that:

(a) Heavier nuclei are denser than lighter nuclei. (b) The density of the nuclear interior is roughly the same for all nuclei. (c) The nuclear density increases linearly with $A$. (d) Nuclear matter is compressible at low pressures.

Question 3

Elastic electron scattering from nuclei measures primarily the:

(a) Matter radius (proton + neutron distribution) (b) Charge radius (proton distribution only) (c) Neutron radius (neutron distribution only) (d) Nuclear skin thickness directly

Question 4

The PREX-2 experiment measured the neutron skin of ${}^{208}\text{Pb}$ using:

(a) Proton elastic scattering (b) Pionic atoms (c) Parity-violating electron scattering (d) Muonic x-ray spectroscopy

Question 5

The atomic mass unit $u$ is defined as:

(a) The mass of a proton (b) The mass of a hydrogen atom (c) $1/12$ the mass of a ${}^{12}\text{C}$ atom (d) $1/16$ the mass of an ${}^{16}\text{O}$ atom

Question 6

A Penning trap determines nuclear masses by measuring:

(a) The deflection radius of ions in a magnetic field (b) The time of flight between two detectors (c) The cyclotron frequency of a trapped ion (d) The x-ray energies from electronic transitions

Question 7

A sudden drop in the two-neutron separation energy $S_{2n}$ at a particular neutron number indicates:

(a) A shape transition from spherical to deformed (b) A nuclear magic number (shell closure) (c) The onset of nuclear fission instability (d) A change in the pairing force strength

Question 8

The ground-state spin and parity of ALL even-even nuclei is:

(a) $1^+$ (b) $0^-$ (c) $0^+$ (d) It depends on the specific nucleus

Question 9

The ground state of ${}^{13}\text{C}$ has $J^\pi = 1/2^-$. The unpaired neutron occupies which orbit?

(a) $1s_{1/2}$ (b) $1p_{3/2}$ (c) $1p_{1/2}$ (d) $1d_{5/2}$

Question 10

The nuclear magneton $\mu_N$ is smaller than the Bohr magneton $\mu_B$ by a factor of approximately:

(a) 137 (the fine structure constant inverse) (b) 207 (the muon-electron mass ratio) (c) 1836 (the proton-electron mass ratio) (d) 2 (a factor of the spin $g$-factor)

Question 11

The Schmidt magnetic moment values for odd-$A$ nuclei:

(a) Agree exactly with all measured nuclear moments (b) Provide upper and lower bounds between which most moments fall (c) Are only valid for nuclei near magic numbers (d) Apply only to proton-odd nuclei, not neutron-odd nuclei

Question 12

A nucleus with a positive electric quadrupole moment ($Q > 0$) has:

(a) A spherical shape (b) A prolate (elongated) shape (c) An oblate (flattened) shape (d) A pear shape (octupole deformation)

Question 13

The spectroscopic quadrupole moment of any nuclear state with $J = 0$ is:

(a) Always positive (b) Always negative (c) Zero, regardless of the intrinsic shape (d) Equal to the intrinsic quadrupole moment

Question 14

In the isospin formalism, the proton and neutron are treated as:

(a) Fundamentally different particles with no symmetry relation (b) Two states of the same particle (the nucleon) with isospin $t = 1/2$ (c) Particles with isospin $t = 1$ and $t = 0$, respectively (d) Identical particles that differ only in mass

Question 15

Mirror nuclei are nuclei with:

(a) Equal numbers of protons and neutrons ($N = Z$) (b) The same mass number but different $Z$ and $N$, with $Z$ and $N$ interchanged (c) The same number of protons but different numbers of neutrons (d) Identical excitation spectra with no corrections needed

Question 16

The dominant source of isospin symmetry breaking in nuclei is:

(a) The quark mass difference $m_d - m_u$ (b) The Coulomb interaction between protons (c) The weak interaction (d) Meson exchange currents

Question 17

The surface thickness of the nuclear density distribution is approximately:

(a) 0.5 fm, and varies strongly with $A$ (b) 2.4 fm, and is approximately constant across the periodic table (c) Equal to the nuclear radius for light nuclei (d) Zero for doubly magic nuclei

Question 18

Which of the following is NOT a method for measuring nuclear charge radii?

(a) Elastic electron scattering (b) Muonic atom x-ray spectroscopy (c) Optical isotope shifts (d) Neutron scattering

Answers

1. (c) — Volume $= (4/3)\pi R^3 = (4/3)\pi r_0^3 A$, directly proportional to $A$. This is the mathematical content of density saturation.

2. (b) — Density saturation means the interior density is approximately constant. The nuclear "matter" has an approximately incompressible interior.

3. (b) — Electrons interact electromagnetically and probe the proton (charge) distribution. To measure the neutron distribution, one needs hadronic or weak probes.

4. (c) — PREX-2 exploited the fact that the $Z^0$ boson couples preferentially to neutrons, making the parity-violating asymmetry in electron scattering sensitive to the neutron distribution.

5. (c) — The unified atomic mass unit $u$ is defined as exactly $1/12$ the mass of a ${}^{12}\text{C}$ atom, including its electrons. This is the modern standard (replacing the older oxygen-based scale).

6. (c) — The Penning trap confines a single ion and measures its cyclotron frequency $\nu_c = qB/(2\pi m)$. Comparing $\nu_c$ for the ion of interest and a reference ion gives the mass ratio.

7. (b) — A shell closure (magic number) means the last occupied shell is full; adding neutrons beyond it requires filling a higher, less-bound shell. The binding of the last neutrons drops sharply.

8. (c) — $J^\pi = 0^+$ for the ground state of every known even-even nucleus. The pairing interaction drives nucleons to couple in $J = 0$ pairs, and even parity follows because $(-1)^{\ell + \ell} = +1$.

9. (c) — ${}^{13}\text{C}$ has 7 neutrons. The shell filling is: $1s_{1/2}$ (2), $1p_{3/2}$ (4), $1p_{1/2}$ (1). The unpaired neutron is in $1p_{1/2}$: $\ell = 1$, $j = 1/2$, parity $= (-1)^1 = -$. Thus $J^\pi = 1/2^-$. Correct.

10. (c) — $\mu_N = e\hbar/(2m_p)$ and $\mu_B = e\hbar/(2m_e)$, so $\mu_N/\mu_B = m_e/m_p \approx 1/1836$.

11. (b) — The Schmidt values define two lines (for $j = \ell + 1/2$ and $j = \ell - 1/2$), and experimental moments generally fall between these lines, though not exactly on them. The deviations arise from core polarization, meson exchange currents, and configuration mixing.

12. (b) — $Q > 0$ means the charge distribution is elongated along the spin axis (prolate), like a rugby ball. $Q < 0$ means oblate (flattened).

13. (c) — The quadrupole operator is a rank-2 tensor. By the Wigner-Eckart theorem and the triangle condition, $\langle J = 0 || Q_2 || J = 0 \rangle = 0$, so $Q = 0$ for any $J = 0$ state regardless of intrinsic deformation.

14. (b) — Heisenberg (1932) proposed treating the proton and neutron as isospin-$1/2$ states of the nucleon, in analogy with spin-up and spin-down.

15. (b) — Mirror nuclei have the proton and neutron numbers exchanged: $(Z, N) \leftrightarrow (N, Z)$. They have the same $A$ and the same $|T_3|$.

16. (b) — The Coulomb interaction, being long-range and acting only between protons, is the dominant isospin-breaking effect in nuclei, contributing MeV-scale energy differences. The nuclear charge-symmetry-breaking and quark mass difference effects are smaller ($\sim 100$ keV).

17. (b) — The surface thickness $t = 4a\ln 3 \approx 2.4$ fm (with $a \approx 0.54$ fm) is approximately independent of $A$. This universality reflects the short range of the nuclear force.

18. (d) — Neutron scattering probes the matter distribution (sensitive to both protons and neutrons), not specifically the charge distribution. The other three methods are standard techniques for charge radii.