Quiz — Chapter 3: The Nuclear Force

Instructions: Select the best answer for each question. Detailed solutions follow at the end.

Question 1. The nuclear force saturates, meaning each nucleon interacts primarily with its nearest neighbors. The most direct experimental evidence for this is:

(a) The nuclear force is spin-dependent (b) The binding energy per nucleon $B/A$ is approximately constant for $A > 12$ (c) The deuteron has no excited states (d) Nucleon-nucleon scattering is isotropic at low energy

Question 2. The singlet ($^1S_0$) $np$ scattering length is $a_s = -23.7$ fm. The negative sign and large magnitude indicate:

(a) The interaction is repulsive in this channel (b) A deeply bound state exists (c) A near-threshold virtual state exists (the system is almost bound) (d) The effective range expansion has broken down

Question 3. The deuteron has spin-parity $J^\pi = 1^+$ and a nonzero electric quadrupole moment $Q_d = 0.286$ fm$^2$. The quadrupole moment proves that:

(a) The deuteron contains more than two nucleons (b) The nuclear force includes a tensor component that mixes $S$ and $D$ waves (c) The proton charge distribution is non-spherical (d) The deuteron is in a pure $D$-state

Question 4. In the square well model of the deuteron, the parameter $KR \approx 1.83$ is only slightly above $\pi/2 \approx 1.57$. This means:

(a) The deuteron is deeply bound with many excited states (b) The deuteron is barely bound and has no excited states (c) The potential well is very shallow (d) The wavefunction is entirely confined within the well

Question 5. Approximately what fraction of the deuteron's probability density lies outside the range of the nuclear potential in the square well model?

(a) 10% (b) 30% (c) 50% (d) 70%

Question 6. Yukawa predicted the existence of a mediating particle for the nuclear force with mass approximately:

(a) 0.5 MeV/$c^2$ (electron mass) (b) 100 MeV/$c^2$ (between electron and nucleon) (c) 939 MeV/$c^2$ (nucleon mass) (d) 80,000 MeV/$c^2$ ($W$ boson mass)

Question 7. The Yukawa potential $V(r) = -g^2 e^{-\mu r}/r$ reduces to which familiar potential in the limit $\mu \to 0$?

(a) The harmonic oscillator potential (b) The Woods-Saxon potential (c) The Coulomb potential (d) The hard sphere potential

Question 8. The pion was discovered in 1947 by Powell, Lattes, and Occhialini. Its mass is approximately 140 MeV/$c^2$ and its Compton wavelength is:

(a) 0.14 fm (b) 1.4 fm (c) 14 fm (d) 140 fm

Question 9. At distances $r > 2$ fm, the dominant contribution to the nucleon-nucleon potential is:

(a) $\omega$ meson exchange (short-range repulsion) (b) One-pion exchange (OPEP) (c) Two-pion exchange ($\sigma$ channel) (d) Direct quark-gluon interactions

Question 10. The short-range repulsion ("hard core") of the nuclear force at $r < 0.5$ fm is primarily attributed to:

(a) One-pion exchange (b) $\omega$ meson exchange (and the quark substructure of nucleons) (c) Gravitational repulsion at short distances (d) The Coulomb interaction between protons

Question 11. The Argonne $v_{18}$ potential achieves $\chi^2/\text{datum} \approx 1.09$ for the $NN$ scattering database. The "18" refers to:

(a) 18 adjustable parameters (b) 18 operator components in the potential (c) 18 partial waves fitted (d) 18 mesons exchanged

Question 12. Chiral effective field theory derives the nuclear force from:

(a) Exact solutions of QCD on a lattice (b) The most general theory of nucleons and pions consistent with QCD symmetries (c) Fitting to nuclear masses only, without scattering data (d) Classical meson field theory without quantum corrections

Question 13. The three-nucleon force is needed because:

(a) Newton's third law fails for nuclear forces (b) Two-body potentials fitted to NN scattering data underbind nuclei with $A \geq 3$ (c) The Pauli exclusion principle does not apply to nucleons (d) The deuteron cannot be explained with two-body forces alone

Question 14. The Argonne $v_{18}$ two-body potential predicts the triton binding energy as $B(^3$H$) = 7.62$ MeV, compared to the experimental value of 8.482 MeV. The discrepancy is resolved by:

(a) Including relativistic corrections to the kinetic energy (b) Adding a three-nucleon force (e.g., Urbana IX) (c) Using a deeper square well potential (d) Including the Coulomb interaction between protons

Question 15. The charge independence of the nuclear force means:

(a) The force is the same for charged and neutral particles (b) The $pp$, $nn$, and $np$ strong interactions are equal in the same isospin state (c) The force does not depend on the separation between nucleons (d) Electromagnetic effects are negligible in nuclei

Question 16. Which of the following is NOT a property of the nuclear force?

(a) Short range ($\sim 1$--2 fm) (b) Spin dependence (c) Inverse-square-law distance dependence (d) Approximate charge independence

Question 17. In chiral EFT, the leading three-nucleon force appears at which order?

(a) Leading order (LO) (b) Next-to-leading order (NLO) (c) N$^2$LO (next-to-next-to-leading order) (d) N$^3$LO

Question 18. The oxygen drip line ($^{24}$O is the heaviest bound oxygen isotope) is correctly predicted only when:

(a) The Coulomb force is included (b) Three-nucleon forces are included (c) Relativistic effects are included (d) The neutron is treated as a charged particle

Solutions

1. (b) The near-constancy of $B/A$ for $A > 12$ means each nucleon interacts with a fixed number of neighbors, not with all other nucleons. This is the definition of a saturating, short-range force.

2. (c) A scattering length that is negative and large in magnitude (much larger than the force range $\sim 2$ fm) signals a near-threshold virtual state. If the force were slightly stronger, this channel would have a true bound state with $a > 0$.

3. (b) A pure $S$-state ($L=0$) has spherical symmetry and $Q = 0$. The nonzero $Q_d$ requires an $L = 2$ ($D$-wave) admixture, which arises from the tensor component of the nuclear force.

4. (b) The condition $KR > \pi/2$ is the minimum for any bound state; $KR > 3\pi/2$ is needed for a second state. With $KR \approx 1.83$, barely above $\pi/2$, the deuteron is just barely bound, and there is no second bound state.

5. (d) The exponential tail of the wavefunction outside the well contains approximately 70% of the total probability, a consequence of the small binding energy relative to the well depth.

6. (b) Yukawa estimated the mediator mass from the force range ($\sim 2$ fm): $m \approx \hbar c / (Rc) \approx 200/2 \approx 100$ MeV/$c^2$.

7. (c) When the mediator mass goes to zero ($\mu \to 0$), $e^{-\mu r}/r \to 1/r$, which is the Coulomb potential (up to constants).

8. (b) $\lambda_C = \hbar c / (m_\pi c^2) = 197.3/140 \approx 1.4$ fm.

9. (b) At long range, only the lightest meson (the pion) contributes significantly. Heavier mesons produce potentials with shorter range ($\propto e^{-m_{\text{heavy}} r/\hbar c}$) that are exponentially suppressed beyond $\sim 1$ fm.

10. (b) The $\omega$ meson (mass 783 MeV/$c^2$, range $\sim 0.25$ fm) provides a strong repulsive core. At the quark level, this repulsion is related to the Pauli exclusion principle for quarks and gluon exchange between overlapping nucleons.

11. (b) The Argonne $v_{18}$ has 18 operator components: central, spin-spin, tensor, spin-orbit, quadratic-$L$, and quadratic-spin-orbit, each in isoscalar and isovector channels, plus charge-dependent and charge-asymmetric terms.

12. (b) Chiral EFT constructs the most general Lagrangian for nucleons and pions consistent with the symmetries of QCD — especially chiral symmetry and its spontaneous and explicit breaking — then organizes contributions in a systematic power counting.

13. (b) The deuteron ($A=2$) is well described by two-body forces. For $A \geq 3$, exact calculations with any realistic NN potential underbind the system, requiring genuine three-body interactions.

14. (b) The 0.86 MeV underbinding of the triton is resolved by adding a three-nucleon force. The triton binding energy is then typically used to calibrate the 3NF parameters.

15. (b) Charge independence means the nuclear force depends only on the total isospin state, not on the individual charges: $V_{pp} = V_{nn} = V_{np}$ when compared in the same isospin channel.

16. (c) The nuclear force does not follow an inverse-square law. It has a finite range and exponential falloff (Yukawa-type), not a $1/r^2$ force law.

17. (c) In chiral EFT, the leading 3NF appears at N$^2$LO. At LO and NLO, only two-nucleon forces contribute.

18. (b) Two-body forces alone predict oxygen isotopes bound well past $^{24}$O. The repulsive contribution of three-nucleon forces in neutron-rich systems correctly places the drip line at $^{24}$O.