Quiz — Chapter 17: Nuclear Reaction Fundamentals

Instructions: Select the best answer for each question. Each question has exactly one correct answer.

Q1. In the reaction notation ${}^{12}\text{C}(d, p){}^{13}\text{C}$, the projectile and ejectile are, respectively:

(a) ${}^{12}\text{C}$ and ${}^{13}\text{C}$ (b) $d$ and $p$ (c) $p$ and $d$ (d) ${}^{12}\text{C}$ and $p$

Q2. The Q-value of a nuclear reaction is defined as:

(a) The kinetic energy of the projectile (b) $(M_{\text{final}} - M_{\text{initial}})c^2$ (c) $(M_{\text{initial}} - M_{\text{final}})c^2$ (d) The binding energy of the compound nucleus

Q3. An exothermic reaction has:

(a) $Q < 0$ and requires a minimum beam energy (b) $Q > 0$ and can proceed at any beam energy (in principle) (c) $Q = 0$ and is purely elastic (d) $Q > 0$ but still requires energy above a Coulomb barrier threshold

Q4. The threshold energy for an endothermic reaction always exceeds $|Q|$ because:

(a) Some energy is lost to radiation (b) The Coulomb barrier must also be overcome (c) Momentum conservation requires the products to carry kinetic energy in the lab frame (d) The nuclear force is repulsive at short range

Q5. In the center-of-mass frame, the total momentum of the system is:

(a) Equal to the projectile momentum (b) Equal to the target momentum (c) Zero (d) Equal to $\mu v_{\text{rel}}$

Q6. For a proton ($M_b = 1\,u$) striking a ${}^{208}\text{Pb}$ target ($M_a = 208\,u$), what fraction of the lab kinetic energy is available in the CM frame?

(a) About 50% (b) About 99.5% (c) About 0.5% (d) About 75%

Q7. The unit "barn" equals:

(a) $10^{-24}\,\text{m}^2$ (b) $10^{-28}\,\text{m}^2$ (c) $10^{-24}\,\text{fm}^2$ (d) $10^{-28}\,\text{fm}^2$

Q8. The Rutherford scattering cross section is proportional to:

(a) $\sin^4(\theta/2)$ (b) $1/\sin^2(\theta/2)$ (c) $1/\sin^4(\theta/2)$ (d) $\cos^4(\theta/2)$

Q9. The total Rutherford cross section (integrated over all angles) is:

(a) $\pi a^2$ (b) $4\pi a^2$ (c) Zero (d) Infinite

Q10. The Rutherford scattering formula gives the same result classically and quantum mechanically because:

(a) Nuclear scattering is always classical (b) The $1/r$ Coulomb potential has this special property (c) Quantum corrections are always negligible for charged particles (d) The Born approximation is exact for Coulomb scattering

Q11. In the partial wave expansion, the angular momentum quantum number $l$ is related to the classical impact parameter $b$ by approximately:

(a) $b \approx l/k$ (b) $b \approx (l + 1/2)/k$ (c) $b \approx l^2/k$ (d) $b \approx k/l$

Q12. At low energies ($kR \ll 1$), nuclear scattering is dominated by:

(a) The highest partial wave (b) $p$-wave ($l = 1$) scattering (c) $s$-wave ($l = 0$) scattering (d) All partial waves equally

Q13. The maximum contribution of a single partial wave ($l = 0$) to the elastic cross section is:

(a) $\pi R^2$ (b) $4\pi/k^2$ (c) $\pi/k^2$ (d) $2\pi/k^2$

Q14. In the optical model, the imaginary part of the potential $W(r)$ accounts for:

(a) The spin-orbit interaction (b) The Coulomb barrier (c) Absorption of flux from the elastic channel into reaction channels (d) The Pauli exclusion principle

Q15. A perfectly absorbing nuclear sphere has $\sigma_{\text{tot}} = 2\pi R^2$, which is twice the geometric cross section. The extra $\pi R^2$ comes from:

(a) Coulomb scattering (b) Shadow (Fraunhofer) diffraction in the forward direction (c) Spin-orbit coupling (d) Nuclear excitation

Q16. The $1/v$ law for neutron capture cross sections ($\sigma \propto 1/v$) arises because:

(a) The Coulomb barrier suppresses fast neutrons (b) Slower neutrons spend more time near the nucleus (c) The nuclear radius increases at low energy (d) The neutron wavelength decreases with velocity

Q17. Ericson fluctuations occur when:

(a) Individual compound-nucleus resonances are well separated (b) Compound-nucleus resonances overlap ($\Gamma > D$) (c) Only elastic scattering is possible (d) The beam energy is below the Coulomb barrier

Q18. The energy autocorrelation function of Ericson fluctuations has a Lorentzian shape with width:

(a) Equal to the level spacing $D$ (b) Equal to the average total width $\Gamma$ of the compound nucleus states (c) Equal to the beam energy spread (d) Equal to the Q-value of the reaction

Q19. In the optical model, absorption occurs mainly at the nuclear surface (rather than in the volume) at low energies because:

(a) The nuclear surface is denser than the interior (b) The Coulomb barrier blocks interior penetration (c) The Pauli exclusion principle blocks nucleon-nucleon collisions in the nuclear interior (d) The imaginary potential is repulsive in the interior

Q20. The invariant mass $\sqrt{s}$ of a system is useful because:

(a) It equals the total kinetic energy (b) It is the same in all reference frames (c) It equals the sum of the rest masses (d) It is always conserved in nuclear reactions

Answer Key