Quiz — Chapter 19: Direct Reactions
Instructions: Select the best answer for each question. Each question has exactly one correct answer.
Q1. The characteristic timescale of a direct nuclear reaction is approximately:
(a) $10^{-16}$ s — comparable to compound nucleus lifetimes (b) $10^{-22}$ s — the transit time of the projectile across the nucleus (c) $10^{-8}$ s — comparable to atomic transition times (d) $10^{-10}$ s — comparable to nuclear gamma-ray lifetimes
Q2. In the (d,p) stripping reaction ${}^{40}\text{Ca}(d,p){}^{41}\text{Ca}$, which particle is transferred to the target?
(a) A proton (b) A neutron (c) A deuteron (d) An alpha particle
Q3. The angular distribution of the outgoing proton in a (d,p) reaction on a closed-shell target is primarily determined by:
(a) The spin of the target nucleus (b) The orbital angular momentum $l$ of the transferred neutron (c) The binding energy of the deuteron (d) The Coulomb barrier height
Q4. For an $l = 0$ neutron transfer in a (d,p) reaction, the angular distribution is expected to:
(a) Peak at $90°$ in the center-of-mass frame (b) Show a strong minimum at $0°$ (c) Peak in the forward direction with no minimum at $0°$ (d) Be isotropic (flat in angle)
Q5. In Butler's plane-wave Born approximation, the (d,p) angular distribution is proportional to $|j_l(qR)|^2$. For $l = 2$, how many minima are expected in the forward hemisphere?
(a) Zero (b) One (c) Two (d) Three
Q6. The DWBA differs from the plane-wave Born approximation primarily because:
(a) It includes relativistic corrections (b) It uses optical model wavefunctions that account for elastic scattering distortions in the entrance and exit channels (c) It treats the transferred nucleon as a composite particle (d) It includes all multi-step processes
Q7. In DWBA theory, the spectroscopic factor $S_{nlj}$ is extracted from the ratio:
(a) $S = \sigma_{\text{exp}} / \sigma_{\text{Rutherford}}$ (b) $S = (d\sigma/d\Omega)_{\text{exp}} / (d\sigma/d\Omega)_{\text{DWBA}}^{sp}$ (c) $S = \Gamma_n / \Gamma_{\text{total}}$ (d) $S = Q_{\text{exp}} / Q_{\text{theory}}$
Q8. A spectroscopic factor of $S = 0.65$ for a single-particle state means:
(a) The state is only 65% of its predicted energy (b) The probability that the nuclear state has a pure single-particle configuration is 65% (c) The nuclear radius is 65% of the standard value (d) 65% of the nucleons participate in the reaction
Q9. Pickup reactions such as $(p,d)$ are complementary to stripping reactions because they:
(a) Measure the same orbits at higher precision (b) Probe single-particle orbits below the Fermi energy (hole states), while stripping probes orbits above it (c) Use the same angular distributions as stripping but at different energies (d) Can only be performed with radioactive beams
Q10. The primary advantage of the $(e,e'p)$ knockout reaction over (d,p) transfer for measuring spectroscopic factors is:
(a) It can be performed at radioactive beam facilities (b) The electromagnetic probe is well understood (QED), reducing reaction mechanism uncertainties (c) It has a larger cross section (d) It does not require knowledge of optical potentials
Q11. The "quenching" of spectroscopic factors refers to the observation that:
(a) Spectroscopic factors decrease with increasing beam energy (b) Measured spectroscopic factors are systematically 55--70% of independent-particle model predictions (c) Direct reaction cross sections are smaller than compound nucleus cross sections (d) Spectroscopic factors vanish for nuclei near the drip lines
Q12. In inverse kinematics at a radioactive beam facility, the roles of beam and target are reversed. This means:
(a) The light particle (e.g., deuteron) is the beam and the heavy nucleus is the target (b) The heavy radioactive nucleus is the beam and the light particle (e.g., deuterium) is the target (c) The reaction proceeds in the opposite direction (exothermic becomes endothermic) (d) The angular distributions are mirrored about $90°$
Q13. Which of the following is NOT an experimental signature that distinguishes direct reactions from compound nucleus reactions?
(a) Forward-peaked, structured angular distributions (b) Smooth energy dependence of the cross section (c) Symmetric angular distributions about $90°$ in the CM frame (d) Sensitivity of the cross section to the specific final state populated
Q14. In a one-nucleon knockout reaction on a ${}^{9}\text{Be}$ target at intermediate energies, the momentum distribution of the $(A-1)$ residue reflects:
(a) The momentum of the beam particle (b) The momentum distribution of the knocked-out nucleon in its bound orbit (c) The recoil of the ${}^{9}\text{Be}$ target (d) The Fermi energy of the residual nucleus
Q15. The discovery that $N = 16$ is a new magic number in ${}^{24}\text{O}$ was enabled primarily by:
(a) Precision mass measurements with Penning traps (b) Coulomb excitation measurements (c) Direct reaction experiments (transfer and knockout) at radioactive beam facilities (d) Electron scattering form factor measurements
Answer Key
| Question | Answer | Explanation |
|---|---|---|
| Q1 | (b) | Direct reactions occur during the transit time of the projectile across the nuclear diameter, $\sim 10^{-22}$ s, much shorter than compound nucleus lifetimes. |
| Q2 | (b) | In (d,p) stripping, the deuteron's neutron is stripped off and captured by the target; the proton continues to the detector. |
| Q3 | (b) | The angular distribution pattern — number of minima, position of the first maximum — is determined by the orbital angular momentum $l$ of the transferred neutron. |
| Q4 | (c) | For $l = 0$, $j_0(0) = 1$, so the angular distribution has maximum cross section at $\theta = 0°$ with no forward minimum. |
| Q5 | (c) | The number of minima in the forward hemisphere equals $l$. For $l = 2$, there are two minima. |
| Q6 | (b) | The DWBA replaces plane waves with distorted waves — solutions of the Schrodinger equation with the optical potential — thereby accounting for elastic scattering in both channels. |
| Q7 | (b) | The spectroscopic factor is the ratio of the measured differential cross section to the DWBA single-particle prediction: $S = (d\sigma/d\Omega)_{\text{exp}} / (d\sigma/d\Omega)_{\text{DWBA}}^{sp}$. |
| Q8 | (b) | $S = 0.65$ means the nuclear state has 65% overlap with a pure single-particle configuration; the rest is distributed among more complex configurations due to correlations. |
| Q9 | (b) | Stripping adds a nucleon above the Fermi energy (particle states); pickup removes one from below (hole states). Together they map the complete single-particle spectrum. |
| Q10 | (b) | The electron interacts electromagnetically (QED), not via the strong force, so the reaction mechanism is known precisely, reducing the dominant systematic uncertainty in spectroscopic factor extraction. |
| Q11 | (b) | Quenching is the systematic observation that $S_{\text{exp}} \approx 0.55$--$0.70 \times S_{\text{IPM}}$, due to correlations (SRC, LRC, tensor) beyond the independent-particle model. |
| Q12 | (b) | Inverse kinematics places the exotic (radioactive) nucleus in the beam and the light particle (H, D, Be) in the target, because the exotic nucleus cannot be made into a target. |
| Q13 | (c) | Symmetric angular distributions about $90°$ are characteristic of compound nucleus reactions, not direct reactions. Direct reactions are forward-peaked. |
| Q14 | (b) | The momentum distribution of the residue is the complement of the knocked-out nucleon's momentum distribution in its bound state, providing information about the spatial structure of the orbit. |
| Q15 | (c) | The $N = 16$ shell closure in ${}^{24}\text{O}$ was established through transfer and knockout reaction experiments at radioactive beam facilities, combined with the observation that ${}^{24}\text{O}$ is the neutron drip line of oxygen. |