Exercises — Chapter 19

Timescales and Kinematics

Problem 19.1 ⭐ A 20 MeV deuteron beam strikes a ${}^{40}\text{Ca}$ target.

(a) Calculate the velocity of the deuteron in units of $c$. Use the nonrelativistic approximation $v/c = \sqrt{2T/m_d c^2}$, where $m_d c^2 = 1875.6$ MeV.

(b) Estimate the nuclear radius of ${}^{40}\text{Ca}$ using $R = 1.25 A^{1/3}$ fm.

(c) Calculate the transit time $\tau = 2R/v$. Compare to a typical compound nucleus lifetime of $\tau_{\text{CN}} \sim 10^{-18}$ s.

(d) Estimate the grazing angular momentum $l_{\text{graze}} = kR$, where $k = \mu v / \hbar$ and $\mu$ is the reduced mass of the $d + {}^{40}\text{Ca}$ system. What does this tell you about the peripheral character of the reaction?

Problem 19.2 ⭐ For the reaction ${}^{48}\text{Ca}(d,p){}^{49}\text{Ca}$:

(a) Calculate the $Q$-value using the atomic masses: $M({}^{48}\text{Ca}) = 47.952523$ u, $M(d) = 2.014102$ u, $M(p) = 1.007825$ u, $M({}^{49}\text{Ca}) = 48.955674$ u.

(b) If the deuteron beam energy is 15 MeV in the lab frame, calculate the kinetic energy of the proton emitted at $0°$ in the lab frame (assume the ${}^{49}\text{Ca}$ is left in its ground state).

(c) The first excited state of ${}^{49}\text{Ca}$ is at $E_x = 2.023$ MeV. What is the proton kinetic energy for this state at $0°$?

Problem 19.3 ⭐ Explain qualitatively why the angular distributions of direct reactions are forward-peaked, while compound nucleus angular distributions are approximately symmetric about $90°$ in the center-of-mass frame. Your answer should reference the timescale and the concept of "memory" of the entrance channel.

Angular Distributions and l-Value Determination

Problem 19.4 ⭐⭐ In Butler's plane-wave theory, the angular distribution of a (d,p) reaction is proportional to $|j_l(qR)|^2$, where $q$ is the momentum transfer and $R$ is an effective nuclear radius.

(a) For an $l = 0$ transfer, the angular distribution goes as $|j_0(qR)|^2 = |\sin(qR)/(qR)|^2$. Sketch this function as a function of $\theta_{\text{CM}}$ for $kR = 5$, noting that $q \approx 2k\sin(\theta/2)$ at low momentum transfer. Where is the first minimum?

(b) For $l = 2$, the spherical Bessel function $j_2(x) = (3/x^2 - 1)\sin(x)/x - 3\cos(x)/x^2$. Show that $j_2(0) = 0$ and that the angular distribution vanishes at $\theta = 0°$ for $l \geq 1$.

(c) Explain why the number of minima in the forward hemisphere is a reliable indicator of $l$ even when the DWBA distortions modify the simple Bessel function pattern.

Problem 19.5 ⭐⭐ The table below gives schematic angular distribution data for the reaction ${}^{90}\text{Zr}(d,p){}^{91}\text{Zr}$ at $E_d = 12$ MeV, populating three different final states. For each state, determine the transferred orbital angular momentum $l$ from the pattern of the angular distribution.

Hint: Look at whether the cross section peaks at $0°$ or away from $0°$, and count the minima.

Problem 19.6 ⭐⭐ For the reaction ${}^{208}\text{Pb}(d,p){}^{209}\text{Pb}$ populating the ground state ($J^\pi = 9/2^+$, $l = 4$ transfer):

(a) What is the parity of the transferred neutron's wavefunction? Verify that $\pi = (-1)^l$ is consistent with the parity change from ${}^{208}\text{Pb}$ ($J^\pi = 0^+$) to ${}^{209}\text{Pb}$ ($J^\pi = 9/2^+$).

(b) The transferred neutron has $l = 4$ and $j = 9/2$. What is the relationship between $l$, $s = 1/2$, and $j$ for this orbit? Is it $j = l + 1/2$ or $j = l - 1/2$?

(c) The shell model (Chapter 6) labels this orbit as $2g_{9/2}$. What do the quantum numbers $n = 2$, $l = 4$, $j = 9/2$ mean? How many neutrons can occupy this orbit?

DWBA Analysis

Problem 19.7 ⭐⭐ In the DWBA, the differential cross section for a (d,p) reaction is:

$$\frac{d\sigma}{d\Omega} = S_{nlj} \cdot \sigma_{\text{DWBA}}^{sp}(\theta)$$

A DWBA calculation for ${}^{48}\text{Ca}(d,p){}^{49}\text{Ca}$ at $E_d = 13$ MeV, with the neutron in the $2p_{3/2}$ orbit ($l = 1$, $j = 3/2$), gives a single-particle cross section at the first maximum ($\theta_{\text{CM}} = 18°$) of $\sigma_{\text{DWBA}}^{sp} = 14.5$ mb/sr. The measured cross section at this angle is $10.2$ mb/sr.

(a) Extract the spectroscopic factor $S$.

(b) The independent-particle shell model predicts $S = 1$ for adding a neutron to the first empty $2p_{3/2}$ orbit above the $N = 28$ shell closure. What is the quenching ratio $R_s = S_{\text{exp}}/S_{\text{IPM}}$?

(c) Suggest two physical mechanisms that could account for the reduction of $S$ below unity.

Problem 19.8 ⭐⭐⭐ A key input to the DWBA is the bound-state wavefunction of the transferred neutron. Consider a neutron bound in a Woods-Saxon potential with parameters $V_0 = 55$ MeV, $r_0 = 1.25$ fm, $a = 0.65$ fm, for a target with $A = 90$.

(a) Calculate the radius of the potential: $R = r_0 A^{1/3}$ fm.

(b) For a $1d_{5/2}$ orbit ($n = 1$, $l = 2$, $j = 5/2$) with separation energy $B_n = 8.5$ MeV, estimate the classical turning point $r_t$ from $V(r_t) + \hbar^2 l(l+1)/(2\mu r_t^2) = -B_n$ (you may solve this graphically or numerically).

(c) The asymptotic form of the bound-state wavefunction outside the potential is $\varphi(r) \propto e^{-\kappa r}/r$, where $\kappa = \sqrt{2\mu B_n}/\hbar$. Calculate $\kappa$ and the decay length $1/\kappa$. What does this tell you about the spatial extent of the wavefunction relevant for peripheral (d,p) reactions?

Problem 19.9 ⭐⭐⭐ The optical potential for deuteron scattering at $E_d = 20$ MeV on ${}^{58}\text{Ni}$ has parameters (Hinterberger et al., 1968):

• Real volume: $V_0 = 91.5$ MeV, $r_0 = 1.15$ fm, $a = 0.81$ fm
• Imaginary surface: $W_D = 12.2$ MeV, $r_W = 1.34$ fm, $a_W = 0.68$ fm

(a) Calculate the real and imaginary potential depths at the nuclear surface ($r = R = r_0 A^{1/3}$).

(b) The elastic scattering angular distribution shows a Fraunhofer-like diffraction pattern. Estimate the angular position of the first minimum using $\theta_{\min} \approx \pi/(kR)$, where $k$ is the center-of-mass wave number.

(c) Explain why the imaginary part of the optical potential is concentrated at the nuclear surface (surface-peaked) for deuterons but volume-distributed for higher-energy protons. What does this reflect about the reaction mechanism?

Problem 19.10 ⭐⭐ Explain the physical meaning of each of the following inputs to a DWBA calculation for an $A(d,p)B$ reaction. For each, state whether it affects primarily the shape or the magnitude of the calculated angular distribution.

(a) The entrance-channel optical potential ($d + A$)

(b) The exit-channel optical potential ($p + B$)

(c) The orbital angular momentum $l$ of the transferred neutron

(d) The separation energy $B_n$ of the neutron in the residual nucleus

(e) The spectroscopic factor $S_{nlj}$

Spectroscopic Factors and Sum Rules

Problem 19.11 ⭐⭐ The Macfarlane-French sum rule states:

$$\sum_f S^{(+)}_{nlj}(A \to f) - \sum_f S^{(-)}_{nlj}(A \to f) = (2j+1) - 2\langle n_{nlj} \rangle_A$$

For the $1g_{9/2}$ neutron orbit in ${}^{90}\text{Zr}$ ($Z = 40$, $N = 50$):

(a) $N = 50$ is a magic number, so the $1g_{9/2}$ orbit is completely filled. What is $\langle n_{1g_{9/2}} \rangle$? What does the sum rule predict for the difference between stripping and pickup spectroscopic factors?

(b) In the pickup reaction ${}^{90}\text{Zr}(p,d){}^{89}\text{Zr}$, the spectroscopic factor for removing a $1g_{9/2}$ neutron to form the $9/2^+$ ground state of ${}^{89}\text{Zr}$ is measured to be $S^{(-)} \approx 7.2$. Compare to the shell-model prediction of $S^{(-)} = 2j + 1 = 10$. What is the quenching ratio?

(c) If we also measure stripping ${}^{90}\text{Zr}(d,p){}^{91}\text{Zr}$ and find that the $1g_{9/2}$ spectroscopic factors to all final states sum to $S^{(+)}_{\text{tot}} = 1.8$, does the sum rule hold? (Remember that the quenching may affect both $S^{(+)}$ and $S^{(-)}$.)

Problem 19.12 ⭐⭐⭐ In the independent-particle model, the spectroscopic factor for removing a proton from a completely filled $1d_{5/2}$ orbit in ${}^{16}\text{O}$ is $S_{\text{IPM}} = 2j + 1 = 6$. The NIKHEF $(e,e'p)$ experiment found that the spectroscopic factor to the $5/2^+$ ground state of ${}^{15}\text{N}$ was $S_{\text{exp}} = 3.3$.

(a) Calculate the quenching ratio $R_s = S_{\text{exp}}/S_{\text{IPM}}$.

(b) The remaining $1d_{5/2}$ proton strength is spread over excited states of ${}^{15}\text{N}$ and into the continuum. If the sum of spectroscopic factors to all discrete $5/2^+$ states below 25 MeV excitation is 4.2, how much strength is "missing" — presumably distributed in the continuum at high excitation?

(c) What types of correlations are responsible for this fragmentation? (Refer to Section 19.4.3.)

Pickup and Knockout Reactions

Problem 19.13 ⭐ Explain why pickup reactions $(p,d)$ are said to probe "hole states," while stripping reactions $(d,p)$ probe "particle states." Use an energy diagram showing the Fermi level of a closed-shell nucleus to illustrate your answer.

Problem 19.14 ⭐⭐ In the $(e,e'p)$ knockout reaction, the missing energy is defined as:

$$E_{\text{miss}} = \omega - T_p - T_{A-1}$$

where $\omega$ is the energy transfer from the electron, $T_p$ is the proton kinetic energy, and $T_{A-1}$ is the recoil kinetic energy of the residual nucleus.

(a) Show that for a proton knocked out from an orbit with separation energy $B_p$, the missing energy is $E_{\text{miss}} = B_p + E_x^*$, where $E_x^*$ is the excitation energy of the residual nucleus.

(b) In the ${}^{12}\text{C}(e,e'p)$ experiment, the missing energy spectrum shows a sharp peak at $E_{\text{miss}} \approx 16$ MeV (the $1p_{3/2}$ proton separation energy) and a broad bump centered around $E_{\text{miss}} \approx 35$–$40$ MeV. What shell-model orbit does the broad bump correspond to?

(c) Why is the $1s_{1/2}$ peak much broader than the $1p_{3/2}$ peak? What does this tell you about the lifetime of the deep hole state?

Problem 19.15 ⭐⭐⭐ In a $(p,2p)$ knockout reaction at 300 MeV, the two outgoing protons are detected in coincidence. In the quasi-free approximation (the knocked-out proton was at rest inside the nucleus), the kinematics are those of free $p$-$p$ scattering.

(a) For free $p$-$p$ scattering at 300 MeV, the two protons emerge at symmetric angles $\theta_1 = \theta_2$ in the lab. Calculate these angles using the relation $\theta_1 + \theta_2 = 90°$ (nonrelativistic limit).

(b) The Fermi momentum of nucleons inside the nucleus ($p_F \approx 250$ MeV/$c$) causes the actual opening angle to vary. Estimate the spread in the sum angle $\theta_1 + \theta_2$ due to the Fermi momentum.

(c) How does the distribution of the sum angle provide information about the momentum distribution of the knocked-out proton?

Inverse Kinematics and Radioactive Beams

Problem 19.16 ⭐⭐ A ${}^{132}\text{Sn}$ beam at 4.8 MeV/nucleon strikes a CD$_2$ target (deuterated polyethylene) in inverse kinematics.

(a) Calculate the total kinetic energy of the ${}^{132}\text{Sn}$ beam in the lab frame.

(b) Calculate the velocity of the center-of-mass of the ${}^{132}\text{Sn} + d$ system (as a fraction of $c$).

(c) In the center-of-mass frame, a proton from the $(d,p)$ reaction is emitted at $\theta_{\text{CM}} = 30°$. Using the Lorentz transformation (or Galilean approximation), estimate the corresponding lab angle. In which direction does the proton go in the lab?

(d) Why must the silicon detector array in this experiment cover backward lab angles?

Problem 19.17 ⭐⭐ Compare the experimental requirements for a "normal kinematics" measurement (${}^{208}\text{Pb}(d,p){}^{209}\text{Pb}$ with a deuteron beam) and an "inverse kinematics" measurement (${}^{132}\text{Sn}(d,p){}^{133}\text{Sn}$ with a tin beam). Discuss:

(a) Target preparation and thickness requirements

(b) Beam intensity requirements (typical stable beams: $\sim 10^{12}$ particles/s; typical radioactive beams: $\sim 10^{4}$–$10^{6}$ particles/s)

(c) Detection strategy (what particles are measured, at what angles)

(d) Energy resolution limitations

Problem 19.18 ⭐⭐⭐ One-nucleon knockout on a ${}^{9}\text{Be}$ target at intermediate energies ($E/A \sim 80$ MeV/nucleon) has been used extensively at NSCL/FRIB. In the eikonal model, the single-particle knockout cross section has contributions from stripping (the nucleon is absorbed by the target) and diffraction (the nucleon and core are elastically scattered apart):

$$\sigma_{sp} = \sigma_{\text{str}} + \sigma_{\text{diff}}$$

(a) For a neutron in an $l = 0$ orbit with separation energy $B_n = 1$ MeV in a light nucleus ($A = 12$), the stripping cross section is $\sigma_{\text{str}} \approx 35$ mb and the diffraction cross section is $\sigma_{\text{diff}} \approx 30$ mb. Calculate the total single-particle knockout cross section.

(b) If the spectroscopic factor is $S = 0.6$, what is the predicted inclusive knockout cross section?

(c) For a more deeply bound state ($B_n = 15$ MeV, same orbit), why would you expect both $\sigma_{\text{str}}$ and $\sigma_{\text{diff}}$ to be smaller? (Think about the spatial extent of the bound-state wavefunction.)

Shell Evolution

Problem 19.19 ⭐⭐ The $N = 20$ shell gap is well established in stable nuclei (e.g., ${}^{40}\text{Ca}$). But in ${}^{32}\text{Mg}$ ($Z = 12$, $N = 20$), knockout experiments show evidence of a deformed ground state, suggesting the $N = 20$ gap has vanished.

(a) In ${}^{40}\text{Ca}$ ($Z = 20$), the proton $1d_{3/2}$ orbit is filled. In ${}^{32}\text{Mg}$ ($Z = 12$), the proton $1d_{5/2}$ orbit is filled but $1d_{3/2}$ is empty. Using the concept of the tensor force (Chapter 3), explain qualitatively how the emptying of the proton $1d_{3/2}$ orbit affects the neutron single-particle energies near $N = 20$.

(b) The knockout reaction ${}^{9}\text{Be}({}^{32}\text{Mg}, {}^{31}\text{Mg} + \gamma)$ populates the states of ${}^{31}\text{Mg}$. If the ground state of ${}^{32}\text{Mg}$ is predominantly a $2p$-$2h$ excitation across the $N = 20$ gap, what would you expect for the momentum distribution width of the knockout residue — broader or narrower than for a nucleus with a good $N = 20$ gap? Explain.

Problem 19.20 ⭐⭐⭐ The doubly magic nucleus ${}^{78}\text{Ni}$ ($Z = 28$, $N = 50$) has long been predicted but is extremely difficult to produce experimentally.

(a) Using the shell model (Chapter 6), list the neutron orbits expected just above $N = 50$ (the first few orbits of the $N = 50$–$82$ shell).

(b) Design a direct reaction experiment to measure the single-particle spectrum of ${}^{79}\text{Ni}$. Specify: (i) the reaction to use, (ii) the beam required, (iii) the target, and (iv) which modern facility could perform this measurement.

(c) Why is this measurement important for nuclear astrophysics, specifically for the $r$-process? (Hint: consider the $N = 50$ waiting point in the $r$-process path.)

Synthesis and Analysis

Problem 19.21 ⭐⭐⭐ A new radioactive beam facility can deliver $10^5$ particles/s of ${}^{56}\text{Ni}$ ($Z = 28$, $N = 28$, doubly magic) at 30 MeV/nucleon.

(a) You want to map the single-particle spectrum above the $N = 28$ shell closure using $(d,p)$ in inverse kinematics. Estimate the cross section to the strongest single-particle state (assume $S \approx 0.7$ and a typical single-particle DWBA cross section of 10 mb/sr at the first maximum).

(b) If the deuterium target is 1 mg/cm$^2$ thick, calculate the number of target atoms per cm$^2$ and the luminosity $\mathcal{L} = I \cdot n_t$.

(c) Estimate the count rate at the first maximum (assuming a detector solid angle of 50 msr centered on the peak).

(d) How many hours of beam time would be needed to accumulate 500 counts at the first maximum? Is this feasible for a week-long experiment?

Problem 19.22 ⭐⭐⭐⭐ (Research) The quenching of spectroscopic factors depends on the nuclear asymmetry $(N-Z)/A$, as reported by Gade et al. (2008). Read the original paper: A. Gade et al., Physical Review C 77, 044306 (2008).

(a) Summarize the systematic trend they observe for knockout reactions on neutron-rich nuclei.

(b) The asymmetry dependence is characterized by plotting $R_s$ vs. $\Delta S$, the difference between the separation energies of the knocked-out nucleon species and the other species. What is the physical meaning of $\Delta S$?

(c) Discuss whether the observed trend is fully explained by nuclear structure theory (correlations) or whether part of it could be an artifact of the reaction theory (eikonal model limitations).

Problem 19.23 ⭐⭐⭐⭐ (Research) The dispersive optical model (DOM) provides a unified framework for describing elastic scattering and single-particle properties, including spectroscopic factors, within a single self-consistent approach.

(a) Explain the key idea of the DOM: the optical model potential and the single-particle Green's function are related by a dispersion relation. How does this connect the real and imaginary parts of the self-energy?

(b) The DOM predicts that the occupancy of the $3s_{1/2}$ proton orbit in ${}^{40}\text{Ca}$ is approximately 0.75, compared to the IPM prediction of 1.0. How does the DOM achieve this reduction without explicit multi-body correlations?

(c) How do DOM predictions for spectroscopic factors compare with $(e,e'p)$ measurements? What are the strengths and limitations of the DOM approach?

Problem 19.24 ⭐ Write a one-paragraph summary explaining, in plain language (suitable for a physics-literate non-specialist), how a $(d,p)$ reaction reveals which orbit a neutron occupies inside a nucleus. Do not use equations.