Exercises — Chapter 17

Q-Values and Conservation Laws

Problem 17.1 ⭐ Determine which of the following reactions are allowed by conservation of charge and baryon number. For those that are allowed, calculate the Q-value using the AME2020 mass excesses.

(a) ${}^{14}\text{N}(\alpha, p){}^{17}\text{O}$

(b) ${}^{7}\text{Li}(p, n){}^{7}\text{Be}$

(c) ${}^{12}\text{C}(n, \alpha){}^{8}\text{Be}$ (note: ${}^{8}\text{Be}$ is unbound but treat it as a state with mass excess $\Delta = 4.942\,\text{MeV}$)

(d) ${}^{16}\text{O}(p, d){}^{14}\text{O}$

Mass excesses (MeV): $\Delta(n) = 8.071$, $\Delta({}^{1}\text{H}) = 7.289$, $\Delta(d) = 13.136$, $\Delta({}^{4}\text{He}) = 2.425$, $\Delta({}^{7}\text{Li}) = 14.908$, $\Delta({}^{7}\text{Be}) = 15.769$, $\Delta({}^{8}\text{Be}) = 4.942$, $\Delta({}^{12}\text{C}) = 0.000$, $\Delta({}^{14}\text{N}) = 2.863$, $\Delta({}^{14}\text{O}) = 8.007$, $\Delta({}^{16}\text{O}) = -4.737$, $\Delta({}^{17}\text{O}) = -0.809$.

Problem 17.2 ⭐ The reaction ${}^{3}\text{He}(n, p){}^{3}\text{H}$ is used in neutron detectors.

(a) Calculate the Q-value. Mass excesses: $\Delta({}^{3}\text{He}) = 14.931\,\text{MeV}$, $\Delta({}^{3}\text{H}) = 14.950\,\text{MeV}$.

(b) Is this reaction exothermic or endothermic?

(c) For thermal neutrons ($T_n = 0.025\,\text{eV}$), what are the kinetic energies of the proton and triton in the CM frame?

(d) The thermal neutron cross section is $\sigma = 5333\,\text{b}$. Compare this to the geometric cross section of ${}^{3}\text{He}$ ($R \approx 1.9\,\text{fm}$). How many times larger is the actual cross section?

Problem 17.3 ⭐ The reaction ${}^{197}\text{Au}(n, \gamma){}^{198}\text{Au}$ is used for gold activation analysis.

(a) Calculate the Q-value. Mass excesses: $\Delta({}^{197}\text{Au}) = -31.141\,\text{MeV}$, $\Delta({}^{198}\text{Au}) = -29.582\,\text{MeV}$.

(b) Explain why $(n, \gamma)$ reactions are almost always exothermic. (Hint: think about the neutron separation energy.)

(c) The gamma ray carries away essentially all of the energy (why?). What is the approximate gamma-ray energy for capture of a thermal neutron?

Problem 17.4 ⭐⭐ For the reaction ${}^{27}\text{Al}(p, n){}^{27}\text{Si}$:

(a) Calculate the Q-value. Mass excesses: $\Delta({}^{27}\text{Al}) = -17.197\,\text{MeV}$, $\Delta({}^{27}\text{Si}) = -12.384\,\text{MeV}$.

(b) Calculate the threshold energy.

(c) What is the kinetic energy available in the CM frame at $T_p = 10\,\text{MeV}$?

(d) At what proton lab energy does $T_{\text{CM}}$ first exceed $|Q|$?

Threshold Energies

Problem 17.5 ⭐⭐ Antiproton production requires the reaction $p + p \to p + p + p + \bar{p}$ (the minimum final state that conserves baryon number).

(a) Calculate the Q-value ($m_p c^2 = 938.272\,\text{MeV}$, $m_{\bar{p}} c^2 = 938.272\,\text{MeV}$).

(b) Using the invariant mass method, derive the threshold energy for antiproton production on a stationary proton target.

(c) The Bevatron at Berkeley was designed with a kinetic energy of 6.2 GeV to produce antiprotons. Verify that this exceeds the threshold.

Problem 17.6 ⭐⭐ Pion production in nucleon-nucleon collisions: $p + p \to p + n + \pi^+$ ($m_{\pi^+}c^2 = 139.57\,\text{MeV}$, $m_n c^2 = 939.565\,\text{MeV}$).

(a) Calculate the Q-value.

(b) Calculate the threshold proton lab kinetic energy.

(c) Compare with the threshold for $p + d \to {}^{3}\text{He} + \pi^0$ ($m_{\pi^0}c^2 = 134.98\,\text{MeV}$, $\Delta({}^{3}\text{He}) = 14.931\,\text{MeV}$). Which has the lower threshold, and why?

Problem 17.7 ⭐ For the endothermic reaction ${}^{12}\text{C}(p, n){}^{12}\text{N}$ with $Q = -18.12\,\text{MeV}$:

(a) Calculate the exact threshold energy.

(b) Compute the approximate threshold using $T_{\text{th}} \approx -Q(1 + M_b/M_a)$. What is the percentage error relative to the exact formula?

(c) For what target mass does the approximate formula become exact?

Center-of-Mass Kinematics

Problem 17.8 ⭐⭐ A 100 MeV proton beam strikes a ${}^{208}\text{Pb}$ target.

(a) Calculate the CM kinetic energy $T_{\text{CM}}$.

(b) Calculate the velocities of the CM frame and of the proton in the CM frame (non-relativistically).

(c) Calculate $\eta = v_{\text{CM}}/v_p^{\text{CM}}$ for elastic scattering.

(d) What is the maximum lab scattering angle for the proton? Can the proton scatter backward in the lab?

Problem 17.9 ⭐⭐ For elastic scattering of neutrons off ${}^{12}\text{C}$ at $T_n = 14\,\text{MeV}$:

(a) Calculate $\eta = M_n / M_{{}^{12}\text{C}} \approx 1/12$.

(b) Show that the lab angle $\theta_{\text{lab}} = 30°$ corresponds to $\theta_{\text{CM}} \approx 33.2°$.

(c) Calculate the Jacobian $d\Omega_{\text{CM}}/d\Omega_{\text{lab}}$ at this angle.

(d) If $(d\sigma/d\Omega)_{\text{CM}} = 200\,\text{mb/sr}$ at $\theta_{\text{CM}} = 33.2°$, what is $(d\sigma/d\Omega)_{\text{lab}}$ at $\theta_{\text{lab}} = 30°$?

Problem 17.10 ⭐⭐⭐ Inverse kinematics. At radioactive beam facilities, the roles of projectile and target are reversed: a heavy radioactive beam bombards a light target (often hydrogen or deuterium).

Consider ${}^{132}\text{Sn}$ (a radioactive doubly-magic nucleus) at 500 MeV/nucleon scattering elastically off a proton target.

(a) Calculate $\eta$ for this system. Is $\eta$ greater or less than 1?

(b) What is the maximum lab scattering angle for the proton?

(c) Show that there is kinematic focusing: the proton recoils are concentrated in a forward cone. Calculate the maximum recoil angle.

(d) Why is inverse kinematics advantageous for studying short-lived isotopes?

Problem 17.11 ⭐⭐⭐ Invariant mass spectroscopy. In the reaction ${}^{9}\text{Be}({}^{13}\text{C}, {}^{8}\text{Be} + n){}^{13}\text{C}$ at 100 MeV/nucleon, one wants to reconstruct the invariant mass of the ${}^{8}\text{Be} + n$ system to look for excited states of ${}^{9}\text{Be}$.

(a) Write the invariant mass of the two-body subsystem ${}^{8}\text{Be} + n$ in terms of their measured four-momenta.

(b) If ${}^{8}\text{Be}$ and $n$ are emitted with relative kinetic energy $E_{\text{rel}} = 0.1\,\text{MeV}$ in their CM frame, what is the invariant mass $\sqrt{s_{8+n}}$ in MeV/$c^2$?

(c) This technique is used to measure the excitation spectrum of unbound nuclei. Explain why invariant mass is the appropriate variable (hint: Lorentz invariance).

Rutherford Scattering

Problem 17.12 ⭐ Alpha particles ($z_1 = 2$) with kinetic energy $T = 5.5\,\text{MeV}$ scatter from gold ($z_2 = 79$, $A = 197$).

(a) Calculate the Coulomb parameter $a$ and the distance of closest approach $d_0 = 2a$.

(b) Calculate $d\sigma/d\Omega$ at $\theta = 10°$, $45°$, $90°$, and $170°$. Express in fm$^2$/sr.

(c) Verify that the ratio $[d\sigma/d\Omega]_{10°} / [d\sigma/d\Omega]_{90°}$ equals $\sin^4(45°)/\sin^4(5°)$.

Problem 17.13 ⭐⭐ At what alpha-particle energy does the Rutherford formula break down for scattering off ${}^{40}\text{Ca}$ ($Z = 20$, $R \approx 4.1\,\text{fm}$) at $\theta = 180°$? (The formula breaks down when the distance of closest approach $d_0$ becomes comparable to the nuclear radius.)

Problem 17.14 ⭐⭐ Rutherford as a measurement tool. A 2 MeV proton beam strikes a thin target containing an unknown element. At $\theta_{\text{lab}} = 170°$, the Rutherford cross section is measured to be $d\sigma/d\Omega = 24.8\,\text{b/sr}$.

(a) Assuming the target is much heavier than the proton ($\theta_{\text{CM}} \approx \theta_{\text{lab}}$), determine $z_2$ from the measured cross section.

(b) Identify the element. (This is the principle behind Rutherford backscattering spectrometry, RBS, widely used in materials science.)

Problem 17.15 ⭐⭐⭐ Derivation: Rutherford cross section in the lab frame. Starting from the CM-frame Rutherford formula and the lab-to-CM angle transformation for elastic scattering, derive the lab-frame differential cross section:

$$\left(\frac{d\sigma}{d\Omega}\right)_{\text{lab}} = \left(\frac{a}{2}\right)^2 \frac{1}{\sin^4\theta_{\text{lab}}} \cdot \frac{\left[\cos\theta_{\text{lab}} + \sqrt{1 - (M_b/M_a)^2\sin^2\theta_{\text{lab}}}\right]^2}{\sqrt{1 - (M_b/M_a)^2\sin^2\theta_{\text{lab}}}}$$

Show that this reduces to the CM formula when $M_a \gg M_b$.

Problem 17.16 ⭐⭐⭐ The Sommerfeld parameter. The Sommerfeld parameter $\eta_S = z_1 z_2 e^2 / (4\pi\epsilon_0 \hbar v)$ (where $v$ is the relative velocity) determines the importance of Coulomb effects.

(a) Show that $\eta_S = a / \lambdabar$ where $a$ is the Coulomb parameter and $\lambdabar = \hbar/\mu v$ is the reduced de Broglie wavelength.

(b) Calculate $\eta_S$ for 5.5 MeV alphas on gold. Is $\eta_S \gg 1$ or $\ll 1$? Interpret this physically.

(c) Show that $\eta_S \gg 1$ is the condition for the classical (Rutherford) result to be valid. Why does the classical limit work so well for nuclear Coulomb scattering?

(d) For what energy would $\eta_S = 1$ for proton-proton scattering? Below this energy, the classical Rutherford formula requires quantum corrections (Mott scattering for identical particles).

Cross Sections

Problem 17.17 ⭐ A detector with area $A_{\text{det}} = 2\,\text{cm}^2$ is placed at distance $r = 25\,\text{cm}$ from a target at angle $\theta = 30°$. The beam intensity is $I = 10^{10}\,\text{protons/s}$, and the target has $n = 5 \times 10^{18}\,\text{atoms/cm}^2$.

(a) Calculate the solid angle subtended by the detector.

(b) If $d\sigma/d\Omega = 50\,\text{mb/sr}$ at $\theta = 30°$, how many scattered particles per second reach the detector?

(c) In a 1-hour run, how many counts are accumulated?

Problem 17.18 ⭐⭐ The total cross section for neutrons on ${}^{56}\text{Fe}$ at $E_n = 1\,\text{MeV}$ is approximately $\sigma_{\text{tot}} = 3.5\,\text{b}$. An iron plate of thickness $x = 10\,\text{cm}$ ($\rho = 7.87\,\text{g/cm}^3$, $A = 56$) is placed in a neutron beam.

(a) Calculate the number density of iron atoms $n$ (atoms/cm$^3$).

(b) Calculate the macroscopic cross section $\Sigma = n\sigma$ (in cm$^{-1}$).

(c) What fraction of neutrons are transmitted through the plate without interaction? (Use $I/I_0 = e^{-\Sigma x}$.)

(d) What thickness of iron would reduce the beam intensity by a factor of 100?

Problem 17.19 ⭐⭐ The $1/v$ law for neutron capture cross sections states that $\sigma(v) = \sigma_0 v_0 / v$, where $\sigma_0$ is the cross section at velocity $v_0$ (conventionally $v_0 = 2200\,\text{m/s}$, corresponding to thermal energy $E_0 = 0.0253\,\text{eV}$).

(a) Show that the $1/v$ law is equivalent to $\sigma(E) = \sigma_0 \sqrt{E_0/E}$.

(b) For ${}^{10}\text{B}(n, \alpha){}^{7}\text{Li}$ with $\sigma_0 = 3840\,\text{b}$, calculate $\sigma$ at $E = 1\,\text{eV}$, $1\,\text{keV}$, and $1\,\text{MeV}$.

(c) At what energy does the $1/v$ law predict $\sigma = 1\,\text{mb}$? Is the $1/v$ law still valid at this energy?

Partial Waves

Problem 17.20 ⭐⭐ For s-wave ($l = 0$) neutron scattering off a hard sphere of radius $R$:

(a) Show that the s-wave phase shift is $\delta_0 = -kR$ (the wavefunction must vanish at $r = R$).

(b) Calculate the s-wave cross section $\sigma_0 = 4\pi\sin^2(kR)/k^2$.

(c) In the low-energy limit ($kR \ll 1$), show that $\sigma_0 \to 4\pi R^2$. Interpret this result: why is it four times the geometric cross section?

(d) For neutrons on ${}^{208}\text{Pb}$ ($R \approx 7.1\,\text{fm}$), at what neutron energy is $kR = 1$? Above this energy, higher partial waves must be included.

Problem 17.21 ⭐⭐⭐ Unitarity limit and resonances. The maximum contribution of a single partial wave $l$ to the elastic cross section is:

$$\sigma_l^{\max} = \frac{4\pi(2l+1)}{k^2}$$

(a) Show this occurs when $\delta_l = \pi/2$ (modulo $\pi$).

(b) For $l = 0$ neutrons at $E = 1\,\text{eV}$, calculate $\sigma_0^{\max}$ in barns. Compare to measured thermal neutron cross sections for ${}^{135}\text{Xe}$ ($\sigma = 2.65 \times 10^6\,\text{b}$). Can this be an s-wave resonance?

(c) For $l = 0$ at arbitrary energy, show that $\sigma_0^{\max} = 4\pi\lambdabar^2$ where $\lambdabar = 1/k$. Compute this for $E = 0.0253\,\text{eV}$ neutrons and compare to the ${}^{135}\text{Xe}$ cross section.

Problem 17.22 ⭐⭐⭐ A nuclear potential scatters $l = 0$ neutrons with S-matrix element $S_0 = \eta_0 e^{2i\delta_0}$.

(a) Write expressions for the elastic cross section $\sigma_{\text{el}}$, reaction cross section $\sigma_{\text{rxn}}$, and total cross section $\sigma_{\text{tot}}$ in terms of $\eta_0$ and $\delta_0$.

(b) Show that $\sigma_{\text{tot}} \leq 4\pi/k^2$ (the unitarity bound).

(c) For what values of $\eta_0$ and $\delta_0$ is $\sigma_{\text{rxn}}$ maximized? What is the maximum reaction cross section for a single partial wave?

(d) Show that $\sigma_{\text{el}} \geq \sigma_{\text{rxn}}$ is always true when only s-waves contribute. (Hint: show $|1 - S_0|^2 \geq 1 - |S_0|^2$ for $|S_0| \leq 1$.)

Problem 17.23 ⭐⭐ Estimate the number of partial waves that contribute to the scattering of:

(a) 14 MeV neutrons off ${}^{56}\text{Fe}$ ($R \approx 4.6\,\text{fm}$)

(b) 50 MeV protons off ${}^{90}\text{Zr}$ ($R \approx 5.5\,\text{fm}$)

(c) 5 MeV alphas off ${}^{197}\text{Au}$ ($R \approx 7.0\,\text{fm}$)

For each, compute $l_{\max} = kR$ and state how many partial waves ($l = 0, 1, \ldots, l_{\max}$) contribute.

Optical Model

Problem 17.24 ⭐⭐ The optical model predicts diffraction minima in elastic scattering at angles $\theta_n \approx (n + 1/2)\pi/(kR)$.

(a) For 30 MeV protons scattering off ${}^{40}\text{Ca}$ ($R \approx 4.1\,\text{fm}$), calculate the expected angular positions of the first three diffraction minima.

(b) The measured first minimum is at $\theta \approx 33°$. Use this to extract $R$ and compare with $R = 1.25 A^{1/3}\,\text{fm}$.

(c) Why do the minima become shallower (less deep) at larger angles? (Hint: think about the absorptive potential.)

Problem 17.25 ⭐⭐⭐ Shadow scattering. A perfectly absorbing sphere ($\eta_l = 0$ for $l \leq l_{\max}$, $\eta_l = 1$ for $l > l_{\max}$) has $\sigma_{\text{rxn}} = \pi R^2$ and $\sigma_{\text{tot}} = 2\pi R^2$.

(a) Show that the elastic cross section is also $\sigma_{\text{el}} = \pi R^2$ by computing $\sigma_{\text{el}} = \sigma_{\text{tot}} - \sigma_{\text{rxn}}$.

(b) Where does the extra elastic scattering come from? Show that it is concentrated in the forward direction, within an angular cone $\theta \lesssim 1/(kR)$.

(c) This is called the "shadow" or Fraunhofer diffraction peak. Calculate its angular width for 100 MeV neutrons on ${}^{208}\text{Pb}$.

(d) Show that in the high-energy limit, the differential elastic cross section at $\theta = 0$ is $(d\sigma/d\Omega)_{\theta=0} = k^2 R^4 / 4$.

Problem 17.26 ⭐⭐ The optical model for neutron-nucleus scattering at $E = 10\,\text{MeV}$ uses a Woods-Saxon potential with $V_0 = 47\,\text{MeV}$, $R = 1.25A^{1/3}\,\text{fm}$, and $a = 0.65\,\text{fm}$.

(a) Calculate the internal wave number $K = \sqrt{2m(E + V_0)}/\hbar$ inside the nucleus.

(b) For $A = 120$ ($R \approx 6.16\,\text{fm}$), calculate $KR$ and estimate the number of internal reflections needed for size resonances.

(c) Size resonances occur when $2KR \approx n\pi$. For what neutron energies (approximately) do the first three size resonances occur?

Ericson Fluctuations

Problem 17.27 ⭐⭐⭐ The cross section for ${}^{60}\text{Ni}(p, \alpha)$ between $E_p = 10$ and $14\,\text{MeV}$ shows Ericson fluctuations with a measured autocorrelation width of $\Gamma \approx 50\,\text{keV}$.

(a) What is the average lifetime of the compound nucleus states at this excitation energy? Use $\tau = \hbar/\Gamma$.

(b) The average level spacing at this excitation energy is estimated to be $D \approx 5\,\text{eV}$. Calculate the ratio $\Gamma/D$. Is the system in the overlapping resonance regime?

(c) How many compound-nucleus levels contribute to the cross section at each energy point?

Synthesis and Research Problems

Problem 17.28 ⭐⭐ The Coulomb barrier. For charged-particle reactions, the Coulomb barrier height is approximately:

$$V_C = \frac{k Z_a Z_b e^2}{R_a + R_b} = \frac{1.440\,Z_a Z_b}{1.25(A_a^{1/3} + A_b^{1/3})}\,\text{MeV}$$

(a) Calculate $V_C$ for the following reactions: - $p + {}^{12}\text{C}$ - $\alpha + {}^{40}\text{Ca}$ - ${}^{12}\text{C} + {}^{12}\text{C}$ - ${}^{16}\text{O} + {}^{208}\text{Pb}$

(b) At what fraction of the Coulomb barrier height does the Rutherford formula begin to break down? (Hint: the formula breaks down when $d_0 \approx R_a + R_b$, which occurs at $E_{\text{CM}} = V_C$.)

(c) For $p + {}^{12}\text{C}$ at the solar core temperature ($T \approx 1.5 \times 10^7\,\text{K}$, $kT \approx 1.3\,\text{keV}$), calculate the ratio $V_C / E_{\text{thermal}}$. This ratio explains why solar nuclear reactions proceed so slowly — the proton must quantum-mechanically tunnel through a barrier that is thousands of times higher than its thermal energy.

Problem 17.29 ⭐⭐⭐ The Q-value systematics. Using the semi-empirical mass formula (Chapter 4), derive an approximate expression for the Q-value of the $(d, p)$ reaction ${}^{A}\text{X}(d, p){}^{A+1}\text{X}$ in terms of the neutron separation energy $S_n$ of the product nucleus.

Show that $Q(d,p) = S_n({}^{A+1}\text{X}) - B_d$, where $B_d = 2.224\,\text{MeV}$ is the deuteron binding energy. Explain physically why $(d, p)$ reactions are exothermic for most nuclei.

Problem 17.30 ⭐⭐⭐ Energy resolution and Q-value spectroscopy. In a ${}^{208}\text{Pb}(d, p){}^{209}\text{Pb}$ experiment at $T_d = 12\,\text{MeV}$ and $\theta_{\text{lab}} = 30°$, the proton energy spectrum shows peaks corresponding to the ground state ($J^{\pi} = 9/2^+$, $E^* = 0$) and first excited state ($J^{\pi} = 11/2^+$, $E^* = 0.779\,\text{MeV}$) of ${}^{209}\text{Pb}$.

(a) Calculate the ground-state Q-value. Mass excesses: $\Delta({}^{208}\text{Pb}) = -21.749\,\text{MeV}$, $\Delta({}^{209}\text{Pb}) = -17.617\,\text{MeV}$.

(b) What is the Q-value for populating the first excited state?

(c) Using the Q-value equation, estimate the proton energy difference $\Delta T_p$ between the ground-state and first-excited-state peaks at $\theta_{\text{lab}} = 30°$. (Hint: differentiate the Q-value equation with respect to $Q$ at fixed $\theta_{\text{lab}}$ and $T_b$.)

(d) If the detector energy resolution is $\delta E = 50\,\text{keV}$ (FWHM), can the two states be resolved?

Problem 17.31 ⭐⭐⭐⭐ Research problem: optical model fitting. The global Koning-Delaroche optical potential has energy-dependent parameters. At $E = 25\,\text{MeV}$ for neutrons on ${}^{90}\text{Zr}$, representative parameters are: $V_0 = 45\,\text{MeV}$, $r_V = 1.24\,\text{fm}$, $a_V = 0.66\,\text{fm}$, $W_D = 10\,\text{MeV}$, $r_D = 1.26\,\text{fm}$, $a_D = 0.58\,\text{fm}$.

(a) Plot the real and imaginary parts of this potential as functions of $r$.

(b) Calculate the reaction cross section using the sharp-cutoff approximation ($T_l = 1$ for $l \leq l_{\max}$, $T_l = 0$ for $l > l_{\max}$) with $l_{\max} = kR$.

(c) Estimate the mean free path $\lambda$ of a neutron inside nuclear matter using $\lambda = \hbar v / W$ where $v$ is the velocity inside the potential well ($E + V_0$ kinetic energy) and $W$ is the imaginary potential depth. Compare with the nuclear radius.

(d) Explain the physical meaning of $\lambda$ being comparable to $R$.

Problem 17.32 ⭐⭐⭐⭐ Research problem: designing an experiment. You want to measure the $(d, p)$ reaction on ${}^{48}\text{Ca}$ at a beam energy where the angular distributions are most sensitive to the orbital angular momentum $l$ of the transferred neutron.

(a) Calculate the Q-value for ${}^{48}\text{Ca}(d, p){}^{49}\text{Ca}$ (ground state, $J^{\pi} = 3/2^-$). Mass excesses: $\Delta({}^{48}\text{Ca}) = -44.215\,\text{MeV}$, $\Delta({}^{49}\text{Ca}) = -41.289\,\text{MeV}$.

(b) Calculate the threshold energy (if endothermic) or confirm it is exothermic.

(c) For a deuteron beam energy of 10 MeV, calculate $T_{\text{CM}}$, the proton energy in the CM frame, and the maximum lab proton angle.

(d) The angular distribution of the protons peaks at $\theta_{\text{CM}} \approx 0°$ for $l = 0$ transfer but shifts to larger angles ($\sim 25°$--$35°$) for $l = 1$ or $l = 2$. What angular range should your detector array cover to distinguish $l$-values?

(e) What angular resolution (in degrees) is needed to resolve the first diffraction minimum in the DWBA angular distribution?

Problem 17.33 ⭐⭐⭐⭐ Research problem: Ericson fluctuation analysis. The excitation function (cross section vs. energy) for ${}^{28}\text{Si}(p, \alpha){}^{25}\text{Al}$ between $E_p = 12$ and $16\,\text{MeV}$ shows rapid fluctuations superimposed on a smooth trend.

(a) You measure the cross section at 200 equally-spaced energy points. Describe how you would compute the energy autocorrelation function $C(\epsilon)$ from this data.

(b) The autocorrelation function has a half-width of $\Gamma = 80\,\text{keV}$. Calculate the compound-nucleus lifetime $\tau$ and compare it to the transit time $\tau_{\text{transit}} = 2R/v$ for a 14 MeV proton traversing ${}^{28}\text{Si}$ ($R \approx 3.8\,\text{fm}$).

(c) If the average level spacing is $D = 2\,\text{keV}$, how many compound-nucleus states contribute to the cross section at each energy? Is this consistent with the overlapping-resonance regime?

(d) The measured variance of the fluctuations is $\text{Var}(\sigma)/\langle\sigma\rangle^2 = 0.05$. Using the relation $C(0) = 1/N_{\text{eff}}$, estimate the effective number of open channels $N_{\text{eff}}$. Is this reasonable for this system?