Exercises — Chapter 18

The Compound Nucleus Model

Problem 18.1 ⭐ The compound nucleus ${}^{64}$Zn$^*$ can be formed by two reactions:

$$p + {}^{63}\text{Cu} \to {}^{64}\text{Zn}^* \qquad \text{and} \qquad \alpha + {}^{60}\text{Ni} \to {}^{64}\text{Zn}^*$$

(a) Using the atomic mass data below, calculate the Q-value for each reaction.

• $M({}^{63}\text{Cu}) = 62.929601\,\text{u}$, $M({}^{1}\text{H}) = 1.007825\,\text{u}$
• $M({}^{60}\text{Ni}) = 59.930791\,\text{u}$, $M({}^{4}\text{He}) = 4.002603\,\text{u}$
• $M({}^{64}\text{Zn}) = 63.929142\,\text{u}$

(b) If the proton beam energy (lab) is $T_p = 10\,\text{MeV}$, what excitation energy $E^*$ does this produce in ${}^{64}$Zn$^*$?

(c) What alpha-particle beam energy (lab) is needed to produce the same $E^*$ in ${}^{64}$Zn$^*$?

(d) According to the Bohr independence hypothesis, what quantity should be the same for both reactions when $E^*$ is matched?

Problem 18.2 ⭐ A compound nucleus has a lifetime of $\tau = 5 \times 10^{-15}$ s.

(a) Calculate the total width $\Gamma$.

(b) If the nuclear radius is $R = 6.5$ fm and a typical nucleon speed inside the nucleus is $v \sim 10^7$ m/s, estimate the nuclear transit time.

(c) How many times does a nucleon "bounce" inside the compound nucleus before it decays? Does this justify the statistical picture?

Problem 18.3 ⭐ A neutron with $E = 6.67$ eV is captured by ${}^{238}$U to form ${}^{239}$U$^*$. The neutron separation energy of ${}^{239}$U is $S_n = 4.806$ MeV.

(a) Calculate the excitation energy $E^*$ of the compound nucleus ${}^{239}$U$^*$.

(b) Using the Fermi gas model with $a = 238/8 = 29.8\,\text{MeV}^{-1}$, estimate the nuclear temperature at this excitation energy.

(c) Estimate the level density $\rho(E^*)$ using the Bethe formula. Compare to the observed average level spacing $D_0 = 20.3$ eV.

The Breit-Wigner Formula

Problem 18.4 ⭐ The first resonance of n + ${}^{238}$U has parameters:

• $E_R = 6.67$ eV, $J = 1/2$, $\ell = 0$
• $\Gamma_n = 1.50$ meV, $\Gamma_\gamma = 23.0$ meV
• Target spin $I = 0$, neutron spin $i = 1/2$

(a) Calculate the total width $\Gamma$ and the lifetime of this resonance.

(b) Calculate the statistical spin factor $g_J$.

(c) Calculate the peak capture cross section $\sigma_\gamma(E_R)$ using the Breit-Wigner formula. Express your answer in barns. Use $\lambdabar_R = \hbar / \sqrt{2 m_n E_R}$.

(d) Calculate the peak elastic scattering cross section $\sigma_{\text{el}}^{(\text{res})}(E_R)$.

Problem 18.5 ⭐⭐ The Breit-Wigner formula predicts a Lorentzian line shape.

(a) Show that the full width at half maximum (FWHM) of $\sigma_{a \to b}(E)$ is equal to $\Gamma$.

(b) Show that the area under the resonance (the "resonance area") is:

$$A_R = \int_{-\infty}^{\infty} \sigma_{a \to b}(E) \, dE = \frac{2\pi^2 \lambdabar_R^2 g_J \Gamma_a \Gamma_b}{\Gamma}$$

(Hint: use the standard integral $\int_{-\infty}^{\infty} dx / (x^2 + a^2) = \pi/a$.)

(c) Explain why the resonance area is useful experimentally even when the energy resolution is not good enough to resolve the resonance shape.

Problem 18.6 ⭐⭐ Consider a nucleus with two resonances in the neutron capture cross section, at energies $E_1 = 10$ eV and $E_2 = 50$ eV, both with $J = 1/2$, $g_J = 1$, $\Gamma_\gamma = 30$ meV, $\Gamma_{n,1} = 2$ meV, and $\Gamma_{n,2} = 10$ meV. The target has $I = 0$.

(a) Calculate the peak capture cross sections for both resonances.

(b) At what energy between the two resonances does the capture cross section reach its minimum? (Assume the Breit-Wigner contributions add incoherently.)

(c) Plot or sketch $\sigma_\gamma(E)$ from 1 eV to 100 eV on a log-log scale.

Problem 18.7 ⭐⭐ Energy dependence of partial widths. For an s-wave ($\ell = 0$) neutron resonance, the neutron width varies as $\Gamma_n(E) = \Gamma_n^0 \sqrt{E/E_R}$, where $\Gamma_n^0 = \Gamma_n(E_R)$.

(a) Show that the Breit-Wigner capture cross section for $E \ll E_R$ (far below the resonance) reduces to the $1/v$ form: $\sigma_\gamma(E) \propto 1/\sqrt{E}$.

(b) The thermal ($E_0 = 0.0253$ eV) neutron capture cross section of ${}^{113}$Cd is $\sigma_\gamma = 20{,}600$ b, and the dominant contribution is from a resonance at $E_R = 0.178$ eV. Using the Breit-Wigner formula, estimate the resonance parameters $\Gamma_n$ and $\Gamma_\gamma$. (Use $J = 1$, $I = 1/2$, $i = 1/2$.)

Problem 18.8 ⭐⭐ Interference with potential scattering. The full elastic scattering cross section includes both resonance and potential scattering. For a single s-wave resonance, the elastic cross section is:

$$\sigma_{\text{el}}(E) = 4\pi\lambdabar^2 \left| \sin\delta_{\text{pot}} \, e^{i\delta_{\text{pot}}} + \frac{\Gamma_n/2}{E_R - E - i\Gamma/2}\right|^2$$

where $\delta_{\text{pot}}$ is the potential (hard-sphere) phase shift.

(a) Show that for $\delta_{\text{pot}} = 0$, this reduces to the pure Breit-Wigner elastic cross section.

(b) For $\delta_{\text{pot}} \neq 0$, show that the interference between potential and resonance scattering produces an asymmetric line shape. This is the nuclear analogue of a Fano resonance.

(c) Sketch the elastic cross section near $E_R$ for $\delta_{\text{pot}} = 0$, $\pi/4$, and $\pi/2$. What happens when $\delta_{\text{pot}} = \pi/2$?

Problem 18.9 ⭐⭐ For a p-wave ($\ell = 1$) neutron resonance, the penetrability is $P_1(E) \propto E^{3/2}$ at low energies, so $\Gamma_n(E) \propto E^{3/2}$.

(a) Show that the contribution of a p-wave resonance to the low-energy capture cross section is $\sigma_\gamma(E) \propto E^{1/2}$ rather than $1/\sqrt{E}$.

(b) Explain why s-wave resonances dominate the thermal neutron cross section.

Nuclear Level Densities

Problem 18.10 ⭐ Using $a = A/8$ MeV$^{-1}$, calculate the Fermi gas level density $\rho(E^*)$ at $E^* = S_n$ for:

(a) ${}^{57}$Fe ($S_n = 7.646$ MeV, $A_{\text{compound}} = 57$)

(b) ${}^{239}$U ($S_n = 4.806$ MeV, $A_{\text{compound}} = 239$)

(c) ${}^{209}$Pb ($S_n = 3.937$ MeV, $A_{\text{compound}} = 209$)

Compare your results to the observed level spacings $D_0 \approx 25$ keV, 20 eV, and 32 keV, respectively. (Note: the comparison requires a spin distribution correction — the observed $D_0$ is for a specific spin, while $\rho$ counts all spins.)

Problem 18.11 ⭐⭐ Nuclear temperature. In the Fermi gas model, the excitation energy and temperature are related by $E^* = aT^2$.

(a) Calculate the nuclear temperature $T$ for ${}^{239}$U at $E^* = S_n = 4.806$ MeV, using $a = 29.8$ MeV$^{-1}$.

(b) What is the thermal energy $kT$ in keV? Compare this to the neutron separation energy.

(c) The average energy of an evaporated neutron from a compound nucleus at temperature $T$ is approximately $\langle E_n \rangle \approx 2T$. What is the average energy of neutrons evaporated from ${}^{239}$U$^*$ at $E^* = S_n$?

(d) Explain why this estimate has limited validity: $E^* = S_n$ is barely above the neutron emission threshold.

Problem 18.12 ⭐⭐⭐ The level density parameter from data. The average s-wave level spacing at the neutron separation energy for ${}^{56}$Fe + n is $D_0 = 24.4$ keV. The compound nucleus is ${}^{57}$Fe with $S_n = 7.646$ MeV. The target spin is $I = 0$, and for s-wave neutrons, $J = 1/2$.

(a) Calculate the total level density $\rho(S_n)$ from $D_0$ using the relation:

$$\frac{1}{D_0} = \rho(S_n) \cdot f(J, \sigma)$$

where $f(J, \sigma) = (2J+1)/(2\sigma^2) \exp[-(J+1/2)^2/(2\sigma^2)]$ is the spin distribution factor and $\sigma^2 \approx 0.0888 a^{1/2} A^{2/3} T$ is the spin cutoff parameter. (This is iterative — start with $a \approx A/8$ and refine.)

(b) Solve the Bethe formula for $a$ given $\rho(S_n)$. You will need to solve numerically. What value of $a$ do you obtain?

(c) Compare your extracted $a$ to $A/8$. Is the agreement good? Why might it differ?

Problem 18.13 ⭐⭐⭐ Exponential growth. The level density of ${}^{239}$U grows exponentially with excitation energy.

(a) Using $a = 29.8$ MeV$^{-1}$, calculate $\rho(E^*)$ at $E^* = 1, 2, 3, 4, 5, 6, 7, 8$ MeV.

(b) Plot $\ln[\rho(E^*)]$ vs $\sqrt{E^*}$. Verify that the plot is approximately linear (as predicted by the Bethe formula, $\ln\rho \approx 2\sqrt{aE^*}$ for large $E^*$).

(c) At $E^* = 8$ MeV, the level density is roughly $10^5$ per MeV. At this density, what is the average spacing between levels? Is it feasible to resolve individual levels at this excitation energy?

Hauser-Feshbach and Statistical Model

Problem 18.14 ⭐⭐ In the Hauser-Feshbach model, the compound nucleus decay probability into channel $c$ is proportional to $T_c$.

(a) A compound nucleus can decay by neutron emission ($T_n = 0.15$), proton emission ($T_p = 0.03$), alpha emission ($T_\alpha = 0.005$), and gamma emission ($T_\gamma = 0.001$). Calculate the branching ratio for each channel.

(b) Explain qualitatively why $T_n > T_p > T_\alpha$ for most compound nuclei. (Hint: consider the barriers each particle must tunnel through.)

(c) Despite $T_\gamma \ll T_n$, the capture cross section can be significant at low energies. Why?

Problem 18.15 ⭐⭐⭐ The Hauser-Feshbach formula for neutron capture at energies above the resolved resonance region involves the neutron transmission coefficient $T_n^{(\ell)}$ and the gamma-ray transmission coefficient $T_\gamma$.

(a) Show that the capture cross section for s-wave neutrons in the statistical model is approximately:

$$\sigma_\gamma \approx \pi\lambdabar^2 T_n^{(0)} \frac{T_\gamma}{\sum_c T_c}$$

when only one spin $J$ contributes.

(b) For a nucleus where $T_n \gg T_\gamma$ (above the neutron emission threshold), show that $\sigma_\gamma \approx \pi\lambdabar^2 T_\gamma / \overline{D}$. (Use $T_n = 2\pi \overline{\Gamma_n} / D$.)

(c) Explain physically why the capture cross section in this regime is determined by $T_\gamma$ (the "bottleneck") rather than $T_n$.

Neutron Capture and Applications

Problem 18.16 ⭐ The thermal neutron capture cross sections of several isotopes are:

(a) Assuming the $1/v$ law, calculate $\sigma_\gamma$ at $E = 1$ eV for each isotope.

(b) Which of these isotopes most closely follows the $1/v$ law over the range 0.01–1 eV? Which deviate most? Explain.

(c) For ${}^{113}$Cd, the dominant resonance is at $E_R = 0.178$ eV. Explain why the $1/v$ law breaks down for cadmium above $\sim 0.1$ eV.

Problem 18.17 ⭐⭐ The resonance integral. Calculate the contribution of a single Breit-Wigner resonance to the resonance integral:

$$I_R = \int_0^\infty \sigma_\gamma(E) \frac{dE}{E}$$

(a) Show that for a narrow, isolated resonance with $\Gamma \ll E_R$:

$$I_R \approx \frac{\pi}{2} \frac{\sigma_\gamma(E_R) \Gamma}{E_R} = \frac{2\pi^2 \lambdabar_R^2 g_J \Gamma_n \Gamma_\gamma}{E_R \Gamma}$$

(Hint: since the resonance is narrow, replace $E$ in the $1/E$ factor with $E_R$.)

(b) Calculate $I_R$ for the 6.67 eV resonance of ${}^{238}$U (parameters from Problem 18.4).

(c) The total measured resonance integral for ${}^{238}$U is $I_\gamma = 275$ b. What fraction comes from the 6.67 eV resonance alone?

Problem 18.18 ⭐⭐⭐ Doppler broadening. At temperature $T$, the thermal motion of target atoms broadens the effective resonance shape. The Doppler-broadened cross section is:

$$\sigma_D(E) = \frac{\sigma_0 \Gamma}{2\Delta} \sqrt{\pi} \, \psi(\xi, x)$$

where $\Delta = \sqrt{4 E_R k_B T / A}$ is the Doppler width, $\xi = \Gamma / (2\Delta)$, $x = 2(E - E_R) / \Gamma$, and $\psi$ is the Voigt function.

(a) For the 6.67 eV resonance of ${}^{238}$U at $T = 300$ K, calculate the Doppler width $\Delta$ and compare it to $\Gamma/2$.

(b) At $T = 1000$ K (reactor operating temperature), recalculate $\Delta$. Is Doppler broadening significant?

(c) Explain qualitatively why Doppler broadening increases the effective resonance integral (and hence the resonance absorption), even though the total area under the cross section curve is conserved. (Hint: consider the $1/E$ weighting in the resonance integral and the effect on the transmission through a thick sample.)

Astrophysical Applications

Problem 18.19 ⭐⭐ The Maxwellian-averaged cross section (MACS) for ${}^{56}$Fe(n,$\gamma$)${}^{57}$Fe at $kT = 30$ keV is $\langle\sigma\rangle_{30} = 11.7$ mb.

(a) If the neutron density in an AGB star is $n_n = 3 \times 10^8$ cm$^{-3}$, calculate the capture rate per ${}^{56}$Fe atom.

(b) What is the mean time between captures? Express in years.

(c) The half-life of ${}^{57}$Fe for beta decay is effectively infinite (it is stable). Is the $s$-process condition ($\tau_\beta \ll \tau_{\text{capture}}$) or the $r$-process condition ($\tau_\beta \gg \tau_{\text{capture}}$) relevant here? What is the physical significance?

Problem 18.20 ⭐⭐ In the $s$-process, the abundance of nucleus $i$ on the $s$-process path satisfies $\sigma_i N_i \approx$ const in the local approximation.

(a) The MACS values at $kT = 30$ keV for three isotopes near the $N = 82$ shell closure are:

Using the local approximation, predict the ratios $N({}^{138}\text{Ba})/N({}^{136}\text{Ba})$ and $N({}^{139}\text{La})/N({}^{138}\text{Ba})$.

(b) Explain why ${}^{138}$Ba (magic $N = 82$) has the smallest cross section and hence the highest $s$-process abundance in this region.

(c) This is direct observational evidence for nuclear shell structure. Discuss.

Problem 18.21 ⭐⭐⭐ The $s$-process path. Starting from ${}^{56}$Fe (seed nucleus), the $s$-process builds heavier elements by successive neutron captures and beta decays.

(a) Trace the $s$-process path from ${}^{56}$Fe through ${}^{57}$Fe, ${}^{58}$Fe, ${}^{59}$Fe ($\beta^-$, $t_{1/2} = 44.5$ d), ${}^{59}$Co, ${}^{60}$Co ($\beta^-$, $t_{1/2} = 5.27$ yr), ${}^{60}$Ni, ... . At each step, decide whether neutron capture or beta decay occurs first (assume $\tau_{\text{cap}} \sim 10$ yr for typical $s$-process conditions).

(b) At ${}^{63}$Ni ($t_{1/2} = 100$ yr), the $s$-process path branches because $\tau_\beta \sim \tau_{\text{cap}}$. Explain why branching nuclei are sensitive probes of neutron density and temperature.

Experimental Methods

Problem 18.22 ⭐ A time-of-flight facility has a flight path of $L = 185$ m.

(a) Calculate the time of flight for a 1 eV neutron, a 100 eV neutron, and a 10 keV neutron.

(b) If the timing resolution is $\Delta t = 1\,\mu$s, what is the energy resolution $\Delta E / E$ at each of these energies?

(c) At what energy does $\Delta E / E = 1\%$? This sets the practical upper limit for resolving individual resonances at this flight path.

Problem 18.23 ⭐⭐ A transmission experiment measures $T(E) = \exp(-n\sigma_{\text{tot}}\ell)$ through a ${}^{238}$U sample.

(a) Calculate the sample thickness $\ell$ (in mm) for which $T = 0.5$ at the 6.67 eV resonance peak ($\sigma_{\text{tot}} \approx 23{,}000$ b). The atom density of metallic uranium is $n = 4.83 \times 10^{22}$ atoms/cm$^3$.

(b) At this thickness, what is the transmission at 1 eV (below the resonance), where $\sigma_{\text{tot}} \approx 15$ b?

(c) Explain the concept of "black resonance" and its implications for extracting resonance parameters from thick-sample data.

Synthesis and Advanced Problems

Problem 18.24 ⭐⭐⭐ Testing the Bohr independence hypothesis. Two reactions form the same compound nucleus ${}^{28}$Si$^*$ at excitation energy $E^* = 25$ MeV:

$$\alpha + {}^{24}\text{Mg} \to {}^{28}\text{Si}^* \qquad \text{and} \qquad p + {}^{27}\text{Al} \to {}^{28}\text{Si}^*$$

The measured cross sections for the exit channels are:

(a) Test the independence hypothesis: for each pair of exit channels, compute the ratio of cross sections from the two entrance channels. If the hypothesis holds, all ratios should be equal to $\sigma_{\text{form}}(\alpha) / \sigma_{\text{form}}(p)$.

(b) Are the ratios consistent? Compute the formation cross section ratio.

(c) If the ratios are not exactly equal, what physical mechanism might cause deviations from the independence hypothesis?

Problem 18.25 ⭐⭐⭐ Research problem: Nuclear data for astrophysics. The $r$-process produces approximately half the elements heavier than iron, but many of the nuclei on the $r$-process path have never been studied experimentally.

(a) Using the Fermi gas model with $a = A/8$ MeV$^{-1}$, estimate the level density at the neutron separation energy for ${}^{132}$Sn + n $\to$ ${}^{133}$Sn (magic $N = 82$, $Z = 50$) and for ${}^{130}$Cd + n $\to$ ${}^{131}$Cd ($Z = 48$, $N = 83$). Use $S_n \approx 2.4$ MeV and 3.8 MeV, respectively.

(b) Estimate the s-wave level spacing $D_0$ for each. Which has the larger spacing and why?

(c) Discuss how the uncertainties in these level densities affect $r$-process abundance predictions. Why is FRIB important for this problem?