Exercises — Chapter 13: Alpha Decay

Section A: Warm-Up and Conceptual Questions

Exercise 13.1 — Conceptual: The Classical Paradox

(a) Explain in 2–3 sentences why classical mechanics cannot account for alpha decay. What specific quantity is the alpha particle's kinetic energy less than?

(b) An alpha particle with $T_\alpha = 5\,\text{MeV}$ is emitted from a nucleus with Coulomb barrier height $V_C(R) = 30\,\text{MeV}$. In what sense does the alpha particle "go through" the barrier? What determines the probability of this occurring?

(c) Why does nature preferentially emit alpha particles rather than individual protons or neutrons from heavy nuclei? (Hint: consider the binding energy of ${}^4\text{He}$.)

Exercise 13.2 — Q-Value Calculations

Using the atomic masses below (AME2020 values), calculate the Q-value for each alpha decay and determine whether it is energetically allowed.

(a) ${}^{210}\text{Po} \to {}^{206}\text{Pb} + \alpha$

(b) ${}^{238}\text{U} \to {}^{234}\text{Th} + \alpha$

(c) ${}^{144}\text{Nd} \to {}^{140}\text{Ce} + \alpha$

(d) ${}^{208}\text{Pb} \to {}^{204}\text{Hg} + \alpha$

For each decay with $Q > 0$, also compute the kinetic energy of the emitted alpha particle and the recoil energy of the daughter.

Exercise 13.3 — Energy Partitioning

(a) Show that for alpha decay of a parent at rest, the fraction of the Q-value carried by the alpha particle is $T_\alpha/Q = M_d/(M_d + M_\alpha) \approx (A-4)/A$.

(b) For ${}^{148}\text{Gd} \to {}^{144}\text{Sm} + \alpha$ ($Q = 3.271\,\text{MeV}$), calculate $T_\alpha$ and $T_d$.

(c) For what mass number $A$ does the recoil energy of the daughter first exceed 0.1 MeV, assuming $Q = 5\,\text{MeV}$?

Exercise 13.3b — Why Alpha Particles?

(a) Compute the Q-value for hypothetical emission of a single proton from ${}^{210}\text{Po}$: ${}^{210}\text{Po} \to {}^{209}\text{Bi} + p$. Use $M({}^{209}\text{Bi}) = 208.980399\,\text{u}$, $M({}^{1}\text{H}) = 1.007825\,\text{u}$, $M({}^{210}\text{Po}) = 209.982874\,\text{u}$.

(b) Compute the Q-value for hypothetical deuteron emission: ${}^{210}\text{Po} \to {}^{208}\text{Pb} + d$. Use $M(d) = 2.014102\,\text{u}$, $M({}^{208}\text{Pb}) = 207.976652\,\text{u}$.

(c) Compare these Q-values to $Q_\alpha = 5.407\,\text{MeV}$ for alpha emission. Why is alpha emission so strongly favored? (Consider the binding energy per nucleon of the emitted particle.)

(d) Despite having $Q > 0$ for alpha emission, ${}^{208}\text{Pb}$ does not observably alpha-decay ($Q_\alpha = 0.517\,\text{MeV}$). Estimate the Gamow factor and explain why.

Section B: Coulomb Barrier and Tunneling

Exercise 13.4 — Barrier Parameters

For the alpha decay ${}^{226}\text{Ra} \to {}^{222}\text{Rn} + \alpha$ ($Q = 4.871\,\text{MeV}$):

(a) Calculate the alpha kinetic energy $T_\alpha$.

(b) Calculate the nuclear radius $R = r_0(A_d^{1/3} + 4^{1/3})$ with $r_0 = 1.2\,\text{fm}$.

(c) Calculate the Coulomb barrier height at the nuclear surface: $V_C(R) = 2Z_d e^2/(4\pi\epsilon_0 R)$. Use $e^2/(4\pi\epsilon_0) = 1.440\,\text{MeV}\cdot\text{fm}$.

(d) Calculate the outer classical turning point $b = 2Z_d e^2/(4\pi\epsilon_0 E_\alpha)$.

(e) What is the ratio $V_C(R)/E_\alpha$? By what factor does the barrier height exceed the alpha energy?

(f) Sketch the potential $V(r)$ for this decay, labeling $R$, $b$, $V_C(R)$, $E_\alpha$, and the classically forbidden tunneling region.

Exercise 13.5 — Sommerfeld Parameter

The Sommerfeld parameter is $\eta = z_\alpha Z_d e^2/(4\pi\epsilon_0\hbar v)$ where $v = \sqrt{2E_\alpha/\mu}$.

(a) Show that $\eta$ can be written as $\eta = z_\alpha Z_d \alpha_{\text{em}} \sqrt{\mu c^2/(2E_\alpha)}$ where $\alpha_{\text{em}} = e^2/(4\pi\epsilon_0\hbar c) \approx 1/137.036$ is the fine-structure constant.

(b) Calculate $\eta$ for ${}^{212}\text{Po} \to {}^{208}\text{Pb} + \alpha$ ($E_\alpha = 8.784\,\text{MeV}$).

(c) Calculate $\eta$ for ${}^{232}\text{Th} \to {}^{228}\text{Ra} + \alpha$ ($E_\alpha = 4.012\,\text{MeV}$).

(d) Why is $\eta$ larger for thorium than for polonium? What does this imply about the tunneling probability?

Exercise 13.6 — Gamow Factor Calculation

(a) For ${}^{210}\text{Po} \to {}^{206}\text{Pb} + \alpha$ ($E_\alpha = 5.304\,\text{MeV}$), calculate: - The reduced mass $\mu$ - The nuclear radius $R$ - The outer turning point $b$ - The ratio $\rho = R/b$ - The Sommerfeld parameter $\eta$ - The Gamow factor $G = 2\eta[\arccos\sqrt{\rho} - \sqrt{\rho(1-\rho)}]$ - The tunneling probability $P = e^{-G}$

(b) Repeat for ${}^{232}\text{Th} \to {}^{228}\text{Ra} + \alpha$ ($E_\alpha = 4.012\,\text{MeV}$).

(c) Compute the ratio $P(\text{Po})/P(\text{Th})$. The measured half-life ratio is $t_{1/2}(\text{Th})/t_{1/2}(\text{Po}) \approx 3.2\times10^{15}$. Does the tunneling probability ratio account for most of this?

Exercise 13.7 — The Thick-Barrier Approximation

(a) Starting from the exact Gamow factor $G = 2\eta[\arccos\sqrt{\rho} - \sqrt{\rho(1-\rho)}]$, expand for $\rho \ll 1$ to derive the thick-barrier approximation:

$$G \approx \pi\eta - 4\eta\sqrt{\rho}$$

(b) For ${}^{226}\text{Ra}$ alpha decay (using your results from Exercise 13.4), compute $G$ using both the exact formula and the thick-barrier approximation. What is the percentage error?

(c) At what value of $\rho$ does the thick-barrier approximation introduce a 10% error in $G$?

Section C: The Geiger-Nuttall Law

Exercise 13.8 — Verifying the Geiger-Nuttall Law

The table below lists even-even alpha emitters with $Z_d$ near 86 (radon neighbors):

(a) Convert all half-lives to seconds and compute $\log_{10}\lambda$ (where $\lambda = \ln 2/t_{1/2}$) for each entry.

(b) Plot $\log_{10}\lambda$ versus $Z_d/\sqrt{E_\alpha}$ for the polonium isotopes ($Z_d = 82$). Verify the approximate linearity. Determine the slope and intercept by a least-squares fit.

(c) Do the radon and radium points fall on the same line as polonium, or is there an element-dependent offset? (The Geiger-Nuttall law is strictly linear within a single element.)

Exercise 13.9 — Sensitivity Analysis

(a) For a generic alpha emitter with $Z_d = 86$ and $E_\alpha = 5\,\text{MeV}$, estimate the Gamow factor $G$.

(b) By what factor does $P = e^{-G}$ change if $E_\alpha$ increases by 0.5 MeV (from 5.0 to 5.5 MeV)?

(c) By what factor does $P$ change if $E_\alpha$ decreases by 0.5 MeV (from 5.0 to 4.5 MeV)?

(d) Use these results to explain, quantitatively, why a 10% change in alpha energy produces a change of several orders of magnitude in the half-life.

Section D: Fine Structure and Nuclear Structure

Exercise 13.10 — Fine Structure of ${}^{241}\text{Am}$

${}^{241}\text{Am}$ has $I^\pi = 5/2^-$ and decays to states in ${}^{237}\text{Np}$ with the following properties:

(a) Verify the alpha energies using $E_\alpha = (Q - E_x)(A_d)/(A_d + 4)$ with $Q = 5.638\,\text{MeV}$.

(b) Determine the allowed values of angular momentum transfer $\ell$ for each transition, using the selection rules $|\Delta I| \leq \ell \leq I_p + I_d$ and $\pi_p = \pi_d(-1)^\ell$. Which transitions are "favored" ($\ell = 0$)?

(c) For the transition to the ground state (parity change required), what is the minimum $\ell$? What does this imply about the angular momentum barrier?

(d) Calculate the ratio of Gamow factors $G(E_\alpha = 5.345)/G(E_\alpha = 5.486)$. How much of the intensity reduction for the 158.5 keV state can be explained by the increased Gamow factor alone?

Exercise 13.11 — Hindrance Factors

The even-even nucleus ${}^{234}\text{U}$ ($I^\pi = 0^+$) alpha-decays to states in ${}^{230}\text{Th}$:

The Q-value is $4.858\,\text{MeV}$.

(a) Calculate the alpha energies for each transition.

(b) For each transition, calculate the Gamow factor. (You may use the thick-barrier approximation.)

(c) Using the tunneling probability ratio alone, predict the intensity ratio $I(2^+)/I(0^+)$ and $I(4^+)/I(0^+)$.

(d) Compare your predictions with the measured values. What accounts for the discrepancy? (Hint: consider the centrifugal barrier for $\ell > 0$.)

Section E: Proton and Cluster Radioactivity

Exercise 13.12 — Proton Radioactivity

For the proton decay ${}^{151}\text{Lu} \to {}^{150}\text{Yb} + p$ ($E_p = 1.233\,\text{MeV}$, $\ell = 5$):

(a) Calculate the Coulomb barrier height $V_C(R)$ at the nuclear surface, using $R = r_0(A_d^{1/3} + 1)$ with $r_0 = 1.2\,\text{fm}$.

(b) Calculate the outer turning point $b = Z_d e^2/(4\pi\epsilon_0 E_p)$.

(c) Calculate the Gamow factor for the Coulomb barrier alone (ignoring the centrifugal term). Compare to a typical alpha decay Gamow factor.

(d) Qualitatively, how does the centrifugal barrier for $\ell = 5$ affect the tunneling probability? (You do not need to evaluate the centrifugal integral.)

Exercise 13.13 — Cluster Radioactivity: ${}^{14}\text{C}$ Emission from ${}^{223}\text{Ra}$

For the cluster decay ${}^{223}\text{Ra} \to {}^{209}\text{Pb} + {}^{14}\text{C}$: - $Q = 31.83\,\text{MeV}$ - $T_{14\text{C}} = Q \cdot M_d/(M_d + M_{14\text{C}}) = 31.83 \times 209/223 = 29.83\,\text{MeV}$

(a) Calculate the Coulomb barrier height $V_C(R)$ using $R = r_0(A_d^{1/3} + A_c^{1/3})$ and the charge product $z_c Z_d = 6 \times 82$.

(b) Calculate the outer turning point $b$.

(c) Calculate the Gamow factor. (Use the thick-barrier approximation.)

(d) Compare the Gamow factor for ${}^{14}\text{C}$ emission to that for alpha emission from ${}^{223}\text{Ra}$ ($Q_\alpha = 5.979\,\text{MeV}$, $T_\alpha = 5.871\,\text{MeV}$). Roughly how many orders of magnitude smaller is the cluster decay tunneling probability?

(e) If the preformation factor for ${}^{14}\text{C}$ inside ${}^{223}\text{Ra}$ is approximately $10^{-8}$, estimate the branching ratio for ${}^{14}\text{C}$ emission relative to alpha emission. Compare to the measured value of $\sim 10^{-10}$.

Exercise 13.13b — Two-Proton Radioactivity

${}^{45}\text{Fe}$ ($Z = 26$) is the first confirmed two-proton emitter. It decays via ${}^{45}\text{Fe} \to {}^{43}\text{Cr} + 2p$ with $Q_{2p} = 1.154\,\text{MeV}$ and $t_{1/2} = 1.75\,\text{ms}$.

(a) Why can't ${}^{45}\text{Fe}$ decay by single-proton emission? (Hint: $Q_p({}^{45}\text{Fe}) = -0.072\,\text{MeV}$. What does the negative sign mean?)

(b) If the two protons are emitted as a correlated "${}^2\text{He}$" pair (diproton emission), what is the effective charge product for the Coulomb barrier? How does this compare to alpha decay?

(c) The measured half-life of 1.75 ms is orders of magnitude shorter than what a simple barrier penetration model predicts for two independent protons. What does this tell you about the correlation between the two protons?

(d) Why is two-proton radioactivity only observed in nuclei near or beyond the proton drip line?

Section F: Synthesis and Challenge Problems

Exercise 13.14 — Predicting Unknown Half-Lives ★

The isotope ${}^{296}\text{Og}$ (oganesson-296) has not been synthesized, but theoretical models predict $Q_\alpha \approx 11.7\,\text{MeV}$.

(a) Calculate the expected alpha kinetic energy.

(b) Calculate the Gamow factor, using $Z_d = 116$, $R = 1.2(292^{1/3}+4^{1/3})\,\text{fm}$.

(c) Estimate the half-life using $f = 10^{21}\,\text{s}^{-1}$ and $S = 0.1$.

(d) Is this half-life long enough to be experimentally detectable with current techniques? (Current superheavy element experiments can detect nuclei with half-lives as short as $\sim 10^{-6}\,\text{s}$.)

Exercise 13.15 — The Geiger-Nuttall Law as a Discovery Tool ★★

An unknown alpha emitter is observed to have $T_\alpha = 6.50\,\text{MeV}$ and $t_{1/2} = 4.2\,\text{s}$.

(a) Assuming the emitter is an even-even nucleus, use the Geiger-Nuttall relationship to estimate $Z_d$. (Use the Viola-Seaborg parametrization: $\log_{10}t_{1/2}(\text{s}) = (aZ_d+b)/\sqrt{E_\alpha} + cZ_d + d$ with $a = 1.66175$, $b = -8.5166$, $c = -0.20228$, $d = -33.9069$.)

(b) What parent nuclide ($Z$, $A$) is consistent with your estimate? (You will need to check which isotopes of this element have $Q_\alpha$ values consistent with $T_\alpha = 6.50\,\text{MeV}$.)

Exercise 13.16 — Connecting Alpha Decay to Stellar Nucleosynthesis ★★

${}^{212}\text{Po}$ is formed in the thorium decay series: ${}^{232}\text{Th} \to \cdots \to {}^{212}\text{Bi} \to {}^{212}\text{Po} \to {}^{208}\text{Pb}$.

(a) The half-life of ${}^{212}\text{Po}$ is $0.299\,\mu\text{s}$ and its alpha energy is $8.784\,\text{MeV}$. Calculate the Gamow factor and verify that the tunneling model gives a half-life within an order of magnitude.

(b) ${}^{232}\text{Th}$ is synthesized in the r-process of stellar nucleosynthesis (Chapter 23). If the age of the solar system is $4.57 \times 10^9\,\text{yr}$ and the half-life of ${}^{232}\text{Th}$ is $1.40 \times 10^{10}\,\text{yr}$, what fraction of the primordial ${}^{232}\text{Th}$ remains today?

(c) Why is the fact that ${}^{232}\text{Th}$ has a half-life comparable to the age of the Earth essential for the Th-Pb geochronological dating method?

Exercise 13.17 — Derivation: The Assault Frequency ★

(a) Model the alpha particle as a particle of mass $m_\alpha$ bouncing back and forth inside a one-dimensional box of width $2R$. If its kinetic energy inside the well is $E_{\text{in}} \approx Q + V_0$ (where $V_0$ is the well depth), derive the assault frequency $f = v/(2R)$.

(b) For $V_0 = 40\,\text{MeV}$, $Q = 5\,\text{MeV}$, $R = 7.5\,\text{fm}$, compute $f$ in s$^{-1}$.

(c) The assault frequency varies by at most a factor of $\sim 3$ across all alpha emitters. Why does this make $f$ relatively unimportant compared to $P$ in determining the half-life? Estimate by how much $G$ changes when $f$ changes by a factor of 3 (in terms of equivalent $\Delta G$).

Exercise 13.18 — Barrier Shape and Tunneling ★★

Consider a simplified model in which the Coulomb barrier is replaced by a rectangular barrier of height $V_0$ and width $L$.

(a) The exact quantum-mechanical transmission coefficient for a rectangular barrier is:

$$T = \left[1 + \frac{V_0^2}{4E(V_0-E)}\sinh^2(\kappa L)\right]^{-1}$$

where $\kappa = \sqrt{2m(V_0-E)/\hbar^2}$. Show that for $\kappa L \gg 1$, this reduces to the WKB result $T \approx 16\frac{E(V_0-E)}{V_0^2}\exp(-2\kappa L)$.

(b) For an alpha particle ($m = 3727\,\text{MeV}/c^2$) with $E = 5\,\text{MeV}$ facing a rectangular barrier of height $V_0 = 25\,\text{MeV}$ and width $L = 50\,\text{fm}$, compute $\kappa L$, $T_{\text{exact}}$, and $T_{\text{WKB}}$. How good is the WKB approximation?

(c) Now compute the WKB tunneling probability for the actual Coulomb barrier ($1/r$ shape) with the same outer parameters ($V(R) = 25\,\text{MeV}$, width from $R = 9\,\text{fm}$ to $b = 59\,\text{fm}$). Is the tunneling probability through the Coulomb barrier larger or smaller than through the rectangular barrier? Explain physically.

Exercise 13.19 — Systematic Trends in Q-Values ★

Using the following measured Q-values for the uranium isotopes:

(a) Plot $Q_\alpha$ versus neutron number $N$ for these isotopes. Describe the trend.

(b) Using the SEMF, explain why $Q_\alpha$ decreases as $N$ increases for a fixed $Z$. Which SEMF term is most responsible?

(c) Estimate the Gamow factor for each isotope and plot $\log_{10}(t_{1/2})$ versus $N$. The measured half-lives range from 9.1 minutes (${}^{228}\text{U}$) to $4.47 \times 10^9$ years (${}^{238}\text{U}$). Does the tunneling model qualitatively reproduce this trend?

Exercise 13.20 — Numerical Project: Full WKB Integration ★★★

Extend the alpha_tunneling.py code to include the centrifugal barrier:

$$V_\ell(r) = \frac{z_\alpha Z_d e^2}{4\pi\epsilon_0 r} + \frac{\ell(\ell+1)\hbar^2}{2\mu r^2}$$

(a) Write a function that computes the WKB integral numerically for arbitrary $\ell$, using scipy.integrate.quad. Note that the inner and outer turning points both shift when $\ell > 0$; find them numerically.

(b) For ${}^{234}\text{U}$ alpha decay to the $2^+$ ($\ell=2$) and $4^+$ ($\ell=4$) states of ${}^{230}\text{Th}$, compute the ratio $P(\ell)/P(\ell=0)$.

(c) Compare your numerical centrifugal barrier suppression factors with the measured relative intensities from Exercise 13.11. Discuss any discrepancies.

(d) Use your code to predict the ratio of intensities for the $0^+$, $2^+$, $4^+$, $6^+$, and $8^+$ members of the ground-state rotational band in ${}^{230}\text{Th}$. At what $\ell$ value does the centrifugal barrier suppress the transition by more than a factor of $10^4$?

Exercise 13.21 — The Inverse Problem: Scattering Off a Nucleus ★★

Gamow immediately recognized that the tunneling probability applies in both directions. If an alpha particle with $E < V_C(R)$ is fired at a nucleus, it has a nonzero probability of tunneling into the nucleus and initiating a nuclear reaction.

(a) For a 5 MeV alpha particle incident on ${}^{197}\text{Au}$, calculate the tunneling probability $P$ through the Coulomb barrier. Use $R = 1.2(197^{1/3} + 4^{1/3})\,\text{fm}$.

(b) The cross section for the reaction is approximately $\sigma \approx \pi R^2 P$ (this is a rough estimate). Compute this cross section in barns ($1\,\text{b} = 100\,\text{fm}^2$).

(c) In stellar interiors, thermonuclear reactions occur at thermal energies of $\sim 1$--$10\,\text{keV}$. What is the tunneling probability for a proton with $E = 10\,\text{keV}$ incident on another proton? (Use the Coulomb barrier for two protons: $V = e^2/(4\pi\epsilon_0 r)$, $R \approx 1.5\,\text{fm}$.) This gives you a sense of why stellar nucleosynthesis requires the enormous densities and long timescales found in stellar cores.

(d) The Gamow peak energy for the p-p reaction in the Sun ($T = 1.5 \times 10^7\,\text{K}$) is approximately $E_G = (bkT/2)^{2/3}$ where $b = \pi\eta\sqrt{2\mu E}/E$ is the barrier penetrability parameter. Estimate $E_G$ and explain why solar fusion occurs preferentially at an energy much higher than $kT$ but still far below $V_C$.

Exercise 13.22 — Dimensional Analysis of the Gamow Factor ★

(a) Show that the Gamow factor $G$ is dimensionless by tracking units through the WKB integral $G = (2/\hbar)\int\sqrt{2\mu(V-E)}\,dr$.

(b) Using dimensional analysis, argue that $G$ must be proportional to $z_\alpha Z_d \sqrt{\mu/E}$, up to dimensionless factors. What combination of fundamental constants appears?

(c) The fact that $G$ involves $e^2/(4\pi\epsilon_0\hbar c) = \alpha_{\text{em}} \approx 1/137$ explains why the Gamow factor is neither astronomically large nor negligibly small for typical nuclear parameters. Estimate $G$ for $Z_d \sim 80$, $E_\alpha \sim 5\,\text{MeV}$ using only $\alpha_{\text{em}}$, $Z_d$, and $\sqrt{\mu c^2/E}$, and verify that you get a number in the range 20--50.

Answers to Selected Exercises

13.2(a): $Q = (209.982874 - 205.974465 - 4.002603) \times 931.494 = 0.005806 \times 931.494 = 5.407\,\text{MeV}$. $T_\alpha = 5.407 \times 206/210 = 5.304\,\text{MeV}$, $T_d = 0.103\,\text{MeV}$.

13.2(d): $Q = (207.976652 - 203.973494 - 4.002603) \times 931.494 = 0.000555 \times 931.494 = 0.517\,\text{MeV}$. This is positive, so the decay is energetically allowed, but the extremely low Q-value gives a Gamow factor so large that the half-life exceeds $10^{40}\,\text{yr}$ — this decay is not observable.

13.4(b): $R = 1.2(222^{1/3} + 4^{1/3}) = 1.2(6.055 + 1.587) = 9.17\,\text{fm}$.

13.4(c): $V_C(R) = 2 \times 86 \times 1.440/9.17 = 247.7/9.17 = 27.0\,\text{MeV}$.

13.5(b): $\mu = 4 \times 208/(4+208) \times 931.5 = 3.910 \times 931.5 = 3,642\,\text{MeV}/c^2$. $v = \sqrt{2 \times 8.784/3642} \cdot c = 0.0695c$. $\eta = 2 \times 82 \times 1.440/(197.3 \times 0.0695) = 236.2/13.71 = 17.23$.

13.6(a): For ${}^{210}\text{Po}$: $\mu = 3,636\,\text{MeV}/c^2$, $R = 8.79\,\text{fm}$, $b = 44.6\,\text{fm}$, $\rho = 0.197$, $\eta = 22.69$, $G = 2 \times 22.69 \times [1.107 - 0.397] = 32.22$, $P = e^{-32.22} = 9.9 \times 10^{-15}$.

13.9(a): $G \approx \pi \times 28.3 - 4 \times 28.3\sqrt{0.15} \approx 88.9 - 43.8 = 45.1$ (using the thick-barrier form with $\eta \approx 28.3$ for $Z_d = 86$, $E_\alpha = 5\,\text{MeV}$).

13.8(a): For ${}^{212}\text{Po}$: $t_{1/2} = 2.99\times10^{-7}\,\text{s}$, $\lambda = \ln 2/t_{1/2} = 2.32\times10^6\,\text{s}^{-1}$, $\log_{10}\lambda = 6.36$. For ${}^{218}\text{Po}$: $t_{1/2} = 186\,\text{s}$, $\lambda = 3.73\times10^{-3}\,\text{s}^{-1}$, $\log_{10}\lambda = -2.43$. For ${}^{226}\text{Ra}$: $t_{1/2} = 1600\,\text{yr} = 5.05\times10^{10}\,\text{s}$, $\lambda = 1.37\times10^{-11}\,\text{s}^{-1}$, $\log_{10}\lambda = -10.86$.

13.10(b): For the ground state ($5/2^+$): parent is $5/2^-$, $\Delta I = 0$, parity change requires $\ell$ odd, minimum $\ell = 1$. For the 59.5 keV state ($5/2^-$): $\Delta I = 0$, no parity change, $\ell$ even, minimum $\ell = 0$ — this is the favored transition! The ground-state transition is actually hindered by the $\ell = 1$ barrier despite being to the lowest-energy state.

13.12(a): $R = 1.2(150^{1/3} + 1) = 1.2(5.313 + 1) = 7.58\,\text{fm}$. $V_C = 70 \times 1.440/7.58 = 100.8/7.58 = 13.3\,\text{MeV}$. The barrier height is about 11 times the proton energy.

13.12(c): For proton emission, $z = 1$ so the Gamow factor is roughly half the alpha decay value for similar $Z_d$. Numerically, $G \approx 15$--$20$, much smaller than typical alpha Gamow factors of 30--50. The key difference is that the centrifugal barrier for $\ell = 5$ adds significantly to the effective barrier.

13.13(a): $R = 1.2(209^{1/3} + 14^{1/3}) = 1.2(5.935 + 2.410) = 10.01\,\text{fm}$. $V_C = 6 \times 82 \times 1.440/10.01 = 708.5/10.01 = 70.8\,\text{MeV}$. The barrier is much higher than for alpha decay because the charge product $z_c Z_d = 492$ is much larger.

13.16(b): $N(t)/N_0 = 2^{-t/t_{1/2}} = 2^{-4.57/14.0} = 2^{-0.326} = 0.798$. About 80% of the primordial ${}^{232}\text{Th}$ remains.

13.17(b): $v = \sqrt{2 \times 45\,\text{MeV}/(3727\,\text{MeV}/c^2)} \cdot c = 0.155c = 4.66 \times 10^{22}\,\text{fm/s}$. $f = v/(2R) = 4.66 \times 10^{22}/(2 \times 7.5) = 3.1 \times 10^{21}\,\text{s}^{-1}$.

13.17(c): If $f$ changes by a factor of 3, then $\Delta\ln\lambda = \ln 3 = 1.10$. To achieve the same $\Delta\ln\lambda$ by changing $G$: $\Delta G = 1.10$. Since typical $G$ values are 30--50, a change of 1.1 in $G$ is a $\sim 3\%$ perturbation — far smaller than the uncertainty from the preformation factor. This confirms that $P = e^{-G}$ is the overwhelmingly dominant factor.

13.18(b): $\kappa = \sqrt{2 \times 3727 \times (25-5)}/197.3 = \sqrt{149080}/197.3 = 386.1/197.3 = 1.957\,\text{fm}^{-1}$. $\kappa L = 1.957 \times 50 = 97.8$. $T_{\text{WKB}} \approx 16 \times \frac{5 \times 20}{625} \times e^{-195.7} \approx 2.56 \times e^{-195.7}$. The tunneling probability is astronomically small ($\sim 10^{-85}$).

13.21(c): For two protons at $E = 10\,\text{keV}$: $V_C(R) = 1.440/1.5 = 0.96\,\text{MeV}$, $b = 1.440/0.010 = 144\,\text{fm}$, $\rho = 1.5/144 = 0.0104$. $\eta = 1 \times 1.440/(197.3 \times \sqrt{2 \times 0.010/469}) = 1.440/(197.3 \times 6.53 \times 10^{-3}) = 1.118$. Wait — we need $v/c = \sqrt{2E/\mu c^2} = \sqrt{2 \times 0.010/469} = 6.53 \times 10^{-3}$. Then $\eta = 1 \times 1 \times 1.440/(197.3 \times 6.53 \times 10^{-3}) = 1.440/1.288 = 1.118$. $G \approx \pi \times 1.118 = 3.51$ (in the extreme thick-barrier limit). Actually for pp, $G = 2\pi\eta \approx 7.0$ (using the proper formula). $P = e^{-7.0} \approx 9 \times 10^{-4}$. This is small but not impossibly so — which is why stellar fusion can occur, given the $\sim 10^{57}$ protons in the solar core and billions of years of time.