Exercises — Chapter 21

Coulomb Barrier and Classical Turning Points

Problem 21.1 ⭐ Calculate the Coulomb barrier height $V_B$ for the following reactions. Use $R = r_0(A_1^{1/3} + A_2^{1/3})$ with $r_0 = 1.2\,\text{fm}$.

(a) $p + p$ (proton-proton)

(b) $D + T$ (deuterium-tritium)

(c) $D + D$ (deuterium-deuterium)

(d) $D + {}^3\text{He}$

(e) $p + {}^{12}\text{C}$ (the CNO cycle)

(f) ${}^{12}\text{C} + {}^{12}\text{C}$ (carbon burning in massive stars)

Rank these reactions by barrier height and comment on which are accessible at stellar vs. terrestrial temperatures.

Problem 21.2 ⭐ For two protons with center-of-mass energy $E = 5\,\text{keV}$ (close to the Gamow peak in the Sun):

(a) Calculate the classical turning point $r_c$.

(b) Calculate the nuclear contact radius $R = r_0(A_1^{1/3} + A_2^{1/3})$.

(c) Compute the ratio $r_c / R$. What does this ratio tell you about the thickness of the barrier that must be tunneled through?

(d) Repeat parts (a)–(c) for $E = 50\,\text{keV}$ and $E = 500\,\text{keV}$. At what energy does $r_c = R$ (the particle classically reaches the nuclear surface)?

Problem 21.3 ⭐ The solar core temperature is $T_c = 1.57 \times 10^7\,\text{K}$.

(a) Compute $kT_c$ in keV.

(b) Compute the most probable energy in the Maxwell-Boltzmann distribution ($E_{\text{mp}} = kT/2$) and the mean energy ($\langle E\rangle = 3kT/2$).

(c) What fraction of a Maxwell-Boltzmann distribution has $E > 10\,\text{keV}$? Express your answer as $\exp(-E/kT)$ and evaluate numerically.

(d) What fraction has $E > 550\,\text{keV}$ (the p-p Coulomb barrier)? This should convince you that classical barrier crossing is impossible.

Gamow Factor and Tunneling

Problem 21.4 ⭐⭐ Compute the Gamow energy $E_G = 2\mu c^2 (\pi Z_1 Z_2 \alpha)^2$ for:

(a) $p + p$ ($\mu = m_p/2$)

(b) $D + T$ ($\mu = 6m_u/5$, where $m_u = 931.5\,\text{MeV}/c^2$)

(c) $p + {}^{12}\text{C}$ ($\mu = 12 m_u/13$)

(d) ${}^{12}\text{C} + {}^{12}\text{C}$ ($\mu = 6 m_u$)

For each, also compute $2\pi\eta$ at $E = 10\,\text{keV}$ and the corresponding Gamow factor $\exp(-2\pi\eta)$.

Problem 21.5 ⭐⭐ The WKB tunneling probability through a pure Coulomb barrier is $P(E) = \exp(-2\pi\eta)$ in the limit $R \ll r_c$.

(a) For $p + p$ at $E = 6\,\text{keV}$ (the Gamow peak), compute $P$.

(b) For D-T at $E = 25\,\text{keV}$ (approximately the Gamow peak for tokamak temperatures), compute $P$.

(c) Why is $P$ for D-T so much larger than for $p + p$ at their respective Gamow peaks? Identify both the $Z_1 Z_2$ effect and the energy effect.

(d) Despite the much smaller tunneling probability, the Sun produces enormous fusion power. Estimate the number of $p + p$ pairs interacting per second in the solar core (assume $n_p \approx 6 \times 10^{31}\,\text{m}^{-3}$, $V_{\text{core}} \approx 1.5 \times 10^{25}\,\text{m}^3$, and $\langle\sigma v\rangle_{pp} \approx 4 \times 10^{-52}\,\text{m}^3/\text{s}$).

Problem 21.6 ⭐⭐⭐ Full derivation: WKB integral for the Coulomb barrier.

Starting from $P = \exp\!\left(-\frac{2}{\hbar}\int_R^{r_c}\sqrt{2\mu(V(r)-E)}\,dr\right)$ with $V(r) = Z_1 Z_2 e^2/(4\pi\epsilon_0 r)$:

(a) Use the substitution $r = r_c \cos^2\theta$ to evaluate the integral. Show that the result is:

$$\int_R^{r_c}\sqrt{\frac{r_c}{r}-1}\,dr = r_c\left[\cos^{-1}\sqrt{R/r_c} - \sqrt{(R/r_c)(1-R/r_c)}\right]$$

(b) Expand for $R/r_c \ll 1$ and show that the leading term gives $P \approx \exp(-2\pi\eta)$.

(c) Evaluate the first correction term and show it is $+4\sqrt{\eta R/r_c}$ inside the exponential. For $p + p$ at $E = 6\,\text{keV}$, by what factor does this correction modify $P$?

The Gamow Peak

Problem 21.7 ⭐⭐ For D-T fusion at $kT = 10\,\text{keV}$ (a typical tokamak target):

(a) Compute the Gamow peak energy $E_0 = (E_G(kT)^2/4)^{1/3}$.

(b) Compute the Gamow peak width $\Delta = 4\sqrt{E_0 kT/3}$.

(c) Evaluate the integrand $\mathcal{F}(E) = \exp(-E/kT - \sqrt{E_G/E})$ at $E = E_0$, $E = E_0 \pm \Delta/2$, and $E = E_0 \pm \Delta$.

(d) Compare the value of $\mathcal{F}(E)$ at the peak to the value at $E = kT$ and at $E = V_B$. This illustrates why neither the thermal peak nor the barrier top contributes significantly to the reaction rate.

Problem 21.8 ⭐⭐ Temperature dependence of the Gamow peak.

(a) Show that $E_0 \propto T^{2/3}$, $\Delta \propto T^{5/6}$, and $\Delta/E_0 \propto T^{1/6}$ (the peak broadens relatively as temperature increases).

(b) Compute $E_0$ and $\Delta$ for $p + p$ at $T = 10^7\,\text{K}$, $1.57 \times 10^7\,\text{K}$ (solar core), and $3 \times 10^7\,\text{K}$.

(c) At what temperature does $E_0$ for $p + {}^{12}\text{C}$ equal $E_0$ for $p + p$ at the solar core temperature? This gives an intuition for why the CNO cycle needs higher temperatures.

Problem 21.9 ⭐⭐⭐ Power-law index.

The reaction rate $\langle\sigma v\rangle$ near a temperature $T_0$ can be approximated as $\langle\sigma v\rangle \propto T^\nu$ where:

$$\nu = \frac{\partial\ln\langle\sigma v\rangle}{\partial\ln T} = \frac{E_0}{kT} - \frac{2}{3}$$

(a) Derive this expression starting from the approximate formula for $\langle\sigma v\rangle$ with constant $S(E_0)$.

(b) Compute $\nu$ for $p + p$, $D + T$, and $p + {}^{12}\text{C}$ (CNO rate-limiting step) at $kT = 1.35\,\text{keV}$.

(c) By how much does the fusion power change if the solar core temperature increases by 1%? By 10%?

(d) Explain physically why the CNO cycle has a much larger $\nu$ than the pp chain.

Astrophysical S-Factor and Cross Sections

Problem 21.10 ⭐ The astrophysical S-factor for ${}^3\text{He} + {}^3\text{He} \to {}^4\text{He} + 2p$ is $S(0) \approx 5.21 \times 10^3\,\text{keV}\cdot\text{b}$.

(a) Compute the Gamow energy $E_G$ for this reaction.

(b) Using $\sigma(E) = S(E)/E \cdot \exp(-\sqrt{E_G/E})$, compute the cross section at $E = 20\,\text{keV}$ and at $E = 200\,\text{keV}$.

(c) Express these cross sections in barns and in $\text{cm}^2$. Compare to the geometric cross section $\pi R^2$ where $R = r_0(3^{1/3} + 3^{1/3})$.

Problem 21.11 ⭐⭐ The S-factor for $p + p \to d + e^+ + \nu_e$ is $S(0) = 4.01 \times 10^{-22}\,\text{keV}\cdot\text{b}$.

(a) Compute the cross section at the Gamow peak energy $E_0 = 6.1\,\text{keV}$.

(b) Express this cross section in $\text{cm}^2$. Compare it to a typical strong interaction cross section (a few barns).

(c) If the solar core has $n_p \approx 6 \times 10^{31}\,\text{m}^{-3}$ and $\langle\sigma v\rangle_{pp} \approx 4 \times 10^{-52}\,\text{m}^3/\text{s}$, compute the pp reaction rate per unit volume. How many reactions occur per second in the entire core (volume $\sim 1.5 \times 10^{25}\,\text{m}^3$)?

(d) Verify that this gives approximately $L_\odot = 3.8 \times 10^{26}\,\text{W}$ when you account for the net energy release per completed pp chain ($\sim 26.2\,\text{MeV}$ deposited in the plasma, with each chain requiring 2 pp reactions).

Problem 21.12 ⭐⭐⭐ Why S(E) is slowly varying. The cross section $\sigma(E)$ for charged-particle reactions varies over many orders of magnitude, while $S(E)$ varies slowly.

(a) For D-T, the S-factor has a resonance at $E \approx 64\,\text{keV}$ (corresponding to the $3/2^+$ level in ${}^5$He). The maximum S-factor is approximately $3 \times 10^4\,\text{keV}\cdot\text{b}$ at the resonance. What is the corresponding cross section $\sigma(64\,\text{keV})$?

(b) Compute $\sigma$ at $E = 10\,\text{keV}$ using $S \approx 1.17 \times 10^4\,\text{keV}\cdot\text{b}$. By what factor has $\sigma$ dropped compared to the resonance peak?

(c) If we had tried to extrapolate $\sigma(E)$ directly from 64 keV to 10 keV (assuming smooth behavior), by how many orders of magnitude would we be wrong compared to the S-factor extrapolation? This illustrates the practical utility of the S-factor.

The pp Chain and CNO Cycle

Problem 21.13 ⭐ For each branch of the pp chain (pp-I, pp-II, pp-III):

(a) Write the complete sequence of reactions.

(b) Verify that the net reaction is $4p \to {}^4\text{He} + 2e^+ + 2\nu_e$ in each case.

(c) Compute the total energy released and the average energy carried away by neutrinos. What fraction of the total energy is "lost" to neutrinos in each branch?

Problem 21.14 ⭐⭐ The pp reaction rate-limiting step.

(a) The mean lifetime of a proton against pp fusion in the solar core is $\tau_{pp} \approx 9 \times 10^9$ years. Compute the reaction rate per proton $\lambda = 1/\tau_{pp}$ in units of $\text{s}^{-1}$.

(b) Using $n_p \approx 6 \times 10^{31}\,\text{m}^{-3}$, compute $\langle\sigma v\rangle_{pp}$ from $\lambda = n_p\langle\sigma v\rangle/2$ (the factor of 2 because identical particles).

(c) Compare your result to the tabulated value $\langle\sigma v\rangle_{pp} \approx 4 \times 10^{-52}\,\text{m}^3/\text{s}$.

(d) The S-factor for $d + p \to {}^3\text{He} + \gamma$ is $S(0) = 2.14 \times 10^{-4}\,\text{keV}\cdot\text{b}$, roughly $10^{18}$ times larger than $S(0)$ for pp. Estimate the lifetime of a deuterium nucleus in the solar core and explain why the Sun's deuterium abundance is negligible.

Problem 21.15 ⭐⭐ The CNO cycle: Nitrogen bottleneck.

In the CNO-I cycle, the rate-limiting step is ${}^{14}\text{N}(p,\gamma){}^{15}\text{O}$ with $S(0) \approx 1.66\,\text{keV}\cdot\text{b}$.

(a) Compute the Gamow energy $E_G$ for $p + {}^{14}\text{N}$ ($Z_1 Z_2 = 7$).

(b) Compute the Gamow peak energy $E_0$ at the solar core temperature.

(c) The S-factor for ${}^{12}\text{C}(p,\gamma){}^{13}\text{N}$ is $S(0) \approx 1.34\,\text{keV}\cdot\text{b}$. Is this step also slow, or is the bottleneck specifically at ${}^{14}$N? Compare the Gamow factors at the respective $E_0$ values.

(d) In CNO equilibrium, the abundance ratios reflect the reaction rates: the slowest step has the most accumulated material. In evolved stars, ${}^{14}$N is indeed the most abundant CNO isotope. Explain why, based on your calculations.

Problem 21.16 ⭐⭐ pp vs. CNO crossover.

(a) The pp luminosity scales as $L_{pp} \propto T^4$ and the CNO luminosity as $L_{\text{CNO}} \propto T^{16}$ (at solar temperatures). If $L_{pp}/L_{\text{CNO}} = 100/1.5 \approx 67$ at $T = T_\odot$, at what temperature do they become equal?

(b) Using the mass-luminosity relation for main-sequence stars ($L \propto M^{3.5}$) and the mass-temperature relation ($T_c \propto M$), estimate the stellar mass at which the CNO cycle takes over.

Lawson Criterion and Terrestrial Fusion

Problem 21.17 ⭐ Fusion fuel comparison.

For the following reactions, compute the Q-value from the nuclear masses and identify the reaction products:

(a) $D + T \to {}^4\text{He} + n$

(b) $D + D \to {}^3\text{He} + n$

(c) $D + D \to T + p$

(d) $D + {}^3\text{He} \to {}^4\text{He} + p$

(e) $p + {}^{11}\text{B} \to 3\,{}^4\text{He}$

Which reactions produce neutrons? Why is this relevant for reactor design?

Problem 21.18 ⭐⭐ Deriving the Lawson criterion.

Starting from the power balance $P_\alpha + P_{\text{ext}} = P_{\text{loss}} + P_{\text{brem}}$ for a D-T plasma at temperature $T$:

(a) Write each term explicitly in terms of $n$, $T$, $\tau_E$, and $\langle\sigma v\rangle$.

(b) For the ignition condition ($P_{\text{ext}} = 0$), derive the Lawson criterion $n\tau_E \geq 12kT/(\langle\sigma v\rangle E_\alpha)$, neglecting bremsstrahlung.

(c) Now include bremsstrahlung losses: $P_{\text{brem}} = C_B n^2 \sqrt{T}$ with $C_B = 5.35 \times 10^{-37}\,\text{W}\cdot\text{m}^3\cdot\text{keV}^{-1/2}$. Show that the ignition condition becomes:

$$n\tau_E \geq \frac{12kT}{\langle\sigma v\rangle E_\alpha - 4C_B\sqrt{T}}$$

(d) Plot $n\tau_E$ vs. $kT$ for $1 < kT < 100\,\text{keV}$. At what temperature is the required $n\tau_E$ minimized? What is the minimum value?

Problem 21.19 ⭐⭐ ITER parameters.

ITER's design targets are: plasma volume $V = 840\,\text{m}^3$, average density $n = 10^{20}\,\text{m}^{-3}$, temperature $kT = 8\,\text{keV}$, energy confinement time $\tau_E = 3.7\,\text{s}$.

(a) Compute the triple product $n\tau_E T$. Compare to the ignition threshold.

(b) Compute the fusion power $P_{\text{fus}} = (n^2/4)\langle\sigma v\rangle E_{\text{fus}} \times V$. Use $\langle\sigma v\rangle \approx 3 \times 10^{-23}\,\text{m}^3/\text{s}$ at $kT = 8\,\text{keV}$.

(c) Compute $Q = P_{\text{fus}}/P_{\text{ext}}$ given that $P_{\text{ext}} = 50\,\text{MW}$.

(d) Of the 17.6 MeV per D-T reaction, 14.1 MeV is carried by the neutron and 3.5 MeV by the alpha. If the alpha is confined and the neutron escapes, what is the alpha heating power $P_\alpha$? Is $P_\alpha > P_{\text{ext}}$?

Problem 21.20 ⭐⭐⭐ NIF ignition analysis.

In the December 2022 NIF shot, 2.05 MJ of laser energy produced 3.15 MJ of fusion energy from a D-T capsule.

(a) How many D-T fusion reactions occurred? (Each produces 17.6 MeV.)

(b) The capsule contained approximately 170 $\mu$g of D-T fuel. What fraction of the fuel was burned? (Assume equal numbers of D and T atoms, total mass 170 $\mu$g.)

(c) The compression lasted roughly 100 ps. Estimate the instantaneous fusion power in watts during the burn.

(d) The laser required $\sim 300\,\text{MJ}$ of electrical energy. Compute the overall engineering gain $G_{\text{eng}} = E_{\text{fus}} / E_{\text{elec}}$. What laser wall-plug efficiency would be needed for $G_{\text{eng}} = 1$?

Advanced and Research Problems

Problem 21.21 ⭐⭐⭐ Electron screening in the laboratory.

At very low energies ($E \lesssim 10\,\text{keV}$), the atomic electrons of the target partially screen the nuclear Coulomb potential. This enhances the measured cross section relative to the bare-nucleus prediction.

(a) Model the screening as a constant energy shift: the effective energy in the Gamow factor is $E + U_e$ instead of $E$, where $U_e$ is the screening potential energy. Show that the enhancement factor is:

$$f_{\text{scr}} = \frac{\sigma_{\text{screened}}}{\sigma_{\text{bare}}} \approx \exp\!\left(\pi\eta \frac{U_e}{E}\right)$$

(b) For $D + D$ with $U_e \approx 25\,\text{eV}$ (the adiabatic limit), compute $f_{\text{scr}}$ at $E = 5\,\text{keV}$ and $E = 50\,\text{keV}$. At what energy does screening become a significant ($>10\%$) correction?

(c) Experimentally, the measured screening potentials are $2\text{–}5\times$ larger than the adiabatic theoretical predictions. This is an unsolved problem in nuclear astrophysics. Discuss possible explanations (environmental effects in metallic targets, thermal motion).

Problem 21.22 ⭐⭐⭐ Plasma beta and magnetic pressure.

The plasma beta is defined as $\beta = nkT / (B^2/2\mu_0)$, the ratio of plasma pressure to magnetic pressure.

(a) For ITER parameters ($n = 10^{20}\,\text{m}^{-3}$, $kT = 8\,\text{keV}$, $B = 5.3\,\text{T}$), compute $\beta$. Include both ion and electron pressure.

(b) In a tokamak, the maximum achievable beta is limited by MHD instabilities (the Troyon limit: $\beta_N \equiv \beta a B / I_p \lesssim 3\%\cdot\text{mT/MA}$). For ITER ($a = 2.0\,\text{m}$, $I_p = 15\,\text{MA}$), what is the maximum beta?

(c) Compute the magnetic field energy stored in the ITER toroidal field coils (approximate the magnetic field volume as $2\pi R \times \pi a^2$ with $R = 6.2\,\text{m}$, $a = 2.0\,\text{m}$). Express in megajoules.

Problem 21.23 ⭐⭐⭐ The triple product through history.

Plot the evolution of the fusion triple product $n\tau_E T$ from the first tokamak experiments to ITER (use data from the chapter). Include: - T-3 (1968): $n \sim 5 \times 10^{18}\,\text{m}^{-3}$, $\tau_E \sim 5\,\text{ms}$, $kT \sim 0.5\,\text{keV}$ - PLT (1978): $n \sim 5 \times 10^{19}$, $\tau_E \sim 30\,\text{ms}$, $kT \sim 2\,\text{keV}$ - TFTR (1994): $n \sim 10^{20}$, $\tau_E \sim 0.3\,\text{s}$, $kT \sim 10\,\text{keV}$ - JET (1997): $n \sim 5 \times 10^{19}$, $\tau_E \sim 0.9\,\text{s}$, $kT \sim 12\,\text{keV}$ - JT-60U (1996): $n \sim 7 \times 10^{19}$, $\tau_E \sim 0.8\,\text{s}$, $kT \sim 12\,\text{keV}$ - ITER (target): $n \sim 10^{20}$, $\tau_E \sim 3.7\,\text{s}$, $kT \sim 8\,\text{keV}$

(a) Compute $n\tau_E T$ for each and plot on a semi-log scale vs. year.

(b) What is the approximate doubling time? How does it compare to Moore's law?

(c) Based on this trend, when would you extrapolate that the ignition triple product ($3 \times 10^{21}$) is reached? How does this compare to the ITER schedule?

Problem 21.24 ⭐⭐⭐ Bremsstrahlung limit on fusion.

Bremsstrahlung radiation losses scale as $P_{\text{brem}} \propto n^2 Z_{\text{eff}}^2 \sqrt{T}$, while fusion power scales as $P_{\text{fus}} \propto n^2 \langle\sigma v\rangle$.

(a) For a pure D-T plasma ($Z_{\text{eff}} = 1$), show that $P_{\text{fus}} > P_{\text{brem}}$ for $kT \gtrsim 4\,\text{keV}$.

(b) For $p + {}^{11}\text{B}$ ($Z_{\text{eff}} \approx 3.5$ accounting for the boron ions), the situation is much worse. Using $\langle\sigma v\rangle$ for $p$-${}^{11}$B (which peaks around $\sim 5 \times 10^{-23}\,\text{m}^3/\text{s}$ at $kT \sim 150\,\text{keV}$), show that bremsstrahlung losses exceed fusion power at all temperatures for a thermal plasma. This is why "aneutronic" $p$-${}^{11}$B fusion is extremely challenging.

(c) What non-thermal approaches might circumvent the bremsstrahlung limit?

Problem 21.25 ⭐⭐ Energy from a gram of fusion fuel.

(a) How many D-T reactions are contained in 1 gram of D-T fuel (equal parts by number)? The average mass per D-T pair is $5 m_u / 2 = 2.5 m_u$.

(b) If all D-T pairs fused, how much energy would be released (in MJ)?

(c) Compare this to 1 gram of ${}^{235}\text{U}$ undergoing fission ($Q \approx 200\,\text{MeV}$) and to 1 gram of coal ($\sim 30\,\text{kJ}$).

(d) Express the D-T energy in terms of the mass-energy equivalent $\Delta mc^2$. What fraction of the total rest mass is converted to energy?

Problem 21.26 ⭐⭐ Tokamak scaling.

The empirical scaling for energy confinement time in H-mode tokamaks is (ITER Physics Basis, IPB98(y,2)):

$$\tau_E = 0.0562 \, I_p^{0.93} B^{0.15} P^{-0.69} n_{19}^{0.41} M^{0.19} R^{1.97} \epsilon^{0.58} \kappa^{0.78}$$

where $I_p$ is in MA, $B$ in T, $P$ (heating power) in MW, $n_{19}$ is density in $10^{19}\,\text{m}^{-3}$, $M$ is ion mass in AMU, $R$ is major radius in m, $\epsilon = a/R$ is inverse aspect ratio, and $\kappa$ is elongation.

(a) For ITER parameters ($I_p = 15\,\text{MA}$, $B = 5.3\,\text{T}$, $P = 50\,\text{MW}$, $n_{19} = 10$, $M = 2.5$ (D-T mix), $R = 6.2\,\text{m}$, $\epsilon = 0.32$, $\kappa = 1.7$), compute $\tau_E$.

(b) If the magnetic field is doubled (to $10.6\,\text{T}$, as proposed for compact HTS tokamaks) and the major radius halved (to $3.1\,\text{m}$), how does $\tau_E$ change? What about the fusion power (proportional to $n^2 V$ where $V \propto R^2 a \propto R^3 \epsilon$)?

(c) Explain qualitatively why private companies pursuing HTS magnets believe compact tokamaks can achieve the same physics performance as ITER in a much smaller device.

Problem 21.27 ⭐⭐⭐ ICF energy budget.

In NIF indirect-drive ICF:

(a) The 192 laser beams deliver 2.05 MJ to the hohlraum. If the hohlraum converts this to X-rays with $\sim 80\%$ efficiency, and the X-rays couple to the capsule with $\sim 15\%$ efficiency, how much energy drives the implosion?

(b) The capsule kinetic energy at stagnation is roughly $20\,\text{kJ}$. What fraction of the drive energy is converted to kinetic energy?

(c) At stagnation, the hot spot has radius $\sim 30\,\mu\text{m}$, temperature $\sim 5\,\text{keV}$, and density $\sim 1000\,\text{g/cm}^3$. Estimate the pressure in the hot spot (using $P = 2nkT$, accounting for electrons). Compare to the pressure at the center of the Sun ($\sim 2.3 \times 10^{16}\,\text{Pa}$).

Problem 21.28 ⭐⭐⭐ (Research) Solar neutrino fluxes.

Using the Standard Solar Model, the predicted neutrino fluxes at Earth are: - pp neutrinos: $\Phi_{pp} = 5.98 \times 10^{10}\,\text{cm}^{-2}\text{s}^{-1}$ - ${}^7$Be neutrinos: $\Phi_{\text{Be}} = 4.93 \times 10^{9}\,\text{cm}^{-2}\text{s}^{-1}$ - ${}^8$B neutrinos: $\Phi_{\text{B}} = 5.46 \times 10^{6}\,\text{cm}^{-2}\text{s}^{-1}$

(a) From the pp flux, estimate the pp reaction rate in the Sun and verify consistency with $L_\odot$.

(b) The pp-III branch produces ${}^8$B neutrinos. Given that the pp-III branching ratio is $\sim 0.02\%$, estimate the total pp chain rate from $\Phi_{\text{B}}$ and compare to part (a).

(c) The Davis chlorine experiment (${}^{37}\text{Cl} + \nu_e \to {}^{37}\text{Ar} + e^-$, threshold 814 keV) was sensitive to ${}^7$Be and ${}^8$B neutrinos but not pp neutrinos. Calculate the expected capture rate in SNU (1 SNU = $10^{-36}$ captures per target atom per second) using the cross sections: $\sigma_{\text{Be}} \approx 2.4 \times 10^{-46}\,\text{cm}^2$, $\sigma_{\text{B}} \approx 1.1 \times 10^{-42}\,\text{cm}^2$.

Problem 21.29 ⭐⭐ (Research) Tritium inventory and breeding.

(a) ITER will consume approximately 0.5 kg of tritium per year (at full power). Given tritium's half-life of 12.3 years, compute the radioactive decay loss per year for a 2 kg inventory.

(b) The world's total tritium supply (primarily from CANDU reactors) is estimated at $\sim 25\,\text{kg}$ and decreasing. If a future fusion power plant produces 1 GW of fusion power, how much tritium does it consume per year?

(c) What tritium breeding ratio (TBR) is needed to compensate for (i) decay losses, (ii) processing losses ($\sim 1\%$), and (iii) startup inventory for new reactors? Argue that TBR $> 1.05$ is likely required.

Problem 21.30 ⭐⭐⭐ (Research) Stellar mass-luminosity and nuclear burning.

The nuclear energy generation rate per unit mass in a star can be written as:

$$\epsilon = \epsilon_0 \rho T_6^\nu$$

where $T_6 = T/(10^6\,\text{K})$ and $\nu$ is the power-law index.

(a) For pp-dominated burning ($\nu \approx 4$), use the mass-radius relation ($R \propto M^{0.8}$) and hydrostatic equilibrium ($T_c \propto M/R$) to derive the mass-luminosity relation $L \propto M^\alpha$. What is $\alpha$?

(b) Repeat for CNO-dominated burning ($\nu \approx 16$). Why do massive stars (CNO-burning) have a steeper mass-luminosity relation?

(c) The main-sequence lifetime is $\tau \propto M/L$. Using your result from (a) and (b), compute $\tau_{\text{MS}}$ for a $10\,M_\odot$ star and a $0.5\,M_\odot$ star, given $\tau_\odot \approx 10^{10}\,\text{yr}$.