Self-Assessment Quiz — Chapter 21

Test your understanding of the core concepts before moving on. Try to answer each question before checking the solutions at the end.

Q1. (Multiple Choice) The Coulomb barrier for two protons is approximately:

(a) 0.5 keV (b) 5 keV (c) 55 keV (d) 550 keV

Q2. (Multiple Choice) The thermal energy $kT$ at the center of the Sun ($T \approx 1.57 \times 10^7\,\text{K}$) is approximately:

(a) 0.14 keV (b) 1.35 keV (c) 13.5 keV (d) 135 keV

Q3. (True/False) Fusion in the solar core occurs because a small fraction of protons have enough energy to classically overcome the Coulomb barrier.

Q4. (Short Answer) Define the Sommerfeld parameter $\eta$ and explain its physical significance. Is $\eta > 1$ or $\eta < 1$ for fusion reactions at stellar energies?

Q5. (Multiple Choice) The Gamow peak energy for p-p fusion in the solar core is approximately:

(a) 0.6 keV (equal to $kT/2$) (b) 6 keV (several times $kT$) (c) 60 keV (well into the Maxwellian tail) (d) 550 keV (at the barrier top)

Q6. (Short Answer) The Gamow peak is the product of two competing factors. Name them and explain why each favors a different energy range.

Q7. (Multiple Choice) The astrophysical S-factor is defined by $\sigma(E) = S(E) E^{-1} \exp(-\sqrt{E_G/E})$. The primary utility of $S(E)$ is that it:

(a) Is always constant, independent of energy (b) Varies slowly with energy, enabling reliable extrapolation to stellar energies (c) Includes only the strong interaction; the Coulomb effects are in the exponential (d) Both (b) and (c)

Q8. (True/False) The S-factor for the pp reaction ($p + p \to d + e^+ + \nu_e$) is about 25 orders of magnitude smaller than for a typical strong-interaction fusion reaction.

Q9. (Multiple Choice) The rate-limiting step of the pp-I chain is $p + p \to d + e^+ + \nu_e$ because:

(a) The Coulomb barrier is highest for this step (b) It requires a weak interaction (proton-to-neutron conversion) (c) Deuterium is extremely rare in the Sun (d) The reaction is endothermic

Q10. (Short Answer) In the pp-I chain, what is the average time a proton waits before undergoing the pp reaction? Why is this long timescale important for the Sun's lifetime?

Q11. (Multiple Choice) The CNO cycle dominates over the pp chain in stars with:

(a) Core temperatures below $10^7\,\text{K}$ (b) Core temperatures above $\sim 1.7 \times 10^7\,\text{K}$ (c) Low metallicity (few C, N, O nuclei) (d) White dwarf cores

Q12. (Short Answer) In the CNO cycle, carbon, nitrogen, and oxygen act as catalysts. What does this mean? Which CNO isotope accumulates to the highest abundance in equilibrium, and why?

Q13. (Multiple Choice) The solar neutrino problem was resolved by:

(a) A more accurate solar model that predicted fewer neutrinos (b) The discovery that nuclear reaction rates in the Sun are lower than calculated (c) The discovery of neutrino oscillations, which convert $\nu_e$ to $\nu_\mu$ and $\nu_\tau$ (d) The realization that Davis's chlorine detector was malfunctioning

Q14. (True/False) Terrestrial fusion research focuses on D-T reactions because they have the lowest Coulomb barrier of any fusion reaction.

Q15. (Short Answer) A tokamak uses two types of magnetic field — toroidal and poloidal — to confine the plasma. Explain why the toroidal field alone is insufficient and what the poloidal field accomplishes.

Q16. (Multiple Choice) In the December 2022 NIF ignition shot, the ratio of fusion energy produced to laser energy on target was approximately:

(a) 0.01 (b) 0.15 (c) 1.5 (d) 15

Q17. (Short Answer) State the Lawson criterion for D-T ignition in terms of $n\tau_E$ and give the approximate numerical threshold. What is the physical meaning of $\tau_E$?

Q18. (Multiple Choice) The fusion gain factor $Q$ for JET's best D-T result (1997) was approximately:

(a) $Q \approx 0.01$ (b) $Q \approx 0.67$ (c) $Q \approx 5$ (d) $Q \approx 10$

Q19. (True/False) In a D-T fusion reactor, the 14.1 MeV neutron carries 80% of the fusion energy and must be captured in a lithium blanket to both extract heat and breed new tritium.

Q20. (Short Answer) A common statement is that "the physics of fusion is largely solved but the engineering is not." Name three specific engineering challenges that remain on the path to commercial fusion power.

Solutions

Q1. (d) $\sim 550\,\text{keV}$. $V_B = Z_1 Z_2 e^2 / (4\pi\epsilon_0 R) \approx 1.44/(2 \times 1.2) \approx 0.6\,\text{MeV}$.

Q2. (b) $kT \approx 1.35\,\text{keV}$. Using $k = 8.617 \times 10^{-5}\,\text{eV/K}$: $kT = 8.617 \times 10^{-5} \times 1.57 \times 10^7 \approx 1353\,\text{eV} \approx 1.35\,\text{keV}$.

Q3. False. The fraction of protons with energy $> V_B$ is $\sim\exp(-550/1.35) \approx 10^{-177}$, essentially zero. Fusion occurs by quantum tunneling through the barrier at energies far below $V_B$.

Q4. $\eta = Z_1 Z_2 e^2/(4\pi\epsilon_0 \hbar v) = Z_1 Z_2 \alpha c / v$, where $v$ is the relative velocity. It measures the strength of the Coulomb interaction relative to the kinetic energy. For fusion at stellar energies, $\eta \gg 1$ (the Coulomb repulsion is much stronger than the kinetic energy), which means the tunneling probability is exponentially suppressed.

Q5. (b) $E_0 \approx 6\,\text{keV}$, which is about $4.5 \times kT$. It sits in the tail of the Maxwell-Boltzmann distribution but far below the barrier top.

Q6. The Gamow peak is the product of: (1) the Maxwell-Boltzmann distribution $\exp(-E/kT)$, which is large at low energy and falls exponentially at high energy (more particles at low energy); and (2) the tunneling probability $\exp(-\sqrt{E_G/E})$, which is small at low energy and rises at high energy (thinner barrier at higher energy). Their product is maximal at an intermediate energy $E_0$.

Q7. (d) Both (b) and (c). $S(E)$ varies slowly because the rapid Coulomb and kinematic energy dependences have been divided out, and it encapsulates the nuclear (strong/weak) interaction physics.

Q8. True. $S_{pp}(0) = 4.01 \times 10^{-22}\,\text{keV}\cdot\text{b}$, while typical strong-interaction S-factors are $10^3\text{–}10^4\,\text{keV}\cdot\text{b}$.

Q9. (b) It requires a weak interaction — one proton must convert to a neutron via $W^+$ exchange. The weak interaction has a coupling constant $\sim 10^{-5}$ relative to the strong interaction, which suppresses the S-factor by many orders of magnitude.

Q10. $\sim 9 \times 10^9$ years (9 billion years). This is comparable to the Sun's main-sequence lifetime of $\sim 10^{10}$ years. The weak-interaction bottleneck ensures that the Sun burns its hydrogen slowly, allowing it to shine stably for billions of years — long enough for complex life to evolve on Earth.

Q11. (b) Above $\sim 1.7 \times 10^7\,\text{K}$. The CNO cycle has a much steeper temperature dependence ($\propto T^{16}$) due to the higher Coulomb barrier ($Z_1 Z_2 = 6$ or 7), so it overtakes the pp chain ($\propto T^4$) at higher temperatures.

Q12. Catalysts are consumed and regenerated within the cycle — the total number of C+N+O nuclei is conserved, but four protons are converted to one helium-4. ${}^{14}$N accumulates because the rate-limiting step is ${}^{14}\text{N}(p,\gamma){}^{15}\text{O}$, which has the smallest S-factor. Material piles up before the slowest step.

Q13. (c) Neutrino oscillations. The SNO experiment (2001–2002) showed that the total neutrino flux (all flavors) matched predictions; only the $\nu_e$ fraction was reduced by oscillations during transit.

Q14. False. D-T does not have the lowest Coulomb barrier — p-p has a lower barrier. D-T is favored because it has the largest $\langle\sigma v\rangle$ at accessible temperatures (due to a resonance in ${}^5$He) and a large Q-value (17.6 MeV).

Q15. The toroidal field alone cannot confine plasma because the field gradient and curvature (stronger field on the inboard side of the torus) cause charge-dependent drifts: ions drift up and electrons drift down (or vice versa). This charge separation creates a vertical electric field that drives the plasma radially outward. The poloidal field twists the field lines helically, so each line samples both the top and bottom of the torus, averaging out the drift and maintaining equilibrium.

Q16. (c) $\sim 1.5$. The ratio was $3.15\,\text{MJ} / 2.05\,\text{MJ} = 1.54$.

Q17. $n\tau_E \gtrsim 4 \times 10^{20}\,\text{m}^{-3}\cdot\text{s}$ (at optimal temperature $kT \approx 13\,\text{keV}$). $\tau_E$ is the energy confinement time — the time over which the plasma would lose its stored energy if all heating were suddenly turned off. It measures how well the magnetic field confines the plasma energy.

Q18. (b) $Q \approx 0.67$. JET produced 16.1 MW of fusion power from 24 MW of heating power.

Q19. True. The neutron carries $14.1/17.6 = 80\%$ of the energy. The lithium blanket serves dual purposes: breeding tritium ($n + {}^6\text{Li} \to T + {}^4\text{He}$) and converting the neutron's kinetic energy to thermal energy for electricity generation.

Q20. Three key challenges (any three of): (1) Materials — first wall and divertor must withstand 14.1 MeV neutron bombardment causing displacement damage and embrittlement. (2) Tritium breeding — achieving TBR > 1 with realistic blanket designs, never demonstrated experimentally. (3) Plasma instabilities — especially disruptions that can dump hundreds of MJ onto the wall in milliseconds. (4) Superconducting magnets — achieving high fields reliably over decades of operation. (5) Reliability and duty cycle — transitioning from pulsed experiments to steady-state power plants with $>80\%$ availability.