Nuclear fusion Study Guide

📖 Core Concepts Nuclear fusion – two or more atomic nuclei combine to form a heavier nucleus, releasing energy from the mass defect (Einstein $E=mc^{2}$). Exothermic vs. endothermic fusion – nuclei lighter than Ni‑62 release energy when fused; heavier nuclei require energy. Lawson triple product – the required product of plasma density (n), temperature (T), and energy‑confinement time (τ) for net gain: $nT\tau$. Coulomb barrier – electrostatic repulsion between positively charged nuclei; quantum tunnelling (Gamow factor) lets particles fuse below the classical barrier. Reactivity $\langle\sigma v\rangle$ – velocity‑averaged product of fusion cross‑section ($\sigma$) and relative speed ($v$); determines fusion rate in a Maxwellian plasma. Binding‑energy curve – peaks at Ni‑62; explains why light nuclei (e.g., D, T, $^{3}$He) are most fusible. Neutronicity – fraction of released energy carried by neutrons: $\displaystyle \frac{E{\text{fus}}-E{\text{ch}}}{E{\text{fus}}}$. Bremsstrahlung loss – X‑ray emission from electron‑ion collisions; a major cooling channel in terrestrial plasmas. --- 📌 Must Remember D‑T reaction: $D+T \rightarrow \,^{4}\!He\;(3.5\;\text{MeV}) + n\;(14.1\;\text{MeV})$, total $17.6\;\text{MeV}$. Coulomb barrier for D‑T ≈ 0.1 MeV → requires $T \gtrsim 1.2\times10^{9}\;$K (≈ 100 keV). Magnetic‑confinement triple product goal: $nT\tau > 1\times10^{21}\;\text{keV·s·m}^{-3}$. Inertial confinement ignition: adequate areal density $\rho R$ plus temperature (≈ 10–100 keV). Peak reactivity temperature (maximizes $\langle\sigma v\rangle/T^{2}$): ≈ 100 MK for D‑T, higher for D‑$^{3}$He and p‑$^{11}$B. Bremsstrahlung fraction: modest for D‑T; becomes dominant for aneutronic fuels (D‑$^{3}$He, p‑$^{11}$B). Pressure penalty for non‑hydrogenic fuel (charge $Z$): ion density reduced by $2/(Z+1)$ at fixed pressure. --- 🔄 Key Processes Fusion rate calculation $R = n{1}n{2}\langle\sigma v\rangle$ (for two species) For a single‑species plasma (e.g., D‑D): $R = \tfrac{1}{2}n^{2}\langle\sigma v\rangle$ (factor‑2 advantage). Quantum tunnelling (Gamow factor) $G = \dfrac{2\pi Z{1}Z{2}e^{2}}{\hbar v}$ → tunnelling probability $\propto e^{-G}$. Magnetic confinement (tokamak) Create toroidal $B$‑field → charged ions gyrate around field lines, limiting cross‑field transport. Maintain $n$, $T$, and $\tau$ to satisfy Lawson criterion. Inertial confinement (laser‑driven) Deliver a short, intense laser pulse → implode fuel capsule → achieve high $\rho R$ and $T$ before plasma expands. Bremsstrahlung loss estimation $P{\text{Brem}} \propto Z{\text{eff}} n{e} T^{1/2}$ (higher $Z{\text{eff}}$ and $T$ increase loss). --- 🔍 Key Comparisons D‑T vs. D‑D Cross‑section: D‑T ≈ 5 barns at 100 keV; D‑D ≈ 0.1 barn. Neutron production: D‑T produces 1 neutron per reaction (14.1 MeV); D‑D yields both neutrons and tritium/He‑3. Magnetic vs. Inertial Confinement Confinement time: magnetic (seconds) → low density; inertial (nanoseconds) → extremely high density. Triple product: magnetic aims for $nT\tau\sim10^{21}$; inertial relies on $\rho R$ instead of $\tau$. Neutronic (D‑T) vs. Aneutronic (D‑$^{3}$He, p‑$^{11}$B) Neutron fraction: high vs. 0. Required temperature: ≈ 10 keV vs. > 200 keV. Bremsstrahlung: modest vs. dominant loss channel. --- ⚠️ Common Misunderstandings “Higher temperature always better.” → Past the optimal $T$ for a given fuel, $\langle\sigma v\rangle/T^{2}$ falls, raising the required triple product. “Fusion can occur at room temperature with enough pressure.” – Coulomb barrier and tunnelling probabilities are negligible without $T\sim10^{8-9}$ K. “Aneutronic fuels are automatically safer.” – They demand much higher $T$ and suffer severe Bremsstrahlung losses, offsetting the neutron‑radiation advantage. “Lawson criterion is a single number.” – The required $nT\tau$ depends on the fuel’s reactivity curve; D‑T’s threshold is far lower than p‑$^{11}$B’s. --- 🧠 Mental Models / Intuition “Hill‑and‑valley” picture – The Coulomb barrier is a hill; quantum tunnelling gives particles a chance to slip through the valley underneath. “Triple product as a fuel‑economy metric” – Think of $n$, $T$, and $\tau$ as “price, speed, and time” you must invest; if any one is low, you must over‑pay on the others. “Neutron vs. photon waste” – Neutrons are like “bullet waste” that damage structures; Bremsstrahlung photons are “leaky light” that cool the plasma. --- 🚩 Exceptions & Edge Cases Heavy‑fuel pressure penalty – For $Z>1$ fuels, ion density drops (factor $2/(Z+1)$), reducing reaction rate even if temperature is high. Single‑species advantage – D‑D (or p‑p) gains a factor‑2 because every ion can pair with any other; mixed‑fuel plasmas lack this symmetry. Optically thick stellar plasma – Bremsstrahlung photons are re‑absorbed, so stellar fusion isn’t limited by X‑ray loss, unlike terrestrial reactors. --- 📍 When to Use Which Choose D‑T when you need the highest reactivity and are prepared to handle neutron flux (most tokamak and ICF designs). Choose D‑$^{3}$He if you can provide $T\gtrsim200\,$keV and want fewer neutrons for reduced shielding. Choose p‑$^{11}$B only for research into truly aneutronic concepts; expect extreme temperature, high Bremsstrahlung, and low power density. Magnetic confinement for steady‑state, continuous‑power concepts; Inertial confinement for pulsed, high‑density experiments (e.g., NIF). --- 👀 Patterns to Recognize “Exponential temperature dependence” – Fusion rates often follow $\exp(-\text{constant}/T^{1/3})$; a small $T$ increase can dramatically boost $\langle\sigma v\rangle$. “Neutron‑rich signatures” – Presence of 14.1 MeV neutrons → D‑T dominant; 2.45 MeV neutrons → D‑D dominant. “Bremsstrahlung rise with $Z{\text{eff}}$” – High‑$Z$ impurity spikes in plasma diagnostics usually precede a drop in net fusion power. --- 🗂️ Exam Traps Mistaking the barrier energy for required plasma temperature – The 0.1 MeV Coulomb barrier for D‑T corresponds to ≈ 1.2 billion K, not 0.1 MeV temperature directly. Confusing “break‑even” with “net‑positive gain” – NIF’s 3.15 MJ output from 2.05 MJ input is energy gain (>1) but not yet a self‑sustaining power plant. Assuming the same triple‑product value for all fuels – D‑T’s $nT\tau$ target is far lower than that for p‑$^{11}$B; using the D‑T number for aneutronic fuels yields an impossible design. Overlooking Bremsstrahlung for high‑$Z$ fuels – Selecting p‑$^{11}$B without accounting for its large X‑ray losses will predict an unrealistically high net power. ---