Reactor physics Study Guide

📖 Core Concepts Criticality – Balance where neutron production = neutron loss; described by the effective multiplication factor k. Effective multiplication factor (k) – Ratio of neutrons in one generation to the previous one. k = 1: critical; k < 1: subcritical; k > 1: supercritical. Reactivity (ρ) – Measure of deviation from criticality: \(\rho = \dfrac{k-1}{k}\). Positive ρ → supercritical, negative ρ → subcritical. Six‑factor formula – Breaks neutron life‑cycle into six probabilities: fast‑fission factor (ε), resonance escape (p), thermal utilization (f), reproduction (η), fast non‑leakage (P\F), thermal non‑leakage (P\T). Overall: \(k = \varepsilon p f \eta PF PT\). Moderation – Slowing fast neutrons to thermal energies (< 1 eV) to raise fission probability in \(\mathrm{^{235}U}\). Light nuclei (H, D, C) are best. Delayed neutrons – Small fraction (≈ 0.65 % for \(\mathrm{^{235}U}\)) emitted seconds after fission; they lengthen the effective neutron lifetime to 0.1 s and make reactor control possible. Reactor poisons – Strong neutron absorbers (e.g., Xe‑135) that can shut down or hinder restart. Enrichment – Raising \(\mathrm{^{235}U}\) fraction from natural 0.7 % to a few percent (LEU) for water‑moderated reactors; higher enrichment for weapons or certain research reactors. --- 📌 Must Remember k‑criticality relation: \(k = 1 \iff \rho = 0\). Reactivity formula: \(\rho = \dfrac{k-1}{k}\). Dollar unit: 1 $ = \beta\) (the delayed‑neutron fraction). Average neutrons per fission (ν): 2 – 3 for \(\mathrm{^{235}U}\) & \(\mathrm{^{239}Pu}\). Six‑factor product: \(k = \varepsilon p f \eta PF PT\). Prompt neutron lifetime: 1 ms; effective lifetime (with delayed neutrons): 0.1 s. Xe‑135 half‑life: ≈ 9 h; causes “iodine pit” after shutdown. Temperature coefficients: Negative → heating ↓ reactivity (under‑moderated). Positive → heating ↑ reactivity (over‑moderated). Void effect in water reactors: Steam formation removes moderation → rapid power drop. Typical enrichment: Natural U: 0.7 % \(\mathrm{^{235}U}\). Light‑water reactors: 3–5 % \(\mathrm{^{235}U}\). Heavy‑water/graphite reactors can use natural U. --- 🔄 Key Processes Neutron life‑cycle (six‑factor) calculation Compute each factor (ε, p, f, η, \(PF\), \(PT\)). Multiply to obtain k; decide criticality. Reactivity insertion with control rods Withdraw rod → ↓ absorption → ↑ ρ. Rate of power rise limited by delayed neutrons (≈ 0.1 s). Xe‑135 buildup & decay During operation: Xe‑135 + neutron → Xe‑136 (absorbs neutrons, doesn’t accumulate). After shutdown: production stops, decay dominates → peak Xe‑135 10–12 h, then decays (half‑life 9 h). Enrichment via centrifuge Feed UF₆ gas → high‑speed rotation → lighter \(\mathrm{^{235}U}\) isotopes concentrate toward the center → extraction. Temperature coefficient effect Rise in coolant temperature → density ↓ → moderation ↓ (negative coeff) → power self‑regulates. --- 🔍 Key Comparisons Prompt neutrons vs. Delayed neutrons Prompt: emitted instantly, τ ≈ 1 ms, ≈ 99.35 % of total. Delayed: emitted seconds later, τ ≈ 15 s, ≈ 0.65 % of total, crucial for control. Light water vs. Heavy water moderators Light water (H₂O): good moderator and strong absorber → requires enrichment. Heavy water (D₂O): similar slowing power, far lower absorption → can run on natural U. Negative vs. Positive temperature coefficient Negative: power drops when temperature rises → inherent safety. Positive: power rises with temperature → risk of runaway. Natural uranium vs. Enriched fuel Natural: 0.7 % \(\mathrm{^{235}U}\), usable in heavy‑water/graphite reactors. Enriched: 3–5 % for LWRs, higher k without excessive moderation. Control rods vs. Chemical poisons Control rods: solid absorbers (Cd, B) inserted/withdrawn mechanically. Chemical poisons: soluble (e.g., soluble boron) dissolved in coolant for fine reactivity tuning. --- ⚠️ Common Misunderstandings “k > 1 means safe” – No; supercriticality without proper control leads to power excursion. “All neutrons are prompt” – Ignoring delayed neutrons underestimates reactor controllability. “More moderator always improves reactivity” – Excess moderator raises neutron absorption, lowering k. “Xe‑135 only matters during shutdown” – It also influences power transients (e.g., after a rapid power increase). “Higher enrichment always better” – Increases proliferation risk and fuel cost; not needed for many designs. --- 🧠 Mental Models / Intuition k as a “gear ratio”: If you think of each neutron as a gear tooth, k = 1 means the gear turns at the same speed each turn; k > 1 speeds up, k < 1 slows down. Reactivity as “budget”: Positive ρ adds “extra neutrons” to the budget; negative ρ subtracts. Delayed neutrons as “brake pads”: Though few, they dramatically increase the time you have to react, preventing the system from “skidding out of control.” Moderator density ↔ “traffic jam”: Too many slow neutrons crowd the core, increasing chances they get captured before causing fission. --- 🚩 Exceptions & Edge Cases Void coefficient sign reversal: In some BWR designs, the void coefficient can be slightly positive at low power, requiring careful startup procedures. High‑temperature reactors (e.g., molten salt): Use different moderators or none; the six‑factor formula still applies but with altered P\F/P\T values. Xe‑135 “burnout” during high‑power operation: At very high flux, Xe‑135 is burned faster than produced, temporarily raising reactivity (the “Xe‑135 dip”). Fast reactors: No moderator; k determined mainly by fast‑fission factor ε and η; delayed neutrons still essential for control. --- 📍 When to Use Which Determine k: Use the six‑factor formula for thermal reactors; use fast‑fission factor and η for fast reactors. Reactivity units: Use pcm for small adjustments (1 pcm = 10⁻⁵ Δk/k); use dollars when comparing to β (critical for safety analysis). Choosing moderator: If enrichment is limited → heavy water or graphite. If coolant must also be moderator → light water. Poison management: For short‑term shutdowns → rely on Xe‑135 decay. For long‑term reactivity control → insert control rods or add soluble boron. Enrichment method selection: Centrifuges are most economical today; laser enrichment reserved for very high purity or small batches. --- 👀 Patterns to Recognize k ≈ 1 & ρ ≈ 0 → system is at steady state; any small positive ρ will cause exponential power rise. Presence of a void in water‑moderated core → sudden drop in reactivity (negative void coefficient). Rapid power increase followed by a dip → likely Xe‑135 buildup (iodine pit effect). High p (resonance escape) + low f (thermal utilization) → indicates too much moderator or absorber in the core. Reactivity insertion > β → prompt critical condition → immediate power surge (dangerous). --- 🗂️ Exam Traps Confusing k and ρ: Remember ρ = (k − 1)/k, not simply k − 1. Assuming all neutrons are prompt: Questions that ask about reactor control always require inclusion of delayed neutrons. Mixing up moderator absorption vs. slowing: Heavy water slows neutrons well and absorbs little; light water absorbs more, demanding enrichment. Neglecting the void effect: In BWR questions, “what happens when steam forms?” – answer: reactivity drops (negative void coefficient). Xenon poisoning timing: “After shutdown, reactivity is highest after hours.” – correct answer  10 h (peak Xe‑135). Dollar vs. pcm: 1 $ = β (≈ 0.0065 for U‑235); 1 $ ≈ 6500 pcm, not 100 pcm. Temperature coefficient sign: Over‑moderated reactors have positive coefficient; many students pick “negative” by default. ---