Nuclear Reactor Physics

A fission event in U-235 releases 2–3 neutrons with kinetic energies around 2 MeV — these are 'fast' neutrons travelling at roughly 2 × 10⁷ m/s. At these high energies, the probability (cross-section) of a neutron being captured by another U-235 nucleus is tiny, around 1 barn (1 b = 10⁻²⁸ m²). Slow the neutron down to 'thermal' energies near 0.025 eV (the kinetic energy of room-temperature particles), and the fission cross-section of U-235 jumps to over 580 barns — more than 500 times larger. The moderator's job is to slow neutrons through repeated elastic collisions without absorbing them. Energy transfer in an elastic collision is maximised when the colliding masses are equal — which is why ordinary water (hydrogen nuclei, mass ≈ 1 u) and graphite (carbon, mass 12 u) are classic moderators. Deuterium oxide (heavy water, D₂O) is even better because deuterium absorbs far fewer neutrons than hydrogen while still slowing them efficiently. Without a moderator, most neutrons escape or are wasted before they can sustain the chain reaction.

The neutron multiplication factor k is defined as the ratio of neutrons in one generation to the previous generation. If k > 1 (supercritical), power rises exponentially and extremely rapidly — exactly what a bomb requires. If k < 1 (subcritical), the reaction dies. A steady reactor needs k = 1 exactly. Control rods are made from materials with extremely high neutron-absorption cross-sections: boron-10 (capture cross-section ≈ 3840 b for thermal neutrons) and cadmium-113 (≈ 20,000 b) are typical choices. Inserting the rods deeper into the core absorbs more neutrons, lowering k and reducing reactor power; withdrawing them raises k and increases power. A vital safety feature arises from 'delayed neutrons': roughly 0.65% of the neutrons from U-235 fission are released seconds to minutes later by radioactive decay of fission fragments. This small delay makes the effective neutron generation time ~0.1 s rather than ~10⁻⁴ s, giving control systems enough time to respond. If a reactor relied only on prompt neutrons, human or mechanical response would be impossibly fast. The RBMK reactor at Chernobyl had a design flaw — a 'positive void coefficient' — meaning steam bubbles in the coolant increased reactivity rather than decreasing it, making control rods temporarily unable to compensate. When operators inserted control rods during a test, a design flaw caused a brief power surge that destroyed the reactor before the rods could damp it.

Fission converts mass to energy at a rate governed by E = mc². For U-235, the total energy per fission is about 200 MeV, of which ~85% appears as kinetic energy of the fission fragments, which is quickly thermalised (converted to heat) within the fuel rods themselves. A coolant — pressurised water, heavy water, liquid sodium, or CO₂ gas depending on reactor type — flows around the fuel rods and carries this thermal energy away. In a Pressurised Water Reactor (PWR), the most common design worldwide, the primary coolant loop is kept at ~155 atm pressure so water remains liquid even above 300 °C. This superheated water flows through a steam generator (the heat exchanger), where it transfers energy to a separate secondary loop of water at lower pressure, producing steam. That steam drives a turbine connected to a generator, converting thermal energy to mechanical to electrical energy. The two loops never mix, so radioactive coolant from the core stays isolated from the turbine system. Thermal efficiency of a PWR is typically 33%, constrained by the Carnot limit: no heat engine can convert heat to work more efficiently than η_Carnot = 1 − T_cold/T_hot.

• k — Neutron multiplication factor (dimensionless): Ratio of neutrons produced in one generation to the previous one. k = 1 for a critical reactor.
• \eta — Reproduction factor (dimensionless): Average number of fission neutrons produced per thermal neutron absorbed in the fuel (~2.07 for U-235).
• f — Thermal utilisation factor (dimensionless): Fraction of thermal neutrons absorbed specifically in the fuel rather than in moderator, coolant, or structural material.
• p — Resonance escape probability (dimensionless): Probability that a neutron slows to thermal energy without being captured at a resonance energy of U-238.
• \varepsilon — Fast fission factor (dimensionless): Slight increase in neutrons from fast-neutron fission of U-238; typically ~1.03.
• P_{NL} — Non-leakage probability (dimensionless): Probability that a neutron does not escape the reactor core before causing a fission.

Problem. A research reactor core contains 10 kg of U-235. If the reactor operates at a steady thermal power of 1.00 MW, estimate how many fission events occur per second, and how long the fuel would last if the entire U-235 mass fissioned at this rate.

Answer. Approximately 3.1 × 10¹⁶ fissions per second; the fuel would theoretically last about 26 years at full power.

• Nuclear Fission and Chain Reactions — Reactor physics directly harnesses the fission chain reaction; understanding fission cross-sections and neutron multiplication is prerequisite to understanding why moderators and control rods are needed.
• Mass–Energy Equivalence (E = mc²) — The 200 MeV of energy released per fission arises from a mass deficit of the products relative to the reactants, directly applying Einstein's famous relation.
• Heat Engines and Carnot Efficiency — The steam turbine in a nuclear plant is a heat engine; its maximum efficiency is set by the Carnot limit, determining how much of the fission heat actually becomes electricity.
• Radioactive Decay and Half-Life — Delayed neutrons — critical for reactor control — come from radioactive decay of fission fragments. Spent fuel rod radioactivity is also governed by decay laws.