Nuclear fission: chain reactions, reactors, fuel cycles

Nuclear fission is the splitting of a heavy atomic nucleus into smaller nuclei, releasing energy. The released energy comes from the nuclear binding energy curve — a plot of binding energy per nucleon against atomic mass — which has a maximum near iron (mass ~56). Nuclei lighter than iron release energy by fusing together; nuclei heavier than iron release energy by fissioning apart.

The most-used fission reaction:

$\text{U-235} + n \rightarrow \text{products} + 2\text{–}3\, n + \sim 200\, \text{MeV}.$

A U-235 nucleus absorbing a neutron splits into two fission products (typical examples: barium and krypton, strontium and xenon — many possible pairs) plus 2–3 free neutrons plus about 200 MeV of energy. The energy appears as kinetic energy of the fragments (which is captured as heat) plus gamma rays plus neutrino energy (which mostly escapes).

Quantitatively: 200 MeV per fission of one U-235 nucleus corresponds to about 80 TJ per kg of fully fissioned U-235 — about $10^7$ times the energy per kg of coal. The structural consequence: a 1 GW nuclear plant consumes about 20–25 tonnes of enriched uranium per year, versus 2–3 million tonnes of coal for the same energy output. The waste volume scales similarly.

A self-sustaining nuclear reactor depends on the neutron multiplication factor $k_{\text{eff}}$ — the number of neutrons in one generation produced per neutron in the previous generation.

• $k_{\text{eff}} < 1$: subcritical. Reaction dies out.
• $k_{\text{eff}} = 1$: critical. Steady-state reaction.
• $k_{\text{eff}} > 1$: supercritical. Reaction grows.

Reactor operation aims for $k_{\text{eff}} = 1$. Excess reactivity is controlled by neutron-absorbing control rods (boron, hafnium, cadmium) that are inserted to push $k_{\text{eff}}$ down, withdrawn to allow it back up.

A critical subtlety enables safe reactor control: about 0.65% of fission neutrons are delayed neutrons released seconds to minutes after the fission event (from decay of certain fission products), not immediately. If the prompt-neutron multiplication alone is just barely subcritical, the reactor still maintains $k_{\text{eff}} = 1$ overall because of the delayed contribution. Power changes happen on the timescale of delayed-neutron production (seconds), not the prompt-neutron timescale ($10^{-5}$ s). This is what makes reactor control by mechanical rods possible. If reactivity exceeds the delayed-neutron fraction, the reactor becomes prompt critical — the timescale collapses to milliseconds, and control becomes impossible (the failure mode of Chernobyl's RBMK in 1986).

U-235 absorbs neutrons most readily when the neutrons are slow (kinetic energy ~0.025 eV, called thermal energies). Fission neutrons emerge fast (~2 MeV). A moderator slows neutrons to thermal energies via elastic collisions with light nuclei.

Three common moderators:

• Light water (H₂O). Effective moderator, abundant, low-cost. Hydrogen absorbs neutrons more than other moderators, so light-water reactors require enriched uranium fuel (3–5% U-235 instead of the natural 0.72%) to compensate.
• Heavy water (D₂O). Deuterium absorbs neutrons much less than hydrogen, so heavy-water reactors can use natural uranium directly. Cost: producing D₂O is energy-intensive (CANDU plants have large heavy-water inventories).
• Graphite. Carbon is a moderately good moderator. Used in early reactors (e.g., the first reactors at Hanford and the Soviet RBMK; the British Magnox and AGR). Graphite-moderated reactors can use natural uranium.

Fast reactors skip moderation entirely. They run on fast neutrons, which require a different fuel mix (~20% fissile material — typically plutonium plus U-238) and a different coolant (sodium or lead — no hydrogen to moderate accidentally). Their advantage: they can breed more fissile material than they consume, by converting U-238 to plutonium. Their cost: substantially more complex coolant systems and metallurgy.

The choice of moderator and coolant shapes nearly everything about reactor design — fuel composition, refueling cycle, safety systems, waste characteristics, and cost.

Roughly 80% of commercial nuclear reactors worldwide are light-water reactors (LWRs) in two main configurations.

Pressurized Water Reactor (PWR). Water in the primary loop is held at ~15 MPa to prevent boiling; it transfers heat through steam generators to a secondary loop where steam is produced for the turbine. Two-loop design separates reactor coolant from turbine steam.

Boiling Water Reactor (BWR). Water in the reactor vessel is allowed to boil; steam is fed directly to the turbine. Single-loop design is simpler but the reactor coolant directly contacts the turbine.

Both use:

• Enriched uranium fuel (3–5% U-235) in zirconium-alloy cladding.
• Light-water moderator and coolant.
• Control rods inserted from above (PWR) or below (BWR).
• Operating temperatures of ~290–320°C (limited by water's pressure-vessel-feasible temperature).
• Thermal efficiencies of ~33% (constrained by low steam temperature).

LWRs benefit from:

• Decades of operational experience. Most reactors built since 1970 are LWRs; the design has been iterated, regulated, and standardized extensively.
• Inherent negative feedback. As reactor temperature rises, water density falls, moderation decreases, $k_{\text{eff}}$ falls — a stabilizing feedback. This is opposite of the RBMK's positive void coefficient that contributed to the Chernobyl accident.
• Established supply chain. Fuel fabricators, reactor vendors, regulators all have LWR-specific competencies.

Limits:

• Low thermal efficiency (~33%) due to water's pressure-vessel temperature ceiling.
• Pressure-vessel embrittlement sets reactor lifetime to 40–80 years with appropriate license extensions.
• Spent fuel contains long-lived actinides (plutonium isotopes, neptunium, americium) that drive the waste-management problem.

The nuclear fuel cycle is the set of processes that turn uranium ore into reactor fuel and (optionally) recycle spent fuel.

Mining and milling. Uranium ore (typically 0.1–1% U) is mined, crushed, and chemically processed into yellowcake (U₃O₈). World reserves are large; supply has been adequate from a handful of countries (Kazakhstan, Canada, Australia, Niger, Russia).

Conversion. Yellowcake is converted to gaseous uranium hexafluoride (UF₆) for enrichment.

Enrichment. Natural uranium is 0.72% U-235 / 99.28% U-238. LWRs need ~3–5% U-235; some research reactors and naval propulsion need ~20% (low-enriched uranium, LEU); weapons need >90% (highly enriched uranium, HEU). The technology is centrifuge separation in nearly all modern facilities. Enrichment capacity is concentrated in a few firms (Urenco, Tenex, CNNC, Orano).

Fuel fabrication. Enriched UF₆ is converted to UO₂ ceramic pellets, loaded into zirconium-alloy tubes, and assembled into fuel bundles.

Reactor operation. A fuel assembly stays in the reactor 4–6 years, gradually transitioning from fresh fuel (mostly U-235) to spent fuel (depleted U-235, accumulated fission products, plutonium and minor actinides).

Spent fuel storage. Removed from the reactor, fuel sits in spent-fuel pools for several years to cool, then can move to dry-cask storage for decades.

Once-through vs reprocessing.

• Once-through cycle (US, Canada, most others): spent fuel is stored indefinitely. Simpler and lower-cost but accumulates the long-lived actinide inventory.
• Reprocessing (France, Russia, UK historically, Japan): spent fuel is dissolved and uranium plus plutonium recovered for MOX (mixed-oxide) fuel. Reduces waste volume and uranium consumption but is expensive and has proliferation implications.

The choice between once-through and reprocessing is partly economic, partly policy. There is no global consensus, and different national programs have made different choices.

Beyond LWRs, several other reactor families operate or are in development.

CANDU (Canada Deuterium Uranium). Heavy-water moderator and coolant; natural uranium fuel; online refueling. Built in Canada, India, Korea, China, Argentina, Romania. Distinctive advantage: no enrichment infrastructure required.

Gas-cooled reactors. Use CO₂ or helium coolant and graphite moderator. Britain's Magnox and AGR designs are the historical examples. Newer designs (HTGRs — high-temperature gas-cooled reactors) target much higher coolant temperatures (~750°C) to enable process-heat applications and higher thermal efficiency.

Sodium-cooled fast reactors. Use liquid sodium coolant, no moderator, with fast neutron spectrum. France's Phénix and Superphénix, Russia's BN-600 and BN-800, and several research reactors. Capable of breeding fuel from U-238 and burning long-lived actinides (reducing waste's hazard timescale).

Lead-cooled fast reactors. Use molten lead or lead-bismuth eutectic. Lower freezing point trade-off and corrosion challenges; Russian submarine experience plus several design programs.

Molten-salt reactors. Use molten fluoride or chloride salt as fuel-carrying coolant. Fluid fuel allows online removal of fission products and addition of fresh fuel. Operational examples have been research reactors; several commercial designs are in licensing.

Small modular reactors (SMRs). A category of designs (LWR, HTGR, fast, molten-salt) at 50–300 MWe scale, intended to be factory-built, transported to site, and assembled with substantially shorter construction time and lower up-front capex per unit than conventional reactors. NuScale (US, light-water), Rolls-Royce SMR (UK, light-water), TerraPower Natrium (sodium fast reactor with molten-salt thermal storage), X-energy (HTGR), and several others.

The structural appeal of SMRs: smaller absolute capital outlay per unit, factory production economies, modular site deployment, simpler licensing for standardized designs. The structural challenge: building the first units, achieving the cost reductions that depend on volume production, and competing with the falling cost of variable renewables plus storage.

The SMR industry has had high attention and slow build-out; the first commercial units are coming online in the late 2020s, with the question of whether the cost trajectories meet projections still empirical.

Spent fuel from an LWR contains:

• ~95% remaining uranium (mostly U-238, depleted U-235).
• ~1% plutonium (a mix of isotopes; some fissile, some not).
• ~4% fission products (medium-lived; most decay to background within ~300 years).
• ~0.1% minor actinides (neptunium, americium, curium; long-lived).

The radiological hazard of spent fuel falls by a factor of 1000 over the first 1000 years (from fission-product decay), then declines slowly over hundreds of thousands of years from the actinide content.

Disposal approaches:

• Deep geological repositories. Bury the waste in stable rock formations 300–1000+ m below the surface, isolated from the biosphere by engineered and natural barriers. Finland's Onkalo (operating from ~2025), Sweden's Forsmark (under construction), and France's Cigeo (in licensing) are the most advanced examples. The US Yucca Mountain project remains administratively suspended; deep boreholes are an emerging alternative.
• Reprocessing. Reduces the volume requiring deep disposal and removes much of the long-lived actinide content (used as MOX fuel in some reactors). Doesn't eliminate the waste problem — fission products still require disposal.
• Transmutation in fast reactors. Long-lived actinides can be fissioned in fast spectra, converting them into shorter-lived fission products. Reduces the hazard timescale from 100,000+ years to ~300 years. Engineering reality has been slower than concepts.

Decommissioning a reactor at end of life is a 40–80-year project covering shutdown, fuel removal, decontamination, dismantlement, and site remediation. Costs typically run $500 million to$2 billion per unit, with most of that incurred decades after the last operation.

The overall waste problem is well-bounded engineering — small volumes of high-radioactivity material — but is politically contested in most countries. The structural argument for nuclear's place in low-carbon generation rests substantially on whether countries can establish deep-geological repository programs.

Cost. Nuclear is capital-intensive. Modern LWR construction in recent Western projects (Olkiluoto, Flamanville, Vogtle, Hinkley Point C) has produced overnight capital costs of $7,000–12,000/kW, with construction times of 10–15 years and substantial cost overruns relative to initial budgets. Asian builds (in South Korea, China, UAE) have completed faster and cheaper. The cost difference reflects construction-management experience, supply-chain depth, regulatory turnover, and labor-force familiarity — not differences in physics.

Levelized cost of nuclear is sensitive to discount rate. At low rates (~3%), recent builds approach competitiveness with other low-carbon sources. At high rates (~7%), they price out unless the capacity value and zero-emission attributes are recognized in the market design. Nuclear's economic position therefore depends partly on the policy and market regime in addition to construction execution.

Safety. Modern LWR designs have probabilistic risk assessments showing core-damage frequencies of $10^{-5}$ to $10^{-6}$ per reactor-year. The three large public accidents (Three Mile Island 1979, Chernobyl 1986, Fukushima 2011) had qualitatively different causes (operator error and design weakness, prompt criticality in an unstable design, station blackout with hydrogen explosion) and qualitatively different consequences. Per-TWh mortality from nuclear (including accidents and lifecycle) is among the lowest of any generation source — well below coal, oil, and natural gas, and comparable to wind and solar — when assessed using the same methodology across sources.

Industry structure. Reactor vendors are concentrated: Westinghouse (US/Japanese ownership), Framatome (France), Rosatom (Russia), KEPCO (Korea), CGN and CNNC (China). Fuel cycle services are more concentrated still: Urenco and Tenex on enrichment, several specialty firms on fabrication. The supply chain runs through skilled metallurgical, nuclear-engineering, and project-management workforces that are not quickly rebuilt where they have eroded.

The structural lesson: nuclear's role in any country's low-carbon mix depends on construction execution, regulatory durability, supply-chain capacity, and electricity-market design. The physics is straightforward; the engineering is mature; the institutional and economic factors are where the variation across countries lies.