# Nuclear chain reaction A **nuclear chain reaction** is a sequence of [[Nuclear_fission|fissions]] in which the [[Neutron|neutrons]] released by one splitting [[Atomic_nucleus|nucleus]] go on to split others. A single fission of ²³⁵[[Uranium|U]] frees two or three spare neutrons and about 172 MeV of energy, and the whole behaviour of the system turns on one dimensionless number: the fraction of those neutrons that survive to cause another fission.[^murphy-fission][^murphy-table157] Call the average number of fissions caused by one fission *k*. If k is less than one the reaction dies, if k is greater than one it grows geometrically, and if k is exactly one it holds — and the entire difference between a power station and a bomb lies in the third decimal place of that number. In the microsim below the reader has a single control, k from 0.80 to 1.20, and watches a live branching model rather than a formula. Each fission emits ν ∈ {2, 3} neutrons with a mean of 2.5 — an illustrative reading of Murphy's "typically 2–3" — and each neutron causes a further fission with [[Probability|probability]] k/2.5, so that on average `N(g+1) = k·N(g)` from one generation to the next.[^murphy-fission][^spec-e48] Three panels read out: a log-scale population trace, cumulative energy at 172.3 MeV per fission, and a histogram of fragment masses drawn from the thermal ²³⁵U yield table, which always grows two humps near A = 95 and A = 140 however the run is seeded.[^murphy-table157][^endf] Every hundredth event is drawn in full with its [[Atomic_number|Z]] and N bookkeeping, so the conservation the equation hides stays visible. The sim is deliberately *not* a reactor: it has no [[Delayed_neutron|delayed neutrons]], no moderator and no temperature, and its k is imposed rather than computed.[^spec-e48] On the [[Energy]] flagship this article serves the *Fission and the chain reaction* section of Part V — Transformation, and it is the root of the nuclear family: [[Nuclear_reactor]] adds the delayed neutrons that make k steerable, and [[Nuclear_fuel_cycle]] follows the uranium that feeds it. ## History Leó Szilárd conceived a neutron-multiplying chain reaction in 1933, before any fission had been observed, and patented the idea the following year; what he lacked was a reaction that released more neutrons than it consumed.[^rhodes] Otto Hahn and Fritz Strassmann found it at the end of 1938, when uranium bombarded with neutrons produced barium, and Lise Meitner and Otto Frisch interpreted the result in the first weeks of 1939 as the nucleus splitting in two with a large release of energy.[^rhodes] The first controlled self-sustaining chain reaction followed on 2 December 1942, in a lattice of [[Graphite|graphite]] and natural uranium built under the stands of a stadium in Chicago.[^rhodes] Nature had done it first. At Oklo in Gabon, uranium ore bodies sustained chain reactions about 1.7 billion years ago, as the depleted ²³⁵U left behind in the ore revealed in 1972.[^oklo] The reason they could is arithmetic on the two [[Half-life|half-lives]]: with ²³⁵U at 0.704 Gyr and ²³⁸U at 4.47 Gyr, the present 0.72 % of ²³⁵U in natural uranium was about 2.9 % that long ago — the [[Enriched_uranium|enrichment]] of modern light-water reactor fuel, and enough for ordinary groundwater to serve as the moderator.[^murphy-fuel][^derived-ncr] ## Process A thermal neutron absorbed by ²³⁵U makes a ²³⁶U nucleus that immediately splits. Murphy writes the canonical example as Eq. 15.2, `²³⁵U + n → ⁹⁰Br + ¹⁴⁴La + 2n`, and the point of writing it out is the bookkeeping: 236 nucleons go in and 236 come out, and the two fragment charges must sum to 92.[^murphy-fission] The number of spare neutrons is not fixed — typically two or three — and neither are the fragments. ### Fuel Only three nuclides are fissile by thermal neutrons in practice: ²³³U, ²³⁵U and ²³⁹[[Plutonium|Pu]]. Of these only ²³⁵U occurs in nature, and it makes up 0.72 % of natural [[Uranium|uranium]] against 99.2745 % of ²³⁸U, a ratio of about 140 to 1.[^murphy-fuel] The reason the ratio is that lopsided is decay: ²³⁵U has a half-life of 0.704 Gyr and ²³⁸U 4.47 Gyr, so the light [[Isotope|isotope]] has been disappearing six times faster, and running the ratio backwards puts the two at equal abundance about 6 billion years ago — before the Earth existed.[^murphy-fuel] The energy density is what makes the [[Fissile_material|fuel]] worth the trouble. Fission of ²³⁵U releases 16.8 million kcal per gram, against about 10 kcal/g for any chemical [[Energy_density|fuel]], a factor near two million.[^murphy-table157] Murphy's Example 15.4.2 converts that into a household number: a person drawing 10,000 W continuously for a year needs 3.155×10¹¹ J, which is 4.5 g of ²³⁵U.[^murphy-table157] At world scale, burning half of the 0.72 % of ²³⁵U in 7.6 Mt of uranium reserves gives 27,300 t of ²³⁵U, about 2×10²¹ J.[^murphy-fuel] ### Enrichment process Natural uranium at 0.72 % will chain-react only with a very good moderator and a great deal of it, so most reactors run on fuel [[Enriched_uranium|enriched]] to 3–5 % ²³⁵U; weapons need at least 20 % and typically about 85 %, and the ²³⁸U-rich residue, at 0.3 % or less, is depleted uranium.[^murphy-fuel] Because the two isotopes are chemically identical, separation is physical and exploits the mass difference alone. In the gaseous route the working substance is UF₆, and [[Graham's_law|Graham's law]], `r₁/r₂ = √(M₂/M₁)`, gives a per-stage separation factor of √(352.04/349.03) = 1.0043 — a 0.43 % edge, which is why a [[Gaseous_diffusion|diffusion]] plant is a cascade of more than a thousand stages.[^averill-graham] The cascade itself belongs to [[Nuclear_fuel_cycle|the fuel cycle]]; here it matters only because k depends on it. ### Reaction products Fission fragments are random but not uniform. The mass split is strongly bimodal, with peaks near A = 95 and A = 140, and the sim samples that distribution from an inverse-CDF table built on evaluated thermal-fission yield data rather than choosing one fixed pair.[^murphy-fission][^endf] The bookkeeping in the sim's every-hundredth event is Murphy's Example 15.4.1: a bromine fragment at Z = 35 forces its partner to Z = 57, lanthanum, and the neutron count then fixes the mass number — ¹⁴⁶La if no neutrons escape, ¹⁴⁴La if two do.[^murphy-fission] The energy comes from the mass. Murphy's Table 15.7 balances the reaction at 236.05259 amu in and 235.86757 amu out, a defect of 0.18502 amu, which at 931.494 MeV per amu is 172.3 MeV — and this is the constant the sim's energy counter uses.[^murphy-table157] It is worth knowing that the quick graphical estimate from the [[Nuclear_binding_energy|binding-energy]] curve gives about 210 MeV for the same reaction, because it credits the fragments with binding they have not yet acquired and ignores the unbound spare neutrons and the later [[Beta_decay|β⁻]] decays.[^murphy-be] Those later decays are why fission products remain a problem long after the chain stops: ⁹⁰Sr and ¹³⁷Cs dominate the waste from about five years to a few hundred.[^murphy-fuel] ## Reactor physics Turning "each fission causes k more" into a machine requires two quantities the branching model leaves out: how long a generation takes, and what actually determines k. The sim supplies neither — it counts generations rather than seconds, and takes k as given — so both have to be put back by hand. ### Prompt neutron lifetime The prompt neutron lifetime ℓ is the mean time from a neutron's birth in fission to its absorption or escape. In a thermal system most of it is spent slowing down and then diffusing at thermal speed, and ℓ is of order 10⁻⁴ s; in a fast, unmoderated system nothing slows the neutrons and ℓ falls to about 10⁻⁷ s.[^lamarsh] That difference of three orders of magnitude is the difference between a device that can be steered and one that cannot. The arithmetic is brutal. On prompt neutrons alone, a population grows as `N(t) = N₀·exp((k−1)·t/ℓ)`, so a thermal system at ℓ = 10⁻⁴ s held at k = 1.001 — one part in a thousand above critical — multiplies its power by e every 0.1 s, a factor of 22,000 in one second.[^derived-ncr][^lamarsh] No mechanical control could follow that, and the fact that real reactors are operated by hand is the clue that something else governs them: see [[Nuclear_reactor]]. ### Mean generation time The mean generation time Λ is the prompt lifetime divided by the multiplication factor, `Λ = ℓ/k`, and it is Λ rather than ℓ that appears in the kinetics equations.[^lamarsh] In the sim, one step of the log population trace is one generation; because the sim has no clock, the horizontal axis is generations, not seconds, and the reader supplies the timescale by choosing a system. At Λ ≈ 10⁻⁴ s the 70 generations in which a k = 1.01 population doubles occupy 7 ms; in a fast assembly at Λ ≈ 10⁻⁸ s the same 70 generations take under a microsecond.[^derived-ncr] ### Effective neutron multiplication factor The effective multiplication factor k_eff is defined as the ratio of neutrons in one generation to those in the previous — equivalently, neutron production divided by the sum of absorption and leakage. The system is subcritical below 1, critical at exactly 1 and supercritical above it, and the standard decomposition writes k_eff as a product of six factors: the neutrons produced per thermal absorption in fuel, the fraction of thermal absorptions that occur in fuel, the resonance escape probability, the fast fission factor, and two non-leakage probabilities.[^lamarsh] Every one of them is a design or operating variable, which is what makes k an output of the machine rather than an input. The sim inverts that relationship deliberately, making k the one control so the consequence is visible in isolation.[^spec-e48] The consequence is geometric: at k = 1.01 the population doubles in about 70 generations, at k = 0.99 it halves in about 69, and one percent decides everything.[^derived-ncr] Ball's introductory treatment makes the same point with pure doubling — one fission becoming 16,384 in fourteen generations at k = 2.[^ball-chain] Murphy describes the chain and its control in principle only, and prints no multiplication equation at all, which is why the forms above are given here as standard rather than as Portal Book results.[^murphy-fuel][^lamarsh] ## Nuclear weapons A weapon runs the same chain with the opposite design goal: k as far above 1 as possible, in a fast unmoderated assembly where Λ is of order 10⁻⁸ s, so that the population climbs through the fifty or sixty generations needed to release a significant fraction of the material's energy in well under a microsecond — before the assembly blows itself apart and k falls below 1. The assembly itself is a geometry problem. A neutron's mean free path in the metal scales as 1/ρ, and criticality requires the radius to be a fixed multiple of that path, so the [[Critical_mass|critical mass]] of a bare sphere scales as `M ∝ ρ·R³ ∝ 1/ρ²`.[^derived-ncr][^serber] Compressing fissile metal to twice its normal density therefore cuts the critical mass by a factor of four, which is the whole principle of an implosion design. A power reactor cannot do any of this: its fuel is a few percent ²³⁵U in a lattice that only works when a moderator is present, so it can melt but cannot explode like a bomb.[^murphy-fuel] ### Predetonation The danger in any weapon design is that the chain starts while the assembly is only barely supercritical, so the material disassembles at a tiny fraction of its yield — predetonation, or the fizzle. The neutron that starts it need not be supplied: spontaneous fission in the material provides a background rate, and in reactor-produced plutonium the ²⁴⁰Pu content raises that rate far enough that a slow gun-type assembly is certain to predetonate. That single fact forced the implosion route for plutonium, and it is the reason weapons-grade plutonium is defined by its ²⁴⁰Pu fraction rather than by its ²³⁹Pu fraction alone.[^serber] The same statistics, benignly, govern reactor startup, where a deliberate neutron source is installed so that the approach to criticality is monitored rather than blind.[^lamarsh] ## Nuclear power plants and control of chain reactions A power reactor holds k_eff at exactly 1 and moves it by a fraction of a percent to change power. Murphy describes control rods in principle — insert a strong absorber and the population falls, withdraw it and the population grows — without an equation, and that is as far as the chain reaction alone can take the story.[^murphy-fuel] The physics that makes the holding possible is that a small fraction of fission neutrons, about 0.65 % in ²³⁵U, are not prompt at all but are emitted seconds later by decaying fission products; they stretch the effective generation time from 10⁻⁴ s to something near a tenth of a second, and a reactor is operated in the band where those [[Delayed_neutron|delayed neutrons]] are needed to reach criticality. The point-kinetics treatment of that band belongs to [[Nuclear_reactor]]. The scale is worth stating. In 2019 the world fleet was 455 plants with 393 GW installed and about 295 GW average output, some 11 % of world electricity.[^murphy-fuel] At 172.3 MeV per fission, one thermal megawatt-day consumes about 1.2 g of ²³⁵U, so a 3 GW-thermal core burns roughly 3.7 kg a day — a fuel logistics problem unlike any other kind of [[Power_(physics)|power]] plant.[^derived-ncr][^murphy-table157] The [[Control_rod|control rods]], the moderator temperature and the accumulating fission-product poisons are all, in the end, terms in k. ## See also - [[Nuclear_fission]] — the single event this page chains together - [[Fission_product_yield]] — the bimodal histogram the sim samples - [[Critical_mass]] - [[Fissile_material]] - [[Nuclear_reactor]] — k made steerable by delayed neutrons - [[Nuclear_fuel_cycle]] — where the enriched uranium comes from - [[Nuclear_binding_energy]] — why the 172 MeV is there to release - [[Delayed_neutron]] ## References [^murphy-fission]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*. Chapter 15: only ²³³U, ²³⁵U and ²³⁹Pu are fissile and only ²³⁵U occurs naturally, p. 270; fragment charges and neutron numbers conserved, pp. 271–273; Eq. 15.2 `²³⁵U + n → ⁹⁰Br + ¹⁴⁴La + 2n` with 236 nucleons throughout, and the typical 2–3 spare neutrons and bimodal fragment masses near A ≈ 95 and ≈ 140, pp. 272–273; Example 15.4.1, the Br/La charge bookkeeping, p. 272. https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^murphy-table157]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*. Chapter 15, Table 15.7: 236.05259 amu in, 235.86757 amu out, Δm = 0.18502 amu = 172.3 MeV, pp. 273–274; Box 15.3, 16.8×10⁶ kcal/g against ~10 kcal/g for chemical fuel, p. 274; Example 15.4.2, 4.5 g of ²³⁵U per person-year at 10,000 W, p. 274. https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^murphy-be]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*. Chapter 15: Table 15.5 binding energies, including ²³⁵U at 7.59 MeV per nucleon, pp. 267–268; the graphical fission estimate giving ≈210 MeV against the exact 172 MeV, and why it overestimates, pp. 274–275. https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^murphy-fuel]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*. Chapter 15: the chain reaction and control rods described in principle, with no multiplication equation given, p. 275; the 2019 fleet at 455 plants, 393 GW installed, 295 GW average and 11 % of world electricity, p. 276; natural uranium at 0.72 % ²³⁵U and 99.2745 % ²³⁸U with half-lives 0.704 Gyr and 4.47 Gyr and equal abundance ≈6 Gyr ago, p. 277; reactor fuel at 3–5 %, weapons at ≥20 % and typically ~85 %, depleted uranium at ≤0.3 %, and the 27,300 t / 2×10²¹ J reserve figure, pp. 277–278; the ⁹⁰Sr and ¹³⁷Cs waste window, p. 281; fuel can melt down but cannot explode like a bomb, p. 282. https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^averill-graham]: Averill, Bruce; Eldredge, Patricia (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, *Gases*: Graham's law `r₁/r₂ = √(M₂/M₁)`, p. 945; the UF₆ cascade with M(²³⁵UF₆) = 349.03 and M(²³⁸UF₆) = 352.04 g/mol, a per-stage factor of 1.0043, and the book's stage count of about 1.15×10³, pp. 948–949. https://open.umn.edu/opentextbooks/textbooks/general-chemistry-principles-patterns-and-applications [^ball-chain]: Ball, David (2011). *Introductory Chemistry*, pp. 752–757: the chain reaction developed as pure doubling, one fission to 16,384 in fourteen generations, and the fissile-material thresholds at p. 757. https://open.umn.edu/opentextbooks/textbooks/introductory-chemistry [^lamarsh]: Lamarsh, John R.; Baratta, Anthony J. *Introduction to Nuclear Engineering*, 3rd ed. Chapters on nuclear reactor theory and reactor kinetics (page to pin): the prompt neutron lifetime of order 10⁻⁴ s in thermal systems and 10⁻⁷ s in fast systems, the mean generation time `Λ = ℓ/k`, the definition of k_eff as production over absorption plus leakage, the six-factor formula, and the startup neutron source. No Energy Portal Book states reactor kinetics; these are standard-text results, and this page cites them as such rather than assigning them a page in a Portal Book. [^serber]: Serber, Robert. *The Los Alamos Primer* (1943 lectures; annotated edition, University of California Press, 1992), page to pin: the fast unmoderated chain, the disassembly that ends it, critical mass and compression, and predetonation from background neutrons — including the spontaneous-fission rate that rules out a gun assembly for reactor-grade plutonium. [^rhodes]: Rhodes, Richard. *The Making of the Atomic Bomb* (Simon & Schuster, 1986), page to pin: Szilárd's 1933 conception and 1934 patent, the Hahn–Strassmann radiochemistry of December 1938, the Meitner–Frisch interpretation of January 1939, and the first self-sustaining chain reaction on 2 December 1942. [^oklo]: International Atomic Energy Agency, reporting on the *Oklo natural fission reactors*, Gabon (page to pin): the anomalous ²³⁵U depletion found in 1972 and the reactor episodes dated to about 1.7 billion years ago. No Portal Book covers Oklo. [^endf]: Cross Section Evaluation Working Group, *ENDF/B-VIII.0 evaluated nuclear data library* (2018), distributed by the National Nuclear Data Center, Brookhaven National Laboratory (page to pin): the thermal ²³⁵U fission-product mass yields from which the sim's inverse-CDF table is baked. Murphy's Fig. 15.15 shows the histogram but publishes no tabulated yields. [^spec-e48]: Matter & Energy Cluster contract, `_registry/plans/ENERGY_SECTIONS.md` row E48: the sim concept, with k ∈ [0.80, 1.20] as the single control; the live branching model drawing ν ∈ {2, 3} with mean 2.5 and fissioning with probability k/2.5; the log population trace, the cumulative energy counter at 172.3 MeV per fission, the fragment-mass histogram from the ²³⁵U yield table, and the Z/N bookkeeping on every hundredth event. The contract carries the pitfall this page repeats: the k slider is not a reactor model — there are no delayed neutrons and no moderator, and it must be labelled. [^derived-ncr]: Computed for this article from the equations on the page and the cited constants: the Oklo enrichment, `(235/238)_then = (235/238)_now · 2^(t/T₂₃₅ − t/T₂₃₈)` = 0.007253 × 2^(1.7/0.704 − 1.7/4.47) = 0.0297, a ²³⁵U fraction of 2.9 % at t = 1.7 Ga, with the half-lives of [^murphy-fuel]; the prompt-only growth `exp((k−1)t/ℓ)` = e^10 = 2.2×10⁴ in one second at k = 1.001 and ℓ = 10⁻⁴ s; doubling and halving times ln2/ln(1.01) = 69.7 and ln0.5/ln(0.99) = 69.0 generations; 70 generations at Λ = 10⁻⁴ s = 7 ms and at Λ = 10⁻⁸ s = 0.7 µs; the critical-mass scaling `M ∝ ρR³` with `R ∝ λ ∝ 1/ρ`, hence `M ∝ 1/ρ²`; 172.3 MeV = 2.76×10⁻¹¹ J, so 1 MW-day = 8.64×10¹⁰ J is 3.13×10²¹ fissions = 1.2 g of ²³⁵U at 3.902×10⁻²² g per atom, and a 3 GW-thermal core burns 3.7 kg/day; the person-year check, 3.155×10¹¹ J ÷ 2.76×10⁻¹¹ J = 1.14×10²² fissions = 4.46 g, against Murphy's printed 4.5 g. ## External links - [*Energy and Human Ambitions on a Finite Planet*](https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet) (Murphy, 2021), Chapter 15 — the fission bookkeeping, Table 15.7 and the fuel arithmetic behind this page - [*General Chemistry: Principles, Patterns, and Applications*](https://open.umn.edu/opentextbooks/textbooks/general-chemistry-principles-patterns-and-applications) (Averill & Eldredge, 2011), Chapter 10 — Graham's law and the UF₆ cascade - [*Introductory Chemistry*](https://open.umn.edu/opentextbooks/textbooks/introductory-chemistry) (Ball, 2011) — the doubling picture of the chain - The Wikipedia pair's *External links* section lists the agency and data-library sites for the nuclear data used here. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Nuclear_chain_reaction.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Nuclear chain reaction* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Nuclear_chain_reaction.html" data-title="Nuclear chain reaction"></div> *Built from `MICROSIM_GUIDE/specs/sims/Nuclear_chain_reaction.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Nuclear_chain_reaction) : [Wikitube](https://en.wikitube.io/wiki/Nuclear_chain_reaction) · pinned revision [1374312317](https://en.wikipedia.org/w/index.php?oldid=1374312317) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Energy]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Energy row E48 · sim pending (matter/Nuclear_chain_reaction).*