The Nuclear Chain Reaction: One Neutron Becomes a Trillion

The loop, and the one number that governs it

A chain reaction is a feedback loop written in neutrons. Step one: a free neutron is absorbed by a fissile nucleus. Step two: the nucleus splits (fission), releasing about 200 MeV of energy and, on average, 2 to 3 new neutrons. Step three: those neutrons scatter through the fuel, and some are absorbed by other fissile nuclei — returning us to step one. The reaction sustains itself only if the loop closes on itself often enough.

The bookkeeping collapses into a single quantity, the multiplication factor k, defined as the number of neutrons in one generation divided by the number in the generation before it. If each fission's 2.4 neutrons produce on average one further fission, then k = 1 and the population is constant. If they produce more than one, k > 1 and the population climbs geometrically: after n generations it is multiplied by kⁿ. This is the whole story of exponential growth. Starting from a single neutron, reaching a trillion (10¹²) descendants takes only about 40 doublings; fissioning the ~2.5×10²⁴ nuclei in a kilogram of uranium takes roughly 80 doublings. In a fast assembly where each generation lasts about ten nanoseconds, those 80 generations elapse in under a microsecond — which is why a supercritical mass releases its energy as an explosion. In a reactor, the same loop is held at exactly k = 1.

Inside a single fission: a liquid drop pulled apart

Why does a nucleus split at all? The best working picture is the liquid-drop model of Bohr and Wheeler (1939). A nucleus behaves like a charged droplet held together by the short-range strong nuclear force — an effective surface tension — while every proton repels every other through the long-range Coulomb force. In a heavy nucleus like uranium (92 protons) these are nearly balanced. When U-235 absorbs a neutron it becomes an excited U-236 compound nucleus, and the added energy sets the drop oscillating. If it deforms into a dumbbell shape, surface tension tries to pull it back, but Coulomb repulsion between the two bulging ends tries to drive them apart. Past a critical deformation — the saddle point, guarding a fission barrier of about 6 MeV — repulsion wins, the neck pinches off at scission, and two fragments fly apart carrying ~168 MeV of kinetic energy.

The energy comes from the nuclear binding-energy curve. Uranium is bound by about 7.6 MeV per nucleon; the mid-mass fragments (typically around mass 95 and 140, e.g. krypton and barium) are bound by about 8.5 MeV per nucleon. That ~0.9 MeV/nucleon gain, over 236 nucleons, is the ~200 MeV released — a hundred million times the energy of a chemical bond. Niels Bohr's key insight explains which nuclei do this with slow neutrons: U-235 has an odd number of neutrons, so adding one delivers a pairing-energy bonus that pushes the compound nucleus's excitation (~6.5 MeV) just above its fission barrier — a zero-energy thermal neutron suffices. U-238 has an even neutron count; adding a neutron yields less excitation (~4.8 MeV), below its barrier, so it fissions only if struck by a fast neutron of ~1 MeV or more.

The multiplication factor and the neutron economy

To engineer k you must track where every neutron goes. In a large thermal reactor the infinite-medium factor is written as the four-factor formula, k∞ = η · ε · p · f:

• η (reproduction factor): neutrons produced per neutron absorbed in fuel, η = ν/(1+α) ≈ 2.07 for thermal U-235, since some absorptions merely capture the neutron (n,γ) without fissioning.
• ε (fast-fission factor): a small bonus (~1.03) from fast neutrons that fission U-238 before slowing down.
• p (resonance-escape probability): the fraction of neutrons that slip past U-238's fierce capture resonances (thousands of barns at eV energies) while slowing down.
• f (thermal utilization): the fraction of thermalized neutrons absorbed in fuel rather than in moderator, coolant, or structure.

A real, finite reactor also leaks neutrons out of its surface, so the effective factor is k = k∞ · P_fast · P_thermal, where the non-leakage probabilities depend on the core's size and shape through its geometric buckling. Operators speak in terms of reactivity, ρ = (k−1)/k, the fractional departure from criticality. Reactivity is the control variable: every rod motion, temperature change, and fuel burn-up is accounted as so many units of ρ added or removed.

Moderators and control rods: dialing k to 1

Fission neutrons are born fast, with an average energy near 2 MeV, but U-235's fission cross section is only ~2 barns for such neutrons and a huge ~585 barns for thermal neutrons at 0.025 eV (cross sections for absorption roughly follow a 1/v law at low energy). The job of the moderator is to slow neutrons to thermal speeds through elastic scattering. Kinematics favors light nuclei: a head-on elastic collision transfers the most energy when the target mass equals the neutron's, so a neutron can lose everything in one hit on a hydrogen nucleus but only a sliver on carbon. It takes on average ~18 collisions with hydrogen (ordinary water), ~25 with deuterium (heavy water), or ~114 with carbon (graphite) to thermalize a fission neutron. Heavy water and graphite are prized because they barely absorb neutrons themselves, letting natural or slightly enriched uranium reach k = 1.

Control rods do the opposite: they are made of voracious neutron absorbers — boron-10, whose (n,α) cross section is ~3840 barns, or cadmium-113 at ~20,000 barns — and pushing them into the core removes neutrons from the economy, lowering k. Withdrawing them raises k. Because the neutron chain responds to reactivity within milliseconds, operators do not adjust k directly; they insert small reactivity steps and let the physics settle, trusting one more effect to keep the response slow enough to manage.

Critical mass: when volume beats surface

Even with k∞ > 1, a chain reaction dies if the fuel is too small, because neutrons escape through the surface before finding a nucleus. This is a geometry contest. Neutron production happens throughout the bulk, scaling with volume (~r³); neutron leakage happens at the boundary, scaling with surface area (~r²). The ratio of loss to production therefore falls as 1/r, so there is always a size above which production wins. That threshold is the critical mass — the smallest amount of fuel in which one neutron, on average, triggers exactly one further fission before escaping.

For a bare sphere of pure metal the numbers are famous: about 52 kg for uranium-235 and about 10 kg for plutonium-239, the plutonium figure smaller because its higher ν ≈ 2.9 and larger fission cross section make it a more prolific neutron source. Surrounding the core with a neutron reflector (a tamper of beryllium, natural uranium, or tungsten) bounces escaping neutrons back and can cut the critical mass by more than half — to ~15 kg for U-235 with a thick reflector. A weapon works by assembling a supercritical geometry faster than the growing reaction can blow it apart: a gun-type design fires one subcritical piece into another; an implosion design crushes a subcritical sphere to higher density with chemical explosives, shrinking every neutron's mean free path. Nature ran the same experiment: at Oklo in Gabon, ~1.7 billion years ago, when natural uranium was ~3% U-235, groundwater-moderated ore bodies sustained slow chain reactions for hundreds of thousands of years.

Delayed neutrons: the seconds that make reactors possible

Here is the subtlety that makes controlled fission power feasible at all. About 99.35% of fission neutrons are prompt, emitted within ~10⁻¹⁴ s of scission. The remaining ~0.65% (the delayed-neutron fraction β ≈ 0.0065 for U-235; only ~0.0021 for Pu-239) are delayed: they are emitted by the beta decay of certain neutron-rich fission fragments — precursors such as bromine-87 and iodine-137 — over half-lives ranging from ~0.2 seconds to ~56 seconds. The neutron does not appear until its precursor decays, so these neutrons arrive seconds after the fission that made them.

Their effect on timing is decisive. The prompt-neutron lifetime in a thermal reactor is only ~10⁻⁴ s; if the chain ran on prompt neutrons alone, a reactivity of just +0.1% would double the power in a fraction of a second — far too fast to control. But delayed neutrons stretch the average generation time to about 0.1 s, so as long as the reactor is kept delayed critical — with reactivity held below β (below "one dollar") — the power rises on a leisurely timescale of tens of seconds, easily managed by rods and feedback. Cross the line to prompt critical (ρ ≥ β), and the reactor no longer needs the delayed neutrons to sustain itself; it goes supercritical on prompt neutrons alone, with a power excursion in milliseconds. That boundary is exactly what the SL-1 accident (1961) and the Chernobyl disaster (1986) crossed, and exactly what a bomb is engineered to cross deliberately. The first proof that the whole scheme worked came on 2 December 1942, when Enrico Fermi's Chicago Pile-1 — a lattice of graphite and natural uranium with cadmium control rods — was brought to k = 1 and held there, the world's first controlled, self-sustaining nuclear chain reaction.