Nuclear Reactor Simulator: Criticality, Doppler Feedback and SCRAM

k-effective: the single number that runs a reactor

A nuclear reactor's entire behaviour reduces to one ratio, the effective neutron multiplication factor k_eff — the average number of neutrons from one fission that go on to cause another fission in the next generation.

k_eff  <  1   subcritical   -- reaction dies out
k_eff  =  1   critical      -- steady, self-sustaining power
k_eff  >  1   supercritical -- power rises generation to generation

reactivity  rho = (k_eff - 1) / k_eff   (dimensionless, often quoted in pcm or dollars)

Reactor operators do not chase k_eff = 1 directly — they manage reactivity ρ, which is zero exactly at criticality. Positive ρ means the reactor is gaining power, negative ρ means it is losing power, and the operator's whole job during normal operation is keeping ρ hovering almost exactly at zero, nudging it with control rods to raise or lower power deliberately.

Control rods and the neutron budget

Control rods are made of strong neutron absorbers (boron, cadmium, hafnium). Inserting them removes neutrons from the chain before they can cause more fissions, pushing k_eff down; withdrawing them lets more neutrons survive to cause fission, pushing k_eff up. Because reactors run on a delicate balance of neutron production against loss (absorption plus leakage), rod position is a direct, fast lever on reactivity — but not the only one: coolant temperature, boron dissolved in the coolant (in PWRs), and fuel burnup all shift k_eff too, which is why control rods alone are constantly compensating for other, slower changes in the neutron budget.

Doppler feedback: the reactor's own safety brake

The single most important safety feature in a power reactor is not mechanical — it is a piece of nuclear physics called Doppler broadening. As fuel temperature rises, thermal motion of the uranium-238 nuclei broadens the energy range over which U-238 strongly absorbs neutrons (the resonance absorption peaks), so more neutrons are captured without causing fission as fuel heats up. That extra absorption is a negative feedback: a power rise heats the fuel, which lowers k_eff within milliseconds — far faster than any mechanical control system could react — which pulls power back down. This prompt negative temperature coefficient is why a properly designed reactor is inherently self-stabilising against small power excursions; reactors that lacked a strongly negative coefficient (part of what went wrong at Chernobyl, where the coefficient could go positive at certain operating points) are far more dangerous to operate.

Delayed neutrons: why reactors are controllable at all

Roughly 99.35% of neutrons from U-235 fission are emitted essentially instantly ("prompt"), but about 0.65% are delayed neutrons, released seconds later by the radioactive decay of certain fission fragments. That small delayed fraction is what makes a reactor controllable with mechanical rods at all: if reactor kinetics depended only on prompt neutrons, the neutron population would double or halve on a sub-millisecond timescale — far too fast for any control system. Because criticality with delayed neutrons included happens at k_eff barely above the prompt-neutron-only threshold, the effective response time of the whole reactor is set by the seconds-to-minutes half-lives of the delayed-neutron precursors, giving operators (and automated systems) a comfortable window to react.

SCRAM: the emergency shutdown

A SCRAM (emergency shutdown) drives every control rod fully into the core as fast as gravity and hydraulics allow, driving k_eff sharply below 1 within a second or two. Because delayed neutrons and, more significantly, ongoing radioactive decay heat from already-produced fission products continue for hours after shutdown, a SCRAM stops the chain reaction almost immediately but does not stop heat production instantly — this residual decay heat, roughly 6-7% of full power immediately after shutdown and decaying over hours and days, is exactly what must keep being removed by cooling systems after an emergency shutdown, and its mismanagement was the proximate cause of the Fukushima Daiichi meltdowns after the reactors themselves had already successfully SCRAMmed.

Frequently asked questions

Try it live

Everything above runs in your browser — open Nuclear Reactor and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.