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critical systems · chain reaction

Nuclear Reactor
Criticality

A chain reaction is a feedback loop. Pull the control rods out, every splitting atom makes two new ones, and the count explodes. Push them in, the feedback dies. Reactors run on the knife-edge between.

neutrons uranium-235 fission flash neutrons 0 · k ≈ 1.00 · state idle
What's happening Each neutron races around the core. If it hits a uranium atom, it splits it and releases 2-3 new neutrons. Control rods absorb whatever enters their zone.
What to try Hit Supercritical. Count explodes, everything magenta-flashes. Hit Subcritical. Chain dies in seconds. Hit Critical and try to keep it there.
Why it matters k = 1 by 0.3% of rod depth. This is why reactor control is automated and why operators are trained for years before touching a rod.

source notes

Model level and refs

  • Mechanism only, not a neutronics solver. Real reactors compute the diffusion equation on a 3D mesh with energy groups and delayed neutron precursors; this sim tracks individual neutron particles with toy kinematics.
  • Fission yields 2-3 neutrons in U-235 (average ~2.4 fast neutrons per thermal fission). Real cross-sections depend on neutron energy; here fission probability is a single slider.
  • Control rods absorb neutrons via boron, cadmium, or hafnium in real reactors. Here absorption is a per-frame probability inside the rod zone.
  • Moderator (water or graphite) is implicit: neutron speed is set to a mid value that maps roughly onto thermal neutron behaviour without explicit slowing-down physics.
  • No delayed neutrons, no temperature feedback, no xenon poisoning. These are the mechanisms that actually make real reactors stable; this sim deliberately strips them to show the raw chain reaction.

deeper dive

Why criticality is a knife-edge

A nuclear reactor and a nuclear bomb are the same mechanism at two different settings. Both rely on the chain reaction: one neutron hits a uranium atom, the atom splits, and 2-3 new neutrons come out. Those new neutrons can hit other atoms and split them too. Whether the chain grows, holds steady, or dies depends on one ratio.

The multiplication factor, k

Call k the average number of new neutrons each old neutron produces before it escapes or gets absorbed. If k is below 1, each generation is smaller than the last and the chain dies out. If k is exactly 1, every generation replaces itself and the reaction holds steady. If k is above 1, each generation is bigger than the last and the count grows exponentially. Reactors run at k = 1.00 to within a fraction of a percent. Go to 1.01 and the power doubles every few seconds. Go to 0.99 and it halves. The interesting regime is a razor-thin strip.

What the control rods actually do

Control rods are made of materials that eat neutrons: boron, cadmium, hafnium. When a rod is deep in the core, it absorbs a lot of neutrons, which means fewer neutrons get to uranium atoms, which means fewer fissions, which means a smaller k. Pulling the rod out reduces the absorption rate and raises k. The whole job of reactor operation is finding the rod position where k = 1 and then holding it there while the fuel composition slowly changes over months.

In the simulation above, one slider controls how far the five rods are inserted. Notice how sensitive k is: 55% insertion and the chain limps along, 50% and it runs steady, 45% and it blows up. In a real reactor the sensitivity is similar, which is why rod movements are measured in millimeters and why control is automated.

Where the same math shows up

Chain reactions are not just physics. Any system where an event produces on average X new events of the same kind obeys this math. Infectious diseases use the exact same model, where X is called R₀. Wildfires, viral social posts, bank runs, nuclear reactors, same equation, same knife-edge around the value 1. Our forest-fire simulation shows cascades at the other end of the same spectrum.

Things to try

Set rods to Critical, wait 5 seconds, then pull the slider 5% toward supercritical. Watch the sparkline. Drop fuel density in the middle of a supercritical run; criticality dies because there are not enough fissile atoms left to sustain k. Push fission probability to 95% and fission yield to 3.0; the reaction becomes hair-trigger, even the critical preset runs hot. Browse the full library for other systems where a small parameter shift flips the whole picture.