Nuclear Chain ReactionBeta

About Nuclear Chain Reaction

This is a two-dimensional Monte Carlo model that follows every neutron as a dot. The core is a disc in the middle of the canvas. Each neutron flies in a straight line until, by chance, it causes a fission, gets absorbed, or scatters. A fission releases two or three neutrons (2.5 on average). A neutron that crosses the edge of the disc is sent back with a probability equal to the reflector setting; otherwise it counts as leakage and is gone.

The cross sections are tuned in screen pixels rather than real barns, but the ratios that matter are kept: fast neutrons are readily captured by U-238, while slowed-down thermal neutrons are very good at splitting U-235. That is why natural uranium fizzles with no moderator and comes alive once the moderator slider is up.

One number, k, decides the outcome

The multiplication factor k is how many neutrons one generation leaves to the next. A thermal fission of U-235 releases about 2.43 neutrons on average (NRC figure), so only about 41% of them have to go on to cause another fission for k to equal 1. The other 59% can leak out or be soaked up by U-238, control rods and structure without the reaction dying. This model uses 2.5 per fission, so its break-even share is 40%.

Small differences in k compound quickly. Starting from one neutron, 100 generations at k = 1.1 give roughly 13,800 neutrons; at k = 0.9 you are left with 0.00003, effectively nothing. The k readout on screen is an estimate: neutrons born in the recent window divided by neutrons removed (fission, absorption and leakage) in the same window. With only a few dozen neutrons around, it jumps all over the place. A delayed neutron counts as born when it is released at the end of its wait, not at the moment of the fission that made it. So when k reads close to 1 the number of moving neutrons really is holding steady, and after you drive the rods in and fissions drop off, k does not fall straight to 0 while waiting delayed neutrons are still being released.

How each slider pushes on k

Core size controls leakage. A small disc has a lot of edge for its area, so neutrons tend to fly out before they meet another nucleus. In this model the critical radius of a bare, highly enriched core with no reflector is about 40% of the largest allowed size. The Critical Mass scenario starts at 22%, which is why it dies out as it stands.

A reflector cuts leakage without changing the size. The Reflector scenario caps the core at 30% and asks you to reach criticality with the reflector alone. The same physics is why criticality-safety rules at fuel facilities worry about water, concrete or even a person standing next to fissile material: all of them can bounce neutrons back.

The moderator turns fast neutrons into thermal ones. In this tool one scatter off the moderator has a 40% chance of thermalizing a neutron on the spot; in real graphite it takes on the order of a hundred collisions, which would be invisible on screen. Switch the slowing-down model to Realistic and a neutron needs 5 moderator collisions to become thermal, with a fresh chance of U-238 capture at each one. In that mode natural uranium plus moderator never reaches criticality and slowly dies out; you have to raise the enrichment to about 3% to keep the chain going, the same reason light-water reactors burn low-enriched fuel. Control rods are six strips that slide down from the top, and any neutron that enters one is almost certainly absorbed, whatever its energy.

Delayed neutrons: why a reactor can be steered by hand

More than 99.3% of fission neutrons appear essentially instantly, but about 0.65–0.7% come out later, when certain fission products decay. The NRC's reactor physics manual gives a prompt neutron lifetime of about 10⁻⁴ s and a delayed neutron lifetime of about 13 s, which averages out to a generation time of roughly 0.1 s. That factor of a thousand is what makes control possible.

Take k = 1.001, only 0.1% supercritical. With prompt neutrons alone the period is 10⁻⁴ ÷ 0.001 = 0.1 s, and the power grows about 22,000-fold in one second. With a 0.1 s generation time the period is 100 s, and after a full minute the power has grown only about 1.8 times. The NRC manual adds a practical figure: about 175 pcm of reactivity gives a 26-second period, or a tenfold rise per minute. If the excess reactivity reaches β (around 0.007), the core is critical on prompt neutrons alone — prompt critical — and the delayed-neutron cushion is gone.

This tool sets the delayed fraction to 25% with a mean delay of 3 s. At the real 0.65% no human hand on a slider could keep up. That is why, in the Reactor scenario, the power answers a rod movement several seconds late, and the dots that sit still inside the core are delayed neutrons waiting to be born.

What Chicago Pile-1 showed on December 2, 1942

Under the west stands of Stagg Field at the University of Chicago, in a former squash court, Enrico Fermi's team built a roughly spherical pile of graphite bricks with lumps of natural uranium metal and uranium oxide set in a lattice. According to the Department of Energy's history, George Weil withdrew the cadmium control rod at 3:25 p.m.; 28 minutes later, at 3:53 p.m., Fermi called for the emergency rod “Zip” to go back in. A liquid-control squad stood on a platform above the pile, ready to flood it with cadmium-salt solution if the mechanical rods failed.

Earlier that day Fermi pulled the rod out a foot at a time and predicted where the recorder pen would level off, and it did. In a subcritical pile a neutron source keeps feeding neutrons in, and the population settles at 1/(1−k) times the source: 20 times at k = 0.95, 100 times at k = 0.99. Each rod step raised the plateau, until k passed 1 and the trace kept climbing. You can watch the same thing in the Natural Uranium scenario by raising the moderator.

What the model leaves out

• It is a flat disc. A real core is three-dimensional, and the surface-to-volume relation changes, so the critical size here does not translate to real kilograms.
• Fuel and moderator are mixed evenly. The lattice effect of placing uranium in separate lumps, as CP-1 did, is not modelled.
• There is no temperature feedback, no xenon or other poisons and no fuel burnup — the effects that actually keep a light-water reactor's power stable.
• “Meltdown” simply means the 20,000-neutron cap was exceeded. No heat or cooling is computed.

Worked examples

How to use it

1. In the Critical Mass scenario, click the canvas to inject neutrons and watch most of them leak out through the edge.
2. Raise the core size or the reflector until the k readout settles above 1.
3. In the Reactor scenario, nudge the rods a little, then wait a few seconds before the next move.

Frequently asked questions