This is a top-down, geometric neutron-transport model — not a well-mixed reactivity equation. A disc of fuel nuclei sits inside a bounded pile of radius R. Each fission releases k neutrons (rounded stochastically, since k is rarely a whole number) that fly off in random directions at a fixed speed. A neutron that reaches an unsplit nucleus triggers another fission; a neutron that reaches the edge of the pile leaks out and is lost for good.
Whether the pile is subcritical, critical or supercritical is not set directly by k alone — it emerges from the race between k and geometry: raise fuel density and neutrons travel less far, on average, before hitting a nucleus, so fewer escape before they can multiply. A low-density, low-k pile stalls out (subcritical); a dense, high-k pile cascades exponentially (supercritical). This is the classic size/density/multiplication trade-off behind real reactor and weapon criticality — distinct from the point-kinetics ODE and delayed-neutron precursor model used elsewhere on this site, which treats the neutron population as a single well-mixed number instead of tracking individual particles and geometric leakage.
- Fuel density — how many fuel nuclei are packed into the fixed-radius pile. Higher density shortens the average neutron flight before impact.
- Multiplication factor k — neutrons released per fission. k<1 alone does not guarantee subcriticality here, and k>1 does not guarantee runaway — geometric leakage can offset it either way.