Delayed Neutrons & Reactor Period

Reactivity & Fuel

Reactivity ρ 0.00 $

Delayed fraction β 0.65 %

Prompt generation time Λ 4.0e-4 s

Precursor decay constant λ 0.077 /s

Presets

Live Readout

Relative power n / n₀

1.00×10⁰

Precursors C

1.00

Reactor period τ

∞

Elapsed time

0.00 s

critical

How it works

Every fission releases neutrons instantly (prompt) except a small fraction β that escape only later, when short-lived fission fragments themselves decay and emit a neutron — the delayed neutrons. This simulator integrates the one-delayed-group point-kinetics equations:

dn/dt = ((ρ − β) / Λ)·n + λ·C
dC/dt = (β / Λ)·n − λ·C

n is the neutron population (reactor power), C the concentration of delayed-neutron precursors, Λ the prompt-neutron generation time, λ the precursor decay constant, and ρ = (k−1)/k the reactivity. Reactivity is shown in dollars, ρ/β — one dollar is exactly the point where β of reactivity has been added and the reactor turns prompt critical: n grows on the prompt timescale Λ (microseconds–milliseconds) instead of the delayed timescale 1/λ (seconds), which is the difference between a controllable power reactor and an uncontrollable excursion. Below 1$ any positive step reactivity still produces a fast initial jump (the "prompt jump") followed by a slow rise paced by precursor decay — this is why control rods can hold a reactor steady even though prompt neutrons alone would double the population in microseconds.

The live reactor period τ (e-folding time of the asymptotic exponential) is solved from the inhour equation ρ(ω+λ) = βω + Λω(ω+λ) for the dominant root ω, giving τ = 1/ω. The power readout is genuinely integrated (RK4, sub-stepped) from the same ODEs, not just approximated by the analytic period.

Reactivity ρ — the step change in criticality, in dollars ($ = ρ/β). Positive = supercritical.
Delayed fraction β — compare a U-235-like core (β ≈ 0.65%, more forgiving) against a Pu-239-like core (β ≈ 0.21%, reaches prompt-critical with much less reactivity).
Prompt generation time Λ — how fast prompt neutrons alone would multiply; short in a fast reactor, long in a thermal one.
Precursor decay constant λ — effective one-group decay rate of the delayed-neutron emitters; sets the pace of the slow rise.

t = 0.00 s · n/n₀ = 1.00

Prompt neutrons (n) Delayed precursors (C)

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