Reactor Kinetics Calculator
Calculate how a nuclear reactor's power level changes over time following a sudden reactivity insertion. Used by reactor physicists and operators to analyze power excursions, startup transients, and control rod worth.
Last updated: September 2026
Formula below · 2 sources (nrc.gov, Wikipedia) · Updated Sep 2026
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About this calculator
Reactor kinetics describes how neutron population — and therefore reactor power — evolves after a change in reactivity. This calculator uses the prompt-jump approximation to the point-reactor-kinetics equation: P(t) = P₀ · (β_eff/(β_eff − ρ)) · exp(λ_eff · ρ · t / (β_eff − ρ)), where P₀ is initial power, ρ is the reactivity insertion (Δk/k), β_eff is the effective delayed neutron fraction (typically 0.0065 for U-235), t is time after insertion, and λ_eff ≈ 0.0767/s is the average delayed-neutron precursor decay constant for U-235. Valid for small positive reactivity insertions below prompt-critical (ρ < β_eff). When ρ ≥ β_eff the reactor becomes prompt-critical and this formula does not apply — the tool returns a warning string instead of a number. This approximation deliberately drops the prompt neutron lifetime ℓ (~10–50 µs in thermal reactors): after the initial 'prompt jump', the transient's timescale is set by the much slower delayed-neutron precursor decay (~1/λ_eff ≈ 13 s), which is why ℓ does not appear in this equation or as an input here — it only matters during the sub-microsecond jump itself, not the transient this calculator models.
How to use
Assume a reactor at 100% power receives a positive reactivity insertion of ρ = 0.001 Δk/k, with β_eff = 0.0065. Calculate power at t = 10 seconds: prefactor = β_eff / (β_eff − ρ) = 0.0065 / 0.0055 ≈ 1.182. Exponent = λ_eff × ρ × t / (β_eff − ρ) = 0.0767 × 0.001 × 10 / 0.0055 ≈ 0.1395. P(10) = 100 × 1.182 × exp(0.1395) ≈ 100 × 1.182 × 1.150 ≈ 135.9% of initial power.
Frequently asked questions
What is reactor period and how does it relate to reactivity?
The reactor period T is the time constant for exponential power change: P(t) = P₀ × e^(t/T). A positive reactivity insertion produces a positive (finite) period and rising power; negative reactivity produces a negative period and falling power. The inhour equation relates period to reactivity, accounting for all six delayed neutron groups. Shorter periods mean faster power changes and less time for operators or automatic systems to respond — which is why reactor protection systems are calibrated to trip the reactor when the period falls below a safe threshold (typically 3–10 seconds).
What is the effective delayed neutron fraction (β_eff) and why is it critical to reactor safety?
Delayed neutrons are emitted by certain fission products (precursors) milliseconds to minutes after fission, rather than instantaneously. β_eff is the fraction of all fission neutrons that are delayed, weighted by their importance to the neutron chain reaction. For U-235 thermal fission β_eff ≈ 0.0065 (0.65%); for Pu-239 it is lower at ~0.0021. This small fraction is enormously important: it slows the effective neutron reproduction rate from microseconds to ~0.1 seconds, giving control systems time to act. A reactor is prompt critical — and potentially uncontrollable — only when inserted reactivity equals or exceeds β_eff.
Why doesn't this calculator ask for the prompt neutron lifetime?
The prompt neutron lifetime ℓ is the average time between a neutron's birth and its absorption to cause the next fission — typically 10–100 µs in thermal reactors, and 0.1–1 µs in fast reactors. It governs how fast the initial 'prompt jump' happens, but for a sub-prompt-critical insertion (ρ < β_eff) that jump settles within microseconds, and the transient this calculator reports at t seconds later is governed by the much slower delayed-neutron precursor decay instead. Because ℓ cancels out of the prompt-jump approximation used here, it is not one of this calculator's inputs — it would only matter for analyzing the sub-microsecond jump itself, or for prompt-critical (ρ ≥ β_eff) transients, which this tool flags as a warning rather than computing a number for.