Reactor Criticality Calculator
Compute the neutron multiplication factor k for a nuclear reactor using the six-factor formula. Essential for reactor design, startup analysis, and safety margin assessment.
About this calculator
A nuclear reactor sustains a chain reaction only when, on average, each fission produces exactly one subsequent fission. This condition is quantified by the effective neutron multiplication factor k. The six-factor formula (here condensed to five inputs with non-leakage lumping thermal and fast leakage) states: k = f × η × p × ε × P_NL, where f is the thermal utilisation factor (fraction of thermal neutrons absorbed in fuel), η is the reproduction factor (neutrons produced per thermal neutron absorbed in fuel), p is the resonance escape probability (fraction of neutrons avoiding resonance capture while slowing down), ε is the fast fission factor (boost from fast fissions in U-238), and P_NL is the non-leakage probability (fraction of neutrons that do not escape the reactor). When k = 1 the reactor is exactly critical; k > 1 is supercritical (power rising); k < 1 is subcritical (power falling). Reactivity ρ = (k − 1) / k is often quoted in pcm (per cent mille).
How to use
For a typical thermal reactor core, suppose: f = 0.90, η = 2.02, p = 0.75, ε = 1.05, P_NL = 0.97. Enter each value into its respective field. The calculator evaluates: k = 0.90 × 2.02 × 0.75 × 1.05 × 0.97. Step by step: 0.90 × 2.02 = 1.818; × 0.75 = 1.3635; × 1.05 = 1.4317; × 0.97 = 1.3887. The result k ≈ 1.389 indicates a strongly supercritical configuration — in a real reactor, control rods or soluble boron would be inserted to reduce k to exactly 1.0 during normal operation.
Frequently asked questions
What is the difference between a critical, subcritical, and supercritical nuclear reactor?
A critical reactor has k = 1, meaning each generation of neutrons is exactly replaced by the next and power remains steady. A subcritical reactor has k < 1; the chain reaction dies out without an external neutron source, which is the safe shutdown condition. A supercritical reactor has k > 1, so neutron population — and therefore power — grows exponentially with each generation. During normal operation, reactors are held precisely at k = 1 through the balance of control rods, coolant temperature feedback, and chemical shim. The distinction between prompt and delayed supercriticality is critical for safety: delayed neutrons give operators time to respond to reactivity changes.
How does the resonance escape probability affect reactor criticality?
As fast neutrons slow down in the moderator, they pass through energy ranges where U-238 has enormous absorption cross-sections called resonance peaks. The resonance escape probability p is the fraction of neutrons that lose energy without being captured at these resonances. A higher p means more neutrons survive to become thermal neutrons that can drive fission. Reactor designers increase p by using a well-moderated lattice that allows neutrons to slow down quickly through the resonance region, and by choosing cladding and structural materials with low resonance absorption. Temperature also affects p through Doppler broadening of resonance peaks, creating a strong negative feedback mechanism that enhances reactor safety.
Why is the non-leakage probability important in small versus large reactor cores?
Neutrons produced near the surface of a reactor core have a significant chance of escaping before causing another fission. The non-leakage probability P_NL captures this geometric effect and approaches 1.0 as the core becomes very large relative to the neutron migration length. Small research reactors and early experimental piles had low P_NL values, requiring higher fuel enrichment to compensate. Large commercial power reactors — with core diameters of several metres — achieve P_NL values of 0.96–0.99, making leakage a minor correction. Reflectors made of beryllium, graphite, or water are placed around the core to scatter escaping neutrons back in and effectively increase P_NL without adding more fuel.