Nuclear Criticality Calculator
Compute the effective neutron multiplication factor (k_eff) for a fissile system using the six-factor formula. Used by nuclear engineers to determine whether a reactor or fissile assembly is subcritical, critical, or supercritical.
About this calculator
Nuclear criticality is governed by the six-factor formula: k_eff = η × f × p × ε × P_NL, where η (eta) is the reproduction factor (neutrons produced per absorption in fuel), f is the thermal utilization factor (fraction of thermal neutrons absorbed by fuel), p is the resonance escape probability (fraction of neutrons avoiding resonance capture), ε (epsilon) is the fast fission factor (extra fissions from fast neutrons), and P_NL is the non-leakage probability (fraction of neutrons that do not escape the system). When k_eff = 1 the chain reaction is self-sustaining (critical); k_eff < 1 means subcritical (reaction dies out); k_eff > 1 means supercritical (power rises). Each factor is dimensionless and between 0 and 1 except ε, which is slightly above 1. Reactor designers tune these parameters through fuel enrichment, moderator choice, and geometry to achieve a controlled critical state.
How to use
Suppose a thermal reactor has η = 2.06, f = 0.71, p = 0.87, ε = 1.03, and non-leakage probability P_NL = 0.96. Enter each value into its respective field. The calculator computes: k_eff = 2.06 × 0.71 × 0.87 × 1.03 × 0.96 = 2.06 × 0.71 = 1.4626; × 0.87 = 1.2725; × 1.03 = 1.3107; × 0.96 ≈ 1.258. Because k_eff > 1, this system is supercritical and would require control rod insertion to return to k_eff = 1.
Frequently asked questions
What does a k_eff value greater than 1 mean for a nuclear reactor?
A k_eff greater than 1 means the reactor is supercritical: each fission generation produces more neutrons than the last, causing power to rise exponentially. In a power reactor this is a transient condition used briefly during startup or power increases, then corrected by inserting control rods. An uncontrolled supercritical excursion is extremely dangerous and must be avoided through engineered safety systems.
How does non-leakage probability affect nuclear criticality in small reactors?
Non-leakage probability P_NL accounts for neutrons that escape the reactor core without causing fission, and it is strongly dependent on reactor size and geometry. Smaller cores have a higher surface-to-volume ratio, so more neutrons leak out, driving P_NL closer to zero and reducing k_eff. This is why small research reactors require higher fuel enrichment or reflectors to compensate for the increased neutron leakage and maintain criticality.
Why is the resonance escape probability important in uranium-fueled reactors?
Uranium-238, which makes up the bulk of natural and low-enriched uranium fuel, has very large neutron absorption cross-sections at specific resonance energies (roughly 1–100 eV). Neutrons slowing down from fission energies must pass through these resonance energies, and a significant fraction can be absorbed by U-238 without causing fission. A high resonance escape probability p indicates that most neutrons survive this slowing-down process and reach thermal energies where fission in U-235 is far more likely, making p a critical efficiency parameter in reactor design.