The effective multiplication factor (k-eff) is the average number of neutrons from one fission that go on to cause another fission. k-eff = 1 is exactly critical; >1 is supercritical (power rising); <1 is subcritical (power falling).

What is reactivity (rho)?

Reactivity rho = (k-eff − 1) / k-eff. It measures how far the reactor is from criticality. A positive rho means the reactor is supercritical; negative means subcritical. It is dimensionless (or expressed in pcm: 1 pcm = 10⁻⁵).

What are "dollars" of reactivity?

Dollars = rho / beta, where beta is the delayed neutron fraction (~0.0065 for U-235 thermal reactors). One dollar of reactivity equals beta. Exceeding one dollar drives the reactor prompt-critical, which is the key safety boundary.

What is the reactor period?

The reactor period T is the time for reactor power to change by a factor of e (~2.718). A positive (shorter) period means rapid power rise; a longer or negative period means slow change or decline.

What delayed neutron fraction should I use?

For U-235 in a thermal reactor, beta ≈ 0.0065. For Pu-239, beta ≈ 0.0021. For U-233, beta ≈ 0.0026. Fast reactors have lower effective beta values due to spectrum effects.

Is this tool suitable for real reactor safety analysis?

No. This is a simplified point-kinetics educational tool. Real criticality safety requires licensed neutronics codes (MCNP, KENO, Serpent) and qualified nuclear engineers.

Reactor Criticality Helper

Compute k-eff, reactivity (rho and dollars), and reactor period from neutron generation and loss rates.

Neutron Balance Inputs

Neutron production rate (neutrons/s or relative units)

Neutron absorption + leakage rate (same units)

Kinetics Parameters (optional)

Delayed neutron fraction β (dimensionless, e.g. 0.0065)

U-235 thermal: 0.0065 | Pu-239: 0.0021 | U-233: 0.0026

Prompt neutron lifetime ℓ* (μs)

Thermal reactors ~50–100 μs | Fast reactors ~0.1–1 μs

Mean delayed neutron lifetime ℓ_d (s)

Typical U-235 composite ~13 s

Effective Multiplication Factor

— k-eff

0 (fully subcritical) 1 (critical) 2 (supercritical)

Reactivity ρ

—

Δk/k (dimensionless)

Reactivity pcm

—

1 pcm = 10^-5

Reactivity $

—

ρ / β

Reactor Period (Inhour approximation)

Prompt period T_p

—

(prompt neutrons only)

Stable period T (with delayed)

—

(single-group delayed approx.)

Formulas used

k-eff = Production / (Absorption + Leakage)

ρ = (k-eff − 1) / k-eff

Reactivity $ = ρ / β

Prompt period T_p = ℓ* / (ρ − β) (only for ρ > β)

Stable period T solves ρ = ℓ*/T + β/(1 + T/ℓ_d) (single-group inhour equation)

Point-kinetics approximation. Educational use only.

Summary

This tool computes the three core criticality parameters used in reactor physics: effective multiplication factor (k-eff), reactivity in both rho and dollars, and reactor period. It uses the point-kinetics approximation with user-supplied neutron production rates, loss rates, delayed neutron fraction, and prompt neutron lifetime. Results are for educational and preliminary engineering use only.

How it works

Enter the neutron production rate and neutron absorption/leakage rate.
The tool computes k-eff as the ratio of neutrons produced to neutrons lost per generation.
Reactivity (rho) is derived from k-eff: rho = (k-eff − 1) / k-eff.
Enter the delayed neutron fraction (beta) to convert reactivity from rho to dollars.
Enter the prompt neutron lifetime to compute the prompt and delayed reactor period.

Use cases

Verify reactor criticality conditions in academic nuclear engineering courses.
Estimate reactivity worth of control rod movements in simplified models.
Convert reactivity units between rho (delta-k/k) and dollars during design review.
Compute reactor period to estimate power doubling time during startup.
Cross-check hand calculations for criticality safety assessments.
Demonstrate neutron balance concepts in training exercises.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu