Neutron Flux Calculator
Calculate neutron flux φ from neutron density and average neutron speed for reactor physics analysis. Useful for estimating reaction rates, power density, and material irradiation exposure inside reactor cores.
Last updated: May 2026
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About this calculator
Neutron flux φ is the central quantity in reactor physics: it represents the total path length all neutrons in a unit volume traverse per unit time, equal to neutron density times mean speed. The formula used here is φ = n × v × 100, where n is neutron density in n/cm³, v is neutron speed in m/s, and the ×100 converts v from m/s to cm/s (1 m/s = 100 cm/s) so the result is in standard units of n/(cm²·s). The fundamental relation is φ = nv with consistent units throughout — the ×100 here is the m/s→cm/s unit conversion. Variables: n is typically 10⁸–10¹⁰ n/cm³ for thermal reactors and 10¹⁴–10¹⁵ n/cm³ for inside high-flux test reactors; v is 2,200 m/s for room-temperature thermal neutrons (Maxwell-Boltzmann peak), increasing as √T for higher temperatures and reaching ~10⁷ m/s for fast neutrons. Edge cases and caveats: this formula gives the scalar flux integrated over all directions, not the angular flux (which requires solving the transport equation). It assumes monoenergetic neutrons; in practice reactors have a distribution from thermal (~0.025 eV) to fast (~MeV), and flux is usually reported as group-wise integrals over energy bins (thermal flux, epithermal flux, fast flux). Reaction rate is R = Σ × φ where Σ is the macroscopic cross-section; this calculator gives φ, not R. The formula has no spatial dependence — real reactors have peak flux at the core center decreasing toward edges per a Bessel-function or cosine spatial distribution. Fluence (n/cm²) is the time integral of flux and is the right metric for cumulative material damage.
How to use
Example 1 — typical commercial PWR. Neutron density n = 5 × 10⁸ n/cm³ at thermal energies, mean speed v = 2,200 m/s (room-temperature Maxwell peak). Step 1: convert v to cm/s — 2,200 m/s × 100 = 2.2 × 10⁵ cm/s. Step 2: φ = n × v = 5 × 10⁸ × 2.2 × 10⁵ = 1.1 × 10¹⁴ n/(cm²·s). This matches the typical commercial-PWR thermal flux of ~10¹³–10¹⁴ n/cm²/s. Example 2 — research reactor with elevated density. n = 2 × 10¹⁰ n/cm³, thermal v = 2,200 m/s. φ = n × v = 2 × 10¹⁰ × 2.2 × 10⁵ = 4.4 × 10¹⁵ n/(cm²·s) — typical of a high-flux research reactor like HFIR or ATR. (Enter v in m/s; the calculator applies the ×100 m/s→cm/s conversion internally.)
Frequently asked questions
What are typical neutron flux values inside different reactor types?
Neutron flux varies by orders of magnitude depending on reactor type and purpose. Commercial pressurized water reactors (PWRs) and boiling water reactors (BWRs) operate with thermal flux around 3 × 10¹³ to 1 × 10¹⁴ n/(cm²·s) in the core center. CANDU heavy-water reactors run similar levels. High-flux research reactors like the HFIR at Oak Ridge or the ATR at Idaho reach 1 × 10¹⁵ n/(cm²·s) — the highest steady-state thermal flux anywhere on Earth. Fast reactors like sodium-cooled designs have fast-neutron flux of ~10¹⁵ n/(cm²·s) and very low thermal flux because there is no moderator. Subcritical assemblies and zero-power critical facilities operate at flux levels down to 10⁵ n/(cm²·s) for safe characterization work. Pulsed reactors (Godiva, TRIGA-pulse) can briefly reach 10¹⁸ n/(cm²·s) during millisecond bursts.
What is the difference between thermal, epithermal, and fast neutron flux?
Neutrons in reactors span six orders of magnitude in energy, and reactor physics divides them into three broad ranges. Thermal neutrons (< 0.5 eV, with a Maxwellian peak near 0.025 eV) are in thermal equilibrium with the moderator — most fissions in thermal reactors are caused by these. Epithermal neutrons (0.5 eV to ~10 keV) are slowing down through the resonance region where U-238 has strong capture peaks (the resonance escape probability is critical here). Fast neutrons (10 keV to ~20 MeV, peak ~2 MeV) are born from fission and have not yet lost much energy to scattering. Each group has its own flux, cross-sections, and reaction rates; modern reactor analysis uses 2-group, 5-group, or even 50-group energy discretizations depending on the level of fidelity required. The Westcott or thermal-only formula φ = nv is strictly valid only for the thermal group at a defined reference temperature.
How is neutron flux related to reactor power output?
Reactor thermal power is the product of fission rate and energy released per fission, which equals macroscopic fission cross-section times thermal flux times core volume times ~200 MeV per fission. As a rough rule, a thermal reactor producing 1 watt of fission power has about 3.1 × 10¹⁰ fissions per second; for an unenriched LWR fuel composition, this corresponds to a thermal flux of about 5 × 10⁷ n/(cm²·s) per watt-per-cm³ of power density. A typical 3,000 MWth commercial PWR has core volume around 30 m³ giving average power density of ~100 W/cm³ and thermal flux ~3 × 10¹³ n/(cm²·s). Power density varies threefold between core center and edges, so peak flux is correspondingly higher than the volume average. These relationships let operators infer flux from instrument power readings or vice versa for safety analysis.
What are common mistakes when calculating neutron flux?
The most common mistake is unit inconsistency between n (often given in n/cm³) and v (often given in m/s) — the formula needs both in consistent length units, which requires multiplying v by 100 to convert m/s to cm/s, or dividing the result accordingly. Another error is conflating flux with reaction rate; flux is just nv, while reaction rate requires multiplying by the macroscopic cross-section Σ = N × σ. People also misuse the thermal-neutron 2,200 m/s value for non-thermal spectra; for fast reactors or fission neutrons fresh from fission events, v ≈ 1.4 × 10⁷ m/s and the thermal formula does not apply. Treating flux as a scalar single number when reactor regions have very different flux levels (factor of 3 between center and edge of a typical core) introduces large errors in calculations of local effects like burnup distribution or activation product inventory. Finally, ignoring time dependence — flux is not constant during reactor startup, shutdown, or load follow — produces incorrect cumulative fluence for activation analysis.
When should I NOT use this calculator?
Skip the simple φ = nv formula when the neutron spectrum is highly non-thermal — fast reactors, accelerator-driven systems, and spallation sources need group-wise flux calculations from full transport codes (MCNP, OpenMC, Serpent). Avoid it for accurate reactor design calculations where spatial flux distribution matters; use diffusion theory or transport-equation solvers instead. Do not use it to estimate reactor power directly — you need the fission cross-section and core volume too. For activation analysis where short bursts or pulsed flux are involved, the time-average flux is misleading; integrate flux over the actual irradiation profile (fluence in n/cm²) before computing activation. The formula also breaks down near strong absorbers or moderators where the flux gradient is large and the local n × v product is not representative of the regional average. For shielding and dose calculations, use angular flux (Ψ) and the appropriate dose-rate kerma factors instead of scalar flux.