3.1208e+16
fissions/s
2,087,063,726,839.4268
n/cm^2/s
2.5626e+25
atoms
5.835000e-22
cm^2
3.1208e+16
fissions/s
2,087,063,726,839.4268
n/cm^2/s
2.5626e+25
atoms
5.835000e-22
cm^2
The Neutron Flux Calculator determines the neutron flux in a nuclear reactor from its thermal power, fuel mass, fission cross section, and energy per fission. Neutron flux (phi, in units of neutrons per cm^2 per second) is the most important parameter characterizing the neutron field in a reactor and is fundamental to reactor design, fuel burnup calculations, and radiation damage assessment.
The fission rate in a reactor is related to power by: fission rate (fissions/s) = Power (W) / Energy per fission (J). The average neutron flux is related to the fission rate by: fission rate = phi * Sigma_f * V = phi * sigma_f * N_atoms, where sigma_f is the microscopic fission cross section and N_atoms is the total number of fissile atoms. This gives phi = fission rate / (N * sigma_f).
Typical thermal neutron fluxes: research reactors: 10^12 - 10^14 n/cm^2/s; power reactors: 10^13 - 10^14 n/cm^2/s; fission bomb peak: ~10^25 n/cm^2/s for microseconds; spallation neutron sources (SNS, ISIS): 10^14 n/cm^2/s pulsed.
Neutron flux determines nuclear transmutation rates (activation analysis, isotope production), fuel depletion rates, and radiation damage in structural materials. Exposure is measured in neutrons per cm^2 (integrated flux = fluence), and structural materials typically fail after receiving 10^21-10^22 n/cm^2 fast neutron fluence.
Neutron activation analysis uses the reaction rate R = phi * sigma_activation * N to determine element concentrations from the activity of activated isotopes — an important nondestructive analytical technique in environmental science, archaeometry, and forensics.
Fission rate = P(W) / (E_f * 1.602e-13 J/MeV). Number of U-235 atoms: N = (mass_kg * 1000 / 235) * 6.022e23. Average flux: phi = fission_rate / (N * sigma_f_cm2). The formula assumes uniform flux and criticality (nu * sigma_f = sigma_total_absorption + leakage).
phi > 10^14: high-flux research reactor (used for isotope production, neutron scattering). phi 10^12-10^13: typical power reactor. phi < 10^10: subcritical assembly or neutron source. The flux determines the rate of all neutron-induced reactions: activation, burnup, transmutation, and material damage.
Inputs
Results
A 1 MW research reactor with 10 kg U-235 has a fission rate of ~3.1e16 fissions/s and average thermal neutron flux of ~8.9e12 n/cm^2/s — sufficient for neutron activation analysis and isotope production.
Inputs
Results
A 1 GW(thermal) power reactor with 1000 kg U-235 undergoes 3.1e19 fissions/s. The average flux over the large fuel volume is ~10^12 n/cm^2/s, consistent with typical power reactor values.
phi = n * v, where n is neutron density (n/cm^3) and v is neutron speed (cm/s). It represents the total neutron path length per unit volume per unit time. Units: n/cm^2/s. Also equals reaction rate per unit macroscopic cross section.
Flux phi (n/cm^2/s) is instantaneous. Fluence (n/cm^2) is the time-integrated flux: fluence = integral of phi dt. Fluence determines total radiation damage, isotope production, and activation.
Irradiating a sample with a known neutron flux to produce radioactive isotopes: R = phi * sigma * N. Measuring the gamma-ray spectrum of activated products identifies and quantifies elements nondestructively with sub-ppm sensitivity.
A reactor is critical (self-sustaining chain reaction) when the effective multiplication factor k_eff = 1. k_eff > 1 (supercritical) means flux is growing; k_eff < 1 (subcritical) means flux is decreasing. Control rods absorb neutrons to control k_eff.
About 99.3% of fission neutrons are emitted immediately (prompt). About 0.7% are delayed (from decay of fission products) by 0.1-80 seconds. Delayed neutrons allow reactor control with mechanical control rods — without them, the reactor would be uncontrollably fast.
Moderators slow fast neutrons (2 MeV) to thermal energies (0.025 eV) to enhance fission cross sections. Common moderators: light water (H2O), heavy water (D2O), graphite. The moderating ratio is sigma_scattering * xi / sigma_absorption (higher is better).
Neutron flux is not uniform in a reactor core. It peaks at the center of a bare reactor (approximately sinusoidal in each dimension for a cylindrical core). Flux flattening using burnable poisons or enrichment zoning is used to maximize power without hotspots.
For many nuclei at thermal energies, the absorption cross section is proportional to 1/v (inversely to neutron speed): sigma = sigma_0 * v_0/v, where v_0 = 2200 m/s is the standard thermal speed. At higher energies, resonances dominate.
Fast neutrons displace atoms from crystal lattice sites (displacement per atom, dpa). After ~1 dpa (about 10^21 fast n/cm^2), steel shows embrittlement, swelling, and creep. This limits reactor pressure vessel lifetime to 40-60 years.
Activity A = sigma_a * phi * N * (1 - e^(-lambda*t)), where sigma_a is the activation cross section, phi is flux, N is target atoms, lambda is the product decay constant, and t is irradiation time. Saturation activity A_sat = sigma_a * phi * N is reached after ~7 half-lives.
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