Neutron Flux and Reaction Rate Calculator

The reaction rate R (reactions per cm³ per second) equals the product of atom number density N (atoms/cm³), microscopic cross-section σ (cm²), and neutron flux φ (n/cm²/s). The product Nσ = Σ is the macroscopic cross-section (cm⁻¹). The formula is R = Σφ = Nσφ. It is the fundamental equation linking the neutron population (flux), the target material (cross-section), and the rate of nuclear reactions in the material.

Neutron flux φ is defined as φ = n × v, where n is the neutron number density (neutrons/cm³) and v is the neutron speed (cm/s). It represents the total path length traveled by all neutrons in 1 cm³ per second. Units are n/cm²/s. For monoenergetic neutrons, φ also equals the number of neutrons crossing a 1 cm² surface per second (from one side). In reactors with a broad energy spectrum, total flux is the integral over all energies.

The microscopic cross-section σ is the effective target area per atom for a given nuclear reaction. It is a quantum mechanical probability, not a literal geometric area. The unit barn (b) was chosen because early nuclear physicists said these cross-sections were 'as big as a barn' (10⁻²⁴ cm²). Common values: U-235 fission at thermal energies is 582 b, thermal neutron scattering in hydrogen is about 20 b. Fast neutron cross-sections are typically much smaller (0.1-10 b) because neutrons pass atoms too quickly to interact effectively.

The macroscopic cross-section Σ = Nσ (units cm⁻¹) is the reaction probability per unit path length of a neutron in the material. The mean free path λ = 1/Σ (cm) is the average distance a neutron travels before undergoing the reaction. A material with Σ = 0.5 cm⁻¹ has λ = 2 cm. Σ values add for each reaction type: the total macroscopic cross-section is Σ_total = Σ_absorption + Σ_scatter.

Reactor power is linked to flux through the fission reaction rate: P = R_f × Q × V, where R_f = Σ_f × φ is the fission rate per cm³, Q is the energy per fission (≈200 MeV = 3.2×10⁻¹¹ J), and V is the core volume (cm³). Rearranging: φ = P / (Σ_f × Q × V). For a typical LWR with power density P/V ≈ 200 W/cm³, Σ_f ≈ 0.35 cm⁻¹, and Q = 200 MeV, the thermal flux is approximately 1.8×10¹³ n/cm²/s.

Thermal neutron flux refers to neutrons that have been moderated to thermal equilibrium with the coolant (energies around 0.025 eV at room temperature). Fast neutron flux refers to high-energy neutrons (above 0.1 or 1 MeV, depending on convention) produced directly by fission. In a thermal reactor, fissions occur predominantly in the thermal flux, where U-235 cross-sections are much larger. The fast flux is important for structural damage (displacements per atom) and for Pu-239 production via U-238 fast capture.

Thermal research reactors (e.g., HFIR at ORNL): thermal flux up to 2×10¹⁵ n/cm²/s in the reflector. Power reactor fuel: thermal flux of 1-5×10¹³ n/cm²/s. Subcritical assemblies or startup sources: 10⁶-10⁸ n/cm²/s. Material test reactors: 10¹⁴-10¹⁵ n/cm²/s. Medical cyclotron neutron beams: 10⁸-10¹² n/cm²/s. The flux levels determine both the reaction rate for production of isotopes and the radiation damage to structural materials.

Atom number density N = (ρ × Nₐ × w_i) / A_i, where ρ is the material density (g/cm³), Nₐ = 6.022×10²³ mol⁻¹ (Avogadro's number), w_i is the weight fraction of isotope i, and A_i is its atomic mass (g/mol). For UO2 at 10.4 g/cm³ (molecular weight 270 g/mol), total uranium density: N_U = (10.4 × 6.022×10²³) / 270 = 2.32×10²² atoms U/cm³. For 3% enriched fuel, N_235 = 0.03 × 2.32×10²² = 6.96×10²⁰ U-235 atoms/cm³.

Neutron flux is measured using several methods: (1) Fission chambers contain a thin fissile coating (U-235 or Pu-239) and measure the fission rate electrically. (2) Activation foils are irradiated in known positions then counted in a gamma spectrometer; the induced activity reveals the flux integral. (3) Self-powered neutron detectors (SPNDs) use beta current from neutron activation of the detector material. (4) Ex-core ion chambers and ex-vessel detectors provide continuous monitoring from outside the pressure vessel.

A one-group flux approximation treats all neutrons as having a single representative energy (usually the thermal peak energy of 0.025 eV), using single-group cross-sections averaged over the neutron energy spectrum. It is reasonably accurate for reactions that are dominated by thermal neutrons in well-moderated systems, and is the basis for simple hand calculations of thermal reaction rates. For fast reactor analysis, transmutation calculations, or systems with resonance absorption, multi-group methods with tens to hundreds of energy groups are required.

Neutron activation analysis (NAA) exploits the neutron reaction rate to identify and quantify elements. A sample is placed in a neutron flux φ for time t, producing radioactive isotopes at rate R = Nσφ. After irradiation, gamma spectroscopy measures the activity A = R × (1 - e^(-λt)), where λ is the decay constant of the product. By measuring A and knowing φ and σ from literature, the original atom density N is determined. NAA can detect trace elements at concentrations of parts per billion, useful in forensics, food safety, and archaeological dating.

Neutron irradiation displaces atoms from their lattice positions (displacements per atom, dpa), hardening and embrittling structural steels. Fast neutrons (above 1 MeV) cause most atomic displacements because they have sufficient energy to initiate collision cascades. Fluence (time-integrated flux, n/cm²) is the accumulated dose metric. PWR pressure vessel steel must remain below a fast fluence limit of about 1-2×10¹⁹ n/cm² to maintain fracture toughness above regulatory limits. Reducing fast flux at the vessel wall is a key reactor design constraint.