Neutron Flux Calculator
Calculate the neutron reaction rate in a nuclear material given neutron density, velocity, microscopic cross-section, and target atom density. Used in reactor physics, shielding design, and neutron activation analysis.
About this calculator
Neutron flux (φ) is defined as the product of neutron density and neutron speed: φ = n × v, where n is the neutron density (n/cm³) and v is the neutron velocity (cm/s). The reaction rate per unit volume is then R = φ × Σ = n × v × σ × N, where σ is the microscopic cross-section in barns (1 barn = 10⁻²⁴ cm²) and N is the target nucleus density (nuclei/cm³). This calculator evaluates R = (n × v × σ × N) / 10²⁴, incorporating the barn-to-cm² conversion. Cross-sections vary enormously by isotope and neutron energy — thermal neutrons interacting with boron-10 have σ ≈ 3,840 barns, while fast neutrons see much smaller values. Reaction rates determine fuel depletion, activation product buildup, and detector response in nuclear systems.
How to use
Suppose you have a neutron density of 10⁸ n/cm³, a neutron velocity of 2,200 m/s (2.2 × 10⁵ cm/s, the thermal velocity), a microscopic cross-section of 100 barns, and a target nucleus density of 5 × 10²² nuclei/cm³. Using R = (n × v × σ × N) / 10²⁴: R = (10⁸ × 2.2 × 10⁵ × 100 × 5 × 10²²) / 10²⁴. Numerator: 10⁸ × 2.2 × 10⁵ = 2.2 × 10¹³; × 100 = 2.2 × 10¹⁵; × 5 × 10²² = 1.1 × 10³⁸. Divide by 10²⁴: R = 1.1 × 10¹⁴ reactions/cm³/s. This is a typical reaction rate for a thermal reactor material.
Frequently asked questions
What is the difference between neutron flux and neutron fluence in reactor physics?
Neutron flux (φ) is an instantaneous quantity representing the number of neutrons passing through a unit area per unit time, expressed in n/cm²/s. Neutron fluence is the time-integrated flux — the total number of neutrons per unit area accumulated over a period, in n/cm². Fluence is the relevant quantity for assessing cumulative radiation damage to reactor materials such as pressure vessel steel, which undergoes embrittlement as fluence increases over decades of operation. Reactor safety evaluations set limits on lifetime fluence to the vessel to ensure structural integrity is maintained throughout the plant's design life.
How does neutron energy affect the cross-section used in reaction rate calculations?
Microscopic cross-sections (σ) are strongly energy-dependent. For many isotopes, thermal neutrons (energy ~0.025 eV, velocity ~2,200 m/s) produce cross-sections orders of magnitude larger than fast neutrons (energy >1 keV). For example, uranium-235 has a fission cross-section of about 585 barns for thermal neutrons but only ~1 barn for fast neutrons — a factor of nearly 600 difference. This is why thermal reactors moderate neutrons to low energies to maximize fission probability. Fast reactors deliberately avoid moderation to exploit different cross-section behavior and breed plutonium from U-238. Always use the cross-section value appropriate for your neutron energy spectrum.
What is neutron activation analysis and why does neutron flux matter for it?
Neutron activation analysis (NAA) is an analytical technique that bombards a sample with neutrons, causing target nuclei to absorb neutrons and become radioactive isotopes. The resulting gamma-ray spectrum identifies and quantifies trace elements with high sensitivity. The reaction rate R = φ × σ × N directly determines how much radioactivity is induced — a higher neutron flux produces stronger signals and allows detection of smaller concentrations. Research reactors with fluxes of 10¹³–10¹⁴ n/cm²/s are commonly used for NAA. The technique is applied in archaeology, environmental monitoring, forensics, and semiconductor quality control to measure elemental concentrations down to parts per billion.