Calculate nuclear and particle reaction cross sections, mean free path, reaction rates, and interaction probability for beam-target experiments.
The reaction cross section (σ) is the effective area a target nucleus or particle presents for a specific interaction. Measured in barns (1 b = 10⁻²⁴ cm²), it quantifies the probability that a projectile will interact with a target — fundamental to nuclear physics, particle physics, and reactor engineering. Because the units are microscopic and energy-dependent, hand calculations are easy to misread. Even a small unit slip can change the predicted rate by orders of magnitude.
This calculator computes reaction rates, mean free path, interaction probability, and attenuation for beam-target configurations. Enter the cross section, beam flux, and target properties to get quantitative predictions for your experiment or reactor calculation.
Whether you're calculating neutron absorption in a reactor fuel element, estimating luminosity-weighted event rates at a particle collider, or computing cosmic ray interaction depths, this tool provides the standard nuclear physics relationships in a convenient interface with reference cross sections for common reactions.
Cross section calculations involve very small numbers (10⁻²⁴) and exponentials that are error-prone by hand. This calculator handles the unit conversions and provides instant sensitivity analysis. It is useful when you need quick checks on rates, attenuation, or target thickness without rebuilding the equations from scratch. That is especially helpful during experiment planning and reactor-shielding estimates.
Reaction rate R = σ × Φ × N. Interaction probability P = 1 - e^(-n×σ×x). Mean free path λ = 1/(n×σ). Attenuation I/I₀ = e^(-n×σ×x). Where σ = cross section, Φ = flux, n = number density, x = thickness.
Result: λ = 35.6 cm, P = 2.8% per cm for a representative thermal target
Using σ = 0.585 barns and number density 0.048 × 10²⁴/cm³ gives a mean free path of 35.6 cm and a 2.8% reaction probability per cm.
The concept of cross section reduces the quantum-mechanical scattering problem to an effective area. For s-wave neutron scattering, the maximum theoretical cross section at a resonance is σ_max = 4πλ², where λ is the de Broglie wavelength — this can be thousands of barns for slow neutrons, explaining why thermal neutron cross sections are often much larger than the geometric nuclear size.
The total cross section is the sum of partial cross sections: σ_total = σ_elastic + σ_inelastic + σ_absorption + σ_fission + ... Each partial cross section has its own energy dependence and resonance structure.
The mean free path λ = 1/(nσ) is the average distance a particle travels between interactions. For thermal neutrons in water, λ ≈ 0.5 cm (hydrogen has σ_scatter ≈ 20 barns and high number density). For gamma rays in lead, λ ≈ 1-2 cm. Shielding design uses these values to calculate the thickness needed to attenuate radiation to safe levels.
At particle colliders, cross sections are measured in pb (picobarns, 10⁻³⁶ cm²) or fb (femtobarns, 10⁻³⁹ cm²). The Higgs boson production cross section at the LHC is about 50 pb. With a luminosity of 10³⁴ cm⁻²s⁻¹, this gives roughly one Higgs boson every two seconds — illustrating the extraordinary precision needed in these experiments.
1 barn = 10⁻²⁴ cm² = 10⁻²⁸ m². Named humorously because nuclear physicists considered it "as big as a barn" compared to nuclear sizes. Typical cross sections range from microbarns to kilobarns.
It represents the effective area a target presents — a larger cross section means higher interaction probability, as if the target were physically larger. The term is conceptual rather than a literal geometric area for most quantum interactions.
It depends on the interaction type (elastic, inelastic, absorption), projectile energy, and nuclear structure. Resonances can make cross sections spike by orders of magnitude at specific energies.
σ_geo = π(R₁ + R₂)² where R = r₀A^(1/3) with r₀ ≈ 1.2 fm. For heavy nuclei, σ_geo ≈ 1-2 barns. Actual cross sections can be much larger (resonances) or smaller (Coulomb barrier).
For neutrons: 1/v law at low energies (thermal), resonances at intermediate energies, and roughly geometric at high energies. Charged particles have Coulomb barrier effects at low energies.
Luminosity L (cm⁻²s⁻¹) × cross section σ (cm²) = event rate (s⁻¹). The LHC delivers ~10³⁴ cm⁻²s⁻¹, so a 1 nb cross section gives 10 events/second.