Nuclear Cross Section Calculator
The nuclear cross section is the fundamental quantity describing the probability of a nuclear reaction. Reaction rate R = sigma * phi * N, where sigma is the cross section (in barns, with 1 barn = 10^-28 m^2), phi is the incident particle flux (particles per m^2 per second), and N is the target nucleus number density (nuclei per m^3). This calculator computes the macroscopic reaction rate and also converts cross sections between barns and SI units. The nuclear geometric cross section estimated from the hard-sphere nuclear radius formula R = 1.2 * A^(1/3) fm is also computed as a reference, though actual cross sections for specific reactions can differ by orders of magnitude due to resonances and quantum effects.
Nuclear reaction rate formula
sigma (m^2) = sigma (barns) * 1e-28
R = sigma * phi * N (reactions/m^3/s)
Rate per nucleus = sigma * phi
1 barn = 10^-28 m^2. R is in reactions per cubic meter per second. Rate per nucleus = sigma (m^2) * phi (n/m^2/s) = reactions per nucleus per second. Macroscopic cross section Sigma = sigma * N.
Notable nuclear cross sections
- U-235 thermal neutron fission: 584 barns.
- Xe-135 thermal neutron absorption: 2.6 million barns (huge resonance, causes reactor poisoning).
- H-1 (proton) thermal neutron scattering: 20 barns.
- C-12 thermal neutron absorption: 0.0035 barns (nearly transparent, used as moderator).
- Geometric cross section of U-238: about 174 millibarns.
Nuclear cross section: frequently asked questions
What is a nuclear cross section?
A nuclear cross section sigma is an effective area that quantifies the probability of a nuclear reaction. It is measured in barns (1 barn = 10^-28 m^2 = 10^-24 cm^2). A large cross section means a high reaction probability. Cross sections are energy-dependent and are tabulated for every isotope and reaction type in evaluated nuclear data files (ENDF).
What is a barn?
The barn (symbol b) is a non-SI unit of area used in nuclear physics: 1 barn = 10^-24 cm^2 = 10^-28 m^2. It was coined during the Manhattan Project with the remark that uranium nuclei are 'as big as a barn' compared to other nuclear targets. 1 millibarn (mb) = 10^-3 b, 1 microbarn (ub) = 10^-6 b.
How is reaction rate calculated from cross section?
Reaction rate R = sigma * phi * N, where sigma is the cross section (m^2), phi is the particle flux (particles per m^2 per second), and N is the number of target nuclei per unit volume (m^-3). Units of R are reactions per m^3 per second. For a specific reaction rate per target nucleus, use sigma * phi.
What is neutron flux?
Neutron flux phi = n * v, where n is the neutron number density (n/m^3) and v is the neutron velocity (m/s). In a nuclear reactor, thermal neutron flux is typically 10^13 to 10^14 n/cm^2/s (10^17 to 10^18 n/m^2/s). The product sigma * phi gives the reaction rate per target nucleus.
What is the geometric cross section of a nucleus?
The nuclear radius follows R = R0 * A^(1/3) where R0 = 1.2 fm (1 fm = 10^-15 m) and A is mass number. The geometric cross section is pi * R^2 = pi * (R0)^2 * A^(2/3). For uranium-238 (A=238): R = 7.44 fm, geometric cross section = 174 mb. Actual reaction cross sections can be much larger due to quantum effects.
Official sources
- NIST: Nuclear Data.
- IAEA: Nuclear Data Services.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.