1.000000e-26
cm²
0.01
barns
10
millibarns
1,000,000
cm
1.000000e-14
per target/particle
1.000000e-26
cm²
0.01
barns
10
millibarns
1,000,000
cm
1.000000e-14
per target/particle
The Nuclear Cross Section Calculator determines the effective target area for a nuclear reaction, given the observed reaction rate, incident particle flux, and number of target nuclei. The nuclear cross section (σ) is one of the most important quantities in nuclear and particle physics, quantifying the probability that a specific nuclear reaction will occur when a projectile particle encounters a target nucleus.
Despite its name, the cross section is not simply the geometric area of a nucleus. It is an effective area that encapsulates all the quantum mechanical complexities of the interaction—the nuclear force, Coulomb barrier, resonance effects, angular momentum conservation, and available final states. A reaction with a large cross section is highly probable; one with a small cross section is rare.
The fundamental relationship is \(R = \sigma \cdot I \cdot N\), where \(R\) is the reaction rate (events per second), \(I\) is the incident flux (particles per cm² per second), and \(N\) is the number of target nuclei. Solving for \(\sigma\) gives the cross section in cm².
Cross sections are conventionally measured in barns (b), where 1 barn = \(10^{-24}\) cm². The name originated as a humorous expression during the Manhattan Project—a uranium nucleus, at about 1 barn, was considered "as big as a barn" compared to smaller particle physics cross sections. Nuclear reactions typically range from millibarns to kilobarns, while particle physics processes can be measured in picobarns or femtobarns.
This calculator also computes the mean free path—the average distance a particle travels through the target material before interacting—which is critical for reactor design, shielding calculations, and particle detector design. The interaction probability per target per incident particle is also provided.
Cross section measurements are the primary experimental output of nuclear physics. They are tabulated in databases like ENDF (Evaluated Nuclear Data File) and EXFOR, and they drive the design of nuclear reactors, medical accelerators, radiation shielding, and astrophysical models of stellar nucleosynthesis.
The cross section is derived from the reaction rate formula:
$$R = \sigma \cdot I \cdot N$$
Solving for the cross section:
$$\sigma = \frac{R}{I \cdot N}$$
where \(R\) is in reactions/s, \(I\) is in particles/cm²·s, and \(N\) is the number of target nuclei.
Barn conversion:
$$\sigma_{\text{barn}} = \frac{\sigma_{\text{cm}^2}}{10^{-24}}$$
Mean free path:
$$\ell = \frac{1}{n \cdot \sigma}$$
where \(n\) is the number density of target nuclei (here approximated as N for a thin target).
Cross sections near 1 barn (10⁻²⁴ cm²) are typical for nuclear reactions. Thermal neutron capture can have cross sections of thousands of barns (e.g., boron-10: 3,840 barns). High-energy scattering cross sections are often in millibarns. If your result is many orders of magnitude from typical values, check the units of flux and reaction rate. A shorter mean free path means more frequent interactions.
Inputs
Results
With these parameters, the cross section is exactly 1 barn—a typical order of magnitude for nuclear reactions. The mean free path of 10⁶ cm (10 km) indicates a thin target where most neutrons pass through without interacting.
Inputs
Results
This reproduces the famous boron-10 thermal neutron capture cross section of 3,840 barns, which is why boron is used in nuclear reactor control rods and neutron shielding.
A barn (b) is a unit of area equal to 10⁻²⁴ cm², roughly the geometric cross-sectional area of a uranium nucleus. The name originated during the Manhattan Project as wartime slang—physicists considered nuclear targets "as big as a barn" compared to the much smaller cross sections they were used to in particle physics. Despite its humorous origin, it became the standard unit.
The cross section includes quantum mechanical effects beyond simple geometry. At low energies, the de Broglie wavelength of the projectile can be much larger than the nucleus, making the effective interaction area larger. Resonance effects can enhance cross sections by orders of magnitude when the projectile energy matches a nuclear energy level. Some thermal neutron cross sections exceed 10⁴ barns.
Cross section (σ) is an effective area with units of cm² or barns—it is an intrinsic property of the reaction. Reaction probability depends on both the cross section and the experimental conditions (flux, target thickness). For a thin target, probability ≈ σ × n × Δx, where n is the number density and Δx is the target thickness.
Cross sections are measured by directing a beam of known flux onto a target of known composition and thickness, then counting the reaction products. Detectors surrounding the target measure the rate of specific reactions. By knowing R, I, and N, the cross section is calculated from σ = R/(IN). Multiple measurements at different energies map out the cross section as a function of energy.
The Evaluated Nuclear Data File (ENDF) is the primary international repository of nuclear cross section data, maintained by Brookhaven National Laboratory. It contains evaluated (quality-checked and recommended) cross sections for thousands of nuclear reactions across all energies, used for reactor design, radiation transport simulations, and medical physics calculations.
Several factors influence cross section magnitude: (1) the projectile energy relative to the Coulomb barrier, (2) resonance effects when energy matches nuclear levels, (3) the angular momentum change required, (4) the nuclear structure of target and projectile, and (5) the specific reaction channel. Neutron reactions typically have larger cross sections than charged particle reactions because neutrons face no Coulomb barrier.
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