5.83500000e-22
cm^2
35.1267
1/cm
0.028468
cm
35.1267
5.554795e-16
1
5.5548e-6
particles/s
10,000,000,000
reactions/s
5.83500000e-22
cm^2
35.1267
1/cm
0.028468
cm
35.1267
5.554795e-16
1
5.5548e-6
particles/s
10,000,000,000
reactions/s
The Cross Section Calculator computes the mean free path, attenuation, and reaction rate for nuclear or particle interactions using the reaction cross section. The cross section sigma is the fundamental measure of the probability of a nuclear or particle interaction, expressed as an effective area (in units of barns, where 1 barn = 10^-24 cm^2).
The concept of cross section originates from classical mechanics where the geometric cross section pi*R^2 determines collision rates. In quantum mechanics, the effective cross section for a reaction can be much larger or smaller than the geometric size of the nucleus, depending on quantum tunneling, resonances, and selection rules. The cross section depends strongly on the projectile energy and the target nucleus.
For a beam of particles incident on a thin target, the fraction of beam particles that undergo a reaction is sigma * N * x, where N is the number density of target atoms (atoms/cm^3), and x is the target thickness. For thicker targets, the attenuation is exponential: I = I0 * exp(-sigma*N*x), analogous to Beer-Lambert law.
1 barn = 10^-24 cm^2 was originally defined as about the cross-sectional area of a uranium nucleus (~150 fm^2), with the facetious remark that hitting a nucleus is like hitting the broad side of a barn. Cross sections for various reactions span many orders of magnitude: hard sphere (nuclear size) ~ 1 barn, slow neutron absorption by U-235 ~ 583 barns, thermal neutron capture by Xe-135 ~ 2.6 million barns (the strongest neutron absorber known), neutrino scattering ~ 10^-44 cm^2 (10^-20 barns).
Cross sections are central to nuclear reactor design, radiation shielding, particle detector design, medical physics (proton therapy), and fundamental particle physics experiments.
Cross section unit: 1 barn = 10^-24 cm^2. Macroscopic cross section: Sigma = sigma * N (cm^-1). Mean free path: lambda = 1/Sigma = 1/(sigma*N) cm. Thin target reaction probability: P = Sigma * x = sigma * N * x. Reaction rate: R = I0 * Sigma * x = I0 * sigma * N * x reactions/s (valid for x much less than mean free path).
Mean free path is the average distance a particle travels before interacting. For thermal neutrons in water (sigma ~ 0.66 barns for H, N ~ 6.7e22/cm^3): mean free path ~ 2.3 cm. U-235 for thermal neutrons (sigma_f ~ 583 barns): much shorter interaction lengths. Reaction rate proportional to both beam intensity and target thickness * density.
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Results
U-235 has a thermal neutron fission cross section of 583.5 barns. In a 1 cm thick U metal target (number density ~4.8e22 atoms/cm^3), the macroscopic cross section is ~28 cm^-1 — extremely high, meaning most neutrons interact within a few mm.
Inputs
Results
Fast neutrons (sigma ~ 2.5 barns on Fe) have a mean free path of ~4.7 cm in iron. A 10 cm iron shield attenuates about 88% of incident neutrons.
An effective area representing the probability of a nuclear reaction. Larger sigma means higher probability. It is NOT the actual geometric area of the nucleus but an effective area determined by quantum mechanics.
1 barn = 10^-24 cm^2 = 10^-28 m^2. It is approximately the cross-sectional area of a uranium nucleus (~175 fm^2 = 1.75 barns). The name was coined (as a joke) at Los Alamos during the Manhattan Project.
Sigma (capital) = sigma (barn) * N (number density) with units of cm^-1. It is the interaction probability per unit path length. The mean free path is 1/Sigma.
Quantum tunneling allows reactions at impact parameters larger than the nuclear radius. Resonances (matching projectile energy to nuclear excited states) can enormously enhance cross sections. Xe-135 absorbs thermal neutrons with sigma = 2.6 million barns.
A sharp increase in cross section at specific projectile energies where the compound nucleus (projectile + target) matches an excited state energy. At resonance, the cross section can be millions of times larger than the geometric cross section.
Total = all interactions. Elastic = projectile scattered without energy transfer to nucleus (direction changes only). Inelastic = nucleus is left in excited state. Reaction = new particles produced (fission, capture, etc.). sigma_total = sigma_elastic + sigma_inelastic + sigma_reaction.
Neutron cross sections for fission (U-235: 583 barns), capture (U-238: 2.73 barns at thermal), and moderation (H: 0.33 barns) determine criticality conditions, fuel burnup rates, and reactor neutron economy.
k = eta * epsilon * p * f: the nuclear multiplication factor in a thermal reactor. Each factor involves cross sections: eta = neutrons per absorption in fuel (uses sigma_fission/sigma_absorption), p = resonance escape probability, f = thermal utilization. k must equal 1 for criticality.
By measuring the attenuation of a known beam through a known target: sigma = -ln(I/I0)/(N*x) for thick targets, or sigma = (I0-I)/(I0*N*x) for thin targets. Modern measurements use accelerator beams, nuclear reactors, and coincidence detection.
1 mb (millibarn) = 10^-3 b = 10^-27 cm^2. 1 fb (femtobarn) = 10^-15 b = 10^-39 cm^2. Particle physics at the LHC measures Higgs production cross sections in femtobarns. Nuclear reactions use barns. The 10^24 range reflects the vast energy spectrum in physics.
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