Cross Section Reaction Calculator Formula
Understand the math behind the cross section reaction calculator. Each variable explained with a worked example.
Formulas Used
Atom Number Density
atom_density = n_densityMacroscopic Cross Section (Sigma)
macro_sigma = n_density * sigma_cm2Mean Free Path
mean_free_path = 1 / (n_density * sigma_cm2)Variables
| Variable | Description | Default |
|---|---|---|
micro_sigma | Microscopic Cross Section (sigma)(barns) | 580 |
density | Material Density(g/cm³) | 19.1 |
atomic_mass | Atomic Mass (A)(g/mol) | 235 |
n_density | Derived value= density * 6.022e23 / atomic_mass | calculated |
sigma_cm2 | Derived value= micro_sigma * 1e-24 | calculated |
How It Works
Nuclear Reaction Cross Section
The macroscopic cross section combines the probability of a single nucleus interacting (microscopic cross section) with the number of target nuclei per volume.
Formulas
N = rho × NA / A (atom number density)
Sigma = N × sigma (macroscopic cross section)
Lambda = 1 / Sigma (mean free path)
where sigma is in barns (1 barn = 10⁻²⁴ cm²), NA = 6.022 × 10²³/mol.
Worked Example
U-235 (sigma_f = 580 barns, density 19.1 g/cm³, A = 235).
- 01N = 19.1 × 6.022 × 10²³ / 235 = 4.89 × 10²² atoms/cm³
- 02Sigma = 4.89 × 10²² × 580 × 10⁻²⁴ = 28.38 cm⁻¹
- 03Mean free path = 1 / 28.38 = 0.0352 cm
Frequently Asked Questions
What is a barn?
A barn (b) = 10⁻²⁴ cm² is the standard unit for nuclear cross sections. The name comes from the phrase "as big as a barn" because early nuclear physicists found certain cross sections surprisingly large.
Why does cross section vary with neutron energy?
Nuclear interactions depend on the relative energy between neutron and nucleus. At thermal energies, many fissile isotopes have large cross sections that follow a 1/v law. At higher energies, resonances create sharp peaks and valleys.
What is mean free path?
The average distance a neutron travels between interactions. Short mean free path means high reaction probability. In U-235 fuel, the fission mean free path is only fractions of a centimeter.
Ready to run the numbers?
Open Cross Section Reaction Calculator