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).
micro_sigma = 580density = 19.1atomic_mass = 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
Ready to run the numbers?
Open Cross Section Reaction Calculator