Nuclear Cross Section Calculator

A nuclear cross section (σ) is a measure of the probability that a specific nuclear reaction will occur between an incident particle and a target nucleus. It is expressed in units of barns (b) or millibarns (mb), where 1 barn = 10⁻²⁴ cm². A larger cross section means a higher likelihood of interaction.

Rutherford scattering describes the elastic Coulomb scattering of charged particles off a target nucleus. The Rutherford differential cross section is dσ/dΩ = (Z₁Z₂e²/4E)² × 1/sin⁴(θ/2), where Z₁ and Z₂ are atomic numbers, E is the center-of-mass energy, and θ is the scattering angle. It serves as the classical baseline against which non-Rutherford effects are measured.

Cross sections are measured in barns (b), millibarns (mb), microbarns (μb), or nanobarns (nb). 1 barn = 10⁻²⁴ cm² ≈ roughly the geometric cross-sectional area of a heavy nucleus. Differential cross sections are given in mb per steradian (mb/sr).

The mean free path (λ) is the average distance a particle travels in a material before undergoing a reaction. It is calculated as λ = A / (ρ × Nₐ × σ), where A is the molar mass, ρ is the density, Nₐ is Avogadro's number, and σ is the total cross section. A larger cross section means a shorter mean free path.

The differential cross section (dσ/dΩ) describes how scattering is distributed as a function of angle — it tells you how many particles are scattered into a specific solid angle. The total cross section (σ) is obtained by integrating the differential cross section over all solid angles (4π steradians) and represents the overall reaction probability regardless of direction.

For Rutherford scattering, the cross section decreases as beam energy increases (σ ∝ 1/E²). For nuclear reactions, cross sections often exhibit resonances — sharp peaks at specific energies where quantum-mechanical effects dramatically enhance the reaction probability. At very high energies, cross sections tend to approach a geometric (hard-sphere) limit.

Areal density (nt) is the number of target atoms per unit area, given by nt = (ρ × t × Nₐ) / A, where ρ is density, t is thickness, Nₐ is Avogadro's number, and A is molar mass. It directly determines reaction probability: P = nt × σ. Thin-target approximations are valid when nt × σ ≪ 1 (probability much less than 100%).

The geometric cross section models the nucleus as a hard sphere with radius R ≈ r₀ × A^(1/3), where r₀ ≈ 1.2 fm and A is the mass number. The total geometric cross section is σ = π × R² and represents the maximum possible cross section at high energies. It provides a useful upper-bound estimate when detailed nuclear structure data is unavailable.