Nuclear Cross Section

Nuclear Cross Section

The cross section $\sigma$ quantifies how likely a nuclear reaction is. Imagine the target nucleus presenting an effective area to an incoming particle — the larger the cross section, the more probable the reaction.

$\sigma \quad [\text{barn}], \quad 1 \text{ barn} = 10^{-28} \text{ m}^2$

The barn is a surprisingly large unit: it was named during the Manhattan Project because uranium nuclei seemed "as big as a barn."

Key Quantities

Reaction rate per target nucleus: $R = \Phi \cdot \sigma \quad [\text{reactions/s}]$ where $\Phi$ is the particle flux (particles/m²/s).

Mean free path — average distance a projectile travels before reacting: $\lambda_{\text{mfp}} = \frac{1}{N \sigma}$ where $N$ is the number density of target nuclei [m⁻³].

Attenuation of a beam passing through material of thickness $x$: $I(x) = I_0 \, e^{-N \sigma x}$

Example: Thermal Neutrons in Water

Water has $N \approx 3.34 \times 10^{28}$ molecules/m³. The total neutron cross section for water near $\sigma \approx 49$ barn gives a mean free path of about 6 mm — thermal neutrons are moderated very efficiently.

Your Task

Implement:

• mean_free_path(N_density, sigma_m2) — mean free path in meters
• attenuation(I0, N_density, sigma_m2, x) — transmitted intensity
• reaction_rate(flux, sigma_m2) — reaction rate per target nucleus