Cross-section notes

If the number of "hits" in a given distance is proportional to the number density I of those still "unhit", then...

dI/dz = -QI, where Q is a cross-section per unit length.

From this follows the law of exponential decay for an un-interacted beam traveling distance z in a target sample...

I = I_o e^-Qz = I_o e^-z/τ, where τ=1/Q is a thickness mean free path.

Thus, the probability of no interaction is P_o = I/I_o = e^-z/τ, and the probability of at least one interaction is P₁₊ = 1-P_o = 1 - e^-z/τ. When (and only when) z is much less than τ, this interaction probability becomes P₁₊ ~ Qz = z/τ.

For example, if the inelastic mean free path for 300kV electrons in silicon is 200nm, the fraction of electrons which have lost energy after going through a 10nm specimen is P₁₊ = 1-e^-10/200 ~ 10/200 = .05.

In place of cross-section per unit length Q, folks who consider scattering a lot often define the cross-section per atom σ = Q/η (e.g. in cm² or barns), where η is the number density of atoms in the target (e.g. in atoms/cc).

For example, since the number density of atoms in silicon (and most solids) is η ~ 5×10²² [atoms/cc], for 300kV electrons the inelastic scattering cross-section per silicon atom is therefore σ ~ (1/200nm)/η = 0.01 Å².

In place of thickness mean-free-path τ, since mass and cross-section are sometimes proportional, nuclear physicists oft use the density mean-free-path τ_ρ = τ ρ (e.g. in g/cm²), where ρ is the mass density of the target (e.g. in g/cc).

For example, since the mass density of silicon is around 2.5[g/cc], the density mean-free-path for 300kV electrons in silicon (and other solids as well) is τ_ρ ~ 50[μg/cm²].