Born Approximation Validity Map

Born gives the exact Rutherford formula at all energies for Coulomb — but at a resonance it predicts nothing. Resonances are non-perturbative. Map shows where the Born series converges for a Yukawa potential V = V₀ e^−r/r.

Validity map — log-log (ka, |V₀|)

Angular distribution dσ/dΩ (normalized)

ka = 3.00 (dimensionless wavenumber)

|V₀| = ξ = 1.00 (dimensionless strength)

ka = 3.00 ξ = 1.00 Born valid: YES dσ/dΩ(90°)/dσ/dΩ(1°) = —