Vol 3 - Landau, Lifshitz - Quantum mechanics - non-relativistic theory (3ed., 1991) · Problem Problem 2 (§127)

Near an extremum of $\theta(\rho)$, the scattering angle becomes insensitive to small changes in impact parameter—this is where rainbow scattering occurs. The key insight is that many different impact parameters bunch up and scatter into nearly the same direction, causing a divergence in the differential cross section. You need to expand $\theta(\rho)$ to second order around the extremum and use the relationship between impact parameter and scattering angle to find how the cross section diverges, revealing the characteristic intensity peak and oscillatory structure of a rainbow.

Near an extremum of $\theta(\rho)$, the scattering angle becomes insensitive to small changes in impact parameter—this is where rainbow scattering occurs. The key insight is that many different impact parameters bunch up and scatter into nearly the same direction, causing a divergence in the differential cross section. You need to expand $\theta(\rho)$ to second order around the extremum and use the relationship between impact parameter and scattering angle to find how the cross section diverges, revealing the characteristic intensity peak and oscillatory structure of a rainbow.