Rainbow singularities in the doubly differential cross section for scattering off a perturbed magnetic dipole

We take the scattering of electrons off a perturbed magnetic dipole as an example for demonstrating chaotic scattering systems with three open degrees of freedom. We explain the connection between the chaotic invariant set, the scattering functions and the doubly differential cross section. The most interesting structures in this cross section are curves of rainbow-type singularities. The ideal magnetic dipole has rotational symmetry; therefore, the dynamics has a conserved angular momentum, and the chaotic set of the three degrees of freedom system can be represented as a stack of chaotic sets of the reduced two degrees of freedom system where the numerical value of the conserved angular momentum serves as stack parameter. However, any real magnet deviates from a perfect dipole; therefore, we explain to which extent the qualitative properties of the cross section are robust against perturbations of the system, e. g., against deviations of the magnetic field from a perfect dipole which in general include the destruction of the rotational symmetry.