Fold, Cusp and Swallowtail Catastrophes in the Evolution of Planar Closed Orbits of Hydrogen in Crossed Electric and Magnetic Fields

摘要

We investigated the morphology of hydrogenic planar orbits in crossed fields. We found that for a certain family of orbits the scaled classical action, the scaled electric field, the scaled energy and the shape of the orbit, which can be described by the initial and final angles, are related through a swallowtail catastrophe.The three scaled parameters generate the control space and the angles the behavioral space. The unfolding of this manifold generates a cusp catastrophe, in agreement with the fact that the scaled classical action of the trajectories as a function of the maximum approach of the electron to the nucleus reaches its maxima and minima values when the orbits are closed. Properties of this catastrophe as hysteresis, inaccessibility and divergence are investigated. The fold catastrophe is also present as the cusp curve generated by the extremal points of the scaled classical action. Manifestations of these catastrophes in the photoabsorption spectra are discussed.