Kepler Map

Abstract

We present a consecutive derivation of mapping equations of motionfor the one-dimensional classical hydrogenic atom in a monochromaticfield of any frequency. We analyze this map in the case of high andlow relative frequency of the field and transition from regular tochaotic behavior. We show that the map at aphelion is suitable forinvestigation of transition to chaotic behavior also in the lowfrequency field and even for adiabatic ionization when the strength ofthe external field is comparable with the Coulomb field. Moreover, theapproximate analytical criterion (taking into account theelectron’s energy increase by the influence of the field) yieldsa threshold field strength quite close to the numerical results. Wereveal that transition from adiabatic to chaotic ionization takesplace when the ratio of the field frequency to the electron Keplerfrequency approximately equals 0.1. For the dynamics and ionization ina very low frequency field the Kepler map can be converted to adifferential equation and solved analytically. The threshold field ofthe adiabatic ionization obtained from the map is only 1.5% lower thanthe exact field strength of static field ionization.