Approximate solutions for the excitation of an atomby a colliding charged particle, developed as perturbations of the state of the system at infinite or asymptotic separation of atom and particle and also as perturbations of the state of the system at vanishing velocity of relative motion, are extended, in the impact parameter form (suitable to collision of heavy particles), to include the interaction of states that are degenerate at asymptotic separation, and to obtain thereby a condition for the Born approximation that depends on the phase relations of degenerate states in the collision and excludes the low velocity region to the Born approximation, except for the excitation of S states.This is a correction of previous theories, whichconclude that the Born solution is a general approximation at low velocities for weak interactions.
A low velocity perturbation solution is established interms of the stationary states of the system and developed to show that, at sufficiently low velocity of relative motion, the atomic states are coupled to the moving particle in the range of interaction.Differences in coupling energy affect the coherence of asymptotically degenerate states in the collision and influence the orientations of final excited states.