The formation of negative hydrogen ions by scattering protons from a metal surface is described with two models: a probability model and an amplitude model. In both models the electron motion is described quantum mechanically and the nuclear motion classically. However, in the probability model the time evolution of the ionization probability is considered, while in the amplitude model the time evolution of the corresponding wave function amplitude is considered. The electron affinity level of an atom close to the metal is lowered by means of image forces and broadened due to resonant transition of an electron between te conduction band of the metal and the valence shell of the atom. The calculated position of the affinity level and the transition rate in both models give rise to maximum negative ionization efficiencies of 4% on W(110), 40% on cesiated tungsten and 15% on cesium.
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