Image-potential states 

When an electron is in front of a conductive surface, an electric field in the vacuum is induced which can be described by introducing a positive charge opposite to the electron within the conductor (Fig 1).

This electric field leads to an attracive, Coulomb-like force (Eq. 1), which itself is associated with an attractive potential (Eq. 2).

Within this potential a Rydberg-like series of quantum states, the so-called image-potential states, are formed. Their binding energy is given in equation 3.

The quantum defect a arises from the small penetration of the image-potential states into the bulk and the corresponding matching conditions. Since the binding energy is quite small (1/16th of the hydrogen atom) and is pinned to the vacuum level, the image-potential states are unoccupied. As you can see in figure 2, the average distance of the wavefunction to the surface increases with increasing quantum number, as well as the overlap to the bulk decreases. The small overlap to the bulk implicates longer lifetimes of the image-potential-state electrons compared to excited electrons in the bulk. Typical values for the lifetime range from several tens of femtoseconds for the first image-potential state up to some picoseconds for image-potential states with higher quantum numbers [Höfer et al., Science 277, 1480].

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