Image potentials

The zeroth-order solution to the above equations is tire Gotiy-Chapman theory dating from the early part of the 20th cenPiry [20], In this solution, the ionic aPnosphere is ignored, as is the mirror image potential for the ion. Equation A2.4.90 can therefore be ignored and equation A2.4.89 reduces to... 

Figure Bl.26.21. Potential energy curves for an electron near a metal surface. Image potential curve no applied field. Total potential curve applied external field = -E. ...

The exact same formula may be derived using the concept of an image potential (obtained by inverting the sign offrs and ts), and the QA name is often used together with the TRIM acronym, which stands for Trust Radius Image Minimization ... 

The reason for the success of this type of data fitting is that for moderately large barriers it becomes unimportant whether escape for the image potential is treated within the framework of the Onsager or the RS model. An indication that the Onsager description is, nevertheless, more appropriate is the intersection of j(/ ) curves for variable temperature at high electric fields. This is a characteristic feature of Onsager processes 24J. 

There remains the estimated value of °(H+/H2)vs. UHV based on binding energies for image potential-induced surface states,49 which is,... 

The classical result for the image potential is -q/4x, independent of the metal, but various theories of the metal which assume an infinite potential barrier for the metal electrons give potentials which are reduced in size near the metal boundary, so that the interaction energy is actually finite24 at x = 0. An interpolation formula which reproduces this behavior is... 

Figure 7.9 Potential energy diagram for electrons in and near a metal to which a high negative potential is applied. Electrons in the valence band of the metal see an attractive potential equal to -eFr (F is the applied field in V/cm) outside the metal behind a barrier formed by the applied field and the image potential.

Fig. 8.1. Field ionization of a hydrogen atom (H). (a) close to a tungsten surface (W), (b) isolated. Conditions and symbols electric field 2 V A Pw image potential of W distorted by the field, Ph potential of the hydrogen atom distorted by the field, X work function, p Fermi level. Broken lines represent potentials in absence of the electric field. Adapted from Ref. [4] by permission. Verlag der Zeitschrift ftir Naturforschung, 1955.

Fig. 2.2. The image potential of an electron near a metal surface. The electron induces positive charge at the metal surface, (a) The effect of the positive surface charge is equivalent to a fictitious image charge behind the metal surface, (b) fhe distance dependence of the image potential.

Fig. 2.3. The image potential of an electron inside a tunneling barrier. The...

The MBA provides another simple explanation of why the image potential is not observable. According to Eq. (2.42), as long as the integral of the distortion potential is a constant (that is, the shaded area in Fig. 2.8 remains constant while varying the tip-sample distance), the effect of distortion is a constant independent of the barrier thickness. Therefore, the effect of barrier lowering due to image force is not observable. 

In this section, we will derive the correction factor for the tunneling current due to the image force. For a free metal surface, the image potential pertinent to this surface, Eq. (2.2), is always prc.sent. The the simple image potential is always an essential part of the free sample, and is always implied in any first-... 

Fig. 4.2. Charge distribution and surface potential in a jellium model, (a) Distribution of the positive charge (a uniform background abruptly drops to zero at the boundary) and the negative charge density, determined by a self-consistent field calculation. (b) Potential energy as seen by an electron. By including all the many-body effects, including the exchange potential and the correlation potential, the classical image potential provides an adequate approximation. (After Bardeen, 1936 see Herring, 1992.)...

Fig. 4.3. Position of the image plane in the jellium model. The surface potential of an electron in the jellium model is calculated using the local-density approximation. By fitting the numerically calculated surface potential with the classical image potential, Eq. (4.7), the position of the image plane is obtained as a function of r, and z. The results show that the classical image potential is accurate down to about 3 bohrs from the boundary of the uniform positive charge background. For metals used in STM, r, 2 — 3 bohr, zo 0.9 bohr. (Reproduced from Appelbaum and Hamann, 1972, with permission.

Payne, M. C., and Inkson, J. C. (1985). Measurement of work functions by tunneling and the effect of the image potential. Surface Science 159, 485-495. 

Consider the deviations of 0max from the pzc quantitatively. Can you associate them with the probable orientation of the adsorbed organic molecules on the surface Or perhaps on image potentials formed by dipoles of the organic with the metal Perform simple calculations to determine whether breakdown of the first approximation assumption of zero interaction of the organic molecule with the field, or image forces, explains the deviation of Vmax of the organic molecule from the pzc. (Bockris)... 

---