Lang and Kohn

In this section, we discuss the concept of classical image force. The validity of this concept has been verified using quantum mechanics in a many-body formalism (Bardeen, 1936 Lang and Kohn, 1970 Appelbaum and Hamann, 1972 Herring, 1992). We will present it in Chapter 4. 

Fig. 4.4. Surface energy predicted by the jellium model. A test case of the accuracy of the jellium model, conducted by Lang and Kohn (1970). Only for four alkali metals, Na, K, Rb, and Cs, are the predictions of surface energy by the jellium model fair. For most metals, with r, < 2.5 bohr, the surface energy predicted by the jellium model is negative, contradicting seriously with experimental facts. On the other hand, the calculated values of surface energies with crystal lattices agree much better with the values measured experimentally. (After Lang and Kohn, 1970).

More recently, Lang and Kohn (1970) treated the jellium model with modern methods and modern computers. In particular, they included exchange terms in the frcc-clcctron approximation, as we have discussed. In lliis context it is interesting that the potential well that holds the electrons in the metal comes predominantly from theexchange potential the electrostatic potential itself gives only... 

Theoretical values of surface energy, obtained by Lang and Kohn (1970), as a function of r, where r, is related to the electron density by W = (4nr /3y. The key is shown at the left. Calculations based on psetidopotential theory were made for bcc and fee structures, with results given as top and bottom, respectively, of a vertical line at the electron density for each metal. For Na, K, Rb, and Cs, the lines would almost be contained within the experimental points and were not drawn. 

The calculations as performed by Lang and Kohn required major computational effort, and meaningful simplified approaches appear not to have been developed yet. Simpler calculations may be possible and one could expect the electrostatic contributions to vary as and the electronic contributions to... 

Lang and Kohn (1971) also calculated the work function itself, correctly finding sizable variations from one face to another. The results agreed with existing data to within 5 to 10 percent. 

Fig. 6. Self-consistent surface charge density in the jellium model for K (solid line) and A1 (dashed line) (Lang and Kohn, 1 y70). Distance is measured in Fermi wavelengths from the positive background edge charge density is measured relative to the bulk density pa-...

The ionization potential IP is the energy necessary to extract one electron from the neutral cluster. For a macroscopic solid this is called the work function, W. In this case Lang and Kohn [33] have shown that W can be expressed as the sum of three terms... 

In most treatments of the surface electron distribution, the so-called "jellium model has been used, for which the atomic cores are smeared into a planar uniform background. The jellium model has been found, e.g., Lang and Kohn (1973), to work well for simple metals and, with regard to the surface electron response, has been employed to treat the behavior of some of the noble metals (Equiluz, 1984). 

Some early attempts to provide a solution to this question for the simpler case of a free charge in vacuum near a metal surface were made by Bardeen (1940), Sachs and Dexter (1950), Cutler and Davis (1964), and most recently by Lang and Kohn (1973), but mainly for the interaction of a charge in vacuum with electrons at or near a metal surface. 

More recently, Lang and Kohn (1973) gave a detailed discussion of this problem, including the question of chemisorption, e.g., of alkali metal atoms on transition metal surfaces such as those of W, Ta, Re. The problem is approached first by considering the profiles of the charge induced by a uniform external electric field for metals of different bulk electron densities. At electrodes, it is to be noted, an additional factor is the potential-dependent surface electron density, q, given, in the case of liquid metals of surface tension y> by q = -(8y/9E),... 

Fig. 4. Model for a metal interface with a charge near to it at r (from Lang and Kohn, 1973).

Interfacial dipole At a metal-vacuum interface, electron tunnelling into vacuum modifies the electron-electron interactions and weakens the electron-ion interaction ( ee and Eec, respectively, in Equation (1.4.64)). These two processes use up energy when a surface is formed. Lang and Kohn quote the following values 0.43 J/m and 0.5 J/m at a magnesium-vacuum interface, although the latter, calculated in perturbation, is likely to be over-estimated... 

Theory of the Inhomogeneous Electron Gas edited by S. Lundqvist and N. H. March Chapter 5 by N. D. Lang is a good one-stop shop for most of the early material stemming from the work of Lang and Kohn. 

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