Metal jellium model

Figure 2. Sketch of an uncharged metal surface (simulated by the jellium model) covered by a macroscopic solvent layer, showing the components of the electric potential drop. 8%M is the surface potential of the metal modified by the solvent layer %s + 6%s is the surface potential of the solvent modified by the contact with the metal %s is the unmodified surface potential of the solvent layer at the external surface.

Schmidder and Henderson282 have studied several solvents and metals, using the jellium model for the metal and the MSA for the solution. Deviations of the Parsons-Zobel plot from linearity in the experimental results72,286-288 at the highest concentration have been attributed to the onset of ion-specific adsorption. However, data at other electrode charges... 

The C, values for Sb faces are noticeably lower than those for Bi. Just as for Bi, the closest-packed faces show the lowest values of C, [except Bi(lll) and Sb(lll)].28,152,153 This result is in good agreement with the theory428,429 based on the jellium model for the metal and the simple hard sphere model for the electrolyte solution. The adsorption of organic compounds at Sb and Bi single-crystal face electrodes28,152,726 shows that the surface activity of Bi(lll) and Sb(lll) is lower than for the other planes. Thus the anomalous position of Sb(lll) as well as Bi(lll) is probably caused by a more pronounced influence of the capacitance of the metal phase compared with other Sb and Bi faces28... 

Figure 5.7. Schematic representation of the definitions of work function O, chemical potential of electrons i, electrochemical potential of electrons or Fermi level p = EF, surface potential %, Galvani (or inner) potential

Figure 6.13. The electron distribution in the model metal jellium gives rise to an electric double layer at the surface, which forms the origin of the surface contribution to the work function. The electron wave function reaches...

A simple metal like lithium or aluminum should best reveal the properties of the jellium model. To be sure, all long range order influence has been switched off, we measured S(q, co) of liquid A1 (T = 1000K). Figure 6 shows the result of a measurement for q = 1.5 a.u. together with theoretical calculations. 

With respect to the thermodynamic stability of metal clusters, there is a plethora of results which support the spherical Jellium model for the alkalis as well as for other metals, like copper. This appears to be the case for cluster reactivity, at least for etching reactions, where electronic structure dominates reactivity and minor anomalies are attributable to geometric influence. These cases, however, illustrate a situation where significant addition or diminution of valence electron density occurs via loss or gain of metal atoms. A small molecule, like carbon monoxide,... 

The ionic profile of the metal was modeled as a step function, since it was anticipated that it would be much narrower than the electronic profile, and the distance dx from this step to the beginning of the water monolayer, which reflects the interaction of metal ions and solvent molecules, was taken as the crystallographic radius of the metal ions, Rc. Inside the metal, and out to dl9 the relative dielectric constant was taken as unity. (It may be noted that these calculations, and subsequent ones83 which couple this model for the metal with a model for the interface, take the position of the outer layer of metal ion cores to be on the jellium edge, which is at variance with the usual interpretation in terms of Wigner-Seitz... 

Since each atom in a f.c.c. array of purely metallic atoms is the same as every other atom (except at the surface), only a representative positive ion needs to be considered. Let it interact with a spherical portion (radius = R) of the electron gas which has a density of one electron per ion. This is called the jellium model. 

The latter effect can be understood within a simple model for metals the jellium model, which is based on the following ideas As is... 

Figure 3.4 Distribution of the electronic density in the jellium model the metal occupies the region x < 0. The unmarked curve is for an uncharged surface, the other two curves are for the indicated surface-charge densities. The distance along the x axis is measured in atomic units (a.u.), where 1 a.u. of length = 0.529 A.

In order to estimate the magnitude of the surface dipole potential and its variation with the charge density, we require a detailed model of the metal. Here we will explore the jellium model further, which was briefly mentioned in Chapter 3. 

Jellium is a good model for sp metals. This group of metals comprises, amongst others, the elements Hg, Cd, Zn, Tl, In, Ga and Pb, all of which are important as electrode materials in aqueous solutions. They possess wide conduction bands with delocalized electrons, which form a quasi-free-electron gas. The jellium model cannot be applied to transition metals, which have narrow d bands with a localized character. The sd metals Cu, Ag and Au are borderline cases. Cu and Ag have been successfully treated by a modified version of jellium [3], because their d orbitals are sufficiently low in energy. This is not possible for gold, whose characteristic color is caused by a d band near the Fermi level. 

The fact that evaporated potassium arrives at the surface as a neutral atom, whereas in real life it is applied as KOH, is not a real drawback, because atomically dispersed potassium is almost a K+ ion. The reason is that alkali metals have a low ionization potential (see Table A.3). Consequently, they tend to charge positively on many metal surfaces, as explained in the Appendix. A density-of-state calculation of a potassium atom adsorbed on the model metal jellium (see Appendix) reveals that the 4s orbital of adsorbed K, occupied with one electron in the free atom, falls largely above the Fermi level of the metal, such that it is about 80% empty. Thus adsorbed potassium is present as K, with 8close to one [35]. Calculations with a more realistic substrate such as nickel show a similar result. The K 4s orbital shifts largely above the Fermi level of the substrate and potassium becomes positive [36], Table 9.2 shows the charge of K on several metals. 

To explain where the surface contribution to the work function comes from, we need a model for the electron distribution in the surface region of a metal. One of the simplest is the jellium model [18]. 

Figure A.9 A potential energy diagram of an atom chemisorbed on the model metal jellium shows the broadening of the adsorbate orbitals in the resonant level model.

The role of the metal electrons in determining properties of the metal/electrolyte solution interface was reviewed by Komyshev, who presented and discussed an alternative approach to the jellium model. As stated by Komyshev, the jellium model, which is correct for the interpretation of capacity measurements, appears to be too rough to explain the data obtained with modem techniques, such as many spectroscopic and miaoscopic techniques (see later discussion). 

Fig. 2-10. Profile of electron density and electronic potential energy across a metal/vacuum interface calculated by using the jellium model of metals MS = jellium surface of metals Xf = Fermi wave length p. - average positive charge density P- s negative charge density V = electron exchange and correlation energy V, - kinetic energy of electrons. [From Lange-Kohn, 1970.]...

Kg. 2-11. Work function, 4>, observed and calculated by using the jellium model as a function of Wigner-Seitz radius, rs, for various metals rs = 3 / (4 n n, = electron... 

Fig. 6-21. Charge distribution profile across a metal/aqueous solution interface (M/S) (a) the hard sphere model of aqueous solution and the jellium model of metal (the jellium-sphere model), (b) the effective image plane (IMP) and the effective excess charge plane x, (c) reduction in distance /lxd,p to the closest approach of water molecules due to electrostatic pressure, o, = differential excess charge on the solution side og = total excess charge on the solution side Oy = total excess charge on the metal side.

Abstract This chapter reviews the methods that are useful for understanding the structure and bonding in Zintl ions and related bare post-transition element clusters in approximate historical order. After briefly discussing the Zintl-Klemm model the Wade-Mingos rules and related ideas are discussed. The chapter concludes with a discussion of the jellium model and special methods pertaining to bare metal clusters with interstitial atoms. 

Keywords Jellium model Metal clusters Wade-Mingos rules Zintl ions Contents... 

In some cases, macroscopic models are used for simplified discussions of certain phenomena without atomic resolution. A macroscopic tip-sample distance should be defined. To avoid confusion, we use the term barrier thickness instead. Throughout the book, the barrier thickness is always denoted by a upper-case letter, such as W or L. In the Sommerfeld model of the free-electron metals, the barrier thickness is the distance between the surface of the metal pieces. In the jellium model (see Chapter 4), the barrier thickness is defined as the distance between the image-force planes. 

The simplest model of a metal surface is the jellium model, which is a Sommerfeld metal with an abrupt boundary. In provides a useful semiquanti-tative description of the work function and the surface potential (Bardeen, 1936). It validates the independent-electron picture of surface electronic structure Essentially all the quantum mechanical many-body effects can be represented by the classical image force, which has been discussed briefly in Section... 

The jellium model for the surface electronic structure of free-electron metals was introduced by Bardeen (1936) for a treatment of the surface potential. In the jellium model, the lattice of positively charged cores is replaced by a uniform positive charge background, which drops abruptly to zero at the... 

In the bulk, the charge density of electrons n equals in magnitude the charge density of the uniform positive charge background +, thus to preserve charge neutrality. The only parameter in the jellium model, r,, is the same as in the Sommerfeld theory of free-electron metals. 

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.

---