Jellium model of metal

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.]...

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.

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.

Similar to the failures of the free-electron model of metals (Ashcroft and Mermin, 1985, Chapter 3), the fundamental deficiency of the jellium model consists in its total neglect of the atomic structure of the solids. Furthermore, because the jellium model does not have band structure, it does not support the concept of surface states. Regarding STM, the jellium model predicts the correct surface potential (the image force), and is useful for interpreting the distance dependence of tunneling current. However, it is inapplicable for describing STM images with atomic resolution. 

The jellium model of the free-electron gas can account for the increased abundance of alkali metal clusters of a certain size which are observed in mass spectroscopy experiments. This occurrence of so-called magic numbers is related directly to the electronic shell structure of the atomic clusters. Rather than solving the Schrodinger equation self-consistently for jellium clusters, we first consider the two simpler problems of a free-electron gas that is confined either within a sphere of radius, R, or within a cubic box of edge length, L (cf. problem 28 of Sutton (1993)). This corresponds to imposing hard-wall boundary conditions on the electrons, namely... 

Jellium model of the metal, 890 and crystal structure, 892 and pseudo potentials, 892 and surface of potential, 893 Jeng, organic adsorption, 975, 979 Jovancicevic, 1125, 1263 Junction c-i, 1081... 

Fig. 6.71. The jellium model of the metal electrode. The positive background charge abruptly disappears at the jellium edge while spillover electrons can be found beyond the edge (shaded area). The continuous line represents the profile of electrons in the interfacial region. The positions of the ion cores are indicated by the arrows.

Through the jellium model of the metal we have explained the effect of the metal electrons on the interfacial properties. We also know that the spillover of electrons creates a separation of charges at the metal edge, and consequently, a surface potential. However, what is the magnitude of this surface potential How important is its contribution to the total potential drop in the interfacial region ... 

The current control at the one-by-one electron accuracy level is feasible in mesoscopic devices due to quantum interference. Though the electric charge is quantized in units of e, the current is not quantized, but behaves as a continuous fluid according to the jellium electron model of metals. The prediction of the current quantization dates back to 1983 when D. Thouless [Thouless 1983] found a direct current induced by slowly-traveling periodic potential in a ID gas model of non-interacting electrons. The adiabatic current is the charge... 

One can develop a particularly simple scheme by using the assumption of spherical symmetry together with the jellium model of solid state or nuclear physics to compute the effective potential for clusters of different sizes. In this model, the electrons are treated as free particles by analogy with the conduction band of the solid and the ionic structure within the cluster is completely neglected. This obviously results in a great simplification of the problem, especially if the system is spherical, and might be thought too drastic an approximation. In fact, the jellium model only applies to a specific class of clusters (which we call metallic), but was of enormous importance to the history of the field as it revolutionised cluster physics. 

Within the jellium model for metal clusters [95,53] as described in the introduction, the positive background potential is in a first approximation normally chosen as a spherical shape of the following form... 

The one-dimensional jellium model of a metal [61-64] has been used quite fre-... 

The jellium model of a metal surface. The ion density Nio terminates abruptly at the surface but the electron density extends beyond it. The net charge density, proportional to N]on - N, gives a dipole layer and a potential, V, which holds the electrons in the metal. The minimum energy required to remove an electron from the metal is the work function, calculations indicate that the work function arises from the exchange energy rather than from this kind of electrostatic dipole. 

According to the jellium model the metal can be considered as metal ions embedded in the electron plasma. The thermodynamic condition for electronic equilibrium between both phases is, from a chemist s point of view, equal values of the electrochemical potentials jUg of the electrons. 

In this work we start with the primitive jellium model, as appropriate for alkaline metals. In the jellium model for metal clusters a fundamental input is the size-dependent ionic density. Fortunately, when one of us started this calculation in 1984 [3], some experimental data about the size dependence of the nearest-neighbor distance were available from EXAFS (extended X-ray absorption fine structure) measurements [19]. Except for fine details the size dependence is very weak. This means that in a first approximation the bulk density of the metal can be used as input for a cluster calculation. A second question is the size dependence of the shape. Since electron micrographs very often show a spherical shape, at least for the larger clusters, a spherical shape will be assumed for all cluster sizes. This means that for monovalent systems the radius R of the jellium cluster is determined by its bulk density... 

In the treatment of Badiali et al. (1981) the jellium model of the metal electron system is used with the jellium edge being assumed to be a plane passing through the centers of surface atoms of the metal. Solvent molecules then lie in contact with the surface at a distance equal to the radii, T, of surface metal atoms and hence are separated from the jellium edge by a distance T. This is not an altogether realistic model and, in fact, does not take into account the "overspill effect associated with the wave function of the metal s electrons at the surface. Another problem is that the solvent is represented by an electron-repulsive dielectric continuum, little related to the properties of water dipoles which are involved at the Hg surface in aqueous systems that have mostly been experimentally studied in double-layer capacitance works. 

The jellium model of a metal surface takes into account, in a simplified though significant way, the problem of electron-electron interaction. The atomic potential is smeared out into a uniform positive background extending over the region z<0. The electron density is taken into account by a dimensionless parameter rg, defined by the equation... 

The so-called jellium model of a metal assumes that the positive charge of atomic cores is smeared out uniformly. One considers the electron density in the field of such a charge distribution. 

Figure 6.4b shows the diagonal part of the screened interaction for Cu(lll) calculated using the one-electron model potential described above. The result of neglecting the surface state in the calculation of the response function is also shown, together with what would be obtained describing the system with a jellium model of density equivalent to that of the 4s valence of Cu (tg = 2.67). The surface state makes a strong contribution to Im [— W] / at the surface. This contribution increases with a> and decreases with qrn [37]. The enhancement of Im [— W] /to at the surface depends on the extent of the wave function in the vacuum and is increased for lower work function [37, 38] or metal overlayers [39]. 

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

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,... 

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