Jellium plane

According to Vitanov et a/.,61,151 C,- varies in the order Ag(100) < Ag(lll), i.e., in the reverse order with respect to that of Valette and Hamelin.24 63 67 150 383-390 The order of electrolytically grown planes clashes with the results of quantum-chemical calculations,436 439 as well as with the results of the jellium/hard sphere model for the metal/electro-lyte interface.428 429 435 A comparison of C, values for quasi-perfect Ag planes with the data of real Ag planes shows that for quasi-perfect Ag planes, the values of Cf 0 are remarkably higher than those for real Ag planes. A definite difference between real and quasi-perfect Ag electrodes may be the higher number of defects expected for a real Ag crystal. 15 32 i25 401407 10-416-422 since the defects seem to be the sites of stronger adsorption, one would expect that quasi-perfect surfaces would have a smaller surface activity toward H20 molecules and so lower Cf"0 values. The influence of the surface defects on H20 adsorption at Ag from a gas phase has been demonstrated by Klaua and Madey.445... 

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... 

Here, n(z z) is the charge density induced at z by a charged plane, of unit surface charge density, located at z, and pb is the bulk jellium density. The Hellmann-Feynman theorem was used. The full moment of n(z z) obeys... 

Thus, only the tail of the electron density (outside the jellium) contributes. The last term above may further be related11 to the position of the image plane zim so, at the point of zero charge,... 

As shown in Fig. 5-20, the effective image plane is located close to but away from the jellium metal edge. CombiiungEqns. 5-27 and 5-28 yields the surface potential Xm as a fiinction of Om in Eqn. 5-29 ... 

Fig. 6-20. Charge distribution profile across an interface between metal and vacuum (MAO (a) ionic pseudo-potential in metal, (b) diffuse electron tailing away from the jellium metal edge, (c) excess charge profile. n(x) s electron density at distance x = electron density in metal x, = effective image plane On = differential excess charge On = 0 corresponds to the zero charge interface.

Table 6-3. The effective image plane position of a metal in vacuum estimated as a function of electron density in metal x, distance at the effective image plane fiom the jellium metal edge rws = Wigner-Seitz radius (a sphere containing one electron) which is related to electron density n, in metal (1 / n, = 4 n / 3 ) au = atomic unit (0.529 A). [From Schmickler, 1993.]...

The interfacial solution layer contains h3 ated ions and dipoles of water molecules. According to the hard sphere model or the mean sphere approximation of aqueous solution, the plane of the center of mass of the excess ionic charge, o,(x), is given at the distance x. from the jellium metal edge in Eqn. 5-31 ... 

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.

Next, we discuss the plane of the closest approach (x = of water molecules to the jellium metal edge (x = 0). At the zero charge interface, this plane of closest approach of water molecules is separated by a distance equal to the radius of water molecules from the metal siuface. As the interfadal excess charge increases, the electrostatic pressure (electrostriction pressiue) reduces the distance of Xdip in prop>ortion to the square of the interfadal charge, a (= om = - os) the electrostatic force in the compact layer is proportional to om x as. The change in Xitp due to the interfadal charge is then given by Eqn. 5-32 ... 

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. 

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.

Figure 2.8. Electron density contours for atomic chemisorption on jellium with electron density that corresponds to A1 metal. Upper row Contours of constant electron density in the plane normal to the surface. Center row Difference in charge density between isolated adatom and metal surface, full line gain and dashed line loss of charge density. Bottom row Bare metal electron density profile. Reproduced from [30].

In the jellium and structureless pseudopotentials models of a metal surface, there is translational symmetry in the plane parallel to the surface and the effective potential in which the electrons move is local. Thus, the structure of the setf-consistent KS orbitals for these models is of the form... 

However, a complete physical Me UPD model does not yet exist. Recently, calculations based on a jellium model with lattices of pseudopotentials for the 2D Meads phase and S were started by Schmickler and Leiva [3.234-3.239]. In addition, local density full potential linearized augmented plane wave calculations were carried out by Neckel [3.240, 3.241). Both approaches are important for a better understanding of Me UPD phenomena on single crystal surfaces taking into account structural aspects. 

From dimensional arguments alone the average HF exchange potential must be proportional to the inverse of r, i.e. proportional to the one-third power of the density. By averaging the HF exchange potential over the plane-wave one-electron eigenfunctions of the jellium model up to the Fermi wavevector... 

Fig. 4 Normalized electronic density profile at the surface of jellium with a bulk density of 15.3 atomic units. The upper dashed curve is for a surface charge density (Jm = —0.1 C m , the lower dashed curve for = 0.1 C m. The full curve is for an uncharged surface. The arrow gives the position of the effective image plane for an uncharged surface.

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