Effective image plane

The plane of the center of mass of induced image charge, oJ,x), is called the effective image plane [Lang-Kohn, 1973], and its position, Xm, is given by Eqn. 5-28 ... 

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.]...

Table 5-4. Ibe effective image plane position, X , estimated on three crystal surface planes of metallic aluminum in vacuum. Electron density increases in order from the (111), (100) to (110). [From Schmickler, 1993.]...

This plane of the center of mass of the excess ionic charge o,(x) is the effective excess charge plane on the solution side, which may be compared with the effective image plane on the metal side. In simple cases, the effective excess charge plane coincides with the outer Helmholtz plane (the plane of closest approach of hydrated ions) as shown in Fig. 5-21. 

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.

Fig. 6-22. Effect of interfadal excess charge, om, on the effective image plane, Xi , the shift of the plane of closest approach of water molecules, the inverse interfacial capadty on the metal side, 1 /Cm and the interfadal capacity on the solution side, Cs. M/vac = metaWacuum interface M/sol = metal/solution interface.

The hard-sphere electrolyte model, presented in section n.4, can be used to estimate the dipole moment in the absence of charge transfer. The model can be improved by noting that, as mentioned before, the effective image plane of a metal usually sits at a distance xim in front of the geometrical surface. From Eq. (73), in which B and A are obtained by comparing Eq. (72) with Eq. (50), we get ... 

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.

A different EB design to effect an image plane of 140 mm in length with a linear mass scale for detection on a photographic plate has been published by Bain-bridge and Jordan. [74] Their paper is especially recommended, as it also nicely illustrates the use of photoplates in these days. 

This allocation of experiments has the effect of making the normalized uncertainty and normalized information contours more axially symmetric (the design isn t quite rotatable there are still only four mirror-image planes of reflection symmetry). However, because no experiments are now being carried out at the center point, the amount of uncertainty is greater there (and the amount of information is smaller there). The overall effect is to provide a normalized information surface that looks like a slightly square-shaped volcano. 

Change lighting effects and vary surface texture Plot image planes against one another as scatter plots Associate up to three single-channel intensity images to produce RGB image... 

Where T = transmission by the lens, F = the ratio of focal length to effective diameter of the lens, and E0 = illumination of the object plane (lx) and that of the image plane E, the formula would be ... 

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