Equipotential surfaces

A still different approach to multilayer adsorption considers that there is a potential field at the surface of a solid into which adsorbate molecules fall. The adsorbed layer thus resembles the atmosphere of a planet—it is most compressed at the surface of the solid and decreases in density outward. The general idea is quite old, but was first formalized by Polanyi in about 1914—see Brunauer [34]. As illustrated in Fig. XVII-12, one can draw surfaces of equipo-tential that appear as lines in a cross-sectional view of the surface region. The space between each set of equipotential surfaces corresponds to a definite volume, and there will thus be a relationship between potential U and volume 0. 

Figure Bl.6.1 Equipotential surfaces have the shape of lenses in tlie field between two cylinders biased at different voltages. The focusing properties of the electron optical lens are specified by focal points located at focal lengthsandy2, measured relative to the principal planes, The two principal rays emanating...

The CHA is used for xps today in most commercial spectrometers. The most common version of the CHA is the 180° device shown in Figure 22. It consists of two concentric hemispherical surfaces of radii R2 and R. These surfaces have a potential difference of AH appHed between them so that the outer surface is negative and the inner surface is positive. The median equipotential surface Rq falls between these surfaces ideally, Rq = The... 

Foi shts placed at from the center of curvature, the electrons passed by this analyzer foUow the equipotential surface described by R. With an acceptance angle 8a shown in Figure 22 and a sht width w, the energy resolution of the CFIA is given by... 

The CHA is shown in schematic cross-section in Fig. 2.5 [2.5]. Two hemispheres of radii ri (inner) and T2 (outer) are positioned concentrically. Potentials -Vi and -V2 are applied to the inner and outer hemispheres, respectively, with V2 greater than Vi. The source S and the focus E are in the same plane as the center of curvature, and Tq is the radius of the equipotential surface between the hemispheres. If electrons of energy E = eVo are injected at S along the equipotential surface, they will be focused at Eif ... 

If electrons are injected not exactly along the equipotential surface, but with an angular spread Aa about the correct direction, then the energy resolution is given by ... 

Inasmuch as the boundary surface is an equipotential surface for both potentials Ui and U2, their difference is also constant on this surface and correspondingly we can write... 

Inasmuch as this system is not used so often as Cartesian, cylindrical, or spherical coordinates, let us describe it in some detail. First of all, we find a condition when a family of non-intersecting surfaces can be a family of equipotential surfaces. Suppose that the equation of the surfaces is... 

This equation describes any level surface of the potential U of the gravitational field y, where x, y are coordinates of a point on the surface, while C is the value of the potential. At the same time, the potential of the attraction field varies on this surface. Our next step is to represent the left hand side of Equation (2.195) in the spherical system of coordinates and then, using Equation (2.192), obtain the equation of the equipotential surface, which coincides with the outer surface of the earth spheroid. As was shown earlier, the potential related to a rotation is... 

Earlier we solved the boundary value problem for the spheroid of rotation and found the potential of the gravitational field outside the masses provided that the outer surface is an equipotential surface. Bearing in mind that, we study the distribution of the normal part of the field on the earth s surface, where the position of points is often characterized by spherical coordinates, it is natural also to represent the potential of this field in terms of Legendre s functions. This task can be accomplished in two ways. The first one is based on a solution of the boundary value problem and its expansion into a series of Legendre s functions. We will use the second approach and proceed from the known formula, (Chapter 1) which in fact originated from Legendre s functions... 

Here R and tp are coordinates of the equipotential surface and they vary from point to point, but the potential remains the same. Next, we will attempt to find the relationship between these coordinates in the explicit form. Taking into account the fact that parameters /20 and m are very small, we will use the following... 

To illustrate Equation (2.253), consider two points a and b located at different equipotential surfaces and assume that they are located so close to each other that the field between them changes linearly along the vector line. At the same time the separation of these surfaces may vary. First, choose the path ab b, where points b and b are at the same plumb line, perpendicular to both surfaces. Fig. 2.8c. Then,... 

Suppose that within the interval a—b the equipotential surfaces are parallel to each other, that is, the field does not change along these surfaces. In such a case the... 

Here is a point inside the volume V. Hence, it is impossible to distinguish between the field caused by a volume distribution of masses and the field generated by masses on the equipotential surface S, provided that the condition (4.6) is met and the observation point is located outside S. As a rule, a three-dimensional body and... 

The potential theory postulates a unique relationship between the adsorption potential ep and the volume of adsorbed phase contained between that equipotential surface and the solid. It is convenient to express the adsorbed volume as the corresponding volume in the gas phase. 

The values of Cy, of course, depend on which equipotential surface is used to represent the ion. Since these surfaces can be arbitrarily chosen, it might be supposed that all the values of Cy can also be arbitrarily chosen. However, the number of ions is always less than the number of bonds. If there are ions in the array, it is only possible to assign arbitrary values of Cy to - 1 bonds, those in the spanning tree described in Section 2.5 below. For the remaining bonds, those that close the loops in the network, a knowledge of the bond topology alone is insufficient to determine Cy. To find these values of Cy, the geometry of the array, i.e. the positions of the ions, must also be known. 

An example for the field cones and equipotential surface is shown in Fig. 3.9 for d = 1.2 mm and rt= 420 A. The vertical line represents a position of 5rt away from the tip. The field lines are drawn so that their density is proportional to the field strength. Field distributions and equipotential surfaces of other tip shapes have also been investigated, particularly as regards the field emission current density distribution,24,31 but will not be discussed here. 

Let AB see Figure 50), be an infinitely extended metal surface, and let there be a charge e at C. The lines of force from C must be at right-angles to the metal AB since this is an equipotential surface by virtue of the conductivity of the metal. These lines of force will not... 

Thus the contribution of the structured ionic cloud to the total potential at the surface of the central ion will not be as it is in the DH theory, and because the electrostatic model requires an equipotential surface to be maintained there, a new model is needed. We therefore approximate an ion to a dielectric sphere of radius a, characterized by the dielectric constant of the solvent D, and having a charge Q, residing on an infinitesimally thin conducting surface. This type of model has been exploited by previous workers (17,18) and may be reconciled with a quantum-mechanical description (18). 

The mutual polarization of ions is equivalent to the redistribution of surface charge on the central ion in response to the nonhomogeneous field of the ionic cloud. We need not speculate here on the physical nature of the equipotential surface, except to emphasize that it refers to a solvated species, and one of our... 

The cell for rotating electrodes, Fig. 7, is usually cylindrical and surrounded by a water jacket for thermostatting purposes, but as long as the cell walls are more than 1 cm or so from the rotating assembly, there are usually no cell edge effects. The auxiliary electrode is very often contained in a separate compartment behind a glass frit in order to avoid contamination problems. A Luggin capillary, where required, can be positioned in various ways unless it is more than 0.5 cm from the electrode, it must be placed under the centre of the disc in order to avoid a non-equipotential surface this can cause some problems with disturbance of the fluid flow. 

Figure 11-11. Displacement reactions of the type AX+BY = AY+BX. a) lost mechanism, schematic, b) Equipotential surfaces and the evolution of phase boundary instability, lost mechanism, c) Wagner mechanism of displacement reactions, schematic.

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