Continuous distribution of charge

As we have seen in Sec. IX,4 the dipole layer may not be treated as a double layer with a continuous distribution of charge. Discrete charge distributions have to be assumed. Gomer (335) criticizes Boudart on this point. He evaluates potential curves for a discrete distribution of dipole... 

The Poisson equation thus cannot be taken over without modification and therein lies a major problem for the theory. The two quantities, i rj and py, relate to a continuous distribution of charge, but there is no such thing as a continuous charge density for discrete charges, here ions. 

If the Poisson equation is modified in this way by using such time-averaged rjr and p, and then is applied to the electrolyte solution, there has been a smearing out of the discrete charges of the ionic atmosphere into a continuous distribution of charge. Only then is it legitimate to use the Poisson equation of electrostatics to discuss the theory of electrolyte solutions, i.e. 

It is reasonable to assume that a spherically symmetrical continuous distribution of charge could reflect in some way the spherically symmetrical distribution of all the discrete ions around the central j-ioti. This requires looking at what a charge density , i.e. the average charge per unit volume at a given point, for discrete ions means. 

The PBE describes a continuous distribution of charge inside the solvent and on the surface—any effect related to the discreteness of charge and the ion volume is ignored. In general, the PBE has to be solved numerically. Analytical solutions exist only for simple geometries and/or symmetrical electrolytes (e.g. Lyklema 1995, Chap. 3.5). Even for an isolated sphere there is no exact analytical solution. 

Calculation of the Coulomb matrix element J s in (15.79) involves not point charges (as in the FMM method) but continuous distributions of charge defined by the basis functions. Therefore, quantum chemists modified the FMM method to deal with interactions involving continuous charge distributions. One such modification for rapid evaluation of the Coulomb matrix elements for large molecules is the continuous fast multipole method (CFMM) [C. A. White et al., Chem. Phys. Lett., 253,268 (1996)]. Another is the Gaussian very fast multipole method (GvFMM) [M. C. Strain, G. E. Scuseria, and M. J. Frisch, Science, 271, 51 (1996)]. 

If we consider a molecule as having a static but continuous distribution of electronic charge around a rigid nuclear framework, then its electrical or electrostatic potential will have a term similar to Eq. (3.2), with Q. being the positive charges of the nuclei, ZA, and a... 

The density p(r) might also be described as the fractional probability of finding the (entire) electron at point r. However, chemical experiments generally do not probe the system in this manner, so it is preferable to picture p(r) as a continuous distribution of fractional electric charge. This change from a countable to a continuous picture of electron distribution is one of the most paradoxical (but necessary) conceptual steps to take in visualizing chemical phenomena in orbital terms. Bohr s orbits and the associated particulate picture of the electron can serve as a temporary conceptual crutch, but they are ultimately impediments to proper wave-mechanical visualization of chemical phenomena. 

For an assembly of positive point nuclei and a continuous distribution of negative electronic charge, we obtain... 

For several metal - - H2 systems, it has been found that the experimental heat of adsorption is a linear function of the surface coverage S4, 131). This implies that for these systems the distribution of dipoles approximates to a continuous layer of charge and that Equation (11) correctly describes the change in the heat of adsorption when an electron is moved through the double charge layer. Originally Boudart (108) used the relation... 

CA 61, 8124(1964) (Satd soln of AN in liquid NH3 is pumped into drill holes and penetrates permeable earth layers. Shaped charge fuse is used for initiation. This method permits one to explode the total chge, because of the continuous distribution of expl in the earth formation)... 

Let us now turn to the thickness of the conducting channel. The concept of thickness is not that obvious, because the actual distribution of charge-carriers decreases continuously from the insulator-semiconductor interface to the semiconductor bulk, so one can more sensibly speak of an effective thickness. The distribution can be estimated by resolving Poisson s equation (Eq. 3) ... 

Although the quantum problem seems to be solved by the hydrodynamics of a continuous distribution of electricity with charge density proportional to mass density, this approach has never been accepted as a serious alternative, largely because of doubts raised by Madelung himself. The most important of these, concerns the self-interaction between the charge elements of an extended electron. 

The solution to Poisson s equation for the depletion layer is discussed further in Chapter 9. The hatched region in Fig. 4.15 represents the gap states which change their charge state in depletion and so contribute to p(x). When there is a continuous distribution of gap states, p(x) is a spatially varying quantity. For the simpler case of a shallow donor-like level, the space charge equals the donor density N-a and the solution for the dependence of capacitance on applied bias, is (see Section 9.1.1)... 

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