Calculated space-charge distributions

Figure 7. Calculated space-charge distributions at various times with fixed ions. Normalized times correspond to the time axis of Figure 1. (Reproduced with permission from Ref. 10. Copyright 1989 M.I.T.)...

The general method for the determination of the flat band potential is based on the Mott-Schottky linear plot based on ca-pacitance/voltage relation. Starting from Eq. (9) the space charge distribution was calculated, and its potential dependence lead to the derivation of a model equivalent to a capacitance, given by ... 

Model. The comparison of theory and experiment in Figure 9 indicates that the simplified model can be used to calculate transients which agree with experiment in gross trends. The model permits quantitative analysis of bulk and contact space-charge effects in PI transient current measurements. In particular, this model is sufficient to calculate measurement-history effects due to mobile ions and bulk electronic space-charge (9). The relaxation of space-charge upon removal of the bias is intrinsically slower than its accumulation. Thus, the sample history is stored in the space-charge distributions. These results will be demonstrated in a future publication. 

Since our calculations agree well with the experimental data, we may conclude that the assumptions made previously are reasonable and that a space charge distribution is operative in the solid side of the silver chloride-aqueous solution interface. This space charge distribution results from a negative potential Vc(0) in the solid and corresponds to a deficiency in silver ion vacancies, the maximum occurring near the equivalence point. This conclusion will be further supported in the following section on AC conductance and capacitance measurements. 

Multipole moments are useful quantities in that they collectively describe an overall charge distribution. In Chapter 0, I explained how to calculate the electrostatic field (and electrostatic potential) due to a charge distribution, at an arbitrary point in space. 

As an example Fig. 6 shows the distribution of the ions for a potential difference of A(j) = 0(00) — 0(—00) = kT/cq between the two bulk phases. In these calculations the dielectric constant was taken as e = 80 for both phases, and the bulk concentrations of all ions were assumed to be equal. This simplifies the calculations, and the Debye length Lj), which is the same for both solutions, can be used to scale the v axis. The most important feature of these distributions is the overlap of the space-charge regions at the interface, which is clearly visible in the figure. 

ESI mass spectra of mixtures are difficult to interpret, because each component produces ions with many different charge states. The most direct and reliable method to solve this problem is to use high-resolution MS and calculate the charge states by measuring the spacing of the isotope peaks. ESI mass spectrometry of (polymeric) mixtures with broad molecular weight distribution benefits from a prior separation that reduces the polydispersity of the analyte. 

The various M-space indices displayed here — and in the equations which follow — are calculated via the wave function charge distribution mapping (2.6) from the appropriate Ay-spacc indices (see BH-I for details). The regime intermediate between the SC and the BO limits requires the solvent electronic polarization matrix [cf. (2.12)]... 

The entire set of molecules contained 782 bonds out of which 111 a-bonds were selected. The parameters were calculated by our methods to build a reactivity space with electronegativity difference, resonance effect parameter, bond polarizability, bond polarity, a-charge distribution, and bond dissociation energy as six coordinates. 

Clearly not Atomic charges are not molecular properties, and it is not possible to provide a unique definition (or even a definition which will satisfy all). It is possible to calculate (and measure using X-ray diffraction) molecular charge distributions, that is, the number of electrons in a particular volume of space, but it is not possible to uniquely partition them among the atomic centers. 

Another approach to providing atomic charges is to fit the value of some property which has been calculated based on the exact wavefunction with that obtained from representation of the electronic charge distribution in terms of a collection of atom-centered charges. In practice, the property that has received the most attention is the electrostatic potential, 8p. This represents the energy of interaction of a unit positive charge at some point in space, p, with the nuclei and the electrons of a molecule (see Chapter 4). 

The potential distribution ( ) in the space-charge region is described, with due account for Eq. (13), by the self-consistent Poisson-Boltzmann equation. Its first integral can be calculated analytically, so that the electric field at the semiconductor surface Ssc = — /dx is expressed as (Garrett and Brattain, 1955 see also Frankl, 1967)... 

A similar expression is to be expected for the distribution of electrons in exhaustion boundary layers. The evaluation of these equations is not easy. Therefore, it is desirable to simplify our assumptions concerning the distribution of the electrons and holes in the boundary layer. This simplification is illustrated by Fig. 3. With a suitable choice of the thickness I of the boundary layer, the simplification will satisfactorily approximate the real case. The distribution of space charge in an inundation boundary layer can be similarly calculated, and will be shown below in parentheses. 

---