Dipole Distributions

This was averaged over the total distribution of ionic and dipolar spheres in the solution phase. Parameters in the calculations were chosen to simulate the Hg/DMSO and Ga/DMSO interfaces, since the mean-spherical approximation, used for the charge and dipole distributions in the solution, is not suited to describe hydrogen-bonded solvents. Some parameters still had to be chosen arbitrarily. It was found that the calculated capacitance depended crucially on d, the metal-solution distance. However, the capacitance was always greater for Ga than for Hg, partly because of the different electron densities on the two metals and partly because d depends on the crystallographic radius. The importance of d is specific to these models, because the solution is supposed (perhaps incorrectly see above) to begin at some distance away from the jellium edge. 

Figure 7.3 shows the radiated intensity S as a function of the observation angle 0 for a dipole 80 nm from the surface. For simplicity, the azimuthal angle of observation is averaged. (This is equivalent to assuming that the excited dipole distribution is azimuthally symmetric about the surface normal.)... 

This was already known from the work of Becker 822) and could be explained by the author 823) by means of these considerations. The increase of the heat of adsorption of the cesium ion, as found by Becker, giving direct proof for the discreteness of the dipole distribution, means a less strong decrease of the heat of adsorption of the cesium atom. When we raise level D of Fig. 6 (Sec. V,8,a) by an amount... 

The field of the gaussian-dipole distribution is calculated from the Green s functions of the two distributions, as a sum in the direct space. (In this calculation the gaussian placed at 0 must be taken into account it plays a... 

In an analogous way with that in Section I.B, we must calculate the dipole sum, or the radiated field, using the dipole distribution... 

The effect of the three internal fields, at 0, ra and 2 , on one molecule is next treated, and after angular averaging the connection between the microscopic and macroscopic response functions established. The polarization of the dipole distribution by re-orientation of the molecules is induced by the static field. Introducing the Boltzmann factor and normalizing the distribution gives, for the number of molecules per unit volume with dipole moment inclined at an angle, 0, to the fields. 

The changes in SCSSD regard the replacement of the permanent dipole moment of each soft solvent sphere by a couple of dipoles, permanent and induced. The permanent dipoles are kept fixed at the orientations minimizing the energy for the solute ground state, while induced dipoles are allowed to be different for each VB structure. A nested iterative procedure is adopted to fix both dipole distributions, and related contribution to the various elements. 

All the models described above for the electrostatic response provide either the charge or dipole distribution due to a potential or field. To complete the description of an electrostatic model these equations must be combined with the appropriate equations that provide the dependence of the potential and field on the charge and dipole distribution. 

Fig. 1 Dipole distribution in (a) water (polar), (b) ethanol (polar) and (c) ethane (non-polar).

A rather novel scheme for modeling molecular polarizabilities as distributed dipole polarizabilities has recently been reported [141]. In this approach, the overall quadrupole induced in a molecule by an external field, as calculated with ab initio methods, is decomposed into induced dipoles distributed to atomic sites. In turn, this yields the dipole polarizability values at those sites. In effect, this relates the overall dipole quadrupole polarizability to a distribution of dipole polarizabihties. 

Fig. 2.6 Dipole distribution. The potential is discontinuous whereas the electric field is continuous on F.

In many problems involving remote sensing, the molecules of interest comprise the outer surface of the particle or have been adsorbed onto it. The presence of the particle affects the inelastic scattering to about the same degree as for the uniformly filled particle. Figure 4.14 shows the angular distribution of the components for the same number of dipoles distributed uniformly over the surface of an otherwise uniform dielectric sphere of refractive index 1.5 for different values of a. The minimum at 90° is rapidly filled in. The scattering in the forward and backward directions appear particularly sensitive to particle size. 

Figure 22.9(a) shows unfilled NR, (b) is for Na -MMT/NR and (c) is for NR/O-MMT. The dramatic variation in SIC with increasing strain is seen in the case of NR/O-MMT nanocomposites. Addition of nanoclay platelets in NR provides a regular polymer network microstructure. The O-MMT and NR are hydrophobic in nature. Hence NR chains are interfacially adsorbed at the outer surface of O-MMT. But in Na-MMT, no such interaction is present due to the changes in dipole distribution. So in the SIC analysis, the NR/O-MMT nanocomposites show sharper crystalline peaks than the other clay nanocomposites. 

The second term on the right of (7.28) i.e. the dipole disorienting factor describes the relaxation of dipoles due to a finite temperamre. The multiplier may be considered as a numerical coefficient k 2/3, as if the distribution function is spherical even in the electric field. In fact, a more precise value was found by Debye by averaging the Fe value over 9 with the field-induced dipole distribution function shown qualitatively in Fig. 7.11. Since the thermal motion of dipolar molecules destroys the field induced polar order, we introduce a thermal relaxation time Td, as the first (linear) approximation of the relaxation rate. In order to find this time, we should exclude from the kinetic equation. 

Gibbs free energies of hydration, before and after ionization (AG y/l) and AGhyj(2)) were obtained by employing the Langevin dipole relaxation method (45-47) incorporated in the Polaris 3.2 program (46). Before and after ionization, solvent relaxation is modelled by evaluating the relaxation of discrete dipoles distributed on a lattice... 

Dipole distribution Ensemble of dipoles of the same nature but having different energies-per-entity (i.e., distributed over energy levels ). 

Combinations of several dipoles in multipoles (see Chapter 7) or in dipole distributions can be analyzed in terms of snccessive valnes of the standard efforts featuring each dipole. 

The addition of converging paths on the node of the current density -j expresses the mounting in parallel of two dipoles distributed in space, a capacitive and an inductive dipole. 

Dipole distribution (several identical dipoles distributed over energy-per-entity levels). More details are given in Section 14.4 in the form of tables. 

The complexity scale of Formal Objects is divided into levels. In this book, the range goes from singletons to dipole distributions. 

An ensemble of identical dipoles but with various values of energy-per-entity (i.e., a distribution of values over a range of discrete values) constitutes a Formal Object called dipole distribution. The energy-per-entity node of this Formal Object is drawn with a flattened pentagon (i.e., a regular pentagon deformed by moving its horizontal side toward the (tenter). 

As a result of the intermolecular origin of the field gradients it is much more difficult to calculate the spectral densities for ions than for the cases considered in Chapter 2. Let us first consider the situation which appears to best rationalize the experimental results, i,e, we assume that there is a random orientation of the water molecules around the relaxing ion. This corresponds to a weakly hydrated ion and would be approximately valid for chloride, bromide and iodide ions. The total auto-correlation function is for the case of random orientational dipole distribution around the relaxing ion obtained by integration to be... 

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