The local field

As soon as we consider the molecular nature of a material, we realise that the internal electric field will vary from point to point as a consequence of the interaction of fields from the dipoles which are induced on each molecule by the applied field, although the space-average electric field over a volume large in comparison with molecular size (this is equivalent to the classical electric field based on a continuum model) may still be uniform. The field acting on an individual polarisable entity like an atom or molecule is called the local field Eh, and it is an important concept in linking observable bulk behaviour of a material with the properties of its constituent atoms or molecules. 

In calculating a local field it is convenient to divide it into two principal components  

Since the total charge per unit area on the electrodes as shown previously is Q + F), we have directly for the magnitude of the field from this source 

The field at A from the polarisation of the material outside the sphere may be further resolved into two parts in terms of the apparent surface charges which it produces at its boundaries. The first part is due to the apparent charges — P and +P per unit area at the surface adjacent to the electrodes and has the value —P/e0. The second part is due to the apparent charges P cos per unit area on the surface of the sphere, where 9 is the angle shown in Fig. 2.5, and it may be evaluated by integrating over surface elements confined between angles 9 and (9 + d0). Such an element has an area 2nr2sm0d9. By symmetry the fields perpendicular to P at the centre of the sphere cancel out and the net field in the direction parallel to P takes the value 

the field at A due to the molecules inside the sphere, will depend on the detailed arrangement of the molecules. For certain special cases, including simple cubic lattices and completely random arrays, 

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The internal field is that microwave field which is generally the object for solution when MaxweU s equations are appUed to an object of arbitrary geometry and placed in a certain electromagnetic environment. The is to be distinguished from the local field seen by a single molecule which is not necessarily the same (22). The dielectric permittivity as a function of frequency can be described by theoretical models (23) and measured by weU-developed techniques for uniform (homogeneous) materials (24). 

Electric field measurement at the boundary of a metal container filled with charged material. Examples include pipelines and storage vessels. The electric field can be used to calculate charge density (3-5.1). Eield meters can also be lowered into containers such as silos to determine the local fields and polarities. Quantitative interpretation of the reading requires correction for field intensification and is sometimes accomplished using computer simulations. 

In condensed phases, the separation between molecules is of the order of the molecular dimensions and the result is that each molecule is polarized not by just the ordinary applied electric field E but by a local field F consisting of E plus the fields of all the other dipoles. Once the local field is known, we can use equation 15.8 to find the polarization, simply by substituting aF for aE. The calculation of F is difficult because the dipoles that contribute to F are themselves determined by F and a self-consistent treatment is necessary. This is achieved by relating F to P, ultimately giving an equation for Xe-... 

We consider first the polarizability of a molecule consisting of two or more polarizable parts which may be atoms, bonds, or other units. When the molecule is placed in an electric field the effective field which induces dipole moments in various parts is not just the external field but rather the local field which is influenced by the induced dipoles of the other parts. The classical theory of this interaction of polarizable units was presented by Silberstein36 and others and is summarized by Stuart in his monograph.40 The writer has examined the problem in quantum theory and finds that the same results are obtained to the order of approximation being considered. 

The double layer is described by its effective thickness, d, and by its field strength E (Fig. 6.15). The adsorbed moleculeJias a dipole moment P. It is well documented100 that the local field strength E can affect strongly not only the chemisorptive bond strength but also the preferred orientation of the adsorbate (Fig. 6.16). 

G2, to G3, and to G4, the effective enhancement was 10%, 36%, and 35% larger than the value estimated by the simple addition of monomeric values. The enhancement included the local field effect due to the screening electric field generated by neighboring molecules. Assuming the chromophore-solvent effect on the second-order susceptibility is independent of the number of chro-mophore units in the dendrimers, p enhancement can be attributed to the inter-molecular dipole-dipole interaction of the chromophore units. Hence, such an intermolecular coupling for the p enhancement should be more effective with the dendrimers composed of the NLO chromophore, whose dipole moment and the charge transfer are unidirectional parallel to the molecular axis. 

Figure 2.13 Relation between the action potential recorded intracellularly from a cat spinal motoneuron following antidromic stimulation (int.) and the local field potential recorded with an extracellular electrode (ext.). (Adapted from Terzuolo, AC and Araki, T (1961) Ann. NY Acad. Sci. 94 547-558). Published by NYAS...

Pnorm = P/( " fv) Ynorm = whereis the volume fraction of QDs, ( ) is the local-field... 

The local field acting on a specific dipole in a lattice is... 

The dipolar contributions to the local field diverges if the terms are replaced by their absolute values. 

Surprisingly, in addition to the chain contribution, F , the local field calculated in the p-approach also contains the uniform part, = AnP /d. This last term noticeably increases the effective number of nearest neighbors so that the CH catastrophe disappears. 

As was mentioned in Section 2, there exists a variety of different theoretical approaches to calculate the local field factor g q). Following Farid et al. [7], the behavior of g q) for large q is connected to the size z of the step in the occupation number function n(k) fork = kF, kF being the Fermi-momentum (see Figure 8). This... 

To determine the behavior of g(q) for large q, we performed measurements of iS lq, ) of Li for 1.1 a.u. < q < 2.6 a.u. and performed for each spectrum a fit of the g(g)-modified c° to the experimental data. Figure 10 shows the result of this semi-empirical determination of g(q) together with the shape of the local-field correction factor after Farid et al. [7] calculated for different values ofz solid line (z = 0.1), dashed line (z = 0.5) and dash-dotted line (z = 0.7). One clearly sees that the curve for the surprisingly small value of z = 0.1 fits our experimental findings best. 

We have shown for the case of Li that the step in the occupation number function is surprisingly small z 0.1 and provided semi-empirically obtained values for the local-field correction factor. For the case of Al, we showed the additional cancellation of self-energy and vertex correction. 

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