Corrections for Local Fields

For the case of electronic polarization, in deriving Equation 23.6, we wrote P o)) = (E/so)X) i i( ) when we should have used Ei in place of the applied field. Therefore Equation 23.6 should be written as 

Values for a are measured for various ionic species and are typically 0(10 ) at optical frequencies for the various ionic species (see Qiapter 24), hence 

So we see that the effect of electronic polarization in the visible region is to increase the dielectric constant by a quantity that is on the order of 1. For the more general case, the solution for a(w) given by Equation 23.5 can be separated into real and imaginary parts by multiplying top and bottom by the complex conjugate. 

There is a small difference in the for o) wq, 2.523 for the corrected versus 2.01 for the uncorrected. This difference will be greater for larger values of Na 0)/so. Also note the 

The electronic dielectric function in the vicinity of o Q. The dashed line is the uncorrected dielectric function and the solid line is the dielectric function corrected for the local field. Both functions tail off to 1 (the dotted line) for mq- 

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Nonequilibrium solvation model for the electric dipole polarizability. b Result corrected for local field effects. c Ref.[27], mean value for T between 283.1 5 and 293.1 5 K. 

Here, N is the number of hyperpolarizable groups per unit volume (number density), F is a factor correcting for local field effects, and 0 is the angle between the permanent dipole /to of the molecule (z direction) and the direction of the poling field (Z direction). The brackets indicate an averaging over all molecular orientations weighted by an orientational distribution function. 

When a non-centrosymmetric solvent is used, there is still hyper-Rayleigh scattering at zero solute concentration. The intercept is then determined by the number density of the pure solvent and the hyperpolarizability of the solvent. This provides a means of internal calibration, without the need for local field correction factors at optical frequencies. No dc field correction factors are necessary, since in HRS, unlike in EFISHG, no dc field is applied. By comparing intercept and slope, a hyperpolarizability value can be deduced for the solute from the one for the solvent. This is referred to as the internal reference method. Alternatively, or when the solvent is centrosymmetric, slopes can be compared directly. One slope is then for a reference molecule with an accurately known hyperpolarizability the other slope is for the unknown, with the hyperpolarizability to be determined. This is referred to as the external reference method. If the same solvent is used, then no field correction factor is necessary. When another solvent needs to be used, the different refractive index calls for a local field correction factor at optical frequencies. The usual Lorentz correction factors can be used. 

Uncertainties in the use of these sensors include the temperature nonuniformity across the sensor s thickness at high heat flux, the edge correction for localized gauges, and the disturbance of the temperature field caused by the presence of the sensor [118,122-124]. 

Solvatochromic effects on the transition energies of molecules in solution often can be related phenomenologically to the solvent s dielectric constant and refractive index. The analysis is similar to that used for local-field correction factors (Sect. 3.1.5). Polar solvent molecules around the chromophore will be ordered in response to the chromophore s ground-state dipole moment (/taa), and the oriented... 

The enhancement of the diamagnetic susceptibility caused by a ring current is not an isotropic effect but occurs only when the plane of the ring is oriented perpendicular to the external magnetic field. Consequently, the presence or absence of a pronounced anisotropy A% can, in principle, be taken as evidence for or against a ring current and can thus be employed as an aromaticity probe. The need to correct for local anisotropy effects of single and multiple bonds, however, and limitations in the experimental determination of have impeded widespread applications of this property. Moreover, for spherical molecules such as fullerenes, this criterion cannot be sensibly applied at all. 

If we consider the optical response of a molecular monolayer of increasing surface density, the fomi of equation B 1.5.43 is justified in the limit of relatively low density where local-field interactions between the adsorbed species may be neglected. It is difficult to produce any rule for the range of validity of this approximation, as it depends strongly on the system under study, as well as on the desired level of accuracy for the measurement. The relevant corrections, which may be viewed as analogous to the Clausius-Mossotti corrections in linear optics, have been the... 

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. 

Compute the dip angle of the magnetic field vector after correction for the drill collar field, it should check with the local magnetic field data. What do you conclude if it does not ... 

The next step to include electron-electron correlation more precisely historically was the introduction of the (somewhat misleading) so-called local- field correction factor g(q), accounting for statically screening of the Coulomb interaction by modifying the polarizability [4] ... 

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. 

N Is the number of molecules per unit volume (packing density factor), fv Is a Lorentz local field correction at frequency v(fv= [(nv)2 + 2]/3, v = u) or 2u). Although generally admitted, this type of local field correction Is an approximation vdilch certainly deserves further Investigation. IJK (resp Ijk) are axis denominations of the crystalline (resp. molecular) reference frames, n(g) Is the number of equivalent positions In the unit cell for the crystal point symmetry group g bjjj, crystalline nonlinearity per molecule, has been recently Introduced 0.4) to get general expressions, lndependant of the actual number of molecules within the unit cell (possibly a (sub) multiple of n(g)). 

For ions in crystals of high symmetry, as in the case of our reference octahedral ABe center, the correction factor is Eioc/Eo = (n + 2)/3 (Fox, 2001), where n is the refractive index of the medium. Although this correction factor is not strictly valid for centers of low symmetry, it is often used even for these centers. Thus, assuming this local field correction and inserting numerical values for the different physical constants, expression (5.21) becomes... 

This expression is exactly coincident with Equation (5.21), which leads to Smakula s formula. Equation (5.22), after inserting numerical values and the local field correction factor for centers of high symmetry. 

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