Surface susceptibility

The theoretical framework developed above is valid in the electric dipole approximation. In this context, it is assumed that the nonlinear polarization Ps 2a)) is reduced to the electric dipole contribution as given in Eq. (1). This assumption is only valid if the surface susceptibility tensor co, m) is large enough to dwarf the contribution from higher... 

The electric quadrupole Q 2co) involves both the gradient of the electromagnetic incident electric field E u)) and the gradient of the electric quadrupole susceptibility tensor Xq 2o), CO, co). This problem is nonetheless solved by the mere addition of supplementary terms in the surface susceptibility tensor. As a result, the surface susceptibility tensor becomes an effective tensor instead of a purely surface specific one [27,38] ... 

The surface susceptibility tensor of a chiral surface possesses different symmetry properties as compared to the surface susceptibility tensor of an isotropic surface. The main difference for a chiral surface arises from the axes OX and the O Y, the two axes in the plane of the surface, which are no longer indistinguishable. The nonvanishing elements of the susceptibility tensor are then [52] ... 

The standard method of treating the electric dipole contribution from the surface region is to view the nonlinear polarization as a dipole sheet which can be described by the surface susceptibility tensor, xf. This is equivalent to considering the three layer system shown in Fig. 3.1 with the dipole sheet placed just below the surface at z = 0+ and allowing the surface layer thickness, d, to approach zero. The surface contribution to the polarization can then be written as... 

When the data in Fig. 3.2 are fit to Eq. (3.10), the results yield A = 0 and f (y) = sin 3 y/ for I (11, ) [67, 68]. Using p-polarized input and output, the data fits to A/B = 1 and f y/) = cos 3 // for I (11, 11). With appropriate choices of the input and output polarizations and separate SH phase measurements, combinations of the four independent surface susceptibility elements and the bulk susceptibilities y and C contributing to A and B for this system were also determined. There is inherent difficulty in separating the bulk surface susceptibilities such that the bulk susceptibility y is always measured in a linear combination with a surface susceptibility element for any experimental geometry [82-84]. 

Although the rotational anisotropy scans are informative, considerably more information can be obtained by separate determination of changes in the isotropic and anisotropic components of the surface susceptibility tensor as done by Koos et al. [122]. The experiments consist of monitoring the SH intensity at a fixed angle of 0 = 30° where / oc a 2 and Tj a b0) 2. The results shown in Fig. 5.16 are displayed in terms of thallium coverage. The data has been fitted to a simple linear Langmuir isotherm model of Heinz [79] where the adsorbate contribution to x(2) varies linearly with coverage such that... 

In order to extract information on the molecular orientation distribution, the relationship between the macroscopic surface susceptibilities and the molecular hyperpolarizabilities, needs to be considered. It is usual to consider the intrinsic non-linear response of each of the molecules as independent of the other molecules so that the interfacial response is an average over the orientational distribution and scales with the molecular density (squared). Even the modification of this response due to local field... 

The surface susceptibility is a 27-element tensor that can generally be reduced... 

The input polarization dependence of p- and s-polarized SH light was measured to elucidate the molecular orientation of L and ML species. The results can be fitted well to the theory and the relative values of three non-zero elements of the surface susceptibility [63-65], which determine the molecular orientation, are obtained from the fitting. Since hyperpolarizability of azobenzene dyes is known to be dominated by a single element which is an element of the hyperpolarizability along the tc-jt moment direction [52], the tilt angle 0 can be estimated from the relative values of the surface susceptibility. 

The surface susceptibility Xsfg is a 27-element tensor that can generally be reduced to a handful of non-vanishing elements after consideration of the symmetry of the system. In particular, the interface between two isotropic bulk media is isotropic in the plane of the interface (it has Coov symmetry), and Xsfg reduces to the following four independent non-zero elements... 

The theoretical framework developed above is valid in the electric dipole approximation. In this context, it is assumed that the nonlinear polarization PfL(2 >) is reduced to the electric dipole contribution as given in Eq. (1). This assumption is only valid if the surface susceptibility tensor x (2 > >, a>) is large enough to dwarf the contribution from higher orders of the multipole expansion like the electric quadrupole contribution and is therefore the simplest approximation for the nonlinear polarization. At pure solvent interfaces, this may not be the case, since the nonlinear optical activity of solvent molecules like water, 1,2-dichloroethane (DCE), alcohols, or alkanes is rather low. The magnitude of the molecular hyperpolarizability of water, measured by DC electric field induced second harmonic... 

Here, a, is the principal value of the tensor of the generalized polarizability of one adsorbed molecule [Eq. (1.35)] and (W ), is the number of the adsorbed molecules per unit area that contribute to the /th component of the polarizability. The quantity on the right of Eq. (1.89) is called surface susceptibility. 

Servant et al. [503, 607] have shown that the spectra of the imaginary parts of the surface susceptibility tensor [Eiq. (1.89)], which are easily obtained from the polarized IR spectra with no measurement of the film thickness, take into account the domainlike composition typical of LB films. However, this approach demands a priori knowledge of the polarizability of the adsorbed molecule—a parameter that can be calculated assuming a certain microscopic model of the molecular packing only. 

Similar to the surface susceptibility method [503, 507], DR fitting allows one to exclude the film thickness and optical constants as input parameters in spectral simulations, a very important advantage in the case of adlayers when there is no way to determine the film thickness accurately [622], As opposed to spectrum fitting, DR fitting is applicable to the cases when (1) properties of the film depend on the substrate (the general case), (2) preliminary information about the film is poor or absent, and (3) when the film can not be perfectly reproduced in a series of experiments. Moreover, as will be shown below, DR fitting is more sensitive to MO. 

The determination of R (see Eq. 7.25) by a measurement of the partition coefficient k in the laboratory, by some type of sorption experiment for the determination of adsorbed solid-phase concentration versus solution-phase concentration in equilibrium (so-called isopleths or isotherm), however, supplies values of R which are not comparable with the actual in field retardation factors in a mineral liner, because the specific surface susceptible to the pollutant is very different in both cases. Therefore, based on diffusion coefBcients in free aqueous solutions, the effective diffusion coefficients of organic substances for water-saturated mineral liners were estimated by using a tortuosity factor F= 1/3 (Muller et al. 1997a). Table 7.6 shows the results for various soil materials. 

The decisive new quantitity introduced by Dignam et. al. is the surface susceptibility... 

With equation 63 the surface susceptibility tensor reads ... 

Any surface or interface breaks the inversion symmetry and is therefore a possible source of second-order effects. Owing to their surface sensitivity, second-harmonic generation measurements have developed into a very useful tool for probing the orientation of organic molecules in well-structured monolayers, such as those obtainable by the Langmuir-Blodgett technique (see Table 6). The surface susceptibility may in general be written as... 

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