Susceptibility tensor properties

From the point of view of tensor properties, the surface nonlmear susceptibility /, J-(.is quite analogous to the... 

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] ... 

Most of the four above-mentioned properties for Raman spectra can be explained by using a simple classical model. When the crystal is subjected to the oscillating electric field = fioc " of the incident electromagnetic radiation, it becomes polarized. In the linear approximation, the induced electric polarization in any specific direction is given by Pj = XjkEk, where Xjk is the susceptibility tensor. As for other physical properties of the crystal, the susceptibility becomes altered because the atoms in the solid are vibrating periodically around equilibrium positions. Thus, for a particular... 

However, perpendicular to the pyridine planes, a small positive n interaction was unambiguously defined 50 cm 1 < e < 130 cm"1. Any value for e outside this range, produced hopelessly inadequate descriptions of the magnetic properties, and it is certainly this small amount of n interaction that is responsible for the low symmetry of the ligand field and the unusual orientation of the susceptibility tensor. This is an example of a sharply fitting parameter. 

Characterization of Molecular Hyperpolarizabilities Using Third Harmonic Generation. Third harmonic generation (THG) is the generation of light at frequency 3co by the nonlinear interaction of a material and a fundamental laser field at frequency co. The process involves the third-order susceptibility x 3K-3 , , ) where —3 represents an output photon at 3 and the three s stand for the three input photons at . Since x(3) is a fourth (even) rank tensor property it can be nonzero for all material symmetry classes including isotropic media. This is easy to see since the components of x(3) transform like products of four spatial coordinates, e.g. x4 or x2y2. There are 21 components that are even under an inversion operation and thus can be nonzero in an isotropic medium. Since some of the terms are interrelated there are only four independent terms for the isotropic case. 

The real and imaginary parts of the refractive index n quantify the scattering and absorption (or amplification) properties of a material- The refractive index is besfl derived from the susceptibility tensor y of the material, defined below, whi j describes the response of a macroscopic "system to incident radiation [212], Spe fically, an incident electric field E(r, t), where r denotes the location in the medium, tends to displace charges, thereby polarizing the medium. The change in dmd(r, the induced dipole moment, from point r to point r + dr is given in terms of th polarization vector P(r, t), defined as... 

The components of a symmetrical second-rank tensor, referred to its principal axes, transform like the three coefficients of the general equation of a second-degree surface (a quadric) referred to its principal axes (Nye, 1957). Hence, if all three of the quadric s coefficients are positive, an ellipsoid becomes the geometrical representation of a symmetrical second-rank tensor property (e.g., electrical and thermal conductivity, permittivity, permeability, dielectric and magnetic susceptibility). The ellipsoid has inherent symmetry mmm. The relevant features are that (1) it is centrosymmetric, (2) it has three mirror planes perpendicular to the... 

The magnetic properties of the H2 molecule as arising from its two electrons are well known, both experimentally and from basic theory. Thus accurate calculations of its magnetic shielding and its dipole magnetizability (susceptibility) tensors are available (e.g. see Refs. 41 and 42). 

To understand and optimize the electro-optic properties of polymers by the use of molecular engineering, it is of primary importance to be able to relate their macroscopic properties to the individual molecular properties. Such a task is the subject of intensive research. However, simple descriptions based on the oriented gas model exist [ 20,21 ] and have proven to be in many cases a good approximation for the description of poled electro-optic polymers [22]. The oriented gas model provides a simple way to relate the macroscopic nonlinear optical properties such as the second-order susceptibility tensor elements expressed in the orthogonal laboratory frame X,Y,Z, and the microscopic hyperpolarizability tensor elements that are given in the orthogonal molecular frame x,y,z (see Fig. 9). 

Bertini et report a detailed study of the magnetic properties of myglobin, in which they determined the axial and orthorhombic terms of the paramagnetic susceptibility tensor using a combination of hyperfine chemical shift measurements and static suscetibility measurements (Evans Method). The determination of the magnetic anisotropy provided a measurement of the residual dipolar couplings and also permitted a separation of the contact and pseudocontact chemical shift contributions of resonances of the Fe(III) ligands. 

In order to obtain a useful material possessing a large second order nonlinear susceptibility tensor % 2) one needs to use molecules with a large microscopic second order nonlinear hyperpolarizability tensor B organised in such a way that the resulting system has no centre of symmetry and an optimized constructive additivity of the molecular hyperpolarizabilities. In addition, the ordered structure thus obtained must not loose its nonlinear optical properties with time. The nonlinear optical (NLO) active moieties which have been synthesized so far are derived from the donor-rc system-acceptor molecular concept (Figure 1). 

As a consequence of the uniaxial symmetry all material properties of nematics have to be represented by tensors. For instance, the dielectric displacement D and E are connected by the dielectric susceptibility tensor e as ) = eoeE = tole E + i —e ) n-E)n]. Thus e depends in general on the local director orientation and is specified by two dielectric constants, ey and e (for E parallel and perpendicular to n, respectively). An analogous representation applies to the electric conductivity tensor [Pg.102]

Munn et have attempted to relate the macroscopic refractive indices and susceptibility tensor for the iodoform and sulfur (Sg) crystals and the crystalline complex CHal.SSg to the molecular polarizabilities and hyperpolarizabilities. The attempt is only partially successful and illustrates the difficulties that will be encountered in projects such as that described in ref. 246. Penhuis and Muim have performed calculations of the linear and non-Unear optical properties of layers of Langmuir-Blodgett films and find that the internal field varies little after the first layer. They have investigated Langmuir-Blodgett... 

Equation 3 can now be employed to calculate the linear and nonlinear susceptibilities in (1) and (2), revealing very specific tensor properties exhibited by poled polymers. With the foregoing model for the potential energy, the thermodynamic averages are sometimes also expressed in terms of the Langevin functions, which are defined as... 

As with the evaluation of the linear susceptibility, one can determine the properties of the nonlinear susceptibility tensor from Eqs. 2 and 3. In regions of low dispersion the x ifk tensor exhibits special symmetry properties that are referred to as Kleinman... 

Material properties are characterized by the permittivity tensor Sy or the dielectric susceptibility tensor Xij describing relations between the field quantities by... 

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