Chirality tensor

Figure 7.17 The sign of the chirality tensor Qu along the molecular axes.

The notion the orientation of the chiral group with respect to the orientation axis of the guest is qualitative at the moment. But, possibly, the introduction of the tensor equations in Section 3.4 will allow us to use the principal axes of the chirality tensor Qj to describe this orientation. 

The chirality interaction tensor Wy is responsible for the interaction of the chiral guest, the chirality of which is described by the chirality tensor Cy, with the anisotropic host Wy = ikLkj- Ly covers the anisotropic host properties. W = Tr JT), is different from zero. The gyki are the orientational distribution coefficients of the guest in the molecular ensemble of the guest-host phase. From the two possible representations to avoid non-diagonal elements in (3.19), the representation in the system of principal axes of the order tensor is chosen instead of the principal axes of the chirality interaction tensor Wy. With Saupe s order parameters the HTP is also a sum of three terms... 

To relate macroscopic properties, especially the results of chirality measurements, to mesoscopic and further to pseudoscalar molecular properties, experimental data should be available in order to develop and check structure-property relations, mechanisms, and models. A set of usable data for the chiral nematic phase consists of the composition of the phase, p, HTP, V.2, Ki, K2, and K3. On the microscopic scale the (HTP)i as well as the microscopic order paramet S, D, A, B, the helicity tensor Qy or the chirality interaction tensor W y, and a chirality tensor Cy or an equivalent quantity should be known for every component. At the moment, the coordinates W y can only be estimated because a variation of the order of the chiral dopant for a constant host order needs to be known for their measurement. Some data are collected in TABLE 1 in order to give a feeling for the size and sign of available quantities [25-33] but, as can be seen, no complete sets of data for a system have yet beoi given in the literature. 

Given the interest and importance of chiral molecules, there has been considerable activity in investigating die corresponding chiral surfaces [, and 70]. From the point of view of perfomiing surface and interface spectroscopy with nonlinear optics, we must first examhie the nonlinear response of tlie bulk liquid. Clearly, a chiral liquid lacks inversion synnnetry. As such, it may be expected to have a strong (dipole-allowed) second-order nonlinear response. This is indeed true in the general case of SFG [71]. For SHG, however, the pemiutation synnnetry for the last two indices of the nonlinear susceptibility tensor combined with 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] ... 

Only for achiral surfaces does the last tensor element vanish altogether. Equation (4) retains a similar form but now accommodates a new tensor element. To date, very few experimental works have been reported on chiral surfaces, although the nonlinear effects are expected to be rather large [51]. 

Quantitative Determination of Electric and Magnetic Second-Order Susceptibility Tensors of Chiral Surfaces... 

Nonvanishing Components of Second-Order Susceptibility Tensor for Second-Harmonic Generation in Electric-Dipole Approximation for Achiral and Chiral Isotropic (i.e. isotropic in the plane of the film) Films0... 

Tensor Achiral Isotropic Film Chiral Isotropic Film... 

The largest components of the tensors / and yare chiral. Furthermore, these components are larger than the chiral component of y ". Hence the strongest magnetic quantities are directly associated with the chirality of... 

As stated before, the coefficients /, g, and h are linear combinations of the components of the tensors yeee, y eem, and ymee. For a system with Ccc, symmetry (i.e., chiral, isotropic surface symmetry) and -polarized second-harmonic light detected in transmission, these coefficients are given by (For the complete set of equations, see previous sections.)... 

As a particular example of materials with high spatial symmetry, we consider first an isotropic chiral bulk medium. Such a medium is, for example, an isotropic solution of enantiomerically pure molecules. Such material has arbitrary rotations in three dimensions as symmetry operations. Under rotations, the electric and magnetic quantities transform similarly. As a consequence, the nonvanishing components of y(2),eee, y 2)-een and y,2)jnee are the same. Due to the isotropy of the medium, each tensor has only one independent component of the xyz type ... 

However, the components of the yj2) e, e tensor are chiral (i.e., only present in a chiral isotropic medium), whereas the components of the tensors y 2) and y(2) meeare achiral (i.e., present in any isotropic medium, chiral or achiral). Hence, only the electric dipole response of chiral isotropic materials is related to chirality. The experimental work on chiral polymers described in Section 4 showed that large magnetic contributions to the nonlinearity are due to chirality. However, such contributions will therefore not survive in chiral isotropic media. In this respect, the electric dipole contributions associated with chirality may prove more interesting for applications. 

From the form of the polarization it is clear that in order to observe any nonlinear optical effect, the input beams must not be copropagating. Furthermore, nonlinear optical effects through the tensor y eee requires two different input frequencies (otherwise, the tensor components would vanish because of permutation symmetry in the last two indices, i.e., ytfl eee = Xijy ) For example, sum-frequency generation in isotropic solutions of chiral molecules through the tensor y1 1 1 has been experimentally observed, and the technique has been proposed as a new tool to study chiral molecules in solution.59,61 From an NLO applications point of view, however, this effect is probably not very useful because recent results suggest that the response is actually very low.62... 

There appears no tensor mean field in QCD as a result of chiral symmetry. So we, hereafter, only consider AV. 

It is evident that methods analogous to the ones developed here could be applied to molecular properties which, instead of being pseudoscalar, belong to some other representation of the skeleton point group (vector, tensor, etc. properties). To treat such properties, one needs only to induce from a different representation of than the chiral one. 

The analysis of IR spectra of different synthetic mixtures of C-labeled and non-labeled enantiomeric actetates poses no problems. After applying an automated baseline correction to the spectra and correcting the absorbance of one enantiomer in the synthetic mixtures by the absorbance of the other enantiomer at this position, the accuracy of the /i.seMJo-enantiomeric system is excellent, specifically, within +3% in comparison to the ee values determined by chiral GC 101). Using commercially available HTS-FTIR systems, high-throughput measurements are easily possible. The analysis can be performed on a Tensor 27 FTIR spectrometer coupled to a HTS-XT system, which is able to analyze the samples on 96- or... 

The handedness, or chirality, inherent in foundational electrodynamics at the U(l) level manifests itself clearly in the Beltrami form (903). The chiral nature of the field is inherent in left- and right-handed circular polarization, and the distinction between axial and polar vector is lost. This result is seen in Eq. (901), where , is a tensor form that contains axial and polar components of the potential. This is precisely analogous with the fact that the field tensor F, contains polar (electric) and axial (magnetic) components intermixed. Therefore, in propagating electromagnetic radiation, there is no distinction between polar and axial. In the received view, however, it is almost always asserted that E and A are polar vectors and that is an axial vector. 

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