Symmetry and polarization

We have mentioned that it is impossible for uniformly oriented nematic liquid crystals to have spontaneous polarization because of their symmetry. Now let us consider the possibility of spontaneous polarization in other liquid crystal phases. For rod-like molecules, it is impossible in any liquid crystal phase to have spontaneous polarization along the liquid crystal director because n and -n are equivalent. 

Cholesteric liquid crystals consist of chiral molecules and therefore do not have reflection symmetry. The symmetry group of cholesteric hquid crystals is 2 [1,3]- A cholesteric liquid crystal is invariant for the two-fold (180°) rotation around n, which rules out the possibility of spontaneous polarization perpendicular to n. It is also invariant for the two-fold rotation around an axis that is perpendicular to the n - (the hehcal axis) plane, which mles out the possibility of spontaneous polarization parallel to n. Therefore there is no ferroelectricity in the cholesteric phase. 

Smectic-A liquid crystals, besides the orientational order as nematics, possess onedimensional positional order. They have a layered structure. The liquid crystal director n is perpendicular to the smectic layers. The symmetry of smectic-A is Do h if the constituent molecule is achiral or D if the constiment molecule is chiral. It is invariant for any rotation around n. It is also invariant for the two-fold rotation around any axis perpendicular to n. The continuous rotational symmetry is around n and therefore there is no spontaneous polarization in any direction perpendicular to n. Hence it is impossible to have spontaneous polarization in smectic-A, even when the constiment molecule is chiral. 

Smectic-C liquid crystals are similar to smectic-A liquid crystals except that the liquid crystal director is no longer perpendicular to the layer but tilted. For the convenience of symmetry discussion, let us introduce a unit vector a which is perpendicular to the layer. The symmetry group is C2h- The two-fold rotational symmetry is around the axis that is perpendicular to the na plane (which contains both n and a). This implies that there is no spontaneous polarization in the na plane. The reflection symmetry is about the ita plane, and therefore there is no spontaneous polarization perpendicular to the na plane either. This rules out the possibility of spontaneous polarization in smectic-C liquid crystals. 

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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.)... 

Investigations of the effects of physical adsorption on the optical spectra of adsorbates have been limited in number. However, a few observations have been reported on perturbations of molecules by surface electric fields. These involve changes in the symmetry and polarization. 

Figure 8-5. Byzantine mosaics from Ravenna, Italy, with one-dimensional space-group symmetry and polar (a) and nonpolar axes (b). Photographs by the authors.

Holman, K.T. Pivovar. A.M. Ward. M.D. Engineering crystal symmetry and polar order in molecular host frameworks. Science 2001. 294. 1907-1911. 

A molecule is chiral if it cannot be superimposed on its mirror image (or if it does not possess an alternating axis of symmetry) and would exhibit optical activity, i.e. lead to the rotation of the plane of polarization of polarized light. Lactic acid, which has the structure (2 mirror images) shown exhibits molecular chirality. In this the central carbon atom is said to be chiral but strictly it is the environment which is chiral. 

The distinction between in-plane A symmetry) and out-of-plane (A" symmetry) vibrations resulted from the study of the polarization of the diffusion lines and of the rotational fine structure of the vibration-rotation bands in the infrared spectrum of thiazole vapor. 

Figure 6.22 shows, for example, that the symmetry species of vibrational fundamental and overtone levels for V3 alternate, being Aj for u even and B2 for v odd. It follows that the 3q, 3q, 3q,. .. transitions are allowed and polarized along the y,z,y,... axes (see Figure 4.14 for axis labelling). 

The Raman and infrared spectra for C70 are much more complicated than for Cfio because of the lower symmetry and the large number of Raman-active modes (53) and infrared active modes (31) out of a total of 122 possible vibrational mode frequencies. Nevertheless, well-resolved infrared spectra [88, 103] and Raman spectra have been observed [95, 103, 104]. Using polarization studies and a force constant model calculation [103, 105], an attempt has been made to assign mode symmetries to all the intramolecular modes. Making use of a force constant model based on Ceo and a small perturbation to account for the weakening of the force constants for the belt atoms around the equator, reasonable consistency between the model calculation and the experimentally determined lattice modes [103, 105] has been achieved. 

SHG has been used to study electrode surface symmetry and order using an approach known as SH rotational anisotropy. A single-crystal electrode is rotated about its surface normal and the modulation of the SH intensity is measured as the angle (9) between the plane of incidence and a given crystal axis or direction. Figure 27.34 shows in situ SHG results for an Au(ll 1) electrode in 0.1 M NaC104 + 0.002 M NaBr, using a p-polarized beam. The results indicate the presence of two distinct onefold... 

Goodwin That is similar to what happens in Fucus. There is a symmetrical cell, and even in the absence of any polarization due to light, it will break symmetry and produce an axis. There is probably a similar sort of stochastic event that triggers some kind of polymerization or pattern. 

Mo2(02CCH2).. Metal compounds with multiple metal-metal bonds such as Mo2(02CCH3)4 of symmetry, have attracted much experimental and theoretical attention focussed on the description of bonding and bond strength (46-48). Their electronic structure has been investigated experimentally by various methods such as resonance Raman, photoelectron spectroscopy, ultraviolet absorption and polarization studies of the matrix isolated sample (49-56). 

Another example of the potential utility of polarized edge spectra for structure determination is found for [MoO J2" (28). This molecule has C2V symmetry and the C2 axes of all of the molecules in the unit cell are collinear. Thus, when the crystal is oriented with the polarization parallel to the S-S interatomic vector, the polarization is perpendicular to the Mo-0 bonds and nearly parallel to the Mo-S bonds. Similarly, the crystal can be oriented with the polarization perpendicular to the Mo-S bonds and nearly parallel to the Mo-0 bonds. For both orientations, excellent agreement was obtained with SCF-X a calculations of the edge structure (j ). 

In Chapter 8, Stavola and Pearton discuss the local vibrational modes of complexes in Si that contain hydrogen or deuterium. They also show how one can use applied stress and polarized light to determine the symmetry of the defects. In the case of the B-H complex, the bond-center location of H is confirmed by vibrational and other measurements, although there are some remaining questions on the stress dependence of the Raman spectrum. The motion of H in different acceptor-H complexes is discussed for the Be-H complex, the H can tunnel between bond-center sites, while for B-H the H must overcome a 0.2 eV barrier to move between equivalent sites about the B. In the case of the H-donor complexes, instead of bonding directly to the donor, H is in the antibonding site beyond the Si atom nearest to the donor. The main experimental evidence for this is that nearly the same vibrational frequency is obtained for the different donor atoms. There is also a discussion of the vibrational modes of H tied to crystal defects such as those introduced by implantation. The relationship of the experimental results to recent theoretical studies is discussed throughout. 

Polar structures may have rotation symmetry and reflection symmetry. However, there can be no rotation or reflection normal to the principal rotation axis. Thus, the presence of the mirror plane normal to the C2 axis precludes any properties in the SmC requiring polar symmetry the SmC phase is nonpolar. 

Figure 8.6 Three-dimensional slice of C2 symmetrical SmC phase, showing tilt cone, polar axis (congruent with twofold symmetry axis), smectic layer planes, tilt plane, and polar plane.

Figure 8.16 Illustration of symmetry of Soto Bustamante-Blinov achiral antiferroelectric smectic LC with finite number of layers. Such systems can be studied using DRLM technique with thin freely suspended smectic films, (a) With even number of bilayers, film has local C2 symmetry, and therefore no net electric polarization, (b) With odd number of bilayers, film has local Cnv symmetry and is therefore polar, with net spontaneous electric polarization in plane of layers.

Taken from the three spontaneous symmetry-breaking events leading to this layer structure [formation of layers with long-range orientational order of the director (Sm), tilt of the director from the layer normal (C), and polar orientation of the molecular arrows (P)], we term phases of this type SmCP. All of the complex textures and EO behavior of NOBOW in the B 2 phase can be understood in terms of various stacking modes of SmCP layers as shown in Figure 8.23. 

Figure 8.24 Illustration of layer structure and symmetries observed for NOBOW thermodynamic phase (majority domains) in freely suspended films, (a) Films of even-layer number have achiral, nonpolar C symmetry, (b) Films of odd-layer number have chiral and polar C2 symmetry, with net polarization normal to tilt plane (lateral polarization).

The Cj symmetry also means that a freely suspended film possessing an even number of layers is nonpolar, as is observed. When the film possesses a finite and odd number of layers, however, the symmetry of the system is C2, as indicated in Figure 8.24b. This structure is both chiral and polar, leading to the observed net polarization in films of NOBOW with odd-layer number. 

Without going into too much detail, it is relatively easy to intuitively understand optical rotation in second-harmonic generation from a chiral thin film by simply considering the nonvanishing polarization components generated in a chiral and achiral film. For example, for an achiral thin film with CXA, symmetry and for the experimental situation shown in Figure 9.5, the nonvanishing components of the polarization can be written as ... 

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