Polar point groups

Crystals with one of the ten polar point-group symmetries (Ci, C2, Cs, C2V, C4, C4V, C3, C3v, C(, Cgv) are called polar crystals. They display spontaneous polarization and form a family of ferroelectric materials. The main properties of ferroelectric materials include relatively high dielectric permittivity, ferroelectric-paraelectric phase transition that occurs at a certain temperature called the Curie temperature, piezoelectric effect, pyroelectric effect, nonlinear optic property - the ability to multiply frequencies, ferroelectric hysteresis loop, and electrostrictive, electro-optic and other properties [16, 388],... 

Classically, there is only one way to obtain a noncentrosymmetric crystal packing. The use of enantio-merically pure dipolar compounds ensures crystallization in one of the 11 enantiomorphic groups, whereof 10 show a piezoelectric effect. Only five (namely, those featuring an unique rotational axis 1, 2, 3, 4. 6) of the enan-tioinorphic groups that are piezoelectric also allow for pyroelectricity (see Table 1). At present, there is no design eoneept with whieh to obtain a polar point group by spontaneous nueleation. 

The point group C2 is a polar point group with the C2-axis being a polar axis allowing a nonzero spontaneous electric polarization Pg. In a molecular picture, this means that the transverse dipole moments are not canceled due to the lack of the mirror plane. In consequence, a spontaneous electric polarization Pg occurs along the polar C2-axis and thus perpendicular to the plane spaimed by n and k ... 

The symmetry of the ferroelectric smectic C phase corresponds to the polar point group C2. When proceeding along the z coordinate perpendicular to the smectic layers, the director n and the polarization vector P directed along the C2 axis rotate that is, a helix of pitch h is formed. 

The n ==> n transition thus involves ground Ai) and exeited Ai) states whose direet produet (Ai x Ai) is of Ai symmetry. This transition thus requires that the eleetrie dipole operator possess a eomponent of Ai symmetry. A glanee at the C2v point group s eharaeter table shows that the moleeular z-axis is of A symmetry. Thus, if the light s eleetrie field has a non-zero eomponent along the C2 symmetry axis (the moleeule s z-axis), the n ==> 71 transition is predieted to be allowed. Light polarized along either of the moleeule s other two axes eannot induee this transition. 

In NFI3, belonging to the point group, there are degenerate E vibrations V3 and V4, shown in Figure 4.17. The transitions 1q and 2q are allowed and polarized along the z(Cb axis because both Vj and V2 are ui vibrations, and Table A. 12 in Appendix A shows that T Tb=Ai. Similarly, 3q and 4q are allowed and polarized in the xy plane because... 

Acetylene (HC=CH) belongs to the point group whose character table is given in Table A.37 in Appendix A, and its vibrations are illustrated in Figure 6.20. Since V3 is a vibration and T T ) = 2"+, the 3q transition is allowed and the transition moment is polarized along the z axis. Similarly, since Vj is a vibration, the 5q transition is allowed with the transition moment in the xy plane. 

The example we consider is the two-photon fluorescence excitation specfrum of 1,4-difluorobenzene, shown in Figure 9.29 and belonging to the >2 point group. The transition between the ground and first singlet excited state is Table A. 3 2 in Appendix A shows that 82 = r(T ) and, therefore, according to Equation (7.122), the electronic transition is allowed as a one-photon process polarized along the y axis which is in-plane and... 

Pyrazine belongs to the D2h point group, and Table A.32 in Appendix A shows that since B u = r(Tj) the Oq band is polarized along the x axis, which is perpendicular to the molecular plane. 

The most important materials among nonlinear dielectrics are ferroelectrics which can exhibit a spontaneous polarization PI in the absence of an external electric field and which can spHt into spontaneously polarized regions known as domains (5). It is evident that in the ferroelectric the domain states differ in orientation of spontaneous electric polarization, which are in equiUbrium thermodynamically, and that the ferroelectric character is estabUshed when one domain state can be transformed to another by a suitably directed external electric field (6). It is the reorientabiUty of the domain state polarizations that distinguishes ferroelectrics as a subgroup of materials from the 10-polar-point symmetry group of pyroelectric crystals (7—9). 

We note that the bilayer smectic phase which may be formed in main-chain polymers with two odd numbered spacers of different length (Fig. 7), should also be polar even in an achiral system [68]. This bilayer structure belongs to the same polar symmetry group mm2 as the chevron structure depicted in Fig. 17b, and macroscopic polarization might exist in the tilt direction of molecules in the layer. From this point of view, the formation of two-dimensional structure of the type shown in Fig. 7, where the polarization directions in neighbouring areas have opposite signs, is a unique example of a two dimensional antiferroelectric structure. 

Section 3.3.4 pointed out that cosolvents alter aqueous ionization constants as the dielectric constant of the mixture decreases, acids appear to have higher pKa values and bases appear (to a lesser extent than acids) to have lower values. A lower dielectric constant implies that the force between charged species increases, according to Coulomb s law. The equilibrium reaction in Eq. (3.1) is shifted to the left in a decreased dielectric medium, which is the same as saying that pKa increases. Numerous studies indicate that the dielectric constant in the region of the polar head groups of phospholipids is 32, the same as the value of methanol. [381,446-453] Table 5.2 summarizes many of the results. 

Polar symmetry (point group Coov or lower) is quite familiar at the molecular level as the symmetry required for the existence of a molecular dipole moment. Molecules possessing higher, nonpolar symmetry, cannot possess a permanent molecular dipole moment even when there are bond dipoles. Until the 1970s, no LC phases were known to possess polar symmetry, in spite of the fact that most mesogens are polar. 

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