Electrical polarization distortional

The soft mode concept can be extended to all distortive phase transitions (transitions with relatively small atomic displacements), even if they are only close to second order. In the case of a ferro-distortive transition, as for example in BaTiOs or KDP, the order parameter is proportional to the spontaneous electric polarization Fj. d F/ dp is not only proportional to co, but also to the dielectric susceptibility. This does not, however, mean that all components of the order parameter eigenvector must contribute to Ps. 

Because of its piezoelectric properties, synthetic CC-quartz is used for frequency control in electrical oscillators and filters and in electromechanical transducers. When mechanically stressed in the correct direction, CC-quartz develops an electric polarization. The opposite is also tme an applied electric field gives rise to a mechanical distortion in the crystal. Thin sections of quartz are cut to dimensions that produce the desired resonance frequency when subjected to an alternating electric field the vibrating crystal then reacts with the driving circuit to produce an oscillation that can be narrowly controlled. Quartz is ideal for this application because it is hard, durable, readily synthesized, and can be tuned to high accuracy, for example, quartz crystal clocks can be made that are stable to one part in 109. 

When a substance is placed in an electric field, such as exists between the plates of a charged capacitor, it becomes to some extent electrically polarized. The polarization results at least in part from a displacement of electron clouds relative to atomic nuclei polarization resulting from this cause is termed electronic polarization. For molecular substances, atomic polarization may also be present, owing to a distortion of the molecular skeleton. Taken together, these two kinds of polarization are called distortion polarization. Finally, when molecules possessing permanent dipoles are present in a liquid or gas, application of an electric field produces a small preferential orientation of the dipoles in the field direction, leading to orientation polarization. 

General Treatment of Fluctuational Processes. The previous treatment is good only as long as we deal with strongly dipolar substances and all other polarizational effects remain negligible. In the majority of substances, besides reorientation of permanent dipoles, one has to consider reorientation of the polarizability ellipsoids as well as statistical-fluctuational processes. In calculating the electric polarization (277), one has to include the term accounting for linear distortional polarizability of the dielectric (non-linear polarizabilities are dealt with below) ... 

A polar bond in a molecule generates an electric field that can have an appreciable value at the position of a nearby resonating nucleus. This electric field distorts the electronic structure around the nucleus and causes a deshielding by diminishing a. Unlike the inductive effect, the electric-field effect can be derived from a polar group that is many bonds removed from the resonating nucleus. For a significant value of ag, the polar bond must be reasonably close to the nucleus, but need not be in van der Waals contact. 

We know that the quadratic-in-field coupling of an electric field to the dielectric tensor craitributes to the free energy density with the term g = —EaE / n. When liquid crystals possess macroscopic electric polarization P (spontaneous or induced by some external, other than electric field factors), then an additional, linear-in-field term gE = PE is added to the free energy density. One of such a source of the macroscopic electric polarization is orientational distortion of a liquid crystal. 

The new polar symmetry allows for the existence of macroscopic polarization, large or small, depending on the magnitude of the strain and molecular dipole moments shown by small arrows. Due to the distortions, the densest packing of our pears and bananas results in some preferable ahgimient of molecular skeletons in such a way that molecular dipoles look more up than down. By definition, the dipole moment of the unit volume is electric polarization. These simple arguments brought R. Meyer to the brilliant idea of piezoelectric polarization [25] ... 

As has been shown, the splay and bend distortions of a nematic create electric polarization. There is also a converse effect the external electric field causes a distortion due to the flexoelectric mechanism. For example, if the banana-shape molecules with transverse dipoles are placed in the electric field, the dipoles are partially aligned along the field and their banana shape induces some bend. This effect takes place even in nematics with zero dielectric anisotropy. 

It is well known that nematic liquid crystals are nonpolar. However, for a certain asymmetrical shape of the molecules, splay or bend deformations of the director field lead to an electrical polarization [87]. This feature is known as the flexoelectric effect. Theoretically, the influence of an electric field on CLCs for the case where the helical axis is oriented parallel to the plane of the sample was first considered by Goossens [88]. Experimentally, the flexoelectric electro-optic effect in CLCs can be observed in conventional sandwich cells with transparent electrodes when the helix axis of the CLC lies parallel to the glass surfaces [89]. In the absence of an electric field, the CLC behaves as a uniaxial material with its optic axis perpendicular to the director and parallel to the helix axis. When an electric field is applied normal to the pitch axis, the helix distorts, as shown in Figure 6.6. Thus, the optical axis is reoriented and the medium becomes biaxial. The deviation direction... 

Let us now consider the case of a polarized-distorted sphere such as the one in Figure 8. We would expect that associated with the polarization process there is a distortion process because dipoles are associated with some real molecular group that has a shape. In other words, when the dipoles orient in the electric field, there is an average projection in the field direction that is different from the average when the field is turned off. At this point it is necessary to postulate a molecular mechanism for the distorted sphere. Consider the sphere to consist of N ellipsoids with a major (2a) to minor (a) axis ratio of 2. The dipole moment = 4D units, which was computed from Eq. (12), , = 1) 7 = room... 

Closely related phenomenon to the piezoelectricity in liquid crystals is the flexoelectricity introduced by R.B. Meyer. Flexoelectricity means a linear coupling between the distortion of the director and the electric polarization. The constituent molecules of the nematic liquid crystals are rotating around their axes, and in absence of electric fields they are nonpolar. However, polar axes can arise in a liquid crystal made up from polar pear- or banana-shape molecules when they are subjected to splay or bend deformations, respectively. In these cases, the polar structures correspond to more efficient packing of the molecules (see Figure 8.17). 

The molecular statistical approach to calculate the flexoelectric coefficients was developed independently by Helfrich and Derzhanski and Petrov. The calculation is based on the requirement to ensure maximum packing condition. The excess number (AN = N+ - N-) of the molecules with dipole moment n determines the electric polarization P= AN/i. Dividing this by the distortion we get the flexoelectric coefficient. 

Figure Bl.5.2 Nonlinear dependence of tire polarization P on the electric field E. (a) For small sinusoidal input fields, P depends linearly on hence its hannonic content is mainly tiiat of E. (b) For a stronger driving electric field E, the polarization wavefomi becomes distorted, giving rise to new hannonic components. The second-hamionic and DC components are shown.

Now let us examine the molecular origin of Molecular polarity may be the result of either a permanent dipole moment p or an induced dipole moment ind here the latter arises from the distortion of the charge distribution in a molecule due to an electric field. We saw in Chap. 8 that each of these types of polarity are sources of intermolecular attraction. In the present discussion we assume that no permanent dipoles are present and note that the induced dipole moment is proportional to the net field strength at the molecule ... 

An electric dipole consists of two equal and opposite charges separated by a distance. AH molecules contain atoms composed of positively charged nuclei and negatively charged electrons. When a molecule is placed in an electric field between two charged plates, the field attracts the positive nuclei toward the negative plate and the electrons toward the positive plate. This electrical distortion, or polarization of the molecule, creates an electric dipole. When the field is removed, the distortion disappears, and the molecule reverts to its original condition. This electrical distortion of the molecule is caHed induced polarization the dipole formed is an induced dipole. 

Carbon atoms in free space have spherical symmetry, but a carbon atom in a molecule is a quite different entity because its charge density may well distort from spherical symmetry. To take account of the finer points of this distortion, we very often need to include d, f,. .. atomic orbitals in the basis set. Such atomic orbitals are referred to as polarization functions because their inclusion would allow a free atom to take account of the polarization induced by an external electric field or by molecule formation. 1 mentioned polarization functions briefly in Section 9.3.1. 

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