Point symmetries, ferroelectrics

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). 

It is known that the crystal symmetry defines point symmetry group of any macroscopic physical property, and this symmetry cannot be lower than corresponding point symmetry of a whole crystal. The simplest example is the spontaneous electric polarization that cannot exist in centrosymmetric lattice as the symmetry elements of polarization vector have no operation of inversion. We remind that inversion operation means that a system remains intact when coordinates x, y, z are substituted by —x, —y, —z. If the inversion center is lost under the phase transition in a ferroic at T < 7), Tc is the temperature of ferroelectric phase transition or, equivalently, the Curie temperature), the appearance of spontaneous electrical polarization is allowed. Spontaneous polarization P named order parameter appears smoothly... 

Multiferroics with composition BaA/F4 where Af = Mg, Mn, Fe, Co, Ni, Zn have orthorhombic structure with 2 mm point symmetry at high temperatures with extrapolated Curie temperature above the melting point. At T = 25-70 K ferroelastic ferroelectric structure displays purely antiferromagnetic or weak ferromagnetic ordering. 

The molecular ion here considered is NO2, in the stable phase of sodium nitrite. This phase is ferroelectric, with lattice belonging to the non-centrosymmetric body-centered orthorombic system, with spatial group m2m [1]. The unit cell contains two formula units both with the same point symmetry. The spatial arrangement of the atoms inside the unit cell is shown in Figure 1. (Notice that the axes are relabeled with respect to the crystallographic convention). 

The helical smectic C state has the point symmetry (< 22), illustrated in Fig. 19, which does not permit a polar vector. It is therefore neither pyroelectric nor ferroelectric. Nor can it, of course, be piezoelectric, which is also easily realized after a glance at Fig. 14 if we apply a pressure or tension vertically, i.e. across the smectic layers (only in this direction can the liquid crystal sustain a strain), we may influence the pitch of the helix but no macroscopic po-... 

It should be noted that, whereas ferroelectrics are necessarily piezoelectrics, the converse need not apply. The necessary condition for a crystal to be piezoelectric is that it must lack a centre of inversion symmetry. Of the 32 point groups, 20 qualify for piezoelectricity on this criterion, but for ferroelectric behaviour a further criterion is required (the possession of a single non-equivalent direction) and only 10 space groups meet this additional requirement. An example of a crystal that is piezoelectric but not ferroelectric is quartz, and ind this is a particularly important example since the use of quartz for oscillator stabilization has permitted the development of extremely accurate clocks (I in 10 ) and has also made possible the whole of modern radio and television broadcasting including mobile radio communications with aircraft and ground vehicles. 

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],... 

The earliest approach to explain tubule formation was developed by de Gen-nes.168 He pointed out that, in a bilayer membrane of chiral molecules in the Lp/ phase, symmetry allows the material to have a net electric dipole moment in the bilayer plane, like a chiral smectic-C liquid crystal.169 In other words, the material is ferroelectric, with a spontaneous electrostatic polarization P per unit area in the bilayer plane, perpendicular to the axis of molecular tilt. (Note that this argument depends on the chirality of the molecules, but it does not depend on the chiral elastic properties of the membrane. For that reason, we discuss it in this section, rather than with the chiral elastic models in the following sections.)... 

It is interesting to point out here that with all of the theoretical speculation in the literature about polar order (both ferroelectric and antiferroelectric) in bilayer chevron smectics, and about reflection symmetry breaking by formation of a helical structure in a smectic with anticlinic layer interfaces, the first actual LC structure proven to exhibit spontaneous reflection symmetry breaking, the SmCP structure, was never, to our knowledge, suggested prior to its discovery. 

Finally, ferroelectricity is manifest in asymmetrical crystals producing domains of spontaneous polarization whose polar axis direction can be reversed in an electric field directed opposite the total dipole moment of the lattice. The two (or more) directions can coexist in a crystal as domain structures comprising millions of unit cells which contain the same electric orientation. The symmetry elements are temperature sensitive in ferroelectric materials [27]. At a particular temperature called the Curie Point the values of the piezoelectric coefficients reach particularly high values. Above the Curie Point the crystal transformation is to a less polar form and the ferroelectric nature disappears. 

Barium titanate, BaTi03, is probably the most widely studied ferroelectric oxide. Extensive studies were conducted on this compound during World War II in the United States, England, Russia, and Japan, but the results were not revealed until after the war. Barium titanium(IV) oxide was found to be a ferroelectric up to a temperature of 120°C., which is its Curie point. Above 120°C., barium titanium(IV) oxide has the cubic perovskite structure, and below this temperature the oxygen and titanium ions are shifted and result in a tetragonal structure with the c axis approximately 1% longer than the a axis. Below 0°C., the symmetry of barium titanate becomes orthorhombic, and below —90°C., it becomes trigonal. 

In the temperature range 193-278 K, the atoms in the BaTi03 crystal undergo further shifts, and the lattice symmetry is reduced to the orthorhombic system and C2v point group, as shown in Fig. 10.4.2(c). This crystalline modification has spontaneous polarization with ferroelectric properties. 

Figure 2.7-8 Vibrational species of thiourea as free molecule, point group C2, and under the constraints of the site symmetry, C of the unit cell at room temperature, well as of the ferroelectric modification, C, Cartesian... 

Espinet and Serrano have previously reported that disrupting the symmetry of a series of dinuclear palladium(II) complexes by forming the mononuclear complexes with a -diketonate ligand yields a advantageous reduction in the melting point [93]. This chemistry was extended to ferroelectric complexes (Figure 46) to yield complexes that switched much faster than their first reported ferroelectric metallo-mesogen [94]. 

The same increase in complexity is found when the point group of a ferroelectric crystal with aligned electric dipoles is considered, as, from a symmetry point of view the two cases are identical. 

The ferroelectric materials show a switchable macroscopic electric polarization which effectively couples external electric fields with the elastic and structural properties of these compounds. These properties have been used in many technological applications, like actuators and transducers which transform electrical signals into mechanical work [72], or non-volatile random access memories [73]. From a more fundamental point of view, the study of the phase transitions and symmetry breakings in these materials are also very interesting, and their properties are extremely sensitive to changes in temperature, strain, composition, and defects concentration [74]. 

There is a structural requirement for ferroelectricity. There are a total of 32 different symmetry point groups, 21 of which do not possess a center of symmetry. Ferro-electrics are part of a small subgroup of noncentrosym-metric crystals. Related properties are piezoelectricity and pyroelectricity. Dielectrics belonging to all but one of the groups of noncentrosymmetric crystals are piezoelectric. Pyroelectric crystals form a further subgroup of 10 types of crystal having especially low symmetry as shown in Table 31.5. 

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