Polar crystal classes

Eleven acentric crystal classes are chiral, i.e., they exist in enantiomorphic forms, whereas ten are polar, i.e., they exhibit a dipole moment. Only five (1,2, 3, 4, and 6) have both chiral and polar symmetry. All acentric crystal classes except 432 possess the same symmetry requirements for materials to display piezoelectric and SHG properties. Both ferroelectricity and pyroelectricity are related to polarity a ferroelectric material crystallizes in one of ten polar crystal classes (1, 2, 3,4, 6, m, mm2, 3m, 4mm, and 6mm) and possesses a permanent dipole moment that can be reversed by an applied voltage, but the spontaneous polarization (as a function of temperature) of a pyroelectric material is not. Thus all ferroelectric materials are pyroelectric, but the converse is not true. 

Actually, it has been shown that the highest second order susceptibility coefficients (macroscopic nonlinearity), for a given chromophore (microscopic nonlinearity), can be reached in the (polar) crystal classes 1, 2, m and mm2, while other polar crystal classes are less favourable [21]. 

Pyroelectrics. Pyroelectric ceramics are materials that possess a uoique polar axis and are spontaneously polarized ia the abseace of an electric field. Pyroelectrics are also a subset of piezoelectric materials. Ten of the 20 crystal classes of materials that display the piezoelectric effect also possess a unique polar axis, and thus exhibit pyroelectricity. In addition to the iaduced charge resultiag from the direct pyroelectric effect, a change ia temperature also iaduces a surface charge (polarizatioa) from the piezoelectric aature of the material, and the strain resultiag from thermal expansioa. 

Ferroelectrics. Among the 32 crystal classes, 11 possess a centre of symmetry and are centrosymmetric and therefore do not possess polar properties. Of the 21 noncentrosymmetric classes, 20 of them exhibit electric polarity when subjected to a stress and are called piezoelectric one of the noncentrosymmetric classes (cubic 432) has other symmetry elements which combine to exclude piezoelectric character. Piezoelectric crystals obey a linear relationship P,- = gijFj between polarization P and force F, where is the piezoelectric coefficient. An inverse piezoelectric effect leads to mechanical deformation or strain under the influence of an electric field. Ten of the 20 piezoelectric classes possess a unique polar axis. In nonconducting crystals, a change in polarization can be observed by a change in temperature, and they are referred to as pyroelectric crystals. If the polarity of a pyroelectric crystal can be reversed by the application on an electric field, we call such a crystal a ferroelectric. A knowledge of the crystal class is therefore sufficient to establish the piezoelectric or the pyroelectric nature of a solid, but reversible polarization is a necessary condition for ferroelectricity. While all ferroelectric materials are also piezoelectric, the converse is not true for example, quartz is piezoelectric, but not ferroelectric. 

Pot a fuller discussion of the phenomenon, and a list of the crystal classes which (according to current theories) may exhibit it, see Wooster (1938) and International Tables (1952). Rotation of the plane of polarization, though difficult to observe and measure except in large isotropic and uniaxial crystals, and therefore of rather limited application in identification or structural investigations, is a phenomenon of great interest that has played an important part in the historical development of chemistry and crystallography. 

The anomalous phase-displacement method detects essentially a polar axis this means that it can detect lack of centrosymmetry in general, for all non-centrosymmetric crystals have polar directions. For instance, although crystal class 4 is not usually referred to as a polar class (for its fourfold axis is not polar), nevertheless, except for the fourfold axis and all directions normal to it, all other directions are polar (see International Tables, 1952, p. 43), and the corresponding regions of the reciprocal lattice will therefore show the intensity differences which have been described. 

If the combination of X-ray and morphological evidence does not determine the space-group uniquely, additional information may be sought by tests for piezo-electric and pyro-electric properties, and by an optical examination for any evidence of rotation of the plane of polarization. (See Chapter VIII.) The results of such tests may settle the matter, since only certain crystal classes have these properties. Only positive results are decisive the apparent absence of piezo-electric or pyro-electric effects may be due to feeble phenomena. 

Among the 20 crystal classes lacking a center of symmetry ten of them contain a unique polar axis and exhibit pyroelectricity in addition to piezoelectricity, i.e. in the unstrained dipolar network of these crystals the dipole moment components remain and add to a resultant polar-axis moment. The term pyroelectricity is assigned because thermal expansion will expand or contract the dipole. The pyroelectric constant is defined by ... 

Crystal system Crystal class Chiral (enantiomorphism) Optical activity (circular dichroism) Polar (pyroelectric) Piezoelectric, SHG... 

For the present purpose it is only necessary to distinguish polar crystals, i.e. those that are spontaneously polarized and so possess a unique polar axis, from the non-polar variety. Of the 32 crystal classes, 11 are centrosymmetric and consequently, non-piezoelectric. Of the remaining 21 non-centrosymmetric classes, 20 are piezoelectric and of these 10 are polar. An idea of the distinction between polar and non-polar structures can be gained from Fig. 2.3 and Eqs (2.70) and (2.71). 

Table 4.6 Non-centrosymmetric crystal classes and polar directions...

Crystal class directions polar axis effect group activity possible... 

Pyroelectric crystals are ones that are spontaneously polarizable (see below) and in which a change in temperature produces a change in that spontaneous polarization. A limited number of pyroelectric crystals have the additional property that the direction of spontaneous polarization can be reversed by application of an electric field, in which case they are known as ferroelectrics. Thus a ferroelectric is a spontaneously polarized material with reversible polarization. Before proceeding much further it is important to appreciate that not all crystal classes can exhibit polar effect. 

As in the case of piezoelectrics, the elementary dipoles will cancel out if the crystallographic unit cell has a centre of symmetry. However, another condition is also needed to produce a spontaneous polarisation, the presence of a unique polar axis, which is a direction in the crystal unrelated by symmetry to any other direction, not even the antiparallel direction. The dipoles lie parallel to the polar axis of the crystal (see Section 5.1.3). Of the 20 piezoelectric crystal classes, only 10 fulfil this criterion and give rise to the pyroelectric effect. The relationship between the appearance of piezoelectricity and pyroelectricity and the symmetry of the crystal is set out in Figure 11.12. 

In polar crystals, the piezoelectric polarization generated as a result of mechanical stress application will contribute to the spontaneous polarization existing previously. In polar-neutral crystals, the polar directions are mutually compensated . As a result of mechanical stress application, singular polar direction appears in such crystals. Piezoelectric polarization is generated in that direction and crystal is piezoelectrically polarized. The only exception among the polar-neutral classes is cubic 432 class, where all piezoelectric coefficients are identically equal zero because of symmetry (Zheludev 1975). 

Pyroelectrics are materials that possess a unique polar axis and are spontaneously polarized in the absence of an electric field. Of the 20 crystal classes that display piezoelectricity, 10 exhibit pyroelectricity. Since pyroelectric ceramics are a subset of piezoelectric materials, they are also piezoelectric in nature. The polarization exhibited by pyroelectric materials is also a function of temperature, and the change in polarization with temperature may be expressed by... 

In a preceding paragraph we have shown that a necessary condition for second order NLO activity in a material is the absence of the symmetry center. This a stringent requirement and it is perhaps the most difficult feature the chemistry of these materials must face with. To get a non-centrosymmetric arrangement of molecules on a macroscopic scale is a difficult task, and even more difficult is to get a non-centrosymmetric polar structure. A polar structure is that in which there is at least one macroscopic direction (polar axis) which is not changed in the opposite direction by symmetry transformations allowed for the material. With reference to crystals, out of the 32 crystal classes only the following 20 correspond to non-centrosymmetric crystals 1,2, m, 222, mm2,4, -4,422,4mm, -42m, 3, 32, 3m, 6, —6, 622, 6mm, —62m, 23, —43m... 

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