Centrosymmetric crystals, polarization

Crystals whose structures are not centrosymmetric are polar because their centers of positive charge are displaced slightly from their centers of negative charge. Examples are crystals with the wurtzite structure which have polar axes along their (0001) directions. Also, crystals with the zincblende structure are polar in their (111) directions. 

These effects have proved important in improving the methods available for resolution of enantiomers by crystallization (267). Furthermore, by studies of the morphological changes induced, one may determine the faces at which the impurities are dominantly attached (270,271). Then, in suitable systems, it is possible to determine the absolute configuration of a polar crystal if one knows that of the impurity (272), or to determine that of the impurity if one knows the structure of the centrosymmetric crystal with which it interacts (270). 

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. 

The easiest combination of two chiral guests in the cylindrical capsule comes by the double encapsulation of racemic species with adequate size, shape and polarity to be accommodated in pairs, giving two diastereoisomeric complexes, the homochiral couple (R) - (R)/(S) - (S) and the heterochiral (R) -(S) combination. ( )-trans-1,2-Cyclohcxancdiol has all the above requisites and showed a 1.2 ratio between the two complexes in favor of the heterochiral combination [60]. This observation may be related to the preference in nature for centrosymmetric crystals or, alternatively stated, the higher melting points of racemates vs. enantiopure compounds where the resolution is driven by the less soluble pair [61,62], In the cylindrical capsule a single couple of chiral molecules is extrapolated from the bulk and the interactions between the two is governed by the shape of the cavity and their goodness of fit within the cavity. 

Figure 2-33. Left The centrosymmetric arrangement of acetanilide molecules of the crystal resulting in centrosymmetric crystal habit Right. The p-acetanilide molecules are aligned in a head-tail orientation resulting in the occurrence of a polar axis of the crystal habit [39], Both are reprinted with permission from 1981 American Chemical Society and D. Y. Curtin and I. C. Paul.

A centrosymmetric stress cannot produce a noncentrosymmetric polarization in a centrosymmetric crystal. Electric dipoles cannot form in crystals with an inversion center. Hence, only the twenty noncentrosymmetric point groups are associated with piezoelectricity (the noncentrosymmetric cubic class 432 has a combination of other symmetry elements which preclude piezoelectricity). The piezoelectric strain coefficients, dj for these point groups are given in Table 8.7, where, as expected, crystal symmetry dictates the number of independent coefficients. For example, triclinic crystals require the full set of 18 coefficients to describe their piezoelectric properties, but mono-chnic crystals require only 8 or 10, depending on the point group. 

The principal structural requirement for second order nonlinear effects in assemblies of molecules is the lack of a centre of symmetry, and considerable efforts have been expended in trying to induce potentially useful molecular entities to crystallize in non-centrosymmetric or polar crystals (Curtin and Paul 1981 Liter et al. 1991). As demonstrated below, this is a necessary, but not sufficient condition for obtaining nonlinear effects. True to form, the variety of crystallization experiments has led to a number of polymorphic structures, and to information about the relationship between the properties of these materials and their structures, as well as useful guidelines for attempting to obtain the desired non-centrosymmetric crystal structures. 

One property that makes octopoles interesting is the lack of dipole moment in the ground state. This should increase the probability of non-centrosymmetric crystallization and prevent detrimental dipolar (aggregate) interaction. In addition, the ratio of off-diagonal versus diagonal P tensor components is higher than that for traditional dipolar systems, which opens up the possibility of less stringent polarization schemes for parametric and electro-optic processes [95]. Furthermore, it has been shown that the efficiency-transparency trade-off favors octopolar molecules over traditional dipolar molecules [96]. 

In the case of a centrosymmetric crystal, the polarization induced by the first nonlinear term in equation 1.2 is p . If a field — is applied, the... 

Centrosymmetric crystals can be transformed into chiral or polar mixed crystals and this enables us to obtain novel means for absolute asymmetric synthesis. The principle is based on selective introduction of a guest molecule into a centrosymmetric host structure, thus reducing the symmetry of the mixed crystal. Crystallization of (E)-cinnamamide (space group P2i/c) in the presence of (E)-... 

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

DABCO or pyrazine monosalts have been found to exhibit a strong dielectric response because H transfers alter the polarisation of the NH—N hydrogen bonds and monocations (Katrusiak Szafratiski, 2006 Szafratiski Katrusiak, 2004 2007). This property occurs for centrosymmetric crystals, and can be rationalized by the formation of polar nano-regions. Because of the H transfers -... 

Figure 9-5. A fragment of a centrosymmetric crystal structure with the molecules (represented as arrows parallel to their molecular dipoles) NH+—N hydrogen-bonded into antiparallel chains along [y] (the anions are neglected for clarity). The ideal crystal structure with antiparallel molecules in neighbouring chains is marked in green (full arrowheads). Due to defects in the fourth and seventh chains, in which 5 and 6 molecules have reversed orientation, respectively, two polar nanoregions are formed. The red and blue colours and open arrowheads mark these nanoregions, the polarisation of which is indicated with large grey arrows...

Figure 15.12 (a) Unstressed centrosymmetric crystal. The arrows represent dipole moments, (b) Applying a stress to such a crystal cannot result in polarization, (t) Unstressed noncentrosymmetric crystal, i.e., piezoelectric. Note that this structure is not ferroelectric because it does not possess a permanent dipole, d) Stressed crystal develops a polarization as shown, (c) Unstressed polar crystal, i.e.. a ferroelectric, possesses a permanent dipole even in the unstressed state. (/) Stressed ferroelectric crystal. The applied stress changes the polarization by AP. 

Recent application of these principles to polymorphic control is shown schematically in Fig. 15 for a hypothetical dimorphic system in which one polymorph is centrosymmetric and the other crystallizes in a polar space group [36]. In the former crystal, the molecules are arranged in antiparallel orientation whereas in the latter they are aligned along a common direction. A tailor-made auxiliary which binds to both crystals would do so at the two (indistinguishable) ends of the centrosymmetric crystal but only at one end of the polar crystal. The latter would therefore grow at the expense of the former and polymorphic control will have been achieved. 

Fig. 15 a Schematic illustration of the packing arrangement in a polar and a centrosymmetric crystal, b The effect of a designed inhibitor on crystal growth. (Adapted from [36] with permission... 

Generally, the piezoelectric effect could exist just in non-centrosymmetrical crystallographic symmetry classes. Mechanical stress/strain as a second-rank symmetrical tensors are basically centrosymmetrical external fields. If the materials crystallographic symmetry include cerrtre of symmetry operation, the resulting symmetry of material subjected to such field is also cerrtrosymmetrical (see Neuman s Law in Nye (1985)). Therefore, piezoelectric effect is excluded. Centrosymmetrical crystal stays centrosymmetrical even after the application of the mechanical stress and no polar direction for the polarization vector might exist in such stmcture. 

Many polar crystals also have valuable optical properties resulting from the interaction of the electrons with the electric field of the radiation. For example, the frequency of a beam of electromagnetic radiation may be doubled by sending the beam through a non-centrosymmetric crystal The incident radiation carries an oscillating electric field through the crystal, a field that deforms the electronic wavefimction, and therefore the electric field, of the crystal. The extent to which the electric field of the crystal is changed increases with the polarizability of the electrons and with the intensity of the radiation. 

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... 

Among non-centrosymmetric crystal classes only the following 11 correspond to polar crystals, which are of particular relevance for second order NLO 1, 2, m, mm2, 3, 3m, 4,4mm, 6, 6mm... 

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