Centrosymmetric crystals

As an illustration, we consider the case of SFIG from the (111) surface of a cubic material (3m. syimnetry). More general treatments of rotational anisotropy in centrosymmetric crystals may be found in the literature [62. 63 and M]- For the case at hand, we may detennine the anisotropy of the radiated SFl field from equation Bl.5.32 in conjunction with the fonn of -)from table Bl.5.1. We fmd, for example, for the p-in/p-out and s-... 

Sipe J E, Moss D J and van Driel H M 1987 Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals Phys. Rev. B 35 1129-41... 

Hoier, R., Bakken, L.N., Marthinsen, K. and Holmestad, R. (1993) Structure factor determination in non-centrosymmetric crystals by a two-dimensional CBED-based parameter refinement method, Ultramicroscopy, 49, 159-170. 

G. Assignment of Absolute Configuration Using Centrosymmetric Crystals.38... 

B. Liquid-Solid Reactions in Centrosymmetric Crystals Osmium... 

The use of chiral tailor-made additive molecules through their effect on the morphology of chiral or centrosymmetric crystals during crystal growth. 

This anisotropic distribution of the occluded additive provides a second independent method of confirming the absolute configuration assigned by means of the morphological changes, once the mechanism of adsorption is known. This principle will be met again in the growth of centrosymmetric crystals. 

We illustrated in Section II why conventional X-ray diffraction cannot distinguish between enantiomorphous crystal structures. It has not been generally appreciated that, in contrast to the situation for chiral crystals, the orientations of the constituent molecules in centrosymmetric crystals may be unambiguously assigned with respect to the crystal axes. Thus, in principle, absolute configuration can be assigned to chiral molecules in centrosymmetric crystals. The problem, however, is how to use this information which is lost once the crystal is dissolved. 

A prerequisite for application of this method is that within the centrosymmetric racemic crystal a specific functional group attached to an R molecule points toward the face fl but not toward fl (Scheme 12). By symmetry, the same functional group attached to an S molecule will emerge at the enantiotopic face f 1, but not at f 1. Crystallization of a centrosymmetric crystal in the presence of a chiral additive R designed so that it will fit in the site of an R molecule on the growing crystal faces fl or f2, but not on the enantiotopic faces fl or f2,... 

We shall illustrate the principle here using four examples of centrosymmetric crystals (/ ,S)-serine (45,78), W-acetylvaline, glycine (47), and glyclygly-cine (79). All four crystal structures appear in monoclinic symmetry (point symmetry Urn). 

This approach may also be applied to racemic bilayers built up from homo-chiral Langmuir-Blodgett monolayers. By measuring the two-dimensional diffraction pattern from such a bilayer it is possible to deduce the molecular chirality of each of the two monolayers in the order they were inserted to construct the bilayer. This approach can be extended to multilayers. Thus, in principle, we close the circle started in Section IV-G-1. It is possible to assign the absolute configuration of chiral molecules in centrosymmetric crystals provided that one can construct the crystal (in this case the multilayer) by adding homochiral layers one by one. 

Point symmetry mmm (shorthand for 2lm, 2Jm, 21m) comprises three perpendicular mirror planes, which automatically generates three perpendicular twofold axes at their lines of intersection, and a center of symmetry at the origin. This point group applies to all orthorhombic centrosymmetric crystals, such as those of space group Pbca. 

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). 

Hauptman H. and J. Karle, Solution of the phase problem I. The centrosymmetric crystal. ACA monograph No. 3. Ann Arbor, (Edwards Brothers, Inc., 1953). 

Figure 6. Successive changes of the phase value of a Fourier wave with index h = 2 moves the region with high potential (black areas) from the origin at X = 0 in the top map towards X = 1/4 in the map at the bottom. This shows that the value of the phase (f) determines the positions with high potential within the unit cell, whereas the amplitude A just affects the intensity. Note, that the maps with a phase shift of (j) = 0° and (f) = 180° have a centre of symmetry at the origin of the unit cell, whereas the other maps have no symmetry centre. From this we can draw another important conclusion if we put the origin of the unit cell on a centre of symmetry we have only two choices for the phase value, = 0° or (j)= 180°. As we will see later, this feature is of great importance for solving centrosymmetric crystal structures.

For centrosymmetric crystals this cannot happen as the surfaces will be the same on both sides of the crack line. In these cases it is postnlated that traces of impurities play an important role in the effect. 

Due to termination of the series, however, p(r) is severely affected by ripples. In addition, especially in the case of non-centrosymmetric crystals, the phase of vector F(S) is not known with precision and this affects a correct reconstruction of the density. Therefore, Fourier summation cannot be used for precise and accurate mapping of electron density. On the other hand, a model is necessary to overcome these limitations and to produce a function that is sufficiently close to the real, quantum mechanical />(r) in all regions of the crystal. 

Vaida, M., Popovitz-Biro, R., Leiserowitz, L., and Lahav, M. (1991). Probing reaction pathways via asymmetric transformations in chiral and centrosymmetric crystals. In Photochemistry in Organized and Condensed Media, ed. V. Ramamurthy. VCH, pp. 248-302. 

Weissbuch, L, Addadi, L., Berkovitch-Yellin, Z., et al. (1984). Spontaneous generation and amplification of optical activity in amino acids by enantioselective occlusion into centrosymmetric crystals of glycine. Nature, 310, 161. ... 

Table 2. Crystal Systems, Laue Classes, Non-Centrosymmetric Crystal Classes (Point Groups) and the Occurrence of Enantiomorphism and Optical Activity 31...

Crystal System Laue Class" Non-Centrosymmetric Crystal Classesa,b Enantiomorphism Optical Activity"... 

As discussed in detail in Ref. 36, for use in optoelectronics only systems crystallizing in non-centrosymmetric crystal lattices are of interest if the use of expensive enantiomers of chiral molecules is to be avoided. This considerably limits the available crystal lattices since most organic achiral molecules crystallize into centrosymmetric space groups. An interesting example of enantioselective inclusion complexation was reported by Gdaniec and coworkers [37]. 

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