Helicity tensor

To relate macroscopic properties, especially the results of chirality measurements, to mesoscopic and further to pseudoscalar molecular properties, experimental data should be available in order to develop and check structure-property relations, mechanisms, and models. A set of usable data for the chiral nematic phase consists of the composition of the phase, p, HTP, V.2, Ki, K2, and K3. On the microscopic scale the (HTP)i as well as the microscopic order paramet S, D, A, B, the helicity tensor Qy or the chirality interaction tensor W y, and a chirality tensor Cy or an equivalent quantity should be known for every component. At the moment, the coordinates W y can only be estimated because a variation of the order of the chiral dopant for a constant host order needs to be known for their measurement. Some data are collected in TABLE 1 in order to give a feeling for the size and sign of available quantities [25-33] but, as can be seen, no complete sets of data for a system have yet beoi given in the literature. 

Figure 4. (A) A tensor plot of the C shielding in alanine with sheet dihedral angles ( - -60°, ij/ = -60°). (Reproduced with permission from ref. 18. Copyright 1997 American Chemical Society.)...

To visualize an object with Dm symmetry, imagine a cylinder whose outside is covered with n slanted striations, as illustrated at the top of Figure 8. The two constructions shown (D symmetry) are enantiomorphs whose sense of chirality is related to the way in which the striations are slanted. As n approaches infinity, the symmetry of the constructions approaches Z) in the limit, infinitely many C2 axes are embedded in a plane perpendicular to the C axis. This is the symmetry of a stationary cylinder undergoing a twisting motion, as indicated by the arrows on the cylinders at the bottom of Figure 8, and of an axial tensor of the second rank.41 It is also the helical symmetry of a nonpolar object undergoing a screw displacement, that is, of an object whose enantiomorphism and sense of chirality are T-invariant. 

The measured crystal optical activity, in general, can be either of molecular origin or due to the chiral helical arrangement of chiral or achiral molecules in the crystal, or both. The two factors are difficult to separate. Kobayashi defined a chirality factor r = (pc — ps)/pc = 1 — pslpc, where pc is the rotatory power per molecule of a randomly oriented crystal aggregate derived from the gyration tensors determined by HAUP, and ps that in solution [51]. It is a measure of the 4 crystal lattice structural contribution to the optical activity and represents the severity of the crystal lattice structural contribution to the optical activity, and represents the severity of the restriction of the freedom of molecular orientation by forming a crystal lattice. Quartz is a typical example of r = 1, as it does not contain chiral molecules or ions and its optical activity vanishes in random orientation (ps = 0). 

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