Vector helicity

Fig. 2.50 Representation of 124 and 125, viewed along the Cu-Cu vector. Helical twist 0 is the P-Cu-Cu-P dihedral angle, in which both phosphorus atoms belong to the same dpp(d)a bridge...

Fig. 2. Depiction of conformal mapping of graphene lattice to [4,3] nanotube. B denotes [4,3] lattice vector that transforms to circumference of nanotube, and H transforms into the helical operator yielding the minimum unit cell size under helical symmetry. The numerals indicate the ordering of the helical steps necessary to obtain one-dimensional translation periodicity.

The quantity x is a dimensionless quantity which is conventionally restricted to a range of —-ir < x < tt, a central Brillouin zone. For the case yj = 0 (i.e., S a pure translation), x corresponds to a normalized quasimomentum for a system with one-dimensional translational periodicity (i.e., x s kh, where k is the traditional wavevector from Bloch s theorem in solid-state band-structure theory). In the previous analysis of helical symmetry, with H the lattice vector in the graphene sheet defining the helical symmetry generator, X in the graphene model corresponds similarly to the product x = k-H where k is the two-dimensional quasimomentum vector of graphene. 

Fig. 2. A portion of an unrolled cylinder with screw helicity. The broken line is parallel to the cylinder axis, and the cylindrical sheet has been cut along a generatrix (full line parallel to the cylinder axis), a and b are the unit vectors of the two-dimensional carbon layer in hybridization.

One verifies that the vector = 1/V2(8 — ), constructed from positive energy solutions of (9-470), (9-471), (9-472), and (9-473), corresponds in the case of a photon of definite energy, to the photon having its spin parallel to its direction of motion, i.e., positive helicity (s-k = + k ). 

Since P must remain normal to z and n, the polarization vector forms a helix, where P is everywhere normal to the helix axis. While locally a macroscopic dipole is present, globally this polarization averages to zero due to the presence of the SmC helix. Such a structure is sometimes termed a helical antiferroelectric. But, even with a helix of infinite pitch (i.e., no helix), which can happen in the SmC phase, bulk samples of SmC material still are not ferroelectric. A ferroelectric material must possess at least two degenerate states, or orientations of the polarization, which exist in distinct free-energy wells, and which can be interconverted by application of an electric field. In the case of a bulk SmC material with infinite pitch, all orientations of the director on the tilt cone are degenerate. In this case the polarization would simply line up parallel to an applied field oriented along any axis in the smectic layer plane, with no wells or barriers (and no hysteresis) associated with the reorientation of the polarization. While interesting, such behavior is not that of a true ferroelectric. 

In the crystal structures, neighboring doublehelices have the same rotational orientation and the same translation of half a fiber repeat as in the PARA 1 model. Only the Ax vector is slightly larger in the calculated interaction (1.077 nm) than in the observed ones 1.062 nm and 1.068 nm in the A type and B type, respectively. This may be due to the fact that in the crystal structures the helices depart slightly from perfect 6-fold symmetry. Also, no interpenetation of the van der Waals surfaces is allowed in the calculations, whereas some of them may occur in the cristallographic structure. It is quite interesting to note that the network of inter double-helices hydrogen bonds found in the calculated PARA 1 model reproduces those found in the crystalline structures. 

With regard to orientation, consider a repeating side-group originating at atom the first atom of the side-group being BL. For certain chain symmetries (helical, for instance) the bond vectors b(A. B )... 

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