Screw axis

Axes of symmetry. An axis about which rotation of the body through an angle of 2njn (where n is an integer) gives an identical pattern 2-fold, 3-fold, 4-fold and 6-fold axes are known in crystals 5-fold axes are known in molecules. In a lattice the rotation may be accompanied by a lateral movement parallel to the axis (screw axis). 

If the space group contains screw axes or glide planes, the Patterson fiinction can be particularly revealing. Suppose, for example, that parallel to the c axis of the crystal there is a 2 screw axis, one that combines a 180° rotation with... 

All tenus in the sum vanish if / is odd, so (00/) reflections will be observed only if / is even. Similar restrictions apply to classes of reflections with two indices equal to zero for other types of screw axis and to classes with one index equal to zero for glide planes. These systematic absences, which are tabulated m the International Tables for Crystallography vol A, may be used to identify the space group, or at least limit die... 

Abstract—The fundamental relations governing the geometry of carbon nanotubes are reviewed, and explicit examples are pre.sented. A framework is given for the symmetry properties of carbon nanotubes for both symmorphic and non-symmorphic tubules which have screw-axis symmetry. The implications of symmetry on the vibrational and electronic structure of ID carbon nanotube systems are considered. The corresponding properties of double-wall nanotubes and arrays of nanotubes are also discussed. 

Of particular importance to carbon nanotube physics are the many possible symmetries or geometries that can be realized on a cylindrical surface in carbon nanotubes without the introduction of strain. For ID systems on a cylindrical surface, translational symmetry with a screw axis could affect the electronic structure and related properties. The exotic electronic properties of ID carbon nanotubes are seen to arise predominately from intralayer interactions, rather than from interlayer interactions between multilayers within a single carbon nanotube or between two different nanotubes. Since the symmetry of a single nanotube is essential for understanding the basic physics of carbon nanotubes, most of this article focuses on the symmetry properties of single layer nanotubes, with a brief discussion also provided for two-layer nanotubes and an ordered array of similar nanotubes. 

FIGURE 7.31 The favored conformation of agarose in water is a double helix with a threefold screw axis. 

The optical isomer of [Co(en)3] referred to in the main text is the (+)NaD isomer, which has a left-handed (laevo) screw axis as shown in Fig. Ba. and according to the convention recommended by ILIPAC is ven tbe symbol A. This is in contrast to its mirror image (Fig. Bb) which has a right-handed (dextro) screw axis and is g ven the symbol A. 

Table 6-1. C2(l molecular poinl group. The electronic stales of the flat T6 molecule are classified according lo the lwo-1 old screw axis (C2). inversion (/). and glide plane reflection (o ) symmetry operations. The A and lt excited slates transform like translations Oi along the molecular axes and are optically allowed. The Ag and Bg stales arc isoniorphous with the polarizability tensor components (u), being therefore one-photon forbidden and Iwo-pholon allowed.

Each polytripeptide chain is twisted around a threefold screw axis and exists in a secondary structure, analogous to the left-handed polyproline II-helix, i.e. with transposition of the peptide bond (pitch 8.4 A, 3 amino acids) (Figs. 2,3). 

FIGURE 35.28 T3fpical pressure buildup along a screw axis of a corotating twin-screw extruders (TSEs). 

Fig. 5. Projection of two molecules of cis-Pt (Bu2dfe)2l2 along the c-axis (omitting the butyl chains). The positions of a twofold axis, a twofold screw axis and one iodine atom of a third molecule axe shown.

Screw rotation. The symmetry element is a screw axis. It can only occur if there is translational symmetry in the direction of the axis. The screw rotation results when a rotation of 360/1V degrees is coupled with a displacement parallel to the axis. The Hermann-Mauguin symbol is NM ( N sub M )-,N expresses the rotational component and the fraction M/N is the displacement component as a fraction of the translation vector. Some screw axes are right or left-handed. Screw axes that can occur in crystals are shown in Fig. 3.4. Single polymer molecules can also have non-crystallographic screw axes, e.g. 103 in polymeric sulfur. 

P2X Z = 2 Dx = 1.59 R = 0.11 for 881 intensities. The acyclic molecule has the planar, zigzag conformation. The terminal OH and N-OH groups are +syn and ap, respectively. There is an interesting system of O-H N hydrogen-bonds which forms a spiral along the screw-axis. Only seven of the hydrogen atoms were located. 

Fig. 18. Diagrammatic representation of the helical polymeric form of the cyclodextrin 32 around the 2X screw axis...

The structure of carbon nanotubes depends upon the orientation of the hexagons in the cylinder with respect to the tubule axis. The limiting orientations are zigzag and arm chair forms, Fig. 8B. In between there are a number of chiral forms in which the carbon hexagons are oriented along a screw axis, Fig. 8B. The formal topology of these nanotube structures has been described [89]. Carbon nanotubes have attracted a lot of interest because they are essentially onedimensional periodic structures with electronic properties (metallic or semiconducting) that depend upon their diameter and chirality [90,91]. (Note. After this section was written a book devoted to carbon nanotubes has been published [92], see also [58].)... 

Because of the special atomic arrangement of the carbon atoms in a carbon nanotube, substitutional impurities are inhibited by the small size of the carbon atoms. Furthermore, the screw axis dislocation, the most common defect found in bulk graphite, is inhibited by the monolayer structure of the Cfjo nanotube. For these reasons, we expect relatively few substitutional or structural impurities in single-wall carbon nanotubes. Multi-wall carbon nanotubes frequently show bamboo-like defects associated with the termination of inner shells, and pentagon-heptagon (5 - 7) defects are also found frequently [7],... 

The molecules are helices having a twofold screw-axis, with an axial rise per disaccharide residue of 9.45 A (945 pm). The axes of the chains are parallel, and about equally spaced, but are not further organized into crystalline arrays. A hydrogen bond was proposed between the OH-3 group of the 2-acetamido-2-deoxy-D-glucose residues and 0-5 of the D-galactose 6-sulfate residues. 

---