Screw axis of symmetry

The 18-residue 5-turn helix has a sixfold screw axis of symmetry, and it would accordingly be expected that these helical molecules would arrange themselves side by side in hexagonal packing in such a way that the unit... 

Figure 7.5 HOCO and LUCO orbital diagrams of polythiophene, PTh. Note that the S symbol is not shown for LUCO as it is covered by a n-orbital. Three unit cells are shown using a screw axis of symmetry, both orbitals are atk=0...

Figure 7.75 Orbital diagram of a seven ring part of an infinite CzS helicene. HOCO is shown atk= n/a using a screw axis of symmetry LUCO at k= 0...

The antiparallel-chain structure (Figure 5-8) has only a screw axis of symmetry, and its modes are distributed as follows A-62 modes, Raman, IR B-61 modes, Raman, IR. As for PGI, extensive IR spectra on isotopic derivatives (NH, CD2 ND, CH2 and ND, CD2) were available [97] as well as Raman spectra on the N-deuterated molecule [98]. The PGI force field [96] was used as a starting point, and refinement required small adjustments in 10 of 70 intramolecular force constants [123]. Amide mode frequencies are given in Table 5-12 the overall rms frequency error is 5.4 cm" b... 

Screw Axis of Symmetry It is the combined effect of rotation and translation, which transforms the array of motifs into self coincidence. The rotation axis is known as screw axis of rotational symmetry or simply screw axis of symmetry (Figs. 5.5 and 5.6). 

Note A screw axis of symmetry is an axis about which a rotation combined with translation parallel to the axis transforms an array of motifs into self coincidence. 

Therefore, reflections will occur only when Fhko does not vanish and it is from the planes hkO, where h is even. Similar arguments may also be applied for screw axis of symmetry and other translational symmetries. Screw axis of symmetry produces absences among reflections from the planes perpendicular to them. For example, 4i symmetry axis parallel to c limits 00/ reflections to those with I = 4n, while 2i-axis parallel to a-axis produces reflections hOO only when h = 2n. These conclusions can be drawn from the calculation of structure factor F ki-... 

From a theoretical point of view, Cui and Kertesz have studied the geometrical and electronic structures of PT and PMeT by energy-band theory where a screw axis of symmetry has been taken into account [68]. These authors confirm that the anti (rod) structure is slighltly more stable for PT while the syn (coil) conformation is preferred for PMeT. Finally, Ferro et al. applied molecular mechanics to obtain a monoclinic model of crystalline PT [84]. Their calculations show the weak dependence of packing energy on the unit cell parameter / , i.e. on the relative length of adjacent PT chains, in agreement with centro-symmetric and translation-ally disordered planar chains. 

Cellulose is a poly-/3-D-l,4-glucoside. The /3-D-glucose units exist in chan-conformations with hydroxyl groups at the equatorial position. Chiral recognition ability of native cellulose is attributed to the crystalline array of parallel chains with a twofold screw axis of symmetry along the chain axes. 

CA of Polysaccharides. Polysaccharides adopt a wide variety of shapes that depend on their composition and their environment. In solution, polymers are almost always random coils that have local regions that might be similar to conformations that are found in the solid state. The chapter by Brant and Christ discusses conformations of polysaccharides in solutions both in terms of these local regions and by the overall shape of the random coil in terms of end-to-end distance, etc. The following discussion concerns only linear (unbranched) molecules, and refers only to regular polymers, i.e., those that have repeated sequences of monomeric residues located by screw-axis (helical) symmetry. 

Plate 7 shows alanine in hypothetical unit cells of two space groups. A triclinic unit cell (Plate 7a) is designated PI, being a primitive lattice with only a onefold axis of symmetry (that is, with no symmetry). P2 j (Plate 7b) describes a primitive unit cell possessing a twofold screw axis parallel to c, which points toward you as you view Plate 7. Notice that along any 2l screw axis, successive alanines are rotated 180" and translated one-half the axis length. A cell in space group T>212121 possesses three perpendicular twofold screw axes. 

A screw axis brings the infinite rod into coincidence with itself after a translation through a distance t accompanied by a rotation through an angle a. The screw axis is of the order n = 360°/a. It is a special case when n is an integer. The iron chain and the beryllium dichloride chain have a fourfold (or fourth-order) screw axis, 42. Their overall symmetry is (a) m 42 m. For the screw axis of the second order, the direction of the rotation is immaterial. Other screw axes may be either... 

Figure 9-55. The molecules in the phenol crystal are connected by hydrogen bonds and are forming spirals with a threefold screw axis. This symmetry element is not part of the three-dimensional space group of the phenol crystal. After Zorky and Koptsik [103],...

In addition, the combination of three-dimensional symmetry elements gives rise to a completely new symmetry operator, the screw axis. Screw axes are rototranslational symmetry elements, constituted by a combination of rotation and translation. A screw axis of order n operates on an object by (a) a rotation of 2n/n counter clockwise and then a translation by a vector t parallel to the axis, in a positive direction. The value of n is the order of the screw axis. For example, a screw axis running parallel to the c-axis in an orthorhombic crystal would entail a counter-clockwise rotation in the a - b plane, (001), followed by a translation parallel to +c. This is a right-handed screw rotation. Now if the rotation component of the operator is applied n times, the total rotation is... 

The ordering of the molecules within one sheet of an amino acid crystal may be straight or zig-zag (Fig. 9.2.2). In the straight arrangement, the molecules are all in the same orientation (translational symmetry) in the zig-zag arrangement, the molecules are ordered on a screw-axis (helical symmetry). 

So far we have discussed the macroscopic symmetry elements that are manifested by the external shape of the three-dimensional patterns, that is, crystals. They can be studied by investigating the symmetry present in the faces of the crystals. In addition to these symmetry elements there are two more symmetry elements that are related to the detailed arrangements of motifs (atoms or molecules in actual crystals). These symmetry elements are known as microscopic symmetry elements, as they can only be identified by the study of internal arrangement of the motifs. As X-ray or electron diffraction can reveal the internal structures, these symmetry arrangements can only be identified by X-ray, Electro or Neutron diffraction. Obviously, they are not revealed in the external shape of the pattern. These symmetry elements are classified as microscopic symmetry elements. There are two such types of synunetry elements (i) glide plane of symmetry and (ii) screw axis of synunetry. 

Like this, if we consider all the possible rotation axes of symmetries and the possible translations along the rotation axis, henceforth known as screw axis, we get eleven different types of Screw axes of symmetries. They are listed in Table 5.1. 

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). 

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. 

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.

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. 

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