Screw axis symmetry

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. 

What has been said here is true but obscures another fundamental property of the Fourier transform, one that complicates matters a bit but not hopelessly so. The Fourier transform fails to directly carry translational relationships from one space to another, in particular, from real space into reciprocal space. This means that the transform does not discriminate between asymmetric units based on the distances between them. The immediate relevance of this is that a set of asymmetric units related by a screw axis symmetry operator (which has translational components) in real space is transformed into diffraction space as though it simply contained a pure rotation axis. The translational components are lost. If our crystal has a 6i axis, we will see sixfold symmetry in the diffraction pattern. If we have 2i2j2i symmetry in real space, the diffraction pattern will exhibit 222 (or more properly, mmm) symmetry. 

The evidence for the existence of screw axis symmetry is manifested in certain subclasses of reflections that are systematically absent. These systematic absences, we will see, fall along axial lines (/tOO, OkO, 00/) in reciprocal space and clearly signal not only whether an axis in real space is a screw axis or a pure rotation axis, but what kind of a screw axis it is, for instance, 4i or 42, 6i or 63. Thus the inherent symmetry of the diffraction pattern (which we call the Laue group), plus the systematic absences, allow us to unambiguously identify (except for a few odd cases) the space group of any crystal. 

Real objects are not infinite. For symmetry considerations, it may be convenient to look only at some portions of the whole, where the ends are not yet in sight, and extend them in thought to infinity. A portion of an iron chain and a chain of beryllium dichloride in the crystal are shown in Figure 8-10. Translation from unit to unit is accompanied by a 90° rotation around the translation axis. A portion of a spiral stairway displaying screw-axis symmetry is shown in Figure 8-lla. The imaginary impossible stairway of Figure 8-llb indeed seems to go on forever. 

Figure 8-11. (a) A portion of a spiral stairway displaying screw-axis symmetry. Photograph by... 

The unitary DSs for a general nanotube involve, therefore, the translation Tj and screw axis symmetries only. 

In most of the studies these reflections, which are usually weak relative to the other main reflections, are assumed to be negligible. The controversy continues because the relative intensities can be influenced by experimental conditions such as the periods of exposure of the diffractometric plates. Furthermore, the disallowed reflections tend to be more intense in electron diffractometric measurements than in x-ray diffraction measurements. Thus, more often than not, investigators using electron diffraction challenge the validity of the assumption of twofold screw axis symmetry. 

It is often said that cellulose has a cellobiose repeating unit. In the minds of some workers, that statement conveys the shape that results from twofold screw-axis symmetry. In this work, we propose that the ideal shape for cellulose does not have a twofold structure and that a range of shapes should occur. To the extent that the cellulose molecule can take various shapes, it is unjustifiably limiting to define cellulose in terms of a particular shape. 

HF forms a zigzag chain in the crystal and, in the chain direction, the unit cell contains 2HF molecules (see Figure 9.3). The internal coordinates at successive HF molecules obey a screw axis symmetry. Applying this symmetry, several authors have suggested that a single HF molecule forms the unit cell and only four internal coordinates (ri, r, a, and P) need be optimized. Application of the combined symmetry operation leads to a great decrease in the amount of computational work necessary to determine the equilibrium geometry. 

All repeating arrangements of regular polymers can be described as helical in the sense that they all have crystallographic screw axis symmetry. The descriptors of helices include n, the (possibly noninteger) number of residues per turn of the helix, and h, the rise per residue. The distance between turns of the helix is known as the pitch, p, which is... 

In spite of the usefulness of group theory in reducing the computational labor and providing a better understanding of the physical reality, polymer quantum calculations rarely have made use of more than translational periodicity. One should mention precursors who have gone beyond translational symmetry McCubbin (16d), Blumen and Merkel (16e). The letters have succeeded in implanting cyclic screw axis symmetry into an ab initio program. 

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