Fourfold rotation axes

The fourfold rotation axes connect the midpoints of opposite faces. The sixfold mirror-rotation axes coincide with threefold rotation axes. They connect opposite vertices and are located along the body diagonals. The symbol 6/4 does not directly indicate the symmetry planes connecting the midpoints of opposite edges, the twofold rotation axes, or the center of symmetry. These latter elements are generated by the others. The presence of a center of symmetry is well seen by the fact that each face and edge of the cube has its parallel counterpart. The tetrahedron, on the other hand, has no center of symmetry. 

Oh. Three mutually perpendicular fourfold rotation axes and four threefold rotation axes which are tilted with respect to the fourfold axes in a uniform manner, and a center of symmetry. Examples ... 

Fig. 18.4. Packing of y-cyclodextrin molecules in the crystal structure of y-cyclodextrin-x (n-pro-panol)-26H20. The y-cyclodextrins are stacked along fourfold rotation axes in head-to-tail and head-to-head mode, indicated by the arrows. Some of the 0(6)H hydroxyles are twofold disordered. Hydrogen-bonding contacts are drawn in solid lines [S67]...

In systems with two- and fourfold rotation axes, the e JT effect, discussed so far, is not operative. In the corresponding (sometimes called tetragonal) point groups, the decomposition of the symmetrized direct product E)" of an irreducible representation E with itself reads... 

It is convenient to focus the discussion on octahedral complexes (Fig. 6.1). These are molecules with high symmetry, possessing, for example, fourfold rotation axes, threefold rotation axes, twofold rotation axes, mirror planes, and a centre of symmetry (Fig. 6.2). Symbols such as a g ( aye one gee ), and tig are used to distinguish the behaviour of different orbitals or sets of orbitals under the various operations associated with the rotation axes, mirror planes etc. of the octahedron. Note the shift from symmetry elements (such as rotation axes, mirror planes) Fig. 6.2, to symmetry operations —the act of rotating, reflecting etc. In group theory one is concerned with symmetry operations, rather than with symmetry elements. Lower case symbols are used to denote orbitals or, more generally, wavefunctions. So, a point to which we shall return, an s orbital of an atom at a centre of an octahedral... 

There are now three cubic classes that do not have a center of symmetry. We should therefore expect these three - 23 (T), 43m (Tjj), and 432 (O) - to be piezoelectric, but only the first two are. The latter has three fourfold rotation axes perpendicular to each other, making the piezoelectric tensor isotropic in three dimensions. The piezoeffect would not then depend on the sign of the stress, which is only possible for all... 

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