Cubic point groups rotational symmetry

The cubic point groups are point groups containing symmetry elements for proper rotation by less than 180° around more than one axis. 

Another arrangement is provided by the cubic symmetry the cubic point groups. This kind of arrangement does not exist for smaller proteins. Only oligomeric proteins with 12n protomers may possess cubic symmetry. This symmetry designated as T (or 23) is characterized by more than one rotational axis greater than twofold. For example, the tetrahedron possesses four threefold axes. The threefold symmetry as previously mentioned is not possible when asymmetric protomers of a protein are at the vertices of a tetrahedron. However, it can be obtained if a group of three subunits are located at the four vertices of a tetrahedron. This involves a number of 12 asymmetric protomers. 

Figure 1 shows the arrangements of metal atoms on (111), (100), and (110) surfaces of a face-centered cubic (fee) metal. It can readily be seen that different sites on a particular surface have different symmetry properties. For example, the top layer of atoms on sites 1, 2, 3, and 3 on the (111) face (where the number denotes the number of metal atoms associated with the site) exhibit 6-fold, 2-fold, 3-fold, and 3-fold rotation axes of symmetry, respectively. At this level of discrimination the point group symmetries of those sites are C6(, C2 , Cj , and Cj , respectively. However, when the arrangement of atoms in the second layer is taken into account (there is another atom under only half of the 3-fold sites) the point group symmetries of the first two sites... 

The values of 2% If n (where n is the order of the axis of rotation) that satisfy eq. (16) and therefore are compatible with translational symmetry, are shown in Table 16.1. It follows that the point groups compatible with translational symmetry are limited to the twenty-seven axial groups with n= 1, 2, 3, 4, or 6 and the five cubic groups, giving thirty-two... 

The simplest way to treat it qualitatively is to consider the way the hydrogenic wave functions transform when reducing the symmetry from f +(3) (the 3D rotation group) to the T symmetry point group of the acceptor in a cubic crystal. 

In the example of the regular tetrahedron, (Figures 4.5a, 4.6), the principle symmetry element is the 4 axis, which is put in the primary position in the point group symbol, and defines the [100] direction of the cubic axial set. The rotation triad (3) lies along the (111) directions... 

In order to discuss the rest of the crystallographic point groups, one further has to consider the dihedral rotation groups D4 and D, and the cubic rotation group O. Their character tables, standard basis functions, and a useful choice of group generators are displayed in Tables 6,7, and 8. In this way the material required for symmetry considerations is directly available. 

The three components of the rotational tensor linear order. These symmetry strains are shown in fig. 4. p = corresponds to the fully symmetric volume strain. Deformations of the local environment lead to deformations of the 4f-charge cloud, microscopically one therefore has a coupling of strains to multipolar operators Op of the 4f shell. These are polynomials in J, and of degree 1 = 2, 4 and 6 which again transform as irreducible point-group representations. In the cubic case the quadrupolar (/ = 2) operators are ... 

The symmetries for the seven basic crystal systems described above assume the full symmetry or holohedry of each of the lattices. When the basis is added, some of these symmetries may be restricted. For example, a face-centered cubic (fee) crystal such as A1 has the full cubic symmetry. However, diamond also has the fee structure with atoms occupying the lattice points as well as every other tetrahedral interstitial point. Its point group is 43m, which implies a rotation-inversion on the fourfold axes. The threefold symmetry is preserved without the threefold rotation-inversion. The twofold symmetry is no longer preserved and the only mirror symmetry is along the 110 planes. 

In another example, at temperatures >393 K, barium titanate has the perovskite structure, which is simple cubic with all of the symmetry elements of the cubic lattice, so its point group is Oh or m3m. As the temperature is reduced to its Curie temperature, the lattice contracts and the oxygen ions on the faces of the cube squeeze the titanium ion in the center of the cube so that it is displaced in one direction while the oxygen ions are displaced in the opposite direction, destroying the inversion symmetry as well as the mirror symmetry about the central plane and the rotational symmetry about several of... 

---