Cube structure symmetry groups

Figure 2.14 Division of the regular 48-vertex orbit of 0[, symmetry (a) into the two 24-point sets of the chiral snub cube structure. The dextro projection (b) is also drawn in perspective (c). Coalescing sets of vertices onto the rotational poles leads to the other orbits O, O12 and Os of the O symmetry group. These are achiral they are also orbits of 0[,.

Td, possesses 32 symmetry, and requires a minimum of 12 asymmetric units the cube and octahedron, which belong to the point group Oh, possess 432 symmetry, and require a minimum of 24 asymmetric units and the dodecahedron and icosahedron, which belong to the point group Ih, possess 532 symmetry, and require a minimum of 60 asymmetric units. The number of asymmetric units required to generate each shell doubles if mirror planes are present in these structures. 

Look at Uie drawings accompanying Problem 3.6. Is it possible to superimpose the cube on the dodecahedron Castieman and coworfcers have recently detected a cation with mje = 28, identified as TigC. It b bebeved that the titanium atoms form a cube with the addition of twelve carbon atoms to complete a pentagonal dodecahedron. Draw the proposed structure. What e its point group symmetry ... 

Figure 9-23. The diamond structure after Shubnikov and Koptsik [33], (a) A unit cell the edges of the cube are the a, b, and c axes (b) Two face-centered cubic sublattices displaced along the body diagonal of the cube (c) Projection of some symmetry elements of the Fd im space group onto a horizontal plane. The vertical screw axes 4 and 43 are marked by appropriate symbols. Used with permission.

Figure 2.30 The Faujasite structure of Fd3m space group symmetry with sets of 024h orbits identified about the tetrahedral vertices of a cube positioned at the centres of the truncated octahedra coloured in red.

We saw in Chapter 2 that the characteristic (minimum) symmetry of a cubic crystal is the set of 3-fold axes parallel to the body-diagonals of the cubic cell. These 3-fold axes also imply 2-fold axes parallel to the cube edges. The NaCl structure has the highest class of cubic symmetry, with 4-fold axes and planes of symmetry. An octahedral ion such as (TlCl ) also has full cubic symmetry (m3m) can occupy the Cl positions in the normal NaCl structure. Groups such as S2, (F—H-F) , and CN have lower symmetry and can form the fully symmetrical NaCl structure only if they are rotating or are randomly oriented with their centres... 

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