The crystallographic space groups

Each space group has a space group symbol that summarises the important symmetry 

CHS BUILDING CRYSTAL STRUCTURES FROM LATTICES AND SPACE GROUPS 

Letter symbol Lattice type Number of lattice points per unit cell Coordinates of lattice points 

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The group (E, J) has only two one-dimensional irreducible representations. The representations of 0/(3) can therefore be obtained from those of 0(3) as direct products. The group 0/(3) is called the three-dimensional rotation-inversion group. It is isomorphic with the crystallographic space group Pi. 

Single-crystal precession data indicate orthorhombic symmetry with the crystallographic space group Fddd. This system is not isostructural with any other known metal oxyfluoride or metal dioxide. The cell dimensions, determined from Guinier data, are a = 8.370 1 A. b = 10.182 1 A. and c = 7.030 1 A. The indexed powder data are given in reference 6. 

Kovalev, O. V. (1993) Representations of the Crystallographic Space Groups Irreducible Representations, Induced Representations and Corepresentations, 2nd edn. Philadelphia, PA Gordon and Breach. 

Combining the five point group symmetry operators with the two translational symmetry operators gives a total of exactly 230 different possible combinations, called the crystallographic space groups. Every crystal... 

Table 4.1 Geometric properties of some periodic minimal surfaces. The "genus" of each three-periodic minimal surface (IPMS) is the genus of a unit cell of the IPMS (with symmetrically distinct sides). The "symmetry" refers to the crystallographic space group for the surface (assuming equivalent sidesl.liie surfaces are tabulated in order of deviation of the homogeneity index from the "ideal" value of 3/4. ...

The first and obvious choice of the crystallographic space group is Pm. 

Zeolites are porous crystalline materials with a complex crystallographic structure. The characteristics of the void (pore) network is completely determined by the crystallographic space group and the composition. 

Table 5.2 The crystallographic space group letter symbols...

The internal symmetry of a crystal structure is described. The concepts of the unit cell and lattice are developed, leading naturally to a definition of symmetry elements and, hence, to the crystallographic space groups. The relevance of space groups and symmetry elements to the formation of the extended networks found in supramo-lecular chemistry is summarized. A discussion of common crystal packing modes follows, and the relevance of space group symmetry is discussed in this context. 

TABLE 2.16 The Symbols and Classification of the Crystallographic Space Groups ... 

FIGURE 2.66 The diagrammatic representations of the crystallographic space group ( 17) P222j after Verma and Srivastava (1982) and U.S. Naval Research Laboratory/ Center for Computational Materials Science (2003). 

Thus, the extension of the crystallographic space groups to those containing the anti-symmetry operation will be called magnetic or colored groups (Verma Srivastava, 1982, Sona Gautham, 1992). 

We have introduced an important concept here - the unit cell. In crystallography, the unit cell represents the budding block from which the infinite three-dimensional crystal lattice is built. If we are to model solid-state systems we must make use of a similar concept, from which we can build an infinite array of rephcas positioned in accordance with the crystallographic space-group symmetry operations. 

The icosahedron (Ih symmetry) is composed of 20 identical equilateral triangles. All 12 vertices are identical and linked to 5 other vertices. Since the icosahedron possesses a fivefold rotation axis, it is not compatible with the crystallographic space groups. So only distorted icosahedra can occur in real crystals. 

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