Space-groups symmetries dimensionality

Fig. 14.3 Polyhedral packing plots for the two-dimensional layers of [RE(P2S6),/2(PS4)P in the series of solids A2RE(P2S6)i/2(PS4), where A=K, Cs RE = Y, La. Rare-earth polyhedra are striped PS4 polyhedra are black phosphorous atoms in P2S6 are shown as black circles. Alkali atoms are not shown for clarity. Although these phases have distinctly different structures based on space group symmetry and atomic positions, the compounds are clearly related upon close inspection of the building blocks.

The crystal structures of (EDT-TTFBr2)2MX4 and (EDO-TTFBr2)2MX4 are quite similar, although the space group symmetry is different in these two systems. However, this difference comes only from the conformation of terminal six-membered rings of the donor molecules, which plays no important role in the physical properties of the present salts. The donor molecules are stacked in a head-to-tail manner to form quasi-one-dimensional columns as shown in Fig. 6a. 

Again, we assume that the solid surface in question is untarnished. Even so, most surfaces are not ideal. They undergo energy-lowering processes known as relaxation or reconstruction. The former process does not alter the symmetry, or structural periodicity, of the surface. By contrast, surface reconstruction is a surface symmetry-lowering process. With reconstruction, the surface unit cell dimensions differ from those of the projected crystal unit cell. It will be recalled that a crystal surface must possess one of 17 two-dimensional space group symmetries. The bulk crystal, on the other hand, must possess one of 230 space group symmetries. 

Objects or patterns which are periodic in one, two, and three directions will have one-, two-, and three-dimensional space groups, respectively. The dimensionality of the object/pattem is merely a necessary but not a satisfactory condition for the dimensionality of their space groups. We shall first describe a planar pattern after Budden [3] in order to get the flavor of space-group symmetry. Also, some new symmetry elements will be introduced. Later in this chapter, the simplest one-dimensional and two-dimensional space groups will be presented. The next Chapter will be devoted to the three-dimensional space groups which characterize crystal structures. 

Figure 8-4. Polar (a) And nonpolar (b) Decorations of Byzantine mosaics from Ravenna, Italy, with one-dimensional space-group symmetry (photographs by the authors).

Biological macromolecules are often distinguished by their helical structures to which one-dimensional space-group symmetries are applicable. Figure 8-15a shows Linus Pauling s sketch of a polypeptide chain, which he drew while he was looking for the structure of alpha-keratin. When he decided to fold the paper, he arrived at the alpha-helix. The solution may have come in a sudden moment,... 

I. Hargittai, G. Lengyel, The Seven One-Dimensional Space-Group Symmetries Illustrated by Hungarian Folk Needlework. ./ Chem. Educ. 1984, 61, 1033-1034. 

I. Hargittai and G. Lengyel, The Seventeen Two-Dimensional Space-Group Symmetries in Hungarian Needlework. J. Chem. Educ. 1985, 62, 35-36. 

Figure 1.41. The three-dimensional view of the crystal structure of tmdW %02o (vanadium oxide layers and nitrogen atoms from tma molecules) with added symmetry elements. The space group symmetry of the material is C2/m.

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