Two-Dimensional Space Groups

Lattice Noncoordinate notation Coordinate (international) notation 

The five unique plane lattices were described above under the assumption that the lattice points themselves have the highest possible symmetry. In this case these five unique lattices will have the symmetries listed in Table 8-2. 

When the point-group symmetries are combined with the plane lattices, 17 two-dimensional space groups can be produced. In such treatment, severe limitations are imposed on the possible point groups that may be combined with lattices to produce space groups. Some symmetry elements, such as the fivefold rotation axis, are not compatible with translational symmetry and from this, forbidden symmetries follow in classical crystallography. This and the lifting of such limitations in modem crystallography will be examined in Chapter 9. 

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Electron-diffraction patterns were recorded for the dry and frozen-hydrate fonns of pustulan from Pustulan papullosa. The frozen-hydrate form crystallizes in a rectangular unit-cell, with a = 2.44 nm and b = 1.77 nm. The chain-axis repeat was not determined. Systematic absences led to the two-dimensional space-group Pgg. Dehydration results in a reversible, partial collapse of the crystals. 

Figure 11.7 The 17 two-dimensional space groups continuation on page 360. [Adapted from I. Hargittai and G. Lengyel, J. Chem. Educ., 1985,62, 35.]...

Mackay, A. L. 1986 Two-dimensional space groups with sevenfold symmetry. Acta crystallogr. A 42, 55-56. 

If the 10 point groups allowed are arranged in nonredundant patterns allowed by the five 2D Bravais lattices, 17 unique two-dimensional space groups, called plane groups, are obtained (Fedorov, 1891a). Surface structures are usually referred to the underlying bulk crystal structure. For example, translation between lattice points on the crystal lattice plane beneath and parallel to the surface (termed the substrate) can be described by an equation identical to Eq 1.10 ... 

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 8-1. Planar decoration with two-dimensional space group after Budden [4], (a) The decoration (b) Symmetry elements of the pattern (c) Some of the glide reflection planes and their effects in the pattern.

The simplest two-dimensional space group is represented in four variations in Figure 8-29. This space group does not impose any restrictions on the parameters a, b, and y. The equal motifs repeated by the translations may occur in the following four different versions (strafing from the upper left and clockwise) they may be completely separated from one another they may consist of disconnected parts they may intersect each other and finally, they may fill the entire plane without gaps and overlaps. Of course, such variations are possible for any of the more complicated two-dimensional space groups as well. 

Figure 8-29. The simplest two-dimensional space group in four variations.

The mathematician George Polya prepared a set of drawings for the 17 two-dimensional space groups with patterns that completely fill the surface without gaps or overlaps (Figure 8-34) [50], A comprehensive and in-depth treatise of tilings and patterns has been published by Griinbaum and Shephard [51],... 

There are 17 two-dimensional space groups arising from five Bravais nets associated with translation over a surface [28]. A Ceo molecule adsorbed onto a surface will therefore be subject to a local symmetry belonging to one of ten possible site symmetries Cev, Ce, C4V, C4, Csv, C3, Cxv, C2, Q, and Ci. None of these site groups support triply degenerate irreps and so the Ti LUMO will be split whenever Ceo... 

Canadian crystallographer Frangois Brisse has designed a series of two-dimensional space-group drawings related to Canada [8-23]. The series was... 

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