M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 62
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The latter are called the crystallographic axes. The angles between the edges of the three-dimensionalunit cell are ␣, , and ␥, but only ␥ is needed for the plane lattice.Figure 8-27 shows three planar networks based on the same planelattice. Two and only two lines intersect in each point of all threenetworks. Accordingly, the parallelograms of all three networks havethe same area.
All of them are unit cells, in fact, primitive cells. Eachof these parallelograms is determined by two sides a and b, and theangle ␥ between them. These are called the cell parameters.The general plane lattice (a) shown in Figure 8-28 is called a parallelogram lattice. The other four plane lattices of Figure 8-28 arespecial cases of the general lattice. The rectangular lattice (b) has aprimitive cell with unequal sides. The so-called diamond lattice (c)has a unit cell with equal sides. A special case of the diamond lattice iswhen the angle between the equal sides of the unit cell is 120 degrees,and this lattice (d) is then called rhombic, or triangular since the short8.5. Two-Dimensional Space Groups399Figure 8-25.
Scheme for establishing the symmetry of planar networks afterCrowe [37].4008 Space-Group SymmetriesFigure 8-26. (a) Plane lattice defined by two non-collinear translations; (b) Illustration of primitive and unit cells on a plane lattice (after Azaroff, [38] used withpermission from McGraw-Hill and L. V. Azaroff).Figure 8-27.
Different networks based on the same plane lattice.Figure 8-28. The five unique plane lattices (a–e, see text).8.5. Two-Dimensional Space Groups401Table 8-2. Symmetries of the Five Unique Plane LatticesSpace groupLatticeNoncoordinatenotationCoordinate(international) notationa) Parallelogram latticeb) Rectangular latticec) Diamond latticed) Hexagonal or triangular latticee) Square lattice(b/a):2(b:a):2·m(a/a):2·m(a/a):6·m(a:a):4·mp2pmm2cmm2p6mmp4mmcell diagonal divides the unit cell into two equilateral triangles. Thislattice may also be considered as having hexagonal symmetry.
Finallythere is the square lattice (e).The five unique plane lattices were described above under theassumption that the lattice points themselves have the highest possiblesymmetry. 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 planelattices, 17 two-dimensional space groups can be produced. In suchtreatment, severe limitations are imposed on the possible point groupsthat may be combined with lattices to produce space groups.
Somesymmetry elements, such as the fivefold rotation axis, are not compatible with translational symmetry and from this, forbidden symmetriesfollow in classical crystallography. This and the lifting of such limitations in modern crystallography will be examined in Chapter 9.8.5.1. Simple NetworksThe 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 ␥. The equal motifs repeated bythe translations may occur in the following four different versions(strating from the upper left and clockwise): they may be completelyseparated from one another; they may consist of disconnected parts;they may intersect each other and; finally, they may fill the entire planewithout gaps and overlaps.
Of course, such variations are possible forany of the more complicated two-dimensional space groups as well.4028 Space-Group SymmetriesFigure 8-29. The simplest two-dimensional space group in four variations.Especially intriguing are those variations which cover the wholeavailable surface without gaps. Of the regular polygons, this ispossible only with the equilateral triangle, the square, and theregular hexagon. For the latter, characteristic examples are shown inFigure 8-30 including some in which the hexagons have only approximately regular shapes.Planar motifs of irregular shape can be used in infinite numbersto construct planar patterns covering the entire available surface.M.
C. Escher is famous—among others— for his periodic drawingswhich fill the plane without gaps and overlaps [39]. Their symmetryaspects have been discussed in detail by crystallographer CarolineMacGillavry. She worked closely with the artist to create a set ofperiodic drawings for instructions in crystallography. The patterns inFigure 8-31 are from her book [40]. One of them has p1 symmetry.The unit cell is the combination of a fish and a boat.
The repetition ofthe flies, butterflies, falcons, and bats in the Escher drawing, also inFigure 8-31, is accomplished by mirror planes. The two-dimensionalspace group is pmm and the mirror planes are indicated separately asthe borders of the primitive cell.Canadian crystallographer François Brisse has designed a seriesof two-dimensional space-group drawings related to Canada anddedicated to the XIIth Congress of the International Union of Crystallography (IUCr), Ottawa, 1981 [44]. One of them is shown inFigure 8-32.
The symbol of the XIIth IUCr Congress was a unit offour stylized maple leaves related by fourfold rotation, and this is theunit cell of the drawing. The maple leaf, Canada’s symbol, is shown8.5. Two-Dimensional Space Groups403(a)(b)(c)(d)Figure 8-30. Networks of regular hexagons covering the surface without gapsor overlaps. (a) Honeycomb. Photograph by and courtesy of Pál Zoltán Örösi,Budapest; (b) Moth compound eye (magnified x 2000). Photograph by and courtesyof J.
Morral, Storrs, Connecticut; (c) Computer-generated graphene-sheet model;(d) Graphite-sheet-like window fence at Topkapi Sarayi in Istanbul (photograph bythe authors).4048 Space-Group SymmetriesFigure 8-31. Top: Caroline H. McGillavry [41] and one of M. C. Escher’s periodicdrawings, of fish and boats with space group p1 [42] (reproduced with permissionfrom the International Union of Crystallography). The unit cell consists of one fishand one boat; Bottom: Another of Escher’s periodic drawings, of flies, butterflies,falcons and bats [43] (reproduced with permission from the International Union ofCrystallography).
The primitive cell is framed by the square whose sides are partsof the mirror planes in the periodic drawing.in a more natural appearance on a stamp. Disregarding the coloring,the two-dimensional space group of the pattern is p4gm, the same asthat of the Portugese tile decoration in Figure 8-32.Khudu Mamedov was another crystallographer who created periodic drawings. His purpose was to immortalize ancient patterns foundin his native Azerbaijan. He and his pupils published a remarkable little book Decorations Remember [46]. The space group of thedrawing Unity of Figure 8-33 is pl, with the basic motif consisting ofan old man and a young warrior.
The repetition of the uniform shapestruly satisfies the requirement of it being a two-dimensional spacegroup. A closer look, however, reveals distinct individuality of facialexpressions, especially for the old men. This diversity of the individuals in the uniformity of the space goup is refreshing. Mamedov’sformer associates recently published a renewed version [47] of the8.5. Two-Dimensional Space Groups405(a)(b)(c)(d)Figure 8-32. (a) Periodic drawing by François Brisse [45] (reproduced with permission); (b) The basic motif is a stylized maple leaf and the unit cell displays fourfoldrotation; it was the symbol of the XIIth Congress of the International Union ofCrystallography, Ottawa, 1981; (c) Canadian stamp with the maple leaf; and (d) APortugese tile decoration (photograph by the authors). The patterns in the tile and inBrisse’s drawing are related by a quarter of a circle rotation.old book.
The volume is luxuriously produced and many new patternshave been added. However, this time computer graphics replaced theoriginal hand-drawings and the diversity of individuality is gone inthe new version of Unity [48].The mathematician George Pólya prepared a set of drawings for the17 two-dimensional space groups with patterns that completely fill thesurface without gaps or overlaps (Figure 8-34) [50]. A comprehensive and in-depth treatise of tilings and patterns has been published byGrünbaum and Shephard [51].4068 Space-Group SymmetriesFigure 8-33. Khudu Mamedov’s periodic drawing Unity [49].8.5.2.
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