M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 58
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Engl. 711–724.118. Ibid.119. Ibid.120. E. M. Brzostowska, R. Hoffmann, C. A. Parish, “Tuning the BergmanCyclization by Introduction of Metal Fragments at Various Positions of theEnediyne. Metalla-Bergman Cyclizations.” J. Am. Chem. Soc. 2007, 129,4401–4409.Chapter 8Space-Group SymmetriesThe beauty of life is,. . . geometrical beauty. . .J. Desmond Bernal [1]8.1. Expanding to InfinityUp to this point, structures of mostly finite objects have beendiscussed.
Thus, point groups were applicable to their symmetries.A simplified classification of various symmetries was presented inChapter 2 (cf., Figure 2-31 and Table 2-2). Point-group symmetriesare characterized by the lack of periodicity in any direction. However,repetition is a fundamental feature in our world, both in nature and inwhat we create. “Whatever can be done once can always be repeated,”this is how Louise B. Young begins the description of shapes andstructures of nature in the book, The Mystery of Matter [2]. Periodicitymay be introduced by translational symmetry. If periodicity is present,space groups are applicable for the symmetry description.
There is aslight inconsistency here in the terminology. Even a three-dimensionalobject may have point-group symmetry. On the other hand, the socalled dimensionality of the space group is not determined by thedimensionality of the object. Rather, it is determined by its periodicity.The following groups are space-group symmetries where the superscript refers to the dimensionality of the object, and the subscript tothe periodicity.G 11G 21G 22G 31G 32G 33M. Hargittai, I.
Hargittai, Symmetry through the Eyes of a Chemist, 3rd ed.,C Springer Science+Business Media B.V. 2009DOI: 10.1007/978-1-4020-5628-4 8, 3713728 Space-Group SymmetriesObjects or patterns which are periodic in one, two, and threedirections will have one-, two-, and three-dimensional space groups,respectively. The dimensionality of the object/pattern is merely anecessary but not a satisfactory condition for the “dimensionality”of their space groups. We shall first describe a planar pattern afterBudden [3] in order to get the flavor of space-group symmetry.Also, some new symmetry elements will be introduced. Later inthis chapter, the simplest one-dimensional and two-dimensional spacegroups will be presented.
The next Chapter will be devoted to thethree-dimensional space groups which characterize crystal structures.A symmetric pattern expanding to infinity always contains a basicunit, a motif, which is then repeated infinitely throughout the pattern.Figure 8-1a presents a planar decoration. The pattern shown is onlypart of the whole as the latter expands, in principle, to infinity!The pattern is obviously highly symmetrical. Figure 8-1b shows thesystem of mutually perpendicular symmetry planes by solid lines.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 glidereflection planes and their effects in the pattern.8.1.
Expanding to Infinity373Some of the fourfold and twofold rotation axes are also indicated inthis figure. A new symmetry element in our discussion is the glidereflection (called also, glide mirror), which is shown by a dashed line.Some of these glide reflections are indicated separately in Figure 8-1c.A glide-mirror plane is a combination of translation and reflection.It is a symmetry element that can be present in space groups only.The glide-reflection plane involves an infinite sequence of consecutive translations and reflections.
Whereas in a simple canon, there isonly repetition of the tune at certain intervals in time, as shown inFigure 8-2a; Figure 8-2b shows a different canon in which repetitionis combined with reflection. Two further patterns with glide-reflectionsymmetry are given in Figure 8-3. They are also thought to extend toinfinity, at least in our imagination.Simple translation is the most obvious symmetry element of thespace groups. It brings the pattern into congruence with itself overand over again.
The shortest displacement through which this translation brings the pattern into coincidence with itself is the elementary translation or elementary period. Sometimes it is also calledthe identity period. The presence of translation is seen well in thepattern in Figure 8-1. The symmetry analysis of the whole pattern wascalled by Budden the analytical approach. The reverse procedure is theFigure 8-2. Top: Canon illustrating simple repetition; Bottom: repetition combinedwith glide reflection.3748 Space-Group SymmetriesFigure 8-3. Illustrations for glide mirrors. (a) Pillow-edge from Buzsák, Hungary(used with permission from Györgyi Lengyel); (b) Function describing simpleharmonic motion (reflection occurs following translation along the t axis by halfa period, T/2).synthetic approach in which the infinite and often complicated patternis built up from the basic motif. Thus, the pattern of Figure 8-1a maybe built up from a single crochet.
There are several ways to proceed.For example, the crochet may be subjected to simple translation,then reflection, and then transverse reftection. The horizontal arrayobtained this way is a one-dimensional pattern. It can be extended toa two-dimensional pattern by simple translation or by glide reflection.Eventually the complete two-dimensional pattern of Figure 8-1 can bereconstructed. In this synthetic approach, instead of the single crochet,any other motif combined from it could be selected for the start. If thecrosslike motif were chosen, which contains eight of these crochets,then only translations in two directions would be needed to build upthe final pattern.
To learn the most about the structure and symmetries8.2. One-Sided Bands375of a pattern, it is advantageous to select the smallest possible motif forthe start.The one-dimensional space groups are the simplest of the spacegroups. They have periodicity only in one direction. They may referto one-dimensional, two-dimensional, or three-dimensional objects,cf., G 11 , G 21 , and G 31 , of Table 2-2, respectively. The “infinite” carbonchains of the carbide molecules.
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