M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 59
Текст из файла (страница 59)
. . =C=C=C=C=C= . . .. . . −C≡C−C≡C−C≡ . . .represent one-dimensional patterns. The elementary translation oridentity period is the length of the carbon–carbon double bond inthe uniformly bonded chain while it is the sum of the lengths of thetwo different bonds in the chain consisting of alternating bonds. Asthe chain of molecules extends along the axis of the carbon–carbonbonds, this axis can be called the translation axis. The carbon-carbonaxis is a singular axis, and it is not polar as the two directions alongthe chain are equivalent. Earlier we have seen the binary array ...A B A B A B ... in a crystal.
The unequal spacings between the atomA and the two adjacent atoms B produced a polar axis (cf., Section 2.6on polarity).8.2. One-Sided BandsFigure 8-4 presents two band decorations; one of them has a polaraxis while the other has a nonpolar axis. An important feature of thesepatterns is that they have a polar singular plane, which is the plane ofthe drawing. This plane is left unchanged during the translation. Suchtwo-dimensional patterns with periodicity in one direction are calledone-sided bands [5].There are altogether seven symmetry classes of one-sided bands.They are illustrated in Figure 8-5 for a suitable motif, a black triangle.A brief characterization of the seven classes is given here, followingtheir notation:l.
(a). The only symmetry element is the translation axis. The translation period is the distance between two identical points of theconsecutive black triangles.3768 Space-Group Symmetries(a)(b)Figure 8-4. Polar (a) And nonpolar; (b) Decorations of Byzantine mosaics fromRavenna, Italy, with one-dimensional space-group symmetry (photographs by theauthors).Figure 8-5. The seven symmetry classes of one-sided bands.8.2. One-Sided Bands377Figure 8-6.
Illustration of the seven symmetry classes of one-sided bands byHungarian needlework [8]. The numbering corresponds to that of Figure 8-5.(1) Edge decoration of table cover from Kalocsa, Southern Hungary; (2) Pillow-enddecoration from Tolna county, Southwest Hungary; (3) Decoration patched ontoa long embroidered felt coat of Hungarian shepherds in Bihar county, EasternHungary; (4) Embroidered edge-decoration of bed-sheet from the 18th century.Note the deviations from the described symmetry in the lower stripes of the pattern;(5) Decoration of shirt-front from Karád, Southwest Hungary; (6) Pillow decoration pattern from Torocko (Rimetea), Transylvania, Romania; (7) Grape-leaf patternfrom the territory east of the river Tisza.3788 Space-Group Symmetries2.
(a)·ã. Here the symmetry element is a glide-reflection plane (ã).The black triangle comes into coincidence with itself after translation through half of the translation period (a/2) and reflection inthe plane perpendicular to the plane of the drawing.3. (a):2. There is translation and twofold rotation axis in this class.The twofold rotation axis is perpendicular to the plane of the onesided band.4. (a):m.
The translation is achieved by transverse symmetry planesin this pattern.5. (a)·m. Here the translation axis is combined with a longitudinalsymmetry plane.6. (a)·ã:m. Combination of glide-reflection plane with transversesymmetry planes characterizes this class. These elements generatenew ones such as twofold rotation. Consequently, there are alternative descriptions of this symmetry class. One of them isby combining twofold rotation with glide reflection—the corresponding notation is (a):2·ã. Another is by combining twofold rotation with transverse reflection for which the notation is (a):2:m.7. (a)·m:m.
This pattern has the highest symmetry achieved by acombination of transverse and longitudinal symmetry planes. Inthis description the twofold axes perpendicular to the plane of thedrawing are generated by the other symmetry elements. An alternative description is (a):2·m.The seven one-dimensional symmetry classes for the one-sidedbands are illustrated by patterns of Hungarian needlework inFigure 8-6 [6]. This kind of needlework is a real “one-sided band.”Figure 8-7 presents a scheme to facilitate establishing the symmetryclass of one-sided bands [7].8.3. Two-Sided BandsIf the singular plane of a band is not polar, the band is two-sided.The one-sided bands are a special case of the two-sided bands.Figure 8-8a shows a one-sided band generated by translation of a leafmotif.
Figure 8-8b depicts a two-sided band characterized by a glidereflection plane. There is a translation by half of the translation periodand then a reflection in the plane of the drawing. The leaf patterns areparalleled by patterns of the triangle in Figure 8-8. A new symmetry8.3. Two-Sided Bands379Figure 8-7. Scheme for establishing the symmetry of a one-sided band afterCrowe [9].element is illustrated in Figure 8-8c, the twofold (or second-order)screw axis, 21 .
The corresponding transformation is a translation byhalf the translation period and a 180◦ rotation around the translationaxis. Bands have altogether 31 symmetry classes, including the sevenone-sided bands. Table 8-1 gives two different notations for the sevenone-sided band classes and for the two two-sided ones shown inFigure 8-8b and c as illustrations.The so-called coordinate, or international, notation refers to themutual orientation of the coordinate axes and symmetry elements[11]. The notation always starts with the letter p, referring to the translation group.
Axis a is directed along the band, axis b lies in the planeof the drawing, and axis c is perpendicular to this plane. The first,second, and third positions of the symbol after the letter p indicatethe mutual orientation of the symmetry elements with respect to thecoordinate axes. If no rotation axis or normal of a symmetry planecoincides with a coordinate axis, the number 1 is placed in the corresponding position in the symbol. The coincidence of a rotation axis,3808 Space-Group SymmetriesFigure 8-8.
One-sided and two-sided bands using a one-sided leaf motif and a onesided black triangle motif. (a) One-sided bands generated by simple translation.The plane of the drawing is a polar singular plane; (b) Two-sided bands generatedfrom the same motifs as before, by introducing a glide-reflection plane. The singularplane in the plane of the drawing is no longer polar. The glide-reflection plane coinciding with the plane of the drawing is labeled ã11 [10]. Note that the two sides ofthe leaves are of different color (black and white); (c) Two-sided bands generatedfrom the same motifs as before, by introducing a screw axis of the second order,21 . Used with permission from Nauka Publishers, Moscow; (d) Pavement in Erice,Italy as an example of a one-sided band; (e) Lamps on the Alexander III Bridge inParis (photographs by the authors).8.4.
Rods, Spirals, and Similarity Symmetry381Table 8-1. Examples of Notations of Band SymmetriesNoncoordinate notationCoordinate(international) notation(a)(a)·ã(a):2(a):m(a)·m(a)·2:m ≡(a):2·ã ≡(a): ã·m(a)·m :m ≡(a):2·m(a)·21(a)· ã11p1p1a1p112pm11p1m1pma2pmm2p21 11p11a2 or 21 , or the normal of a symmetry plane, m or ã, with one of thecoordinate axes is indicated by placing the symbol of this element inthe corresponding position in the notation.8.4. Rods, Spirals, and Similarity SymmetryThe “infinite” carbide molecule is, of course, of finite width.
It isindeed a three-dimensional construction with periodicity in one direction only. Thus, it has one-dimensional space-group symmetry (G 31 ).It is like an infinitely long rod. For a rod, the axis is a singular axis,and it has no singular plane. All kinds of symmetry axes may coincidewith the axis of the rod, such as a translation axis, a simple rotationaxis, or a screw-rotation axis. Of course, these symmetry elements,except the simple rotation axis, may characterize the rod only if itexpands to infinity.
As regards symmetry, a tube, a screw, or variousrays are as much rods as are the stems of plants, vectors, or spiral stairways. A conspicuous example is the nanotubes and nanorods that arefinding broadening applications in current science and technology dueto their ability of providing novel mechanical, electrical, and thermalproperties, and other uses, such as hydrogen storage [12]. A computerdrawing and an analogy in artistic expression are shown in Figure 8-9.The symmetries of the structural diversity in the nanoworld have beendiscussed extensively [13]. Many of our examples throughout thisbook would also qualify for belonging to the nanoworld.We have stressed above the necessity of considering our objectsextending to infinity at least in one direction in order to qualify3828 Space-Group SymmetriesFigure 8-9.
Nanotubes; (a) Computer graphic by Zoltán Varga, Budapest; (b) Decoration in the Royal Palace in Bangkok, Thailand (photograph by the authors).for applying space-group symmetry description to them. However,real objects are, of course, not infinite. For symmetry considerations,it may be convenient to look only at some portions of the whole,where the ends are not yet in sight, and extend them in thought toinfinity.
A portion of an iron chain and a chain of beryllium dichloride in the crystal are shown in Figure 8-10. Translation from unit tounit is accompanied by a 90◦ rotation around the translation axis. Aportion of a spiral stairway displaying screw-axis symmetry is shownin Figure 8-11a. The imaginary impossible stairway of Figure 8-11bindeed seems to go on forever.A screw axis brings the infinite rod into coincidence with itself aftera translation through a distance t accompanied by a rotation throughan angle ␣.
The screw axis is of the order n = 360◦ /␣. It is a specialcase when n is an integer. The iron chain and the beryllium dichloridechain have a fourfold (or fourth-order) screw axis, 42 . Their overallsymmetry is (a)·m·42 :m. For the screw axis of the second order, thedirection of the rotation is immaterial. Other screw axes may be either8.4. Rods, Spirals, and Similarity Symmetry383Figure 8-10. Rods with 42 screw-axis. (a) Iron-chain in the vicinity of the RoyalPalace in Madrid (photograph by the authors); (b) Beryllium dichloride chain in thecrystal.Figure 8-11. (a) Spiral staircase (and its shadow) in Cambridge, England,displaying screw-axis symmetry (photograph by the authors); (b) “Impossiblestairway” (drawn after a movie poster advertising Glück im Hinterhaus).3848 Space-Group Symmetriesleft-handed or right-handed.
The pair of left-handed and right-handedhelices of Figure 2-39 is an example.The scattered leaf arrangement around the stems of many plantsis a beautiful occurrence of screw-axis symmetry in nature. Thestem of Plantago media shown in Figure 8-12a certainly does notextend to infinity. It has been suggested, however, that for plants theplant/seed/plant/seed ... infinite sequence, at least in time, providesenough justification to apply space groups in their symmetry description. Let us consider now the relative positions of the leaves aroundthe stem of Plantago media.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
