M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 10
Текст из файла (страница 10)
SimpleColumnar (C)2. Combination1. Simple2. CombinationPlane (P)1. Regular developed in plane2. Irregular number of branchesa. Elementary needleb. Bundle of needlesa. Pyramidb. Bulletc. Hexagonala. Bulletsb. Columnsa. Simple plateb. Branches in sector formc. Plate with simple extensionsd. Broad branchese. Simple stellar formf. Ordinary dendritic formg. Fernlikeh. Stellar form with plates at endsi. Plate with dendritic extensions2.3. Combined SymmetriesMain groupsTable 2-1. Part of Nakaya’s General Classification of Snow CrystalsaSubgroupsTypesa.
Three-branchedb. Four-branchedc. Others5152Main groupsTable 2-1. (Continued)Subgroups3. Twelve branchesColumn/plane combinations (CP)4. Malformed manyvarieties5. Spatial assemblage ofplane branches1. Column with plane at bothends2. Bullets with platesTypesa. Fernlikeb. Broad branchesa. Spatial hexagonalb. Radiatinga. Column with platesb. Column with dendritesa. Bullets with platesb. Bullets with dendrites1. Ice2. Rimed3.
MiscellaneousaAfter U. Nakaya, Snow (in Japanese), Iwanami-Shoten Publ. Co., Tokyo, 1938 (latest printing, 1987).2 Simple and Combined Symmetries3. IrregularColumnar with extended side planes (S)Irregular snow particles (I)2.4. Inversion53Figure 2-27. From Nakaya’s general classification of snow crystals [33].2.4. InversionWhat is the symmetry of the 1,2-dibromo-1,2-dichloro-ethanemolecule as shown in Figure 2-28? There is no symmetry plane and norotation axis. However, any two atoms of the same kind are related bya line connecting them and going through the midpoint of the centralbond. This midpoint is the only symmetry element of this moleculeand it is called the symmetry center or inversion point.
The application of this symmetry element interchanges the atoms, or more generally, any two points located at the same distance from the center alongthe line going through the center. This interchange is called inversion.The notation of inversion symmetry is i.An inversion may also be represented as the consecutive application of two simple symmetry elements, namely a twofold rotation and mirror-reflection, or vice versa. For the molecule inFigure 2-28, this could be described, for example, in the followingway: (a) rotate the molecule by 180◦ about the C–C bond as therotation axis and (b) apply a symmetry plane perpendicular to andbisecting the C–C bond; or (a) apply a twofold rotation axis perpendicular to the ClCCCl plane and going through the midpoint of theC–C bond and then (b) apply a mirror plane coinciding with theClCCCl plane.
These operations are indicated in Figure 2-28 and in542 Simple and Combined SymmetriesFigure 2-28. The 1,2-dibromo-1,2-dichloroethane molecule. Its center of symmetryis the midpoint of the C–C bond. An inversion is equivalent to the consecutive application of twofold rotation and reflection.both examples the results are invariant to the order in which the twooperations are performed.The sphere is a highly symmetrical object which possesses a centerof symmetry. Conjugate locations on the surface of a sphere arerelated by an inversion through the center of symmetry.
The geographical consequences of such an inversion are emphasized in a newspaper article on New Zealand by the famous journalist, the lateJames Reston in his Letter from Wellington. Search for End of theRainbow [36]:Nothing is quite the same here. Summer is fromDecember to March. It is warmer in the NorthIsland and colder in the South Island. The peopledrive on the left rather than on the right. Even thesky is different—dark blue velvet with stars of theSouthern Cross—and the fish love the hooks.Madrid, Spain, corresponds approximately to Wellington, NewZealand, by inversion.2.5. Singular Point and Translational Symmetry55The notation of the symmetry center or inversion center is 1̄while the corresponding combined application of twofold rotation and mirror-reflection may also be considered to be just onesymmetry transformation.
The symmetry element is called a mirrorrotation symmetry axis of the second order, or twofold mirror-rotationsymmetry axis and it is labeled 2̃. Thus, 1̄ ≡ 2̃.The twofold mirror-rotation axis is the simplest among the mirrorrotation axes. There are also axes of fourfold mirror-rotation, sixfoldmirror-rotation, and so on. Generally speaking, a 2n-fold mirrorrotation axis consists of the following operations: a rotation by(360/2n)◦ and a reflection through the plane perpendicular to the rotation axis.
The symmetry of the snowflake involves this type of mirrorrotation axis. The snowflake obviously has a center of symmetry. Thesymmetry class m·6:m contains a center of symmetry at the intersection of the six-fold rotation axis and the perpendicular symmetryplane. In general, for all m·n:m symmetry classes with n even, thepoint of intersection of the n-fold rotation axis and the perpendicular symmetry plane is also a center of symmetry. When n is oddin an m·n:m symmetry class, however, there is no center of symmetrypresent.2.5. Singular Point and Translational SymmetryThe midpoint of a square is unique, there is no other point equivalentto it (Figure 2-29); it is called a singular point.
A corner of the samesquare is not singular, the symmetry transformations of the squarereproduce it, and there are altogether four equivalent corner points ofthe square. The same argument applies if the point happens to be onone of the symmetry axes of the square.
An arbitrarily chosen point ina square will have 7 other equivalent points because of the symmetrytransformations of the square, so altogether there will be eight equivalent points. In an asymmetric figure each point is singular and themultiplicity of each point is one.The symmetry classes characterizing figures or objects which haveat least one singular point are called point groups. The center ofthe circular pattern of the pavement in Figure 2-30a is a singularpoint. Another pattern is displayed by the pavement in Figure 230b, consisting of identical arcs. If it is supposed that this pavement562 Simple and Combined SymmetriesFigure 2-29.
Singular point and the equivalence of points in a square.is a fragment of an infinitely large one, there is no singular pointin it. Assuming an infinite extent for this pavement pattern isnatural because of its periodicity. The absence of a singular pointleads to regularity expressed in infinite repetition which characterizes translational symmetry.
This kind of symmetry precludesthe presence of singular points though does not preclude the presence of a singular line or plane. The symmetry classes characterizing entities with translational symmetry are called space groups.One-dimensional space groups describe the symmetries involvinginfinite repetition or periodicity in one direction, two-dimensionalspace groups those involving periodicity in two directions andthree-dimensional space groups describe the symmetry classes whenperiodicity is present in all three directions. Figure 2-31 andTable 2-2 summarize the possible cases considering dimensionalityand periodicity.(a)(b)Figure 2-30.
Pavements in L’Aquila, Italy (photographs by the authors). Thesystem of concentric circles (a) Has point-group symmetry and the pattern of arcs;(b) If extended to infinity, has space-group symmetry.2.6. Polarity57Figure 2-31. Dimensionality and periodicity in point groups and space groups. Thisfigure is consistent with Table 2-2.2.6. PolarityA line is polar if its two directions can be distinguished and a plane ispolar if its two surfaces are not equivalent. This definition of polarityhas, of course, nothing to do with charge separation.
A polar line hasa “head” and a “tail” and a polar plane has a “front” and a “back.” Avertical line on the surface of the Earth is polar with respect to gravityand a sheet of paper with one of its sides painted is polar with respectto its color.An axis is polar if its two ends are not brought into coincidence bythe symmetry transformations of the symmetry group of its figure. Ananalogous definition applies to the two sides of a polar plane.If a symmetry group includes a center of symmetry, polarity isexcluded because in a centrosymmetric figure a directed line orsegment of a face changes direction by the inversion.
In the case of theabsence of a center of symmetry, there will be at least one directed lineor face which is not accompanied by parallel counterparts reversed indirection.The significance of polar axes can be demonstrated, for example,in crystal morphology. A few examples will be mentioned herefollowing Curtin and Paul’s review of the chemical consequences ofthe polar axis in organic crystal chemistry [37]. Figure 2-32a shows acentrosymmetric acetanilide crystal. The faces occur in parallel pairsand the crystal is non-polar.
On the other hand, the p-chloroacetanilidecrystal shown in Figure 2-32b is noncentrosymmetric and some of the58aTable 2-2. Dimensionality (m) and Periodicity (n) of Symmetry Groups G mn after EngelhardtPeriodicityn=0n=1n=2n=3Dimensionalityno periodicityperiodicity inperiodicity inperiodicity inone directiontwo directionsthree directionsG 00m = 1, One-dimensionalG 10G 11m = 2, Two-dimensionalG 20G 21G 22G 31G 32m = 3, Three-dimensionalG 30aW. Engelhardt, Matematischer Unterricht 1963, 9(2), 49.G 332 Simple and Combined Symmetriesm = 0, Dimensionless2.6.
Polarity59Figure 2-32. Two crystals from Groth’s Chemische Kristallographie [38].(a) Centrosymmetric rhombic bipyramidal acetanilide; (b) Noncentrosymmetricrhombic pyramidal p-chloroacetanilide.faces occur without parallel ones at the opposite end of the crystal.This crystal has a polar axis parallel to its long direction.The morphological symmetry differences between the acetanilideand p-chloroacetanilide crystals originate from their internalstructures.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
