M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 4
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It used101 IntroductionFigure 1-10. Wolfgang Krätschmer and Donald Huffman re-enacting their firstproduction of buckminsterfullerene, in 1999 in Tucson, Arizona (photograph by theauthors) [27].to be a fundamental dogma of crystallography that fivefold symmetryis a noncrystallographic symmetry. We shall return to this questionin Section 9.3. There have been many attempts to cover the surfacewith regular pentagons without gaps and overlaps and some examples [29] are shown in Figure 1-11. Then, Roger Penrose found twoelements that, by appropriate matching, could tile the surface withlong-range pentagonal symmetry though only in a nonperiodic wayFigure 1-11. Attempts of pentagonal tiling by (a) Dürer (after Crowe) [32];(b) Shubnikov (after Mackay); and (c) Kepler (after Danzer et al.) [33, 34].1 Introduction11Figure 1-12.
Roger Penrose, 2000 (photograph by the authors) and a Penrose tiling.(Figure 1-12) [30]. This pattern was extended by Alan Mackayinto the third dimension and he even produced a simulated diffraction pattern that showed 10-foldedness (Figure 1-13) [31]. It wasabout the same time that Dan Shechtman was experimenting withmetallic phases of various alloys cooled with different speeds andobserved 10-foldedness in an actual electron diffraction experiment(Figure 1-14) for the first time.
The discovery of quasicrystals hasadded new perspective to crystallography and the utilization ofsymmetry considerations.Figure 1-13. Alan L. Mackay, 1982 (photograph by the authors) and his simulated“electron diffraction” pattern of three-dimensional Penrose tiling [31] (photographcourtesy of Alan Mackay, London).121 IntroductionFigure 1-14. Dan Shechtman (photograph by the authors) and his electron diffraction pattern with 10-fold symmetry (photograph courtesy of Dan Shechtman, Haifa).While considering the symmetries of individual molecules orextended structures, we should not lose sight of the place of symmetryconsiderations in the large picture of studying nature.
In this, weturn to the chemical engineer turned theoretical physicist EugeneP. Wigner (1902–1995). He worked out fundamental relationships ofprofound importance for the place of symmetry with respect to thelaws of nature and observable physical phenomena. In this discussion, the term physics stands for physical sciences that include chemistry. In his Nobel lecture, Wigner stated that the symmetry principles“provide a structure and coherence to the laws of nature just as thelaws of nature provide a structure and coherence to a set of events,”the physical phenomena [35].
David J. Gross, a recent Nobel laureatephysicist summarized Wigner’s teachings in a simple diagram [36]:Symmetry principles → Laws of nature → Physical phenomenaWigner (Figure 1-15) was well known for his legendary politenessand modesty that was perceived by some as somewhat forced andartificial. However, there was nothing forced or artificial when heshowed modesty in formulating the principal task of physics andstressed the limitations in its ambitions:Physics does not endeavor to explain nature.
Infact, the great success of physics is due to a restriction of its objectives: it only endeavors to explain1 Introduction13Figure 1-15. Eugene P. Wigner with one of the authors in 1969, in front of the(then) Department of Physics at the University of Texas at Austin (photograph byunknown photographer).the regularities in the behavior of objects. Thisrenunciation of the broader aim, and the specification of the domain for which an explanation can besought, now appears to us an obvious necessity.
Infact, the specification of the explainable may havebeen the greatest discovery of physics so far. . . .The regularities in the phenomena which physical science endeavors to uncover are called thelaws of nature. The name is actually very appropriate. Just as legal laws regulate actions andbehavior under certain conditions but do not tryto regulate all actions and behavior, the laws ofphysics also determine the behavior of its object ofinterest only under certain well-defined conditionsbut leave much freedom otherwise [37].To emphasize the pioneering character of Wigner’s contribution, wequote another Nobel laureate theoretical physicist, Steven Weinberg,according to whom “Wigner realized, earlier than most physicists,the importance of thinking about symmetries as objects of interestin themselves” [38].
Wigner had formulated his views on symmetriesin the 1930s when physicists talked about symmetries in the contextof specific theories of nuclear force. “Wigner was able,” Weinbergcontinues, “to transcend that and he discussed symmetry in a way,which didn’t rely on any particular theory of nuclear force” [39].141 IntroductionAnother Nobel laureate physicist, Gerard ‘t Hooft traced back toWigner the notion that symmetry can break in many different waysand that “Both symmetry and symmetry breaking are examples ofpatterns that we see in Nature” [40].Beyond Wigner’s statements of general validity, the question maybe asked whether “chemical symmetry” differs from any other kind ofsymmetry? Furthermore, whether symmetries in the various branchesof the physical sciences can be distinguished as to their characteristic features and whether they could be hierarchically related? Thesymmetries in the great conservation laws of physics [41] are, ofcourse, present in any chemical system.
The symmetries of moleculesand their reactions are part of the fabric of biological structure. Leftand-right symmetry is so important for living matter that it may bematched only by the importance of “left-and-right” symmetry in theworld of fundamental particles, including the violation of parity, as ifa circle is closed, but that is, of course, an oversimplification.When we stress the importance of symmetry, it is not equivalentwith declaring that everything must be symmetrical.
In particular,when the importance of left-and-right symmetry is stressed, it istheir relationship, rather than their equivalence, that has outstandingsignificance.It has already been referred to that symmetry considerationshave continued their fruitful influence on the progress of contemporary chemistry. This is so for contemporary physics as well. Itis almost surprising that fundamental conclusions with respect tosymmetry could be made even during recent decades.
It was relatedby C. N. Yang that Paul A. Dirac considered Albert Einstein’s mostimportant contributions to physics “his introduction of the conceptthat space and time are symmetrical” [42]. The same Dirac alsohad the prescience to write as early as 1949 that “I do not believethat there is any need for physical laws to be invariant under reflections” [43]. Then, in 1956, Tsung Dao Lee and Chen Ning Yang(Figure 1-16) suggested a set of experiments to show that conservationof parity may be violated in the weak force of nuclear interations [44]. Indeed, three different experiments almost simultaneously confirmed Lee and Yang’s supposition within months [45]. Thediscovery happened “swiftly” during an “exciting period” [46].
It hadlong-range effects and since then broken symmetries have receivedincreasing attention [47]. The term “relates to situations in which1 Introduction15Figure 1-16. Tsung Dao Lee and Chen Ning Yang at the time of the Nobel ceremonies in December 1957 in Stockholm (courtesy of the Manne Siegbahn Institutethrough Ingmar Bergström, Stockholm).symmetries which we expect to hold are valid only approximatelyor fail completely” [48]. The three basic possibilities are incompletesymmetry, symmetry broken by circumstances, and spontaneouslybroken symmetry.“Symmetry is a stunning example of how a rationally derivedmathematical argument can be applied to descriptions of natureand lead to insights of the greatest generality” [49].
But what issymmetry? We may not be able to answer this question satisfactorily, at least not in all its possible aspects. According to thecrystallographer (and symmetrologist) E. S. Fedorov—as quoted byA. V. Shubnikov—“symmetry is the property of geometrical figuresto repeat their parts, or more precisely, their property of coincidingwith their original position when in different positions” [50]. To this,the symmetrologist (and crystallographer) A. V. Shubnikov addedthat while symmetry is a property of geometrical figures, obviously,“material figures” may also have symmetry.
He further stated thatonly parts which are in some sense equal among themselves canbe repeated, and noted the two kinds of equality, to wit, congruentequality and mirror equality. These two equalities are the subsets ofthe metric equality concept of Möbius, according to whom “figuresare equal if the distances between any given points on one figure areequal to the distances between the corresponding points on anotherfigure” [51]. According to the geometer H. S. M. (Donald) Coxeter,161 Introduction“When we say that a figure is ‘symmetrical’ we mean that there isa congruent transformation which leaves it unchanged as a whole,merely permuting its component elements” [52].Symmetry also connotes harmony of proportions, which is a rathervague notion, according to Hermann Weyl [53].
This very vagueness,at the same time, often comes in handy when relating symmetry andchemistry, or generally speaking, whenever the symmetry concept isapplied to real systems. Mislow and Bickart [54] communicated anepistemological note on chirality in which much of what they haveto say about chirality, as this concept is being applied to geometrical figures versus real molecules, solvents, and crystals, is true aboutthe symmetry concept as well. They argue that “it is unreasonableto draw a sharp line between chiral and achiral molecular ensembles: in contrast to the crisp classification of geometric figures, oneis dealing here with a fuzzy borderline distinction, and the qualifying‘operationally’ should be implicitly or explicitly attached to ‘achiral’or ‘racemic’ whenever one uses these terms with reference toobservable properties of a macroscopic sample.” Further, they quoteScriven [55]: “when one deals with natural phenomena, one enters‘a stage in logic in which we recognize the utility of imprecision.’”The human ability to geometrize non-geometrical phenomena greatlyhelps to recognize symmetry even in its “vague” and “fuzzy” variations.
In accordance with this, Weyl referred to Dürer who “considered his canon of the human figure more as a standard from which todeviate than as a standard toward which to strive” [56].Symmetry in its rigorous sense helps us to decide problems quicklyand qualitatively. The answers lack detail, however [57]. On the otherhand, the vagueness and fuzziness of the broader interpretation ofthe symmetry concept allow us to talk about degrees of symmetry,to say that something is more symmetrical than something else.
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