M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 75
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The crystal arises from the local interactionsbetween individual atoms. He furthermore said that a regular structureshould mean a structure generated by simple rules, but the list of rulesconsidered to be simple and “permissible” should be extended. Theserules would not necessarily form groups. Furthermore, Mackay foundthe formalism of the International Tables for X-Ray Crystallography4849 Crystalsto be too restrictive and quoted Bell, the historian of mathematics, onthe rigidity of the Euclidean geometry formalism: “The cowboys havea way of trussing up a steer or a pugnacious bronco which fixes thebrute so that it can neither move nor think.
This is the hog-tie and it iswhat Euclid did to geometry” [122].Mackay had a long list covering a whole range of transitions fromclassical crystallographic concepts to what is termed the modernscience of structure at the atomic level. This list is reproduced inTable 9-6. There is resonance of several of Mackay’s ideas withother directions in modern chemistry, where the non-classical, theTable 9-6. Mackay’s List of Transition from the Classical Concepts of Crystallography to the Modern Concepts of a Science of StructureaClassical conceptsModern conceptsAbsolute identity ofcomponentsAbsolute identity of theenvironment of each unitOperations of infinite range“Euclidean” space elements(plane sheets, straight lines)Unique dominant minimum infree energy configurationspaceInfinite number of units.CrystalsAssembly by incrementalgrowth (one unit at a time)Substitution and nonstoichiometryQuasi-identity and quasiequivalenceLocal elements of symmetry of finite rangeCurved space elements.
Membranes, micelles,helices. Higher structures by curvature oflower structuresOne of many quasi-equivalent states;metastability recording arbitrary information(pathway); progressive segregation andspecialization of information structureFinite numbers of units. Clusters; “crystalloids”Assembly by intervention of other components(“crystalase” enzyme). Information-controlledassembly. Hierarchic assemblySingle level of organizationHierarchy of levels of organization.
Small span(with large span of level)of each levelRepetition according toRepetition according to program. Cellularsymmetry operationsautomataCrystallographic symmetryGeneral symmetry operations (equal “programoperationsstatements”)Assembly by a single pathway Assembly by branched lines in configurationin configuration spacespace. Bifurcations guided by “information”,i.e., low-energy events of the hierarchy belowaA. L. Mackay, “De Niva Quinquangula: On the pentagonal snowflake.” Kritallografiya (Sov. Phys.
Crystallogr.) 1981, 26, 910–919 (517–522).9.7. Beyond the Perfect System485non-stoichiometrical, the non-stable, the non-regular, the non-usual,the non-expected are gaining importance. For crystallography it seemsto be a long way yet to perform all the suggested transitions but thebreakthroughs so far have been fascinating and promising.
Impressive progress has been reported in the studies of liquids, amorphousmaterials, metallic alloys as regards the description of their structuralregularities.Liquid structures, for example, cannot be characterized by any ofthe 230 three-dimensional space groups and yet it is unacceptable toconsider them as possessing no symmetry whatsoever. Bernal notedpresciently that the major structural distinction between liquids andcrystalline solids is the absence of long-range order in the former[123]. A generalized description should also characterize liquid structures and colloids, as well as the structures of amorphous substances.It should also account for the greater variations in their physicalproperties as compared with those of the crystalline solids. Bernal’sideas have greatly encouraged further studies in this field whichis usually called generalized crystallography.
Referring to Bernal’sgeometrical theory of liquids, Belov noted in Bernal’s obituary: “...his last enthusiasm was for the laws of lawlessness” [124].The paradoxical incompleteness and inadequacy of perfectsymmetry, compared with less-than-perfect symmetry, are wellexpressed in a short poem entitled “Gift to a Jade” by the Englishpoet Anna Wickham [125]:For love he offered me his perfect world.This world was so constricted and so smallIt had no loveliness at all,And I flung back the little silly ball.At that cold moralist I hotly hurledHis perfect, pure, symmetrical, small world.The structures intermediate between the perfect order of crystalsand the complete disorder of gases are not merely rare exceptions.On the contrary, they are often found in substances which are verycommon in our environment or are widely used in various technologies. They include plastics, textiles, and rubber, among others.
Glass isan especially fascinating material whose amorphous atomic networkwas discussed by Zachariasen a long time ago, but his teachings arestill considered to be valid [126]. Figure 9-61 shows Zachariasen’s4869 Crystalstwo-dimensional representations of crystalline and amorphous structures of compounds of the same composition, A2 O3 . Guinier discussedthe intermediate structures between the extreme types of perfect orderand disorder, and he declared a quarter of a century ago that “Thedistinction into two well separated classes is an oversimplificationof the complex reality” [127].
He envisaged a continuous passagefrom the exact scheme of neighboring atoms in a crystal to the veryflexible arrangement in an amorphous body. The term paracrystal wascoined for domains with approximate long-distance order in the rangeof a few tens to a few hundreds of atomic diameters. Today we callthe discipline dealing with such structures nanoscience.
Figure 9-62provides a schematic representation of a paracrystal lattice with oneatom per unit cell. The blackened areas indicate the regions wherean atom is likely to be found around the atom fixed at the origin. Atgreater distances the neighboring sites first overlap, then merge, andthus eventually the long-range order vanishes completely. Guinier’steachings have further enriched recent studies of complex intermetallic structures [128].(a)(b)Figure 9-61.
Zachariasen’s [129] representation of the atomic arrangement in thecrystal (a) and glass (b) of A2 O3 .One of the most fascinating examples of non-periodic regulararrangements was described by Mackay in a paper titled—in obviousreference to Kepler’s treatise about the six-cornered snowflake—Denive quinquangula (on the pentagonal snowflake) [131].
A regular but“non-crystalline” structure is built from regular pentagons in a plane.It starts with a regular pentagon of given size (zeroth-order pentagon).Six of these pentagons are combined to make a larger one (first-order9.7. Beyond the Perfect System487Figure 9-62. Paracrystal lattice with one atom per unit cell adapted from Guinier[130]. Used with permission.pentagon). As is seen in Figure 9-63, the resulting triangular gaps arecovered by pieces from cutting up a seventh zeroth-order pentagon.This indeed yields five triangles plus yet another regular pentagon ofthe order –1. This construction is then repeated on an ever increasingscale as indicated in Figure 9-63. The hierarchic packing of pentagonsbuilds up like a pentagonal snowflake.
Attempts of pentagonal tilingof the plane were already quoted in the Introduction, and will bereferred to again in the next Section.Figure 9-63. Tiling with regular pentagons after Mackay [132].Mackay called attention to yet another limitation of the 230 spacegroup system. It covers only those helices that are compatible withthe three-dimensional lattices. All other helices that are finite in oneor two dimensions are excluded. Some important virus structures withicosahedral symmetry are among them. Also, there are very small4889 Crystalsparticles of gold that do not have the usual face-centered cubic latticeof gold. They are actually icosahedral shells.
The most stable configurations contain 55 or 147 atoms of gold. But icosahedral symmetryis not treated in the International Tables and crystals are only definedfor infinite repetition.Crystals are really advantageous for the determination of the structure of molecules. A crystal provides an amplification which multiplies the scattering of the X-rays from a single molecule by thenumber of molecules in the array, perhaps by 1015 . It also minimizesthe damage to individual molecules by the viewing radiation.
Thespots are emphasized in the diffraction pattern and the backgroundis neglected. The damaged molecules transfer their scattering contribution to the background as do those which are not repeated withregular lattice periodicity. However, defects and irregularities may beimportant and may well be lost in present-day sophisticated structuralanalyses.It is perhaps worth pointing out that every crystal is in fact defective, even if its only defect is that it has surfaces. However, if a crystalis only a ten-unit-cell cube, about half of the unit cells lie in thesurface and thus have environments very different from those of theother half.
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