M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 64
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Chem. 535–536.50. G. Pólya, “Über die Analogie der Kristallsymmetrie in der Ebene.” Z. Krist.1924, 60, 278–282.51. B. Grünbaum, G. C. Shephard, Tilings and Patterns, W. H. Freeman, New York,1987.52. Pólya, Z. Kristallogr. 278–282.53. Shubnikov, Koptsik, Symmetry in Science and Art.54. See, H. Giger, “Moirés.” In Symmetry: Unifying Human Understanding.Ed. I. Hargittai. Pergamon Press, Oxford, 1986, pp. 329–361; W. Witschi,“Moirés.” In Symmetry, Unifying Human Understanding, I.
Hargittai, ed., Pergamon Press, New York and Oxford, 1986, pp. 363–378.55. Ibid.56. Giger, Symmetry, Unifying Human Understanding, pp. 329–361.Chapter 9Crystals...all the works of the crystallographers...demonstrate that there is only varietyeverywhere where they suppose uniformity...Georges Leclerc [Comte de] Buffon [1]... But I must speak again about crystals, shapes,colors.
There are crystals as huge as the colonnadeof a cathedral, soft as mould, prickly as thorns;pure, azure, green, like nothing else in the world,fiery black; mathematically exact, complete, likeconstructions by crazy, capricious scientists, orreminiscent of the liver, the heart ... There arecrystal grottos, monstrous bubbles of mineral mass,there is fermentation, fusion, growth of minerals,architecture and engineering art ... Even in humanlife there is a hidden force towards crystallization. Egypt crystallizes in pyramids and obelisks,Greece in columns; the middle ages in vials;London in grinny cubes ... Like secret mathematical flashes of lightning the countless laws ofconstruction penetrate the matter.
To equal natureit is necessary to be mathematically and geometrically exact. Number and phantasy, law andabundance—these are the living, creative strengthsof nature; not to sit under a green tree but to createcrystals and to form ideas, that is what it means tobe at one with nature!M. 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 9, 4134149 CrystalsThese are the words of Karel Čapek the Czech writer after hisvisit to the mineral collection of the British Museum [2]. He addeda drawing (Figure 9-1) to his words to express his humility in front ofthese miracles of nature. Figure 9-2 displays crystal images from themineral collection of Eötvös University, Budapest.Figure 9-1.
Karel Čapek’s drawing after his visit to the mineral collection of theBritish Museum (reproduced with permission) [3].The word crystal comes from the Greek krystallos, meaning clearice. The name originated from the mistaken belief that the beautifultransparent quartz stones found in the Alps were formed from waterat extremely low temperatures. By the 18th century, the name crystalwas applied to other solids that were also bounded by many flat facesand had generally beautiful symmetrical shapes.
Crystals have alsobeen considered to be mystical. A sad angel looks hopelessly at thehuge polyhedron in Albrecht Dürer’s Melancholia, which is a truncated rhombohedron (Figure 9-3). There has been some discussion asto whether Dürer meant a particular mineral by it and, if so, whichone. It was concluded that this polyhedron “is simply an exercise inaccurate draughtsmanship and that the art historians have made ratherheavy weather of its explanation ...
The integral proportions show thatno particular mineral was intended” [4]. The polyhedron was evengiven a name, melancholyhedron and has been claimed to be formed9 Crystals(a)(d)415(b)(c)(e)Figure 9-2. Crystals (a–c, e) Photographs by and courtesy of the associates of theMineral Collection of Eötvös University, Budapest; (d) Photograph by the authorsfrom the collection of Aldo Domenicano and Anna Rita Campanelli, Rome.by twelve arsenic atoms around a nickel atom in the solid-state structure of nickel arsenide [5].Space-group symmetries have played an outstanding role inEscher’s graphic art.
So what he wrote about crystals is also ofinterest [7]:Long before there were men on this globe, allthe crystals grew within the earth’s crust. Thencame a day when, for the very first time, a humanbeing perceived one of these glittering fragments4169 CrystalsFigure 9-3. The polyhedral arsenic skeleton in the NiAs crystal, resembling thepolyhedron in Dürer’s Melancholia [6].of regularity; or maybe he struck against it with hisstone ax; it broke away and fell at his feet; then hepicked it up and gazed at it lying there in his openhand.
And he marveled.There is something breathtaking about the basiclaws of crystals. They are in no sense a discovery ofthe human mind; they just “are”—they exist quiteindependently of us. The most that man can do isto become aware, in a moment of clarity, that theyare there, and take cognizance of them.The symmetry of the shapes of the crystals is their most easilyrecognizable feature. The Russian crystallographer E.
S. FedorovFigure 9-4. Different shapes of sodium chloride crystals as a consequence of theinfluence of impurities.9.1. Basic Laws417remarked that “the crystals glitter with their symmetry.” Obviously,this outer symmetry is a consequence of the inner structure. However,with the same inner structure, crystals may grow into different forms.Besides, under natural conditions, crystals seldom grow into theirwell-known regular forms. Under different conditions, in the presence of different impurities, for example, different forms may grow.Figure 9-4 shows the influence of impurities upon the form of sodiumchloride crystals.9.1. Basic LawsIt has been recognized already in the earliest stages in the history ofcrystallography [8] that the most important characteristic of the outersymmetry of the crystals is not really the form itself but rather twophenomena expressed by two rules.
One is the constancy of the anglesmade by the crystal faces. The other is the law of rational interceptsor the law of rational indices.As early as 1669 the Danish crystallographer Steno made a detailedstudy of ideal and distorted quartz crystals (Figure 9-5). He tracedtheir outlines on paper and found that the corresponding angles ofdifferent sections were always the same regardless of the actual sizesand shapes of the sections. Thus, all quartz crystals, however muchdistorted from the ideal, could result from the same fundamental modeof growth and, accordingly, corresponded to the same inner structure.Instruments were developed to measure the angles made by thecrystal faces.
In 1780 the contact goniometer, Figure 9-6a, was alreadyin usage. Later, for more precise measurement of the interfacialangles, the reflecting goniometer was introduced (Figure 9-6b).Another interesting phenomenon observed early in crystals is theircleavage. It is characteristic that they break along well-defined planes.Figure 9-5. Sections of ideal and distorted quartz crystals.4189 Crystals(a)(b)Figure 9-6. Goniometers; (a) Contact goniometer from Haüy [9]; (b) Scheme ofreflecting goniometer.Haüy noticed that the cleavage rhombs from any calcite crystal alwayshad the same interfacial angles. Thus, he suggested that all calcitecrystals could be built of these fundamental cleavage rhombs.
Thisis illustrated in Figure 9-7 which is from Haüy’s Traité de Cristallographie [10]. From the units shown in Figure 9-7, it is possible tobuild straight edges corresponding to the faces of a cube, as well asinclined edges corresponding to the faces of an octahedron.
Edgesinclined at other edges may also be built. Let the dimensions of thecleavage unit be a and b (Figure 9-8); then tan Θ 1 = b/a and tan Θ 2 =b/2a, and generally tanΘ = mb/na, where m and n are rational integers. By extension into the third dimension, we may have a referenceface making intercepts a, b, c on three axes. The intercepts made byany other face must be in proportion of rational multiples of theseintercepts.
This is called the law of rational intercepts.9.1. Basic Laws419(a)(b)Figure 9-7. Cleavage rhombs and the stacking of cleavage rhombs from Haüy [11].4209 CrystalsFigure 9-8. Inclined edges from cleavage units and illustration for the law ofrational intercepts.Usually the crystal faces are described by the reciprocals of themultiples of the standard intercepts, hence the name “the law ofrational indices.” In Figure 9-8 three lines are adopted as axes whichmay also be directions of the crystal edges. A reference face ABCmakes intercepts a, b, c on these axes. Another face of the crystal,e.g., DEC, can be defined by intercepts a/h, b/k, c/l. Here h, k, l aresimple rational numbers or zero.
They are called Miller indices. Theintercept is infinite if a face is parallel to an axis, and h or k or l will bezero. For orthogonal axes the indices of the faces of a cube are (100),(010), and (001). The indices of the face DEC in Figure 9-8 are (231).The simple cleavage model of Haüy indeed revealed a lot about thestructure of crystals. However, it was not generally applicable sincecleavages do not always lead to cleavage forms that can necessarily fillspace by repetition, and, as is known, there are only a limited numberof space-filling polyhedra.The characterization of the regularities in the outer form of the crystals has led to the recognition of three-dimensional periodicity in theirinner structure.
This was long before the possibility of determiningthe atomic arrangements in crystals by diffraction techniques hadmaterialized.9.1. Basic Laws421It was 200 years before Dalton and 300 years before X-ray crystallography that Kepler discussed the atomic arrangement in crystals. Inhis Strena seu de nive sexangula he presented arrangements of closepacked spheres [12]. These are reproduced in Figure 9-9.
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