M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 69
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Further layers can be added in the samefashion. The third layer is shown in Figure 9-30, and this is what isknown as the Mackay icosahedron. It is an example of icosahedralpacking of equal spheres. The layers of spheres succeed each otherin cubic close packing sequence on each triangular face. Each spherewhich is not on an edge or vertex touches only six neighbors, threeabove and three below. Each such sphere is separated by a distance of5% of its radius from its neighbors in the plane of the face of the icosahedron.
The whole assembly can be distorted to cubic close packingin the form of a cuboctahedron. This distortion may be envisaged asa reversible process by the kind of transformation discussed earlier.The Mackay icosahedron has “made tremendous impact on particle,cluster, intermetallics, and quasicrystal researchers. . .” [52].Figure 9-30. Icosahedral packing of spheres showing the third shell [53]. This ispopularly called “the Mackay icosahedron.”Herbert Hauptman, a mathematician turned crystallographer andchemistry Nobel laureate for 1985, has devoted a lot of attention toclose packing of spheres in the icosahedron.
Figure 9-31 shows oneof his beautiful stained-glass models.9.5. Dense Packing449Figure 9-31. Herbert Hauptman (photograph by the authors) and one of his stainedglass models—an icosahedron—with densely packed spheres (photograph courtesyof Herbert Hauptman, Buffalo, New York).While the most symmetrical arrangement of 12 neighbors, viz.,the icosahedral coordination, does not lead to the densest possiblepacking, others do.
The cuboctahedron and its “twinned” version,alone or in combination, lead to infinite sphere packing with thesame high density (0.7405). Both coordination polyhedra are shownin Figure 9-32. The “twinned” polyhedron is obtained by reflectingone half of a cuboctahedron cut parallel to a triangular face across theplane of section.9.5.3. Connected PolyhedraThere are, of course, more complex forms of closest packing thanthose considered so far. Besides, the species to be packed need notbe identical.
Thus, close packing of atoms of two kinds could beconsidered. Close-packed structures with atoms in the intersticesare also important. The interstice arrays may have very differentFigure 9-32. Cuboctahedron and “twinned” cuboctahedron.4509 Crystalsarrangements in various structures. A shorthand notation of someconfigurations has been worked out [54] to facilitate the description of more complicated systems, which is illustrated in Figure 9-33.Suppose, for example, that in a compound with composition AX2 ,each atom A is bonded to four X atoms and that all four X atoms areequivalent. Each X atom must then be bonded to two A atoms. Thelines of the squares in Figure 9-33 do not represent chemical bonds;rather, these squares stand for polyhedral arrangements.
Among theAXn polyhedral groups the most common are the AX4 tetrahedra andAX6 octahedra. They may appear in various orientations in the crystalstructures. Similar structural features have already been discussed forthe polyhedral molecular geometries. Whereas in molecules only two,or at most a few, polyhedra were joined, here we deal with their infinite networks.Figure 9-33. Shorthand notation for some common structural units after Wells [55];(a) Notations for tetrahedron; (b) Notations for octahedron.Many crystal structures may be built from the two most important coordination polyhedra, the tetrahedron and octahedron.
Theymay share vertices, edges, or faces. The ways how the polyhedra are connected introduce certain geometrical limitations withimportant consequences as to the variations of the interatomicdistances and bond angles. Examples are shown in Figure 9-34 andFigures 9-36–9-39 for a variety of ways to connect tetrahedral andoctahedral units. Tetrahedra share two vertices or/and three verticesin Figure 9-34. For one of these, decorations analogous to its projection is shown in Figure 9-35. Octahedra share adjacent vertices and9.5. Dense Packing451Figure 9-34. Connected tetrahedra [56]. (a) All tetrahedra share two vertices; (b)and (c) All tetrahedra share three vertices; (d) and (e) Some tetrahedra share two,others share three vertices.Figure 9-35.
Decorations, analogous to the pattern of Figure 9-34d, but extendingin two dimensions (photographs by the authors).4529 CrystalsFigure 9-36. Connected octahedra: (a) and (b) Two representations of four octahedra sharing adjacent vertices and forming a tetramer; (c) Infinite chain of octahedra connected at adjacent vertices; (d) Infinite chain of octahedra connected atnonadjacent vertices.form a tetramer in two representations in Figure 9-36a and b. Twomore examples show infinite chains of octahedra sharing adjacent(Figure 9-36c) and nonadjacent (Figure 9-36d) vertices. Octahedrasharing two, four, or six edges are presented in Figure 9-37. Anexample of octahedra sharing faces and edges is seen in Figure 9-38.Finally, a composite structure from tetrahedra and octahedra is shownin Figure 9-39.Figure 9-37.
Octahedra sharing edges: (a) Two edges; (b) Four edges; (c) Six edges.9.5. Dense Packing453Figure 9-38. Joined octahedra: sharing faces and edges in the Nb3 S4 crystal [57].The tetrahedra and octahedra are important building blocks ofcrystal structures. The great variety of structures combining thesebuilding blocks, on one hand, and the conspicuous absence of someof the simplest structures, on the other hand, together suggest thatthe immediate environment of the atoms is not the only factor whichdetermines these structures. Indeed, the relative sizes of the participating atoms and ions are of great importance.9.5.4. Atomic SizesThe interatomic distances are primarily determined by the positionof the minimum in the potential energy function describing the interactions between the atoms in the crystal.
The question is then, whatare the sizes of the atoms and ions? The extension of electron densityfor an atom or an ion is not rigorously defined; no exact size can beFigure 9-39. A composite structure (kaolin) built from joined tetrahedra and octahedra (reproduced with permission) [58].4549 Crystalsassigned to it. Atoms and ions change relatively little when forminga strong chemical bond, and even less for weak bonds. For thepresent discussion of crystal structures, the atomic and ionic radiishould, when added appropriately, yield the interatomic and interionicdistances characterizing these structures.Covalent and metallic bondings suppose a strong overlap of theoutermost atomic orbitals and so the atomic radii will be approximately the radii of the outermost orbitals. The atomic radii are empirically obtained from interatomic distances [59].
For example, thelength of the bond C–C is 154 pm in diamond, Si–Si is 234 pm indisilane, and so on. The consistency of this approach is shown by theagreement between the Si–C bond lengths determined experimentallyand calculated from the corresponding atomic radii. The interatomicdistances appreciably depend on the coordination. With decreasingcoordination number, the bonds usually get shorter. For coordinations8, 6, and 4, the bonds get shorter by about 2, 4, and 12%, respectively,as compared with the coordination number of 12.The covalent bond is directional and multiple covalent bonds areconsiderably shorter than the corresponding single ones.
For carbonas well as for nitrogen, oxygen, or sulfur, the decrease on going froma single bond to a double and a triple bond amounts to about 10 and20%, respectively.Establishing the system of ionic radii is even a less unambiguousundertaking than that for atomic radii. The starting point is a systemof analogous crystal structures. Such is, for example, the structure ofsodium chloride and the analogous series of other alkali halide facecentered crystals.
In any case the ionic radii represent relative sizes,and if the alkali and halogen ions are chosen for starting point, then theionic radii of all ions represent the relative sizes of the outer electronshells of the ions as compared with those of the alkali and halogenions.Consider now the sodium chloride crystal structure shown inFigure 9-40. It is built from sodium ions and chloride ions, and itis kept together by electrostatic forces. The chloride ions are muchlarger than the sodium ions. As equal numbers of cations and anionsbuild up this structure, the maximum number of neighbors will be thenumber of the larger chloride ions that can be accommodated aroundthe smaller sodium ion. The opposite would not work: although moresodium ions could surround a chloride ion, the same coordination9.5.
Dense Packing(a)455(b)Figure 9-40. The sodium chloride crystal structure in two representations. Thespace-filling model is from W. Barlow [60].could not be achieved around the sodium ions. Thus, the coordination number will obviously depend on the relative sizes of the ions. Inthe simple ionic structures, however, only such coordination numbersmay be accomplished that make a highly symmetrical arrangementpossible. The relative sizes of the sodium and chloride ions allowsix chloride ions to surround each sodium ion in six vertices of anoctahedron.
Figure 9-41 shows the arrangement of ions in cube-facelayers of alkali halide crystals with the sodium chloride structure. AsFigure 9-41. The arrangement of ions in cube-face layers of alkali halide crystalswith the sodium chloride structure. Adaptation from Pauling [61]. Copyright (1960)Cornell University. Used by permission of the publisher, Cornell University Press.4569 CrystalsFigure 9-42. Cesium chloride crystal structure.the relative size of the metal ion increases with respect to the size ofthe halogen anion, greater coordination may be possible. Thus, forexample, the cesium ion may be surrounded by eight chloride ionsin eight vertices of a cube in the cesium chloride crystal as shown inFigure 9-42.9.6. Molecular CrystalsA molecular crystal is built from molecules and is easily distinguishedfrom an ionic/atomic crystal on a purely geometrical basis.
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