M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 65
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Incidentally, close packing of spheres was invoked and illustrated in Dalton’sworks (Figure 9-10) in relation to gas absorption [13]. A close-packedarrangement of cannon balls expressing close packing is shown inFigure 9-11. The fundamental importance of Kepler’s idea is that hecorrelated, for the first time, the external forms of solids with theirinner structure. Kepler’s search for harmonious proportions is thebridge between his epoch-making discoveries in heavenly mechanicsand his less widely known but similarly seminal ideas in what is calledtoday crystallography. As Schneer has remarked [14], the renaissanceera has provided a stimulating background for the beginnings of thescience of crystals.It is to be noted that even after the discovery of Haüy’s model, attention was focused on the packing in crystals. The aim was to find thosearrangements in space that are consistent with the properties of thecrystals.The symmetry of the form of the crystal is a consequence of itsstructure.
The same high symmetry of the form, however, may beeasily achieved for a piece of glass by artificial mechanical intervention. By acquiring the same outer form, as is typical for a piece ofdiamond, the piece of glass will not acquire all the other propertiesFigure 9-9. Closely packed spheres by Kepler [15].4229 CrystalsFigure 9-10.
Closely packed spheres by Dalton [16].that the diamond possesses. The difference in value has long ago beenrecognized. In the India of the sixth century portrayed by Kama Sutraof Vatsayana, one of the arts which a courtesan had to learn was mineralogy. If she were paid in precious stones, she had to be able to distinguish real crystals from paste [17].It is primarily the structure, and, accordingly, the outer andinner symmetry properties of the crystal, that determines its manyoutstanding physical properties. The mechanical, electrical, magnetic,and optical properties of crystals are all in close conjunction with theirsymmetry properties [18].Figure 9-11. Arrangements of close packing: Cannon balls in the Castel Sant’Angelo in Rome (photograph by the authors).9.2.
The 32 Crystal Groups423In an actual crystal the atoms are in permanent motion. However,this motion is much more restricted than that in liquids, let alonegases. As the nuclei of the atoms are much smaller and heavier thanthe electron clouds, their motion can be well described by small vibrations about the equilibrium positions. In our discussion of crystalsymmetry, as an approximation, the structures will be regarded asrigid. However, in modern crystal molecular structure determinationatomic motion must be considered [19]. Both the techniques of structure determination and the interpretation of the results must includethe consequences of the motion of atoms in the crystal.9.2.
The 32 Crystal GroupsAlthough the word crystal in its every-day usage is almost synonymous with symmetry, in classical crystallography there are severerestrictions on crystal symmetry. While there are no restrictions inprinciple for the number of symmetry classes of molecules, this is notso for the crystals. All crystals, as regards their form, belong to oneor another of only 32 symmetry classes.
They are also called the 32crystal point groups. Figures 9-12 and 9-13 show them by examplesof actual minerals and by stereographic projections with symmetryelements, respectively.Stereographic projection starts by representing the crystal through aset of lines perpendicular to its faces. The introduction of this methodof representation followed soon after the invention of the reflectinggoniometer. Let us place the crystal in the center of a sphere andextend its face normals to meet the surface of the sphere as seen inFigure 9-14a. A set of points will occur on the surface of the sphererepresenting the faces of the crystal. Join now all the points in thenorthern hemisphere to the South Pole, and mark the points on theequatorial plane where these connecting lines intersect this plane.This will create a representation of the faces on the upper half of thecrystal within a single circle as seen in Figure 9-14b.
Performing asimilar operation for the points of the equator (Figure 14-c) and forthe points in the southern hemisphere (Figure 14-d), we arrive at therepresentation of the whole crystal within the circle (Figure 14-e). Thepoints from the northern hemisphere are marked by dots, and those4249 CrystalsFigure 9-12. Representation of the 32 crystal point groups by actual minerals (afterBuerger; Dana; and Zorky) [20].from the southern hemisphere by small circles.
Some examples forsimple polyhedra are shown in Figure 9-15.9.3. RestrictionsThe restrictions we discuss in this Section are valid for classical crystallography, but are no longer so in a broader domain of science aboutcrystals, called often generalized crystallography. Our discussion willconsider the broadening meaning of crystallography with emphasis onthe discovery of the so-called quasicrystals. However, it is instructiveto examine the origins of the restrictions in classical crystallography.To have 32 symmetry classes for the external forms of crystals is a9.3. RestrictionsFigure 9-13.projections.425Representation of the 32 crystal point groups by stereographicdefinite restriction, and it is the consequence of periodicity in the innerstructure.
The translation periodicity limits the symmetry elementsthat may be present in a crystal. The most striking limitation is theabsence of fivefold rotation. Consider, for example, planar networksof regular polygons (Figure 9-16). Those with threefold, fourfold, andsixfold symmetry cover the available surface without any gaps, whilethose with fivefold, sevenfold, and eightfold symmetry leave gaps onthe surface.
Figure 9-17 presents a planar network of octagons. It isevident that the regular octagons cannot cover the surface withoutgaps, and there are smaller squares among the octagons.Let us examine now the possible types of symmetry axes in spacegroups [22]. Figure 9-18 shows a lattice row with a period t. An n-fold4269 CrystalsFigure 9-14.
The preparation of stereographic representation.rotation axis, Cn , is placed in each lattice point. Since n rotations, eachby an angle , must lead to superposition, it does not matter in whichdirection the rotations are performed. Two rotations by about twoaxes but in opposite directions are shown in Figure 9-18. The two newlattice points produced this way are labeled p and q. These two newpoints are equidistant from the original row, and hence the line joiningthem is parallel to the original lattice row. The length of the parallelline joining p and q must be equal to some integer multiple m of theperiod t.
Were it not, then the line joining the two new lattice points pand q would not be a translation of the lattice and the resulting arraywould not be periodic.9.3. Restrictions427Figure 9-15. Representation of simple highly symmetrical shapes: (a) Cube;(b) Tetrahedron; (c) Octahedron; (d) Rhombic dodecahedron.Using Figure 9-18, it is possible to determine the possible valuesthat the rotation angle can have in the lattice,mt = t + 2t cosm = 0, ±1, ±2, ±3, . .
.where +m or −m is taken depending on the direction of the rotation:Figure 9-16. Planar networks of regular polygons with up to eight-fold symmetry.4289 CrystalsFigure 9-17. Octagonal planar network: Hungarian needlework [21].Figure 9-18. Illustration to the determination of the possible throws that rotationaxes can have in space groups. After Azaroff [23]; Copyright (1960) McGraw-Hill,Inc.; used with permission.cos =m−12Only the solutions corresponding to the range−1 ≤ cos ≤ 1need be considered, and these are shown in Table 9-1. Five solutions are possible, and, accordingly, only five kinds of rotation axesare compatible with a lattice. Thus, not only fivefold symmetry isnot allowed in crystal structures in classical crystallography, but allperiods larger than six are impossible.
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