M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 66
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Naturally, this applies to theplanar networks as well.9.3. Restrictions429Table 9-1. Allowed Rotation Axes n in a LatticePossible values of m − 1cos (◦ )−1− 120+ 12+1–2–10+1+21801209060360 or 0n23461The permissible periods of mirror-rotation axes have the same limitations as those of the proper rotation axes.Let us examine now the limitations on the screw axes. In a latticethe screw axes must be parallel to a translation direction. After n rotations by an angle and n translations by the distance T, that is, aftern translations along the screw axis, the total amount of translationdistance in the direction of this axis must be equal to some multipleof the lattice translation mt,nT = mtwhere n and m are integers. Rearranging this equation,T =mt,nwhere m, of course, may be 0, 1, 2, 3, etc., but n may only be 1, 2, 3,4, or 6.
It is then possible to determine the permissible values of thepitch of the screw axes in lattices. They are summarized in Table 9-2,taking also into consideration that (3/2)t = t + (1/2)t, (5/4)t = t + (1/4)t,etc. There are only eleven screw axes that are allowed in a lattice, nm ,according to Table 9-2. The subscript in the notation is the m of theexpression T = (mt)/n. The proper rotation axes may be consideredto be special cases of the screw axes, with m = 0 and m = n. The 11screw axes are shown in perspective in Figure 9-19.
It is seen there thatsome pairs are identical except for the direction of the screw motion.Such screw axes are enantiomorphous. The enantiomorphous screwaxis pairs are the following:31 and 3241 and 4361 and 6562 and 64 .430Table 9-2. Possible Values of the Pitch T of an n-Fold Screw AxisA. Possible values of Tn=10t,1t,2t, . . .n=20t,(1/2)t,(2/2)t,(3/2)t, . . .n=30t,(1/3)t,(2/3)t,(3/3)t,n=40t,(1/4)t,(2/4)t,(3/4)t,n=60t,(1/6)t,(2/6)t,(3/6)t,B. Possible values of T (redundancies eliminated)n=1n=2(1/2)t,n=3(1/3)t,(2/3)tn=4(1/4)t,(2/4)t,(3/4)tn=6(1/6)t,(2/6)t,(3/6)t,C. Notation of screw axes in a latticen=221n=33132n=4414243n=6616263(4/3)t, . .
.(4/4)t,(4/6)t,(5/4)t, . . .(5/6)t,(4/6)t,(5/6)t6465(6/6)t,(7/6)t, . . .9 Crystals9.3. Restrictions431Figure 9-19. The eleven screw axes. The simple twofold, threefold, fourfold andsixfold axes are also shown for completeness. After Azaroff [24]; copyright (1960)McGraw-Hill, Inc.; used with permission.432Glide type9 CrystalsTable 9-3. Possible Glide PlanesSymbolTranslation componentAxialaa/2Axialbb/2Axialcc/2Diagonalna/2 + b/2; b/2 + c/2; or c/2 + a/2da/4 + b/4; b/4 + c/4; or c/4 + a/4DiamondaaTranslation component is one-half of the true translation along the face diagonalof a centered plane lattice.Finally, the only remaining symmetry element is considered, theglide-reflection plane.
It causes glide reflection as a result of reflectionand translation. The translation component T of a glide plane is onehalf of the normal translation of the lattice in the direction of the glide.A glide along the a axis is T = (1/2)a and this is called an a glide.Similarly, a diagonal glide can have T = (1/2)a + (1/2)c. The differentpossible glides are summarized in Table 9-3.The fact that the crystal has a lattice framework imposes strict limitations on the symmetry of its outer form. On the other hand, the question arises as to whether it is possible to derive any information aboutthe crystal lattice from the knowledge of the symmetry of its outerform.The 32 crystal point groups can be classified by symmetry criteria.They are usually grouped according to the highest ranking rotationaxis that they contain.
The resulting groups are called crystal systems.There are altogether seven of them and they are listed in Table 9-4.The crystal point groups have to be combined with all possible spacelattices in order to produce the space groups.9.4. The 230 Space GroupsThere are 14 infinite lattices, called Bravais lattices, in threedimensional space. They are shown in Figure 9-20. These latticesare the analogs of the five infinite lattices in two-dimensional space(Figure 8-28). The Bravais lattices are presented as systems of pointsat vertices of parallelepipeds. The corresponding parallelepipeds arecapable of filling space without gaps or overlap.
The representationof the lattices by systems of points is especially useful as it makesit possible to join the lattice points in any desired way conformingTriclinic1 (or 1)MonoclinicLattice typeNumbering inFigure 20a = b = c␣ =  = ␥ = 90◦P12 (or 2)a = b = c␣ = ␥ = 90◦ = PC (or A)23Orthorhombic222 (or 222)a = b = c␣ =  = ␥ = 90◦PC(or B or A)IF4567Trigonal(rhombohedral)3 (3)a=b=c␣ =  = ␥ = 90◦R8Hexagonal6 (6)a = b = c␣ =  = 90◦ , ␥ = 120◦P9Tetragonal4 (or 4)a = b = c␣ =  = ␥ = 90◦PI1011CubicFour 3 (or 3)a=b=c␣ =  = ␥ = 90◦PIF1213149.4. The 230 Space GroupsSystemTable 9-4. Characterization of Crystal SystemsMinimal symmetry (diagnosticRelations between edges andsymmetry elements)angles of unit cell4334349 CrystalsFigure 9-20. The 14 Bravais lattices.with the symmetry requirements. In this way, not only the originalparallelepipedal forms but any other possible figures may be used asbuilding units for the space lattice.The 14 Bravais lattices are enumerated in Table 9-4 as the followingtypes: primitive (P, R), side-centered (C), face-centered (F), and bodycentered (I).
The numbering of the Bravais lattices in Table 9-4corresponds to that in Figure 9-20. The lattice parameters are alsoenumerated in the table. In addition, the distribution of lattice typesamong the crystal systems is shown.The actual infinite lattices are obtained by parallel translations ofthe Bravais lattices as unit cells. Some Bravais cells are also primitive cells, others are not. For example, the body-centered cube is aunit cell but not a primitive cell. The primitive cell in this case is anoblique parallelepiped constructed by using as edges the three directed9.4.
The 230 Space Groups435segments connecting the body center with three nonadjacent verticesof the cube.The three-dimensional space groups are produced by combiningthe 32 crystallographic point groups with the Bravais lattices. Sincethe symmetry elements in a space lattice can have translation components, indeed not only the 32 groups but also the analogous groups,which have screw axes and glide planes, have to be considered.
Thereare altogether 230 three-dimensional space groups! Their completedescription can be found in Volume A of the International Tables forCrystallography [25]. Only a few examples are discussed here.There are only two combinations possible for the triclinic system.They are named P1 and P1. For the monoclinic system three pointgroups are to be considered and two lattice types. Combining P andI lattices, on one hand, and point group 2 and symmetry 21 on theother hand, the four possible combinations are P2, P21 , I2, and I21 .The latter two, however, are equivalent; only their origins differ.The description of the symmetry elements of the space groups issimilar to that of the point groups [26]. The main difference is thatthe order by which the symmetry elements of the space groups arelisted may be of great importance, except for the triclinic system.The order of the symmetry elements expresses their relative orientation in space with respect to the three crystallographic axes.
Forthe monoclinic system, the unique axis may be the c or the b axis.For the P2 space group, the complete symbol may be P112 or P121.The ordering of symbols for the orthorhombic system is especiallyimportant. The symmetry elements are usually listed in the order abc.The space groups which belong to the crystal class 2mm are properlypresented as Pmm2, c being the unique axis.In the tetragonal system, the c axis is the fourfold axis.
Thesequence for listing the symmetry elements is c, a, [110], since thetwo crystallographic axes orthogonal to c are equivalent. For example,the three-dimensional space group notation P4m2 has the followingmeaning: the unique axis in a primitive tetragonal lattice is a 4axis, the two a 4 axes are parallel to m, and the [110] direction hastwofold symmetry. A similar sequence is used for listing the symmetryelements of the hexagonal system, for which the c axis again is theunique axis and the other two are equivalent. P denotes the primitivehexagonal lattice while R denotes the centered hexagonal lattice inwhich the primitive rhombohedral cell is chosen as the unit cell.4369 CrystalsAll three crystallographic axes are equivalent in the cubic system.The order of listing the symmetry elements is a, [111], [110].
Whenthe number 3 appears in the second position, it merely serves to distinguish the cubic system from the hexagonal one.It may be of interest to add some new symmetry to a group or todecrease its symmetry and examine the consequences. If the addition produces a new group, it is called a supergroup of the originalgroup. If eliminating symmetry leads to a new group, it is usually asubgroup of the original one. For example, the point group 1 is obviously a subgroup of all the other 31 groups as it has the lowest possiblesymmetry. On the other hand, the highest symmetry cubic group canhave no supergroups.It is important to distinguish between the symmetry of the latticeand the symmetry of the actual building elements of the crystal—the atoms, ions, or molecules.
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