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Crystallographic Point Groups APPENDIX A Crystallographic Point Groups In this appendix we examine briefly the symmetry properties of nonmagnetic crystals. According to the axiom of material invariance, the macroscopic symmetry of all nonmagnetic crystals may be described by an isotropy group {S}. Accordingly, under the transformations of the material frame of reference X = SX, SST = STS = 1, det S = ± 1, (A.l) the constitutive functionals must remain form-invariant for all members of the symmetry group {S}. Local properties of crystals are restricted by the point group. The symmetry operators, that act at a fixed point 0 and leave invariant all distances and angles in a three-dimensional space, are called the point group. The symmetry operators that have these properties are rotations about axes through 0, and products (combinations) of rotations and inversions. Of course, such products include reflections in planes through o. If the group contains only rotations, it is called a proper rotation group. This is isomorphic with the group 0+(3) of all 3 x 3 orthogonal matrices. Operators, whose matrices have determinant (-1), are called improper rota­ tions. They are products of proper rotations and inversion. We note that the inversion commutes with all rotations. Every subgroup of 0+(3) is a proper point group. Proper point groups of finite order are classified as: Cyclic (Cn = n); Dihedral (Dn = n22, n even, Dn = n2, n odd); Tetrahedral (T = 23); and Octahedral (0 = 432). A crystallographic point group is restricted by a requirement that an operator must be compatible with the translational symmetry of a crystalline solid. Hence, the appropriate symmetry operations are identity = E, inversion = C, reflection in certain planes = (1, rotations = Cnr• The rotation Cnr is an anticlockwise rotation through 2n/n radians about the axis indicated by r. The eleven proper point groups are listed in Table A.l, together with their 374 Appendix A. Crystallographic Point Groups Table A.t. Crystallographic pure rotation groups. Cyclic groups Symmetry elements C, = 1 E C2 = 2 E, C2z C3 = 3 E, C3., C3z C4 =4 E, C4 ., Ci., C2z C6 = 6 E, C6 ., Ci., C3., C3., C2z Dihedral groups D2 = 222 E, C2x, C2y, C2z D3 = 32 E, C3., C3., Cl" Cl2, Cl3 D4 = 422 E, C4 ., Ci., C2x, C2y, C2 ., C2a, C2b D6 = 622 E, C6z, C6z, C3z' C3"z, C2z, C~r' Ci, Tetrahedral group T= 23 Octahedral group 0=432 symmetry elements. In this table the first column (C1 , C2 , ••• , 0) denotes the Schonflies notation, and the second column (1, 2, ... ,432) denotes the inter­ national notation. In addition to purely rotational symmetry, the space lattice possesses symmetries of reflections in various planes (det S = -1). In order to include such symmetry operations, we multiply the proper point group {P} by {E, C}. This produces a new set of eleven point groups that are subgroups of 0(3). If the point group {P} has an invariant subgroup {H} of index 1 2, then {P} = {H} + C{P - H} (A. 2) is also a point group. This process gives ten more point groups. The possible crystallographic point groups are 32 in number, as listed in Table A.2. By examination of the metrical properties, crystal classes are divided into seven crystal systems. Each system possesses one and the same metrical property. If hi denotes the lattice bases then the length oflattice bases \hl\ = a, \h2\ = b, \h3\ = c, and angles ex = angle(h2, h3)' p = angle(h3' hd, and y = angle(h 1 , h3), for each crystal system, are the same. This is called a holohedry of the space lattices. In Table A.l, j = 1,2,3,4; m = x, y, z; p = a, b, c, d, e,J; and r = 1,2,3; and the labels of the symmetry operations can be identified from Figures A.1-A.3. In Figures A.l and A.2 the labels of the symmetry operations are placed on the figure in the position to which the letter E is taken by that operation. 1 The index of a subgroup is the integer obtained by dividing the order of the group by that of the subgroup. Figure A.t. Symmetry elements: triclinic, monoclinic, rhombic, and tetragonal systems, , , , (4Z ... , y. -------------: __ ------1.._ , , (2a , , , , / , , , , , , (2,• (2y Figure A.2. Symmetry elements: ,, trigonal and hexagonal systems, 2" C2 •1,' E 3" ~, ~ \ (6, \ , ' , ......', .. '........ x '" •'" -', 3' ciz (2, : ('2, 2" •l' Figure A.3. Symmetry elements: cubic system, 3 '-""--______---V w -..I 0- :> '"0 '"0 (1) Table A.2. The 32 conventional crystal classes. 6- ><' ?> Class (J System number Class name Symmetry transformations Order .... '< ...'" Triclinic Pedial C.l I e:. 2 0' 2 Pinacoidal ciT I,C OQ .... Monoclinic 3 Sphenoidal C2 2 I,D3 2 '" 4 Domatic C,m I, R3 2 ~(") 5 Prismatic C2h 2/m I, C,R3' D3 4 'ti 9. Orthorhombic 6 Rhombic-disphenonidal D2222 I, D., D2, D3 4 ...= 7 Rhombic-pyromidal C2v 2mm I, R., R2, D3 4 Cl .... 8 Rhombic-dipyramidal D2h mmm I, C, D., D2, D3, R., R2, R3 8 0 s:: '"0 Tetragonal 9 Tetragonal-pyramidal C4 4 I, D3, R. T3, R2 T3 4 '" 10 Tetragonal-disphenoidal C24 I, D3, D. T3, D2 T3 4 11 Tetragonal~dipyramidal C4h 4/m I, D3, D. T3, D2 T3, R. T3, R2 T3, C, R3 8 12 Tetragonal-trapezahedral D4422 I, D., D2, D3, CT3, R. T3, R2 T3, R3 T3 8 13 Ditetragonal-pyramidal C4v 4mm I, R., R2, D3, T3, R. T3, R2 T3, D3 T3 8 14 Tetragonal-scalenohedral D2v 42m I, D., D2, D3, T3, D. T3, D2 T3, D3 T3 8 15 Ditetragonal-dipyramidal D 4/mmm I, D., D , D , CT , R. T , R2 T , R3 T , C, R., R , R , T , D. T , 4h 2 3 3 3 3 3 2 3 3 3 16 Trigonal 16 Trigonal-pyramidal C3 3 J, SI' S2 3 17 Rhombohedral E3 3 J, SI, S2, C, CSI' CS2 6 18 Trigonal-trapezohedral D3 32 J, SI, S2, DI, DISI, DIS2 6 19 Ditrigonal-pyramidal C3v 3m J, SI' S2' R I , RISI, R I S2 6 20 Hexagonal-scalenohedral D3v 3m J, SI' S2' C, CSI' CS2, R I , RISI, R I S2, DI, DISI, DIS2 12 Hexagonal 21 Hexagonal-pyramidal C6 6 J, SI, S2, D3, D3 SI, D3S2 6 :> 22 Trigonal-dipyramidal C3h 6 J, SI' S2' R3, R3SI , R 3S2 6 "0 23 Hexagonal-dipyramidal C6h 6/m J, SI, S2, R3, R3SI , R 3S2, C, CSI, CS2, D3, D3 SI, D3S2 12 'g ::s 24 Hexagonal-trapezohedral D6 622 J, SI, S2' D3, D3 SI, D3S2, DI, DISI, DIS2, D2SI, D2S2, D2 12 e: >I 25 Dihexagonal-pyramidal C6v 6mm J, SI' S2, D3, D3 SI, D3S2, R I , RISI, R I S2, R 2, R2SI , R2S2 12 26 Ditrigonal-dipyramidal D3h 62m J, SI' S2, R 3, R3SI , R3S2, R I , RISI, R I S2, D2, D2SI, D2S2 12 ~ (") 27 Dihexagonal-dipyramidal D6h 6/mm J, SI' S2, C, CSI' CS2, DI, DISI, DIS2, D2, D2SI, D2S2, R I , RISI, 24 '<... R I S , R , R SI , R S , R , R SI , R S , D , D S , D S til 2 2 2 2 2 3 3 3 2 3 3 I 3 2 ....a Cubic 28 Tetartoidal T23 J, DI, D2, D3, C3j, Clj 12 0' OQ 29 Diploidal T" m3 J, DI, D2, D3, C, R I , R2, R 3, C3j, Clj, S6j, S6j 24 ... "0 30 Gyroidal 0432 J, D .. D2, D3, C2p , C3j, Clj, C4m, Cim 24 '" 31 Hextetrahedral Td43m J, DI, D2, D3, (i,p, C3j, Clj, S4m, Sim 24 ~ 32 Hexoctohedral Oh m3m J, DI, D2, D3, C ' C , Clj, C4m, Cim, C, R I , R2, R3, (i,p, S6j, S6j, '"0 2P 3l 48 0 S4m, Sim 5' .... 0 ... 0 ~ "0til VJ -.l -.l 378 Appendix A. Crystallographic Point Groups The transformation matrices are given by 0 1 C = 0-1l I ~ (~ ° ~). 0 ~). ro 0 -1 0 0 0 1 -1 1 Rl = ~l R, ~(~ ~ ). R, ~(~ ~). r 0 n 0 0 -1 0 0 0 -1 (-I 1 o ,03 = 0 -1 ~). O2 = ~ 0) C D'~(~ 0 -1 0 -1 0 0 ~). 0 0 1 T, ~(~ 0 !). T, ~(~ 1 ~). T'~(! 0 1 0 0 n 1 0 M, ~(~ 0 M, ~(! 0 (A.3) 0 !). 1 ~). ( -1/2 ~/2 0) -~/2 0) Sl = -f/2 -1/2 0 , S2 = ~/2l/2 -1/2 0, o 1 r o 1 where I is the identity and C is the central inversion. Rl , R2, R3 are reflections in the planes whose normals are along the Xl = X-, X2 = Y-, and X3 = z­ directions, respectively. 0 1 , O2, 0 3 are rotations through n radians about the Xl -, X 2 -, and x3-axes, respectively. Tl is a reflection through a plane which bisects the X2- and X3 -axes and contains the xl-axis. T2 and T3 are analogously defined. Ml and M2 are rotations through 2n/3 clockwise and anticlockwise, about an axis making equal acute angles with the axes Xl' x 2 , and X3' Sl and S2 are rotations through 2n/3 clockwise and anticlockwise, respectively, about the X3 = z-axis. APPENDIX B Crystallographic Magnetic Groups As noted in Section 5.4, the symmetry properties of magnetic materials must include a time-inversion operator which reverses the spin of each atom. The situation is visualized simply by considering a chain of equally spaced atoms on a line (Figure B.1). Disregarding their spin, we see that the X2 -axis is a twofold symmetry axis, and in addition, the X2 X3 -plane is a reflection plane (Figure B.
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