230种空间群.pdf-资料库

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Symmetry Operations and Space Groups Crystal Symmetry 32 point groups of crystals compatible with 7 crystal systems crystallographers use Hermann-Mauguin symmetry symbols Carl Hermann Charles-Victor Mauguin German 1898 - 1961 French 1878 - 1958 Symmetry Elements Center of Symmetry: 1 there are 5 types in point symmetry 1. center of symmetry (or inversion) 2. rotation (or proper) axis 3. mirror 4. rotation–inversion axis 5. identity : point : line : plane : line : no element 1 n m n 1 a point in the molecule through which if another point on the molecule is taken, will meet an identical point on the molecule an equal distance away 1 Center of Symmetry: 1 Rotation Axis: n all points (x, y, z) (–x, –y, –z) if 1 is placed at the origin y 1 z + (x, y, z) x (–x, –y, –z) – n is an integer which gives the degrees of rotation: 2 n or 360o n n is the number of times molecule is rotated, at an identical appearance, each time stopping before returning to the starting point 2 3 4 6 n is the foldness of the rotation axis only 2, 3, 4, and 6-fold axes allowed in crystal symmetry 9/18/2013 1

9/18/2013 Rotation Axis: 4 360o 4 = 90o Mirror: m plane within the molecule that, when acting as a mirror, reflects the molecule into itself 90o 180o 270o 360o 4 Rotation-Inversion n Representation of Symmetry rotation followed by inversion this is a different definition than Schoenflies system Arthur Moritz Schönflies – German 1891 rotation followed by reflection point symmetry often represented symbolically in the form of points on a circle (projection of a sphere) a point above plane is a filled circle: a point below plane is an open circle: two points directly on top of each other: starting with one point, find other points generated by symmetry 32 point groups compatible with 7 crystal systems Triclinic 1 1 (Ci) 1 (C1) highest symmetry in each crystal system is called: Laue Group (Schoenflies symbol) have center of symmetry Monoclinic 2 2 m (Cs) (C2) (C2h) monoclinic convention: symmetry located wrt b axis 2/m 2: 2-fold axis along b m: mirror perpendicular to b 2/m: 2-fold axis along b, perpendicular to a mirror 2

Monoclinic Orthorhombic 2 m 2/m 2 (C2v) mm2 xyz 3 symbols refer to: 222 (D2) mmm (D2h) axes along a, b, or c mirrors perpendicular to a, b, or c Rhombohedral (Trigonal) Tetragonal 3 (C3) 2 3 (S6) 3m (C3v) 4 (C4) 4 (S4) 4/m (C4h) 32 (D3) 3m (D3d) 4mm (C4v) 42m (D2d) 422 (D4) 4/mmm (D4h) 3 32 3 3m 3m Tetragonal Hexagonal 4 4 4mm 42m 4/m 422 6 (C6) 6 (C3h) 6/m (C6h) 6mm (C6v) 6m2 (D3h) 622 (D6) 4/mmm 6/mmm (D6h) 9/18/2013 3

9/18/2013 Hexagonal Cubic 6 6 6mm 6m2 6/m 622 6/mmm Lattices 14 Bravais lattices have Laue symmetry Lattices 14 Bravais lattices have Laue symmetry all have a center of symmetry center of symmetry very important in crystallography: centrosymmetric or noncentrosymmetric 23 (T) m3 (Th) 43m (Td) 432 (O) m3m (Oh) 23 432 m3 m3m 43m 14 Bravais Lattices triclinic monoclinic P1 P2/m C2/m orthorhombic Pmmm Cmmm Immm Fmmm rhombohedral tetragonal R3m P4/mmm I4/mmm hexagonal cubic P6/mmm Pm3m Im3m Fm3m Translational Symmetry in repeating lattices, two additional symmetry elements translational elements 1. screw axis rotation and translation: nr rotation by 360o/n; followed by translation of r/n along that axis (a, b or c) 2-fold screw axis most common: 21 2. glide plane reflection and translation: a, b, c, n or d reflection across plane; followed by translation of 1/2 (usually) unit cell parallel to plane along a, b, c, face diagonal (n), or body diagonal (d) 4

, c b a Screw Axis - 21 , + , – ½ ‛ , c b a Glide Plane - a ab plane ½ ‛ a-glide c http://www.cut-the-knot.org/Curriculum/Geometry/GlideReflection.shtml Space Groups Plane Groups translational elements + point symmetry  space groups in 2-D, referred to as plane groups there are 17 distinct ways of packing repeating object in 2-D wallpaper patterns p1 pm p2 pg p2mm Plane Groups Plane Groups p2mg p2gg cm c2mm 9/18/2013 5

Plane Groups Plane Groups p4 p4mm p3 p31m p4gm p3m1 Plane Groups p6 p6mm Plane Groups p1 pg p2gg pm cm c2mm pmg pmm p2 p4 p4mm p4gm p3 p3m1 p31m p6 p6mm Space Groups translational elements + 32 crystal point groups; 230 space groups 230 distinct ways of packing repeating object in 3-D Space Groups Centrosymmetric space groups P21 Pc P2 Pm Cc P2/m P21/m C2/m P2/c C2 Cm P21/c C2/c P1 P1 triclinic 1 1 monoclinic 2 m 2/m orthorhombic 222 mm2 mmm I222 P2221 P21212 P212121 C2221 C222 F222 P222 I212121 Pmm2 Pmc21 Pcc2 Pma2 Pca21 Pnc2 Pmn21 Pba2 Pna21 Pnn2 Ccc2 Amm2 Abm2 Ama2 Aba2 Fmm2 Cmm2 Cmc21 Fdd2 Imm2 Iba2 Pmmm Pnnn Pccm Pban Pmma Pnna Pmna Pcca Pbam Pccn Pnma Cmcm Cmca Cmmm Cccm Cmma Ccca Fmmm Fddd Immm Ibam Ibca Pbcm Pnnm Pmmn Pbcn Pbca Imma Ima2 9/18/2013 6

Space Groups P42 P43 I4 I41 I4/m P4222 I41/a P42212 tetragonal 4 4 4/m 422 4mm 42m P4 P4 P4/m P422 P4322 P4mm P42mc P42m P42b P41 I4 P42/m P4212 P43212 P4bm P42bc P42c P4n2 P4/n P4122 I422 P42cm I4mm P421m I4m2 P42/n P41212 I4122 P42nm I4cm P421c I4c2 4/mmm P4/mmm P4/mcc P4/nbm P4/nnc P4cc I41md P4m2 I42m P4/mbm P4/mnc P42/nnm P4/nmm P4/nnc P42/mbc P42/mnm P42/nmc P42/ncm I4/mmm I4/mcm I41/amd P42/mmc P42/mcm P42/nbc P4nc I41cd P4c2 I42d I41/acd Space Groups P32 P3 P3 P312 P3m1 P31m trigonal/rhombohedral P31 3 R3 3 P3112 32 P321 P31m P3c1 3m 3m P31c P3m1 hexagonal 6 6 6/m 622 6mm P61 P65 P6 P6 P6/m P622 P6mm P62m P6m2 P63/m P6122 P6cc P62c P6c2 6m2 6/mmm P6/mmm P6/mcc P63/mcm P63/mmc P62m P62c R3 P3121 P31c P3c1 P3212 R3m R3m P3221 R32 R3c R3c P62 P64 P63 P6522 P6222 P63cm P63mc P6422 6m2 P6322 P6m2 P6c2 cubic 23 m3 432 43m m3m Space Groups Symmetry P23 Pm3 P432 P4132 P43m Pm3m Fd3m F23 Pn3 P4232 I4132 F43m Pn3n Fd3c I23 Fm3 F432 I43m Pm3n Im3m P213 Fd3 F4132 P43n Pn3m Ia3d I213 Im3 I432 F43c Fm3m Pa3 P4332 I43d Fm3c 7 crystal systems: point symmetry of external lattice 14 Bravais lattices: translational symmetry of lattice points 32 point groups: point symmetry of external crystal 230 space groups: translational symmetry inside crystal molecules Space Groups Cambridge Structural Database (CSD) all compounds crystallize in one or more of these space groups usually possible to find P1, but always try to find the highest possible symmetry. structures observed in all 230 space groups ~95% of all structures: monoclinic, triclinic, orthorhombic ~83% of all structures: P21/c, P1, P212121, C2/c, P21, Pbca http://www.ccdc.cam.ac.uk/ 592938 structures 40000 35000 30000 25000 20000 15000 10000 5000 s e r u t c u r t S f o r e b m u N 0 1965 1975 1985 Year 1995 2005 9/18/2013 7

Space Group Frequency Space Group Nomenclature P21/c 200000 150000 Pī 100000 s e r u t c u r t S d e h s i l b u P f o r e b m u N space group name comes from Bravais lattice symbol, modified for translational symmetry easy to understand the components of many names, especially monoclinic and orthorhombic: P21/c (P 2-1 on c) primitive unit cell (1 lattice point) 2-fold screw axis along b (unique axis) c glide (translation along c axis) in ac plane (┴ to b) primitive unit cell (1 lattice point) b glide (translation along b axis) in bc plane (┴ to a) c glide (translation along c axis) in ac plane (┴ to b) a glide (translation along a axis) in ab plane (┴ to c) P212121 C2/c 50000 P21 Pbca 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 Space group number Pbca Standard and Non-standard Settings Equivalent Positions sometimes a space group that is not on the list of 230 is given in a publication some space groups can be derived which are identical with another space group  choice depends on convention P21/a identical with P21/c switching a and c label in monoclinic does not change the symmetry P21/n alternate setting of P21/c  closer to 90o preferred c΄ c n a Pnam same as Pnma switch b and c label space groups used to locate symmetry related atoms in unit cell for example, if a benzene ring is located on a mirror: locate 3 C and 3 H, H H H H C C C C C C C C H others at symmetry equivalent positions C H H H H asymmetric unit is the smallest part that generates the rest of the unit cell contents by all symmetry operations of space group Equivalent Positions, Asymmetric Unit and Z Equivalent Positions of Symmetry Elements equivalent positions are divided into: general positions special positions asymmetric unit along with general and special positions allows an interpretation of Z (number of molecules in unit cell), and possible molecular symmetry axis 2 2 2 21 21 21 plane m m m a ║to a b c a b c ┴ to a b c b position x, y, z x, y, z x, y, z x + ½, y, z x, y + ½, z x, y, z + ½ x, y, z x, y, z x, y, z x + ½, y, z plane ┴ to a b b c c n n n d d d c a c a b a b c a b c position x + ½, y, z x, y + ½, z x, y + ½, z x, y, z + ½ x, y, z + ½ x, y + ½, z + ½ x + ½, y, z + ½ x + ½, y + ½, z x, y + ¼ , z + ¼ x + ¼, y, z + ¼ x + ¼, y + ¼, z 9/18/2013 8

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