9/18/2013. Symmetry Operations and Space Groups. Crystal Symmetry. Symmetry Elements. Center of Symmetry: 1. Rotation Axis: n. Center of Symmetry: 1
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1 Symmetry Operations and rystal Symmetry 32 point groups of crystals compatible with 7 crystal systems crystallographers use ermann-mauguin symmetry symbols arl ermann German harles-victor Mauguin French there are 5 types in point symmetry. center of symmetry (or inversion) : point 2. rotation (or proper) axis : line 3. mirror : plane 5. identity Symmetry Elements 4. rotation inversion axis : line n : no element n m enter of Symmetry: 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 enter of Symmetry: Rotation Axis: n all points () ( x, y, z) if is placed at the origin y z + () n is an integer which gives the degrees of rotation: 2 or n 360 o n is the number of times molecule is rotated, each time stopping at an identical appearance, before returning to the starting point n x ( x, y, z) n is the foldness of the rotation axis only 2, 3, 4, and 6-fold axes allowed in crystal symmetry
2 Rotation Axis: 4 Mirror: m 360 o = 90 4 o plane within the molecule that, when acting as a mirror, reflects the molecule into itself o 4 Rotation-Inversion n Representation of Symmetry rotation followed by inversion this is a different definition than Schoenflies system Arthur Moritz Schönflies German 89 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 Monoclinic 2 ( ) ( i ) highest symmetry in each crystal system is called: Laue Group 2 ( 2 ) m ( s ) 2/m ( 2h ) monoclinic convention: symmetry located wrt b axis 2: 2-fold axis along b m: mirror perpendicular to b (Schoenflies symbol) have center of symmetry 2/m: 2-fold axis along b, perpendicular to a mirror 2
3 Monoclinic Orthorhombic 2 2 m 2/m mm2 ( 2v ) 222 (D 2 ) mmm (D 2h ) xyz 3 symbols refer to: axes along a, b, or c mirrors perpendicular to a, b, or c Rhombohedral (Trigonal) Tetragonal 3 ( 3 ) 2 3 (S 6 ) 3m ( 3v ) 4 ( 4 ) 4 (S 4 ) 4/m ( 4h ) 32 (D 3 ) 3m (D 3d ) 4mm ( 4v ) 42m (D 2d ) 422 (D 4 ) 4/mmm (D 4h ) m 3m Tetragonal exagonal 4 4 4/m 6 ( 6 ) 6 ( 3h ) 6/m ( 6h ) 4mm 42m 422 6mm ( 6v ) 6m2 (D 3h ) 622 (D 6 ) 4/mmm 6/mmm (D 6h ) 3
4 exagonal ubic 6 6 6/m 23 (T) m3 (T h ) 43m (T d ) 6mm 6m (O) m3m (O h ) 6/mmm m3 m3m 43m Lattices 4 Bravais Lattices triclinic monoclinic 4 Bravais lattices have Laue symmetry P P2/m 2/m orthorhombic Pmmm mmm Immm Fmmm rhombohedral tetragonal R3m P4/mmm I4/mmm hexagonal cubic P6/mmm Pm3m Im3m Fm3m Lattices Translational Symmetry in repeating lattices, two additional symmetry elements 4 Bravais lattices have Laue symmetry all have a center of symmetry center of symmetry very important in crystallography: centrosymmetric or noncentrosymmetric translational elements. screw axis rotation and translation: n r rotation by 360 o /n; followed by translation of r/n along that axis (a, b or c) 2-fold screw axis most common: 2 2. glide plane reflection and translation: a, b, c, n or d reflection across plane; followed by translation of /2 (usually) unit cell parallel to plane along a, b, c, face diagonal (n), or body diagonal (d) 4
5 Screw Axis - 2 a c b,, +, Glide Plane - a ½ a c b, ½ a-glide c ab plane translational elements + point symmetry space groups in 2-D, referred to as plane groups there are 7 distinct ways of packing repeating object in 2-D p p2 wallpaper patterns pm pg p2mm p2mg cm p2gg c2mm 5
6 p4 p4mm p3 p3m p4gm p3m p pg p2gg pm cm c2mm pmg pmm p2 p4 p6 p6mm p4mm p4gm p3 p3m p3m p6 p6mm translational elements + 32 crystal point groups; 230 space groups 230 distinct ways of packing repeating object in 3-D triclinic P P entrosymmetric space groups monoclinic 2 P2 P2 2 m Pm Pc m c 2/m P2/m P2 /m 2/m P2/c P2 /c 2/c orthorhombic 222 P222 P222 P2 2 2 P F222 I222 I2 2 2 mm2 Pmm2 Pmc2 Pcc2 Pma2 Pca2 Pnc2 Pmn2 Pba2 Pna2 Pnn2 cc2 Amm2 Abm2 Ama2 Aba2 Fmm2 mm2 mc2 Fdd2 Imm2 Iba2 Ima2 mmm Pmmm Pnnn Pccm Pban Pmma Pnna Pmna Pcca Pbam Pccn Pbcm Pnnm Pmmn Pbcn Pbca Pnma mcm mca mmm ccm mma cca Fmmm Fddd Immm Ibam Ibca Imma 6
7 tetragonal 4 P4 P4 P4 2 P4 3 I4 I4 4 P4 I4 4/m P4/m P4 2 /m P4/n P4 2 /n I4/m I4 /a 422 P422 P42 2 P4 22 P4 2 2 P P P P I422 I4 22 4mm P4mm P4bm P4 2 cm P4 2 nm P4cc P4nc P4 2 mc P4 2 bc I4mm I4cm I4 md I4 cd 42m P42m P42c P42 m P42 c P4m2 P4c2 P42b P4n2 I4m2 I4c2 I42m I42d 4/mmm P4/mmm P4/mcc P4/nbm P4/nnc P4/mbm P4/mnc P4/nmm P4/nnc P4 2 /mmc P4 2 /mcm P4 2 /nbc P4 2 /nnm P4 2 /mbc P4 2 /mnm P4 2 /nmc P4 2 /ncm I4/mmm I4/mcm I4 /amd I4 /acd trigonal/rhombohedral 3 P3 P3 P3 2 R3 3 P3 R3 32 P32 P32 P3 2 P3 2 P3 2 2 P3 2 2 R32 3m P3m P3m P3c P3c R3m R3c 3m P3m P3c P3m P3c R3m R3c hexagonal 6 P6 P6 P6 5 P6 2 P6 4 P6 3 6 P6 6/m P6/m P6 3 /m 622 P622 P6 22 P P P P mm P6mm P6cc P6 3 cm P6 3 mc 6m2 P6m2 P6c2 P62m P62c 6m2 P6m2 P6c2 P62m P62c 6/mmm P6/mmm P6/mcc P6 3 /mcm P6 3 /mmc cubic 23 P23 F23 I23 P2 3 I2 3 m3 Pm3 Pn3 Fm3 Fd3 Im3 Pa3 432 P432 P F432 F4 32 I432 P P4 32 I m P43m F43m I43m P43n F43c I43d m3m Pm3m Pn3n Pm3n Pn3m Fm3m Fm3c Fd3m Fd3c Im3m Ia3d Symmetry 7 crystal systems: point symmetry of external lattice 4 Bravais lattices: translational symmetry of lattice points 32 point groups: point symmetry of external crystal 230 space groups: translational symmetry inside crystal molecules ambridge Structural Database (SD) all compounds crystallize in one or more of these space groups usually possible to find P, 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: P2 /c, P, P2 2 2, 2/c, P2, Pbca Number of Structures structures Year 7
8 Number of Published Structures Pī P2 /c P /c P2 Space Group Frequency Pbca Space group number Space Group Nomenclature space group name comes from Bravais lattice symbol, modified for translational symmetry easy to understand the components of many names, especially monoclinic and orthorhombic: P2 /c (P 2- on c) primitive unit cell ( lattice point) 2-fold screw axis along b (unique axis) c glide (translation along c axis) in ac plane ( to b) Pbca primitive unit cell ( 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) Standard and Non-standard Settings 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 P2 /a identical with P2 /c switching a and c label in monoclinic does not change the symmetry P2 /n alternate setting of P2 /c c c n closer to 90 o preferred a Pnam same as Pnma switch b and c label Equivalent Positions space groups used to locate symmetry related atoms in unit cell for example, if a benzene ring is located on a mirror: locate 3 and 3, asymmetric unit is the smallest part that generates the rest of the unit cell contents by all symmetry operations of space group others at symmetry equivalent positions Equivalent Positions, Asymmetric Unit and Z 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 Equivalent Positions of Symmetry Elements axis to position 2 a 2 b 2 c 2 a x + ½, y, z 2 b x, y + ½, z 2 c + ½ plane to n b x + ½, y, z + ½ m a n c x + ½, y + ½, z m b d a x, y + ¼, z + ¼ m c d b x + ¼, y, z + ¼ a b x + ½, y, z plane to position a c x + ½, y, z b a x, y + ½, z b c x, y + ½, z c a + ½ c b + ½ n a x, y + ½, z + ½ d c x + ¼, y + ¼, z 8
9 Equivalent Positions from entering Transforming oordinates for centered groups, add the following to each P general position: A x, y + ½, z + ½ x + ½, y + ½, z F x + ½, y + ½, z x + ½, y, z + ½ x, y + ½, z + ½ I x + ½, y + ½, z + ½ R x + 2/3, y + /3, z + /3 x + /3, y + 2/3, z + 2/3 x ¼ = (x + ¼) = (x ¼ + ½) = (x + ½) = x ½ = x + ½ (by adding ) y + ¼ = y ¼ + ½ = y + ½ Transforming oordinates Transforming oordinates x ¼ = (x + ¼) = (x ¼ + ½) = (x + ½) = x ½ = x + ½ (by adding ) y + ¼ = y ¼ + ½ = y + ½ x y, P2 /a x + ¼ = (x ¼) + ½ = x + ½ + ½ = x (by subtracting ) rename: x ¼as x y ¼as y z as z. x ¼, y ¼, z 2. x ¼, y +¼, z 3. x + ¼, y + ¼, z 4. x + ¼, y ¼, z x + ½, y + ½, z x + ½, y + ½, z Transforming oordinates Special Positions related by a change in sign related by a change in sign x + ½, y + ½, z x + ½, y + ½, z z y, 2 molecules/cell finally, change to preferred setting P2 /c; switch x and z x, y + ½, z + ½ x, y + ½, z + ½ general positions P2 /c if an object is located at = 0, 0, 0; only unique point generated by symmetry is at 0, ½, ½ also true for: 0, 0, ½ 0, ½, 0 ½, 0, ½ ½, ½, 0 ½, 0, 0 ½, ½, ½ 9
10 note: Special Positions 0, 0, 0 0, ½, ½ 0, 0, ½ 0, ½, 0 ½, 0, ½ ½, ½, 0 ½, 0, 0 ½, ½, ½ an object (molecule) at a special position has to have the same symmetry as the special position Special Positions an atom on a special position has at least one fixed coordinate; part of the atom generates the rest: one fixed position (axis to plane) for an atom on a mirror two fixed positions (other axes) for an atom on a rotation axis three fixed positions for an atom on an inversion center in P2 /c, a center of symmetry Z = 4 for an object on a general position in P2 /c ac mirror (x, 0, z) or (x, ½, z) Z = 2 for an object on a special position in P2 /c asymmetric unit is ½ of the molecule International Tables for rystallography International Tables for rystallography site symmetry general positions Z special positions 0
11 General Positions for Other P2 /c: Z = P2 /c: Z = 4 P2 /c: Z = 2 atom on special position no atom on special position
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