NOTE: Symbolism In Schönflies notation, what does the symbol S2 mean? Point Group of the Space Group Crystal System Space Group Molecules Solids Symmetry Operation Schönflies Notation International Notation Proper Rotation (by 2π/n) Cn (C2, C3, C4, …) “n” (2, 3, 4, …) Identity E = C1 1 Improper Rotation Sn = h  Cn (S3, S4, S5, …) Inversion (x,y,z)  (–x,–y,–z) i = S2 Mirror plane  Principal Axis h = S1 /m (n/m is the designator: 4/m) Mirror plane  Principal Axis v , d (= S1) m = NOTE: S2 = h  C2 x y , (z) In Schönflies notation, what does the symbol S2 mean? (x,y,z) S2 = inversion C2 rotation followed by sh  C2 axis In International notation, what does the symbol mean? x y (z) 2-fold (C2) rotation followed by inversion ( ) (x,y,z) , Why are the symbols S2 and not used?

Symbolism: Crystal Systems What rotational symmetries are consistent with a lattice (translational symmetry)? C1 C2 (2π/2) C3 (2π/3) C4 (2π/4) C6 (2π/6) Crystal System Minimum Symmetry Primitive Unit Cell Lattice Types Triclinic None a  b  c;      Monoclinic One 2-fold axis (b-axis) a  b  c;  =  = 90,   90 Orthorhombic Three orthogonal 2-fold axes a  b  c;  =  =  = 90 Tetragonal One 4-fold axis (c-axis) a = b  c;  =  =  = 90 Cubic Four 3-fold axes a = b = c;  =  =  = 90 Trigonal One 3-fold axis a = b = c;  =  =  a = b  c;  =  = 90,  = 120 Hexagonal One 6-fold axis (c-axis)  = angle between b and c  = angle between a and c  = angle between a and b c a b

“Centered Lattices” Symbolism: Bravais Lattices 7 Crystal Systems = 7 Primitive Lattices (Unit Cells): P Crystal System Minimum Symmetry Primitive Unit Cell Lattice Types Triclinic None a  b  c;      P Monoclinic One 2-fold axis (b-axis) a  b  c;  =  = 90,   90 P C Orthorhombic Three orthogonal 2-fold axes a  b  c;  =  =  = 90 P C (A) I F Tetragonal One 4-fold axis (c-axis) a = b  c;  =  =  = 90 P I Cubic Four 3-fold axes a = b = c;  =  =  = 90 P I F Trigonal One 3-fold axis a = b = c;  =  =  a = b  c;  =  = 90,  = 120 R Hexagonal One 6-fold axis (c-axis) (rhombohedral) “Centered Lattices”  = angle between b and c  = angle between a and c  = angle between a and b ? c a b I F C B A Body- (All) Face- Base-

Symbolism: Point Groups Schönflies Notation Type Symbol Features Uniaxial n Single rotation axis Cn nh + mirror plane  Cn axis nv + n mirror planes || Cn axis Low Symmetry 1 Asymmetric (NO symmetry) s Mirror plane only i Inversion center only Dihedral n Rotation axis Cn + n C2 axes  Cn axis nd nh Polyhedral T, Th , Td Tetrahedral; 4 C3 axes (cube body-diagonals) O, Oh Octahedral; 4 C3 axes + 3 C4 axes (cube faces) I, Ih Icosahedral; 6 C5 axes

Symbolism: Crystallographic Point Groups Allowed Rotations = C1 C2 C3 C4 C6 32 Point Groups Crystal System Schönflies Symbol International Symbol Order / Inversion? Full Abbrev. Directions Triclinic 1 1 1 / No (Holohedral) i 2 / Yes Monoclinic s 1m1 or 11m m [010] or [001] 2 / No 2 121 or 112 2 2h 12/m1 or 112/m 2/m 4 / Yes Orthorhombic 2v 2mm [100][010][001] 4 / No 2 222 2h 2/m 2/m 2/m mmm 8 / Yes Tetragonal 4 4 [001]{100}{110} 4 4h 4/m 2d 8 / No 4v 4mm 4 422 4h 4/m 2/m 2/m 4/mmm 16 / Yes Yes: Laue Groups b c m2m or mm2 a b c c a b a+b a–b

Symbolism: Crystallographic Point Groups (cont.) System Schönflies Symbol International Symbol Order / Inversion? Full Abbrev. Directions Trigonal 3 3 [001]{100}{210} 3 / No 6 6 / Yes 3v 3m or 3m1 6 / No 3 32 or 321 (Holohedral) 3d 12 / Yes Hexagonal 6 6 3h 6h 6/m 3h 12 / No 6v 6mm 6 622 62 6h 6/m 2/m 2/m 6/mmm 24 / Yes Cubic T 23 {100}{111}{110} Th Td 24 / No O 432 Oh 48 / Yes c a b 31m 312 c a b a b c