Symbolism Symmetry Operation Schnflies Notation International Notation Proper Rotation (by 2//n) Cn (C2/, C3, C4, ) n (2/, 3, 4, ) Identity Improper Rotation E = C1 Sn = h Cn (S3, S4, S5, ) 1 n 1 n (3, 4, 5,) Inversion (x,y,z) (x,y,z) i = S2/ 1 Mirror plane Principal Axis h = S1 Mirror plane Principal Axis v , d (= S1) /m (n/m is the designator: 4/m) 2/ m= y In Schnflies notation, what does the symbol S2 mean? (x,y,z) x In International notation, what does the symbol 2/ mean? y (x,y,z) x 1 Symbolism: Crystal Systems What rotational symmetries are consistent with a lattice (translational symmetry)? Crystal System Minimum Symmetry Primitive Unit Cell 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, = 12/0 Hexagonal One 6-fold axis (c-axis) a = b c; = = 90, = 12/0 c a b Lattice Types = angle between b and c = angle between a and c = angle between a and b 2/ Symbolism: Bravais Lattices 7 Crystal Systems = 7 Primitive Lattices (Unit Cells): Including Centering: 14 Bravais Lattices 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, = 12/0 R (rhombohedral) P Hexagonal One 6-fold axis (c-axis) a = b c; = = 90, = 12/0 P Centering? c a b = angle between b and c = angle between a and c = angle between a and b 3 Symbolism: Point Groups Schnflies Notation Type Symbol Features Uniaxial
n Single rotation axis Cn Low Symmetry Dihedral Polyhedral nh + mirror plane Cn axis nv + n mirror planes || Cn axis 1 Asymmetric (NO symmetry) s Mirror plane only i Inversion center only n Rotation axis Cn + n C2/ axes Cn axis nd + n mirror planes || Cn axis nh + mirror plane Cn axis 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 4 Symbolism: Crystallographic Point Groups Allowed Rotations = C1 C2 C3 C4 C6 32 Point Groups International Symbol Crystal System
Schnflies Symbol Full Abbrev. Triclinic 1 1 1 1 1 s 1m1 or 11m m 2/ 12/1 or 112/ 2/ 2/ / No 2/h 12//m1 or 112//m 2//m 4 / Yes 2/v 2/mm 2/mm 2/ 2/2/2/ 2/2/2/ 4 / No 2/h 2//m 2//m 2//m 44 mmm 44 8 / Yes 42/m 4/m 42/m 4/m
(Holohedral) Monoclinic (Holohedral) Orthorhombic (Holohedral) Tetragonal i 4 4 4h Directions Order / Inversion? 1 / No 2/ / Yes [010] or [001] [100][010][001] [001]{100}{110} 2/ / No 4 / No 4 / No 4 / No 8 / Yes 8 / No 2/d 4v 4mm 4mm 8 / No 4 42/2/ 42/2/ 8 / No 5 Symbolism: Crystallographic Point Groups (cont.) International Symbol Crystal System Schnflies Symbol Full Abbrev. Directions
Trigonal 3 3 3 3 3 [001]{100}{2/10} Hexagonal 3v 3m or 3m1 3m or 3m1 6 / No 3 3m 3 m1 32/ or 32/1 332/ m or 3 m1 or 32/1 6 / No 3d 6 3h 6h 12/ / Yes 6 6 6 m2/ 6/m 6 6 [001]{100}{2/10} Cubic 6 / No 6 / No 6 m2/ 6/m 12/ / Yes 12/ / No 3h (Holohedral) 3 / No 6 / Yes
6 (Holohedral) Order / Inversion? 6v 6mm 6mm 12/ / No 6 62/2/ 62/ 12/ / No 6h / m 32//m 6/m2/ 2//m m3 6/mmm 2/4 / Yes T 43m 2/3 43m 2/3 12/ / No 2/4 / Yes Th Td {100}{111}{110} 4 / m 3 2/ / m m3m 2/4 / No 6 Symbolism: Symmetry Operations (4h = 4/mmm = 4/m 2/m 2/m) Schnflies International Coordinates (1) E 1
x, y, z (2/) C2/ = C42/ 2/ 0,0,z (3) C4 (4) Schnflies International Coordinates (9) i x, y, z x, y, z (10) h 4+ 0,0,z y, x, z (11) S 43 0,0,z; 0,0,0 y, x, z C43 4 0,0,z y, x, z (12/) S4 0,0,z; 0,0,0 y, x, z (5) C2/ 2/ 0,y,0 x, y, z (13) v
m x,0,z x, y, z (6) C2/ 2/ x,0,0 x, y, z (14) v m 0,y,z x, y, z (7) C2/ 2/ x,x,0 y, x, z (15) d m x,,z y, x, z (8) C2/ 2/ x,,0 y, x, z (16) d m x,x,z y, x, z m x,y,0 x, y, z 4/mmm y x + 7 Symmorphic Space Groups Point Group = Space Group = Symmorphic Space Groups (73): Nonsymmorphic Space Groups (157):
8 Symmorphic Space Groups Space Group: I4/mmm Asymmetric Unit: BaFe2As2 Ba (2a): Fe (4d): As (4e): 0 0 0 0 0 0 0.3544 9 Symmorphic Space Groups What is the empirical formula of this compound? How many formula units are in the unit cell? Asymmetric Unit: Dy Dy Ni B B (4e) (4d) (8j) (16l) (16m) 0 0 0.273 0.274 0.269 0 0.5 0.5 0.111 0.269 0.203 0.25 0 0 0.152 10 11