The 32 Point Groups 32 Crystallographic Point Groups 圃 cHc【cA dd mirror plane to the above 1l basic point groups, the dding mirror plane intersect ation, no new symmetry 6,6, Tetragonal S4. C4h D 2)Mirror plane is vertical to the main rotation axis, Py axis, and is Cubie agonal to the neighbo 令 Attention: adding F、toDa、 T and o is equal to dk、Tb and O respectively, Oa is equal to Oh. Identifying Point Groupss= Identifying Point Groups(1)The sandry is acedia Ye Pefecfy Ootheca a We can use a flow chart such mey(事 in this process are. mine the symmetry is Isthere at leof ene Iso rotting 2. Determine if there is a 卤面 incipal rotation axis. 3. Determine if there otation axes perpendicular to he principal axis. 4. Determine if there 5. Assign point group Identifying Point Groups(2, mris n Identifying Point Groups(3) ardr gee nis of rotation t axis of roation C Are theren Cax Is there a pane of symmetry o perpendicular to De perpend calar to the ipd us C parallel to the principe10 The 32 Point Groups C1 C2 C3 C4 C6 D2 D3 D4 D6 T O 1 1 +Ph Cs C2 h C3 h C4 h C6h D2h D3 h D4h D6h Th Oh 2 2 +Pv - - C2 v C3 v C4 v C6v - - - - - - - - - - - - 2 6 +Pd - - - - - - - - - - D2d D3 d - - - - Td - - 2 9 +C Ci - - C3 i - - - - - - - - - - - - - - - - 3 1 n S4 3 2 Add mirror plane to the above 11 basic point groups, the adding mirror plane intersect at one point with other symmetry elements, and in addition, no new symmetry types are formed, thus there are three ways: 1)Mirror plane is horizontal with the main rotation axis, Ph 2)Mirror plane is vertical to the main rotation axis, Pv 3)Mirror plane is vertical to the main rotation axis,and is diagonal to the neighboring 2-fold axis), Pd vAttention: adding Pv to Dn、T and O is equal to Dnh、Th and Oh respectively, Od is equal to Oh . 32 Crystallographic Point Groups Crystal System Number of Point Groups Herman-Mauguin Point Group Schoenflies Point Group Triclinic 2 1,`1 C1, Ci Monoclinic 3 2, m, 2/m C2, Cs , C2h Orthorhombic 3 222, mm2, mmm D2, C2v, D2h Trigonal 5 3,`3, 32, 3m, `3m C3, S6, D3, C3v, D3d Hexagonal 7 6,`6, 6/m, 622, 6mm, `62m, 6mm C6, C3h, C4h, D6, C6v, D3h , D6h Tetragonal 7 4,`4, 4/m, 422, 4mm,`42m, 4/mmm C4, S4, C4h, D4, C4v, D2d, D4h Cubic 5 23, m3, 432, `432, m`3m T, Th, O, Td, Oh We can use a flow chart such as this one to determine the point group of any object. The steps in this process are: 1. Determine the symmetry is special. 2. Determine if there is a principal rotation axis. 3. Determine if there are rotation axes perpendicular to the principal axis. 4. Determine if there are mirror planes. 5. Assign point group. Identifying Point Groups Identifying Point Groups (1) Identifying Point Groups (2) Identifying Point Groups (3)