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How to read. Title. and understand…. Page. Left system. crystal system. Left point group. point group symbol. Left space group1. space group symbol international (Hermann-Mauguin) notation. Left space group2. space group symbol
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How to read Title and understand…
Left system crystal system
Left point group point group symbol
Left space group1 space group symbol international (Hermann-Mauguin) notation
Left space group2 space group symbol Schönflies notation
Left symmetry diagram diagram of symmetry operations positions of symmetry operations
Left positions diagram diagram of equivalent positions
Left origin origin position vs. symmetry elements
Left asymmetric unit definition of asymmetric unit (not unique)
Left Patterson Patterson symmetry Patterson symmetry group is always primitive centrosymmetric without translational symmetry operations
Right positions equivalent positions
Right special positions special positions
Right subgroups subgroups
Right absences systematic absences systematic absences result from translational symmetry elements
Right generators group generators
Individual items Interpretation of individual items
Left system crystal system
Systems 7 (6) Crystal systems Triclinic a ¹ b ¹ c a, b, g ¹ 90º Monoclinic a ¹ b ¹ c a=g = 90º, b ¹ 90º Orthorhombic a ¹ b ¹ c a=b=g = 90º Tetragonal a = b ¹ c a=b=g = 90º Rhombohedral a = b = c a=b=g Hexagonal a = b ¹ c a=b= 90º , g = 120º Cubic a = b = c a=b=g = 90º
Left point group point group symbol
Point groups describe symmetry of finite objects (at least one point invariant) Point groups Set of symmetry operations: rotations and rotoinversions (or proper and improper rotations) mirror = 2-fold rotation + inversion Combination of two symmetry operations gives another operation of the point group (principle of group theory)
Point groups describe symmetry of finite objects (at least one point invariant) Point groups general _ _ _ _ _ _ _ _ _ _ _ __ Schönflies International Examples Cn N 1, 2, 4, 6 Cnv Nmm mm2, 4mm Cnh N/m m, 2/m, 6/m Cni , S2n N 1, 3, 4, 6 Dn N22 222, 622 Dnh N/mmm mmm, 4/mmm Dnd N2m, Nm 3m, 42m, 62m T, Th , Td 23, m3, 43m O, Oh 432, m3m Y, Yh 532, 53m
32 crystallographic point groups (crystal classes) 11 noncentrosymmetric Point groups crystallographic _ _ _ _ _ _ _ _ Triclinic 1 1 Monoclinic 2 m, 2/m Orthorhombic 222 mm2, mmm Tetragonal 4, 422 4, 4/m, 4mm, 42m, 4/mmm Trigonal 3, 32 3, 3m, 3m Hexagonal 6, 622 6, 6/m, 6mm, 62m, 6/mmm Cubic 23, 432 m3, 43m, m3m
Trp RNA-binding protein 1QAW Trp 11-fold NCS axis (C11)
Xylose isomerase 1BXB Xyl 222 Tetramer 222 NCS symmetry (D2)
Left space group space group symbols
Space groups Space groups describe symmetry of infinite objects (3-D lattices, crystals) Combination of point group symmetry with translations - Bravais lattices - translational symmetry elements
Bravais lattices but the symmetry of the crystal is defined by its content, not by the lattice metric
Selection of unit cell - smallest - simplest - highest symmetry Choice of cell
Left symmetry diagram diagram of symmetry operations positions of symmetry operations
Left origin origin position vs. symmetry elements
Left asymmetric unit definition of asymmetric unit (not unique) Va.u. = Vcell/N rotation axes cannot pass through the asymm. unit
Left positions diagram diagram of equivalent positions
Right positions equivalent positions these are fractional positions (fractions of unit cell dimensions)
Higher symmetry axes include lower symmetry ones 4 includes 2 6 “ 3 and 2 41 and 43 “ 21 42 “ 2 61 “ 31 and 21 65 “ 32 and 21 62 “ 32 and 2 64 “ 31 and 2 63 “ 3 and 21 Multiple symmetry axes
