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Title How to read and understand…
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Left system crystal system
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Left point group point group symbol
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Left space group1 space group symbol international (Hermann-Mauguin) notation
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Left space group2 space group symbol Schönflies notation
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Left symmetry diagram diagram of symmetry operations positions of symmetry operations
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Left positions diagram diagram of equivalent positions
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Left origin origin position vs. symmetry elements
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Left asymmetric unit definition of asymmetric unit (not unique)
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Left Patterson Patterson symmetry Patterson symmetry group is always primitive centrosymmetric without translational symmetry operations
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Right positions equivalent positions
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Right special positions special positions
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Right subgroups subgroups
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Right absences systematic absences systematic absences result from translational symmetry elements
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Right generators group generators
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Individual items Interpretation of individual items
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Left system crystal system
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Systems 7 (6) Crystal systems Triclinic a b c , , 90 º Monoclinic a b c 90 º, 90 º Orthorhombic a b c 90 º Tetragonal a b c 90 º Rhombohedral a b c Hexagonal a b c 90 º, 120 º Cubic a b c 90 º
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Left point group point group symbol
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Point groups describe symmetry of finite objects (at least one point invariant) 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)
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Point groups general Point groups describe symmetry of finite objects (at least one point invariant) Schönflies International Examples C n N 1, 2, 4, 6 C nv Nmm mm2, 4mm C nh N/m m, 2/m, 6/m C ni, S 2n N 1, 3, 4, 6 D n N22 222, 622 D nh N/mmm mmm, 4/mmm D nd N2m, Nm 3m, 42m, 62m T, T h, T d 23, m3, 43m O, O h 432, m3m Y, Y h 532, 53m _ _ _ _ _ _ _ _ _ _ _ __
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Point groups crystallographic 32 crystallographic point groups (crystal classes) 11 noncentrosymmetric 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 _ _ _ _ _ _
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Trp Trp RNA-binding protein 1QAW 11-fold NCS axis (C 11 )
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Xyl Xylose isomerase 1BXB
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Xyl 222 Xylose isomerase 1BXB Tetramer 222 NCS symmetry (D 2 )
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Left space group space group symbols
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Space groups Combination of point group symmetry with translations - Bravais lattices - translational symmetry elements Space groups describe symmetry of infinite objects (3-D lattices, crystals)
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Bravais lattices but the symmetry of the crystal is defined by its content, not by the lattice metric
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Choice of cell Selection of unit cell - smallest - simplest - highest symmetry
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Space group symbols
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321 vs. 312
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Left symmetry diagram diagram of symmetry operations positions of symmetry operations
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Symmetry operators symbols
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Left origin origin position vs. symmetry elements
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Origin P212121
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Origin P212121b
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Origin C2
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Origin C2b
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Left asymmetric unit definition of asymmetric unit (not unique) V a.u. = V cell /N rotation axes cannot pass through the asymm. unit
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Asymmetric unit P21
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Left positions diagram diagram of equivalent positions
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Right positions equivalent positions these are fractional positions (fractions of unit cell dimensions)
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2-fold axes
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P43212 symmetry
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P43212 symmetry 1
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P43212 symmetry 2
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P43212 symmetry 2b
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Multiple symmetry axes Higher symmetry axes include lower symmetry ones 4 includes 2 6 “ 3 and 2 4 1 and 4 3 “ 2 1 4 2 “ 2 6 1 “ 3 1 and 2 1 6 5 “ 3 2 and 2 1 6 2 “ 3 2 and 2 6 4 “ 3 1 and 2 6 3 “ 3 and 2 1
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P43212 symmetry 3
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P43212 symmetry 4
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P43212 symmetry 4b
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P43212 symmetry 5
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P43212 symmetry 6
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P43212 symmetry 7
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P43212 symmetry 8
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P43212 symmetry 8b
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Right special positions special positions
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Special positions 0
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Special positions 1
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Special positions 2
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Special positions 3
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Special positions 3b
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Special positions on non-translational symmetry elements (axes, mirrors or inversion centers) degenerate positions (reduced number of sites) sites have their own symmetry (same as the symmetry element)
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Right subgroups subgroups
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Subgroups reduced number of symmetry elements cell dimensions may be special cell may change
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Subgroups 0
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Subgroups 1a
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Subgroups 1b
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Subgroups PSCP Dauter, Z., Li M. & Wlodawer, A. (2001). Acta Cryst. D57, 239-249. After soaking in NaBr cell changed, half of reflections disappeared
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Right generators group generators
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Right absences systematic presences (not absences) systematic absences result from translational symmetry elements
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Absences 1
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Absences 2
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Personal remark My personal remark: I hate when people quote space groups by numbers instead of name. For me the orthorhombic space group without any special positions is P2 1 2 1 2 1, not 19
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