Title How to read and understand…

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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  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

Left point group point group symbol

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)

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 _ _ _ _ _ _ _ _ _ _ _ __

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 _ _ _ _ _ _

Trp Trp RNA-binding protein 1QAW 11-fold NCS axis (C 11 )

Xyl Xylose isomerase 1BXB

Xyl 222 Xylose isomerase 1BXB Tetramer 222 NCS symmetry (D 2 )

Left space group space group symbols

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)

Bravais lattices but the symmetry of the crystal is defined by its content, not by the lattice metric

Choice of cell Selection of unit cell - smallest - simplest - highest symmetry

Space group symbols

321 vs. 312

Left symmetry diagram diagram of symmetry operations positions of symmetry operations

Symmetry operators symbols

Left origin origin position vs. symmetry elements

Origin P212121

Origin P212121b

Origin C2

Origin C2b

Left asymmetric unit definition of asymmetric unit (not unique) V a.u. = V cell /N rotation axes cannot pass through the asymm. unit

Asymmetric unit P21

Left positions diagram diagram of equivalent positions

Right positions equivalent positions these are fractional positions (fractions of unit cell dimensions)

2-fold axes

P43212 symmetry

P43212 symmetry 1

P43212 symmetry 2

P43212 symmetry 2b

Multiple symmetry axes Higher symmetry axes include lower symmetry ones 4 includes 2 6 “ 3 and and 4 3 “ “ “ 3 1 and “ 3 2 and “ 3 2 and “ 3 1 and “ 3 and 2 1

P43212 symmetry 3

P43212 symmetry 4

P43212 symmetry 4b

P43212 symmetry 5

P43212 symmetry 6

P43212 symmetry 7

P43212 symmetry 8

P43212 symmetry 8b

Right special positions special positions

Special positions 0

Special positions 1

Special positions 2

Special positions 3

Special positions 3b

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)

Right subgroups subgroups

Subgroups reduced number of symmetry elements cell dimensions may be special cell may change

Subgroups 0

Subgroups 1a

Subgroups 1b

Subgroups PSCP Dauter, Z., Li M. & Wlodawer, A. (2001). Acta Cryst. D57, After soaking in NaBr cell changed, half of reflections disappeared

Right generators group generators

Right absences systematic presences (not absences) systematic absences result from translational symmetry elements

Absences 1

Absences 2

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 P , not 19