How to read and understand… Title

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crystal system Left system

point group symbol Left point group

space group symbol international (Hermann-Mauguin) notation Left space group1

space group symbol Schönflies notation Left space group2

diagram of symmetry operations positions of symmetry operations Left symmetry diagram

Left positions diagram diagram of equivalent positions Left positions diagram

origin position vs. symmetry elements Left origin

definition of asymmetric unit (not unique) Left asymmetric unit

Patterson symmetry group is always primitive centrosymmetric without translational symmetry operations Left Patterson

equivalent positions Right positions

Right special positions

subgroups Right subgroups

systematic absences systematic absences result from translational symmetry elements Right absences

group generators Right generators

Interpretation of individual items Individual items

crystal system Left system

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 Systems

point group symbol Left point group

describe symmetry of finite objects (at least one point invariant) 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

Schönflies International Examples Cn N 1, 2, 4, 6 Cnv Nmm mm2, 4mm Point groups describe symmetry of finite objects (at least one point invariant) _ _ _ _ _ _ _ _ _ _ _ __ 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 Point groups general

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 Point groups crystallographic

Trp RNA-binding protein 1QAW 11-fold NCS axis (C11) Trp

Xylose isomerase 1BXB Xyl

Xylose isomerase 1BXB Tetramer 222 NCS symmetry (D2) Xyl 222

space group symbols Left space group

describe symmetry of infinite objects (3-D lattices, crystals) Space groups describe symmetry of infinite objects (3-D lattices, crystals) Combination of point group symmetry with translations - Bravais lattices - translational symmetry elements Space groups

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

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

Rhombohedral cell 1

Rhombohedral cell 2

Rhombohedral reciprocal lattice 1

Rhombohedral reciprocal lattice 2

Rhombohedral reciprocal lattice 3

Space group symbols

321 vs. 312

diagram of symmetry operations positions of symmetry operations Left symmetry diagram

Symmetry operators symbols

origin position vs. symmetry elements Left origin

Origin P212121

Origin P212121b

Origin C2

Origin C2b

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

Asymmetric unit P21

Left positions diagram diagram of equivalent positions Left positions diagram

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

2-fold axes

3-fold axis 1

3-fold axis 2

Various positions 1

Various positions 2

Various positions 3

Various positions 4

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 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

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 0

Special positions 1

Special positions 2

Special positions 3

Special positions 3b

on non-translational symmetry elements 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) Special positions

subgroups Right subgroups

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

Subgroups 0

Subgroups 1a

Subgroups 1b

Subgroups 3a

Subgroups 3b

Subgroups 2a

Subgroups 2b

Dauter, Z., Li M. & Wlodawer, A. (2001). Acta Cryst. D57, 239-249. After soaking in NaBr cell changed, half of reflections disappeared Subgroups PSCP

PSCP orthorhombic diffraction 1

PSCP orthorhombic diffraction 2

PSCP hexagonal diffraction

group generators Right generators

Generators 1

Generators 2

Generators 3

Generators 4

Generators 5

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

Absences 1

Absences 2

Patterson symmetry group is always primitive centrosymmetric without translational symmetry operations Left Patterson

My personal remark: P212121, not 19 I hate when people quote space groups by numbers instead of name. For me the orthorhombic space group without any special positions is P212121, not 19 Personal remark