Title

How to read

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

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 , , 90º

Monoclinic a b c 90º, 90º

Orthorhombic a b c 90º

Tetragonal a b c 90º

Rhombohedral a b c

Hexagonal a b c 90º , 120º

Cubic a b c 90º

Left point group

point group symbol

Point groups

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

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

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Point groups crystallographi

c

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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_ _

_ _

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Trp

Trp RNA-binding protein 1QAW

11-foldNCS axis (C11)

Xyl

Xylose isomerase 1BXB

Xyl 222

Xylose isomerase 1BXB

Tetramer222 NCSsymmetry (D2)

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

Rhombohedral cell 1

Rhombohedral cell 2

Rhombohedral reciprocal lattice 1

Rhombohedral reciprocal lattice 2

Rhombohedral reciprocal lattice 3

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)

Va.u. = Vcell/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

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

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

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

Subgroups

reduced number of symmetry elements

cell dimensions may be special

cell may change

Subgroups 0

Subgroups 1a

Subgroups 1b

Subgroups 3a

Subgroups 3b

Subgroups 2a

Subgroups 2b

Subgroups PSCP

Dauter, Z., Li M. & Wlodawer, A. (2001). Acta Cryst. D57, 239-249.

After soaking in NaBr cell changed, half of reflections disappeared

PSCP orthorhombic diffraction 1

PSCP orthorhombic diffraction 2

PSCP hexagonal diffraction

Right generators

group generators

Generators 1

Generators 2

Generators 3

Generators 4

Generators 5

Right absences

systematic presences (not absences)

systematic absences result from translational symmetry elements

Absences 1

Absences 2

Left Patterson

Patterson symmetry

Patterson symmetry group is always primitive centrosymmetric without translational symmetry operations

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

P212121, not 19