title how to read and understand…. page left system crystal system
Post on 20-Dec-2015
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TRANSCRIPT
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
_
_ _
_ _
_ _
_
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
