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How to read
and understand…
Title
Page
crystal
system
Left system
point
group
symbol
Left point
space
group
symbol
international
(Hermann-Mauguin)
notation
Left space
space
group
symbol
Schönflies
notation
Left space
diagram
of symmetry
operations
positions
of symmetry
operations
Left symmetry
diagram of
equivalent
positions
Left positions
origin position vs.
symmetry elements
Left origin
definition of
asymmetric unit
(not unique)
Left
asymmetric
Patterson
symmetry
Patterson symmetry group is always
primitive
centrosymmetric
without translational symmetry operations
Left Patterson
equivalent
positions
Right positions
special
positions
Right special
subgroups
Right
systematic
absences
systematic
absences
result from
translational
symmetry elements
Right absences
group generators
Right
Interpretation of
individual items
Individual
crystal
system
Left system
7 (6) Crystal systems
Triclinic
a  b  c
a, b, g  90º
Monoclinic
a  b  c
a = g = 90º, b  90º
Orthorhombic
a  b  c
a = b = g = 90º
Tetragonal
a = b  c
a = b = g = 90º
Rhombohedral
a = b = c
a = b = g
Hexagonal
a = b  c
a = b = 90º , g = 120º
Cubic
a = b = c
a = b = g = 90º
Systems
point
group
symbol
Left point
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
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
Cni , S2n
N/m
_
N
m, 2/m, 6/m
_ _ _ _
1, 3, 4, 6
Dn
N22
222, 622
Dnh
N/mmm
_
_
N2m, Nm
mmm, 4/mmm
_
_
_
3m, 42m, 62m
Dnd
T , Th
O , Oh
Y , Yh
,
Td
_
23, m3, 43m
432, m3m
__
532, 53m
Point groups
32 crystallographic point groups
(crystal classes)
11 noncentrosymmetric
Triclinic
1
_
1
Monoclinic
2
m, 2/m
Orthorhombic
222
Tetragonal
4, 422
Trigonal
3, 32
Hexagonal
6, 622
Cubic
23, 432
mm2, mmm
_
4, 4/m, 4mm,
_
42m, 4/mmm
_
_
3, 3m, 3m
_
6, 6/m, 6mm,
_
62m, 6/mmm
_
m3, 43m, m3m
Point groups
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
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
Rhombohedral
Rhombohedral
reciprocal
Rhombohedral
reciprocal
Rhombohedral
reciprocal
Space group symbols
321 vs. 312
diagram
of symmetry
operations
positions
of symmetry
operations
Left symmetry
Symmetry
operators
origin position vs.
symmetry elements
Left origin
Origin P212121
Origin
Origin C2
Origin C2b
definition of
asymmetric unit
(not unique)
Va.u. = Vcell/N
rotation axes cannot pass
through the asymm. unit
Left
asymmetric
Asymmetric
diagram of
equivalent
positions
Left positions
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
Various
Various
Various
P43212 symmetry
P43212 symmetry 1
P43212 symmetry 2
P43212 symmetry 2b
Higher symmetry axes
include lower symmetry ones
4
6
41 and 43
42
61
65
62
64
63
includes
“
“
“
“
“
“
“
“
2
3
21
2
31
32
32
31
3
and 2
and
and
and
and
and
21
21
2
2
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
special
positions
Right special
Special positions 0
Special
Special
Special
Special
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
subgroups
Right
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
orthorhombic
PSCP
orthorhombic
PSCP
hexagonal
group generators
Right
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
Patterson symmetry group is always
primitive
centrosymmetric
without translational symmetry operations
Left Patterson
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
Personal