introduction to point and space group symmetry - hu...
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Introduction to point and space group symmetry
Humboldt-Universität zu Berlin, Institut für Physik, AG TEM Newtonstrasse 15, D-12489 Berlin E-mail: [email protected] Web: http://crysta.physik.hu-berlin.de/ag_tem/
Holm Kirmse
Workshop on Electron Crystallography, Nelson Mandela Metropolitan University, South Africa, October 14-16, 2013
Embedding Crystallography
Mathematics Lattice, Symmetry operations, Group theory
2
Physics Transparency
Conductivity
Birefringence
Piezoelectricity
Pyroelectricity
Chemistry Bonding
Composition
Structure
Cleavability
Polarity
CRYSTALLO-GRAPHY
Topics of Crystallography
3
Geometric Crystallography
Structure analysis X ray and electron
diffraction
Crystal growth
Crystal chemistry Crystal physics Crystal defects
Geometrical Crystallography, e.g., Au
4
Crystal system Coordinate system
Symmetry element
Projection
view [100] Indices Crystal classes (cubooktahedral)
Crystal shape
(111)
facet
of
hedron
octa-
(100)
Road map Introduction
• What is a crystal?
• Definition of the 7 crystal systems
• Indexing planes and directions
• Bravais lattices
• Stereographic projection
• Symmetry operations of point groups
• The 32 point groups
• From point groups to layer groups
• Symmetry operations of layer groups
• The 17 layer groups
• Transition to third dimension: space groups
• Symmetry operations of space groups
• Example for determination of a space group
• Notations 5
What is a crystal?
• Characteristics of an ideal crystal:
– Flat regular surfaces/facets
– Characteristic symmetry
– Chemically homogeneous object
– Anisotropic properties
– Periodic arrangement of constituents like atoms or molecules along the three spatial directions
6
Rock crystal, SiO2
Definition of a crystal
– A crystal is a homogeneous anisotropic solid with periodic arrangement of constituents like atoms or molecules in all three spatial dimensions.
– Since 1991: A crystal is defined as a solid exhibiting discrete diffraction spots. This definition includes real crystals with defects, incommensurate crystals and quasicrystals.
7
Coordinate system
• 3-dimensional lattices are described by three not necessarily orthogonal directions x, y, z having lattice parameters a, b, c.
• The angle between the three directions are denoted as a, b, g.
• As a convention a right-handed coordinate system is used.
9
a b
c
a b
g x
z
y
The 7 crystal systems
Crystal system Lattice parameters
Triclinic a ≠ b ≠ c a ≠ b ≠ g ≠ 90°
Monoclinic a ≠ b ≠ c a = g = 90° b ≠ 90°
Orthorhombic a ≠ b ≠ c a = b = g = 90°
Tetragonal a = b ≠ c a = b = g = 90°
Trigonal a = b = c a = b = g ≠ 90°
a = b ≠ c a = b = 90° g = 120°
Hexagonal a = b ≠ c a = b = 90° g = 120°
Cubic a = b = c a = b = g = 90°
10
Indexing of crystallographic directions
• Two-dimensional lattice
12
Crystallographic direction: straight line crossing two arbitrary points of the lattice
a
b
[010] [120] [110] [210]
Iuvw = 0·a + 1·b + 0·c Description analogous to vectors
Indexing of crystallographic directions
• Three-dimensional lattice
13
Triple product: (a × b) · c > 0, i.e. non-coplanar
Point in space: Iuvw = u·a + v·b + w·c ; u,v,w Z
Infinite periodic arrangement of points within the 3-d space
c
a
b
Indexing of crystallographic planes
• Derivation of MILLER indices:
14
O
a
A
c
b
C
B
p:n:mc
OC:b
OB:a
OA
000
m : n : p = 2 : 4 : 3
3
1:4
1:2
1
p
1:n
1:m
1
h : k : l = 6 : 3 : 4 (6 3 4) plane
WEISS indices:
MILLER indices
The 14 Bravais lattices (A. Bravais, 1850)
20
Crystal system
Cente-ring
Symbol Lattice
Triclinic Prim. aP
Monocl.
Prim. mP
Face cent.
mA
Orthorh.
Prim. oP
Body cent.
oI
Basal-pl. cent.
oC
Face cent.
oF
Crystal system
Cente-ring
Symbol Lattice
Tetrag.
Prim. tP
Body cent.
tI
Trigonal Rhom-bohedr.
hR
Trig. + Hexag.
Prim. hP
Cubic
Prim. cP
Body cent.
cI
Face cent.
cF
From inner structure to morphology: description of crystals
• Correspondence between morphology and structure
• Any crystal face (morphology) is oriented parallel to a lattice plane (structure).
• The symmetry of the outer shape of a crystal is higher or equal to the symmetry of its inner structure.
22
Example: Galenite (PbS):
Crystal projections: stereographic projection
23
• Projection method:
Pole sphere
Observation spot
Face normal
Pole of face
North pole of pole sphere
Projection plane Projected pole
South pole of pole sphere
Crystal projections: stereographic projection
25
• Examples for stereograms:
Hexahedron Octahedron
Rhombic dodecahedron Polyhedron:
Stereo- gram
111
010
001
010001
111111
111 101
011
110 011
110
101011
011
100
100
111
111111
111
101
110
110 101
( )
( )
( )
( )
( )
( ) ( )
( ) ( )
( )
( ) ( )
( )
( )
( )
( ) ( ) ( )
( )
( )
( ) ( )
( )
( ) ( )
( )
Symmetry of crystals
Point symmetry operations:
27
Symmetry operation with at least one point of the object remains at its original position. The corresponding symmetry element is called point symmetry element.
Trivial point symmetry operations : ROTATION, INVERSION, REFLECTION
Combined point symmetry operations : ROTATION INVERSION, REFLECTION ROTATION
Point symmetry elements in 3-d space: Rotation axis, inversion centre, mirror plane, rotation inversion axis, reflection rotation plane
Point symmetry operations
28
• Rotation:
4-fold rotation Rotation angle: 90° Symbol: 4 Graphic symbol:
Stereogram: Shape: Impact on structure motive:
Tetragonal pyramid
Point symmetry operations
29
• Rotation Number of positions
Angle Symbol Stereogram Shape
1 360° 1 - Pedion
2 180° 2 Sphenoid
3 120° 3 Trigonal pyramid
4 90° 4 Tetragonal pyramid
6 60° 6 Hexagonal pyramid
Point symmetry operations
30
• Inversion
Stereogram: Shape: Impact on structure motiv:
Symbol: 1 Graphic symbol:
Pinakoid
Inversion centre
Point symmetry operations
31
• Reflection
Stereogram: Shape:
Doma
Impact on structure motiv:
Mirror plane
Symbol: m (mirror) = 2 + 1 Graphic symbol:
Point symmetry operations
32
• Rotation inversion
4-fold rotation inversion Symmetry operation: 90° + Inversion Symbol: 4 Graphic symbol:
Stereogram: Shape: Impact on structure motiv:
Tetragonal disphenoid
Point symmetry operations
33
• Rotation inversion Number of positions
Angle Symbol Stereogram Shape
1 360° 1 Pinakoid
2 180° 2 Doma
3 120° 3 Rhombo-hedron
4 90° 4 Tetragonal disphenoid
6 60° 6 Hexagonal dipyramid
The 32 point symmetry groups
35
X X Xm Xm X2 m
tric
linic
1
mo
no
clin
ic
ort
ho
rh.
2
trig
on
al
3
tetr
ago
nal
4
hex
ago
nal
6
cub
ic
23
X m
X m
1
2
3
4
6
3m
232
2m
1
m
2
m
4
m
6
6m
3
3m
2
2m1 m
2m1 212 2mmmm
1
2mm
m3
mm4
mm6
3m
23m2
2mmm2
m
23
m24
m26
m34
222
32
422
622
432
m
2
m
2
m
2
m26mm
3
m
2
m
2
m
4
m
2
m
2
m
6
m
23m
4
Notation of point symmetry groups
• Notation following Hermann-Maugin
36
HA: main axis, NA: minor axis, ZA: intermediate axis
Crystal system 1st position 2nd position 3rd position
Triclinic x - -
Monoclinic y - -
Orthorhombic x y z
Trigonal z (HA) x (NA) -
Tetragonal z (HA) x (NA) xy (ZA)
Hexagonal z (HA) x (NA) xy (ZA)
Cubic [100] [111] [110]
n: n-fold rotation axis, n: n-fold rotation inversion axis, m: mirror plane, : n-fold rotation axis with mirror plane
m
n
Example for the determination of a point symmetry group
49
Crystal system: tetragonal
Crystal class: ditetragonal-pyramidal
Symbol following Hermann-Maugin:
Symmetry elements
Stereographic projection
Ditetragonal pyramid
mm4
8 (+1) facets
Morphology / shape of crystal
The two-dimensional lattice
• Infinite periodic arrangement of points (i.e. atoms, ions, or molecules) at a plane
51
a
b
|a| = a |b| = b |a x b| > 0 (i.e. non-collinear)
Expression for a single point: Iuv = u·a + v·b; u,v Z
Symmetry operation: Translation
54
• Shift of a motive (asymmetric unit: atoms, ions, molecules) by translation vector t
t
No longer pure point symmetry only
Space filling
New symmetry operations
New symmetry operation: Glide reflection g
56
Simultaneous application of translation and reflection:
Step 1: Translation by t = ½ a0
Step 2: Reflection
Symmetry elements of layer groups (wallpaper groups)
Reflection m Rotation
Glide reflection g
Motive Angle Multiplicity Symbol
360° 1
180° 2
120° 3
90° 4
60° 6
Symbol
Glide component: t/2
t
Lattice types
Primitive: p Centered: c
The 17 layer groups (wallpaper groups) oblique
rectangular
quadratic
hexagonal
p 1 p 211
p 1m1 p 1g1
c 1m1 p 2mm
p 2mg p 2gg c 2mm
p 4
p 4gm
p 4mm
p 3 p 3m1
p 31m p 6 p 6mm
The 17 layer groups (wallpaper groups) p 1 p 211
p 1m1 p 1g1
c 1m1 p 2mm
p 2mg p 2gg c 2mm
p 4
p 4gm
p 4mm
p 3 p 3m1
p 31m p 6 p 6mm
oblique
rectangular
quadratic
hexagonal
The three-dimensional lattice
• Infinite periodic arrangement of points within the 3-d space
63
c
a
b
Space groups
65
• Space groups for describing symmetry relations in 3-d space.
• Space groups include all symmetry operations of a 3-dimensional, infinitely extended, and perfect crystal structure.
• The notation of the space group is done after Hermann-Mauguin.
• Number of Space groups in the 3-d space: 230
• But! By definition there is an infinite number of space groups!
• There are 73 types of space groups comprising the identical (point) symmetry elements as the point symmetry group but extended by the translation operation: These are the symmorphic space groups.
Symmetry operation: Glide reflection
68
n d
(a+b)/4, (a+c)/4, (b+c)/4, (a+b+c)/4
(a+b)/2, (a+c)/2, (b+c)/2, (a+b+c)/2
Symmetry operation: Screw rotation
69
• 2-fold rotation • 2-fold screw rotation
= ½ ao
• Translation period of a screw axis np: = p/n
2 21
Symmetry operation: Screw rotation
70
= 1/4 = 2/4 = 3/4
41 43
42
• 41 and 43 are enantiomorphous screw axis
• 41 dextrorotatory, 43 laevorotatory, 42 without rotary sense like 2
• 4-fold screw rotation:
Symbols for orientation of rotation axis along the viewing direction
Symbols of rotation axes and screw rotation axes
71
Symbols for inclined rotation axis
Example for a space group
• Marcasite (FeS2)
73
½ +
½ +
½ -
½ -
½
+
-
+
-
Projection onto basal plane
Example for a space group
• Marcasite (FeS2)
74
½ +
½ +
½ -
½ -
½
+
-
+
-
Space group:
¼, ¾ ¼, ¾
¼, ¾ ¼, ¾ ¼, ¾
n
2
m
2
n
2P 11
Notation of space groups
• Notation following Hermann-Maugin
76
HA: main axis, NA: minor axis, ZA: intermediate axis
Crystal system 1st position 2nd position 3rd position 4th position
Triclinic Lattice type x - -
Monoclinic Lattice type y - -
Orthorhombic Lattice type x y z
Trigonal Lattice type z (HA) x (NA) -
Tetragonal Lattice type z (HA) x (NA) xy (ZA)
Hexagonal Lattice type z (HA) x (NA) xy (ZA)
Cubic Lattice type [100] [111] [110]
n: n-fold rotation axis, n: n-fold rotation inversion axis, np: n-fold screw axis, m: mirror plane, : n-fold rotation axis with mirror plane
m
n
Introduction
What is a crystal?
Definition of the 7 crystal systems
Indexing planes and directions
Bravais lattices
Stereographic projection
Symmetry operations of point groups
The 32 point groups
From point groups to layer groups
Symmetry operations of layer groups
The 17 layer groups
Transition to third dimension: space groups
Symmetry operations of space groups
Example for determination of a space group
Notations
Road map
77