Introduction to point and space group symmetry
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1 Workshop on Electron Crystallography, Nelson Mandela Metropolitan University, South Africa, October 14-16, 2013 Introduction to point and space group syetry Hol Kirse Huboldt-Universität zu Berlin, Institut für Physik, AG TEM Newtonstrasse 15, D Berlin E-ail: Web:
2 Ebedding Crystallography Cheistry Bonding Coposition Structure Cleavability Polarity CRYSTALLO- GRAPHY Physics Transparency Conductivity Birefringence Piezoelectricity Pyroelectricity Matheatics Lattice, Syetry operations, Group theory 2
3 Topics of Crystallography Geoetric Crystallography Structure analysis X ray and electron diffraction Crystal growth Crystal cheistry Crystal physics Crystal defects 3
4 Geoetrical Crystallography, e.g., Au Crystal syste Coordinate syste (111) facet of octahedron (100) view [100] Indices Crystal classes (cubooktahedral) Syetry Projection eleent Crystal shape 4
5 Road ap Introduction What is a crystal? Definition of the 7 crystal systes Indexing planes and directions Bravais lattices Stereographic projection Syetry operations of point groups The 32 point groups Fro point groups to layer groups Syetry operations of layer groups The 17 layer groups Transition to third diension: space groups Syetry operations of space groups Exaple for deterination of a space group Notations 5
6 What is a crystal? Characteristics of an ideal crystal: Flat regular surfaces/facets Characteristic syetry Cheically hoogeneous object Anisotropic properties Periodic arrangeent of constituents like atos or olecules along the three spatial directions Rock crystal, SiO 2 6
7 Definition of a crystal A crystal is a hoogeneous anisotropic solid with periodic arrangeent of constituents like atos or olecules in all three spatial diensions. Since 1991: A crystal is defined as a solid exhibiting discrete diffraction spots. This definition includes real crystals with defects, incoensurate crystals and quasicrystals. 7
8 Coordinate syste 3-diensional lattices are described by three not necessarily orthogonal directions x, y, z having lattice paraeters a, b, c. The angle between the three directions are denoted as a, b, g. As a convention a right-handed coordinate syste is used. z c b a b x a g y 9
9 The 7 crystal systes Crystal syste Lattice paraeters Triclinic a b c a b g 90 Monoclinic a b c a = g = 90 b 90 Orthorhobic 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
10 Indexing of crystallographic directions Two-diensional lattice b a Crystallographic direction: straight line crossing two arbitrary points of the lattice [010] [120] [110] [210] I uvw = 0 a + 1 b + 0 c Description analogous to vectors 12
11 Indexing of crystallographic directions Three-diensional lattice Infinite periodic arrangeent of points within the 3-d space c b a Triple product: (a b) c > 0, i.e. non-coplanar Point in space: I uvw = u a + v b + w c ; u,v,w Z 13
12 Indexing of crystallographic planes Derivation of MILLER indices: a A c C O B (6 3 4) plane WEISS indices: b OA a 0 : OB b 0 : OC c 0 :n:p : n : p = 2 : 4 : 3 1 : 1 n : 1 p 1 2 : 1 4 : 1 3 h : k : l = 6 : 3 : 4 MILLER indices 14
13 The 14 Bravais lattices (A. Bravais, 1850) Crystal syste Sybol Lattice Crystal syste Centering Centering Sybol Lattice Triclinic Pri. ap Pri. tp Monocl. Pri. Face cent. P A Tetrag. Trigonal Body cent. Rhobohedr. ti hr Pri. op Trig. + Hexag. Pri. hp Orthorh. Body cent. Basal-pl. cent. oi oc Cubic Pri. Body cent. cp ci Face cent. of Face cent. cf 20
14 Fro inner structure to orphology: description of crystals Correspondence between orphology and structure Any crystal face (orphology) is oriented parallel to a lattice plane (structure). The syetry of the outer shape of a crystal is higher or equal to the syetry of its inner structure. Exaple: Galenite (PbS): 22
15 Crystal projections: stereographic projection Projection ethod: North pole of pole sphere Pole sphere Pole of face Face noral Projection plane South pole of pole sphere Projected pole Observation spot 23
16 Crystal projections Projection onto pole sphere Stereographic projection Observation spot 24
17 Crystal projections: stereographic projection Exaples for stereogras: Polyhedron: Hexahedron Octahedron Rhobic dodecahedron ( 100) ( 1 1 1) ( 111) ( 101) ( 1 10) ( 110) Stereogra ( 010) ( 001) ( 001 )( 010 ) ( 11 1) ( 1 11) ( 111) ( 111) ( 01 1) ( 101) ( 011) ( 101) ( 011) ( 011) ( 100 ) ( 111) ( 111) ( 101 ) ( 110) ( 110 ) 25
18 Syetry of crystals Point syetry operations: Syetry operation with at least one point of the object reains at its original position. The corresponding syetry eleent is called point syetry eleent. Trivial point syetry operations : ROTATION, INVERSION, REFLECTION Cobined point syetry operations : ROTATION INVERSION, REFLECTION ROTATION Point syetry eleents in 3-d space: Rotation axis, inversion centre, irror plane, rotation inversion axis, reflection rotation plane 27
19 Point syetry operations Rotation: 4-fold rotation Rotation angle: 90 Sybol: 4 Graphic sybol: Ipact on structure otive: Shape: Stereogra: Tetragonal pyraid 28
20 Rotation Point syetry operations Nuber of positions Angle Sybol Stereogra Shape Pedion Sphenoid Trigonal pyraid Tetragonal pyraid Hexagonal pyraid 29
21 Inversion Point syetry operations Sybol: 1 Graphic sybol: Ipact on structure otiv: Inversion centre Shape: Stereogra: Pinakoid 30
22 Point syetry operations Reflection Sybol: (irror) = Graphic sybol: Ipact on structure otiv: Mirror plane Shape: Stereogra: Doa 31
23 Point syetry operations Rotation inversion 4-fold rotation inversion Syetry operation: 90 + Inversion Sybol: 4 Graphic sybol: Ipact on structure otiv: Shape: Stereogra: Tetragonal disphenoid 32
24 Point syetry operations Rotation inversion Nuber of positions Angle Sybol Stereogra Shape Pinakoid Doa Rhobohedron Tetragonal disphenoid Hexagonal dipyraid 33
25 cubic hexagonal tetragonal trigonal onoclinic orthorh. triclinic The 32 point syetry groups X X X X X X2 X
26 Notation of point syetry groups Notation following Herann-Maugin n: n-fold rotation axis, n: n-fold rotation inversion axis, n : irror plane, : n-fold rotation axis with irror plane Crystal syste 1 st position 2 nd position 3 rd position Triclinic x - - Monoclinic y - - Orthorhobic 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] HA: ain axis, NA: inor axis, ZA: interediate axis 36
27 Exaple for the deterination of a point syetry group Morphology / shape of crystal Stereographic projection Syetry eleents Ditetragonal pyraid 8 (+1) facets Crystal syste: tetragonal Crystal class: ditetragonal-pyraidal Sybol following Herann-Maugin: 4 49
28 The two-diensional lattice Infinite periodic arrangeent of points (i.e. atos, ions, or olecules) at a plane b a a = a b = b a x b > 0 (i.e. non-collinear) Expression for a single point: I uv = u a + v b; u,v Z 51
29 Two-diensional patterns Black sea Europe Istanbul Asia Sea of Marara Topkapi palace
30 Syetry operation: Translation Shift of a otive (asyetric unit: atos, ions, olecules) by translation vector t t No longer pure point syetry only Space filling New syetry operations 54
31 Cobination of translation and reflection Translation t Reflection 55
32 New syetry operation: Glide reflection g Siultaneous application of translation and reflection: Step 1: Translation by t = ½ a 0 Step 2: Reflection 56
33 Syetry eleents of layer groups (wallpaper groups) Rotation Motive Angle Multiplicity Sybol Reflection Glide reflection g t Sybol Lattice types Glide coponent: t/2 Priitive: p Centered: c
34 The 17 layer groups (wallpaper groups) oblique p 1 p 211 quadratic p 4 p 4 rectangular p 11 p 1g1 p 4g c 11 p 2 hexagonal p 3 p 31 p 2g p 2gg c 2 p 31 p 6 p 6
35 The 17 layer groups (wallpaper groups) oblique p 1 p 211 quadratic p 4 p 4 rectangular p 11 p 1g1 p 4g c 11 p 2 hexagonal p 3 p 31 p 2g p 2gg c 2 p 31 p 6 p 6
36 Quadratic Syste: p 4g Topkapi palace p 4g
37 The three-diensional lattice Infinite periodic arrangeent of points within the 3-d space c b a 63
38 Space groups Space groups for describing syetry relations in 3-d space. Space groups include all syetry operations of a 3-diensional, infinitely extended, and perfect crystal structure. The notation of the space group is done after Herann- Mauguin. Nuber of Space groups in the 3-d space: 230 But! By definition there is an infinite nuber of space groups! There are 73 types of space groups coprising the identical (point) syetry eleents as the point syetry group but extended by the translation operation: These are the syorphic space groups. 65
39 Syetry operation: Glide reflection a, b c n e 67
40 Syetry operation: Glide reflection (a+b)/2, (a+c)/2, (b+c)/2, (a+b+c)/2 (a+b)/4, (a+c)/4, (b+c)/4, (a+b+c)/4 n d 68
41 Syetry operation: Screw rotation 2-fold rotation 2-fold screw rotation = ½ a o Translation period of a screw axis n p : = p/n 69
42 4-fold screw rotation: Syetry operation: Screw rotation = 1/4 = 2/4 = 3/4 4 1 and 4 3 are enantioorphous screw axis 4 1 dextrorotatory, 4 3 laevorotatory, 4 2 without rotary sense like
43 Sybols of rotation axes and screw rotation axes Sybols for orientation of rotation axis along the viewing direction Sybols for inclined rotation axis 71
44 Exaple for a space group Marcasite (FeS 2 ) Projection onto basal plane ½ - ½ ½ + - ½ - ½ + 73
45 Exaple for a space group Marcasite (FeS 2 ) Space group: 2 n 2 2 n 1 1 P ½ - ¼, ¾ ¼, ¾ ½ ½ + - ½ - ½ + ¼, ¾ ¼, ¾ ¼, ¾ 74
46 Notation of space groups Notation following Herann-Maugin n: n-fold rotation axis, n: n-fold rotation inversion axis, n p : n-fold screw axis, n : irror plane, : n-fold rotation axis with irror plane Crystal syste 1 st position 2 nd position 3 rd position 4 th position Triclinic Lattice type x - - Monoclinic Lattice type y - - Orthorhobic 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] HA: ain axis, NA: inor axis, ZA: interediate axis 76
47 Road ap Introduction What is a crystal? Definition of the 7 crystal systes Indexing planes and directions Bravais lattices Stereographic projection Syetry operations of point groups The 32 point groups Fro point groups to layer groups Syetry operations of layer groups The 17 layer groups Transition to third diension: space groups Syetry operations of space groups Exaple for deterination of a space group Notations 77
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