DIFFRACTION METHODS IN MATERIAL SCIENCE PD Dr. Nikolay Zotov Email: zotov@imw.uni-stuttgart.de Lecture 4_2
OUTLINE OF THE COURSE 0. Introduction 1. Classification of Materials 2. Defects in Solids 3. Basics of X-ray and neutron scattering 4. Crystal Symmetry 5. Diffraction studies of Polycrystalline Materials 6. Microstructural Analysis by Diffraction 7. Diffraction studies of Thin Films 8. Diffraction studies of Nanomaterials 9. Diffraction studies of Amorphous and Composite Materials 2
OUTLINE OF TODAY S LECTURE Lattices Symmetry elements and symmetry operations Crystal Classes Crystal Systems Space Groups 3
Symmetry and Lattices in Art, Architecture, Natural Life M.C. Escher (1898-1972) Periodic Lattice Periodic repetition of points in space 4
Why is Symmetry Important Systematic description of objects, molecules, lattices Description/ Prediction of Atomic Structures Interpretation of Diffraction Data Description of Physical Properties of Crystalline Materials Optical Mechanical Electric and Magnetic 5
Symmetry Operations Symmetry operation is a transformation, which leaves an object the same. Point-Symmetry operations: Identity 1; (E) Rotational Axes n; (n-fold rotation axis) Mirror Plane m; (s, s v, s h ) Inversion -1; (i) Rotoinversion Axes n; (S n ) Point symmetry means that a given point remains fixed (untransformed) 6
Angles of rotation f n m = m360/n Crystallographic Rotational Axes 2-fold f = 180 o 3-fold f = 120, 240 o 4-fold f = 90, 180, 270 o 6-fold f = 60,120,180,240,300 o Crystallographic Rotations n = 1,2,3,4,6!!! 7
Crystallographic Rotational Axes z Symmetry operation X cos(f) -sin(f) 0 x Y = sin(f) cos(f) 0 y Z 0 0 1 z Axis Z x y 3 4 6 8
Non-Crystallographic Rotational Axes 5-fold Quasi-crystals 7-fold 8-fold 10-fold Quasi-crystals 9
Mirror Planes m Symmetry operation X -1 0 0 X m bc plane Y = 0 1 0 Y Z 0 0 1 Z 1 0 0 m ac plane 0-1 0 0 0 1 c b 1 0 0 m ab plane 0 1 0 0 0-1 a m bc 10
Centre of Symmetry Symmetry operation X -1 0 0 X Y = 0-1 0 Y Z 0 0-1 Z Typical indicator in single crystals - Pairs of paralell planes 11
Rotoinversion axis Rotation + inversion 3-fold 4-fold 6-fold 12
Rotoinversion axis Rotation + inversion Symmetry operation X -1 0 0 cos(f) -sin(f) 0 X Y = 0-1 0 sin(f) cos(f) 0 Y Z 0 0-1 0 0 1 Z Axis z c b a Matrix of inversion 4-fold rotoinversion axis parallel to c f = 90 o 0 1 0-1 0 0 0 0-1 13
Crystal System 32 Point Groups (Crystal Classes) Orientation rules!!! Triclinic 1, -1 Monoclinic 2; m, 2/m; b 2 Orthorhombic 222, mm2, 2/m 2/m 2/m ( a b c) Tetragonal 4, -4, 4/m, 422, 4mm, -42m, 4/m 2/m 2/m ( c b [110] ) Hexagonal 3, -3, 32, 3m, -32/m, 6, -6, 6/m, 622, 6mm, -6m2, 6/m 2/m 2/m ( c a [1-10] ) Cubic 23, 2/m 3, 432, -43m, 4/m -3 2/m ( a [111] [1-10] ) 14
Examples of Monoclinic Point Groups (Stereographic Projections) The comma defines the handedness 15
Examples of Orthorhombic Point Groups (Stereographic Projections) 2/m 2/m 2/m Full symbol 16
Examples of Tetragonal Point Groups (Stereographic Projections) 17
Examples of Hexagonal Point Groups (Stereographic Projections) Full symbol 6/m 2/m 2/m 18
Examples of Cubic Point Groups (Stereographic Projections) 19
2 m m 20
3 2 21
-6 m 2 22
Crystal Structure Periodic Lattice Translational Symmetry Basis (Atoms, Molecules in the Unit cell) Point Symmetry Crystal 23
Combination of Translations and Point Symmetry Operations 1. New Symmetry Operations 2. New types of lattices 3. Crystal Structures 24
New Symmetry Operations Translations+point symmetry operations Glide plane (g) Reflection from a plane plus translation (g) along a given direction parallel to the plane The vector g is not a translation of the lattice! Symmetry operation X 1 0 0 X 0 Y = 0 1 0 Y + 1/2 Z 0 0-1 Z 0 ab b g b 25
Glide Planes Symbol Translational Component a, b, c a/2, b/2, c/2 n One from (a+b)/2, (b+c)/2, (a+c)/2 d One from (a+b)/4, (b+c)/4, (a+c)/4 26
New Symmetry Operations Translations+point symmetry operations Srew rotation n j Rotation n with angle (360j/n) plus translation along the axis j = 1,2,, n-1 2 1, 3 1, 3 2, 4 1, 4 2,4 3, 6 1, 6 2, 6 3 ; Translational Component: v = j/n t Symmetry operation 2-fold c 2 1 : 2 axis; Translation parallel to c; f = 180 o X -1 0 0 X 0 Y = 0-1 0 Y + 0 Z 0 0 1 Z 1/2 c Crystallographic screw rotations n = 2,3,4,6!!! 27
3-fold Screw Axes 2 2 3 -t 1 3 1 t t Translational part 2 0 3 1 3 2 0 Symmetry operation X -0.5-0.86 0 X 0 Y = 0.86-0.5 0 Y + 0 Z 0 0 1 Z 1/3t 28
6-fold Screw Axes 29
Comparison of normal and screw rotational axes right-hand screw axes left-hand screw axes 30
Geometrical Symbols of Symmetry Elements perpendicular In-plane Perpendicular Inplane In-plane 31
Crystal Systems Pyrite FeS 2 Rutile TiO 2 Beryl 32
Crystal Systems Topas Azurite 33
D-Spacing in Different Crystal Systems d hkl = 1/G(hkl); G(hkl) = ha* + kb* + lc* G hkl2 = (hkl) g* (hkl) T Cubic a* = bc/v = 1/a b* = ac/v = 1/a c* = ab/v = 1/a 1/a 2 0 0 a* = ß* = g* = 90 o ; g* = 0 1/a 2 0 0 0 1/a 2 1/a 2 0 0 h G 2 = (h k l) 0 1/a 2 0 k ; G = (h 2 + k 2 + l 2 ) 1/2 /a 0 0 1/a 2 l d hkl = a/(h 2 + k 2 + l 2 ) 1/2 34
D-Spacings in Different Crystal Classes 35
New Crystal Lattices To better describe the Crystal symmetry! 14 Bravis Lattices Trigonal (rhomboedrisch) hexagonal 36
What is a Group? Group (G) is a set of objects together with an operation (rule) how to combine two elements from the set to get a new element. The elements of a group fulfill the following properties: # closure the operation on the elements of the group always produces another element of the group (closed set); A.B = C, C is also element of G # the group contains an identity element E, which leaves every element of the group unchanged, when it is conbined with the identity. A.E = E.A = A # for every element A of the group there is an inverse elemen A -1, so that A.A -1 = E In point, plane and space groups the elements are the symmetry transformations! 37
Space Groups Combination of symmetry elements located at diffrent points of the unit cell There are only 230 different space group symmetries (space groups) in 3D (There are only 17 plane groups in 2D) 38
Space Groups Symbols L a b c L P primitive lattice F face-centered lattice I body-centered lattice A,B,C base-centered lattice Examples a b c - Point group symbol along the main symmetry directions discused before P 2/m P m m 2 I 4 F m -3 m 39
Plane Groups 40
Monoclinic Space Group PG Symmetry Operations Mirror plane in the projection plane 41
Orthorhombic Space Group Symmetry Operations International Tables of Crystallography 42
Tetragonal Space Group Symmetrie- Operationen n 43
Hexagonal Space Group Symmetrie operationen 44
Triclinic P1 P 1 Monoclinic P2, P2 1, C2, Pm, Pc, Cm, Cc P2/m, P2 1 /m, C2/m, P2/c, P2 1 /c, C2/c Orthorhombic P222, P222 1, P2 1 2 1 2, P2 1 2 1 2 1, C222 1, C222, F222, I222, I2 1 2 1,2 1, Pmm2, Pmc2 1, Pcc2, Pma2, Pca2 1, Pnc2, Pmn2 1, Pba2, Pna2 1, Pnn2, Cmm2, Cmc2 1, Ccc2, Amm2, Abm2, Ama2, Aba2, Fmm2, Fdd2, Imm2, Iba2, Ima2 Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma, Cmcm, Cmca, Cmmm, Cccm, Cmma, Ccca, Fmmm, Fddd, Immm, Ibam, Ibca, Imma Tetragonal P4, P4 1, P4 2, P4 3, I4, I4 1 P 4, I 4 P4/m, P4 2 /m, P4/n, P4 2 /n, I4/m, I4 1 /a P422, P42 1 2, P4 1 22, P4 1 2 1 2, P4 2 22, P4 2 2 1 2, P4 3 22, P4 3 2 1 2, I422, I4 1 22 P4mm, P4bm, P4 2 cm, P4 2 nm, P4cc, P4nc, P4 2 mc, P4 2 bc, I4mm, I4cm, I4 1 md, I4 1 cd P 4 2m, P 4 2c, P 4 2 1 m, P 4 2 1 c, P 4 m2, P 4 c2, P 4 b2, P 4 n2, I 4 m2, I 4 c2, I 4 2m, I 4 2d P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P4 2 /mmc, P4 2 /mcm, P4 2 /nbc, P4 2 /nnm, P4 2 /mbc, P4 2 /mnm, P4 2 /nmc, P4 2 /ncm, I4/mmm, I4/mcm, I4 1 /amd, I4 1 /acd Trigonal P3, P3 1, P3 2, R3 P 3, R 3 P312, P321, P3 1 12, P3 1 21, P3 2 12, P3 2 21, R32 P3m P 3 1m, P 3 1c, P 3 m1, P 3 c1, R 3 m, R 3 c Hexagonal P6, P6 1, P6 5, P6 2, P6 4, P6 3 P 6 3D Space Groups P6/m, P6 3 /m P622, P6 1 22, P6 5 22, P6 2 22, P6 4 22, P6 3 22 P6mm, P6cc, P6 3 cm, P6 3 mc P 6 m2, P 6 c2, P 6 2m, P 6 2c P6/mmm, P6/mcc, P6 3 /mcm, P6 3 /mmc Cubic P23, F23, I23, P2 1 3, I2 1 3 Pm 3, Pn 3, Fm 3, Fd 3, Im 3, Pa 3, Ia 3 P432, P4 2 32, F432, F4 1 32, I432, P4 3 32, P4 1 32, I4 1 32 P 4 3m, F 4 3m, I 4 3m, P 4 3n, F 4 3c, I 4 3d Pm 3 m, Pn 3 n, Pm 3 n, Pn 3 m, Fm 3 m, Fm 3 c, Fd 3 m, Fd 3 c, Im 3 m, Ia 3 d 45
Classification according to Crystal Systems 46
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Determination of Point Symmetry from Space Group Symbol Replace all symmetry elements with translational component(s) with point group elements I 2 1 3 23 P 4 3 2 1 2 422 Pccn = P 2 1 /c 2 1 /c 2/n 2/m 2/m 2/m m m m The orientation of the symmetry operations remains the same! 50
Additional Sources http://pd.chem.ucl.ac.uk/pdnn/symm2/pntgrp1.htm Point Groups http://it.iucr.org/ International Tables of Crystallograophy Changes 23.11.2017 No Practical 24.11.2017 No Lecture 14.12.2017 Practical in 3P2 25.01.2018 Practical in 3P2 51