1 / 26

General Introduction to Symmetry in Crystallography

General Introduction to Symmetry in Crystallography. A. Daoud-Aladine (ISIS). Outline. Crystal symmetry. Translational symmetry Example of typical space group symmetry operations Notations of symmetry elements. (geometrical transformations). Representation analysis using space groups.

alicia
Download Presentation

General Introduction to Symmetry in Crystallography

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. General Introduction to Symmetry in Crystallography A. Daoud-Aladine (ISIS)

  2. Outline Crystal symmetry • Translational symmetry • Example of typical space group symmetry operations • Notations of symmetry elements (geometrical transformations) Representation analysis using space groups • Reducible (physical) representation of space groups • Irreducible representations of space groups (group properties)

  3. a3 Motif: “molecule” of crystallographic point group symmetry “1” Rn a2 a1 Motif + Lattice = Space group: P 1 Crystal symmetry : Translational symmetry

  4. = h  t = {h|(0,0,1)} t Wigner-Seitz notation Crystal symmetry Space group operations: definition 2 1’ g 1 O h = m ( h point group operation) 1’ 2 1 Space group: P m

  5. for Crystal symmetry : Type of space group operations: rotations h = 1, 2, 3, 4, 6 4 3 Rotations of angle j=2p/n 1 2 e=g4={1|000} g={4+|100} g2={2|110} g3={4-|010} (1) x,y,z (2) –y+1,x,z (3) –x+1,-y+1,z (4) y,-x+1,z Space group: P 4

  6. for Crystal symmetry : Space group operations: rotations h = 1, 2, 3, 4, 6 4 3 Rotations of angle j=2p/n 1 2 2 e=g4={1|000} g={4+|000} g2={2|000} g3={4-|000} (1) x,y,z (2) –y,x,z (3) –x,-y,z (4) y,-x,z 4 3 Space group: P 4

  7. Space group: P Crystal symmetry : Space group operations: improper rotations h = 2 4 3 1 e=g4={1|000} g={ |101} g2={2|110} g3={ |101} (1) x,y,z (2) y+1,-x,-z+1 (3) –x+1,-y+1,z (4) –y+1,x,-z+1

  8. Space group: P Crystal symmetry : Space group operations: improper rotations h = 2 4 3 1 3 e=g4={1|000} g={ |101} g2={2|110} g3={ |101} (1) x,y,z (2) y+1,-x,-z+1 (3) –x+1,-y+1,z (4) –y+1,x,-z+1 4 2

  9. Crystal symmetry Space group operations: mirror 2 1’ 1 O 1’ 2 1 Space group: P m

  10. 2 e={1|000} g={2|00½} (1) x,y,z (2) -x,-y,z+1/2 Crystal symmetry Space group operations: screw axis h: rotation of order n g = a3 p Glide component a2 a1 t = tn + (p/n) ai 2 e={1|000} g={2|11½} (1) x,y,z (2) -x+1,-y+1,z+1/2 g2={1|001} 1 Space group: P 21

  11. h: mirror m ( ) Crystal symmetry Space group operations: glide planes a2 g = a,b,c,n,d 1 Glide component // m t = tn + a1/2 a a2/2 b a3/2 c ai/2 + aj/2 n ai/4 + aj/4 d 2 a3 a1 e={1|000} g={m|01½} (1) x,y,z (2) x,-y+1,z+1/2 g2={1|001} Space group: P c

  12. Crystal symmetry : International tables symbols Improper rotations Mirrors Rotations

  13. b(Pbnn) c(Pbnm) a(Pbnm) a(Pnma) b(Pnma) c(Pnma) c (Pnma)

  14. (zero block symmetry operators)

  15. Outline Crystal symmetry • Translational symmetry • Example of typical space group symmetry operations • Notations of symmetry elements (geometrical transformations) Representation analysis using space groups • Reducible (physical) representation of space groups • Irreducible representations of space groups (group properties)

  16. Problem : The multiplication table is infinite zero-block pure translations a3 {1|000} {2|00½} {1|100} {1|010} {1|001}… {1|000} {1|000} {2|00½} {1|100} {1|010} {1|001}… {2|00½} {2|00½} {1|001} {2|10½} {2|01½} {2|003/2}… {1|100} {1|100} {2|10½} {1|200} {1|110} {1|101}… {1|010} {1|010} {2|01½} {1|110} {1|020} {1|011}… {1|001} {1|001} {2|003/2} {1|101} {1|011} {1|002}… …. a2 a1 2 2 1 How to construct in practice finite reducible and irreducible representations? Space group: P 21

  17. Reducible representations Matrix representation of g M(g) a3 3 a2 a1 2 Si 1 Space group: P 21

  18. Reducible representations a3 3 a2 a1 2 Si 1 Space group: P 21

  19. Reducible representations a3 3 a2 a1 2 Si 1 Space group: P 21

  20. Reducible representations a3 3 a2 a1 2 Si 1 Space group: P 21

  21. Irreducible representations: translations More generally, Bloch functions: • One-dimensional matrix representationof the translations on the basis of Bloch functions • Infinite number of representations labelled by k

  22. f’(r) is a Bloch function fhk(r) Irreducible representations: other symmetries (1) ?? (2) (3)

  23. m  Gk k  -k ! Irreducible representations: the group of k ?? if yes  g Gk k -k

  24. Irreducible representations of Gk Tabulated (Kovalev tables) or calculable for all space group and all k vectors for finite sets of point group elements h

  25. Example: space group Pnma, k=(0.28, 0, 0)

  26. Conclusion • Despite the infinite number of • the atomic positions in a crystal • the symmetry elements in a space group… • …a representation theory of space groups is feasible using Bloch functions associated to k points of the reciprocal space. This means that the group properties can be given by matrices of finite dimensions for the • - Reducible (physical) representations can be constructed on the space of the components of a set of generated points in the zero cell. • Irreducible representations of the Group of vector k are constructed from a finite set of elements of the zero-block. • Orthogonalization procedures can be employed to construct • symmetry adapted functions

More Related