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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.

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General Introduction to Symmetry in Crystallography

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  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

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