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

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

Crystal symmetry : Translational symmetry Motif: “molecule” of crystallographic point group symmetry “1” Motif + Lattice = Space group: P 1

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

Crystal symmetry : Type of space group operations: rotations h = 1, 2, 3, 4, 6 Rotations of angle  /n e=g 4 ={1|000} g={4 + |100} g 2 ={2|110} g 3 ={4 - |010} Space group: P 4 (1) x,y,z (2) –y+1,x,z (3) –x+1,-y+1,z (4) y,-x+1,z for

Crystal symmetry : Space group operations: rotations h = 1, 2, 3, 4, 6 Rotations of angle  /n e=g 4 ={1|000} g={4 + |000} g 2 ={2|000} g 3 ={4 - |000} Space group: P 4 (1) x,y,z (2) –y,x,z (3) –x,-y,z (4) y,-x,z for

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

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

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

Crystal symmetry Space group operations: screw axis Space group: P 2 1 g = t = t n + (p/n) a i a1a1 a2a2 a3a3 p h: rotation of order n 1 2 e={1|000} g={2|11½} (1) x,y,z (2) -x+1,-y+1,z+1/2 g 2 ={1|001} Glide component e={1|000} g={2|00½} (1) x,y,z (2) -x,-y,z+1/2 2

Crystal symmetry Space group operations: glide planes g = a,b,c,n,d t = t n + Glide component // m h: mirror m ( ) a 1 /2 a a 2 /2b a 3 /2c a i /2 + a j /2n a i /4 + a j /4d Space group: P c a1a1 a2a2 a3a3 e={1|000} g={m|01½} (1) x,y,z (2) x,-y+1,z+1/2 g 2 ={1|001} 1 2

Crystal symmetry : International tables symbols Rotations Mirrors Improper rotations

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

(zero block symmetry operators)

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

Space group: P 2 1 a1a1 a2a2 a3a3 1 2 {1|000} {2|00½} {1|100} {1|010} {1|001}… 2 Problem : The multiplication table is infinite {1|000} {1|000}{2|00½}{1|100}{1|010} {1|001}… {2|00½} {2|00½}{1|001} {2|10½} {2|01½} {2|00 3/2 }… {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}… …. zero-block pure translations How to construct in practice finite reducible and irreducible representations?

Space group: P 2 1 a1a1 a2a2 a3a3 1 2 Reducible representations SiSi Matrix representation of g M(g) 3

Space group: P 2 1 a1a1 a2a2 a3a SiSi Reducible representations

Space group: P 2 1 a1a1 a2a2 a3a SiSi Reducible representations

Space group: P 2 1 a1a1 a2a2 a3a SiSi Reducible representations

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

Irreducible representations: other symmetries ?? (1) (2) (3)  ’(r) is a Bloch function  hk (r)

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

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

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

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