General Introduction to Symmetry in

CrystallographyA. 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”

001

r 0

2r

Motif + Lattice =

Space group: P 1

a1 a2

a3

Rn

02r

n2r

Crystal symmetry

Space group operations: definition

1

2

1

2

h = m ( h point group operation)O

g1’

1’

Space group: P m

Wigner-Seitz notation

= {h|(0,0,1)}= h

t

t

Crystal symmetry :

Type of space group operations: rotations

h = 1, 2, 3, 4, 6Rotations of angle /n

e=g4={1|000} g={4+|100}g2={2|110}g3={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

for1

2

34

Crystal symmetry :

Space group operations: rotations

h = 1, 2, 3, 4, 6Rotations of angle /n

e=g4={1|000} g={4+|000}g2={2|000}g3={4-|000}

Space group: P 4

(1) x,y,z(2) –y,x,z(3) –x,-y,z (4) y,-x,z

for1

2

34

4

3

2

Crystal symmetry :

Space group operations: improper rotations

h = 6,4,3),(2,1 m

4

3

2

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

4

Space group: P 4

1

h = 6,4,3),(2,1 m

1

24

4

3

3

2

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

4

Crystal symmetry :

Space group operations: improper rotations

Space group: P 4

Crystal symmetry

Space group operations: mirror

1

2

1

2

O

1’

1’

Space group: P m

12 mh

Crystal symmetry

Space group operations: screw axis

Space group: P 21

g = 54321321211 6,6,6,6,6,4,4,4,3,3,2

t = tn + (p/n) ai

a1a2

a3

p

h: rotationof order n

1

2e={1|000} g={2|11½}

(1) x,y,z(2) -x+1,-y+1,z+1/2

g2={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 = tn +

Glide component // m

h: mirror m ( )2

a1/2 aa2/2 b a3/2 cai/2 + aj/2 nai/4 + aj/4 d

Space group: P c

a1

a2

a3

e={1|000} g={m|01½}

(1) x,y,z(2) x,-y+1,z+1/2

g2={1|001}

1

2

Crystal symmetry :

International tables symbols

Rotations Mirrors Improperrotations

c(Pnma)

a(Pnma

)

a(Pbnm)

b(P

bnn

)

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 21

a1a2

a3

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|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}…

….

zero-block pure translations

How to construct in practicefinite reducible and irreducible representations?

Space group: P 21

a1a2

a3

1

2

Reducible representations

lir

Si

Matrix representation of g M(g)

3

Space group: P 21

a1a2

a3

2

3

1

Si

lir

Reducible representations

Space group: P 21

a1a2

a3

2

3

1

Si

lir

Reducible representations

Space group: P 21

a1a2

a3

2

3

1

Si

lir

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 Gk

k-k

if yes g Gk

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

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