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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. - PowerPoint PPT Presentation
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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: 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
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?
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 of Gk
Tabulated (Kovalev tables) or calculable for all space group and all k vectors for finite sets of point group elements h
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