1. Symmetry of position: periodic order
Lattice :
Set of points (nodes):
Ruvw = u a +v b + w c
(a, b, c) basis, (u, v, w) integers. a b c g
Unit cell :
Volume with no gaps or overlaps, gal parallelepipedic (a,b,c)
Primitive (one node), multiple (symmetry) : elementary (unit cell)
Conventionnal unit cells :
P : Primitive
F : Face-centred
I : Body-centred
A,B,C : Base-centred

2. Point symmetry of lattices
Only 1-, 2-, 3- 4-, 6-fold symmetries
are compatible with periodicity
Every symmetry axe An is normal to a lattice plane
Symmetry of this plane An
BB’ lattice vector
BB’=T-2Tcosa =mT
cosa =p/2 An A2
A’2 T T
a=p
a=p
a=2p /n B B’
An(T)
A-n(-T) a -a An T
A’n

3. Towards Penrose tilling
Tilings
No gaps or overlaps
Kepler ( ) in 1619 : « Harmonices Mundi »
Only symmetry compatible with translation :
1, 2, 3, 4, 6 2 3 5 8 1 4 6
Towards Penrose tilling

4. Stacking of 2D lattices preserving symmetry (Ex. square)
In 2D
4 systems (systems)
5 latttice modes
Oblic : p
Rectangular : p
Rectangular : c
Square : p
Hexagonal : p
In 3D
Stacking of 2D lattices preserving symmetry (Ex. square) P I

5. Bravais lattices In 3D 7 systems (symmetry) 14 lattice modes P I F C _
Triclinic
a  b  c
a  b  g 1 b
Monoclinic
a  b  c
a = g = 90°; b
2/m
Orthorhombic
a  b  c
a = b = g = 90°
2/mmm
In 3D
7 systems (symmetry)
14 lattice modes
Tetragonal
a = b  c
a = b = g = 90°
4/mmm _
Rhomboedric
a = b = c
a = b = g 3m
Hexagonal
a = b  c
a=b=90°;g =120°
6/mmm _
Cubic
a = b = c
a = b = g =90°
m3m

6. Crystallographic point groups
32 crystal classes
Orthorhombique
Monoclinique
Triclinique
Trigonal
Tétragonal
Hexagonal
Cubique
Crystallographic point groups 1 2 3 4 6
222 32
422
622
7 crystal systems
Holohedral :
with the lattice symmetry
Ex : Tétragonal (4/mmm)
... hemihedral, tetarto-hedral
Chiral groups (Direct sym)
Centrosym groups (Laue class)
Improper groups (ind sym.– inv) _ _ _ _ _ 1
2=m 3 4
6=3/m
2/m
4/m
6/m
2mm 3m
4mm
6mm _ _ _ _ _ 3m
42m (4m2)
62m (6m2)
mmm 4/
mmm 6/
mmm 23
432 _ _ _ m3
43m
m3m

7. Relations between the 7 systems
Hexagonal
Cubic
Tetragonal
Group/subgroup
Symmetry breaking
Phase transitions
Trigonal
Orthorhombic
Monoclinic 4 2 L L
Triclinic L
L+e L
L-e L 6 L 3

8. Mauritz Cornelis Escher
Space groups
Mauritz Cornelis Escher
Dutch graphic artist
( ) .
Groupe P4 (chiral)

9. New symmetries
Groupe P4gm
Glide planes
Reflections
Glide planes

10. O : Rotation, Reflection
New symmetry
Glide plane (M,t)
After two reflections M, periodicity T
t=T/2
Combination (O, t)
O : Rotation, Reflection
T : translation T
T/2 M
Notation :
a, b, c, n, d, g
Screw axis (AN, t)
After N translations t periodicity: mc
t = mc/N 21 41 42 61 64
Notation : Nm
(AN, mc/N)

11. Symmetry operations Rotations Roto-reflections Screw axis Reflection
Glide plane

12. Directions (primary, etc.)
Space groups
230 space groups
7 crystalline systems
Notations
Directions (primary, etc.)
Lattice mode
Generators
Point Group
Without translation
I41/amd
Tetragonal
Body centered 4 m

13. Asymmetric unit

14. Symmetry _ _ _ _ Linear Symmetry Point groups Symmetry of position
Rotations
Roto-reflections
Conventionnally
Rotations (An)
Reflections (M)
Inversion (C)
Roto-inversions (An)
Point groups
7 Curie
Symmetry of position
Translations
T= u a + v b + w c
Symmetry allowed
1, 2, 3, 4, 6 ( 3, 4, 6)
M, C
14 Bravais lattices
32 Crystal classes
7 crystal systems
Translations
Rotations
Roto-reflections +
Screw axis
Glide plane
230 Space group
( 7 systems ) _ _ _ _