1. Crystalline state Symmetry in nature Symmetry in arts and industry
Description of symmetry – basic concepts
Crystallography of two dimensions
Crystallography of three dimensions

2. Symmetry elements in 3D Simple symmetry elements – point operations
Rotation axes 1, 2, 3, 4, 6
Reflection planes m
Inversion center – stred súmernosti i,
Compound symmetry elements – point operations
Rotoinversions – inverzné rotačné osi
Rotoreflections – zrkadlové rotačné osi

3. Rotoinversion
(= i)

4. Fourfold inversion axis
new symmetry element

5. translation component
Glide planes in 3D
Compound symmetry elements – with translations
a, b, c
a/2, b/2, c/2 n
(a ± b)/2, etc. d
(a ± b)/4, etc. (a ± b ± c)/4
printed symbol
translation component

6. Screw axes – skrutkové rotačné osi
Compound symmetry elements – with translations
Screw axes
printed symbols
graphical symbols

7. Screw axes
repetition of points

8. Combination of intersecting axes
- every two rotations around intersecting axes can be replaced
by one appropriate rotation
the angles and the orientation of the axes are arbitrary
strong limitation for crystal structures
the rotation angles can acquire only the values 0°, 60°, 90°, 120°, 180°
the same holds for the resulting rotation
which axis combinations are allowed?
there are 35 triplets of axes
but only the following combinations are allowed
222, 223, 224, 226, 233, 234

9. Axial combinations
Three axes Aα , Bβ and Cγ with rotations α, β and γ,
angles between the axes u, v and w Aα Bβ Cγ u v w
axes α β γ w u v
222
180°
90°
223
120°
60°
224
45°
226
30°
233
54°44'
70°32'
234
35°16'

10. Six permissible nontrivial combination of rotations

11. Six crystallographic axial symmetries

12. 32 point groups in 3D permissible axes and their combinations
1, 2, 3, 4, 6, 222, 223, 224, 226, 233, 234,
combination with mirror planes and inversion center 1

13. 32 point groups in 3D

14. Seven crystal systems
The presence (or absence) of rotation axes allows to clasify the crystal structures
The characteristic symmetry indicates the minimal symmetry
that is always present in each crystal system

15. Crystal systems and the point groups
red – possess only rotation axes - enantiomorphic
magenta – possess a center of inversion – centrosymmetric
bold – referred to as polar

16. Polar groups
Those point groups for which every operation leaves more than one
common point unmoved are known as the polar point groups.
1, 2, 3, 4, 6, m, mm2, 3m, 4mm and 6mm
Polar direction – polárny smer
Direction which is not symmetry equivalent to its opposite direction.
Polar direction can only exist in 21 non-centrosymmetric point groups.
20 of them are piezoelectric point groups – crystals with this symmetry
exhibit piezoelectricity.
Exception: group 432 – center of symmetry not present,
but piezoelectricity cannot occur.
Unique direction – jedinečný smer
Direction that is just one and that is not repeated by any symmetry operation.
All unique direction is a polar direction, but only some polar directions
are unique.
Unique directions are present only in 8 of 10 polar groups:
2, 3, 4, 6, mm2, 3m, 4mm and 6mm
Groups 1 and m are excluded

17. Stereographic projection
How to represent three-dimensional angular relations in plane?
Stereographic projection is a quantitative method for presenting three-dimensional
orientation relationships between crystallographic planes and directions on a two-
dimensional figure.

18. Point groups in stereographic projection
TETRAGONAL SYSTEM

19. Point groups in stereographic projection

20. 14 Bravais lattices
special centering
of hexagonal lattice

21. Hexagonal & rhombohedral indicesar br cr

22. Cubic & rhombohedral indicesar br cr

23. Transformation of indices - example
LSMO – space group R-3c (167)
hexagonal indexing
conversion to pseudo-cubic lattice and indexing
a ~ nm
hexagonal → rhombohedral
rhombohedral → cubic
012 → 020
(if a ~ then 010)
104 → 220

24. Symbols of space groups
Lijk – International notation
Schönflies notation
point group
L = lattice – capital letter for 3D lattice L
P – primitive
I – body centered
C – centered
F – face centered
R – trigonal
ijk = symmetry elements of space group
for the different symmetry directions

25. International Tables for Crystallography

26. fractional coordinates –
Wyckoff symbols
x, y, z – coordinates of a point
expressed in units a, b, c
fractional coordinates –
frakčné súradnice

27. 230 space groups Comprehensive derivation:
M. J. Buerger: Elementary Crystallography, MIT Press, 1978, pp
Uneven distribution of crystal structures
70% of elements belong to 4 groups
face-centered cubic
body-centered cubic
hexagonal close-packed
diamond cubic
60% of organic crystalline compounds have one of six space groups

28. Examples of structures
fcc – face-centered cubic four points of space lattice/cell
0,0,0; 1/2, 1/2, 0; 0, 1/2, 1/2; 1/2, 0, 1/2
Al, Cu, Ag, Pd, Pt, Ir
bcc – body-centered cubic two points of space lattice/cell
0,0,0; 1/2, 1/2, 1/2
Fe, Li, Na, K, Rb, Ba, V, Cr,
crystals with the same lattice may have very different structure
iron

29. Examples of structures
α-manganese
g x = 0.089; z = 0.278
g x = 0.356; z = 0.042
c x = 0.356
important!!
1/2 is not 0.5

30. Examples of structures
hcp – hexagonal close-packed one points of space lattice/cell
0,0,0
two atoms/point
1/3, 2/3, 1/4; /3, 1/3, 3/4
Be, Mg, Co, Zn, Zr, Ru 0, 0, 0; /3, 2/3, 1/2
diamond structure fcc lattice, four points of space lattice/cell
two atoms/point
3/8, 3/8, 3/8; 7/8, 3/8, 3/8
0,0,0; 1/4, 1/4, 1/4
C, Si, Ge
GaAs, InP different atoms at two positions

31. Examples of structures
barium titanate BaTiO3 lattice?
points/cell?
motif?
primitive
one/cell
5 atoms
barium
oxigen
titanium

32. How to use the data? X-ray diffraction identification of phases
lattice parameters, space group
position of atoms
calculation of theoretical diffraction pattern