1. Symmetry for Quasicrystals
References:
F. Samavat et. al., Open Journal of Physical Chemistry, 2012, 2, 7-14.

2. Definition of Quasicrystals (QCs)
Materials
With perfect long-range order, but with no 3D translational periodicity.
Sharp diffraction spots
non-crystallographic
rotational symmetry
Old definition of Crystals
Definition till 1991: A crystal is a solid where the atoms form a periodic arrangement.

3. New Definition for Crystal
International Union of Crystallography, “Report of the Executive Committee for 1991”, Acta Cryst., A48, (1992), 922.
“ … By crystal, we mean any solid having an essentially discrete diffraction diagram, and by aperiodic crystal we mean any crystal in which three dimensional lattice periodicity can be considered to be absent”
Diffraction Pattern  crystals !

4. Periodicity Order Crystals Quasicrystals Amorphous Crystals 
Quasicrystals X 
Amorphous X X
Crystals
Quasicrystals
Translation, t
inflation, 
Rotation
1, 2, 3, 4, 6
Rotation
1, 2, 3, 4, 5, 6, 8, 10, 12
τ : scaling ratio

5. Quasiperiodic in 3D (no periodic direction)
Types of QCs
Quasiperiodic in 2D (polygonal or dihedral QCs, one periodic direction  the quasiperodic layers)
Octagonal QCs: local 8-fold symmetry [P & I]
Decagonal QCs: local 10-fold symmetry [P]
Dodecagonal QCs: local 12-fold symmetry [P]
Quasiperiodic in 3D (no periodic direction)
Icosahedral QCs: (axes:12x5-fold, 20x3-fold, 30x2-fold) [P, I & F]
new type (reported in Nature, Nov.2000)
“Icosahedral" QCs with broken symmetry (stable binary Cd5.7Yb)

6. Chris J. Pickard and R. J. Needs, Nature Materials 9,624–627
Octagonal QCs
Chris J. Pickard and R. J. Needs, Nature Materials 9,624–627

7. Decagonal QCs

8. Dodecagonal QCs

9. Icosahedral QCs http://en.wikipedia.org/wiki/File:Icosahedron.gif
Schematic drawings of the unit cell of fcc Zr2Ni structure (a) and examples of icosahedral clusters around Zr and Ni atoms in the unit cell (b).
J. Saida et al., Intermetallics, V. 10, Issues 11–12, November 2002, Pages 1089–1098
Icosahedral QCs

10. Simulations of some diffraction patterns
F. Samavat et. al., Open Journal of Physical Chemistry, 2012, 2, 7-14.
A simulation from an icosahedral quasicrystal

11.
 4
 3
 2 

12. Example of 1D QCs

13.  Cut and Project Fibonacci sequence (1D QCs)
Harald Bohr, Acta Mathematicae, 45, 580 (1925)
Make a cut in a 2D space and project the mathematical points onto a 1D space, a line, and get a 1D quasicrystal
Ignore anything outside of the two lines 
E.g. :
Fibonacci number
Choose 
 tan irrational number (why?)
Make cuts in a 6D space and project in 3D space  3D QCs

14. Aperiodic
Periodic
Aperiodic crystal
Periodic crystal
~ approximant (called)

15. Fibonacci number (series, sequence)
Fibonacci Rabbits:
Fibonacci’s Problem:
If a pair of new born rabbits are put in a pen, how many
pairs of rabbits will be in the pen?
Assumptions:
1. Can produce once every month
2. Always produce one male and one female offspring
3. Can reproduce once they are one month old
4. The rabbits never die

16. 1st month
continue
Birth
2nd month
Grow up
3rd month
4th month
5th month
6th month
Month 1 2 1 3 2 4 3 5 6 8 7 8 21
# of pairs 13 ?

17. Fibonacci number
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, …..
The sequence Fn of Fibonacci numbers is defined by the recurrence relation
Golden ratio

18. B A B B A B A B BA B BA BA B BA B BA BAB BA BAB BAB BA BAB BA BAB
1-D QC

19. Chattopadhyay et al., 1985a and Bendersky, 1985
Type of quasicrystal
QP+
Metric
Symmetry
System
First
Report
Icosahedral
3 D 
(5)
AlMn
Shechtman et al.
1984
Cubic 3D 3
VNiSi
Feng et al
1989
Tetrahedral
AlLiCu
Donnadieu
1994
Decagonal 2D
10/mmm
Chattopadhyay et al., 1985a and Bendersky, 1985
Dodecagonal
12/mmm
NiCr
Ishimasa et al.
1985

20. Type of quasicrystal QP+ Metric Symmetry System
First
Report
Octagonal 2D 2
8/mmm
VNiSi,
CrNiSi
Wang et al.
1987
Pentagonal 
(5)
AlCuFe
Bancel
1993
Hexagonal 3
6/mmm
AlCr
Selke et al.
1994
Trigonal 1D
AlCuNi
Chattopadhyay et al.,
Digonal
222
AlCuCo
He et al.
1988

21. Ho-Mg-Zn Quasicrystal
from