Symmetry for Quasicrystals

References:

http://www.jcrystal.com/steffenweber/qc.html

F. Samavat et. al., Open Journal of Physical Chemistry, 2012, 2, 7-14.

http://en.wikipedia.org/wiki/Icosahedral_symmetry

http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/advanced-chemistryprize2011.pdf

MaterialsWith perfect long-range order, but with no 3D translational periodicity.

Definition of Quasicrystals (QCs)

Sharp diffraction spots non-crystallographicrotational symmetry

Old definition of Crystals

Definition till 1991: A crystal is a solid where the atoms form a periodic arrangement.

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”

New Definition for Crystal

Diffraction Pattern crystals !

τ : scaling ratio

Crystals

Quasicrystals

Amorphous

Periodicity Order

X

X X

Crystals

Translation, t

Rotation1, 2, 3, 4, 6

Quasicrystals

inflation, Rotation

1, 2, 3, 4, 5, 6, 8, 10, 12

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)

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

Octagonal QCs

http://nanopatentsandinnovations.blogspot.tw/2011/10/quasicrystals-discovery-wins-novel.html

Decagonal QCs

http://www.pnas.org/content/108/5/1810/F6.expansion.html

Dodecagonal QCs

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

http://en.wikipedia.org/wiki/File:Icosahedron.gif

Simulations of some diffraction patterns

A simulation from an icosahedral quasicrystal

F. Samavat et. al., Open Journal of Physical Chemistry, 2012, 2, 7-14.

2 3 4

http://www.lassp.cornell.edu/lifshitz/quasicrystals.html

Example of 1D QCs

Cut and Project

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

Choose tan irrational number (why?)2

51tan

E.g. :

Fibonacci numberMake cuts in a 6D space and project in 3D space 3D QCs

Fibonacci sequence (1D QCs)

Aperiodic

Periodic

2

51tan

7

11tan

Aperiodic crystal

Periodic crystal~ approximant (called)

Fibonacci number (series, sequence)

Fibonacci Rabbits:

Fibonacci’s Problem: If a pair of new born rabbits are put in a pen, how manypairs of rabbits will be in the pen?

Assumptions: 1. Can produce once every month2. Always produce one male and one female offspring3. Can reproduce once they are one month old4. The rabbits never die

1st month Birth

Grow up

continue

2nd month

3rd month

4th month

5th month

6th month

Month# of pairs

1

1

2

1

3

2

4

3

5

5

6

8

7

?138

21

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

1,0 ; 1021 FFFFF nnn

...618034.12

51lim

1

n

n

n F

FGolden ratio

1n

n

F

F

B A B B A B A B B A B B A

1-D QC

B A B B A B A B

BA B BA BA B BA B BA

BAB BA BAB BAB BA BABAB BAB

Type of

quasicrystalQP+ Metric Symmetry System

First

Report

Icosahedral 3 D

(5)AlMn

Shechtman et al.

1984

Cubic 3D 3 VNiSiFeng et al

1989

Tetrahedral 3D 3 AlLiCuDonnadieu

1994

Decagonal 2D

(5)10/mmm AlMn

Chattopadhyay

et al., 1985a and

Bendersky, 1985

Dodecagonal 2D 3 12/mmm NiCrIshimasa et al.

1985

53m

3m4

3m

Type of

quasicrystalQP+ Metric Symmetry System

First

Report

Octagonal 2D 2 8/mmmVNiSi,

CrNiSi

Wang et al.

1987

Pentagonal 2D

(5)AlCuFe

Bancel

1993

Hexagonal 2D 3 6/mmm AlCrSelke et al.

1994

Trigonal 1D 3 AlCuNi

Chattopadhyay

et al.,

1987

Digonal 1D 2 222 AlCuCoHe et al.

1988

m5

m3

Ho-Mg-Zn Quasicrystal from

http://cmp.physics.iastate.edu/canfield/photos.html