Symmetry for Quasicrystals

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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

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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

Definition of Quasicrystals (QCs)

Materials

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

non-crystallographic

rotational symmetry

Sharp diffraction spots

Old definition of Crystals

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

New Definition for Crystal

International Union of Crystallography, “Report of the Executive Committee for 1991”, ActaCryst., 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 !

Periodicity

Order

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

Types of QCs

Quasiperiodicin 2D (polygonal or dihedral QCs, one periodic direction  the quasiperodiclayers)

Octagonal QCs: local 8-fold symmetry [P & I]

Decagonal QCs: local 10-fold symmetry [P]

Dodecagonal QCs: local 12-fold symmetry [P]

Quasiperiodicin 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)

Octagonal QCs

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

Decagonal QCs

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

Dodecagonal QCs

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

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

Simulations of some diffraction patterns

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

A simulation from an icosahedral quasicrystal

Cut and Project

Fibonacci sequence (1D QCs)

HaraldBohr, ActaMathematicae,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

 tan irrational number (why?)

Choose 

E.g. :

Make cuts in a 6D space and project in 3D space  3D QCs

Fibonacci number

Aperiodic

Periodic

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 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

continue

Birth

1st month

Grow up

2nd month

3rd month

4th month

5th month

6th month

1

2

4

3

6

5

7

Month

8

# of pairs

1

2

3

5

8

1

13

?

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

Golden ratio

A

B

B

A

B

A

B

B

BA

B

BA

BA

B

BA

B

BA

BAB

BA

BAB

BAB

BA

BAB

BA

BAB

1-D QC

Ho-Mg-Zn Quasicrystal

from

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