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.wiki

pedia.org/wiki/Icosahedral_symmetry http//www.no

belprize.org/nobel_prizes/chemistry/laureates/2011

/advanced-chemistryprize2011.pdf

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.

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 !

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

t scaling ratio

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 (axes12x5-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 9,624627

Decagonal QCs

http//nanopatentsandinnovations.blogspot.tw/2011/

10/quasicrystals-discovery-wins-novel.html

Dodecagonal QCs

http//www.pnas.org/content/108/5/1810/F6.expansio

n.html

http//en.wikipedia.org/wiki/FileIcosahedron.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

1112, November 2002, Pages 10891098

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

http//www.lassp.cornell.edu/lifshitz/quasicrystal

s.html

?

? 2

? 3

? 4

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?)

Make cuts in a 6D space and project in 3D space ?

3D QCs

Aperiodic Periodic

Aperiodic crystal Periodic crystal

approximant (called)

Fibonacci number (series, sequence)

Fibonacci Rabbits

Fibonaccis 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

Birth

continue

Grow up

7

Month

of pairs

13

?

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

BA

BAB

BAB

B

A

B

B

A

B

A

B

B

A

B

B

A

1-D QC

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 3D ?3 AlLiCu Donnadieu 1994

Decagonal 2D ? (?5) 10/mmm AlMn Chattopadhyay et al., 1985a and Bendersky, 1985

Dodecagonal 2D ?3 12/mmm NiCr Ishimasa et al. 1985

Type of quasicrystal QP Metric Symmetry System First Report

Octagonal 2D ?2 8/mmm VNiSi, CrNiSi Wang et al. 1987

Pentagonal 2D ? (?5) AlCuFe Bancel 1993

Hexagonal 2D ?3 6/mmm AlCr Selke et al. 1994

Trigonal 1D ?3 AlCuNi Chattopadhyay et al., 1987

Digonal 1D ?2 222 AlCuCo He et al. 1988

Ho-Mg-Zn Quasicrystal from http//cmp.physics.ia

state.edu/canfield/photos.html