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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
http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/advanced-chemistryprize2011.pdf
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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.
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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 !
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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
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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)
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Chris J. Pickard and R. J. Needs, Nature Materials 9,624–627
Octagonal QCs
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http://nanopatentsandinnovations.blogspot.tw/2011/10/quasicrystals-discovery-wins-novel.html
Decagonal QCs
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http://www.pnas.org/content/108/5/1810/F6.expansion.html
Dodecagonal QCs
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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
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Simulations of some diffraction patterns
A simulation from an icosahedral quasicrystal
F. Samavat et. al., Open Journal of Physical Chemistry, 2012, 2, 7-14.
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2 3 4
http://www.lassp.cornell.edu/lifshitz/quasicrystals.html
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Example of 1D QCs
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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)
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Aperiodic
Periodic
2
51tan
7
11tan
Aperiodic crystal
Periodic crystal~ approximant (called)
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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
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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
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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
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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
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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
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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
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Ho-Mg-Zn Quasicrystal from
http://cmp.physics.iastate.edu/canfield/photos.html