Symmetry for Quasicrystals

1
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
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
?
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
5
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)
6
Octagonal QCs
Chris J. Pickard and R. J. Needs, Nature
Materials 9,624627
7
Decagonal QCs
http//nanopatentsandinnovations.blogspot.tw/2011/
10/quasicrystals-discovery-wins-novel.html
8
Dodecagonal QCs
http//www.pnas.org/content/108/5/1810/F6.expansio
n.html
9
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
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
http//www.lassp.cornell.edu/lifshitz/quasicrystal
s.html
?
? 2
? 3
? 4
12
Example of 1D QCs
13
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
14
Aperiodic Periodic
Aperiodic crystal Periodic crystal
approximant (called)
15
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
16
Birth
continue
Grow up
7
Month
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
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
19
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
20
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
21
Ho-Mg-Zn Quasicrystal from http//cmp.physics.ia
state.edu/canfield/photos.html