QUASICRYSTALS: The end of the beginning
Cesar Pay Gómez
Outline
• History of Quasicrystals
• “Where are the atoms?”
• Past, present and future
Dan Shechtman
The Nobel Prize in
Chemistry 2011 is
awarded to
Dan Shechtman
for the discovery of
quasicrystals.
Crystal
Before QCs
A homogenous solid formed by a repeating, three-dimensional pattern of atoms, ions, or
molecules and having fixed distances between constituent parts.
Crystal
Electrons
X-rays
Glass
?
2-, 3-, 4-, 6-fold
A
A'
B
B'
5 fold symmetry unit ?
5-fold symmetry!
Non-periodic!
Shechtman et al. Phys. Rev. Lett., 53, 1951 (1984)
The discovery
Crystal
Before QCs
A homogenous solid formed by a repeating, three-dimensional pattern of atoms, ions, or
molecules and having fixed distances between constituent parts.
After QCs
Any solid having an essentially discrete diffraction diagram. The word essentially means that
most of the intensity of the diffraction is concentrated in relatively sharp Bragg peaks, besides
the always present diffuse scattering. By 'aperiodic crystal' we mean any crystal in which
three-dimensional lattice periodicity can be considered to be absent.
Quasicrystals
•
•
•
•
•
Long-range ordered, aperiodic crystals with sharp diffraction peaks.
Exhibit crystallographically forbidden symmetries (such as 5-, 8-, 10- or 12-fold rotational symmetry)
Lack periodicity (no unit cell) in 3 dimensions.
The diffraction patterns cannot be indexed with 3 integers (6 are needed for icosahedral QCs).
The structures can be described as projections from a high dimensional space.
Al-Cu-Fe: Stable, ~cm
Quasicrystals
•
•
•
•
•
Long-range ordered, aperiodic crystals with sharp diffraction peaks.
Exhibit crystallographically forbidden symmetries (such as 5-, 8-, 10- or 12-fold rotational symmetry)
Lack periodicity (no unit cell) in 3 dimensions.
The diffraction patterns cannot be indexed with 3 integers (6 are needed for icosahedral QCs).
The structures can be described as projections from a high dimensional space.
Penrose pattern
C
B
36゜
D
A
AC AB
= τ
=
CD AC
1
τ
1+τ
τ＝(1+√5)/2 ～1.618
C
72゜
B
A
D
τ＝(1+√5)/2 ～1.618
τ+1= τ2
5-fold symmetry
Non-periodic!
Shechtman et al. Phys. Rev. Lett., 53, 1951 (1984)
τ-1=1/τ
5-fold symmetry
Non-periodic
Self-similarity (irrational)
Quasicrystals
•
•
•
•
•
Long-range ordered, aperiodic crystals with sharp diffraction peaks.
Exhibit crystallographically forbidden symmetries (such as 5-, 8-, 10- or 12-fold rotational symmetry)
Lack periodicity (no unit cell) in 3 dimensions.
The diffraction patterns cannot be indexed with 3 integers (6 are needed for icosahedral QCs).
The structures can be described as projections from a high dimensional space.
Dihedral Quasicrystals
1
τ
1+τ
τ＝(1+√5)/2 ～1.618
5-fold symmetry
Non-periodic!
Shechtman et al. Phys. Rev. Lett., 53, 1951 (1984)
4Å
4Å
Periodic
direction
Quasicrystals
•
•
•
•
•
Long-range ordered, aperiodic crystals with sharp diffraction peaks.
Exhibit crystallographically forbidden symmetries (such as 5-, 8-, 10- or 12-fold rotational symmetry)
Lack periodicity (no unit cell) in 3 dimensions.
The diffraction patterns cannot be indexed with 3 integers (6 are needed for icosahedral QCs).
The structures can be described as projections from a high dimensional space.
63.43°
1
τ
116.57°
1+τ
τ＝(1+√5)/2 ～1.618
5-fold symmetry
Non-periodic!
Shechtman et al. Phys. Rev. Lett., 53, 1951 (1984)
3D Space filling by two
rhombohedra
Icosahedral Quasicrystals
Quenched
Al-Mn alloy
Icosahedron
Shechtman et al. Phys. Rev. Lett., 53, 1951 (1984)
QC families
Bergman cluster
(Frank-Kasper type)
Mackay cluster
Tsai cluster
(Yb-Cd type)
Approximants
•
•
•
Conventional crystals with periodic long-range order and 3D unit cells.
Should have similar compositions and local atomic arrangements (clusters) as the quasicrystals.
The structures can be solved by standard diffraction techiques.
1/1 approximant
YbCd6
2/1 approximant
Yb13Cd76
”Where are the atoms?”
Structure of i-YbCd5.7 QC
H. Takakura, C. Pay Gómez, A. Yamamoto, M. de Boissieu, A. P. Tsai,
Nature Materials. 2007, 6, 58
Building blocks and linkages in Yb-Cd type approximants
1/1 approximant
YbCd6
2/1 approximant
Yb13Cd76
b
c
subshells
Yb-Cd type
Atomic cluster
H. Takakura*, C. Pay Gómez, A. Yamamoto, M. de Boissieu, A. P. Tsai
Nature Materials. 2007, 6, 58
C. Pay Gómez*, S. Lidin
Angew. Chem., Int. Ed. Engl. 2001, 40, 4037
Tsai
i-YbCd5.7
Bergman (FK-type)
C. Pay Gómez*, S. Lidin
Angew. Chem., Int. Ed. Engl. 2001, 40, 4037
Qisheng Lin, John D. Corbett*,
Proc. Nat. Acad. Sci. 2006, 103, 13589
QC families
Bergman cluster
(Frank-Kasper type)
Mackay cluster
Tsai cluster
(Yb-Cd)
Conclusions
• Due to the discovery of QCs, the definition of crystal had to be
changed.
• QCs have long-range order but lack periodicity in 3D space.
• Approximants are ”normal” crystals containing the same
atomic clusters as QCs.
• Icosahedral QCs can be described as periodic structures in 6D
space.
Thank you!