QUASICRYSTALS: The end of the beginningCesar 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.

2-, 3-, 4-, 6-fold

5 fold symmetry unit ?

AB

A'B'

ElectronsX-rays

Crystal

Glass

5-fold symmetry!Non-periodic!

Shechtman et al. Phys. Rev. Lett., 53, 1951 (1984)

?

Crystal

A homogenous solid formed by a repeating, three-dimensional pattern of atoms, ions, or molecules and having fixed distances between constituent parts.

Before QCs

The discovery

Crystal

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.

After QCs

A homogenous solid formed by a repeating, three-dimensional pattern of atoms, ions, or molecules and having fixed distances between constituent parts.

Before QCs

Al-Cu-Fe: Stable, ~cm

• 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.

Quasicrystals

5-fold symmetryNon-periodic!

1 τ 1+τ

τ ＝ (1+√5)/2 ～ 1.618

36 ゜

72 ゜

A

BC

A

D

B

C

D

ABAC

ACCD

= = τ

τ ＝ (1+√5)/2 　 ～ 1.618

τ+1= τ2 　 　 τ-1=1/τ Self-similarity (irrational)

5-fold symmetry

Non-periodic

Penrose pattern

Shechtman et al. Phys. Rev. Lett., 53, 1951 (1984)

• 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.

Quasicrystals

5-fold symmetryNon-periodic!

1 τ 1+τ

τ ＝ (1+√5)/2 ～ 1.618

Shechtman et al. Phys. Rev. Lett., 53, 1951 (1984)

• 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.

Quasicrystals

Dihedral Quasicrystals

Periodic direction

4Å

4Å

5-fold symmetryNon-periodic!

1 τ 1+τ

τ ＝ (1+√5)/2 ～ 1.618

Shechtman et al. Phys. Rev. Lett., 53, 1951 (1984)

• 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.

Quasicrystals

3D Space filling by two rhombohedra

63.43° 116.57°

Icosahedral Quasicrystals

Shechtman et al. Phys. Rev. Lett., 53, 1951 (1984)

Icosahedron

Quenched Al-Mn alloy

Bergman cluster(Frank-Kasper type)

Mackay cluster

Tsai cluster(Yb-Cd type)

QC families

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 approximantYbCd6

2/1 approximantYb13Cd76

”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

b

c

H. Takakura*, C. Pay Gómez, A. Yamamoto, M. de Boissieu, A. P. TsaiNature Materials. 2007, 6, 58

C. Pay Gómez*, S. LidinAngew. Chem., Int. Ed. Engl. 2001, 40, 4037

Building blocks and linkages in Yb-Cd type approximants1/1 approximant

YbCd6

2/1 approximantYb13Cd76

Yb-Cd type Atomic cluster

subshells

Tsaii-YbCd5.7

Bergman (FK-type)

Qisheng Lin, John D. Corbett*,Proc. Nat. Acad. Sci. 2006, 103, 13589

C. Pay Gómez*, S. LidinAngew. Chem., Int. Ed. Engl. 2001, 40, 4037

Bergman cluster (Frank-Kasper type)

Mackay cluster

Tsai cluster(Yb-Cd)

QC families

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!