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.

2-, 3-, 4-, 6-fold 5 fold symmetry unit ? A B A' B' Electrons X-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 symmetry Non-periodic! 1 τ 1+τ τ ＝ (1+√5)/2 ～ ゜ 72 ゜ A B C A D B C D AB AC CD = = τ τ ＝ (1+√5)/2 ～ τ+1= τ2τ2 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 symmetry Non-periodic! 1 τ 1+τ τ ＝ (1+√5)/2 ～ 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

5-fold symmetry Non-periodic! 1 τ 1+τ τ ＝ (1+√5)/2 ～ 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° °

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 approximant YbCd 6 2/1 approximant Yb 13 Cd 76

”Where are the atoms?” Structure of i-YbCd 5.7 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. Tsai Nature Materials. 2007, 6, 58 C. Pay Gómez*, S. Lidin Angew. Chem., Int. Ed. Engl. 2001, 40, 4037 Building blocks and linkages in Yb-Cd type approximants 1/1 approximant YbCd 6 2/1 approximant Yb 13 Cd 76 Yb-Cd type Atomic cluster subshells

Tsai i-YbCd 5.7 Bergman (FK-type) Qisheng Lin, John D. Corbett*, Proc. Nat. Acad. Sci. 2006, 103, C. Pay Gómez*, S. Lidin Angew. 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!