What is a Quasicrystal? July 23, 2013 What is a Quasicrystal? Figure : rotational symmetry Rotational symmetry An object with rotational symmetry is an object that looks the same after a certain amount of rotation. What is a Quasicrystal? Rotational symmetry An object with rotational symmetry is an object that looks the same after a certain amount of rotation. Figure : rotational symmetry What is a Quasicrystal? Crystals are ordered ) a diffraction pattern with sharp bright spots, Bragg peaks. Crystallography: X-ray diffraction experiment. The first x-ray diffraction experiment was performed by von Laue in 1912. What is a Quasicrystal? Crystallography: X-ray diffraction experiment. The first x-ray diffraction experiment was performed by von Laue in 1912. Crystals are ordered ) a diffraction pattern with sharp bright spots, Bragg peaks. What is a Quasicrystal? All the crystals were found to be periodic from 1912 till 1982. Atoms in a solid are arranged in a periodic pattern. A paradigm before 1982 Crystallographic restriction: If atoms are arranged in a pattern periodic, then 9 only 2,3,4 and 6-fold rotational symmetries for diffraction pattern of periodic crystals. What is a Quasicrystal? Atoms in a solid are arranged in a periodic pattern. A paradigm before 1982 Crystallographic restriction: If atoms are arranged in a pattern periodic, then 9 only 2,3,4 and 6-fold rotational symmetries for diffraction pattern of periodic crystals. All the crystals were found to be periodic from 1912 till 1982. What is a Quasicrystal? A paradigm before 1982 Crystallographic restriction: If atoms are arranged in a pattern periodic, then 9 only 2,3,4 and 6-fold rotational symmetries for diffraction pattern of periodic crystals. All the crystals were found to be periodic from 1912 till 1982. Atoms in a solid are arranged in a periodic pattern. What is a Quasicrystal? Discovery of quasicrystals in 1982 Dan Shechtman (2011 Nobel Prize winner in Chemistry) Figure : Al6Mn What is a Quasicrystal? Q. What are the appropriate mathematical models? (Quasi)crystals Definition for Crystal Till 1991: a solid composed of atoms arranged in a pattern periodic in three dimensions. After 1992 (International Union of Crystallography) any solid having an discrete diffraction diagram. What is a Quasicrystal? (Quasi)crystals Definition for Crystal Till 1991: a solid composed of atoms arranged in a pattern periodic in three dimensions. After 1992 (International Union of Crystallography) any solid having an discrete diffraction diagram. Q. What are the appropriate mathematical models? What is a Quasicrystal? Delone sets d A Delone set Λ in R is a set with the properties: uniform discreteness: 9r > 0 such that for any y 2 Rd Br (y) \ Λ contains at most one element: relative denseness: 9R > 0 such that for any y 2 Rd BR (y) \ Λ contains at least one element: What is a Quasicrystal? P d Dirac Comb δΛ := x2Λ δx for Λ ⊂ R . d P autocorrelation restricted to [−L; L] : δx−y : x;y2Λ\[−L;L]d autocorrelation measure 1 X γ = lim δx−y : L!1 vol[−L; L]d x;y2Λ\[−L;L]d a diffraction measure^γ describes the mathematical diffraction. γ^ is a positive measure:γ ^ =γ ^d +γ ^c : a quasicrystal: a Delone set Λ withγ ^d 6= 0. Mathematical diffraction theory What is a Quasicrystal? d P autocorrelation restricted to [−L; L] : δx−y : x;y2Λ\[−L;L]d autocorrelation measure 1 X γ = lim δx−y : L!1 vol[−L; L]d x;y2Λ\[−L;L]d a diffraction measure^γ describes the mathematical diffraction. γ^ is a positive measure:γ ^ =γ ^d +γ ^c : a quasicrystal: a Delone set Λ withγ ^d 6= 0. Mathematical diffraction theory P d Dirac Comb δΛ := x2Λ δx for Λ ⊂ R . What is a Quasicrystal? autocorrelation measure 1 X γ = lim δx−y : L!1 vol[−L; L]d x;y2Λ\[−L;L]d a diffraction measure^γ describes the mathematical diffraction. γ^ is a positive measure:γ ^ =γ ^d +γ ^c : a quasicrystal: a Delone set Λ withγ ^d 6= 0. Mathematical diffraction theory P d Dirac Comb δΛ := x2Λ δx for Λ ⊂ R . d P autocorrelation restricted to [−L; L] : δx−y : x;y2Λ\[−L;L]d What is a Quasicrystal? a diffraction measure^γ describes the mathematical diffraction. γ^ is a positive measure:γ ^ =γ ^d +γ ^c : a quasicrystal: a Delone set Λ withγ ^d 6= 0. Mathematical diffraction theory P d Dirac Comb δΛ := x2Λ δx for Λ ⊂ R . d P autocorrelation restricted to [−L; L] : δx−y : x;y2Λ\[−L;L]d autocorrelation measure 1 X γ = lim δx−y : L!1 vol[−L; L]d x;y2Λ\[−L;L]d What is a Quasicrystal? a quasicrystal: a Delone set Λ withγ ^d 6= 0. Mathematical diffraction theory P d Dirac Comb δΛ := x2Λ δx for Λ ⊂ R . d P autocorrelation restricted to [−L; L] : δx−y : x;y2Λ\[−L;L]d autocorrelation measure 1 X γ = lim δx−y : L!1 vol[−L; L]d x;y2Λ\[−L;L]d a diffraction measure^γ describes the mathematical diffraction. γ^ is a positive measure:γ ^ =γ ^d +γ ^c : What is a Quasicrystal? Mathematical diffraction theory P d Dirac Comb δΛ := x2Λ δx for Λ ⊂ R . d P autocorrelation restricted to [−L; L] : δx−y : x;y2Λ\[−L;L]d autocorrelation measure 1 X γ = lim δx−y : L!1 vol[−L; L]d x;y2Λ\[−L;L]d a diffraction measure^γ describes the mathematical diffraction. γ^ is a positive measure:γ ^ =γ ^d +γ ^c : a quasicrystal: a Delone set Λ withγ ^d 6= 0. What is a Quasicrystal? L − L = L implies X γ = δL = δx x2L Poisson summation formula says that 1 X γ^ = δ j det Aj x x2L∗ Example: a lattice d d L ⊂ R a lattice, i.e., L = A(Z ); A is d × d invertible matrix. L∗ = fy : eix·y = 1; 8x 2 Lg What is a Quasicrystal? Poisson summation formula says that 1 X γ^ = δ j det Aj x x2L∗ Example: a lattice d d L ⊂ R a lattice, i.e., L = A(Z ); A is d × d invertible matrix. L∗ = fy : eix·y = 1; 8x 2 Lg L − L = L implies X γ = δL = δx x2L What is a Quasicrystal? Example: a lattice d d L ⊂ R a lattice, i.e., L = A(Z ); A is d × d invertible matrix. L∗ = fy : eix·y = 1; 8x 2 Lg L − L = L implies X γ = δL = δx x2L Poisson summation formula says that 1 X γ^ = δ j det Aj x x2L∗ What is a Quasicrystal? 1 Λ − Λ is uniformly discrete. 2 Λ is an almost lattice: 9 a finite set F :Λ − Λ ⊂ Λ + F : 3 Λ is harmonious: ∗ d ix·y 8 > 0; Λ = fy 2 R : je − 1j ≤ g is relatively dense: Theorem If Λ is a Delone set, then the above three conditions are equivalent. Meyer sets Let Λ be a Delone set. What is a Quasicrystal? 2 Λ is an almost lattice: 9 a finite set F :Λ − Λ ⊂ Λ + F : 3 Λ is harmonious: ∗ d ix·y 8 > 0; Λ = fy 2 R : je − 1j ≤ g is relatively dense: Theorem If Λ is a Delone set, then the above three conditions are equivalent. Meyer sets Let Λ be a Delone set. 1 Λ − Λ is uniformly discrete. What is a Quasicrystal? 3 Λ is harmonious: ∗ d ix·y 8 > 0; Λ = fy 2 R : je − 1j ≤ g is relatively dense: Theorem If Λ is a Delone set, then the above three conditions are equivalent. Meyer sets Let Λ be a Delone set. 1 Λ − Λ is uniformly discrete. 2 Λ is an almost lattice: 9 a finite set F :Λ − Λ ⊂ Λ + F : What is a Quasicrystal? Theorem If Λ is a Delone set, then the above three conditions are equivalent. Meyer sets Let Λ be a Delone set. 1 Λ − Λ is uniformly discrete. 2 Λ is an almost lattice: 9 a finite set F :Λ − Λ ⊂ Λ + F : 3 Λ is harmonious: ∗ d ix·y 8 > 0; Λ = fy 2 R : je − 1j ≤ g is relatively dense: What is a Quasicrystal? Meyer sets Let Λ be a Delone set. 1 Λ − Λ is uniformly discrete. 2 Λ is an almost lattice: 9 a finite set F :Λ − Λ ⊂ Λ + F : 3 Λ is harmonious: ∗ d ix·y 8 > 0; Λ = fy 2 R : je − 1j ≤ g is relatively dense: Theorem If Λ is a Delone set, then the above three conditions are equivalent. What is a Quasicrystal? Cut and Project method What is a Quasicrystal? Cut and Project method What is a Quasicrystal? Model Sets a model set (or cut and project set) is the translation of Λ = Λ(W ) = fπ1(x): x 2 L; π2(x) 2 W g: d R : a real euclidean space G: a locally compact abelian group projection maps d d d π1 : R × G ! R ; π2 : R × G ! G d L: a lattice in R × G with π1jL is injective and π2(L) is dense. W ⊂ G is non-empty and W = W 0 is compact. What is a Quasicrystal? A model set is a Meyer set. Schlottman, 1998 A model set Λ has a purely discrete diffraction spectrum. (^γ =γ ^d ). Meyer, 1972 A Meyer is a subset of some model sets. Strungaru, 2005 A Meyer set has a discrete diffraction spectrum. (^γd 6= 0). Some results What is a Quasicrystal? Schlottman, 1998 A model set Λ has a purely discrete diffraction spectrum. (^γ =γ ^d ). Meyer, 1972 A Meyer is a subset of some model sets.