What is a Quasicrystal?

Transcription

1 July 23, 2013

2 Rotational symmetry An object with rotational symmetry is an object that looks the same after a certain amount of rotation.

3 Rotational symmetry An object with rotational symmetry is an object that looks the same after a certain amount of rotation. Figure : rotational symmetry

4 Crystallography: X-ray diffraction experiment. The first x-ray diffraction experiment was performed by von Laue in 1912.

5 Crystallography: X-ray diffraction experiment. The first x-ray diffraction experiment was performed by von Laue in Crystals are ordered a diffraction pattern with sharp bright spots, Bragg peaks.

6 A paradigm before 1982 Crystallographic restriction: If atoms are arranged in a pattern periodic, then only 2,3,4 and 6-fold rotational symmetries for diffraction pattern of periodic crystals.

7 A paradigm before 1982 Crystallographic restriction: If atoms are arranged in a pattern periodic, then 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.

8 A paradigm before 1982 Crystallographic restriction: If atoms are arranged in a pattern periodic, then 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 Atoms in a solid are arranged in a periodic pattern.

9 Discovery of quasicrystals in 1982 Dan Shechtman (2011 Nobel Prize winner in Chemistry) Figure : Al 6 Mn

10 (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.

11 (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?

12 Delone sets A Delone set Λ in R d is a set with the properties: uniform discreteness: r > 0 such that for any y R d B r (y) Λ contains at most one element. relative denseness: R > 0 such that for any y R d B R (y) Λ contains at least one element.

13 Mathematical diffraction theory

14 Mathematical diffraction theory Dirac Comb δ Λ := x Λ δ x for Λ R d.

15 Mathematical diffraction theory Dirac Comb δ Λ := x Λ δ x for Λ R d. autocorrelation restricted to [ L, L] d : x,y Λ [ L,L] d δ x y.

16 Mathematical diffraction theory Dirac Comb δ Λ := x Λ δ x for Λ R d. autocorrelation restricted to [ L, L] d : autocorrelation measure 1 γ = lim L vol[ L, L] d x,y Λ [ L,L] d δ x y. x,y Λ [ L,L] d δ x y.

17 Mathematical diffraction theory Dirac Comb δ Λ := x Λ δ x for Λ R d. autocorrelation restricted to [ L, L] d : autocorrelation measure 1 γ = lim L vol[ L, L] d x,y Λ [ L,L] d δ x y. x,y Λ [ L,L] d δ x y. a diffraction measure ˆγ describes the mathematical diffraction. ˆγ is a positive measure: ˆγ = ˆγ d + ˆγ c.

18 Mathematical diffraction theory Dirac Comb δ Λ := x Λ δ x for Λ R d. autocorrelation restricted to [ L, L] d : autocorrelation measure 1 γ = lim L vol[ L, L] d x,y Λ [ L,L] d δ x y. x,y Λ [ L,L] d δ x y. a diffraction measure ˆγ describes the mathematical diffraction. ˆγ is a positive measure: ˆγ = ˆγ d + ˆγ c. a quasicrystal: a Delone set Λ with ˆγ d 0.

19 Example: a lattice L R d a lattice, i.e., L = A(Z d ), A is d d invertible matrix. L = {y : e ix y = 1, x L}

20 Example: a lattice L R d a lattice, i.e., L = A(Z d ), A is d d invertible matrix. L = {y : e ix y = 1, x L} L L = L implies γ = δ L = x L δ x

21 Example: a lattice L R d a lattice, i.e., L = A(Z d ), A is d d invertible matrix. L = {y : e ix y = 1, x L} L L = L implies γ = δ L = x L δ x Poisson summation formula says that ˆγ = 1 det A x L δ x

22 Meyer sets Let Λ be a Delone set.

23 Meyer sets Let Λ be a Delone set. 1 Λ Λ is uniformly discrete.

24 Meyer sets Let Λ be a Delone set. 1 Λ Λ is uniformly discrete. 2 Λ is an almost lattice: a finite set F : Λ Λ Λ + F.

25 Meyer sets Let Λ be a Delone set. 1 Λ Λ is uniformly discrete. 2 Λ is an almost lattice: a finite set F : Λ Λ Λ + F. 3 Λ is harmonious: ɛ > 0, Λ ɛ = {y R d : e ix y 1 ɛ} is relatively dense.

26 Meyer sets Let Λ be a Delone set. 1 Λ Λ is uniformly discrete. 2 Λ is an almost lattice: a finite set F : Λ Λ Λ + F. 3 Λ is harmonious: ɛ > 0, Λ ɛ = {y R d : e ix y 1 ɛ} is relatively dense. Theorem If Λ is a Delone set, then the above three conditions are equivalent.

27 Cut and Project method

28 Cut and Project method

29 Model Sets a model set (or cut and project set) is the translation of Λ = Λ(W ) = {π 1 (x) : x L, π 2 (x) W }. R d : a real euclidean space G: a locally compact abelian group projection maps π 1 : R d G R d, π 2 : R d G G L: a lattice in R d G with π 1 L is injective and π 2 (L) is dense. W G is non-empty and W = W 0 is compact.

30 Some results

31 Some results A model set is a Meyer set.

32 Some results A model set is a Meyer set. Schlottman, 1998 A model set Λ has a purely discrete diffraction spectrum. (ˆγ = ˆγ d ).

33 Some results A model set is a Meyer set. Schlottman, 1998 A model set Λ has a purely discrete diffraction spectrum. (ˆγ = ˆγ d ). Meyer, 1972 A Meyer is a subset of some model sets.

34 Some results A model set is a Meyer set. Schlottman, 1998 A model set Λ has a purely discrete diffraction spectrum. (ˆγ = ˆγ d ). Meyer, 1972 A Meyer is a subset of some model sets. Strungaru, 2005 A Meyer set has a discrete diffraction spectrum. (ˆγ d 0).

35 Pisot and Salem numbers Definition A Pisot number is a real algebraic integer θ > 1 whose conjugates all lie inside the unit circle. A Salem number is a real algebraic integer θ > 1 whose conjugates all lie inside or on the unit circle, at least one being on the circle.

36 Pisot and Salem numbers Definition A Pisot number is a real algebraic integer θ > 1 whose conjugates all lie inside the unit circle. A Salem number is a real algebraic integer θ > 1 whose conjugates all lie inside or on the unit circle, at least one being on the circle. Remark The set S of all Pisot numbers is infinite and has a remarkable structure: the sequence of derived sets does not terminate. S, S, S,...

37 Quasicrystals corresponding to Pisot or Salem numbers Example θ = and θ = L = {(a + bθ, a + bθ ) : a, b Z} is a Lattice in R 2. Λ = {a + bθ : a + bθ < 1} is a model set with θλ Λ.

38 Quasicrystals corresponding to Pisot or Salem numbers Example θ = and θ = L = {(a + bθ, a + bθ ) : a, b Z} is a Lattice in R 2. Theorem Λ = {a + bθ : a + bθ < 1} is a model set with θλ Λ. Given a Pisot or Salem number θ, there exists a model set Λ such that θλ Λ. Given a model set Λ, if θ is a positive real number with θλ Λ, then θ is a Pisot or Salem number.

39 Riemann Hypothesis Riemann zeta function: ζ(s) = n=1 1 n s.

40 Riemann Hypothesis Riemann zeta function: ζ(s) = n=1 1 n s. Riemann hypothesis: the non-trivial zeros should lie on the critical line 1/2 + it.

41 Riemann Hypothesis Riemann zeta function: ζ(s) = n=1 1 n s. Riemann hypothesis: the non-trivial zeros should lie on the critical line 1/2 + it. Z: the set of imaginary parts of the complex zeros. Z is not uniformly discrete.

42 Riemann Hypothesis Riemann zeta function: ζ(s) = n=1 1 n s. Riemann hypothesis: the non-trivial zeros should lie on the critical line 1/2 + it. Z: the set of imaginary parts of the complex zeros. Z is not uniformly discrete. Truth of hypothesis implies that the Fourier transform of Z is cm,p δ ± log p m, where p is a prime and m is a positive integer.

43 A generalized quasicrystal An aperiodic set Λ is a generalized quasicrystal if it is locally finite, and has a points in every sphere of some radius R. it has a discrete Fourier transform.

44 A generalized quasicrystal An aperiodic set Λ is a generalized quasicrystal if it is locally finite, and has a points in every sphere of some radius R. it has a discrete Fourier transform.

45 References Y. Meyer (1995) Quasicrystals, Diophantine approximations, and algebraic numbers M. Senechal (1995) Quasicrystals and Geometry R. Moody (1997) Meyer sets and their duals. F. Dyson (MSRI Lecture Notes 2002) Random Matrices, Neutron Capture Levels, Quasicrystals and Zeta-function zeros