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

∃ 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

∃ 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

∃ 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: ∃r > 0 such that for any y ∈ Rd

Br (y) ∩ Λ contains at most one element.

relative denseness: ∃R > 0 such that for any y ∈ Rd

BR (y) ∩ Λ contains at least one element.

What is a Quasicrystal? P d Dirac Comb δΛ := x∈Λ δx for Λ ⊂ R . d P autocorrelation restricted to [−L, L] : δx−y . x,y∈Λ∩[−L,L]d autocorrelation measure 1 X γ = lim δx−y . L→∞ vol[−L, L]d x,y∈Λ∩[−L,L]d

a diffraction measureˆγ describes the mathematical diffraction.

γˆ is a positive measure:γ ˆ =γ ˆd +γ ˆc .

a quasicrystal: a Delone set Λ withγ ˆd 6= 0.

Mathematical diffraction theory

What is a Quasicrystal? d P autocorrelation restricted to [−L, L] : δx−y . x,y∈Λ∩[−L,L]d autocorrelation measure 1 X γ = lim δx−y . L→∞ vol[−L, L]d x,y∈Λ∩[−L,L]d

a diffraction measureˆγ describes the mathematical diffraction.

γˆ is a positive measure:γ ˆ =γ ˆd +γ ˆc .

a quasicrystal: a Delone set Λ withγ ˆd 6= 0.

Mathematical diffraction theory

P d Dirac Comb δΛ := x∈Λ δx for Λ ⊂ R .

What is a Quasicrystal? autocorrelation measure 1 X γ = lim δx−y . L→∞ vol[−L, L]d x,y∈Λ∩[−L,L]d

a diffraction measureˆγ describes the mathematical diffraction.

γˆ is a positive measure:γ ˆ =γ ˆd +γ ˆc .

a quasicrystal: a Delone set Λ withγ ˆd 6= 0.

Mathematical diffraction theory

P d Dirac Comb δΛ := x∈Λ δx for Λ ⊂ R . d P autocorrelation restricted to [−L, L] : δx−y . x,y∈Λ∩[−L,L]d

What is a Quasicrystal? a diffraction measureˆγ describes the mathematical diffraction.

γˆ is a positive measure:γ ˆ =γ ˆd +γ ˆc .

a quasicrystal: a Delone set Λ withγ ˆd 6= 0.

Mathematical diffraction theory

P d Dirac Comb δΛ := x∈Λ δx for Λ ⊂ R . d P autocorrelation restricted to [−L, L] : δx−y . x,y∈Λ∩[−L,L]d autocorrelation measure 1 X γ = lim δx−y . L→∞ vol[−L, L]d x,y∈Λ∩[−L,L]d

What is a Quasicrystal? a quasicrystal: a Delone set Λ withγ ˆd 6= 0.

Mathematical diffraction theory

P d Dirac Comb δΛ := x∈Λ δx for Λ ⊂ R . d P autocorrelation restricted to [−L, L] : δx−y . x,y∈Λ∩[−L,L]d autocorrelation measure 1 X γ = lim δx−y . L→∞ vol[−L, L]d x,y∈Λ∩[−L,L]d

a diffraction measureˆγ describes the mathematical diffraction.

γˆ is a positive measure:γ ˆ =γ ˆd +γ ˆc .

What is a Quasicrystal? Mathematical diffraction theory

P d Dirac Comb δΛ := x∈Λ δx for Λ ⊂ R . d P autocorrelation restricted to [−L, L] : δx−y . x,y∈Λ∩[−L,L]d autocorrelation measure 1 X γ = lim δx−y . L→∞ vol[−L, L]d x,y∈Λ∩[−L,L]d

a diffraction measureˆγ describes the mathematical diffraction.

γˆ is a positive measure:γ ˆ =γ ˆd +γ ˆc .

a quasicrystal: a Delone set Λ withγ ˆd 6= 0.

What is a Quasicrystal? L − L = L implies X γ = δL = δx x∈L Poisson summation formula says that

1 X γˆ = δ | det A| x x∈L∗

Example: a lattice

d d L ⊂ R a lattice, i.e., L = A(Z ), A is d × d invertible matrix. L∗ = {y : eix·y = 1, ∀x ∈ L}

What is a Quasicrystal? Poisson summation formula says that

1 X γˆ = δ | det A| x x∈L∗

Example: a lattice

d d L ⊂ R a lattice, i.e., L = A(Z ), A is d × d invertible matrix. L∗ = {y : eix·y = 1, ∀x ∈ L} L − L = L implies X γ = δL = δx x∈L

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∗ = {y : eix·y = 1, ∀x ∈ L} L − L = L implies X γ = δL = δx x∈L Poisson summation formula says that

1 X γˆ = δ | det A| x x∈L∗

What is a Quasicrystal? 1 Λ − Λ is uniformly discrete. 2 Λ is an almost lattice:

∃ a finite set F :Λ − Λ ⊂ Λ + F .

3 Λ is harmonious:

∗ d ix·y ∀ > 0, Λ = {y ∈ R : |e − 1| ≤ } 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:

∃ a finite set F :Λ − Λ ⊂ Λ + F .

3 Λ is harmonious:

∗ d ix·y ∀ > 0, Λ = {y ∈ R : |e − 1| ≤ } 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 ∀ > 0, Λ = {y ∈ R : |e − 1| ≤ } 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:

∃ 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:

∃ a finite set F :Λ − Λ ⊂ Λ + F .

3 Λ is harmonious:

∗ d ix·y ∀ > 0, Λ = {y ∈ R : |e − 1| ≤ } 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:

∃ a finite set F :Λ − Λ ⊂ Λ + F .

3 Λ is harmonious:

∗ d ix·y ∀ > 0, Λ = {y ∈ R : |e − 1| ≤ } 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 ) = {π1(x): x ∈ L, π2(x) ∈ W }.

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 π1|L 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. (ˆγ =γ ˆd ). Meyer, 1972 A Meyer is a subset of some model sets. Strungaru, 2005 A Meyer set has a discrete diffraction spectrum. (ˆγd 6= 0).

Some results

What is a Quasicrystal? Schlottman, 1998 A model set Λ has a purely discrete diffraction spectrum. (ˆγ =γ ˆd ). Meyer, 1972 A Meyer is a subset of some model sets. Strungaru, 2005 A Meyer set has a discrete diffraction spectrum. (ˆγd 6= 0).

Some results

A model set is a Meyer set.

What is a Quasicrystal? Meyer, 1972 A Meyer is a subset of some model sets. Strungaru, 2005 A Meyer set has a discrete diffraction spectrum. (ˆγd 6= 0).

Some results

A model set is a Meyer set. Schlottman, 1998 A model set Λ has a purely discrete diffraction spectrum. (ˆγ =γ ˆd ).

What is a Quasicrystal? Strungaru, 2005 A Meyer set has a discrete diffraction spectrum. (ˆγd 6= 0).

Some results

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

What is a Quasicrystal? Some results

A model set is a Meyer set. Schlottman, 1998 A model set Λ has a purely discrete diffraction spectrum. (ˆγ =γ ˆd ). Meyer, 1972 A Meyer is a subset of some model sets. Strungaru, 2005 A Meyer set has a discrete diffraction spectrum. (ˆγd 6= 0).

What is a Quasicrystal? Remark The set S of all Pisot numbers is infinite and has a remarkable structure: the sequence of derived sets

S, S0, S00,...

does not terminate.

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.

What is a Quasicrystal? 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

S, S0, S00,...

does not terminate.

What is a Quasicrystal? Theorem 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.

Quasicrystals corresponding to Pisot or Salem numbers

Example √ √ 1+ 5 0 1− 5 θ = 2 and θ = 2 . 0 2 L = {(a + bθ, a + bθ ): a, b ∈ Z} is a Lattice in R . Λ = {a + bθ : |a + bθ0| < 1} is a model set with

θΛ ⊂ Λ.

What is a Quasicrystal? Quasicrystals corresponding to Pisot or Salem numbers

Example √ √ 1+ 5 0 1− 5 θ = 2 and θ = 2 . 0 2 L = {(a + bθ, a + bθ ): a, b ∈ Z} is a Lattice in R . Λ = {a + bθ : |a + bθ0| < 1} is a model set with

θΛ ⊂ Λ.

Theorem 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.

What is a Quasicrystal? 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 X cm,pδ± log pm ,

where p is a prime and m is a positive integer.

Riemann Hypothesis

Riemann zeta function: ∞ X 1 ζ(s) = . ns n=1

What is a Quasicrystal? 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 X cm,pδ± log pm ,

where p is a prime and m is a positive integer.

Riemann Hypothesis

Riemann zeta function: ∞ X 1 ζ(s) = . ns n=1

Riemann hypothesis: the non-trivial zeros should lie on the critical line 1/2 + it.

What is a Quasicrystal? Truth of hypothesis implies that the Fourier transform of Z is X cm,pδ± log pm ,

where p is a prime and m is a positive integer.

Riemann Hypothesis

Riemann zeta function: ∞ X 1 ζ(s) = . ns n=1

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.

What is a Quasicrystal? Riemann Hypothesis

Riemann zeta function: ∞ X 1 ζ(s) = . ns n=1

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 X cm,pδ± log pm ,

where p is a prime and m is a positive integer.

What is a Quasicrystal? 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.

What is a Quasicrystal? 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.

What is a Quasicrystal? 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

What is a Quasicrystal?