Recent Progress in Mathematical Diffraction · Recent Progress in Mathematical Diffraction Uwe...

$: Recent Progress in Mathematical Diffraction · Recent Progress in Mathematical Diffraction Uwe Grimm Department of Mathematics & Statistics The Open University Milton Keynes, UK (joint$
Recent Progress inMathematical Diffraction

Uwe Grimm

Department of Mathematics & StatisticsThe Open UniversityMilton Keynes, UK

(joint work with Michael Baake & others)

Workshop on time-frequency analysis and aperiodic order, Trondheim, June 2015 – p.1

MenueDiffraction theoryPP spectra

Poisson formulaModel sets

SC spectraThue–MorseCantorPlanar example

AC spectraBernoulliRudin–ShapiroBernoullisationRandom tilings

Dynamical versusdiffraction spectra


DiffractionOptical diffraction

L A S E R

Diffraction pattern

interference of scattered waves

structure analysis

X-ray, electron or neutron diffraction

information on order and symmetry


Crystallographic restrictionA lattice in 2 or 3 dimensions can only have non-trivialrotational symmetry axes of order 2, 3, 4, or 6.



Why is that true?



Why is that true?

Consider two points of minimal distance in a 2d lattice



Why is that true?


Order 3



Why is that true?


Order 3

120o



Why is that true?


Order 3

120o 120o



Why is that true?


Order 4

90o



Why is that true?


Order 4

90o 90o



Why is that true?


Order 5

72o



Why is that true?


Order 5

72o 72o



Why is that true?


Order 6

60o 60o



Why is that true?


Order 8

45o 45o



Same result holds for three dimensions, but moresymmetries can be realised in lattices of dimension ≥ 4.


DiffractionQuasicrystals 5-fold

3-fold

2-fold

from: D. Shechtman, I. Blech, D. Gratias and J.W. Cahn (1984), Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53, 1951–1953


Diffraction theory

Wiener diagram

g∗−−−→ g ∗ g

Fy

yF

g|.|2−−−→ |g|2

commutative for integrable function g (with g(x) := g(−x) )

Kinematic diffraction:diagonal map g 7→ |g|2

Mathematical diffraction theory:use path via autocorrelationfor translation bounded measures


Why measures?Measures

are natural mathematical objects to describedistributions (of scatterers or radiation) is space

provide a unified generalisation of continuous (density)and discrete (tiling) approaches

ensure that quantities are mathematically well-defined

Absolutely continuous measure µ (with density ) on Rd

µ(f) =

∫

Rd

f(x) dµ(x) =

∫

Rd

f(x) (x) dx

Pure point measure µ on Rd

µ(f) =(∑

i∈Iwi δxi

)(f) =

∑

i∈Iwi f(xi)


Why measures?

Singular continuous measure

gives no weight to any single point

support is uncountable set of zero Lebesgue measure

‘trivial’ cases (e.g. measures on lines in the plane)

probability measureµ for middle-thirdsCantor set withcontinuousdistribution functionF (x) = µ([0, x])

0 11/3 2/3

1

1/2

1/4

3/4

x

F(x)


Diffraction theory

Structure: translation bounded measure ω

assumed ‘amenable’

Autocorrelation: γ = γω = ω ⊛ ω := limR→∞

ω|R ∗ ω|Rvol(BR)

Diffraction: γ = γpp + γsc + γac (relative to Lebesgue measure)

pp: Bragg peaks

ac: diffuse scattering with density

sc: whatever remains ...

General setting: ω y γ = ω ⊛ ω y γ 6y ω


Pure point spectra: Periodic crystals

Point measures: δx , δS :=∑

x∈S δx

Poisson summation formula: δΓ = dens(Γ ) δΓ ∗

for lattice Γ , dual lattice Γ ∗

Periodic crystals: ω = µ ∗ δΓ (µ finite)

⊲ γ = dens(Γ ) (µ ∗ µ) ∗ δΓ⊲ γ =

(dens(Γ )

)2 ∣∣µ∣∣2 δΓ ∗

(pure point diffraction: Bragg peaks on dual lattice)


Pure point spectra: Periodic example I

Periodic Dirac comb

ω = δZ + 12 δZ+ 1

2

+ 14 δZ+{ 1

4, 34}

This can be written as

ω = µ ∗ δZ with µ = δ0 + 14δ 1

4

+ 12δ 1

2

+ 14δ 3

4

We have

ω =(δ0 + 1

4 δ− 1

4

+ 12 δ− 1

2

+ 14 δ− 3

4

)∗ δZ = ω


Pure point spectra: Periodic example I

Using δx ∗ δy = δx+y, one obtains the autocorrelation

γ = ω ∗ ω =(

118 δ0 + 3

4 δ 1

4

+ 98 δ 1

2

+ 34 δ 3

4

)∗ δZ

The diffraction measure is

γ =(

118 + 9

8(−1)k + 32 cos(πk2 )

)∗ δZ

= 4 δ4Z + δ4Z+2 + 14 δ4Z+{1,3} .


Pure point spectra: Periodic example IITwo-dimensional crystal with lattice of periods Z

2 and twoscatterers at positions (0, 0) and (a, b), so

Λ = Z2 ∪(Z

2 + (a, b))

with Dirac comb

ω = δΛ = (δ(0,0) + δ(a,b)) ∗ δZ2

Autocorrelationγ = (δ(0,0) + δ(a,b)) ∗ (δ(0,0) + δ−(a,b)) ∗ δZ2

=(2 δ(0,0) + δ(a,b) + δ−(a,b)

)∗ δ

Z2

Diffractionγ =

(2 + 2 cos

((2π(ka+ ℓb)

))δZ2

=(2 cos

(π(ka+ ℓb)

))2δZ2 for k, ℓ ∈ Z


Pure point spectra: Periodic example II

Diffraction

γ =(2 + 2 cos

((2π(ka+ ℓb)

))δZ2

=(2 cos

(π(ka+ ℓb)

))2δZ2 for k, ℓ ∈ Z

Example with (a, b) = (13 ,

1√3)


Pure point spectra: Aperiodic example

Aperiodic structure as a limit of periodic structures withgrowing periods:

Coloured point set Pi+1 obtained from Pi by changing colourat all positions n ≡ (2 · 4i − 1) mod 4i+1, with P0 = Z.

Consider Z = Pred ∪ Pblue and

ω = δPred=

∑

n∈Pred

δn



Autocorrelation γ =∑

m∈Za(m)δm with

a(m) =2

3

(1 − 1

2r+1

)

for m = (2ℓ+ 1)2r with r ≥ 0 and ℓ ∈ Z.

Diffraction γ =∑

k∈Z[ 12] I(k)δk with

I(k) =

{49 k ∈ Z

19·4r−1 k = (2ℓ+ 1)/2r with ℓ ∈ Z and r ≥ 1



Period doubling substitution

:r 7→ rb

b 7→ rr

Fixed point sequences under 2:

r|r 7−→ rb|rb 7−→ rbrr|rbrr 7−→ rbrrrbrb|rbrrrbrb7−→ rbrrrbrbrbrrrbrr|rbrrrbrbrbrrrbrr 7−→ . . .

Point setsPred =

⋃

i≥0

(2 · 4iZ + (4i − 1)

)∪ {−1}

Pblue =⋃

i≥0

(4iZ + (2 · 4i−1 − 1)

)


Incommensurate composite structureFor irrational α > 0 and u ∈ R

2 consider

ω = δZ2 + δu+Γ = δ

Z2 + δu ∗ δΓ ,

where Γ = αZ × Z ⊂ R2

Example withu = (1

3 ,12) and

α = τ = 12(1 +

√5)


Incommensurate composite structureAutocorrelation

γ = δZ2 +

1

αδΓ +

1

α(δu + δ−u) ∗ (λ⊗ δZ)

Diffraction

γ = δZ2 +

1

α2δΓ ∗ +

2

αcos(2πk2u2) (δ0 ⊗ δZ)

with the dual lattice Γ ∗ = ( 1αZ) × Z

Total intensity of Bragg peak at position (0, n) (n ∈ Z)

γ({(0, n)}

)= 1 +

1

α2+

2

αcos(2πnu2) ≥

(1 − 1

α

)2 ≥ 0


Incommensurate composite structureDiffraction for u = (1

3 ,12) and α = τ


The Fibonacci sequence



Liber Abaci (1202) by Leonardo of Pisa (Fibonacci)


The Fibonacci sequenceSubstitution rule and matrix

:ℓ 7→ ℓs

s 7→ ℓM =

(1 1

1 0

)

One-sided fixed point w = (w) by iteration of on w(0) = ℓ:

ℓ 7→ ℓs 7→ ℓsℓ 7→ ℓsℓℓs 7→ ℓsℓℓsℓsℓ 7→ . . . 7→ w(n) n→∞−−−−→ w

Fibonacci numbers:

|w(n)| = fn+2 with cardℓ(w(n)) = fn+1 and cards(w

(n)) = fn

where f0 = 0, f1 = 1 and fn+1 = fn + fn−1

Golden ratio:

limi→±∞

fi+1

fi=

1 ±√

5

2=

{τ

τ ′


Pure point spectra: Model sets

Fibonacci substitution: ℓ 7→ ℓs, s 7→ ℓ (λPF = τ = 1+√

52 )

Fibonacci point set: Λ ={x ∈ Z[τ ] | x′ ∈ (−1, τ − 1]

}

Pairs (x, x′) form lattice:

s ℓ s ℓ ℓ s ℓ ℓ s ℓ s ℓ



Silver mean substitution: a 7→ aba, b 7→ a (λPF = 1 +√

2 )

Silver mean point set: Λ ={x ∈ Z[

√2 ] | x′ ∈ [−

√2

2 ,√

22 ]}

Pairs (x, x′) form lattice:



CPS:

Rd π←−−− R

d × Rm

πint−−−−→ R

m

∪ ∪ ∪ dense

π(L)1−1←−−−− L −−−−→ π

int(L)

‖ ‖L

⋆−−−−−−−−−−−−−−−−−−−−→ L⋆

Model set: Λ = {x ∈ L | x⋆ ∈ W } (assumed regular)

with W ⊂ Rm compact, λ(∂W ) = 0

Diffraction: γ =∑

k∈L⊛ |A(k)|2 δk

with L⊛ = π(L∗) (Fourier module of Λ)

and amplitude A(k) = dens(Λ)vol(W ) 1W (−k⋆)


Fibonacci diffraction


Example: Ammann–Beenker


Example: Ammann–BeenkerL = Z[ξ] L ∼ Z

4 ⊂ R2 × R

2 O: octagon

ξ = exp(2πi/8) φ(8) = 4 ⋆-map: ξ 7→ ξ3

ΛAB ={x ∈ Z1 + Zξ + Zξ2 + Zξ3 | x⋆ ∈ O

}

1

ξ

ξ2

ξ3

1⋆

ξ⋆

ξ2⋆

ξ3⋆



physical space internal space


Example: Ammann–BeenkerDiffraction intensity:

Bragg peaks at positions k1 + k2ξ2 ∈ 1

2Z[ξ] ⊂ C

(where ξ = exp(2πi/8)) with intensities

I((k1, k2)

)=

1(4π2(k′2+k′1)(k

′2−k′1)

)2(

cos(k′2π) cos(λk′1π)

− cos(k′1π) cos(λk′2π) − k′1k′2

sin(k′2π) sin(λk′1π)

+k′2k′1

sin(k′1π) sin(λk′2π)

)2

with λ = 1 +√

2 and algebraic conjugation ′ :√

2 7→ −√

2


Interlude: Homometry

Problem: distinct structures with identical autocorrelation

Example 1: δ6Z∗

5∑

j=0

cj δjj 0 1 2 3 4 5

cj 11 25 42 45 31 14

cj 10 21 39 46 35 17

same correlations up to order 5 (Grünbaum & Moore)



Problem: distinct structures with identical autocorrelation

Example 1: δ6Z∗

5∑

j=0

cj δjj 0 1 2 3 4 5

cj 11 25 42 45 31 14

cj 10 21 39 46 35 17

same correlations up to order 5 (Grünbaum & Moore)

Example 2: homometric models sets with distinct windows

windows covariogram


Dirac combs on integer lattices

Dirac comb (on Zd): ω =

∑

n∈Zd

w(n) δn

Autocorrelation: γ =∑

m∈Zd

η(m) δm

Autocorrelation coefficients:

η(m) = limN→∞

1

(2N+1)d

∑

n∈[−N,N ]d

w(n)w(n−m)

Diffraction: γ = γpp + γsc + γac


Thue–Morse diffraction

Substitution: :1 7→ 11

1 7→ 11( 1 = −1 )

Iteration and fixed point:

1 7→ 11 7→ 1111 7→ 11111111 7→ . . . −→ v = (v) = v0v1v2v3 . . .

v2i = vi and v2i+1 = vi

⊲ recursion for autocorrelation coefficients:

η(2m) = η(m) and η(2m+1) = −12

(η(m) + η(m+1)

)

for all m ∈ Z, with η(0) = 1

⊲ diffraction is purely singular continuous


Thue–Morse diffractionAbsence of pure point part:

γ = µ ∗ δZ

with µ = γ∣∣[0,1)

and η(m) =

∫ 1

0e2πimy dµ(y)

(Herglotz-Bochner)

Wiener’s criterion: µpp = 0 ⇐⇒ Σ(N) = o(N)

where Σ(N) =N∑

m=−N

(η(m)

)2

Argument: Σ(4N) ≤ 32Σ(2N) (by recursion for η)

⊲ µ = µcont = µsc + µac

Define: F (x) = µ([0, x]

)for x ∈ [0, 1], where F = Fac + Fsc


Thue–Morse diffractionAbsence of absolutely continuous part:

Functional relation:dF(x2

)+dF

(x+1

2

)= dF (x)

dF(x2

)−dF

(x+1

2

)= − cos(πx) dF (x)

valid for Fac and Fsc separately (µac ⊥ µsc)

Define: ηac(m) =∫ 10 e2πimx dFac(x)

y same recursion as for η(m), but ηac(0) free

Riemann-Lebesgue lemma: limm→±∞ ηac(m) = 0⊲ ηac(0) = 0 ⊲ ηac(m) ≡ 0 ⊲ Fac = 0

(Fourier uniqueness thm)

Theorem: µ = µsc and γ is purely sc.


Thue–Morse diffraction

Exponential sum: gn(k) =2n−1∑ℓ=0

vℓ e−2πikℓ

Recursion: gn+1(k) =(1 − e−2πik2n)

gn(k)

with g0(k) = 1

(follows from v(n+1) = v(n)v(n))

Riesz product: γ =∏

n≥0

(1 − cos(2n+1πx)

)

(vague convergence)

⊲ approach can be generalised to other sequencesand higher-dimensional block substitutions


Thue–Morse measure

0 0.5 1

0

0.5

1


Comparison: Cantor measure

0 11/3 2/3

1

1/2

1/4

3/4

x

F(x)


Thue–Morse measure

0 0.5 1

0

10

20

0 0.5 1

0

10

20

0 0.5 1

0

10

20

0 0.5 1

0

0.5

1

0 0.5 1

0

0.5

1

0 0.5 1

0

0.5

1

(k = 4) (k = 5) (k = 6)


Thue–Morse scalingGrowth rate β(k) = limn→∞

log(fn(k))n log(2) (if it exists)

extinctions for all k = m2r with m ∈ Z and r ≥ 0

Lebesgue-almost all k ∈ R do not contribute

growth rate β(k) = log(3/2)log(3) ≈ 0.585 for k = m

3·2r for m ∈ Z

and not divisible by 3 and r ≥ 0

growth rate β(k) = log(5/4)2 log(2) ≈ 0.161 for k = m

5·2r for m ∈ Z

and not divisible by 5 and r ≥ 0

explicit results for k = pq·2r with q ≥ 3 odd, gcd(p, q) = 1

and r ≥ 0

there are values of k where β(k) does not exist

uncountably many values of k in the supporting setWorkshop on time-frequency analysis and aperiodic order, Trondheim, June 2015 – p.29

Period doubling

Block map: ψ : 11, 11 7→ a, 11, 11 7→ b


Period doubling

Block map: ψ : 11, 11 7→ a, 11, 11 7→ b

1111111111111111 1111111111111111

abaaabababaaabab abaaabababaaabaa


Period doubling

Block map: ψ : 11, 11 7→ a, 11, 11 7→ b

1111111111111111 1111111111111111

abaaabababaaabab abaaabababaaabaa

y period doubling: ′ :a 7→ ab

b 7→ aaXTM−−−→ XTM

ψ

yyψ (2:1)

Xpd′−−−→ Xpd

π

yyπ (a.e. 1:1)

Sol × 2−−−→ Sol

↑coincidence

⊲ model set(Dekking)


Planar example


Planar example

Block inflation:

Substitution on v ∈ {1, 1}Z2

(with m,n≥0, 0≤r,s≤2)

(v)3m+r,3n+s =

{vm,n, r ≡ s ≡ 0 mod 2,

vm,n, otherwise,

Autocorrelation coefficients η(m,n) satisfy

η(3m+ r, 3n+ s) =

min(1,r)∑

k=0

min(1,s)∑

ℓ=0

α(r,s)k,ℓ η(m+ k, n+ ℓ)

with α(r,s)k,ℓ ∈ {−2

9 , 0,19 ,

13 , 1}. In particular, η(3m, 3n) = η(m,n).


Planar exampleη(3m, 3n) = η(m,n)

η(3m, 3n+1) = −29η(m,n) + 1

3η(m,n+1)

η(3m, 3n+2) = 13η(m,n) − 2

9η(m,n+1)

η(3m+1, 3n) = −29η(m,n) + 1

3η(m+1, n)

η(3m+1, 3n+1) = −29

(η(m+1, n) + η(m,n+1)

)+ 1

9η(m+1, n+1)

η(3m+1, 3n+2) = −29

(η(m,n) + η(m+1, n+1)

)+ 1

9η(m+1, n)

η(3m+2, 3n) = 13η(m,n) − 2

9η(m+1, n)

η(3m+2, 3n+1) = −29

(η(m,n) + η(m+1, n+1)

)+ 1

9η(m,n+1)

η(3m+2, 3n+2) = 19η(m,n) − 2

9

(η(m+1, n) + η(m,n+1)

)

⊲ diffraction purely singular continuous


Planar exampleDistribution function and density

F (x, y) = xy∑

m,n∈Z

η(m,n) sinc(2πmx) sinc(2πny)

f (N)(x, y) =∂2F (N)

∂x ∂y(x, y) =

N−1∏

ℓ=0

ϑ(3ℓx, 3ℓy

)

with ϑ(x,y) = 1

9

(1+2 cos(2πx)+2 cos(2πy)−2 cos(2π(x+y))−2 cos(2π(x−y))

)2

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

50

100

150

0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8


Planar exampleA pure point diffractive factor:

Continuous mapping ψ : {1, 1}Z2 −→ {1, 1}Z

2

defined by

(ψw)m,n = wm,nwm+1,nwm,n+1wm+1,n+1

Induced inflation rule for factor (with a ∈ {1, 1})

1 1 a

1 1 1

a 7−→ 1 1 1

Fixed points are model sets (with 3-adic internal space)⊲ pure point diffractive⊲ Fourier module Z[13 ] × Z[13 ]

⊲ recovers point part of dynamical spectrum


Interlude: Pinwheel tiling

2

1



Autocorrelation is circularly symmetric,

γΛ = δ0 +∑

r∈D\{0}η(r)µr =

∑

r∈Dη(r)µr,

with µr the normalised uniform distribution on rS1 and µ0 = δ0

R.V. Moody, D. Postnikoff and N. Strungaru, Circular symmetry of pinwheel diffraction,Ann. H. Poincaré 7 (2006) 711–730

⊲(γΛ)pp

=(dens(Λ)

)2δ0 = δ0

⊲ diffraction intensity on rings (singular component)

⊲ also absolutely continuous component?


Interlude: Pinwheel tilingpinwheel radial intensity (numerical)

k

I(k)

0 1 2 3 4


Interlude: Pinwheel tilingpinwheel radial intensity (numerical) square lattice powder diffraction

k

I(k)

0 1 2 3 4(central intensity suppressed; relative scale chosen such peaks at k = 1 match)


AC spectra: Coin tossing sequenceSequence: i.i.d. random variables Wn ∈ {±1}

with probabilities p and 1−p

Entropy: H(p) = −p log(p) − (1−p) log(1−p)

Autocorrelation: γB =∑

m∈Z

ηB(m)δm with

ηB(m) := limN→∞

1

2N+1

N∑

n=−NWnWn+m

(a.s.)=

{1, m = 0

(2p−1)2, m 6= 0

(strong law of large numbers)

Diffraction: γB

(a.s.)= (2p− 1)2δ

Z+ 4p(1 − p)λ


Rudin–Shapiro sequence

Recursive definition:

w(−1) = −1, w(0) = 1, and

w(4n+ ℓ) =

{w(n), for ℓ ∈ {0, 1}(−1)n+ℓw(n), for ℓ ∈ {2, 3}

⊲ autocorrelation γRS = δ0⊲ diffraction γRS = λ

⊲ homometric with coin tossing for p = 12

but zero entropy!


BernoullisationSequence: S ∈ {±1}Z (assumed ergodic)

with Dirac comb ωS =∑

n∈ZSn δn

and autocorrelation γS

Bernoullisation: ω :=∑

n∈ZSnWn δn

(Wn ∈ {±1}

)

Autocorrelation: γ(a.s.)= (2p− 1)2 γS + 4p(1 − p) δ0

(strong law of large numbers)

Application: Rudin–Shapiro, with γS = γRS = δ0⊲ γ = δ0 independently of p

⊲ diffraction γ ≡ λ

⊲ homometric, irrespective of entropy


Diffraction of random tilingsPatch of a square rhombus random tiling,and diffraction from this finite patch:

Expectation: mixed spectrum with trivial Bragg peak at 0,and a mixture of singular and absolutely continuous parts


Dynamical spectrum(X,Z, µ) measure-theoretic dynamical systemHilbert space H = L2(X, µ) with product

〈g | h〉 =

∫

Hg(x)h(x) dµ(x)

Shift map S induces unitary operator U on HUf := f ◦ S (Uf)(x) = f(Sx) for x ∈ X

For f ∈ H, the map n 7→ 〈f | Unf〉 defines a positive definitefunction on Z. Spectral measure of f

〈f | Unf〉 =

∫ 1

0e2πint dσf (t) = σf (n)

Dynamical spectrum (as a set)⋃f∈L2(X,µ) supp(σf )


Dynamical versus diffraction spectrumThe crucial link (Baake, Lenz, van Enter 2013):Consider ξ : X −→ R with w 7→ w0.Then σξ = σ

Xwhich is the fundamental diffraction, which

means that the diffraction measure is γ = σX∗ δ

Z.

Proof:

σξ(n) = 〈ξ | Unξ〉 =

∫

X

ξ(x) ξ(Snx) dµ(x)

= limN→∞

1

2N + 1

N∑

m=−Nξ(Smw)ξ(Sm+nw) (unique ergodicity)

= limN→∞

1

2N + 1

N∑

m=−Nwmwm+n = η(n) = σ

X(n)

generalises to subshift factors encoded by continuousfunctions


Summary & outlookDiffraction as useful toolContinuous spectraaccessibleHomometry more difficultInsensitivity to entropyExtension tohigher dimensionsGeneralisation beyondlattice systemsRandomness withinteractionScalingRelation todynamicalspectrum


Recent review articlesM Baake & U Grimm, Mathematicaldiffraction of aperiodic structures,Chem. Soc. Rev. 41 (2012) 6821,arXiv:1205.3633

M Baake &U Grimm,

Kinematic diffractionfrom a mathematicalviewpoint,

Z. Krist. 226 (2011) 711,arXiv:1105.0095


For yet more see...

MB & UG

Aperiodic OrderVol. 1: A Mathematical Invitation

Cambridge University Press (2013)


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K. Mahler, — dto. — Part II: On the translation properties of a simple class ofarithmetical functions, J. Math. Massachusetts 6 (1927) 158–163

M. Keane, Generalized Morse sequences, Z. Wahrscheinlichkeitsth. Verw. Gebiete 10(1968) 335–353

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