Overview

Historical review from early daysof aperiodic crystals to the state of the art

1. Introduction Standards, Incommensurate, Aperiodic

2. Description Modulation, Intergrowth, Quasicrystals

3. Symmetry Euclidean, Non-Euclidean, Groups

4. Crystal structures Historical and more recent results

5. Structure determination Methods, Programs

6. The Web SIG, Databases, Tools

7. The inverse crystallographic problem Rational indices

8. The geometry problem Scaling transformations

Zao (Miyagi), 17.09.06, Aperiodic06 A. Janner

– p. 1/41

Measurement

Commensurate: common standards

Space Time

length: human body duration: sun, moon1 foot = 12 inches 1 year = 12 months

1 month = 30 days

1 f + 8 i = (1 + 23) f = 20 i 5 m + 7 d = ( 5

12+ 7

360) y = 157 d

Standards −→ Units −→ Integers −→ Rationals

– p. 2/41

Incommensurate

IncommensuratePythagora of Samos (570 - 490 BC)

(ape10, sqrt(2)-incommensurate)(SX=0,SY=150,F=300,U=115)

ds

L

sd

d =√

2s d2

s2= 2 = p

2

q22q2 = p2

p odd no

p even → q odd no

s, d Incommensurate

L = 4 s + 2 d = [4, 2]

Two units: s and d

– p. 3/41

Superstructure

Superstructure

(ape2, superstructure )()

a

b = 5a0 b* a*

Displacive Modulation Satellite Main reflection

Density Modulation

0 b* a*

Intergrowth (Composite)

0 b* a*

b = n a b∗ = 2πb

, a∗ = 2πa

– p. 4/41

Aperiodic

Aperiodic

(ape3, inc intergroth, quasicrystal )()

Intergrowth incommensurate

Quasicrystal

Invariant scaling (Inflation): a -> A = ab, b -> B = aba

a b a b a a b a b a a b a b

A B A B A A

b = α a, α irrational

α =√

2

b =√

2 a,

S =

1 2

1 1

, S−1 =

−1 −21 −1

– p. 5/41

Kyoto

From aperiodic to periodic1972 Kyoto, PM de Wolff: world-lines for modulated atomic positions

A.J.: vibrating crystalt

x

x

4

32

1

1 2 3 4– p. 6/41

Superspace

Superspace: the basic idea

� � � � � � � � � � �� � � � � � � � � � �

� � � � � � � � � � �� � � � � � � � � � �

�� ���

� �

��� �� ���

� �� ���� �

� ���

� �� ���

� � � � � � � � � � � � � �� � � � � � � � � � � � � �

� � � � � � � � � � � � � �� � � � � � � � � � � � � �

� � � � � � � � � � � � � �

���

��

�

!"#

$ $$ $�%

B

B

A

Direct space

A*

B*

B*

a*

b*

Reciprocal space

A*

A

a

b

(ape4)

Basic structure

Modulation

Modulated crystal

Superspace

Lattices

– p. 7/41

Indexing

Indexing

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

n2 n3 n1 2 n3[ n+] = cb

n1 n1 n

a

n2

+

] = a b[ 2

0 0 1 0 1 1 2 1 3 1 3 2 � � � � � � � � � � � �

b

a

+

1n

0 0 0

0 0 1

0 0 2

1 0 0 3 0 0 4 0 0

0 1 1

1 0 2 2 0 2 3 0 2 4 0 2 5 0 2 6 0 2

0 2 1 0 3 1 0 4 1

2 0 0 5 0 0 6 0 0

��

��

��

� � � � � � � � �

b

a

c

Superspace Projection in space

Crenel modulation

Intergrowth (intercalate)

(ape5)

Lattice Z-moduleΣ Μ– p. 8/41

CutProjection

Cut - Projection

(ape6, quasicrystal cutprojection)

x

y

0 1

1 0

-1 -1

0 -10 0

0 2

1 11 2

1 31 4

2 22 3

2 4

2 52 6

3 43 5

3 6

x: Parallel space = External spacey: Perpendicular space = Internal space

– p. 9/41

CutProjection

Cut - Projection

(ape6a, quasicrystal cutprojection)

x

y

0 1

1 0

-1 -1

0 -10 0

0 2

1 11 2

1 31 4

2 22 3

2 4

2 52 6

3 43 5

3 6

x: Parallel space = External spacey: Perpendicular space = Internal space

Atomic surface ������

– p. 9/41

Internal space

Space + Internal space = Superspace

Ib

Ia

Ib

IaA

B

a b

(ape7)

10

01

11

00

10

01

11

00

a b

00 10 01 11

A = (a,aI ), B = (b,bI )

MI = {aI , bI} Σ = {A, B} M = {a, b}Internal space Superspace Space

– p. 10/41

Pentagonal

Pentagonal Z-module bases

0100

1000

0001

0010

1000

0010

0001

0100

Pentagonal basis in space Pentagonal basis in internal space

– p. 11/41

Pentagonal quasicrystal

Pentagonal Quasicrystal

Positions in space Positions in internal spaceClusters Atomic surface

– p. 12/41

Ico quasicrystal

Icosahedral Quasicrystal

0 0 1 0 0 0

0 1 0 0 0 0

1 0 0 0 0 00 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

0 0 1 0 0 0

0 0 0 -1 0 0

0 0 1 0 0 0

0 0 -1 0 0 0

Indexed Icosahedron

fivefold view

twofold view

threefold view

Superspace lattice

6-dimensional cubic lattice

Space Z-module

6 icosahedral vectors(as indexed)

Internal space Z-module

6 icosahedral vectors(not indicated)

Icosahedron

Clusters, Atomic surfaces

– p. 13/41

Dodecahedron

Indexed dodecahedron

0 1 0 -1 0 1

0 1 0 -1 -1 0

-1 0 0 -1 -1 0

-1 0 -1 -1 0 0

0 1 1 0 -1 0

0 0 -1 0 1 1

fivefold view

twofold view

threefold view

Generating point

1 1 1 0 0 0

Space

Dodecahedral clusters

Internal space

Atomic surfaces

– p. 14/41

Triacontahedron

Indexed triacontahedron

-1 1 1 -1 1 1

1 1 -1 -1 -1 -1

0 0 2 0 0 0

0 2 0 0 0 0

1 -1 -1 -1 -1 1

-1 1 -1 -1 -1 -1

0 0 2 0 0 0

0 0 0 -2 0 0

-1 1 1 1 -1 1

-1 1 -1 1 1 1

0 0 2 0 0 0

0 0 -2 0 0 0

fivefold view

twofold view

threefold view

Superspace

6-dimensional cube a

Space

Icosahedron ri = a√

1 + τ2

Dodecahedron rd = a√

3

S1/τ2 [2 2 2 0 0 0]

=[1 1 1 -1 1 -1]

Generating points

Icosahedral vertices:2 0 0 0 0 0

Dodecahedral vertices1 1 1 -1 1 -1

Clusters, Atomic surfaces– p. 15/41

Euclidean

Euclidean symmetries

(ape10a)

x

y

x x

y

t

Translation Rotation Glide reflection

Modulation shift Reflection + shift s

s = [000 12] R5 =

0

B

B

B

B

B

@

0 0 0 −11 0 0 −10 1 0 −10 0 1 −1

1

C

C

C

C

C

A

0

@

my

s

1

A = {my1 | [000 12 ]}

– p. 16/41

Cryst.Scaling

Crystallographic Scaling

Scaling with scaling factor λ

1D (linear) Xλ(x, y, z) = (λx, y, z)2D (planar) Pλ(x, y, z) = (λx, λy, z)3D (radial) Iλ(x, y, z) = (λx, λy, λz)Higher dimensional .......

– p. 17/41

Cryst.Scaling

Crystallographic Scaling

Scaling with scaling factor λ

1D (linear) Xλ(x, y, z) = (λx, y, z)2D (planar) Pλ(x, y, z) = (λx, λy, z)3D (radial) Iλ(x, y, z) = (λx, λy, λz)Higher dimensional .......

Crystallographic transforming a lattice into a lattice

SλΛ = Λ Sλ integral invertible

in general: SλΛ = Σ Σ ⊆ Λ or Λ ⊆ ΣSλ rational invertible

– p. 17/41

hexagonal scaling

Crystallographic Scaling

(sca1)

1 0

0 1

2 0

0 2

1 0

0 12 1

-1 1

S =

2 0

0 2

S =

2 −11 1

λ = 2, ϕ = 0 λ =√

3, ϕ = 300

– p. 18/41

Star Pentagon

Pentagonal Case

(sgk1b)

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

-1 -1 -1 -1

-2 0 -1 -1

1 -1 1 0

0 1 -1 1

-1 -1 0 -2

2 1 1 2

Polygrammal Scaling

Star Pentagon:Schäfli Symbol {5/2}

Scaling matrix: (planar scaling)

S{5/2} =

2̄ 1 0 1̄

0 1̄ 1 1̄

1̄ 1 1̄ 0

1̄ 0 1 2̄

Scaling factor:-1/τ2 = −0.3820...

(τ = 1+√

5

2= 1.618...)

– p. 19/41

QCR Pentagr.Scal.

Pentagonal quasicrystal: Pentagrammal symmetry

W. Steurer, J.Phys: Condens. Matter 3 (1991) 3397-3410

(deca02a,Pentragamma AlMn,U1=60,N0=70)

– p. 20/41

Linear Scaling

Pentagonal Case

(sgk1c)

x

y

A

P

Q

B

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

-1 -1 -1 -1

-2 0 -1 -1

1 -1 1 0

0 1 -1 1

-1 -1 0 -2

2 1 1 2

1 -1 2 -1

1 2 0 3

-3 -2 -1 -3

3 0 1 2

-2 1 -2 -1

-1 2 -1 1

-1 -1 1 -2

2 1 0 3

-3 -1 -2 -3

3 0 2 1

Pentagonal Case

(sgk1d)

τ

τ

1 x

y

A

P

Q

B

1 0 0 0

0 0 0 1

-1 -1 -1 -1

-2 0 -1 -1

1 -1 1 0

0 1 -1 1

-1 -1 0 -2

2 1 1 2

1 2 0 3

-3 -2 -1 -3

3 0 1 2

-2 1 -2 -1

-1 -1 1 -2

2 1 0 3

-3 -1 -2 -3

3 0 2 1

Linear Scaling

Scaling transformation:Yλ(x, y) = (x, λy)

Scaling matrix:

Y1/τ3 =

0 1 1̄ 1

1 1̄ 2 1̄

1̄ 2 1̄ 1

1 1̄ 1 0

Scaling factor:1/τ3 = 0.2361...

(τ = 1+√

5

2= 1.618...)

Linear Scaling

The linear scaling Y1/τ3appears alongthe pentagonal edgewith the scaling ratios

τ : 1 : τ

– p. 21/41

Linear Scaling

Pentagonal Case

(sgk1d)

τ

τ

1 x

y

A

P

Q

B

1 0 0 0

0 0 0 1

-1 -1 -1 -1

-2 0 -1 -1

1 -1 1 0

0 1 -1 1

-1 -1 0 -2

2 1 1 2

1 2 0 3

-3 -2 -1 -3

3 0 1 2

-2 1 -2 -1

-1 -1 1 -2

2 1 0 3

-3 -1 -2 -3

3 0 2 1

Linear Scaling

Scaling transformation:Yλ(x, y) = (x, λy)

Scaling matrix:

Y1/τ3 =

0 1 1̄ 1

1 1̄ 2 1̄

1̄ 2 1̄ 1

1 1̄ 1 0

Scaling factor:1/τ3 = 0.2361...

(τ = 1+√

5

2= 1.618...)

Linear Scaling

The linear scaling Y1/τ3appears alongthe pentagonal edgewith the scaling ratios

τ : 1 : τ

– p. 21/41

QCR lin.scaling

Pentagonal quasicrystal: Linear scaling with fivefold RotationsW. Steurer, J.Phys: Condens. Matter 3 (1991) 3397-3410

(deca03a,Linear scaling AlMn,U1=60,N0=70)

1τ

τ

1τ

τ

1

τ3

– p. 22/41

Groups

Crystallographic symmetry groups

Point groups Morphology Diffraction

32 Crystal classes (1826) Crystal forms (1611-1804) von Laue ZnS (1912)

(3+1)D Rotations Calaverite (1895-1989) γ Na2CO3 (1964)

(3+3)D Rotations Dodecahedron, ... Icosahedral QCR (1984)

1D,2D,3D Scalings Quasicrystals (1985)

Scale-Rotations Snow flakes (1997) Steurer, Haibach (2001)

Space groups Structure Diffraction

3D Fedorov (1895) Periodic crystals Bragg reflections

(3+n)D Superspace (1979) Modulated, Intergrowth Main reflections + Satellites

(3+2)D Ssp gr. (+scaling) 5-,10-,8-,12-gonal QCR 1D,2D Scaling symmetry

(3+3)D Ssp gr. (+scaling) Icosahedral QCR 3D Scaling symmetry

Multimetrical sp gr.(1991) Ice P63/mmcLz 12

LyLx 12

– p. 23/41

History

Aperiodic crystals: Historical turning points

AuTe2 1902 Non-indexable growth forms Penfield, Ford, Smith

1931 Singular point at [5 29 -3] Goldschmidt, Palache, Peacock

1985 4d indexing Dam, Janner, Donnay

Na2CO3 1964 Non-indexable satellites Brouns, Visser, de Wolff

1972 4d space group van Aalst, de Wolff

1974 Harmonic approximation de Wolff et al.

2005 Additional phase transitions Arakcheeva, Chapuis

(TTF)7I5−x 1976 Local symmetry only Johnson, Watson

1980 Incommensurate basic structure Janner, Janssen

SiC, ZnS 1981 Polytypes as modulated crystals Yamamoto

FexS 1982 Higher-order harmonics Yamamoto, Nakazawa

LaTi1−xO3 2000 Family of modular structures Elcoro, Pérez-Mato, Withers

fivefold sym. 1974 Planar tiling with inflation symmetry Penrose

1982 Optical diffraction of Penrose tiling Mackay

Al-Mn 1984 Icosahedral QCR Shechtman, Blech, Gratias, Cahn

Al73Mn21Si6 1988 6d Fourier analysis Gratias, Cahn, Mozer

Deca Al-Mn 1988 Penrose-like QCR Yamamoto

Al78Mn22 1989 5d structure refinement Steurer

1992 Decagrammal scaling symmetry Janner, Steurer – p. 24/41

Calaverite 1

Calaverite AuTe2

Non indexable facets Incommensurate q = [α 0 γ]

Goldschmidt, Palache, Peacock α = −0.409... γ = 0.450...

satellite [0001]

– p. 25/41

Calaverite 2

Calaverite AuTe2

Goldschmidt, Palache, Peacock Dam, Janner, Donnay

– p. 26/41

Na2Co3

γ Na2CO3

Non indexable satellites Incommensurate modulations

Brouns, Visser, de Wolff van Aalst et al., Hogervorst et al.

– p. 27/41

FeS

Polytype Fe1−xS

Modulation: higher-order harmonics

Yamamoto, Nakazawa 1982

– p. 28/41

Family

Family of modular structures

Close-packed LaO3 modules: Variation of the modulation wave vectorq = γc

B

C

A

A A

A

x 3

x 4

LaO3 LaO3 LaO3 LaO3 LaO3

C B C AA

γ = 1/3

(ape12b)

LaTi1−xO3: Elcoro, Pérez-Mato, Withers (2000)K5Yb(MoO4)4 Arakcheeva, Chapuis (2006)

– p. 29/41

Family

Family of modular structures

Close-packed LaO3 module: Variation of the modulation wave vector q = γc

B

C

A

A A

A

x 4

LaO3 LaO3 LaO3 LaO3 LaO3

x 3

A B C A B

zγ = 2/5

(ape12c)

LaTi1−xO3: Elcoro, Perez-Mato, Withers (2000)K5Yb(MoO4)4 Arakcheeva, Chapuis (2006)

– p. 29/41

Penrose

Roger Penrose: Pentaplexity

Tiling with inflation: dart and kite Optical diffraction

Penrose 1974 Mackay 1982

– p. 30/41

AlMn Fourier-Scaling

Al78Mn22: Fourier Map of a Decagonal Plane

W. Steurer, J.Phys: Condens. Matter 3 (1991) 3397-3410

– p. 31/41

AlMn Fourier-Scaling

Al78Mn22: Pentagonal Lattice Points

Comparison of the Fourier map with model positions (Low Indices only)

(deca01a,Indexed Fourier points AlMn,U1=60,N0=70)

-1-1-1-1 0-2-2 0 -3-4-4-3-2-2-4-1

0-2-1-1 1-3-2 0 -2-5-4-3 1 0-1 1 -1-3-4-1

1 0 0 0

-2-2-3-2

-1-3-3-2-1 0-2-1 0-1-3 0

0-4-3-2 0-1-2-1 1-2-3 0 -2-4-5-3

0-1-1-2 1-2-2-1 1 1-1 0 -1-2-4-2

2 0-1 0 -1-2-3-3 0-3-4-2 0 0-3-1 -2-3-6-3

2 0 0-1 0-3-3-3 0 0-2-2 1-1-3-1 -2-3-5-4

1-1-2-2 2-2-3-1 -1-4-5-4 2 1-2 0 -1-1-4-3 0-2-5-2

1-1-1-3 2 1-1-1 -1-1-3-4 0-2-4-3

0 1-3-2 3 0-1-1 0-2-3-4 1-3-4-3

1 0-3-2 2-1-4-1 -1-3-6-4

1 0-2-3 2-1-3-2 -1-3-5-5 2 2-2-1 0-1-5-3

2-1-2-3 0-4-5-5 3 1-2-1 0-1-4-4 1-2-5-3

– p. 31/41

AlMn Fourier-Scaling

Al78Mn22: Planar Scaling (Star Pentagon)

Scaling factor: λ = −1/τ2

(deca02a,Pentragamma AlMn,U1=60,N0=70)

– p. 31/41

AlMn Fourier-Scaling

Al78Mn22: Linear Scalings combined with Fivefold Rotations

Pentagrammal ratios (τ : 1 : τ ) and Off -center decagon

(deca03a,Linear scaling AlMn,U1=60,N0=70)

1τ

τ

1τ

τ

1

τ3

– p. 31/41

Al-Pd-Mn

Decagonal Al-Pd-Mn

Atomic occupation domains and clusters

0 ≤ z ≤ 0.5 0.5 ≤ z ≤ 1Yamamoto 1993

– p. 32/41

Al-Pd-Mn

Decagonal Al-Pd-Mn

Linear and planar crystallographic scalings

1

τ

1

0 ≤ z ≤ 0.5 0.5 ≤ z ≤ 1Yamamoto 1993

– p. 32/41

Ico Al-Mn-Si

Icosahedral Al73Mn21Si6

6d Fourier analysis Indexed positions in a 2-fold plane

0 2 0 2 2 2 0 2 0 2 2 2

0 0 0 20 2 0 0

0 0 -2 0 0 0 2 0

-1 2 0 1 1 2 0 1

-1 0 0 1 1 0 0 1

-1 0 -2 -1

-1 0 0 -1

1 0 0 -1

-1 0 -2 -1

0 0 0 0 2 0 0 0

0 2 2 0

0 0 2 2

n1 n2 n3 n4 = (n1 + τn3, n2 + τn4)Gratias, Cahn, Mozer (1988)

– p. 33/41

Ico Al-Mn-Si

Icosahedral Al73Mn21Si6

6d Fourier analysis Indexed positions in a 2-fold plane

0 2 0 2 2 2 0 2 0 2 2 2

0 0 0 20 2 0 0

0 0 -2 0 0 0 2 0

-1 2 0 1 1 2 0 1

-1 0 0 1 1 0 0 1

-1 0 -2 -1

-1 0 0 -1

1 0 0 -1

-1 0 -2 -1

0 0 0 0 2 0 0 0

0 2 2 0

0 0 2 2

n1 n2 n3 n4 = (n1 + τn3, n2 + τn4)Gratias, Cahn, Mozer (1988)

– p. 33/41

Local order

Icosahedral local order

Bergman cluster Atomic clusters Mackay cluster

Tamura, PhD Thesis, Grenoble (1993)

Puyraimond, Quiquandon, Gratias, Tillard, Belin, Quivy, Calvayrac (2002)

Icosahedron[100000], 12V, 5fold

Dodecahedron[111-11-1], 20V, 3fold

Icosahedron[200000]. 12V, 5fold

44 Atoms

Icosahedron[100000], 12V, 5fold

Icosahedron[200000]. 12V, 5fold

Icosidodecahedron[000202], 30V, 2fold

54 Atoms

– p. 34/41

Methods Programs

Structure determination: Methods and programs

Superspace approach 1982 Yamamoto REMOS

1985 Petr̆íc̆ek JANA

Direct methods 1987 Hao, Liu, Fan

1993 Lam, Beurskens, van Smaalen

Patterson method 1987 Steurer (decagonal QCR)

1988 Gratias, Cahn, Mozer (ico QCR)

Reciprocal space symmetry 1988 Mermin, Lifshsitz, Rabson, et al.

"Trial-and-error" method 1988 Yamamoto, Ishihara

Contrast variation 1989 Janot, de Boissieu, Dubois, Pannetier

Maximum entropy method 1997 Weber, Yamamoto

Haibach, Cervellino, Steurer

2003 van Smaalen, Palatinus, Schneider

Atomic surface modelling 2002 Cervellino, Haibach, Steurer

Charge flipping 2004 Palatinus

2005 Oszlányi, Sütő– p. 35/41

Web

On the Web: SIG Aperiodic

Special Interest Group on Aperiodic Crystalswww-xray.fzu.cz/sgpi/apright.html

Software Yamamoto page

JANA2000

Superspace tools

CCP4

Links Incommensurate structures

Quasicrystals

Databases Bilbao server

Lausanne server

Caracas page

Research Groups

References– p. 36/41

Sym2Struct

From Symmetry to Structure

Classical: A crystal is Euclidean and periodic

New: A crystal has indexable Bragg reflections

Approach: Find the symmetries−→ Solve the structure

230 Space groups:Superspace groups:Multimetrical space groups:

} Indexed Bragg reflectionsConditions for reflectionsConditions for Wyckoff positions

1964 de Wolff Standard 3d space groups only

1982 Yamamoto REMOS (3+1)d superspace groups

1985 Petr̆íc̆ek JANA general modulations

– p. 37/41

Struct2Symm

From Structure to Symmetry

Approach: Solve the structure−→ Find the symmetriesDirect methods, Charge flipping:Tiling models:Indexed atomic positionsFamily of structures

} Space groupScaling symmetryCommon superspace group

Which structure?

Indexed atomic positions Scaling relations

Clusters of indexed polyhedra

Internal positions (same indices) Polyhedral atomic surfaces

Superspace positions (same indices) Indexed polytopes

Indexed Bragg reflections Correlated reflections

– p. 38/41

Felix Klein

The Klein View of Geometry

Felix Klein (Erlanger 1872)

Geometry:A set S, the points of the geometryA group G of transformations of S

No Axioms only Theorems

Euclidean Geometry

Set Group Transformations Invariant objects

S = R3 Euclidean group E(3) Transl. T, Rot. R, Refl. M Distance d (AB)Angle ϕ (ABC)

Similarity Geometry

Set Group Transformations Invariant objects

S = R3 Similarity group S(3) T, R, M, Isotropic scaling Sk Angle ϕ (ABC)Distance ratio d(AB)/d(CD)

Problem: Linear and planar scaling in R3

Crystallographic Geometry ?

– p. 39/41

end

Thank you for the attention

– p. 40/41

Inverse problem

The Inverse Crystallographic Problem

Indexed positions(Rational indices)

Internal positions(same indices)

Superspace positions(same indices)

Equivalent positions(by scale-rotations)

Indexed model(with scale-rotations)

StructureAtomic positions

Rational indices

DiffractionCorrelated reflections

?

?

Polyhedra(with rational indices)

Clusters(local order)

Atomic surfaces(polyhedral)

Polytopes(in superspace)

Scaling relations(crystallographic)

Indexed model(with scale-rotations)

– p. 41/41

OverviewMeasurementIncommensurateSuperstructureAperiodicKyotoSuperspaceIndexingCutProjectionCutProjection

Internal spacePentagonalPentagonal quasicrystalIco quasicrystalDodecahedronTriacontahedronEuclideanCryst.ScalingCryst.Scaling

hexagonal scalingStar PentagonQCR Pentagr.Scal.Linear ScalingLinear Scaling

QCR lin.scalingGroupsHistoryCalaverite 1Calaverite 2Na2Co3FeSFamilyFamily

PenroseAlMn Fourier-ScalingAlMn Fourier-ScalingAlMn Fourier-ScalingAlMn Fourier-Scaling

Al-Pd-MnAl-Pd-Mn

Ico Al-Mn-SiIco Al-Mn-Si

Local orderMethods ProgramsWebSym2StructStruct2SymmFelix KleinendInverse problem