structure of quasicrystals and related phases - iitk.ac.in

STRUCTURE OF QUASICRYSTALS AND 

RELATED PHASES 

ANANDH SUBRAMANIAM 

Guest Scientist (Alexander Von Humboldt Fellow) 

Electron Microscopy Group 

Max-Planck-Institut für Metallforschung 

STUTTGART 

Ph: (+49) (0711) 689 3683, Fax: (+49) (0711) 689 3522 

anandh@mf.mpg.de 

http://www.geocities.com/anandh4444/ 

November 2004

OVERVIEW 

DEFINITION 

DISCUSSION 

OUTLINE 

PROJECTION FORMALISM 

CLUSTER BASED CONSTRUCTION 

Mg-Zn Mg Zn-(Y, (Y, La) SYSTEMS 

“Babuji” 1899-1983

HYPERBOLIC 

EUCLIDEAN 

SPHERICAL 

METAL 

SEMI-METAL 

SEMI-CONDUCTOR 

INSULATOR 

AMORPHOUS 

GAS 

SPACE 

nD + t 

BAND STRUCTURE 

QUASICRYSTALS 

SIZE 

UNIVERSE 

PARTICLES 

ATOMIC 

ENERGY 

STATE / VISCOSITY 

SOLID LIQUID 

STRUCTURE 

RATIONAL 

APPROXIMANTS 

NANO-QUASICRYSTALS NANOCRYSTALS 

STRONG 

WEAK 

ELECTROMAGNETIC 

GRAVITY 

FIELDS 

NON-ATOMIC 

CRYSTALS 

LIQUID 

CRYSTALS

VARIOUS SPACES INVOLVED 

1D, 2D, 3D 4D, 5D, 6D 7D, ....., ND 

PHYSICAL 

SPACES 

PARALLEL SPACE 

QC 

HYPERSPACES 

GENERALIZED 

HYPERSPACES 

(E || ) PERPENDICULAR 

SPACE (E) REAL SPACE RECIPROCAL SPACE

QUASICRYSTALS (QC) 

ORDERED PERIODIC QC ARE 

ORDERED 

STRUCTURES 

WHICH ARE 

NOT 

PERIODIC 

CRYSTALS 

QC 

AMORPHOUS 

CRYSTALS 

(XAL) 

 

MODULATED 

STRUCTURES 

(MS) 

 

INCOMMENSURATELY 

MODULATED STRUCTURES 

(IMS) 

QC Can be thought of as IMS which cannot be constructed with a single “unit cell” 

but can be thought of as covering with a single prototile

SYMMETRY 

XAL QC 

t 

R C R CQ 

t translation 

inflation 

RC rotation 2, crystallographic 

3, 4, 6 

RCQ RC + 5, other 8, 10, 12 

DIMENSION OF QUASIPERIODICITY (QP) 

QP 

HIGHER DIMENSIONS 

QP/P 

QP/P 

QC can be thought of as crystals in higher 

dimensions 

(which are projected on to lower 

dimensions) 

QC can have quasiperiodicity 

along 1,2 or 3 dimensions 

QP XAL 

1 4 

2 5 

3 6 

QC are characterized 

by inflationary 

symmetry and can 

have disallowed 

crystallographic 

symmetries

THE FIBONACCI SEQUENCE 

Fibonacci 1 1 2 3 5 8 13 21 34 ... 

Ratio 1/1 2/1 3/2 5/3 8/5 13/8 21/13 34/21 ... 

Ratio 

2.2 

2 

1.8 

1.6 

1.4 

1.2 

1 

Convergence of Fibonacci Ratios 

1 2 3 4 5 6 7 8 9 10 

n 

= ( 1+5)/2 

WHERE IS THE ROOT OF THE EQUATION x2 –x –1 = 0

Rational 

Approximants 

A 

B 

B A 

B A B 

B A B B A 

B A B B A B A B 

B A B B A B A B B A B B A 

1-D QC 

Schematic diagram showing the structural analogue of 

the Fibonacci sequence leading to a 1-D QC 

Deflated 

sequence 

 

Penrose tiling 

a 

b 

ba 

bab 

babba

LIST OF QC.ppt

[1] I.R. Fisher et al., Phil Mag B 77 (1998) 1601 

FOUND! 

THE MISSING PLATONIC SOLID 

[2] Rüdiger diger Appel, Appel http://www.3quarks.com/GIF-Animations/PlatonicSolids/ 

[1] 

[2] 

Mg-Zn-Ho

DISCUSSION

STRUCTURE OF QUASICRYSTALS 

QUASILATTICE APPROACH 

(Construction of a quasilattice followed by the decorationof the lattice by atoms) 


TILINGS AND COVERINGS 

CLUSTER BASED CONSTRUCTION 

(local symmetry and stagewise construction are given importance) 

 

 

TRIACONTAHEDRON (45 Atoms) 

MACKAY ICOSAHEDRON (55 Atoms) 

BERGMAN CLUSTER (105 Atoms)

E2 

HIGHER DIMENSIONS ARE NEAT 

REGULAR 

PENTAGONS 

GAPS 

S2 

E3 

SPACE FILLING

PROJECTION METHOD 

QC considered a crystal in higher dimension 

Additional basis vectors needed to index the diffraction pattern 

2D 1D 

E Window 

Slope = Tan () 

 

e 2 

Irrational QC 

e 1 

Rational RA (XAL) 

E ||

Projection 

Plane 

KINDS OF STRUCTURES OBTAINED BY 


Irrational Tiles Irrational dimensions 

Strip 

Irrational Rational 

Arrangement Quasiperiodic 

Quasicrystal (QC) 

Rational Tiles Rational dimensions 

Arrangement Quasiperiodic 

Quasiperiodic Superlattice 

(QPSL) 

Tiles Irrational dimensions 

Arrangement Periodic 

QC Approximant 

Tiles Rational dimensions 

Arrangement periodic 

QPSL Approximant

Real Space Reciprocal Space 

1. Rational Lengths L and S arranged 

periodically 

2. Rational lengths L and S arranged in a 

Fibonacci chain 

3. Irrational length L and S arranged 

periodically 

4. Irrational length L and S arranged in a 

Fibonacci chain 

Periodic 

Periodic with satellites 

Peak positions periodic 

Intensities aperiodic 

Aperiodic 

Diffraction properties of various distributions of 

scatterers.

Progressive lowering of dimension starting with an 

N-fold fold symmetry in ND space 

N-fold symmetry 

Hypercubic Lattice viewed 

along [111....1]N 1s 

 

N-D 

Quasiperiodic tiling 2D RA 

 

Sequence of numbers 

Sequence of ‘a’s and ‘b’s 

Polynomial Equation 

 

Convergence of sequence 

Length of ‘a’/length of ‘b’ 

Root of Polynomial Eq. 

1D 

0D 

Approximants 

 

Repeating 

Sequence 

Rational 

Number

ND-0D.ppt

GENERALIZED PROJECTION METHOD 

[A] a1, a2, a3, ..., aN : a set of vectors in E|| 

[B] b1, b2, b3, ..., bN : a set of vectors in E 

W : the acceptance region or window in E 

{n1, n2, n3, ..., nN} are a set of integers in N dimensional space such that 

n1a1 + n2a2 + n3a3 + ... + nNaN is accepted as a point in E|| if and only if: 

n1b1 + n2b2 + n3b3 + ... + nNbN W 

Linear deformations of E do not affect the pattern produced in E||, 

i.e. if E is m dimensional and T is a non singular m m matrix, then: 

n1(Tb1) + n2(Tb2) + ... + nN(TbN) TW, 

if and only if n1b1 + n2b2 + ... + nNbN W 

The pattern in E|| will have a period n1a1 + n2a2 + n3a3 + ... + nNaN 

for any {n1, n2, n3, ..., nN} such that n1b1 + n2b2 + n3b3 + ... + nNbN = 0

2D AND 3D 

QUASILATTICS 

AND THEIR 

APPROXIMANTS 

(QC & RA)

RATIONAL APPROXIMANTS TO THE PENROSE TILING WITH 

ORTHOGONAL BASIS VECTORS 

Fourier transform of the lattice a set of 

10-fold spots are marked with circles. 

{1/1 1/1} RA to the 


Lattice with rectangular 

unit cell ABCD

RA to the 


{1/1 2/1} {3/2 1/1} 

{ 2/1}

RATIONAL APPROXIMANTS WITH APPROXIMATIONS ALONG 

BASIS VECTORS 72 APART 

Fourier transform of the lattice with remnant 

of the 10-fold symmetry marked by circles. 

{1/1 1/1}e RA to the 


Lattice with rectangular unit cell ABCD 

and parallelogram cell EFGH

5-fold [1 

0] 

ICOSAHEDRAL QUASILATTICE 

3-fold [2+1 

0] 

2-fold [+1 1]

{1/1 } P PENTAGONAL QUASILATTICE 

B’[p/q, , ] 

E || 

5 

2 

V 3 

4 

 

q 

 

 

 

0 

 

0 

3 

6 4 

1 

V 1 

3 

q 

1 

 

6 

 

 

q 

1 

V 2 

2 

5 

q 

1 

 

 

q 

 

1 

p 

0 

 

 

2

6 3 

4 

{1/1 } T TRIGONAL QUASILATTICE 

B' = 

E || 

2 2 2 

 

 

 

 

2 1 2 1 

 

2 2 1 1 

 

5 

V 1 

5 

2 

1 

V 2 

4 

6 

V 3

Three dimensional covering with triacontahedra 

Lord, E. A., Ranganathan, S., and Kulkarni, U. D., Current Science, 78 (2000) 64

(a) (b) 

Bergman cluster Mackay double icosahedron 

Important clusters underlying the structure of 

quasicrystals and their approximants. 

(a) Bergman, G., Waugh, J. L. T., and Pauling, L., Acta Cryst., 10 (1957) 2454 

(b) Ranganathan, S., and Chattopadhyay, K., Annu. Rev. Mater. Sci., 21 (1991) 437 

 

= 1

The structure of the Al3Mn decagonal phase 

Hiraga, K., Kaneko, M., Matsuo, Y., and Hashimoto, S., Phil. Mag. B67 (1993) 193 

= 2

(a) (b) 

Cluster of three dodecahedra Four vertex-connected vertex connected icosahedra 

Arrangement of sub-units in complex hexagonal 

phases 

(a) Singh, A., Abe, E., and Tsai, A. P., Phil. Mag. Lett., 77 (1998) 95 

(b) Kreiner, G., and Franzen, H. F., J. Alloys and Compounds, 221 (1995) 15 

= 3

IQC ( = 1) DQC ( = 2) 

 

Mackay Approximant Taylor Approximant 

 

Little Approximant Robinson Approximant 

IQC ( = 1) HQC ( = 3) 

Key: shows a twinning operation 

Relation between IQ C and its approxim ants with 

D Q C, its approxim ants and H Q C via the twinning 

operation

A quadrant of the stereogram of the decagonal phase 

with indices derived by the twinned icosahedron 

model

Stereogram of the Taylor phase obtained by twinning 

of the Mackay approximant to the icosahedral phase

Quadrant of the stereogram corresponding to I3 

cluster

= 2 

= 1 

Icosahedral Quasicrystal = 3 

Decagonal 

Hexagonal 

Quasicrystal 

= 1 

Quasicrystal 

Digonal Pentagonal Cubic R.A.S. Trigonal Hexagonal 

Quasicrystal Quasicrystal Mackay Bergman Quasicrystal R.A.S. 

Orthorhombic 

R.A.S. 

Taylor Little 

Robinson 

Monoclinic 

R.A.S. 

= 108 o 

R.A.S. 

Orthorhombic 

R.A.S 


R.A.S. 

= 90 o 

Trigonal 

R.A.S. 

Unification scheme based on the twinning of the 

icosahedral cluster 

Orthorhombic 

R.A.S. 


R.A.S. 

120 o

EXPERIMENTAL 

Mg-Zn Mg Zn-(Y, (Y, La) 

SYSTEMS

METASTABLE PHASES IN Mg-BASED ALLOYS 

QUASICRYSTALS 

RATIONAL APPROXIMANTS & RELATED STRUCTURES 

METALLIC GLASSES 

NANOCRYSTALS & NANOQUASICRYSTALS

MILESTONES IN Mg-BASED QUASICRYSTAL RESEARCH 

Mg-Zn-Al First Mg-Based QC 

(Icosahedral) 

P. Ramachandrarao, G.V.S. Sastry 

1985 

Mg-Zn-Al-Cu Quaternary System N.K. Mukhopadhyay, G.N. Subbanna, S. 

Ranganathan, K. Chattopadhyay 

1986 

Mg-Zn-Ga Stable QC W. Ohashi, F. Spaepen 

1987 

Mg-Zn-RE Icosahedral QC Z. Luo, S. Zhang, Y. Tang, D. Zhao 

1993 

Mg-Al Cubic QC P. Donnadieu, A. Redjaimia 

1995 

Mg-Zn-RE Decagonal QC T.J. Sato, E. Abe, A.P. Tsai 

1997 

Mg-Zn-RE QC without underlying 

atomic clusters 

E. Abe, T.J. Sato, A.P. Tsai 

1999

IMPORTANT PHASES IN THE Mg-Zn-RE SYSTEMS 

Composition e/a Phase, Symmetry Comments 

Mg3Zn6RE 2.1 Icosahedral, Fm53 

aR = 0.519 

RE = Y, Gd, Tb, Dy, Ho, Er 

dia (0.352, 0.360) 

Mg40Zn58RE2 2.02 Decagonal, 10/mmm RE = Y, Dy, Ho, Er, Tm, Lu 

Mg24Zn65RE10 (S) 2.1 Hexagonal superlattice, P63/mmc 

a = 1.46 nm, c = 0.86 nm 

Mg24Zn65RE10 (M) 2.1 Hexagonal superlattice, P63/mmc 

a = 2.35 nm, c = 0.86 nm 

Mg24Zn65Y10 (L) 2.1 Hexagonal superlattice, P63/mmc 

a = 3.29 nm, c = 0.86 nm 

Mg12ZnY 2.07 ? 

Mg3Zn3Y2 2.25 cF16, Fm3m 

dia < 0.355 

RE = Y, Sm, Gd 

Related to IQC 

RE = Sm, Gd 


RE = Sm 


aS : aM : aL = 3 : 5 : 7 

Mg7Zn3 2 oI142, Immm 1/1 RA to IQC 

Mg4Zn7 2 mC110, B2/m Related to DQC 

MgZn2 2 hp12, P63/mmc Related to S, L & M phases

(a) 

SEM micrograph of as-cast Mg 51 Zn 41 Y 8 alloy showing 

(a) Eutectic Microstructure (b) Four-fold dendrite 

(b)

5-FOLD 

5-FOLD FOLD TO 6-FOLD 6 FOLD 

SEM micrograph of as-cast Mg 51 Zn 41 Y 8 alloy showing 

distorted 5-fold dendrite growing into hexagonal shape 

DEVELOPING 

INTO 6-FOLD 

Initial stages of 

growth

BFI 

As-cast Mg 37 Zn 38 Y 25 alloy showing the formation of a cubic phase (a = 7.07 Å): 

[111] 

[110] [113]

[112] 

[111] [011] 

SAD patterns from a BCC phase (a = 10.7 Å) in as-cast Mg 4 Zn 94 Y 2 alloy showing important zones

High-resolution micrograph 

SAD pattern BFI 

As-cast Mg 37 Zn 38 Y 25 alloy showing a 18 R modulated phase

[1 

0] 

[0 0 1] 

SAD patterns from as-cast Mg 23 Zn 68 Y 9 showing the formation of FCI QC 

[1 1 1] 

[ 1 3 + ]

Uniform deformation along the arrow of the [0 0 1] 2-fold pattern from IQC giving 

rise to a pattern similar to the [ 1 3 + ] pattern

BFI 

TEM micrograph of as-cast Mg 4 Zn 94 Y 2 alloy showing the formation of nanocrystalline Mg 3 Zn 6 Y phase 

BFI 

SAD 

Mg 4 Zn 94 Y 2 as-cast alloy heat treated at 350 o C for 20 hrs (corresponding to the MgZn 5.51 phase) 

SAD

BFI from as-cast Mg 46 Zn 46 La 8 alloy showing patterns from APBs

BFI 

[113] 

Melt-spun Mg 50 Zn 45 Y 5 alloy showing the formation of a cubic phase (a = 6.63 Å) 

[001] 

[111]

Comparison of the [001] two-fold of the FCI QC (a) with the two-fold from other 

phase in the MgZnY (b), (c) and MgZnLa (d) systems

CONCLUSIONS 

A variety of Quasiperiodic and Rational Approximant structures can be realized using the Strip 

Projection Method, which serves to unify these structures using higher dimensions 

Structures with diverse kinds of symmetries can be generated using the Twinned Icosahedron 

Model, which further can be used to construct a unified framework based on the orientations of the 

icosahedron and the lowering of symmetry 

The Mg-Zn-RE systems serves a new ‘model system’ for the study of quasicrystals and related 

phases 

Study of quasicrystals is fun 

ACKNOWLEDGEMENTS 

Dr. Eric A Lord 

Prof. S. Ranganathan 

Dr. K. Ramakrishnan 

Dr. Sandip Bysakh 

Dr. Steffen Weber

APPLICATIONS OF QUASICRYSTALS 

WEAR RESISTANT COATING (Al-Cu (Al Cu-Fe Fe-(Cr)) (Cr)) 

NON-STICK NON STICK COATING (Al-Cu (Al Cu-Fe) Fe) 

THERMAL BARRIER COATING (Al-Co (Al Co-Fe Fe-Cr) Cr) 

HIGH THERMOPOWER (Al-Pd (Al Pd-Mn Mn) 

IN POLYMER MATRIX COMPOSITES (Al-Cu (Al Cu-Fe) Fe) 

SELECTIVE SOLAR ABSORBERS (Al-Cu (Al Cu-Fe Fe-(Cr)) (Cr)) 

HYDROGEN STORAGE (Ti-Zr (Ti Zr-Ni) Ni)

PENROSE TILING 

Inflated tiling

DIFFRACTION PATTERN 

5-fold SAD pattern 

from as-cast 

Mg 23 Zn 68 Y 9 alloy 

1 

2 3 4

structure of quasicrystals and related phases - iitk.ac.in

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