ICOSAHEDRAL CRYSTALS, QUASI-CRYSTALS : NEW FORMS OF INCOMMENSURATE CRYSTAL PHASES

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ICOSAHEDRAL CRYSTALS, QUASI-CRYSTALS : NEW FORMS OF INCOMMENSURATE CRYSTAL

PHASES

T. Janssen

To cite this version:

T. Janssen. ICOSAHEDRAL CRYSTALS, QUASI-CRYSTALS : NEW FORMS OF INCOMMEN- SURATE CRYSTAL PHASES. Journal de Physique Colloques, 1986, 47 (C3), pp.C3-85-C3-94.

�10.1051/jphyscol:1986308�. �jpa-00225718�

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JOURNAL DE PHYSIQUE

Colloque C3, suppl6ment au n 0 7 , Tome 47, juillet 1986

ICOSAHEDRAL CRYSTALS, QUASI-CRYSTALS : NEW FORMS OF INCOMMENSURATE CRYSTAL PHASES

T. JANSSEN

Instituut voor Theoretische Natuurkunde, Universiteit, Nijmegen.

Toernooiveld, NL-6525 ED-Nijmegen, The Netherlands, and Laboratoire de Diblectrique, Universitb de Dijon, B.P. 138, F-21004 Dijon, France

RBsum6

-

La symgtrie d'un quasi-cristal est considgr6e c o m e celle d'un cas particulier d'une phase cristalline incommensurable. Dans ce cadre, la d6fi- nition du concept de quasi-cristal est discut6e. Utilisant les msmes techniques que pour les phases incommensurables, on peut calculer le facteur de structure et dgriver les symgtries possibles. En particulier les quasi- cristaux icosa6driques sont discutEs.

Abstract

-

The symmetry of a quasi-crystal is considered to be that of a special case of an incommensurable crystal phase. In this context the defini- tion of this notion of quasi-crystal is discussed. Using the same techniques as for incommensurate phases one can calculate the structure factor and derive the possible symmetries. In particular, i c o s a h e d r d q u a s i - c r y s t a l s are discussed.

I

-

INTRODUCTION

Since the first report on A10.S6Mn0- 14 that aroused interest at the end of 1984 a number of other materials has been found that share the two characteristic features of the AlMn alloy, namely the presence of sharp diffraction spots, indicating long range order, and that of non-crystallographic point group symmetry of the diffraction pattern. The latter indicates that the structure:does not have three-

dimensional space group symmetry. Since, however, the diffraction vectors may be written as linear combinations with integral coefficients of a finite set of vectors the structure is quasi-periodic.

This property is exactly the characteristic property for an incommensurate crystal phase. Therefore, what has been called a quasi-crystal is in fact a special case of such a phase. Here we shall discuss the difference with other IC phases and use or generalize the techniques developed for the latter to describe their symmetry, structure and properties.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986308

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C3-86 JOURNAL DE PHYSIQUE

The symmetry group o f an I C phase i s n o t a 3-dimensional space group b u t a c r y s t a l - l o g r a p h i c space group i n more t h a n 3 dimensions. Symmetry groups t h a t a r e t r a n f o r - mation groups i n a higher-dimensional space a r e not uncommon i n p h y s i c s .

For example, t h e symmetry group of t h e (bound s t a t e s o f ) hydrogen atom i s 0 ( 4 ) , t h e symmetry g r o u p o n a non-rigid molecule may be a p o i n t group i n more t h a n 3 dimen-- s i o n s . The a d d i t i o n a l dimensions i n t r o d u c e d t o give t h e symmetry group of a incommensurately modulated phase have t h e advantage t h a t one c a n g i v e them a simple i n - t e r p r e t a t i o n : t h e y a r e t h e independent phases o f t h e modulation w a v e ( s ) .

I1 - SYMMETRY

Incommensurate c r y s t a l phases a r e d e f i n e d a s s t r u c t u r e s w i t h a d i f f r a c t i o n p a t t e r n o f s h a r p s p o t s a t p o s i t i o n s t h a t a r e l i n e a r combinations with i n t e g r a l c o e f f i c i e n t s of

4

o r more r a t i o n a l l y independent v e c t o r s . Mackay / I / h a s i n t r o d u c e d t h e n o t i o n of q u a s i - l a t t i c e t o d e s c r i b e such a s i t u a t i o n .

A q u a s i - l a t t i c e , i n o u r d e f i n i t i o n , i s t h e s e t of v e c t o r s t h a t a r e l i n e a r combina- t i o n s w i t h i n t e g r a l c o e f f i c i e n t s o f a f i n i t e s e t of v e c t o r s . The dimension o f t h e q u a s i - l a t t i c e i s t h e dimension of t h e space g e n e r a t e d ( o v e r t h e r e a l numbers) by t h e s e v e c t o r s . I t s r a n k i s t h e minimal number o f r a t i o n a l l y independent v e c t o r s t h a t g e n e r a t e t h e q u a s i - l a t t i c e (over t h e i n t e g e r s ) . I f rank and dimension a r e e q u a l a q u a s i - l a t t i c e i s a l a t t i c e i n t h e u s u a l sense. Using t h i s concept one may say t h a t a n incommensurate c r y s t a l phase i s c h a r a c t e r i z e 3 by t h e Fact t h a t i t s d i f f r a c t i o n v e c t o r s g e n e r a t e a q u a s i - l a t t i c e ~f rank g r e a t e r t h a n i t s dimension.

Two c l a s s e s of incommensurate c r y s t a l phases have a l r e a d y e x t e n s i v e l y been s t u d l e d : modulated s t r u c t u r e s and composite c r y s t a l s . For both c l a s s e s one may d i s t i n g u i s h , i n p r a c t i c e though n o t i n p r i n c i p l e f o r composite systems, a l a t t i c e o f main r e f l e c - t i o n s s e t a p a r t f r m t h e o t h e r d i f f r a c t i o n s p o t s c a l l e d s a t e l l i t e s . From t h e d e f i n i - t i o n o f incommensurate c r y s t a l phase, however, it i s s e e n t h a t a q u a s i - c r y s t a l is an IC.phase t o o , but without l a t t i c e of rr.ain r t f l L , . . c t i o n s . A s i s w e l l known, f o r a n o n c r y s t a l l o g r a ~ h i c p o i n t group such ^hl e t t i c e eoes r c t ex<t i.. Each d i f f r a c t i o n v e c t o r H may be w r i t t e n a s

(hi i n t e g e r ) .

vhen d>O t h e r e i s not 3-dimensional l a t t i c e o f t r a n s l a t i o n v e c t o r s v such t h a t H.v = 0 (mod 2 i'r) f o r a l l H and v. Hence an I C phase does n o t have 3-dimensional space group symmetry. However, it can be shown t h a t i t s symmetry group i s a (3+d)- dimensional c r y s t a l l o g r a p h i c space group / 2 , 3 / . The way t o s e e t h i s , i s t o c o n s i d e r t h e v e c t o r s ( 1 ) a s belonging t o t h e p r o j e c t i o n o f a l a t t i c e i n 3+d dimensions.

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To find the components one considers the point group K of the diffraction pattern. For any el-ement R from this group one has

3+d

R ai = C

r

( R ) ji a; (3)

j=1

The matrices

r(~)

form a representation of a finite point group which is, therefore, equivalent to a group of orthogonal transformations. The basis (a:, a:i) of the 3+d dimensional lattice is chosen such that these orthogonal transformations with respect to this basis are represented by the matrices

q ~ ) .

This fixes the basis up to one or more constants, depending on the number of irreducible components of

r ( ~ ) .

The incommensurate crystal phase has a density function f(r) with Fourier decompo- sition

f (I) = C F(H) exp (iH.r), H E M*

where the summation is over the quasi-lattice M* generated by (1). If the diffraction pattern is considered as projection of a (3+d)-dimensional diffraction pattern, the function

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is the restriction of a function in (3+d) dimensions :

f(r,t) = C F(H) exp [i(~.r+~=.t)] ( 5

HEW*

This function has lattice translation symmetry, where the lattice is generated by the basis {(ai,aIi)) dual to {(a\,a*Ii))

.

AS an example the function f(r) of one variable given by

f(r) = C 6 (r-n-r.-u (q.n)), U(X) = u (x+l)

J j ( 6 )

n j

is the density function for a one-dimensional incommensurately displacively modulated crystal phase. Its embedding is given in Fig.1, first for a sinusoidal modulation u(x) = sin (2 x), then for a structure which corresponds to a modulated structure with discommensurations.

Fig. 1

-

Embedding of a 1-dimensional modulated crystal in 2-dimensional superspace.

Left : sinusoidal modulation ; Right : discommensurations.

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C3-88 JOURNAL DE PHYSIQUE

Because t h e d e n s i t y i n 3+d dimensions h a s l a t t i c e p e r i o d i c i t y i t s symmetry group i s a (3+d)-dimensional space group.

We s h a l l denote t h e " r e a l " 3-dimensional spr.ce by VE, t h e space of a d d i t i o n a l c o o r d i n a t e s by V I '

I11

-

ONE- AND TWO- DIMENSIONAL QUASI-CRYSTALS

An example o f a q u a s i - c r y s t a l i n one dimension i s t h e Fibonacci c h a i n , a double i n f i n i t e s e r i e s o f i n t e r v a l s o f e i t h e r of two k i n d s , wit'n l e n g t h r a t i o

T = (d 5 -1 ) / 2 , and f o r which t h e end p o i n t o f t h e n-th i n t e r v a l i s given by

where f r a c ( x ) i s t h e f r a c t i o n a l p a r t of x . The end p o i n t s a r e o f t h e form m+nT and belong, t h e r e f o r e , t o a q u a s i - l a t t i c e w i t h rank 2 . It can be embedded a s a f u n c t i o n i n 2 dimensions :

xn ( t ) = n ( 3 ~ - 1 ) + (T-1) [ f r a c ( n T + t )

-

1/21. ( 8 ) Another, e q u i v a l e n t , embedding i n 2 dimensions i s g i v e n i n F i g . 2. There t h e 2-dimensional s t r u c t u r e c o n s i s t s s f l i n e elements o f l e n g t h c ( l + T ) , p a r a l l e l t o VI w i t h midpoints a t t h e l a t t i c e p o i n t s of a l a t t i c e w i t h b a s i s

where c i s an a r b i t r a r y c o n s t a n t .

Fig. 2 - &bedding o f t h e Fibonacci c h a i n .

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A 2-dimensional example of a q u a s i - c r y s t a l i s formed by t h e v e r t i c e s of a Penrose p a t t e r n . These v e r t i c e s a r e a t p o s i t i o n s C m.e., where m . a r e i n t e g e r s and e l , e 2 ,

J J J

e3,e,, f o u r o f t h e f i v e v e c t o r s p o i n t i n g from t h e c e n t r e of a r e g u l a r pentagon t o t h e c o r n e r s . A s has been shown i n

/4/

t h e 2-dimensional s t r u c t u r e i s t h e i n t e r s e c - t i o n o f t h e 2-dimensional hyperplane w i t h a 4-dimensional p e r i o d i c s t r u c t u r e w i t h 5 d i s c r e t e 2-dimensional pentagons, p a r a l l e l t o V I i n each u n i t c e l l . This was shown u s i n g a theorem by de B r u i j n

/5/.

I n t h e s e two c a s e s b o t h t h e p o i n t s e t and i t s F o u r i e r t r a n s f o r m a r e q u a s i - l a t t i c e s .

I V

-

STRUCTURE FACTOR

The geometric s t r u c t u r e f a c t o r f o r a s e t of p a r t i c l e s a t p o s i t i o n s r i s given by t h e e x p r e s s i o n

S (H) = C e x p ( i ~ . m ) ( 1 0 )

It i s t h e F o u r i e r t r a n s f o r m o f a f u n c t i o n t h a t i s t h e sum o v e r d e l t a - f u n c t i o n s l o c a - t e d a t t h e p o s i t i o n s of t h e p a r t i c l e s . For an incommensurate c r y s t a l phase, and a f o r t i o r i f o r a q u a s i - c r y s t a l , t h i s F o u r i e r t r a n s f o r m i s t h e p r o j e c t i o n o f t h e F o u r i e r t r a n s f o r m i n 3+d dimensions o f a p e r i o d i c f u n c t i o n . For a d i s p l a c i v e l y modulated s t r u c t u r e w i t h p o s i t i o n s

n ⁺ + u ( q . n ) ,

j j (17)

where n d e n o t e s t h e u n i t c e l l , r . t h e p o s i t i o n o f t h e j - t h p a r t i c l e i n t h e u n i t J

c e l l of t h e b a s i c s t r u c t u r e and u ( x ) t h e p e r i o d i c displacement f u n c t i o n , t h i s s t r u c t u r e f a c t o r i s

Analogously one can c a l c u l a t e t h e s t r u c t u r e f a c t o r f o r t h e Fibonacci c h a i n . I n t h i s c a s e t h e 2 b a s i s v e c t o r s of t h e r e c i p r o c a l l a t t i c e i n 2 dimensions a r e ( 1 , - r ) / ( 2 - T ) and ( ~ , 1 ) / ( 2 - T ) . Hence t o t h e 1-dimensional r e c i p r o c a l l a t t i c e v e c t o r .

H=(m+n ~ ) / ( 2 - T) corresponds t h e 2-dimensional v e c t o r w i t h second component HI = ( - m + n ) / ( 2 - ^T).Then t h e s t r u c t u r e f a c t o r S(H) i s

w i t h y=(n-rn+n~)/(2-s). The i n t e n s i t i e s S ( H ) o f t h e d i f f r a c t i o n v e c t o r s a r e given i n Fig. 3. One n o t i c e s t h e absence o f a l a t t i c e o f main r e f l e c t i o n s , season why t h e Fibonacci c h a i n may be c o n s i d e r e d a s a q u a s i - c r y s t a l . Another s t r i k i n g f e a t u r e i s t h e s c a l i n g p r o p e r t y .

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JOURNAL DE PHYSIQUE

F i g . 3 - I n t e n s i t i e s o f t h e d i f f r a c t i o n s p o t s o f t h e Fibonacci c h a m .

I n a s i m i l a r way, t h e s t r u c t u r e f a c t o r f o r t h e Penrose p a t t e r n may be c a l c u l a t e d ( S i n c e d i f f e r e n t p a t t e r n s may be o b t a i n e d from t h e same 4-dimensicunal p e r i o d i c s t r u c t u r e by a n o t h e r c h o i c e of o r i g i n t h e d i f f r a c t i o n p a t t e r n s a r e t h e same).

Because t h e 4-dimensional s t r u c t u r e has 5 d i s c r e t e elements i n t h e u n i t c e l l , w i t h c e n t r e s e q u i d i s t a n t a l o n g t h e d i a g o n a l o f t h e u n i t c e l l , t h e s t r u c t u r e f a c t o r i s

1 4

S ( H ) = C r d t exp C ~ T ~ ( H ~ . X ~ ( P ) + H , . ~ ) ]

,

( 1 4 )

norm ~ " 1 @ @

where H i s t h e 4-dimensional v e c t o r p r o j e c t e d on H o f t h e form ( 1 ) and xs ( p ) t h e 4-dimensional p o s i t i o n v e c t o r o f t h e c e n t r e o f t h e p-th element i n t h e u n i t c e l l . I n t r o d u c i n g a =I.-2.2.-1 f o r ~=1.2.1.4. r e s ~ e c t i v e l v and

4

a

Ak =

-

t h +-T- l+T

5 3k (h3k+2+h3k-2 ) -

T-

(h3k+1+h3k-l )

'

t h e e x p r e s s i o n f o r t h e s t r u c t u r e f a c t o r becomes

The d i f f r a c t i o n p a t t e r n i s shown i n F i g . 4. The i n t e n s i t i e s a r e p r o p o r t i o n a l t o t h e diameter o f t h e s p o t s . Spots with an i n t e n s i t y lower t h a n 1% of t h e c e n t r a l spot have been o m i t t e d . Also h e r e t h e r e i s no l a t t i c e of main r e f l e c t i o n s .

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Fig. 4 - D i f f r a c t i o n p a t t e r n o f t h e Penrose t i l i n g .

For t h i s t y p e o f q u a s i - c r y s t a l which i s an i n t e r s e c t i o n of a hyperplane w i t h a p e r i o d i c s t r u c t u r e c o n s i s t i n g of d i s c r e t e elements p a r a l l e l t o V t h e e x p r e s s i o n

I f o r t h e s t r u c t u r e f a c t o r becomes

S ( H ) = C I d t exp 1271 i ( H , H ) . ( r j , r I j + t ) ]

I ⁽¹⁷⁾

j Pi

where r . i s t h e (3+d)-dimensional p o s i t i o n v e c t o r o f t h e c e n t r e o f t h e j - t h e l e - s J

ment i n t h e u n i t c e l l , H t h e (3+d)-dimensional r e c i p r o c a l l a t t i c e v e c t o r , HI i t s component i n VI and a I j (HI) an a d d i t i o n a l "atomic s c a t t e r i n g f a c t o r " :

a I j ( x I ) = L . d t exp ( 2 r i H I . t ) , ( 19)

J

where t h e i n t e g r a t i o n i s o v e r t h e p r o j e c t i o n f i o f t h e j - t h element on VI. I n j

terms of t h e components h . o f H and t h e 3+d c o o r d i a n t e s x . . o f t h e c e n t r e o f t h e J 1

j - t h element :

V - THRZE-DIMENSIONAL QUASI-CRYSTALS. ICOSAHEDRAL CRYSTALS.

Although i n p r i n c i p l e composite c r y s t a l s might e x i s t f o r which one cannot d i s t i n - g u i s h a l a t t i c e of main r e f l e c t i o n s ( f o r example i f i t s subsystems a r e completely

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C3-92 JOURNAL DE PHYSIQUE

on t h e same f o o t i n g ) m a t e r i a l s which should be c a l l e d q u a s i - c r y s t a l s by our d e f i - n i t i o n observed till now, do so because t h e y show a n o n - c r y s t a l l o g r a p h i c p o i n t group symmetry. The l i s t of a l l p o s s i b l e of t h e s e groups i s f i n i t e , i f one r e q u i r e s t h a t t h e rank o f t h e q u a s i - c r y s t a l i s l e s s t h a n o r e q u a l t o ( s a y ) s i x . I n t h i s c a s e t h e groups a r e

D5, D8, D l O , I , D5x C 2 , D xC 8 2 ' 1 0 2 ' 12 2 D xC D xC and IxC 2 '

For each of t h e s e groups one can determine t h e i n v a r i a n t q u a s i - l a t t i c e s , t h e corresponding 3id-dimensional B r a v a i s c l a s s e s and t h e space groups belonging t o t h e s e B r a v a i s c l a s s e s /4/.

For example, f o r t h e i c o s a h e d r a l group t h e r e a r e 3 Bravais c l a s s e s i n 6 dimensions (d=3) and 16 non-equivalent 6-dimensional space groups. The 3 Bravais c l a s s e s a r e c e n t e r i n g s of t h e l a t t i c e by

1 1

$al

, 1

a , ) ,

. I

( a 2 , - - ¹

c 2 a 2 ) .

3

^(a3,-: ^{a h )}

where a l . . , a 6 a r e 6 v e c t o r s p o i n t i n g t o t b e f a c e s o f a r e g u l a r dodecahedron :

and c i s a n a r b i t r a r y c o n s t a n t ; choosing c = l , t h e b a s i s ( 2 : ) g e n e r a t e s a hypercubic l a t t i c e . One o b t a i n s a s t r u c t u r e o f s p h e r e s a t t h e v e r t i c e s o f a q u a s i - l a t t i c e by p u t t i n g d i s c r e t e elements a t t h e l a t t i c e p o i n t s o f t h e 6-dimensional l a t t i c e and i n t e r s e c t i n g t h i s p e r i o d i c s t r u c t u r e w i t h a 3-dimensional hyperplane. As elements one may choose t h e t o p o l o g i c a l product o f two s p h e r e s , t h e p r o j e c t i o n on VE b e i n g given by a sphere of r a d i u s RE and t h a t on VI by one w i t h r a d i u s RI. The r e s u l t i n g packing f r a c t i o n P i n 3 dimensions i s t h e n

when N i s t h e number of "atoms" p e r u n i t c e l l . For t h e 3 i c o s a h e d r a l l a t t i c e s P i s given i n t h e f o l l o w i n g t a b l e .

( Centering : P o i n t s un u n i t c e l l : N : : P )

R~ R~

.

(--- i ... L --- ¹

---

²

---

)

(

1

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So a simple i c o s a h e d r a l q u a s i - p e r i o d i c packing s t i l l g i v e s a r a t h e r l o o s e s t r u c t u - r e ( c f . simple cubic i n 3 dimensions : P=0.52).

I n t h i s way one may c o n s i d e r o t h e r q u a s i - c r y s t a l s w i t h more t h a n one atom p e r u n i t c e l l . An i n t e r e s t i n g s t r u c t u r e i s t h a t one o b t a i n s by p l a c i n g an atom A ( t o p o l o g i -

111111

c a l product of 2 s p h e r e s ) i n t h e o r i g i n 000000 and i n

---

and atoms B i n

1 11111 222222

0 0 0 0 0 , 0--- 22222 and t h e 1 0 e q u i v a l e n t p o s i t i o n s . For a simple i c o s a h e d r a l l a t t i c e (which now becomes I - c e n t e r e d ) t h i s s t r u c t u r e has t h e f o l l o w i n g p r o p e r t i e s . I n t h e i n t e r s e c t i o n each atom A i s on t h e average surrounded by 12 atoms B i n a dodecahedral c o n f i g u r a t i o n . The symmetry of t h e p a t t e r n i s i c o s a h e d r a l . The concen- t r a t i o n of atoms A i s 1/7=0.14. C l e a r l y t h i s reminds o f t h e A l 0 . 8 6 ~ ~ 0 . 1 4

But of c o u r s e , t h i s model s t r u c t u r e i s o n l y one o u t o f many.

The s t r u c t u r e f a c t o r f o r t h e l a t t e r s t r u c t u r e i s

Ch. h .

S ( H ) = ~ ( s h . a * ) = [ l + ( - 1 ) '1 [ a i b C ( - l )

'I

1 1 1

( 2 4 ) i

where a . a r e t h e 6 b a s i s v e c t o r s ( 2 2 ) of t h e i c o s a h e d r a l q u a s i - l a t t i c e and a nd b depend on HI. The f i r s t term i s t h e e x p r e s s i o n of an I - c e n t e r i n g . The same pro- p e r t i e s o f dodecahedral average surrounding has a s t r u c t u r e w i t h 7 p a r t i c l e s p e r u n i t c e l l iri a Simple i c o s a h e d r a l l a t t i c e .

V I - CONCLUDING REMARKS

As a d e f i n i t i o n of q u a s i - c r y s t a l we have proposed

,

: an incommensurate c r y s t a l phase without c,dnspicuous l a t t i c e o f main r e f l e c t i o n s . I n t h a t s e n s e it i s a t h i r d family o f I C p$ases, b e s i d e s modulated and composite c r y s t a l s , although t h e b o r d e r l i n e betwee t h e l a t t e r and q u a s i - c r y s t a l s i s n o t n e c e s s a r i l y sharp. I n p a r t i c u l a r , a l l I C ph

f

s e s w i t h n o n - c r y s t a l l o g r a p h i c p o i n t group symmetry should b e c a l l e d quasi-

/ c r y s t a l s .

One can u s e t h e same t e c h n i q u e s a s f o r I C phases t o d e s c r i b e t h e symmetry of quasi- c , r y s t a l s . The symmetry groups a r e c r y s t a l l o g r a p h i c s p a c e groups i n more t h a n 3 dimensions.

A q u a s i - c r y s t a l i s t h e i n t e r s e c t i o n o f a p e r i o d i c s t r u c t u r e i n more t h a n 3 dimen- s i o n s with a 3-dimensional hyperplane.

A s c a n be s e e n from Fig. 2 t h e higher-dimensional s t r u c t u r e corresponding t o t h e Fibonacci c h a i n c o n s i s t s of d i s c r e t e elements. The same i s t r u e f o r t h e embedding o f t h e Penrose t i l i n g s and f o r t h e model AB6 s t r u c t u r e i n s e c t i o n 5. I f t h i s f e a - t u r e i s c h a r a c t e r i s t i c f o r q u a s i - c r y s t a l s , t h e y ressemble i n a c e r t a i n s e n s e modu- l a t e d s t r u c t u r e s with discommensurations (Fig. 1 )

. f

n t h e dynamics t h i s would mean a gap i n t h e phason spectrum.

Other p h y s i c a l p r o p e r t i e s l i k e s p e c t r a o f l a t t i c e v i b r a t i o n s and e l e c t r o n s show s i m i l a r i t y with t h o s e o f I C phases. For example t h e v i b r a t i o n s p e c t r a o f t h e

(11)

C3-94 JOURNAL DE PHYSIQUE

Fibonacci chain shows t h e same h i e r a r c h y a s t h a t of t h e modulated s p r i n g model/6/.

A f u r t h e r i n v e s t i g a t i o n of t h e s e p o i n t s i s i n p r o g r e s s .

REFERENCES

/1/ Mackay, A.L., Physica (1982) 609.

/2/ De Wolff, P.M., Acta C r y s t . (1977) k93.

/3/ Janner, A . , and J a n s s e n , T . , Phys. Rev.

B15

(1977) 643.

/4/

J a n s s e n , T . , "Crystallography o f q u a s i - c r y s t a l s " , Acta C r y s t . A ( t o a p p e a r ) . / 5 / D e B r u i j n , N . G . , Proc. Kon. Ned. Ac. Wet. (1981 ) 39.

/6/-De Lange, C. and J a n s s e n , T., J. Phys. (1981) 5269.