QUASI-CRYSTAL AND CRYSTAL IN AlMn AND AlMnSi. MODEL STRUCTURE OF THE ICOSAHEDRAL PHASE

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QUASI-CRYSTAL AND CRYSTAL IN AlMn AND AlMnSi. MODEL STRUCTURE OF THE

ICOSAHEDRAL PHASE

P. Guyot, M. Audier, R. Lequette

To cite this version:

P. Guyot, M. Audier, R. Lequette. QUASI-CRYSTAL AND CRYSTAL IN AlMn AND AlMnSi.

MODEL STRUCTURE OF THE ICOSAHEDRAL PHASE. Journal de Physique Colloques, 1986, 47

(C3), pp.C3-389-C3-404. �10.1051/jphyscol:1986340�. �jpa-00225752�

JOURNAL DE PHYSIQUE

Colloque C3, supplement au n o 7, Tome 47, juillet 1986

QUASI-CRYSTAL AND CRYSTAL IN AlMn AND AlMnSI. MODEL STRUCTURE OF THE ICOSAHEDRAL PHASE

P. GUYOT, M. AUDIER( and R. LEQUETTE*

L.T.P.C.M. (U.A. CNRS N o 29)

-

E.N.S.E.E.G., Domaine

Universitaire, BP N o 75, F-38402 Saint-Martin-d'Heres Cedex, France

'ARTEMIS (U.A. CNRS N o 3961, B P n o 68, F-38402 Saint-Martin-dlH&res Cedex,.France

Resume

-

I1 e x i s t e une grande s i m i l a r i t e entre l a phase cubique a dans AlMnSi e t l a phase icosaedrique i-AlLrnSi ou AlYn. Une etude par microscopie electro- nique en transmission (T.E.11.) d'un ruban hypertrempe 00 l e s deux phases co- e x i s t e n t , e t a b l i t clairement leurs r e l a t i o n s . On montre que l e s deux s t r u c - tures peuvent Ctre d e c r i t e s d p a r t i r d ' u n i t e s de bases communes -un double icosaedre A1-Hn- connectees en orientation p a r a l l e l e suivant leurs axes d ' o r - dre 3 par des l i a i s o n s Fln octaedriques. Une connection deterministe des ico- saedres dans l a phase i e s t coherente avec u n pavage Penrose d 3 dimensions.

Un s q u e l e t t e i de grandes dimensions e s t c o n s t r u i t par ordinateur, oO l a de- coration du pavage par l e s icosaedres, s a t i s f a i s a n t d des contraintes s t e r i - ques, e s t resolue par un algorithme de theorie des graphes. Des projections du modele sur des plans normaux d des axes de symetrie e t l e s diagrammes de d i f f r a c t i o n calcul@s, apres comparaison avec des r e s u l t a t s T.E.!:. haute reso- lution e t de d i f f r a c t i o n , indiquent q u ' i 1 contient 1 'e s s e n t i e l de l a structu- r e de l a phase i dans ces a l l i a g e s .

Abstract

-

There i s a close s i m i l a r i t y between the cubic a-phase i n AlllnSi and the icosahedral phase i-AlllnSi or Allln. A transmission electron microscopy

(T.E.14.) study of a melt-spun ribbon where the two phases coexist, e s t a b l i s - hes c l e a r l y t h e i r relationships. I t i s shown t h a t both s t r u c t u r e s can be des- cribed with common basic u n i t -a double Al-lln icosahedra- connected in paral- l e l orientation along t h e i r 3-fold axes by octahedral tln bonds. A detenninis- t i c connection of the icosahedra in the i-phase appears t o be consistent with a 3 dim-Penrose t i l i n g . A large s i z e i-skeleton i s then computer generated, wnere tne decoration by icosahedra of the t i l i n g , prescribed by s t e r i c cons- t r a i n t s , i s solved by a graph theory algorithm

.

Projections of the model on planes perpendicular t o symmetry axes and ca1cu:ated d i f f r a c t i o n patterns, orc?compared x i t h hi.jh resolution T.E.M. and d i f f r a c t i o n data, indicate that i t contains the main point of the i-phase s t r u c t u r e in these alloys.

I

-

INTRODUCTION

One expects the physical properties of quasi-crystals, of which only l i t t l e i s known today, t o r e l y f o r lon wave length excitations on t h e i r long range orientational order or translationnay quasiperiodicity, and f o r short wave length ones on t h e i r local atomic arrangements.

Soon a f t e r the discovery of the icosahedral phase (i-phase) i n Al-Mn by Shecthtman e t a l . / I / , the long range problem was solved by the very elegant and powerful1 tech- nique of cut and prciection of periodic 6-dimensional hyperlattices /2/ /3/ /4/, leading t o a 3 dim.-generalization of the Penrose t i l i n g ( 3 DPT). The agreement of the Fourier transform of such a q u a s i l a t t i c e w i t h experimental electron d i f f r a c t i o n data was s u f f i c i e n t l y good t o assign a second order r o l e t o the local atomic order.

( ' o

n leave at Cegedur-Plchiney Research Centre. P-38340 Voreppe. France

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986340

C3-390 J O U R N A L DE PHYSIQUE

However s i g n i f i c a n t i n t e n s i t y v a r i a t i o n s observed i n d i f f r a c t i o n experiments w i t h v a r y i n g s c a t t e r i n g lengths ( X r a y s /6/, neutrons /7/, see a l s o t h i s conference) e v i - denced t h e e x i s t e n c e o f an atomic o r d e r i n g on t h e q u a s i - l a t t i c e s i t e s .

Solve t h e l o c a l o r d e r i s e q u i v a l e n t t o decorate w i t h atoms t h e p r o l a t e and o b l a t e rhombohedra1 t i l e s o f t h e q u a s i - l a t t i c e , w i t h a s p e c i a l d i f f i c u l t y i n h e r e n t t o t h e matching r u l e s c o n s t r a i n i n g t h e t i l i n g : i t i s n o t obvious thatan u n i o u e p a i r o f e l e - mentary t i l e s can be defined. On t h e o t h e r hand i t i s c l e a r t h a t t h e d e c o r a t i o n i s e a s i e r t o perform i n t h e 3 dim-space, i n o r d e r t o reap advantage from t h e elementary r u l e s o f chemistry ; a f u r t h e r c l i m b o f t h e atoms i n t h e 6 dim-hyperspace f o r s 4 n p l e r F o u r i e r t r a n s f o r m c a l c u l a t i o n s i s p o s s i b l e .

I t a l s o appears s i m p l e r and w i t h a p h y s i c a l ground t o decorate t h e 3 DPT by po- lyatomic c l u s t e r s which preserve i t s icosahedral symmetry elements : icosahedron, dodecahedron, rhombic t r i a c o n t a h e d r o n

...

The s i z e o f t h e u n i t s i m p l i e s t o s t a r t w i t h a b a s i c bare t i l i n g o f convenient s i z e ( f i x e d by t h e edge l e n g t h AR o f t h e rhombohe- d r a l t i l e s ) . The a n a l y s i s o f t h e r e s u l t i n g modelling, i n a f u r t h e r step, may be done i n terms o f atoms l o c a t i o n s , and consider, whether o r n o t a d e f l a t e d atomic t i l i n g e x i s t s .

A n a t u r a l choice o f t h e dressing c l u s t e r s can be adressed t o c r y s t a l l i n e phases, where t h e r e are known t o e x i s t b u t packed p e r i o d i c a l l y . On t h e b a s i s o f t h e i n t r i n s i c s t a b i l i t y o f t h e d u s t e r s , t h e d i f f e r e n c e between c r y s t a l s and q u a s i - c r y s t a l s i s t h e n r e l a t e d t o t h e i r c o n n e c t i v i t y ( c o o r d i n a t i o n number, bond l e n g t h ) and i t s long range propagation ( p e r i o d i c ; a p e r i o d i c b u t d e t e r m i n i s t i c ; a t random). To t h a t respect, l i q u i d , amorphous, q u a s i c r y s t a l l i n e and c r y s t a l l i n e s t a t e s may e n t e r i n t o a s i n g l e f i l i a t i o n , where t h e f o r m a t i o n k i n e t i c s i s suspected t o p l a y an important r o l e . I n i t i a l l y , we f o l l o w e d a p a r t o f t h i s procedure, by t r i a l and e r r o r , borrowing i c o - sahedral u n i t s and bonds o f t h e c r y s t a l l i n e cubic a-phase i n AlMnSi o r AlFeSi. I n t r o - ducing a simple breaking o f p e r i o d i c i t y i n t h e connection o f t h e icosahedra, we pro- posed a r e a l i s t i c s t r u c t u r e o f t h e i-phase i n AlMn /7/. Independently, t h e develop- ment o f t h e c u t and p r o j e c t i o n technique l e d E l s e r and Henley /8,9/ t o a s i m i l a r r e s u l t , a f t e r d e s c r i b i n g t h e cubic c r y s t a l through a 6 d i m - s t r i p o f r a t i o n a l slope, approximant o f t h e golden number $I = ( 1

+

& ) / 2 .

I n t h e present work, we g i v e a more complete s t r u c t u r a l approach o f t h e .i-phase. We f i r s t i n t r o d u c e t h e c u b i c a-phase, as observed d u r i n g t h e c r y s t a l l i z a t i o n o f a melt- spun AlFeSi glass. We show next, i n a melt-spun AlMnSi r i b b o n where a and i-phase c o e x i s t , t h e c l o s e r e l a t i o n s between c r y s t a l and q u a s i c r y s t a l . The relevance t o use t h e icosahedral u n i t s o f t h e a phase t o b u i l d t h e i-phase i s o u t l i n e d . F i n a l l y a q u a s i c r y s t a l i s computer generated w i t h a s i z e c l o s e t o t h e c o r r e l a t i o n l e n g t h expe- r i m e n t a l l y determined i n AlMn

-

^{a 20 nm}

-

^/5/. The decoration o f t h e 3 DPT i s solved by a graph algorithm, and beyond t h e basic icosahedron v a r i o u s l a r g e r atomic u n i t s are a l s o t e s t e d . P r o j e c t i o n s o f t h e model on planes perpendicular t o symnetry axes and F o u r i e r transforms a r e compared w i t h h i g h r e s o l u t i o n e l e c t r o n microscopy and d i f f r a c t i o n data.

I 1

-

THE a-PHASE

The cubic a phase has f i r s t a s t r o n g s i m i l a r i t y w i t h t h e i-phase i n i t s as grown den- d r i t i c morphology : i n F i g . l a ) , a a c r y s t a l has been formed a f t e r h e a t i n g a m e l t - spun A17 FeljSi17 glass

/lo/,

whereas i n F i g . l b ) , t h e q u a s i - c r y s t a l s are as spun from a m e l t ~PfjMn. A comparison o f t h e i r r e s p e c t i v e s t o i c h i o m e t r y , as determined by X-EDS a n a l y s i s i n a scanning transmission e l e c t r o n microscope (STEM) /10//11/ i s a l s o i n s - t r u c t i v e : A18Fe2Si f o r a, A14Mn f o r i, i .e. t h e r a t i o o f t r a n s i t i o n metal t o alumi- nium atoms has a cormon value o f 20 %. F i n a l l y a s t r o n g c o r r e l a t i o n i n e l e c t r o n d i f - f r a c t i o n p a t t e r n s was n o t i c e d /7/ /12/, somewhat d e t a i l e d l a t e r i n t h e study of an as spun A174 gMn2 S i 5 8 ribbon, where i - a n d a-phases are found t o c o e x i s t , announces a r e l a t i o n s h i p be?wee~ t h e s t r u c t u r e s o f t h e two phases.

F i g . 1

-

T.E.M. micrograohs. The a o r i-phases are formed by a f i r s t order t r a n s f o r - mation : from t h e amorphous s t a t e f o r a, F i g . l a )

-

a r e j e c t i o n o f pure A1 ( w h i t e

? a r t s ) a t t h e p e r i p h e r y o f t h e c r y s t a l i s c l e a r l y evidenced-, from t h e 1 iq u i d s t a t e f o r i, F i g . l b )

-

t h e i-phase A14Mn i s embedded i n a f.c.c. m a t r i x , which i s a super- s a t u r a t e d s o l i d s o l u t i o n o f Mn i n A1 /11/.

The s t r u c t u r e of t h e a phase, e s t a b l i s h e d by Cooper and Robinson /13/ and Cooper /14/, i s cubic, e i t h e r Pm3 i n AlMnSi o r Im3 i n AlFeSi, w i t h a u n i t c e l l parameter a

, = 12.68

1

and a, = 12.56

fi

r e s o e c t i v e l y . The s t r u c t u r e , s c h e m a t i c a l l y drawn i n F i g . 2, c o n s i s t s i n a packing o f double icosahedra (an A1 icosahedron, w i t h a vacant center, surrounded by a p a r a l l e l double s i z e d Mn icosahedron). The Mn icosahedra a r e

F i g . 2

-

a) Schematic view of t h e a phase ; o n l y t h e Mn icosahedra are drawn ; b) face-to-face octahedral bonds along t h e < I l l > d i r e c t i o n s .

connected along t h e < I l l > d i r e c t i o n s through f o u r o f t h e i r 3 - f o l d axes ; he face- t o - f a c e connections forms a Mn octahedral bond, o f l e n g t h a a d / 2 = 10.98

k

^{i n}

C3-392 JOURNAL DE PHYSIQUE

a-AlMnSi, which maintains t h e icosahedra i n t h e same o r i e n t a t i o n . I n s i d e t h i s skele- ton, t h e A1 icosahedra are connected through < I l l > chains o f t h r e e bonding A1 octa- hedra, whereas o t h e r s A1 atoms cap t h e Mn icosahedral v e r t i c e s . I t i s i n t e r e s t i n g t o note t h a t t h e vacancy i n t r a n s i t i o n metal a t t h e center o f t h e A 1 icosahedron i s a l s o discarded i n t h e i-phase from MBssbauer data /IS/.

I 1 1

-

RELATIONSHIPS BETWEEN a- AND i-PHASE I N A174.2%$i-5.8

I t i s now known t h a t a small a d d i t i o n o f S i s t a b l i z e s t h e i-phase and confers i t a h i g h e r q u a s i - c r y s t a l 1 in e p e r f e c t i o n as compared w i t h t h e i-phase i n AlMn /16/. Fur- thermore a small p r o p o r t i o n o f a phase, which c o e x i s t s . w i t h t h e i-phase i n t h e r i b - bons, confirms (see a l s o /17/) t h e t i g h t s t r u c t u r a l connection t h a t we i n i t i a l l y suggested when t h e two phases were s e p a r a t e l y formed /7/ /12/.

Fig. 3a) i s a b r i g h t f i e l d micrograph o f t h e t i p o f a d e n d r i t e where i s l a n d s o f i- phase c o e x i s t w i t h t h e a-phase. Convenient t i l t s around [ O 1 O l a and [ I O 1 l a a l l o w t o e s t a b l i s h t h e o r i e n t a t i o n r e l a t i o n s h i p s between t h e two phases ( s i m p l y summarized i n F i g . 39

,

where t h e icosahedron -which 2, 3 and 5 f o l d axes are those o f t h e i- hase-, i s i n s c r i b e d i n t h e a cube), namely : 1100la//[A2I i, 11111 // [A31 i, [ @ ! o ~ I ~ / / I A ~ I ~ ([5021a o r [13,0,5bat .9Oor.13O from 1A31i), [ ~ o B I ~ ? / [ A s I ~ (1305la o r 1508Ia a t .75O o r .28O from [A51 i ) .

We a l s o n o t e ( n o t i n d i c a t e d i n t h e Fig. 3 ) t h a t t h e more i n t e n s e spots a r e i n n e a r l y coTncidence : [035Ia and [ 1000001 i

,

^[ 5321 o r * [ 006Ia and ~ 1 1 0 0 0 0 1 ~ , [0311a and [1100011i, using f o r t h e i-phase r e f l e x i o n s t h e 6 icosahedral i n a i c e s o f Bancel e t a1

.

^/5/.

High r e s o l u t i o n imaging i n a x i a l i l l u m i n a t i o n along [100Ia//[A21 i and [503Ia//[A5] 1 are shown i n Fig. 4, where o r i e n t a t i o n r e l a t i o n s h i p and degree o f coherency can be d i r e c t l y appreciated. I n F i g . 4b), two 36O r o t a t e d a v a r i a n t s , a1 and a11, seem t o be issued from t h e i-phase. Using t h e a phase as a standard i t i s then p o s s i b l e t o c a l i b r a t e t h e i-phase q u a s i - l a t t i c e constant, as discussed i n next s e c t i o n . I V

-

MODEL STkUCTU2E OF THE i-PHASE I N AlFln AND h1:lnSi

I V - 1

-

P r i n c i p l e :

k l o g i c a l i d e a t o ~:ioael t h e i-phase s t r u c t u r e i s t o paclc icosanecra m a i n t a i n i n g t h e sace orientation i n o r a e r t o propagate a t long range t h e i r o r i e n t a t i o n a l order. A packing v e r t e x - t o - v e r t e x o r eage-to-edge, as i n i t i a l l y proposed by Shechtnan and Alech

/la/,

see a l s o /lJ/,suffers from a r o t a t i o n freedom arouna t h e connection, which makes t h e r u l e somewhat "ad hoc". A t t h e o p p o s i t e t h e face-to-face connection as i n t h e f i r s t c o u s i n a-phase, i s r i g i d .

Applying t h i s % f o l d a x i s connection, we f i r s t b u i l t by hands /7/ /11/ a double i c o - sahedra skeleton according t o t h e f o l l o w i n g r u l e s : f o r s t e r i c c o n s t r a i n t s no octa- hedral bonds on a d j a c e n t icosahedron faces a r e allowed ; th e f i r s t - n e i g h b o u r s coor- d i n a t i o n number must be lower than 8, which i n s u r e s t h e t r a n s l a t i o n a l p e r i o d i c i t y o f t h e a-phase ; th e d i s t a n c e between f i r s t - n e i g h b o u r i n g icosahedra along a 3 - f o l d a x i s iaay be s l i g h t l y a t variance from t h i s i n t h e a-phase. A l o c a l icosahe- d r a l symmetry o f t h e model was obtained, which comparison o f atoms p r o j e c t i o n s w i t h h i g h r e s o l u t i o n micrographs, and o f dense atomic plane spacings w i t h d i f f r a c t i o n data /5/ was s a t i s f a c t o r y /li/.

The r e s u l t s of5.111, i n terms o f t h e d i f f r a c t i o n i n d e x a t i o n /12/, a l l o w t o determine t h e s i z e a i o f t h e Mn icosahedron edge i n t h e i-phase :

i/ w i t h i n t h e p r e c i s i o n o f t h e d i f f r a c t i o n p a t t e r n o f F i g . 3b), t h e (005) r e - f l e x i o n l i e s a t h a l f - d i s t a n c e of (111010)j and (110000)i, from which we deduce a i = 2/5 a = 5.07 A.

ii/ t h e i - f r i n g e s o f F i g . 4a), pependicular t o t h e 3 - f o l d a x i s , correspond t o (110001)i ; th e i r spacing, a i ($

+

1)/2~'3, as measured from t h e d i r e c t l y imaged cubic c e l l a, i s equal t o 3.66

1

; then a i = 5.10

A.

F i g . 3

-

Transmission e l e c t r o n micrograph o f c o e x i s t i n g a- and i-phases i n a m e l t -

?:lnzoSi ^s~ (a), and r e l a t e d d i f f r a c t i o n p a t t e r n s a f t e r convenient t i l t :C!e!17tbf t o ( f 5

-

see t e x t f o r d e t a i l s

-

(g) summarizes t h e o r i e n t a t i o n r e l a t i o n - ships. The weak spots o f t h e p a t t e r n

-

(e) are formed by double d i f f r a c t i o n by t h e i-phase of t h e a d i f f r a c t e d beams : due t o the a p e r i o d i c i t y o f t h e i-phase, t h i s e f f e c t gives r i s e t o rows o f spots.

JOURNAL DE PHYSIQUE

F i g . 4a)-High r e s o l u t i o n o f t h e a- and i-phase i n t h e o v e r l a p r e g i o n : [ 1001, N [A21

This e s t i m a t i o n i s i n good agreement w i t h t h e Cooper and Robinson measurements i n t h e a phase where a, = a i x (2.493

+

.009). We conclude t h a t a and i-phases i n AltlnSi a r e two d i f f e r e n t p a c ~ i n g s o f t h e sane Mn icosahedron. The d i s t a n c e icosahe- dron center4111 v e r t i c e i s 2 4.85

A .

IV-2

-

Computer model 1 i n g :

The l o c a l a n a l y s i s o f t h i s s k e l e t o n showed t h a t i t was c o n s i s t e n t w i t h 3 DPT rhom- bohedral t i l e s , o f edge l e n g t h A2 = a i $2 m / 2 /12/, shown i n F i g . 5. Four o f t h e e i g n t v e r t i c e s o f both t i l e s a r e occupied by a double icosahedron, on t h e oppo- s i t e v e r t i c e s o f t h e rhombic faces ; whereas t h e o b l a t e rhombohedron i s always empty, two types o f p r o l a t e ones a r e d i s t i n g u i s h e d : one which contains one icosahedron on i t s l a r g e diagonal, i n t h e r a t i o 1: @ f r o m th e occupied v e r t e x t o t h e vacant one ( F i s . 5), t h e o t h e r being empty. The opposite faces o f these rhombohedra are n o t e q u i v a l e n t , which precludes t h e i r t r a n s l a t i o n a l p e r i o d i c packing.

On t h i s basis, an extended s k e l e t o n has been computer generated, d e c o r a t i n g t h e t i l e s o f a 3 DPT,built by t h e c u t and p r o j e c t i o n method /2/, w i t h icosahedra accor-

Fig. 4b)

-

High r e s o l u t i o n of t h e a and i-phase i n t h e overlapp region:C5031a// [A51i.

Fig. 5

-

Icosahedra decoration of t h e p r o l a t e (a) and o b l a t e (b) rhonibohedral t i l e s .

J O U R N A L DE PHYSIQUE

0 o Z * & O ; * T + o o 4 4

*,* *

+

* * * *

+ 0 o + 4 ' O w 4 + 4 ' O w

*

+ o 0 +

* * * ^*

^{+ )}

'*+++,&&++ ++%**++,, &++%*+C+*+* ++ ++ ++, &+++*$+g++&+S,, "C+ +%*+++, && ++

* a + + # o o q + + a + o o + a . g p o o q + + + + o o + a . g p . + + o o + * + + $ o o + ~ + + + $ $ * : o o:*:+&+o o + & $ * ' i o o : * % $ + o o+&+:*:+*+o o o # t 2 * = o o + * :

>~P*Q+$%++++&++*+Q++ 0 ++P*&+$~,++,,~$+*+.+Q++ ++4'p"C++%+Q++ ++P+.+@+++~++++&++

4 + + i 0 0 1 . + + 0 O + * * + + + * ' O + ~ ' O * * + + * ~ Q + O 0 + 4 ' 0 + o + * - u r * a + o c +$++&++

',

^{+&%+ * +& ++ +++++* ++ o +& ++} ++*+&+++,,o'+%+ 0 o +&++ o ++ %*& ++

',

^{++%+ o}

.'+Q#+ 0 ++P*Q++ 0 ++P+ 0 0 +QP+ 0 ++P+ 0 0 +P$

* :

^{0 +Q++ 0 ++P+ 0 0 +QP+ 0} 0 +Q++ O .

+ * * c + o + * * * * + o o + 4 * , * * + 0 o + * + + % 0 + 4 * * * + o o + * + o o + * ' O *

,

'++ +%*+++$#W++++,, &+ +* *+ ++, &+

+*

^*$++C%

*+

, +, &+ +%

*+

++, &*++*+++C* Q+ , +, ++ ^+%

*+

^++, "C$+$+, c L O O + L * U ) * * + O o + * + o o + * * * 4 + o o + * * * * + * * + * + o O + * * C * +

0 0 #++ o ++%To 0 +&%+ 0 0 +&++ 0

++i:

0 0 +#++ 0 ++%*&++ 0 +&%+ o 0 +&++ 0 ++%+ c ++O++P+O O+Q++o++P*Q++*++P+O O+Q+*o++P+O O+QP+O++P+Q++O++#+o o + Q + . b * O C * + O O ' * * C + * + * * * * + O o + * * * * + o o + * * w * + * * + * + o o + * ++ 0 ++ *+*++* &# ++ ++ $*& ++ ++

** *+

^{, +, &#} ++ ⁰ ++ & b

*+ +*

&& +* *+++*+# ++ * ++ 1;0

*+

^{++, &+ +-}

0 0+9++,4c++#+* 0 Q 9 + 0 0 Q++ 4 + + 9 + 0 0 9 + + 4 + + 9 O + * - ( P P 0 4 + 9 + + + * 2 0 C

-

^{o o +}

* sa

⁴ + o o

i+*

+ o o i + e % d * + o o = + % $ a = o+&+i+e.= o o +

*

^{++& o}

'*+ +,'++,+***++++J

*+

^++,&+ +Bq+ +,,'+$++*Q+++*

*+

^{++,@+ +%} *@+$**+ P+, &+

+* *+

^{++ &+ +} _{++ 0}

+ 4 - o C + * + * U ) * * + o o + * * * * + * * c + * + o o + * * * * + o o + * + o o + * * *

>-+++O++*++&++o++&+* o+&++o++**&++o++&+o o + & + + o + + 6 + o o + # + + o o + & * + o -

O O++*+O O+Q++O++P+o * +Q9+ 0 V++++o++P+ 0 0 +Q++ 0 ++9+ 0 0 ++P+ 0

o + * + o O + * - O * * + O o + 4 + o o + * * c + 4 + o o + 4 ' O o c a + o o + a + o + + + + . o + ~ + + o + + + ~ * + + + , * + + + o O + & + + O + + + ~ * + + + , ~ + + + o + + + ~ * + + , , ~ + + + o

0 0 0 0 O O + Q + O 0 0 A D 0 o o + * + o 0 0 o + * + o o

F i g . 6

-

P r o j e c t i o n s of t h e icosahedra s k e l e t o n on planes perpendicular t o a 2-fold a x i s , a ) , and a 5-fold. a x i s , b ) .

9

= Iln atom.

+

= A1 atom. In a ) , t h e p r o l a t e rhom- bohedron, of edge length AR, decorated by 5 icosahedra, has been represented.

F i g . 7

-

High r e s o l u t i o n micrograph o f (A1FlnSi)i taken along a 5 - f o l d a x i s , w i t h a r e s o l u t i o n lower t h a n 6

a.

An i c o s a h e d r a s k e l e t o n , o f edge l e n g t h a i 44 = 34.95

h.,

superimposes e x a c t l y on t h e micrograph ( s e e t e x t ) .

C3-398 JOURNAL DE PHYSIQUE

ding t o t h e r u l e s p r e v i o u s l y described. The s k e l e t o n c o n t a i n s 1095 icosahedra, i . e . 26230 atoms. I t s s i z e i s about 20 nm., c l o s e t o t h e c o r r e l a t i o n l e n ~ t h i n t h e i-phase Aliln /5/. Various a d d i t i o n n a l decorations o f t h e s k e l e t o n have been attempted

( 5

IV-3).

The two-steps a l g o r i t h m used t o b u i l d t h e model i s g i v e n i n Appendix.

IV-3

-

A n a l y s i s o f t h e computer s i m u l a t i o n :

To v i s u a l i z e t h e s t r u c t u r e , t n e p r o j e c t i o n s o f t h e icosahedra s k e l e t o n on planes n o r - mal t o t h e icosahedron symmetry axes have been made. Examples a r e g i v e n i n F i s . 6.

As emphasized i n /7/ and /12/, these p r o j e c t i o n s show t h a t t h e atoms l i e i n planes p a r a l l e l t o t h e p r o j e c t i o n a x i s , d i s t r i b u t e d p e r i o d i c a l l y o r q u a s i - p e r i o d i c a l l y - i n t h e sense o f Levine and S t e i n h a r d t /20/, i . e . w i t h incommensurate i n t e r v a l s - perpen- d i c u l a r l y t o t h e 5, 3 o r 2 f o l d - a x i s l y i n i n t h e p r o j e c t i o n planes. The alignements are a t b e s t seen l o o k i n g t h e f i g u r e s a t gyancing angle. F i g . 7 i s a medium r e s o l u - t i o n micrograph o f t h e i-phase taken w i t h an i n c i d e n t beam p a r a l l e l t o a 5 f o l d - a x i s w i t h an o b j e c t i v e a p e r t u r e which c u t s t h e r e s o l u t i o n below ^.L 6

fi.

The agreement w i t h a p r o j e c t e d s k e l e t o n i n f l a t e d o f $4 w i t h r e s p e c t t o t h i s described i n g.IV-2, i s indeed s t r i k i n g .

On t h e o t h e r hand, t h e icosahedra s k e l e t o n i s e v i d e n t l y a loose s t r u c t u r e , o f s t o i - chiometry A1-Mn, and although t h e average icosahedra c o o r d i n a t i o n number, near 5.5, i s h i g h e r than t h e value 3.4 obtained by S t e r n e t a l . ( t h i s conference) f o r a random packing, o t h e r s atoms A1 and Mn must be added i n o r d e r t o achieve t h e c o r r e c t s t o i - chiometry (% A14Pin) and d e n s i t y , 3.7

,

which i s c l o s e t o t h e a phase d e n s i t y (see Audier and Guyot, t h i s conference).

To t h a t respect, two attempts have been made :

i/ t h e a d d i t i o n o f 30 A1 atoms i n t h e m i d d l e o f each icosahedron edge, d e f i n i n g t h e 54-atom i4ackay icosahedron, as proposed by E l s e r and Henley /8/. But i f t h e s t o i c h i o m e t r y i s almost c o r r e c t (A142 I.lnl2), t h e d e n s i t y i s s t i l l t o small.

ii/ t h e surrounding o f each icosahedron by a triacontahedron, which s i t e s can be occupied by e i t h e r :In o r A1 atoms. F i g . 8 shows such a s t r u c t u r e , w i t h o n l y F.1n atoms a t t h e v e r t i c e s of t h e t r i a c o n t a h e d r a . I t can be seen t h a t two a d j a c e n t tri- contahedra e i t h e r share a face, o r overlapp, which makes t r i c k y t h e e s t i m a t i o n o f t h e d e n s i t y .

The r e s p e c t i v e m e r i t s o f t h e models a r e now examined w i t h r e s p e c t t o t h e i r F o u r i e r transforms. Cuts o f t h i s F o u r i e r transform by planes perpendicular t o a 2 f o l d and a 5 fold-axes a r e shown i n F i g . 9 f o r t h e icosahedra s k e l e t o n (a), t h e P.lackay icosa- hedra (b), and icosahedra-fin t r i a c o n t a h e d r a ( c ) . The s c a t t e r i n g l e n g t h s have been taken w i t h o u t s c a t t e r i n g angle v a r i a t i o n ( f y , / f ~ l = 1.77, f o r t h e s i m u l a t i o n o f t h e e i t h e r e l e c t r o n o r X-rays d i f f r a c t i o n ) . Theke p a t t e r n s appear s i m i l a r t o t h e observed e l e c t r o n d i f f r a c t i o n p a t t e r n s , w i t h however c e r t a i n d i f f e r e n c e s f o r each o f them ; f o r example the i n t e n s i t y r a t i o I (100000)/1 (110000) i s lower than u n i t y f o r t h e icosahedra and t r i a c o n t a h e d r a models, whereas i t i s l a r g e r than u n i t y f o r t h e Flackay icosahedra and t h e experimental p a t t e r n s . C a l c u l a t i o n s o f neutron d i f f r a c t i o n ( f ? d n / f ~ 1 =

-

1.11) have a l s o been made, l e a d i n g t o r e f l e x i o n i n t e n s i t y v a r i a t i o n s i n q u a l i t a t i v e agreement w i t h t h e r e s u l t s o f Dubois e t a l . /6/.

iiowever, even i f t h e e s s e n t i a l f e a t u r e s o f t h e d i f f r a c t i o n p a t t e r n s (symmetry sec- quence o f peaks and s c a l i n g along t h e symmetry axes a r e c o n v e n i e n t l y reproduc& by t h e models, a complete comparison o f t h e ' i n t e n s i t i e s w i t h experiments, w i t h o u t p o s i - t i o n n i n g a l l t h e atoms i s premature. And again, s t o i c h i o m e t r y and d e n s i t y must remove any F o u r i e r space degeneracy. This i s l o n g we1 1 knownfrom model 1 in g t h e amorphous s t a t e . .

.

F i n a l l y we s t r e s s t h e d i f f e r e n c e between our model, which i s a 3 DPT decorated by icosahedra, and an elementary 3 DPT decorated by

atoms.

A f t e r d e c o r a t i n g t h e 3 DPT o f edge l e n g t h AR, s m a l l e r t i l e s of edge l e n g t h a r = Ar a-2, decorated by atoms, may be considered as described i n /12/. But t h e v e r t i c e s o f t h i s a r t i l i n g do n o t form a

F i g . 8

-

P r o j e c t i o n on a plane perpendicular t o a 2 f o l d - a x i s o f t h e icosahedra- t r i a c o n t a h e d r a s t r u c t u r e .

t w i c e d e f l a t e d 3 DPT f o r a simple reason : t h e o v e r a l l s e l f - s i m i l a r i t y r a t i o o f a 3 OPT i s b a s i c a l l y $-3, /2/ /3/ /21/.

Furthermore, depending on which way t h e d i f f r a c t i o n pa terns a r e indexed, d i f f e r e n t values o f ar = ~ ~ 6 - 2 a r e obtained i n i - A l % : e i t h e r 4.85

k

f r a n fl2/ which i s r e l a t e d t o a Hendricks-Tel l e r a n a l y s i s of t h e s t r u c t u r e /22/, o r 4.61, according t o E l s e r /3/.

On t h e basis o f t h e present r e s u l t s , we b e l i e v e t h a t i n AlMnSi, a and i-phases have i n common t h e same Al-Mn double icosahedron, w i t h t h e dimensions p r e v i o u s l y given.

However i n t h e a-phase, the hln atoms do n o t 1 i e e x a c t l y a t t h e v e r t i c e s o f rhombo- hedral t i l e s : they a r e s l i g h t l y (2. 5.4 %) displaced beyond t h e v e r t i c e s o f rhombo- hedra o f edge l e n g t h 4.6

A,

as s c h e m a t i c a l l y drawn i n F i g . 10. .So jt would be i n t h e i-phase, i f one keep t h e same t i l e s . The S i atoms are I n s u b s t i t u t ~ o n on t h e A1 sub- l a t t i c e

.

C3-400 3 0 U R N A L D E PHYSIQUE

Fig. 9 - 2 fold (left) and 5 fold (right) calculated diffraction patterns of the model, a) Al-Mn icosahedra. b) Mackay icosahedra. c) Al-Mn icosahedra + Mn tria- contahedra. The area of the spots is proportionnal to the calculated intensity. The spot letter a holds for (100000), d for (110000), c for (lliolo), h for (110000)/5/.

C3-401

Fig. 10 - In (AlHnSi)a, the Mn atoms do not l i e exactly on the vertices of rhombo- hedra A.S X prolate rhombohedron in dotted l i n e .

APPENDIX

COMPUTER SIMULATION OF THE MODEL

The algorithm to build the computer simulation of the quasi-crystal atomic model is separated in two distinct steps :

1- Building the 3 OPT with the cut-projection method / 2 / . 2- Decorating this tiling with icosahedra.

1- Building the quasi-periodical tiling :

During the construction of the tiling the computer keeps two lists in memory :a list of rhombohedra and a list of rhombic faces external to the tiling. Each algorithmic step contains the following operations :

- choose a face in the list.

- compute by the projection method the new rhombohedron of the tiling on this face

- add the faces of this rhombohedron to the faces list, deleting thosewhich are already present.

All the computations are made in the 6-dimensional space associated with the projec- tion method. Vertices are integer sextuplets, rhombohedra are three-dimensionnal cubes and are represented by a 6-dimension integer point and three axes. At each step we choose the oldest face in the list, so the tiling grows compactly.

A two dimensionnal illustration with the Penrose tiling is given on the figure below.

Here the "rhombohedra" are the rhombs and the "faces" are the edges.

2- Decoration of the tiling :

The Duneau-Katz 3DPT gives us a skeleton to build the net of icosahedra connected with octahedra. The tiling is decorated with the three rhombohedric motif : the oblate rhombohedron and the two prolate ones. There are two groups of icosahedra on the tiling : vertex icosahedra which stand on vertices of the tiling and internal icosahedra which are found inside some prolate rhombohedra. Note that every octahe- dron connects a vertex icosahedron and an internal icosahedron-two icosahedra of the same groupe cannot be connected.

Half the vertices are decorated with a vertex icosahedron. The rule is that no

C3-402 JOURNAL DE PHYSIQUE

a d j a c e n t v e r t i c e s can be b o t h decorated. I f we remember t h a t v e r t i c e s a r e t h e p r o j e c - t i o n of i n t e g e r s e x t u p l e t s and t h a t two v e r t i c e s a r e a d j a c e n t i f t h e y d i f f e r on one and o n l y one c o o r d i n a t e by u n i t y , we have a p r o c e d u r e t o choose t h e d e c o r a t e d v e r t i - ces. Namely, add t h e s i x i n t e g e r c o o r d i n a t e s and ( w i t h o u t l o s s o f g e n e r a l i t y ) we d e c o r a t e t h e v e r t e x if t h i s sum i s even.

I

gne stev in the construction of the tiling

l i t of faces new rhombohedron faces l

-

icancelled faces stupdating

-

added faces

F i n d i n g t h e i n t e r n a l i c o s a h e d r a i s n o t as easy because t h e r e i s no g l o b a l r u l e l i k e t h e c o o r d i n a t e sum. We have o n l y a l o c a l e x c l u s i o n r u l e which d e s c r i b e s when t w o a d j a c e n t p r o l a t e rhombohedra cannot b o t h have an i n t e r n a l icosahedron. A l l rhombic faces a r e d e c o r a t e d w i t h two d i a g o n a l l y opposed v e r t e x icosahedra. L e t us c a l l a f a c e a c r i t i c a l f a c e if t h e two i c o s a h e d r a a r e on t h e l o n g d i a g o n a l . Two p r o l a t e rhombohedra s h a r i n g a c r i t i c a l f a c e cannot b o t h c o n t a i n an i n t e r n a l icosahedron.

Since, o t h e r w i s e , t h e icosahedron on t h e s h a r p v e r t e x o f t h e two rhombohedra would b e connected w i t h t h e two i n t e r n a l icosahedra. Consequently two c o n n e c t i n g o c t a h e d r a would be on two a d j a c e n t f a c e s of t h e i c o s a h e d r o n

-

a c o n f i g u r a t i o n f o r b i d d e n i n t h e model.

1 r

\

m

⁷

internal icosahedron

-

The i n t e r n a l i c o s a h e d r o n problem can be r e s t a t e d u s i n g granh t h e o r y . Consider t h e graph whose v e r t i c e s a r e t h e p r o l a t e rhombohedra. Two v e r t i c e s a r e connected w i t h an edge i f t h e two rhombohedra s h a r e a c r i t i c a l f a c e , t h u s t h e y cannot b o t h c o n t a i n an

i n t e r n a l icosahedron. The s e t o f p r o l a t e rhombohedra c o n t a i n i n g an i n t e r n a l icosa- hedron i s a s t a b l e s e t o f t h i s graph ( a s t a b l e s e t i s a s e t of v e r t i c e s o f a graph c o n t a i n i n g no edges). The problem o f f i n d i n g a s t a b l e s e t w i t h maximum c a r d i n a l i t y i s very d i f f i c u l t , b u t i n our problem we can use a v e r y simple a l g o r i t h m w i t h good r e s u l t s . The a l g o r i t h m i s c a l l e d greedy a l g o r i t h m because i t scans t h e v e r t i c e s ad- d i n g them t o t h e s t a b l e s e t whenever i t i s p o s s i b l e and never reverses on a decision.

This a l g o r i t h m i s executed d u r i n g t h e c o n s t r u c t i o n o f t h e t i l i n g : f o r each new pro- l a t e rhombohedron added t o t h e t i l i n g we add an i n t e r n a l icosahedron i f t h e rhombo- hedron does n o t share a c r i t i c a l f a c e w i t h a rhombohedron already present i n t h e t i l i n g and c o n t a i n i n g an i n t e r n a l icosahedron.

Using t h i s a l g o r i t h m we b u i l t a model o f 1000 rhombohedra and t h e decoration gave 1095 icosahedra, t h e coordinates o f those icosahedra were recorded i n d i s k f i l e s . Using t h i s data we made a l l t h e computations : F o u r i e r transforms, p r o j e c t i o n o f t h e atoms, c o o r d i n a t i o n numbers,

...

T

occ~pied

unoccupied prolate rhomboedm

A

ACKNOWLEDGMENTS

L

This work has been financed by an ATP-CNRS no 90-4029. The C.I.C.G. o f Grenoble i s a l s o acknowledged f o r s u p p o r t i n g t h e s i m u l a t i o n computations, as w e l l as R. Rey- F l a n d r i n and G. Regazzoni f o r s p i n n i n g t h e a l l o y s a t PECHINEY-CEGEDUR, CRV.

REFERENCES

/1/ Shechtman, D., Blech, I., Gratias, D. and Cahn, J.W., Phys. Rev. L e t t .

53

(1984) 1951.

~ u n e a u , M. and Katz, A., Phys. Rev. Lett., 54 (1985) 2688.

Elser, V., Acta C r y s t a l l o g r . t o be ~ u b l i s h e C

Kalugin, P.A., Kitaev, A.Y. and L e v i t o v , L.C., J. Phys. ( P a r i s ) ,

46

(1985) L 601.

-

Bancel, P.A., Heiney, P.A., Steohens, P.W., Goldman, A.I. and Horn, P.M., Phys Rev. L e t t . 54 (1985) 2422.

Dubois, J . c Janot, C. and Pannetier, J., submitted t o Phys. L e t t e r s . Guyot, P. and Audier, M., P h i l . Mag. B,

52

(1985) L15.

Elser, V. and Henley, C.L., Phys. Rev. L e t t . 55 (1985) 2883 Henley, C.L., J . Non-Cryst. S o l i d s

2

(1985) T.

Legresy, J.M., Audier, M., Simon, J.P. and Guyot, P., Acta M e t a l l . (1986) i n press.

C3-404 JOURNAL DE PHYSIQUE

/11/ Guyot, P., J. Microsc. Spectrosc. E l e c t r o n . 10 (1985) 333.

/12/ Audier, M. and Guyot, P., P h i l . Mag. 5, 53 ( m 8 6 ) L43.

/13/ Cooper, M. and Robinson, K., Acta C r i s t a n o g r . , (1986) 614.

/14/ Cooper, M., Acta C r i s t a l l o g r . , 93 (1967) 1106.

/15/ Swartzendruber, L.J., ~ h e c h t m a n 7 D .

,

Bendersky, L. and Cahn, J.W., Phys. Rev B, 32 (1985) 1383.

/16/ C h e z C.H. and Chen, H.S., ATT B e l l Lab. p r e p r i n t .

/17/ Koskenmaki, D.C., Chen, H.S. and Rao, K.V., ATT B e l l Lab. p r e ~ r i n t . /18/ Shechtman, D. and Blech, I.A., Met. Trans. 16A (1985) 1005.

/19/ Stephens, P.U. and Goldman, A.I., preprint.-

/20/ Levine, D. and Steinhardt, P.J., Phys. Rev. L e t t . 53 (1984) 2477.

/21/ Cahn, J.I.I., Shechtman, D. and Gratias, D., t o be pri6lished.

/22/ Hendricks, S. and T e l l e r , E., J. o f Chem. Phys.

10

(1942) 147.