DECAGONAL PHASE

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Submitted on 1 Jan 1986

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DECAGONAL PHASE

L. Bendersky

To cite this version:

L. Bendersky. DECAGONAL PHASE. Journal de Physique Colloques, 1986, 47 (C3), pp.C3-457-C3-

464. �10.1051/jphyscol:1986346�. �jpa-00225758�

DECAGONAL PHASE

L. BENDERSKY

Center for Materials Research, The Johns Hopkins University, Baltimore, MD 21218, U.S.A. and

Institute for Materials Science and Engineering, National Bureau of Standards, Gaithersburg, M D 20899, U.S.A.

A b s t r a c t

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S t u d i e s o f phase f o r m a t i o n i n r a p i d l y s o l i d i f i e d A1-Mn a l l o y s ( c o m p o s t t i o n range 18-22 a t % Mn) show t h a t an i c o s a h e d r a l phase i s r e p l a c e d

by a n o t h e r n o n c r y s t a l l o g r a p h i c phase, a decagonal phase. The decagonal phase i s a n o t h e r exampl e o f q u a s i c r y s t a l f o r m i n g i n A1 - t r a n s i t i o n metal systems. It has 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 (10/m o r lO/mmm) t o g e t h e r w i t h long-range o r i e n t a t i o n a l and p o s i t i o n a l o r d e r and one-dimensional t r a n s l a t i o n a l symmetry. The space group P105/m i s suggested f o r t h e decagonal phase. S i m i l a r i t y and d i f f e r e n c e o f A1-Mn and A1-Pd decagonal phases i s discussed.

I

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INTRODUCTION

Recently, t h e d i s c o v e r y o f an i c o s a h e d r a l A1-Mn phase, which d i f f r a c t s e l e c t r o n s l i k e a s i n g l e c r y s t a l , i.e., has d i s c r e t e d i f f r a c t i o n p a t t e r n s and a t t h e same t i m e 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 m35, was announced by Shechtman e t a l . [1,2]. The i c o s a h e d r a l symmetry i s i n c o n s i s t e n t w i t h 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 c r y s t a l s s i n c e f i v e - f o l d r o t a t i o n axes a r e p r e s e n t which i s d i s a l l o w e d f o r c r y s t a l s : t r a n s l a t i o n a l p e r i o d i c i t y p e r m i t s o n l y a r e s t r i c t e d number o f r o t a t i o n axes, namely 1-, 2-, 3-, 4-, and 6 - f o l d . As a r e s u l t o f t h i s , o n l y 32 c r y s t a l - l o g r a p h i c p o i n t groups a r e a l l o w e d [3]. T h i s c o n t r a d i c t i o n can be r e s o l v e d assuming a s t r u c t u r e w i t h q u a s i p e r i o d i c o r a l m o s t p e r i o d i c p r o p e r t i e s . A F o u r i e r t r a n s f o r m ( d i f f r a c t i o n p a t t e r n ) o f a q u a s i p e r i o d i c o r a l m o s t p e r i o d i c f u n c t i o n y i e l d s a continuum o f t r u e d e l t a f u n c t i o n s (Bragg peaks) [4,5]. A q u a s i p e r i o d i c s t r u c t u r e , o r q u a s i c r y s t a l , can have any p o i n t group symmetry i n i t s d i f f r a c t i o n p a t t e r n . The most famous example o f a d i s c r e t e q u a s i p e r i o d i c s t r u c t u r e i s two- dimensional Penrose t i l i n g [6,7] where two t y p e s o f rhombus (analogous t o two u n i t c e l l ) a r e t i l e d a p e r i o d i c a l l y . The F o u r i e r t r a n s f o r m o f t h e t i l i n g has 1 0 - f o l d symmetry. The l a t t i c e o f such a Penrose t i l i n g was shown t o be q u a s i - p e r i o d i c . The t h r e e - d i m e n s i o n a l a n a l o g o f Penrose t i l i n g w i t h two t y p e s o f rhombohedra t i l e d a p e r i o d i c a l l y y i e l d s i c o s a h e d r a l symmetry [8-101 and t h e under- l a y i n g q u a s i l a t t i c e p r o v i d e s a F o u r i e r t r a n s f o r m s i m i l a r t o t h e e x p e r i m e n t a l d i f f r a c t i o n p a t t e r n s o f i c o s a h e d r a l phase. A p o w e r f u l and g e n e r a l method t o g e n e r a t e q u a s i p e r i o d i c s t r u c t u r e s ( l a t t i c e s ) i s t h e Cut and P r o j e c t i o n Method [11-131 where a s l i c e o f a h i g h e r - d i m e n s i o n a l p e r i o d i c s t r u c t u r e ( l a t t i c e ) i s p r o j e c t e d o n t o an i r r a t i o n a l lower-dimensional hyperplane. By t h i s t e c h n i q u e s t r u c t u r e s w i t h any n o n c r y s t a l l o g r a p h i c 3-D p o i n t group can be o b t a i n e d by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986346

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c h o o s i n g t h e a p p r o p r i a t e h i g h e r dimension c r y s t a l l o g r a p h i c p o i n t group and p r o j e c t i o n plane. The 2-0 and 3 - 0 Penrose t i l i n g s a r e o b t a i n e d f r o m 5- and 6-dimensional hypercubes. 14-, and 1 8 - f o l d axes [14]. P r o j e c t i o n from s i x dimensions, f o r example, a l s p e r m i t s The number o f 3-D p o i n t groups G

S

( 1 ) w i t h no 7-, 9-, p e r i o d i c i t y i s i n f i n i t e [3] s i n c e any r o t a t i o n a x i s i s allowe!. They a r e : ( 1 ) i c o s a h e d r a l m35 and c u b i c m3m and t h e i r subgroups, and ( 2 ) N, N2, N, N/m, Nm, Nm, N/mmm, where N = 1, 2,

. . . ^-.

The second t y p e o f p o i n t group can belong i n general t o t h e G~ f a m i l y , i.e., can have 1-D p e r i o d i c i t y w i t h a t r a n s l a t i o n v e c t o r p a r a l l e l l o t h e r o t a t i o n a x i s N.

The i d e a o f q u a s i p e r i o d i c i t y p r o v i d e s us w i t h an i n f i n i t e number o f l o n g - r a n g e o r d e r e d s t r u c t u r e s w i t h d i f f e r e n t symmetries. Which o f them can be r e a l i z e d i n n a t u r e ? T h i s q u e s t i o n i s p r o b a b l y r e l a t e d t o t h e r e a l i z a t i o n o f c e r t a i n symmetries o f s h o r t - r a n g e bonding o r d e r and t h e p o s s i b i l i t y t o propagate t h i s o r d e r t o a l a r g e e x t e n t . The e x i s t e n c e o f t h e i c o s a h e d r a l phase has a good reason s i n c e t h e s h o r t - r a n g e i c o s a h e d r a l o r d e r (SRIO) i s e x t e n s i v e l y p r e s e n t i n m e t a l l i c systems.

SRIO i s b e l i e v e d by many i n v e s t i g a t o r s t o be a f e a t u r e o f m e t a l l i c g l a s s e s and undercooled l i q u i d s [15,16]. Other t y p e s of s t r u c t u r e s w i t h SRIO a r e Frank-Kasper phases [ I 7 1 d e s c r i b e d as a c o m b i n a t i o n o f i c o s a h e d r a l w i t h c e r t a i n o t h e r p o l y t e t r a - h e d r a l c o n f i g u r a t i o n s o f atoms. P o s s i b i l i t i e s f o r extended i c o s a h e d r a l o r d e r t o e x i s t have been s t u d i e d t h e o r e t i c a l l y by a few groups 118-201, and now b o t h e x p e r i m e n t a l and t h e o r e t i c a l evidence e x i s t .

I n t h i s paper a n o t h e r example o f q u a s i c r y s t a l where SRIO i s p r o b a b l y i m p o r t a n t i s discussed. T h i s i s a decagonal ( o r T) phase e x h i b i t i n g d i s c r e t e d i f f r a c t i o n 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 10/m ( o r lO/mmm) which i s n o t a subgroup o f t h e i c o s a h e d r a l p o i n t group [21]. The phase was f i r s t observed and s t u d i e d i n r a p i d l y s o l i d i f i e d A1-Mn a l l o y s [22] and i n some o t h e r a l u m i n u m - t r a n s i t i o n metal a l l o y s [23]. The q u a s i c r y s t a l l i n e n a t u r e o f t h e decagonal phase i s c o n f i r m e d b y d i f f e r e n t r e s e a r c h groups [24-26,281 where c o n c l u s i o n s a r e based on r e s u l t s o f t r a n s m i s s i o n e l e c t r o n microscopy. I n t h i s paper I p r e s e n t some d e t a i l s c o n c e r n i n g t h e c r y s t a l - l o g r a p h y o f t h e decagonal phases. The r e l a t i o n s h i p o f t h i s phase t o t h e i c o s a h e d r a l phase i s a l s o discussed. Most o f t h e d i s c u s s e d r e s u l t s a r e o b t a i n e d f o r A1-Mn phase, o t h e r w i s e i t w i l l be mentioned.

I 1

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CRYSTALLOGRAPHY OF THE DECAGONAL PHASE

Fig. 1 shows a s e r i e s o f s e l e c t e d - a r e a 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 (SADP1s) from t h e decagonal phase, o b t a i n e d by a t i l t i n g experiment. There a r e one u n i q u e t e n - f o l d ( A ) and t e n e q u i v a l e n t t w o - f o l d ( B ) d i f f r a c t i o n p a t t e r n s , where t h e a n g l e between A and B zone axes i s 90" and t h e a n g l e between a d j a c e n t B zone axes i s 36". Another t y p e o f t w o - f o l d d i f f r a c t i o n p a t t e r n (C) has a zone a x i s 18"

away from t h e zone a x i s B, w i t h a 36" a n g l e between a d j a c e n t C axes. Figs. 2(a) and 2 ( b ) compare convergent-beam d i f f r a c t i o n p a t t e r n s (CBDP's) f r o m t h e decagonal phase and t h e i c o s a h e d r a l phase a t t h e t e n - f o l d zone-axis o r i e n t a t i o n s . The advantage o f CBDP's compared t o c o n v e n t i o n a l ( p a r a l l e l e l e c t r o n beam) SADP1s i s t h e presence o f h i g h e r - o r d e r Laue zone (HOLZ) ( 2 ) l i n e s which r e s u l t from s i g n i f i c a n t e l e c t r o n s c a t t e r i n g i n t o d i f f r a c t i o n maxima i n t h e HOLZ's due t o t h e beam convergence T h e r e f o r e , t h r e e - d i m e n s i o n a l c r y s t a l l o g r a p h i c i n f o r m a t i o n from t h e c r y s t a l a x i s p a r a l l e l t o t h e i n c i d e n t beam i s a v a i l a b l e . The w h o l e - p a t t e r n symmetries a r e 1 0 ( o r 10mm) and 5m, r e s p e c t i v e l y , which i s c o n s i s t e n t w i t h t h e r e s u l t s o f Fig. 1 f o r t h e decagonal phase and w i t h t h e p o i n t group m35 f o r t h e i c o s a h e d r a l phase C271.

(1) We use V a i n s t e i n ' s n o t a t i o n f o r p a r t i a l l y p e r i o d i c group G:, when n g i v e s t h e d i m e n s i o n a l i t y o f t h e space i n which t h e group i s d e f i n e d and t t h e d i m e n s i o n a l i t y o f t h e p e r i o d i c l a t t i c e component.

( 2 ) HOLZ's a r e t h e r e c i p r o c a l - l a t t i c e planes p a r a l l e l t o t h e z e r o - o r d e r p l a n e c o n t a i n i n g t h e o r i g i n and normal t o t h e e l e c t r o n beam d i r e c t i o n . Where t h e Ewald sphere i n t e r s e c t t h e s e HOLZ's c o n c e n t r i c r i n g s appears i n CBDP's and r i n g s i z e corresponds t o t h e HOLZ p e r i o d i c i t y .

i c o s a h e d r a l s t e r e o g r a p h i c p r o j e c t i o n . A i s a u n i q u e t e n - f o l d a x i s , C and B

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t w o - f o l d , D and E,

E' -

p s e u d o - f i v e and p s e u d o - t h r e e - f o l d (see r e f . 2 2 ) . S p m e t r y and pseudo-symmetry axes o f t h e decagonal phase can be o b t a i n e d by m i r r o r p l a n e

r e f l e c t i o n o f t h e i c o s a h e d r a l symmetry axes.

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Fig. 2. Convergent-beam e l e c t r o n d i f f r a t i o n p a t t e r n s f r o m t h e ( a ) decagonal and t h e ( b ) i c o s a h e d r a l phases. Whole-pattern symmetries a r e 10 ( o r l h m ) and 5m, r e s p e c t i v e l y .

Another d i f f e r e n c e between t h e CBDP's o f t h e s e two phases i s t h e s p a c i n g o f HOLZ r i n g s w h i c h c o r r e s onds t o s p a c i n g o f r e c i p r o c a l l a t t i c e l a y e r s . For t h e decagonal

P

phase a 111.24 nm- p e r i o d i c i t y a l o n g t h e t e n - f o l d a x i s was measured, where f o r t h e i c o s a h e d r a l phase t h e s p a c i n g a l o n g t h e f i v e - f o l d a x i s i s n o n p e r i o d i c 1271.

The same 1.24 nm p e r i o d i c i t y a l o n g t h e t e n - f o l d a x i s can be measured i n t h e SADP's B and C ( F i g . 1 ) . I n c o n t r a s t t o t h e i c o s a h e d r a l phase, t h e decagonal phase i s I - D p e r i o d i c .

CBDP's o b t a i n e d a t o r i e n t a t i o n s C and B a r e shown i n Fig. 3. B o t h p a t t e r n s have w h o l e - p a t t e r n symmetry m, where t h e m i r r o r p l a n e i s p e r p e n d i c u l a r t o t h e t e n - f o l d a x i s . C o n s i d e r a t i o n o f t h e d i f f e r e n t non-icosahedral p o i n t groups ( s i n c e 1-D p e r i o d i c i t y i s f o r b i d d e n f o r m35 and 235) w i t h r o t a t i o n symmetry f i v e o r t e n shows t h a t t h e o n l y p o i n t group which s a t i s f i e s t h e observed e x p e r i m e n t a l r e s u l t s i s 10/m. However, t h e presence o f a h i g h d e n s i t y o f d e f e c t s , c o l l i n e a r w i t h t h e t e n - f o l d a x i s ( F i g . 4) c o u l d p o s s i b l y d e s t r o y a m i r r o r p l a n e p a r a l l e l t o t h i s a x i s on CBDP's a t B and C. Because o f t h i s p o s s i b i l i t y t h e p o i n t group lO/mmm may n o t be r u l e d o u t .

Fig. 5 compares SADP's C and B f o r A1 -Mn and A1 -Pd decagonal phases. These two phases were found t o be v e r y s i m i l a r e x c e p t t h a t t h e A1-Pd phase (and a l s o A1-Fe decagonal phase [25,26]) has l a r g e r p e r i o d i c i t y c 5 1.65 nm. F o r b o t h phases odd r e f l e c t i o n s (OOOOe), e = 2n + 1, a l o n g c * a x i s a r e v e r y weak, compared t o t h e even r e f l e c t i o n s a t SADP c. A t B p a t t e r n , r e f l e c t i o n s a l o n g c* a x i s have a l m o s t

u n i f o r m i n t e n s i t y . A p o s s i b l e e x p l a n a t i o n f o r t h i s can be t h a t t h e odd r e f l e c t i o n s a r e k i n e m a t i c a l l y f o r b i d d e n , and t h e i r appearance i s due t o t h e dynamic n a t u r e o f e l e c t r o n s c a t t e r i n g . I n c r y s t a l s , k i n e m a t i c a l l y f o r b i d d e n r e f l e c t i o n s a r e caused b y t h e presence e i t h e r o f a screw a x i s o r o f g l i d e planes. T h i s i s used f o r space-group d e t e r m i n a t i o n . For t h e i c o s a h e d r a l phase t h e space group i s e q u i v a l e n t t o t h e p o i n t group s i n c e no t r a n s l a t i o n a l p e r i o d i c i t y i s present. However, f o r t h e decagonal phase a d i f f e r e n t space group can e x i s t due t o t h e 1-D p e r i o d i c i t y . For t h e proposed 10/m p o i n t group o f t h e A1-Mn and Al-Pd decagonal phases t h e space group P1051m w i t h t e n - f o l d screw a x i s w i l l s a t i s f y t h e c o n d i t i o n o f f o r b i d d e n odd r e f l e c t i o n s . For P105/m space group, a s t r u c t u r e p r o j e c t e d normal t o t h e 105 a x i s w i l l show h a l f - p e r i o d i c i t y c12, and t h i s i n d e e d can be seen on a h i g h - r e s o l u t i o n m i c r o g r a p h o b t a i n e d by Guyot and A u d i e r f o r t h e A1-Mn decagonal phase 1241

Fig. 4. Image o f t h e decagonal phase a t t w o - f o l d o r i e n t a t i o n . Note d e f e c t s c o l l i n e a r w i t h t e n - f o l d a x i s .

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Fig. 5. S e l e c t e d - a r e a d i f f r a c t i o n p a t t e r n s a t o r i e n t a t i o n s C and B f o r A1-Mn and Al-Pd decagonal phases. "Fundamental" s p o t s a r e p o i n t e d w i t h arrows.

Both t h e A1-Mn and A1-Pd decagonal phases have v e r y s i m i l a r pseudo-hexagonal ( p a t t e r n B) and pseudo-square ( p a t t e r n C ) fundamental r e f l e c t i o n s ( F i g . 5). The c parameter (1.24 nm f o r A1-Mn and 1.65 nm f o r Al-Pd) i s n o t a r b i t r a r y b u t depends upon t h e parameter o f t h e q u a s i p e r i o d i c p l a n e (which i s t h e same f o r b o t h phases) t h r o u g h a r e l a t i o n s h i p w i t h t h e "fundamental" r e f l e c t i o n s . The meaning o f t h a t can be t h a t b o t h phases have t h e same b u i l d i n g b l o c k ( w i t h 10- o r 5 - f o l d symmetry a x i s ) b u t d i f f e r e n t s t a c k i n g sequence. The presence o f a p s e u d o - f i v e - f o l d d i f f r a c t i o n p a t t e r n ( F i g . 6) -63 degrees a p a r t from t h e u n i q u e t e n - f o l d a x i s (63.4O i s t h e a n g l e between v e r t i c e s o f an icosahedron) i n d i c a t e s t h a t t h e b u i l d i n g b l o c k possesses an i c o s a h e d r a l symmetry, r e f l e c t e d by a m i r r o r p l a n e (see Fig. 1).

D i f f r a c t i o n p a t t e r n s from t h e decagonal phase show t h e presence o f s t r e a k i n g ( s e e Fig. 1, SADP's B and C). However, t i l t i n g experiments demonstrate t h e e x i s t e n c e o f sheets o f i n t e n s i t y i n r e c i p r o c a l space, normal t o t h e u n i q u e t e n - f o l d a x i s w i t h a s p a c i n g between sheets c o r r e s p o n d i n g t o 111.24 (nm)-1. The t e n - f o l d SADP shows v e r y i n t e n s e background, i n d i c a t i n g t h a t t h e z e r o - o r d e r Laue-zone r e f l e c t i o n s and t h e sheet o f i n t e n s i t y a r e i n t h e same p l a n e f o r t h i s o r i e n t a t i o n . The presence o f p l a n a r i n t e n s i t y i n r e c i p r o c a l space can be due t o t h e presence o f one-dimensional o b j e c t s i n r e a l space, a l i g n e d normal t o t h e s e i n t e n s i t y planes. Two p o s s i b i l i t i e s can be c o n s i d e r e d : (1) The f i r s t i s t h e presence

w i t h f i v e - f o l d p a t t e r n o f t h e i c o s a h e d r a l phase (A).

o f c y l i n d r i c a l domains w i t h domain w a l l s c o n t a i n i n g t h e t e n - f o l d a x i s . For c r y s t a l o r i e n t a t i o n s w i t h a l a r g e a n g l e between t h e beam d i r e c t i o n and t h e t e n - f o l d a x i s , t h e domain boundaries s h o u l d be imaged as l o n g l i n e a r d e f e c t s , and indeed, t h i s i s always observed ( F i g . 4 ) . These domain s t r u c t u r e s c o u l d be analogous t o a n t i - phase domain i n c r y s t a l s o r t o s t a c k i n g f a u l t s o f q u a s i p e r i o d i c packing. ( 2 ) A l t e r n a t i v e l y , t h e one-dimensional c r y s t a l o f t h e decagonal phase can be c o n s i d e r e d as l i n e a r c h a i n s , o r d e r e d p e r p e n d i c u l a r t o t h e c h a i n s a c c o r d i n g t o t h e Penrose t i l i n g o r a n o t h e r t y p e o f p l a n a r q u a s i p e r i o d i c i t y p r e s e r v i n g t e n - f o l d symmetry.

M o d u l a t i o n o f t h e c h a i n s i n a n u n c o r r e l a t e d way ( s o f t e n i n g i n p l a n e ) can g i v e r i s e t o t h e p l a n a r d i f f u s e s c a t t e r i n g .

ACKNOWLEDGEMENTS

T h i s work was supported by t h e Defense Advanced Research P r o j e c t s Agency. T h e i r s u p p o r t i s g r e a t f u l l y acknowledged. I t h a n k J. W. Cahn, D. G r a t i a s , M. J. Kaufman, and R. J. Schaefer f o r h e l p f u l d i s c u s s i o n s , and F. S. B i a n c a n i e l l o and D. C a r r i c k f o r t h e a l l o y and specimen p r e p a r a t i o n .

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