THE STRUCTURE OF ICOSAHEDRAL MnAlSi AND MnAl

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THE STRUCTURE OF ICOSAHEDRAL MnAlSi AND MnAl

E. Stern, Y. Ma, K. Bauer, C. Bouldin

To cite this version:

E. Stern, Y. Ma, K. Bauer, C. Bouldin. THE STRUCTURE OF ICOSAHEDRAL MnAlSi AND MnAl. Journal de Physique Colloques, 1986, 47 (C3), pp.C3-371-C3-377. �10.1051/jphyscol:1986338�.

�jpa-00225750�

THE STRUCTURE OF ICOSAHEDRAL MnAlSi AND

MnAl

E.A. STERN,

Y.

MA,

K.

BAUER and C.E. BOULDIN*

Dept. of Physics, FM-15, University of Washington, Seattle, WA 98195, U.S.A.

' ~ a t i o n a l Bureau

of

Standards, Gaithersburg, MD 20899, U.S.A.

A b s t r a c t - Extended x-ray a b s o r p t i o n f i n e s t r u c t u r e measurements show t h a t i n i c o s a h e d r a l ( i ) (MnAlSi) and (MnA1) t h e r e i s a s t r u c t u r a l u n i t c o n s i s t i n g of a cage of Mn atoms o n t h z v e r t i c e s of a n i c o s a h e d r o n w i t h a n a v e r a g e d i s t a n c e between Mn atoms of 5 A. The i c o s a h e d r a a r e connected w i t h a n a v e r a g e of 3.4

+

1 . 0 n e i g h b o r i n g i c o s a h e d r a . It i s s u g g e s t e d t h a t t h e i - p h a s e grows by randomly n u c l e a t i n g t o g e t h e r Mn i c o s a h e d r a a l o n g t h e i r 2 0 t h r e e f o l d d i r e c - t i o n s a s a l l o w e d by s t e r i c c o n s t r a i n t s , b u t m a i n t a i n i n g o r i e n t a t i o n . The d i f f r a c t i o n p a t t e r n o f a computer s i m u l a t i o n of s u c h a model, c a l l e d a randomly connected i c o s a h e d r a (RCI) model, i s c a l c u l a t e d and compared w i t h experiment. E x c e l l e n t agreement o c c u r s i n t h e peaks when t h e c e n t e r - t o - c e n t e r d i s t a n c e between i c o s a h e d r a i s 10.96

1

f o r i (MnA1). The d i f f r a c t i o n w i d t h s f o r t h e RCI model a r e f i n i t e and v a r y from peak t o peak, i n d i c a t i n g a n e s s e n t i a l d i f f e r e n c e from models o b t a i n e d by p r o j e c t i o n from h i g h e r dimen- s i o n a l s p a c e . The H e n d r i c k s - T e l l e r model c a l c u l a t i o n i s found t o b e a r e a s o n a b l e a p p r o x i m a t i o n t o t h e RCI d i f f r a c t i o n p a t t e r n . It i s a r g u e d t h a t t h e RCI model of a q u a s i c r y s t a l i s f a v o r e d o v e r ones which p r o j e c t from h i g h e r dimens i o n .

I

-

INTRODUCTION

S i n c e t h e d i s c o v e r y of t h e i c o s a h e d r a l p h a s e ( i - p h a s e ) of MnAl by Shechtman a t a l .

[I],

t h e r e h a s b e e n a n i n t e n s e e f f o r t t o d e t e r m i n e t h e a t o m i c arrangement t h a t can produce s u c 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 symmetry y e t s h a r p d i f f r a c t i o n p e a k s . The models proposed c a n b e c l a s s i f i e d i n t o two c a t e g o r i e s . One i s b a s e d on P e n r o s e a p e r i o d i c l a t t i c e s which can b e c o n s t r u c t e d by p r o j e c t i o n from a s i x - d i m e n s i o n a l s p a c e [2-91. The o t h e r i s a random c o n n e c t i o n o f i c o s a h e d r a (RCI) [ l o ] .

I n t h i s p a p e r we p r e s e n t extended x-ray a b s o r p t i o n f i n e s t r u c t u r e (FXAFS) r e s u l t s which c l e a r l y i n d i c a t e f o r i-(MnAlSi) a s t r u c t u r a l u n i t of a c a g e of Mn atoms a t t h e v e r t i c e s of a n i c o s a h e d r o n [Ma, S t e r n , and Bouldin, u n p u b l i s h e d ] . T h i s r e s u l t and t h o s e from o t h e r e x p e r i m e n t s s u g g e s t a n RCI model w i t h c o n n e c t i o n s a l o n g t h r e e f o l d i c o s a h e d r a d i r e c t i o n s s o t h a t a l l i c o s a h e d r a a r e o r i e n t e d t h e same. A c a l c u l a t i o n of t h e d i f f r a c t i o n p a t t e r n e x p e c t e d f o r o u r RCI model i s p r e s e n t e d , and i t i s p o i n t e d o u t t h a t l i n e s h a p e s can b e used t o d i s t i n g u i s h between o u r RCI and p r o j e c - t i o n models. The e x p e r i m e n t a l r e s u l t s f a v o r o u r RCI model.

I1 - EXAFS RESULTS

We u s e EXAFS [ l l ] measurements of t h e Mn K-edge t o compare t h e s h o r t - r a n g e s t r u c t u r e of t h e i - p h a s e w i t h t h a t of t h e a-phase. EXAFS measures t h e r a d i a l d i s t r i b u t i o n a b o u t Mn atoms, n o t o n l y f o r t h e f i r s t A1 s h e l l b u t a l s o , a s we s h a l l show i n t h i s c a s e , f o r Mn s h e l l s up t o 5

A

d i s t a n t . We r e p o r t EXAFS measurements of i c o s a h e d r a l Alg4.6Mnl5.4, A17gMnl5.4Si5.6, and t h e A17gMnl5.4Si5.6 sample a n n e a l e d f o r 44 h o u r s a t 500°C i n vacuum. A powder x-ray d i f f r a c t i o n s c a n of t h e a n n e a l e d sample i n d i -

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986338

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cated that it was in the a-phase. We also measured orthorhombic Al6Mn to use as another standard. All the icosahedral samples were made by the melt spun method.

EXAFS measurements were made at the NSLS beamline X-11 in transmission using a Si(ll1) double crystal monochromator. Measurements were made at room temprature and at 12 K. We report here only results for 12 K; the temperature dependence will be given elsewhere.

Using standard analysis techniques, we obtain the magnitude of the Fourier transforms of k3X (k) of these samples and that of i(MnA1) as shown in Fig. 1. Care was taken to transform all of the samples over the same k-range, 2.311.7 51-' , so that the magnitude of their peaks can be directly compared. Not shown in Fig. 1 are the results obtained on a sample formed by annealing the i-phase at

550°C

for one hour. The magnitude of the transform of this sample has peaks between that of the i-phase and the 44-hour annealed sample, indicating a monotonic change in peak height with annealing. The two MnAlSi phases overlap their first three distinct peaks, but what is particularly striking is the peculiar sequence of the relative heights of these peaks. In the icosahedral phase the first and second peaks have much larger reductions than the third. This is contrary to the standard situation, in which the first shell is the one least reduced and the more distant ones are progressively reduced by usual reduction mechanisms such as thermal and structural disorder. From the k-dependence of their amplitudes and phases we determine that the first peak consists of only (Al,Si) atoms and contains the same number of atoms in all phases within 10%.

It

is not possible to distinguish between A1 and Si atoms in our analysis because their atomic numbers are so close. The first shell around Mn in orthorhombic [12] MnAl6 is used as the Mn-Al standard, while the a-phase [13]

is used as the standard for the second and third peaks.

Analysis of the k-dependence of the back transform of the second and third shells shows that they consist mainly of Mn atoms. From the known structure [13]

of a(MnAlSi), its second shell corresponds to five Mn atoms at an average distance

Fig. 1 - Magnitude of the transform of k3X (k) over the

range 2.3 A-1

<

k

<

11.7 . The a-phase (long dash)

and i-phase (solid) of MnAlSi and the i-phase (short

dash) of MnAl are shown.

d e v i a t i o n of 0.03

A.

To i n t e r p r e t t h e s e d i s t a n c e s i t i s u s e f u l t o summarize t h e s t r u c t u r e of

a(MnA1Si). Neglecting t h e (A1,Si) atoms, t h e Mn atoms form a s k e l e t o n c o n s i s t i n g of two types of somewhat i r r e g u l a r i c o s a h e d r a w i t h t h e Mn atoms a t t h e 12 v e r t i c e s of t h e i c o s a h e d r a . One t y p e of icosahedron is l o c a t e d a t t h e c o r n e r s of a cube and t h e o t h e r i s a t t h e c e n t e r of t h e cube. The i c o s a h e d r a a r e a l l o r i e n t e d t h e same and a r e connected t o e i g h t neighbors of t h e o p p o s i t e t y p e d i r e c t e d a l o n g t h e t h r e e f o l d c u b i c axes which a r e a l o n g t h e (111) d i r e c t i o n s of t h e cube. The neighboring icosa- hedra do n o t s h a r e v e r t i c e s b u t a r e d i s p l a c e d from one a n o t h e r w i t h t h e i r t h r e e f o l d f a c e s p a r a l l e l and forming two o p p o s i t e f a c e s of a d i s t o r t e d octahedron.

The f i v e Mn atoms a t 5.04

i

a r e on t h e n e a r e s t v e r t i c e s of t h e same Pln i c o s a - hedron a s t h e c e n t e r Mn atom and averaged over t h e two t y p e s of i c o s a h e d r a . Four of t h e o t h e r f i v e Mn atoms a r e on neighboring Mn i c o s a h e d r a of t h e o p p o s i t e t y p e through t h e connecting octahedron bonds, w h i l e t h e f i f t h i s on a neighboring Mn icosahedron of t h e same t y p e which h a s no d i r e c t connection w i t h t h e c e n t r a l icosa- hedron. Thus, i n i(MnA1Si) t h e peak a t 4.5

A

i n t h e t r a n s f o r m (5.04

1

t r u e

d i s t a n c e ) i s a measure of t h e r i g i d i t y of t h e Mn icosahedron cage, whereas t h e peak a t 4.0

1

( t r u e d i s t a n c e 4.46 A) is a measure of t h e r i g i d i t y of t h e connection between t h e Mn i c o s a h e d r a .

Q u a l i t a t i v e l y , i t i s c l e a r from F i g . 1 t h a t t h e Mn i c o s a h e d r a remain t h e same i n t h e i c o s a h e d r a l phase w h i l e t h e connections between them a r e changed. A q u a n t i - t a t i v e measure of t h e changes can b e o b t a i n e d by i s o l a t i n g each peak and backtrans- forming t o k-space. The I n - r a t i o of amplitudes and d i f f e r e n c e of phases of each peak a r e taken between t h e i c o s a h e d r a l and a-phases. The r e s u l t f o r t h e t h i r d peak i s t h a t t h e i c o s a h e d r a l phase r e t a i n s 5 f 0.25 Mn neighbors a t t h e same average d i s t a n c e w i t h i n t h e experimental u n c e r t a i n t y of k0.02

1

and i s v e r y s l i g h t l y more d i s o r d e r e d by A U ~ = 0.0006

k2.

The second peak i n d i c a t e s t h a t t h e i c o s a h e d r a l phase has a s m a l l e r number of neighboring i c o s a h e d r a t h a n t h e a-phase.

These r e s u l t s i n d i c a t e t h a t t h e Mn i c o s a h e d r a remain p r a c t i c a l l y unchanged i n t h e i-phase, b u t t h e i r connections a r e modified. The f a c t t h a t t h e c o o r d i n a t i o n f o r t h e t h i r d peak remains a t 5 and t h e sample peaks monotonically t r a n s f o r m t o t h a t of t h e a-phase w i t h a n n e a l i n g s u g g e s t s t h a t t h e Mn i c o s a h e d r a i n t h e i - p h a s e s t i l l do not touch one a n o t h e r ( i . e . , do n o t s h a r e v e r t i c e s ) because touching would i n c r e a s e t h e c o o r d i n a t i o n number. The c o o r d i n a t i o n of t h e second peak i s c o n s i s t e n t w i t h a d e c r e a s e i n t h e number of neighboring i c o s a h e d r a from 8 i n t h e a-phase t o 3 . 4 f 1 . 0 i n t h e i-phase w i t h only s m a l l changes i n d i s t a n c e .

The t r a n s f o r m of i(A1Mn) i n Fig. 1 shows only two of t h e f i r s t t h r e e d i s t i n c t peaks i n t h e (MnAlSi) phases and a t somewhat s h o r t e r d i s t a n c e s . The t h i r d peak i n d i c a t e s t h e presence of 4 . 8 2 0.5 Mn a t an average d i s t a n c e of 4.95 -+ 0.05 w i t h an i n c r e a s e d d i s o r d e r of A U ~ = 0.003

A2

r e l a t i v e t o t h e a-phase. This r e s u l t i s c o n s i s t e n t w i t h a cage of Mn atoms a t t h e v e r t i c e s of an icosahedron being a l s o t h e s t r u c t u r a l u n i t i n i(MnA1). The f i r s t peak c o n s i s t s of only A 1 atoms. The peak t h a t i s missing corresponds t o t h e Mn atoms connecting t h e Mn i c o s a h e d r a . From t h i s we conclude t h a t t h e i(MnA1) phase h a s i t s i c o s a h e d r a connected d i f f e r e n t l y , b u t because t h e second peak i s l o s t i n t h e background we cannot q u a n t i f y how t h i s change occurs. Growth of t h e v a r i o u s phases from t h e melt a l o n g t h r e e f o l d a x e s [15; D.C.

Koskenmaki, H.S. Chen, and K.V. Rao, unpublished] s u g g e s t s t h a t t h e Mn i c o s a h e d r a a r e connected along t h r e e f o l d axes i n t h e two i-phases a s t h e y a r e i n t h e a-phase.

I n summary, a n a l y s e s of EXAFS measurements on i(MnAlSi), i(MnA1), and a(MnA1Si) g i v e s t r o n g experimental evidence t h a t s t r u c t u r a l u n i t s i n t h e form of a cage of Mn atoms a t t h e v e r t i c e s of s l i g h t l y nonre u l a r i c o s a h e d r a e x i s t i n a l l t h r e e w i t h an average Mn-Mn n e a r e s t d i s t a n c e of 5.04

%

I n t h e two (MnAlSi) phases and 4.95

A

i n i(MnA1). The d i f f e r e n c e s between t h e t h r e e a r e i n t h e i n t e r c o n n e c t i o n s of t h e Mn-icosahedra and t h e i r connections t o t h e A 1 atoms. The Mn cage a s a s t r u c t u r a l

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u n i t i s c o n s i s t e n t w i t h models s u g g e s t e d by E l s e r and Henley [ 6 ] and Guyot and Audier [ 9 ]

.

The number o f randomly connected n e i g h b o r i n g i c o s a h e d r a must v a r y from a n a l l o w e d maximum of 8. The f a c t t h a t t h e a v e r a g e number i s 3 . 4 i n d i c a t e s q u i t e l a r g e f l u c t u a t i o n s i n t h e number o f n e i g h b o r i n g i c o s a h e d r a . Our model i s t h u s one i n which t h e i-phase grows by randomly n u c l e a t i n g t o g e t h e r Mn i c o s a h e d r a a l o n g any of t h e i r 20 t h r e e f o l d d i r e c t i o n s a s allowed by l o c a l s t e r i c c o n s t r a i n t s of n o t over- l a p p i n g . The i c o s a h e d r a remain o r i e n t e d . T h i s model i s a m o d i f i c a t i o n o f t h e o r i g i n a l one by Shechtman and B l e c h [ l o ] , which h a s been f u r t h e r developed by Stephens and Goldman ( I )

111 - DIFFRACTION PATTERN OF RCI MODEL

The model was computer g e n e r a t e d by s t a r t i n g w i t h a n i c o s a h e d r o n a t t h e c e n t e r and a t t a c h i n g i c o s a h e d r a o n e by one a l o n g t h e t h r e e f o l d d i r e c t i o n s randomly. The added i c o s a h e d r a c o u l d a t t a c h t o t h e t h r e e f o l d d i r e c t i o n s of any of t h e i c o s a h e d r a p r e s e n t w i t h e q u a l p r o b a b i l i t y . An a t t a c h m e n t would remain i f i t s a t i s f i e d t h e s t e r i c c o n s t r a i n t t h a t i t d i d n o t o v e r l a p any o t h e r i c o s a h e d r a . I f i t d i d o v e r l a p , i t would b e removed and a new a t t a c h m e n t a t t e m p t e d randomly. The c e n t e r - t o - c e n t e r d i s t a n c e of any a t t a c h e d i c o s a h e d r a was f i x e d a t u n i t y , w h i l e t h e dimension o f t h e i c o s a h e d r a a l o n g a t h r e e f o l d d i r e c t i o n ( f a c e - t o - f a c e l e n g t h ) was a l s o s e t e q u a l t o u n i t y .

By t h i s means, c l u s t e r s o f 1 , 0 0 0 , 2,000, 3,000, and 4,000 i c o s a h e d r a were assembled. The d i f f r a c t i o n p a t t e r n o f t h e c e n t e r s of t h e s e c l u s t e r s were c a l c u l a t e d n u m e r i c a l l y by

where r j a r e t h e l o c a t i o n s o f t h e i c o s a h e d r a c e n t e r s .

+

The r e s u l t f o r a c l u s t e r of 1,000 i c o s a h e d r a i s shown i n F i g . 2 f o r

2

a l o n g a twofold a x i s . To improve t h e s t a t i s t i c s , t h e p a t t e r n i s a v e r a g e d o v e r t h e 1 5 twofold d i r e c t i o n s .

+

-+

F i g . 2. P l o t of I ( k ) v e r s u s k a l o n g t h e twofold a x i s f o r a computer s i m u l a t i o n o f a 1 , 0 0 0 c l u s t e r .

( I ) By P.W. Stephens and A . I . Goldman ( u n p u b l i s h e d ) .

t h e a v e r a g e l o c a t i o n of atoms about e a c h c e n t e r , i - e . , t h e s t r u c t u r e f a c t o r . The s t r u c t u r e f a c t o r depends on t h e d e t a i l s o f t h e t y p e s of atoms and t h e i r r e l a t i v e l o c a t i o n s . I n t h i s p a p e r o u r c o n c e r n i s t o d i s t i n g u i s h between t h e p r o j e c t i o n and RCI models and t o d e t e r m i n e which o n e b e s t f i t s t h e e x p e r i m e n t a l r e s u l t s . The s t r u c t u r e f a c t o r g i v e s s i m i l a r c o n t r i b u t i o n s t o b o t h models and cannot b e used t o d i s t i n g u i s h between them. The d i s t i n c t i o n o c c u r s i n t h e l i n e s h a p e s . The l o c a t i o n of t h e d i f f r a c t i o n l i n e s and t h e i r s h a p e s a r e d e t e r m i n e d by t h e p o s i t i o n of t h e i c o s a h e d r a c e n t e r s a l o n e , and we c o n c e n t r a t e on t h a t p o r t i o n of t h e problem h e r e . The i n t e n s i t y of t h e l i n e s h a p e s c a n t h e n b e c a l c u l a t e d by m u l t i p l y i n g by t h e s t r u c - t u r e f a c t o r .

It h a s been s u g g e s t e d ^(l) t h a t t h e H e n d r i c k s - T e l l e r (HT) model [15] c a n b e used t o c a l c u l a t e peak p o s i t i o n s and w i d t h s . The HT model n e g l e c t s a l l s h o r t - r a n g e c o r r e l a t i o n s between t h e i c o s a h e d r a . There i s g e n e r a l l y some s h o r t - r a n g e c o r r e l a - t i o n p r e s e n t , and i t i s e x p e c t e d t h a t t h e HT model w i l l t h e r e f o r e o v e r e s t i m a t e t h e l i n e w i d t h b e c a u s e i t u n d e r e s t i m a t e s t h e c o r r e l a t i o n . An HT c a l c u l a t i o n was made f o r o u r RCI model. The peak p o s i t i o n s a g r e e w i t h t h e computer s i m u l a t i o n o n e s , b u t t h e HT l i n e w i d t h s were t o o l a r g e f o r t h e w i d e s t peaks. However, t h e y g i v e good agreement f o r t h e narrower peaks and a r e t h e o n l y r e l i a b l e means of e x p e c t e d l i n e w i d t h s f o r t h e v e r y n a r r o w e s t p e a k s , where f i n i t e - s i z e e f f e c t s broaden t h e computer

s i m u l a t i o n v a l u e s .

Table 1 shows t h e summary of peak p o s i t i o n s and w i d t h s c a l c u l a t e d by t h e computer s i m u l a t i o n f o r a c l u s t e r o f 3 , 0 0 0 and t h e HT model, which is f o r a n i n f i n i t e c l u s t e r . The e x p e r i m e n t a l v a l u e s [ 1 6 ] a r e a l s o l i s t e d f o r i(MnA1). A c r u d e c o r r e c t i o n f o r f i n i t e - s i z e e f f e c t s f o r t h e 3,000 c l u s t e r was made by numeri- c a l l y p l o t t i n g t h e number d e n s i t y o f t h e 3,000 c l u s t e r a s a f u n c t i o n of d i s t a n c e from t h e c e n t e r and t h e n F o u r i e r t r a n s f o r m i n g t h e smoothed d e n s i t y c u r v e . A convo- l u t i o n of t h i s F o u r i e r t r a n s f o r m w i t h t h e i n f i n i t e c l u s t e r g i v e s t h e f i n i t e c l u s t e r c a s e . The h a l f - w i d t h half-maximum (HWHM) o f t h e F o u r i e r t r a n s f o r m of t h e smoothed d e n s i t y f u n c t i o n was s u b t r a c t e d i n q u a d r a t u r e from t h e l i n e w i d t h s o f t h e 3,000 c l u s t e r t o o b t a i n a n e s t i m a t e of t h e i n f i n i t e c l u s t e r l i n e HWHM, a s l i s t e d i n T a b l e 1.

T a b l e 1. E x p e r i m e n t a l and t h e o r e t i c a l d i f f r a c t i o n peaks f o r i(MnA1). The e x p e r i m e n t a l p e a k s and t h e i r i n d e x i n g f o l l o w Ref.

16. Column 1 g i v e s t h e i n d e x i n g of t h e peak; column 2 g i v e s t h e e x p e r i m e n t a l v a l u e s of peak wave number; column 3 g i v e s t h e c a l - c u l a t e d v a l u e s o f peaks i n u n i t s of k a , where a = 10.96

1

i s t h e c e n t e r i c o s a h e d r a s p a c i n g . The c a l c u l a t e d peak HWHM i s g i v e n i n column 4 f o r t h e computer s i m u l a t i o n and i n column 5 f o r t h e HT model. The e x p e r i m e n t a l HWHM is i n column 6.

1 2 3 4 5 6

k

(1-l)

ka HWHM

2')

Peak Experiment C a l c u l a t e d S i m u l a t e d HT Experiment

cL)

By P.W. Stephens and A . I . Goldman [ u n p u b l i s h e d ] .

JOURNAL

DE

PHYSIQUE

I V

-

DISCUSSION AND SUMMARY

As T a b l e 1 i n d i c a t e s , t h e RCI peaks a l l h a v e f i n i t e w i d t h s which a r e a l l l e s s t h a n t h e measured v a l u e s . There a p p e a r s t o be a n o t h e r b r o a d e n i n g mechanism i n v o l v e d b e s i d e s t h e randomness of c o n n e c t i o n s a s assumed i n t h e RCI model. The peak p o s i t i o n s a g r e e w i t h t h e e x p e r i m e n t a l v a l u e s f o r a c e n t e r - t o - c e n t e r i c o s a h e d r a d i s t a n c e of 10.96

A.

Mention s h o u l d be made of some of t h e peaks n o t l i s t e d i n Table 1, which a r e weak and b r o a d i n b o t h t h e computer s i m u l a t i o n and HT c a l c u l a - t i o n s , s u c h a s t h e o n e s l a b e l e d (110001) and (111010) i n R e f . 1 5 . I n t h e computer s i m u l a t i o n t h e b e h a v i o r o f t h e s e peaks v a r i e d i n q u i t e a n anomalous manner a s t h e c l u s t e r s i z e i n c r e a s e d from N = 1 , 0 0 0 t o N = 4 , 0 0 0 . The i n t e n s i t y o f t h e peaks i n c r e a s e d more s l o w l y t h a n N, and t h e i r s h a p e s changed from a n a p p r o x i m a t e l y symmet- r i c s i n g l e peak t o a n asymmetric, even double-humped, one. The HT c a l c u l a t i o n s h o w e d a b r o a d b u t s i n g l e - p e a k e d s h a p e f o r t h e s e l i n e s . We d o n o t u n d e r s t a n d t h e N depen- dence of t h e s e p e a k s and p l a n t o s t u d y them f u r t h e r .

The e x p e r i m e n t a l l y measured f i n i t e l i n e w i d t h h a s a n a t u r a l e x p l a n a t i o n i n t h e RCI model, w h i l e t h e p r o j e c t i o n method r e q u i r e s a n a d hoc d e f o r m a t i o n and growth f a u l t i n g . However, t h e RCI model w i d t h s do n o t a g r e e w i t h t h e m e a s u r e m e n t s , s u g g e s t - i n g t h e need f o r a n o t h e r l i n e b r o a d e n i n g mechanism i n t h e e x p e r i m e n t s . The R C I model i s p h y s i c a l l y more p l e a s i n g b e c a u s e one e x p e c t s n u c l e a t i o n and growth t o proceed a s i n t h e RCI model. F u r t h e r e x p e r i m e n t a l measurements o f l i n e w i d t h s on i-phase samples w i t h l e s s s t r a i n a r e r e q u i r e d b e f o r e d e f i n i t e c o n c l u s i o n s can b e drawn a s t o which model i s c o r r e c t . A complete d e t a i l e d d e t e r m i n a t i o n o f t h e a t o m i c p o s i t i o n s a w a i t s a r e f i n e m e n t o f t h e s t r u c t u r e f a c t o r t o match e x p e r i m e n t a l i n t e n - sities. The c a l c u l a t i o n p r e s e n t e d h e r e shows t h a t i c o s a h e d r a l u n i t s i n t e r c o n n e c t e d a l o n g t h e t h r e e f o l d a x e s i n a random f a s h i o n w i t h a n a v e r a g e c e n t e r - t o - c e n t e r

s p a c i n g o f 10.96

1

i s c o n s i s t e n t w i t h t h e i c o s a h e d r a l d i f f r a c t i o n p a t t e r n . A s shown by EXAFS, t h e i c o s a h e d r a l u n i t s c d n t a i n a Mn c a g e on t h e v e r t i c e s of a n i c o s a h e d r o n w i t h a n a v e r a g e s e p a r a t i o n of 5

1

between Mn atoms.

There i s a n a p p a r e n t paradox i n f i n i t e l i n e w i d t h s f o r t h e RCI model. The l o c a t i o n of t h e c e n t e r s of e a c h i c o s a h e d r o n c a n b e r e p r e s e n t e d a s a l i n e a r combina- t i o n of 1 0 u n i t v e c t o r s a l o n g t h e 1 0 t h r e e f o l d i c o s a h e d r a l d i r e c t i o n s . These 1 0 u n i t v e c t o r s can b e o b t a i n e d by a p r o j e c t i o n from a 10-dimensional s p a c e down t o t h r e e dimensions. Thus, t h e c e n t e r s o f t h e i c o s a h e d r a c a n b e r e p r e s e n t e d a s p o i n t s on a s i m p l e c u b i c l a t t i c e i n 1 0 dimensions. I f t h e s e p o i n t s a r e randomly occupied by i c o s a h e d r a , t h e n t h e e x p e c t e d d i f f r a c t i o n p e a k s a r e s h a r p and w i l l t h e r e f o r e p r o j e c t a s s h a r p i n t h r e e dimensions.

The paradox c a n b e r e s o l v e d by r e a l i z i n g t h a t t h e p o i n t s a r e n o t occupied randomly. About e a c h p o i n t a r e 20 n e a r e s t n e i g h b o r s , b u t o n l y a s m a l l f r a c t i o n (of t h e o r d e r o f 4 ) a r e occupied b e c a u s e o f s t e r i c c o n s t r a i n t s . The f i n i t e w i d t h s can b e u n d e r s t o o d by a two-dimensional a n a l o g y . Consider a t r i a n g u l a r l a t t i c e o c c u p i e d by l a r g e atoms s o t h a t s t e r i c c o n s t r a i n t s f o r c e t h e o c c u p a t i o n of o n l y o n e - t h i r d of t h e sites l o c a l l y , a s i s t h e c a s e f o r Kr g a s commensurately adsorbed on g r a p h i t e . The t r i a n g u l a r l a t t i c e c a n be s u b d i v i d e d i n t o t h r e e e q u i v a l e n t s u b l a t t i c e s A, B , and C. Sharp d i f f r a c t i o n peaks o c c u r when t h e o c c u p i e d f r a c t i o n o f s i t e s a r e a l l of t y p e A. However, i f t h e sites a r e occupied i n domains where one t y p e of s i t e - - A, B , o r C--is occupied i n each domain, t h e n t h e d i f f r a c t i o n peaks w i l l have a f i n i t e w i d t h c o r r e s p o n d i n g t o t h e a v e r a g e s i z e o f domains.

The RCI model e x p l a i n s i n a s i m p l e p h y s i c a l manner t h e s t a b i l i t y o f t h e i - p h a s e d u r i n g r a p i d c o o l i n g . During r a p i d c o o l i n g t h e growth i s dominated by l o c a l o r d e r which f a v o r s c o n n e c t i o n s a l o n g t h e t h r e e f o l d d i r e c t i o n s . S i n c e t h e r e a r e 20 s u c h t h r e e f o l d f a c e s and l o c a l c o n s t r a i n t s l i m i t t h e number of c o n n e c t i o n s t o , a t most, 8 and s i g n i f i c a n t l y fewer when t h e c o n n e c t i o n s a r e made randomly, t h e r e i s a n over- whelming p r o b a b i l i t y t h a t t h e i c o s a h e d r a l p h a s e w i l l b e produced. To produce t h e o r d e r e d a-phase r e q u i r e s t h e c o n n e c t i o n t o o n l y t h e 8 f a c e s a l o n g t h e (111) d i r e c - t i o n s f o r a l l i c o s a h e d r a . T h i s c o h e r e n c e r e q u i r e s t h e o p e r a t i o n o f s u b t l e l o n g - r a n g e f o r c e s which do n o t have t i m e t o make t h e i r i n f l u e n c e f e l t d u r i n g r a p i d c o o l i n g . Only a n n e a l i n g f o r l o n g t i m e s p e r m i t s t h e c o o p e r a t i v e r e a r r a n g e m e n t of i c o s a h e d r a l c o n n e c t i o n s r e q u i r e d f o r t h e long-range o r d e r i n t o t h e c r y s t a l l i n e a-phase.

i-phase, we favor it over the projection model. If future experiments confirm the RCI model as the correct one, its ascendancy over the projection model can be called a triumph of physics over mathematics.

The samples used in this experiment were kindly supplied by D. Shechtman, R.J. Schaefer, and F.S. Biancaniello of the National Bureau of Standards. We are most appreciative of discussions with J.W. Cahn. Measurements were made on heamline X-11 at the National Synchrotron Light Source, Brookhaven National Laboratory, which is supported by the U.S. Department of Energy, Division of Materials Sciences and Division of Chemical Sciences. The research was supported partially by the U.S.

Department of Energy under Contract DE-A505-80-ER10742.

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