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COMPUTER SIMULATION OF FIELD-ION IMAGES OF QUASICRYSTALS
A. Sakai, J. Hashimoto, T. Sakurai, K. Osawa, T. Ninomiya
To cite this version:
A. Sakai, J. Hashimoto, T. Sakurai, K. Osawa, T. Ninomiya. COMPUTER SIMULATION OF
FIELD-ION IMAGES OF QUASICRYSTALS. Journal de Physique Colloques, 1986, 47 (C7), pp.C7-
469-C7-475. �10.1051/jphyscol:1986779�. �jpa-00225974�
JOURNAL DE PHYSIQUE
Colloque C7, suppl6ment au n o 11, Tome 47, Novembre 1986
COMPUTER SIMULATION OF FIELD-ION IMAGES OF QUASICRYSTALS
A. SAKAI, J. HASHIMOTO, T. SAKURAI, K. OSAWA* and T. NINOMIYA*
The Institute for Solid State Physics, The University o f Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106, Japan
Department o f Physics, The University o f Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan
Abstract
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Field-ion (FI) images of the A1-Mn quasicrystals were calculated using the three-dimensional Penrose tiling. The resulting images have a perfect icosahedral symmetry and show ring structures around some symmetry poles. FI images of two structural models were obtained and compared with the observed FI images of the A1-Mn quasicrystals.I
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INTRODUCTIONSince the first discovery of the quasicrystalline phase of the quenched A1-Mn alloys in 1984 /I/, there have been a large number of experimental and theoretical studies on the structure of these newly found materials which have no translational symmetry but show discrete diffraction patterns of fivefold rotational symmetry. Based on the detailed electrondiffraction experiments on the A1-Mn quasicrystals /1,2,3/, it now seems to be well established that the symmetry of the A1-Mn quasicrystals is consistent with an icosahedral orientational
symmetry. Once the spatial symmetry is specified, the next step is to determine the unit cell and its atomic structure. At present, many answers are proposed to solve this problem, but none of them is
satisfactory to explain the observed electron diffraction intensities.
A direct atomic-scale imaging of the A1-Mn quasicrystals is thus crucially important for the structural studies of this new alloy phase. High resolution electron microscopy has been used by some investigators /2,3/, and they have succeeded to demonstrate the lack of translational symmetry and the existence of pentagonal patterns and to estimate the size of the unit cell. However, the resolution is not sufficient to fix the atomic positions within the unit cell(s).
Recently Melmed and Klein /4/ have studied the real-space
structure of the quasicrystalline phase of the A1-Mn alloys using the field-lon microscope (FIM) which is principally capable to resolve individual atoms of the specimen /5/. In the case of alloys, however, the correspondence between the field-ion (FI) images and the real atomic structure is not necessarily straightforward, since only one constituent of alloys are imaged quite frequently in the FIM.
Computer simulation of FI images is known to be useful in such cases, and we here present some calculated FI images of the A1-Mn
quaslcrystals and compare them with the FIM observations.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986779
JOURNAL DE PHYSIQUE
I1
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MODEL STRUCTUREST h e r e a r e two b a s i c r e q u i r e m e n t s f o r t h e s t r u c t u r a l m o d e l s o f q u a s i c r y s t a l s : t h e symmetry a n d t h e u n i t c e l l s t r u c t u r e . C o n c e r n i n g t h e s y m m e t r y , i t seems t o b e g e n e r a l l y a c c e p t e d t h a t t h e o b s e r v e d i c o s a h e d r a l symmetry o f t h e A1-Mn q u a s i c r y s t a l s c a n b e e x p l a i n e d b y t h e s o - c a l l e d t h r e e - d i m e n s i o n a l P e n r o s e t i l i n g / 6 , 7 / w h i c h d e s c r i b e s t h e m a t h e m a t i c a l method t o f i l l t h e e n t i r e s p a c e u s i n g two r h o m b i c u n i t c e l l s . T h e g e n e r a t e d P e n r o s e l a t t i c e h a s n o t r a n s l a t i o n a l s y m m e t r y , b u t t h e c a l c u l a t e d 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 f r o m t h i s q u a s i l a t t i c e a r e f o u n d t o b e i n good a g r e e m e n t w i t h t h e o b s e r v e d d i f f r a c t i o n p a t t e r n s , s h o w i n g t h e c o r r e c t i c o s a h e d r a l symmetry /7,8/
T h e r e a r e , h o w e v e r , u n s e t t l e d a r g u m e n t s a b o u t t h e d i s t r i b u t i o n o f A 1 a n d Mn a t o m s w i t h i n t h e t w o P e n r o s e u n i t c e l l s . Most o f t h e
p r o p o s e d m o d e l s , h o w e v e r , a g r e e t h a t Mn a t o m s o c c u p y t h e c o r n e r p o s i t i o n s o f t h e u n i t c e l l s so t h a t t h e p o s i t i o n s o f A1 a t o m s a r e t h e m a t t e r o f c o n t r o v e r s y . W e e m p l o y e d two m o d e l s /9/ f o r t h e a t o m i c s t r u c t u r e o f t h e P e n r o s e u n i t c e l l s o f t h e A1-Mn q u a s i c r y s t a l s a s shown i n F i g . 1. I n model I , A1 a t o m s a r e l o c a t e d i n t h e m i d d l e o f e a c h e d g e o f t h e u n i t c e l l s , a n d i n m o d e l 11, t h e y a r e o n t h e
f a c e - c e n t e r e d p o s i t i o n s ( f o r s i m p l i c i t y , w e d i d n o t i n c l u d e t h e body c e n t e r e d a t o m s i n t h e a c u t e u n i t c e l l s ) . I t s h o u l d b e n o t e d t h a t t h e s e two m o d e l s may n o t b e t h e c o r r e c t o n e s i n c e t h e y d o n o t g i v e t h e c o r r e c t s t o i c h i o m e t r y a n d f a i l t o r e p r o d u c e e x a c t l y t h e o b s e r v e d e l e c t r o n d i f f r a c t i o n i n t e n s i t i e s /9/. However, t h e s i m u l a t i o n o f t h e F I l m a g e s o f t h e s e m o d e l s w i l l b e u s e f u l t o t e s t t h e s e n s i t i v i t y o f F I i m a g e s t o t h e a t o m i c s t r u c t u r e o f q u a s i c r y s t a l s a n d t o s u p p l e m e n t t h e i n t e r p r e t a t i o n o f t h e e x p e r i m e n t a l l y o b s e r v e d F I i m a g e s .
I11
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METHOD OF CALCULATIONF i r s t , a t h r e e - d i m e n s i o n a l P e n r o s e l a t t i c e was g e n e r a t e d u s i n g t h e g e n e r a l i z e d d u a l method
/ l o / ,
a n d Mn a t o m s were p l a c e d o n e a c h l a t t i c e p o i n t . W e u s e d t h e f o l l o w i n g s e t o f s i x u n i t v e c t o r s f o r t h e P e n r o s e t i l i n gw h e r e t = ( 5 l I 2
+
1 ) / 2 . A 1 a t o m s were d i s t r i b u t e d a c c o r d i n g t o e i t h e r model I cr m o d e l 11.F I i m a g e s were t h e n c a l c u l a t e d u s i n g t h e t h i n s h e l l m e t h o d /11/:
a t h i n spherical s h e l l o f r a d i u s R a n d t h i c k n e s s d i s d r a w n w i t h i t s c e n t r e a t t h e o r i g i n , and a l l t h e a t o m i c p o s i t i o n s w i t h i n t h e s h e l l a r e s t e r e o - p r o j e c t e d o n t o t h e v i r t u a l s c r e e n t o f o r m t h e s i m u l a t e d F I i m a g e s o n i t
I n o u r c a l c u l a t i o n , R a n d d w e r e f i x e d t o 5 2 . 6 a 0 a n d 0 . 1 0 5 a 0 , , r e s p e c t i v e l y , w h e r e a r e p r e s e n t s t h e l e n g t h o f t h e e d g e t h e r h o m b l c u n i t c e l l s . T h i s c o r n g i n a t i o n o f R a n d d i s c l o s e t o t h e b e s t - f i t v a l u e s o b t a i n e d by Moore a n d R a n g a n a t h a n /11/ i n t h e i r s i m u l a t i o n o f t h e F I i m a g e s o f FCC m e t a l s . I f w e t a k e t h e v a l u e a. = 0 . 4 8 nm, t h e n R a n d d become 25.2 nm a n d 0.050 nm r e s p e c t i v e l y ( N o t e t h a t R
c o r r e s p o n d s t o t h e l o c a l r a d i u s o f t h e t i p r a d i u s i n t h e FIM e x p e r i m e n t s )
.
V I
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RESULTS AND DISCUSSIONC a l c u l a t i o n s of F I i m a g e s w e r e c a r r i e d o u t f o r t h e f o l l o w i n g f i v e c a s e s ,
( i ) o n l y Mn ( c o r n e r ) a t o m s a r e i m a g e d , ( i i ) o n l y A1 a t o m s i n model I a r e i m a g e d , ( i i i ) o n l y A1 a t o m s i n model I1 a r e i m a g e d ,
( i v ) b o t h A1 a n d Mn a t o m s i n m o d e l I a r e i m a g e d , ( v ) b o t h A1 a n d Mn a t o m s i n m o d e l I1 a r e imaged.
F i r s t , w e show i n F i g . 2 t h e F I i m a g e s o b t a i n e d f r o m t h e s k e l t o n l a t t i c e ( Mn a t o m s o n l y , c a s e ( i ) ) . The p r o j e c t i o n a x e s u s e d i n F i g . 2 ( a ) - ( c ) a r e t h e f i v e f o l d , t h r e e f o l d , a n d t w o f o l d symmetry a x e s , r e s p e c t i v e l y . A l s o shown i n F i g . 2 a r e t h e c o r r e s p o n d i n g
s t e r e o g r a p h i c p r o j e c t i o n s o f t h e i c o s a h e d r a l s t r u c t u r e t a k e n f r o m R e f . / l 2 / . I t i s e v i d e n t t h a t t h e s i m u l a t e d F I i m a g e s h a v e a c o m p l e t e i c o s a h e d r a l s y m m e t r y , a n d a l l r i n g s t r u c t u r e s i n t h e i m a g e s p e r f e c t l y c o r r e s p o n d t o t h e p o s i t i o n s o f t h e symmetry a x e s i n t h e s t e r e o g r a p h i c p r o j e c t i o n s . T h i s a g r e e m e n t g u a r a n t e e s t h a t t h e g e n e r a t e d P e n r o s e l a t t i c e h a s a c o r r e c t s p a t i a l symmetry.
I n F i g . 3 ( a ) - ( e ) , w e c o m p a r e F I i m a g e s o b t a i n e d f o r c a s e s ( i - v . T h e c e n t e r p o l e i n t h e s e i m a g e s is t h e f i v e f o l d a x i s , p a r a l l e l t o o n e o f u n i t v e c t o r s d e s c r i b e d i n Eq. ( 1 ) .
The m o s t s t r i k i n g f e a t u r e i n t h e i m a g e s o f F i g . 3 [ b ) - ( e ) i s t h e e x i s t e n c e o f n o t i c e a b l e r i n g s t r u c t u r e s a t t h e f i v e f i v e f o l d p o l e s a r o u n d t h e c e n t e r . The t w o f o l d p o l e s w h i c h a r e l o c a t e d j u s t b e t w e e n t w o f i v e f o l d p o l e s a l s o show c l e a r r i n g p a t t e r n s b u t t h e i r e x t e n t i s l e s s marked c o m p a r e d t o t h e r i n g s a t t h e f i v e f o l d p o l e s . I n c o n t r a s t w i t h f h e s e p o l e s , o n l y weak s t r u c t u r e s c a n b e s e e n a t t h e t h r e e f o l d p o l e s w h i c h r a t h e r a p p e a r a s a b l a n k r e g i o n i n t h e s i m u l a t e d F I i m a g e s . S i n c e t h e r i n g p a t t e r n s i n t h e F I i m a g e s a r e f o r m e d b y t h e i m a g e s o f t h e e d g e a t o m s o f e a c h n e t p l a n e , t h e c o n c e n t r i c r i n g p a t t e r n s i n t h e s e i m a g e s c l e a r l y i n d i c a t e t h e e x i s t e n c e o f a s e t o f p a r a l l e l a t o m i c p l a n e s , l o c a l l y a t l e a s t , i n t h e s e q u a s i c r y s t a l l i n e a l l o y s . T h e r e f o r e , t h e s p a t i a l s t r u c t u r e o f t h e s e m o d e l e d
q u a s i c r y s t a l s i s t h u s " c r y s t a l - l i k e " e v e n i n s p i t e o f t h e l a c k o f t r a n s l a t i o n a l s y m m e t r y , a s a l r e a d y i n d i c a t e d b y t h e e l e c t r o n m i c r o s c o p y . T h e q u a s i c r y s t a l l i n e p h a s e is j u s t t h e i n t e r m e d i a t e b e t w e e n t h e c r y s t a l l i n e a n d t h e a m o r p h o u s s t a t e s o f m a t t e r .
N o s i g n i f i c a n t d i f f e r e n c e s c a n b e f o u n d among t h e i m a g e s i n F i g . 3 ( b ) - ( e ) , e x c e p t t h e p o p u l a t i o n o f t h e a t o m i c s i t e s . T h r o u g h t h e s e i m a g e s , t h e m a j o r p o l e s h a v e n e a r l y t h e same c o n t r a s t . The
s i m i l a r l i t y o f t h e s e i m a g e s , p a r t i c u l a r l y t h e c l o s e r e s e m b l a n c e o f F i g . 3 ( b ) a n d ( c ) , s u g g e s t s t h a t t h e g r o s s f e a t u r e s o f t h e F I i m a g e s a r e r a t h e r i n s e n s i t i v e t o t h e p o s i t i o n s o f A 1 a t o m s w i t h i n t h e u n i t c e l l s . I t i s , t h e r e f o r e , d i f f i c u l t t o d e t e r m i n e t h e a t o m i c s t r u c t u r e s o f t h e u n i t c e l l s f r o m t h e c o n t r a s t o f t h e F I i m a g e s . A l t h o u g h t h i s r e s u l t is n o t c o n c l u s i v e s i n c e w e t e s t e d o n l y two u n i t c e l l
s t r u c t u r e s , i t may b e l i k e l y t h a t a h i g h r e s o l u t i o n FIM o b s e r v a t i o n w i t h s i n g l e a t o m r e s o l u t i o n i s t o b e r e q u i r e d t o d e t e r m i n e a t o m i c p o s i t i o n s w i t h i n t h e u n i t c e l l s .
I n F i g . 3 ( f ) , we i n c l u d e t h e e x p e r i m e n t a l F I m i c r o g r a p h o f A1-Mn q u a s i c r y s t a l o b t a i n e d b y Melmed a n d K l e i n /4/. T h i s o b s e r v e d i m a g e s h o w s f i v e f i v e f o l d a n d f i v e t w o f o l d p o l e s w i t h n e a r l y e q u a l c o n t r a s t o n them. On t h e c o n t r a r y , t h e b i g r i n g p a t t e r n s i n t h e s i m u l a t e d F I
i m a g e s i n F i g . 3 ( b ) - ( e ) a p p e a r e x c l u s i v e l y a t t h e f i v e f o l d p o l e s , a s d e s c r i b e d b e f o r e . A l s o , t h e s t r u c t u r e s d u e t o t h e t w o f o l d a n d
t h r e e f o l d p o l e s n e a r t h e c e n t e r a r e h a r d l y s e e n i n t h e o b s e r v e d F I i m a g e s . S u c h d i f f e r e n c e s b e t w e e n t h e e x p e r i m e n t a l a n d c a l c u l a t e d F I i m a g e s may b e p a r t l y d u e t o t h e s m a l l s i z e o f t h e t i p r a d i u s ( R ) i n t h e s i m u l a t i o n s i n c e t h e s t e e p c u r v a t u r e o f t h e s p h e r i c a l s h e l l t e n d s t o a m p l i f y t h e c o n t r a s t o f some m a j o r p o l e s . However, w e a l s o p o i n t
C7-472 JOURNAL DE PHYSIQUE
o u t t h a t t h e o b s e r v e d F I i m a g e shown i n F i g . 3 ( f ) h a s much s i m i l a r i t y t o t h e s i m u l a t e d i m a g e o f Mn l a t t i c e i n F i g . 3 ( a ) w h i c h show t e n p o l e s a r o u n d t h e c e n t e r w i t h n e a r l y e q u a l c o n t r a s t . I t is t h u s s u g g e s t e d t h a t o n l y Mn atoms a r e imaged i n t h e a c t u a l FIM
o b s e r v a t i o n s . T h i s s e l e c t i v e v i s i b i l i t y o f Mn a t o m s ( a n d t h e i n v i s i b i l i t y o f A1 a t o m s ) i n FIM i s n o t u n e x p e c t e d a c c o r d i n g t o t h e e m p i r i c a l r u l e t h a t t h e a l l o y e l e m e n t o f h i g h e r e v a p o r a t i o n f i e l d (
i . e . Mn i n t h e A1-Mn a l l o y s ) i s p r e f e r e n t i a l l y imaged i n t h e F I m i c r o g r a p h s .
VI
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CONCLUSIONSC o m p u t e r - s i m u l a t e d F I i m a g e s o f A1-Mn q u a s i c r y s t a l were o b t a i n e d b a s e d o n t h e t h r e e - d i m e n s i o n a l P e n r o s e t i l i n g . The r e s u l t i n g i m a g e s were f o u n d t o show a n i c o s a h e d r a l symmetry a n d marked r i n g p a t t e r n s a t t h e f i v e f o l d p o l e s . T h e c o n t r a s t o f F I i m a g e s is l i t t l e i n f l u e n c e d b y t h e A1 p o s i t i o n s w i t h i n t h e P e n r o s e u n i t c e l l s . C o m p a r i s o n w i t h t h e o b s e r v e d F I i m a g e s i n d i c a t e s t h a t o n l y Mn a t o m s a r e v i s i b l e i n t h e FIM o b s e r v a t i o n s .
A c k n o w l e d g e m e n t s
W e a c k n o w l e d g e u s e f u l d i s c u s s i o n s w i t h D r . K . Kimura.
R e f e r e n c e s
/1/ S h e c h t m a n , D., B l e c h , I . , G r a t i a s , D. a n d Cahn, J . W . , P h y s . Rev. L e t t .
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( 1 9 8 4 ) 1951./2/ P o r t i e r , R., S h e c h t m a n , D . , B l e c h , I . , G r a t i a s , D.
a n d Cahn, J . W . , J . M i c r o s c . S p e c t r o s c . E l e c t r o n . 1 0 ( 1 9 8 5 ) 1 0 7 . /3/ H i r a g a , K . , H i r a b a y a s h i , M . , I n o u e , A. a n d ~ a s u m o t o , T . ,
J . Phys. S o c . J p n . 54 ( 1 9 8 5 ) 4077.
/4/ Melmed, A.J. a n d ~ l s n , R., P h y s . Rev. L e t t . 56 ( 1 9 8 6 ) 1 4 7 8 . /5/ M u l l e r , E.W. a n d T s o n g , T.T., F i e l d I o n M i c r o s c o p y ( E l s e v i e r ,
New York, 1 9 6 9 ) .
/ 6 / Mackay, A.L., Sov. P h y s . C r y s t a l l o g r .
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( 1 9 8 1 ) 5 1 7 , a n d P h y s i c a ( 1 9 8 2 ) 609./ 7 / L e v i n e , D. a n d S t e i n h a r d t , P . J . , P h y s . Rev. L e t t .
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( 1 9 8 4 ) 2477./8/ E l s e r , V . , P h y s . Rev.
B32
( 1 9 8 5 ) 4892./9/ H i r a g a , K . , ~ i n z o k u g a k k a i h ~
25
( 1 9 8 6 ) 1 2 1 ( i n J a p a n e s e ) ./ l o /
S o c o l o r , J.E.S., S t e i n h a r d t , P . J . a n d L e v i n e , D . , P h y s . Rev.B32
( 1 9 8 5 ) 5547.
/11/ Moore, A . J . a n d R a n g a n a t h a n , S . , P h i l . Mag.
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( 1 9 6 7 ) 723./12/ Cahn, J . W . , S h e c h t m a n , D. a n d G r a t i a s , D., J . M a t e r . Res.
1 ( 1 9 8 6 ) 1 3 .
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MODEL I
MODEL I1
0 0
F i g . 1
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Two m o d e l s f o r t h e a t o m i c s t r u c t u r e o f t h e P e n r o s e u n i t c e l l s o f t h e A1-Mn a l l o y s /3,9/ ( b o d y - c e n t e r e d a t o m s i n t h e a c u t e c e l l s a r e n o t i n c l u d e d ) .JOURNAL DE PHYSIQUE
F i g . 2
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S i m u l a t e d F I i m a g e s o f t h e Mn q u a s i l a t t i c e v i e w e d a l o n g v a r i o u s symmetry d i r e c t i o n s : ( a ) f i v e f o l d d i r e c t i o n , ( b ) t h r e e f o l d d i r e c t i o n , ( c ) t w o f o l d d i r e c t i o n . S t e r e o g r a p h i c p r o j e c t i o n s o f i c o s a h e d r a l s t r u c t u r e /12/ are a l s o shown for c o m p a r i s o n .F i g . 3
