QUASILATTICES IN IE3 AND THEIR PROJECTION FROM LATTICES IN IEn

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QUASILATTICES IN IE3 AND THEIR PROJECTION FROM LATTICES IN IEn

P. Kramer

To cite this version:

P. Kramer. QUASILATTICES IN IE3 AND THEIR PROJECTION FROM LATTICES IN IEn.

Journal de Physique Colloques, 1986, 47 (C3), pp.C3-75-C3-83. �10.1051/jphyscol:1986307�. �jpa-

00225717�

QUASILATTICES IN I E ~ AND THEIR PROJECTION FROM LATTICES IN IEn

P. KRAMER

Institut fur Theoretische Physik der Universitiit Tiibingen, 0-7400 M b i n g e n , F.R.G.

A b s t r a c t - T h e o r e t i c a l s t u d i e s a r e r e p o r t e d on t h e s t r u c t u r e o f n o n - p e r i o d i c q u a s i l a t t i c e s a s s o c i a t e d w i t h t h e i c o s a h e d r a l g r o u p . The c o n s t r u c t i o n i s b a s e d on d u a l i z a t i o n a n d p r o j e c t i o n o f

l a t t i c e s a n d on i n d u c t i o n a n d s u b d u c t i o n o f g r o u p r e p r e s e n t a - t i o n s .

I - INTRODUCTION

T h i s r e p o r t d e a l s w i t h t h e o r e t i c a l s t u d i e s / I - 5 / o f n o n - p e r i o d i c s p a c e f i l l i n g s a n d q u a s i l a t t i c e s o b t a i n e d by p r o j e c t i o n o f h i g h - d i m e n s i o n a l h y p e r c u b i c l a t t i c e s t o IE3. T h e s e s t u d i e s a r e c o n c e n t r a t e d on g r o u p s a n d i n p a r t i c u l a r on t h e i c o s a h e d r a l g r o u p ~ ( 5 ) . The work was s t i m u - l a t e d by t h e p a r a d i g m o f n o n - p e r i o d i c 2 - d i m e n s i o n a l p a t t e r n s g i v e n by P e n r o s e /6/ , by t h e a l g e b r a i c t h e o r y o f t h e s e p a t t e r n s d e v e l o p e d by d e B r u i j n / 7 / , a n d 3 b y t h e p r o p o s a l o f Mackay / 8 / t o u s e two rhombo- h e d r a l c e l l s i n IE a s a b a s i s f o r n o n - p e r i o d i c 3 - 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 y .

I n a b s t r a c t t e r m s , t r a d i t i o n a l c r y s t a l l o g r a p h y d e a l s w i t h t h e a c t i o n o f s p a c e g r o u p s on E u c l i d e a n s p a c e . The s p a c e g r o u p s h a v e a n o n - t r i v i a l t r a n s l a t i o n s u b g r o u p a n d h e n c e g i v e r i s e t o p e r i o d i c l a t t i c e s . What t h e n i s t h e g r o u p t h e o r y a n d c r y s t a l l o g r a p h y o f n o n - p e r i o d i c q u a s i l a t t i c e s i n IE3, a n d how d o e s i t r e l a t e t o t r a d i t i o n a l c r y s t a l - l o g r a p h y ?

T h e s e q u e s t i o n s r e q u i r e f u l l a t t e n t i o n t h r o u g h t h e r e m a r k a b l e e x p e r i - m e n t a l d i s c o v e r y o f s o l i d m a t t e r p h a s e s o f AltMn by S h e c h t m a n , R l e c h , G r a t i a s a n d Cahn / 9 / . We c a n n o t g i v e h e r e a n a c c o u n t o f t h e r e l e v a n t c o n t r i b u t i o n s f o l l o w i n g t h i s w o r k , compare / l o / , a n d r e t u r n t o a b r i e f s u r v e y o f t h e o r e t i c a l work c a r r i e d o u t i n c o o p e r a t i o n w i t h H a a s e , Kramer, L a l v a n i a n d Mackay.

I1

THE HYPERCUBIC LATTICE I N IEL1 AND I T S PROJECTION TO I E q .

*

I n t h e E u c l i d e ~ n s p a c e IEn c o n s i d e r a n o r t h o n o r m a l b a s i s & I , . . ,

a n d _* i t s d u a l 2 ; . . . ^{, & ,}

b . * b = 6 i j , i , j = I ,.., ^{n .}

-1 -j

The h y p e r c u b i c l a t t i c e Y i s t h e l a t t i c e i n v a r i a n t u n d e r t h e t r a n s - l a t i o n g r o u p T whose e l e m e n t s a r e g e n e r a t e d by t h e s t a r r e d b a s i s . I t s c e l l s a r e t h e f u n d a m e n t a l d o m a i n s o r t r a n s v e r s a l s on IEn u n d e r t h e a c t i o n of T . By u s e o f t h e b a s i s bl, ...&, t h e p o i n t s o f a c e l l w i t h i n d e x s y s t e m

( k l , . . k n ) , k j = + 1 , + 3 , . .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986307

C3-76 JOURNAL DE PHYSIQUE

may be c h a r a c t e r i z e d a s

1 1

{ y I Z ( k i - 2 ) - ( z o b i < z k i , i = l , . . , n } .

Upon c h o o s i n g a f i x e d s h i f t v e c t o r y a n d d e c o m p o s i n g - 2 a s y = x + y ,

- we speak-of a c e l l r e f e r r e d t o t h e s h i f t v e c t o r y a s -

1 1

{x ₁ ? ( k i - 2 ) 2 ( x l t y ) * b i < ?ki, i = I , . . ^{, n )} .

C o n s i d e r f o r 1 c q < n a n o r t h o g o n a l d e c o m p o s i t i o n o f t h e s p a c e E n , IEn ⁺ IEy t 1.E;-9, I E q 1 I E y q

a n d o f t h e v e c t o r s

Y = XI ⁺ 2 2 ' h = hl + li2'

-

The c e l l w i t h g i v e n i n d e x s y s t e m i n t e r s e c t s w i t h IE? i f t h e r e i s a p o i n t 51, - Y = 51 + 1 s u c h t h a t

1 1

Z(ki-2) < ( g l t y ) - b i _- 7 k i , ^{i =} I ,. ^{. , n .}

I n t r o d u c i n g t h e p r o j e c t i o n s

kI1' .knl

o f t h e b a s i s v e c t o r s from IEn t o lEq a n d t h e numbers yi= ( y * b i ) , t h e s e e q u a t i o n s d e t e r m i n e i n IE9 a n n - g r i d Y1 whose c e l l s a r e -

f o r m e d by i n t e r s e c t i o n s o f n h y p e r p l a n e s o r t h o g o n a l t o t h e v e c t o r s j u s t s p e c i f i e d a n d s h i f t e d by t h e numbers y i ,

1 1

- ( k . - 2 ) _{2 1} 2 ~~~b~~ ^t. yi < Zki, i = I ,.., ^{n .}

A d u a l l a t t i c e o r q u a s i l a t t i c e Z1 i s f o r m e d i n TEq by a s s o c i a t i n g t o a n y h y p e r f a c e w i t h f i x e d i n d e x j of a c e l l o f Y1 a d u a l e d g e c o r r e s - p o n d i n g t o t h e v e c t o r b j l .lJpon d e c o m p o s i n g t h e s h i f t v e c t o r a s Y = LI ⁺ y2

-

t h e p a r t o f t h i s v e c t o r i n IEq a m o u n t s t o a n o v e r a l l s h i f t o f t h e q u a s i l a t t i c e w h e r e a s t h e c o m p l e m e n t a r y p a r t d e t e r m i n e s t h e s t r u c t u r e o f t h e q u a s i l a t t i c e .

T h i s c o n s t r u c t i o n d e s c r i b e d i n / 2 / i s m o d i f i e d i n / 1 1 / . T h e r e we d e - t e r m i n e i n t h e s p a c e ]En t h e d u a l s t o t h e p - b o u n d a r i e s o f a c e l l w h i c h i n t e r s e c t s w i t h t h e s u b s p a c e IEq a n d t h e n c o n s t r u c t t h e s t a n - d a r d d u a l ( n - p ) - b o u n d a r i e s i n IEn which form a q u a s i l a t t i c e i n E n . T h i s q u a s i l a t t i c e Z i s t h e n p r o j e c t e d t o IEq. T h i s m o d i f i e d a p p r o a c h u s e s s t a n d a r d c r y s t a l l o g r a p h i c d u a l i z a t i o n i n IEn t h r o u g h o u t .

GROUP ANALYSIS FOR THE PROJECTION

The l a t t i c e Y i n IEn h a s a s i t s f u l l s p a c e g r o u p t h e s e m i d i r e c t p r o - TAR d u c t ( n group

w h e r e R ( n ) i s t h e h y p e r o c t a h e d r a l p o i n t g r o u p . T h i s l a t t e r g r o u p con- t a i n s a l l p e r m u t a t i o n s o f t h e s y m m e t r i c g r o u p ~ ( n ) a n d a l l r e f l e c - t i o n s

bx -+ E i b 2 , E i = t l .

-1

Given a s u b g r o u p H < R ( n ) , t h e r e p r e s e n t a t i o n o f R ( n ) i n IEn s u b d u c e s

i r r e d u c i b l e r e p r e s e n t a t i o n s p a c e s o f H which may s e r v e a s c a n d i d a t e s

f o r lE4. The p r o j e c t i o n from E n t o IEq t h e n commutes w i t h t h e a c t i o n

o f H. T h i s c o n s t r u c t i o n i s a p p l i e d i n / 2 / t o v a r i o u s g r o u p s a n d l e a d s

T a b l e 1 Examples f o r t h e p r o j e c t i o n IEn +IEq + IEn-q

n q s p a c e p o i n t s u b g r o u p IEq t IEn-9 g r o u p g r o u p H

n n - l T ( n ) ~ Q ( n ) a ( n ) S ( n ) IE"-I+ IEI

3 T ( 3 ) ~ O h Oh Cv(3) lE2 t I E I

4 3 T ( ~ ) A Q ( L ) a ( 4 ) T~ S ( L ) E~ t I E I

I V - GROUP AND SUBGROUP ANALYSIS FOR QUASILATTICES ASSOCIATED W I T H

THE I COSAHEDRAL GROUP

S i n c e t h e i c o s a h e d r a l g r o u p ~ ( 5 ) i s n o t c o m p a t i b l e i n lE3 w i t h t r a n s - l a t i o n a l symmetry, t h i s g r o u p i s a n i n t e r e s t i n g c a n d i d a t e f o r t h e g r o u p H c o n s i d e r e d i n s e c t i o n 111. I n / 1 2 / we i n v e s t i g a t e s p a c e s En which a r i s e i n t h i s way. The d i h e d r a l s u b g r o u p s D(m), m = 5 , 3 , 2 o f

~ ( 5 ) a r e employed t o e s t a b l i s h t h e c h a i n o f p o i n t g r o u p s D(m) < A(5) < R ( 6 0 / ( 2 m ) ) , m = 5 , 3 , 2 .

R e p r e s e n t a t i o n t h e o r y o f t h e s e g r o u p s shows t h a t t h e i r r e p 1 3 1 ~ 1 2 o f A ( 5 ) c o n t a i n s t h e n o n - t r i v i a l I -dimens; o n a l r e p r e s e n t a t i o n s o f t h e d i h e d r a l g r o u p s which we d e n o t e by 5 . T h i s r e p r e s e n t a t i o n o f D(m) i n d u c e s a n o r t h o g o n a l r e p r e s e n t a t i o n o f d i m e n s i o n

n = I ~ ( 5 ) 1 / I ~ ( m ) l = 60/(2m) = 6 , 1 0 , 15.

By c o n s t r u c t i o n , t h i s r e p r e s e n t a t i o n y i e l d s a n embedding o f ~ ( 5 ) i n t o

~ ( 6 0 / ( 2 m ) ) . Moreover t h e c o n s t r u c t i o n a s s u r e s t h a t t h e r e s u b d u c t i o n from .Q(60/(2m)) c o n t a i n s t h e 3 - d i m e n s i o n a l r e p r e s e n t a t i o n ) 3 1 $ ) o f

~ ( 5 ) . I f t h e 3 - d i m e n s i o n a l s u b s p a c e f o r t h i s r e p r e s e n t a t i o n i s chosen

f o r t h e p r o j e c t i o n , one o b t a i n s t h r e e t y p e s o f q u a s i l a t t i c e s a s s o -

c i a t e d w i t h t h e i c o s a h e d r a l g r o u p . I n / 1 2 / we a n a l y z e a l l t h e compo-

s i t e and e l e m e n t a r y c e l l s f o r t h e s e q u a s i l a t t i c e s . F o r t h e c a s e s

n = 6 , 1 0 a n d 1 5 , t h e number o f e l e m e n t a r y rhombohedra i s 2 , 5 and 14

r e s p e c t i v e l y .

C3-78 JOURNAL DE PHYSIQUE

T a b l e 2 S u b d u c t i o n of i r r e d u c i b l e r e p r e s e n t a t i o n s from A ( 5 ) t o ~ ( m ) .

A ( 5 ) :

151 1411 1321 131:l 131f1

D ( 5 ) : o 1 0 1 0 0

" 0 0 0 1 1

1 0 1 1 1 0

2 0 1 1 0 1

D ( 2 ) : o 1 1 2 0 0

g o 1 1 1 1

1 0 1 1 1 1

' l o I 1 1 I

V - THE QUASILATTICE ASSOCIATED WITH D( 5 ) < A ( 5 ) < n ( 6 )

T h i s q u a s i l a t t i c e i s c o n s t r u c t e d i n / 2 / a n d a n a l y z e d i n more d e t a i l i n /3/. We b r i f l y d e s c r i b e t h e main f e a t u r e s . The h p e r c u b i c l a t t i c e i s g i v e n i n IE8, a n d t h e embedding ~ ( 5 ) < A ( 5 ) ' 0 ( 6 $ i n t e r m s o f t h e a n a l y s i s g i v e n i n /12/ i s o b t a i n e d by i n d u c i n g from ~ ( 5 ) . I n t h e r e d u c t i o n t o ~ ( 5 ) one f i n d s t h e two i r r e p s

i n t h e n o t a t i o n o f T a b l e 2. These 3 - d i m e n s i o n a l r e p r e s e n t a t i o n s y i e l d two o r t h o g o n a l s u b s p a c e s

o f t h e i n i t i a l s p a c e IE 6 . The p r o j e c t i o n o f t h e h y p e r c u b i c l a t t i c e from 1E t o IE? y i e l d s a h e x a g r i d whose p l a n e s a r e p a r a l l e l t o s i x p a i r s o f f a c e s o f t h e r e g u l a r d o d e c a h e d r o n . The e l e m e n t a r y d u a l c e l l s a r e two rhombohedra d i s c u s s e d a l r e a d y by Kowalewski /13/ i n r e l a t i o n t o t h e r h o m b i c t r i a c o n t a h e d r o n f o u n d 5y K e p l e r / 1 4 / . Mackay / 8 / i n t r o - d u c e d t h e s e c e l l s a s c a n d i d a t e s f o r a q u a s i l a t t i c e i n IE3.

The q u a s i l a t t i c e c o n t a i n s c o m p o s i t e c e l l s i n t h e form o f t h r e e zono- h e d r a w i t h 1 2 , 20 and 30 rhombus f a c e s / 3 / . A n o t h e r c h r a c t e r i s t i c s o f t h e q u a s i l a t t i c e i s t h e e x i s t e n c e o f i n f i n i t e 2 - d i m e n s i o n a l l a y e r s c o n s i s t i n g o f p a c k e d rhombohedra1 c e l l s . Any c e l l i n t h i s l a y e r h a s f o u r e d g e s v e r t i c a l t o a p a i r o f f a c e s o f t h e r e g u l a r d o d e c a h e d r o n . S i x s e t s o f p a r a l l e l s y s t e m s o f l a y e r s c o n t i n u e t h r o u g h t h e f u l l q u a s i l a t t i c e .

The q u a s i l a t t i c e can be d e s c r i b e d a s a s e t o f

i n i e r p e n e t r a t i n g p e r i o d i c l a t t i c e s . Each s u b l a t t i c e i s b a s e d o n 3 o f t h e s i x v e c t o r s p e r p e n d i c u l a r t o t h e f a c e s o f t h e d o d e c a h e d r o n . I t c o n t a i n s t h e c o r r e s p o n d i n g rhombus c e l l s whose c e n t e r i s d i s t o r t e d from t h e a v e r a g e p e r i o d i c p o s i t i o n .

The s h i f t v e c t o r Y i s a n a l y z e d i n / 3 / w i t h r e s p e c t t o t h e s t r u c t u r e o f

t h e q u a s i l a t t i c e . T h i s a n a l y s i s i s c a r r i e d o u t i n t h e complementary

s p a c e I E ' a n d i t i s c o n s i d e r a b l y e x t e n d e d i n / 1 1 / . I t i s shown t h a t

t k e v e c t & r L2 ^{i s} c o n f i n e d t o t h e i n t e r i o r o f a K e p l e r z o n e , d e f i n e d

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

a n i n f i n i t e g r a p h K . The c o n t i n u a t i o n o f t h i s g r a p h i s c o m p l e t e l y d e t e r m i n e d by t h e d i a g n o s i s o f t h i s v e r t e x i n t h e K e p l e r z o n e , a n d s o t h e g r a p h c o n t a i n s a l l t h e i n f o r m a t i o n on t h e m a t c h i n g r u l e s f o r t h e q u a s i l a t t i c e . An example o f a g r a p h i s g i v e n i n F i g . 1 .

F i g . 1 . P a r t o f a g r a p h K i n t h e K e p l e r z o n e . The K e p l e r zone i s t h e i n t e r i o r o f a t r i a c o n t a h e d r o n . T h i s t r i a c o n t a h e d r o n i s shown h e r e i n a p r o j e c t i o n a l o n g a 2 - f o l d a x i s . The open c i r c l e s mark p o i n t s w h i c h form t h e v e r t i c e s o f t h e g r a p h K a n d a r e c o n n e c t e d by e d g e l i n e s . The g r a p h c o n t a i n s 32 v e r t i c e s w h i c h a r e p r o - j e c t i o n s o f v e r t i c e s o f t h e h y p e r c u b e i n lE6 . The b r o k e n l i n e i n d i c a t e s t h e c o n t i n u a t i o n o f t h e i n f i n i t e g r a p h K t o a new s e t o f p o i n t s . To t h i s g r a p h t h e r e c o r r e s p o n d s a q u a s i l a t t i c e w i t h i c o s a h e d r a l p o i n t symmetry. I t c o n t a i n s a c e n t r a l t r i a c o n t a - h e d r o n whose f a c e s a r e c o v e r e d by 30 r h o m b i c d o d e c a h e d r a .

The d i f f r a c t i o n from t h e q u a s i l a t t i c e h a s b e e n s t u d i e d i n a n a p p r o x i - m a t i o n where t h e r e i s one s c a t t e r i n g c e n t e r a t e a c h v e r t e x . A s i m p l e l o n g - r a n g e a p p r o x i m a t i o n i s o b t a i n e d from t h e d e s c r i p t i o n i n t e r m s o f 20 s u b l a t t i c e s / 3 , 4 , 5 / . The i n t e r f e r e n c e of c o n t r i b u t i o n s from d i f f e r e n t s u b l a t t i c e s y i e l d s s t r o n g maxima o f t h e i n t e n s i t y a t p o s i - t i o n s d e t e r m i n e d by t h e F i b o n a c o i numbers.

I n t h e same a p p r o x i m a t i o n o f o n e c e n t e r p e r v e r t e x , p r o j e c t i o n s of t h e q u a s i l a t t i c e h a v e b e e n computed a l o n g t h e 5-, 3-, a n d 2 - f o l d a x i s . R e s u l t s a r e shown i n F i g s . 2 - 4 . These computed r e s u l t s show many f e a t u r e s o f e x p e r i m e n t a l r e s u l t s o b t a i n e d by Urban a n d

c o l l a b o r a t o r s / I 5 1 .

C3-80 JOURNAL DE PHYSIQUE

P i g . 2 . P r o j e c t i o n s o f p o i n t s on v e r t e x p o s i t i o n s o f t h e q u a s i - l a t t i c e i n t o a p l a n e a l o n g t h e 5 - f o l d a x i s .

t i O C C U 0 3 C 0 0 o n o o o 0-0 o o o 0 - o o o o o 3 " ~ o " : ? Q , 0 G " @ ~ o o ~ G o " ~ Q , ~ c O ~ 9 ; 9 ;

G

"

^orno

L O C 0 9 3 0 0 5 0 G 0 0 0 0 0 0 0 0

*

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o n

00 O o O , )

O O O ^ r O e ~ i i \

F i g . ? P r o j e c t i o n o f p o i n t s on v e r t e x p o s i t i o n s o f t h e q u a s i -

l a t t i c e i n t o a p l a n e a l o n g t h e ? - f o l d a x i s .

Eig.4.. Pr ojection of points on vertex positions of the quasi- lattice into a plane along the 2-fold axis.

REFERENCES

f\/ Kramer,P.,Acta Cryst. A38 (1982) 257.

/ 2 / Kramer,P. and Neri.R., Acta Cryst. A^O (1984) 580 / 3 / Kramer,P., Z. Naturf. ^Oa (1985) 775

Ikl Kramer,P., Phys.Lett. 111A (1985) 133 ,

/ 5 / Kramer,P. and Kramer,L. Z. Natuff. iOa (1985) 1162 /6/ Penrose, R., Math. Intelligencer 2 (1979) 32

/ ? / de Bruijn, N.G., Indagationes Mathematicae /3 (1981) 39 and 53 / 8 / Mackay,A.L.,Sov. Phys. crystallogr. 1J3(1981) 517

/9/ Shechtman, D., Blech, I., Gratias.D. and Cahn, J.W., Phys. Rev. Lett. j>2 (198/1) 1951

/10/Mackay, A.L., and Kramer, P., Nature JTH) (1985) 17 /11/Kramer,P. submitted for publication

/12/Haase, R.W., Kramer,L., Kramer,P. and Lalvani,H., submitted for publication

/13/Kowalewski,G., Der Keplersche Korper und andere Bauspiele, K.F. Koehlers Antiquarium, Leipzig 1938

/1A/Kepler, J. , Strena seu de Nive Sexangula, in: Ges. Werke vol.4, ed. by M.Caspar, Munich 1938

/15/Urban, K., private communication

C3-82 JOURNAL DE PHYSIQUE

COMMENTS AFTER THE P . KRAMER TALK :

M . V . JARIC.-

Suggested "standard" orientations of the icosahedra in the physical, R] , and complementary, R? , spaces

PHYSICAL PARALLEL

PROJECTED PHASON

PERPENDICULAR

SPACE

Since several recent papers on icosahedral quasicrystals all use

different orientations of the reference icosahedra in the two

three-dimensional complements of the six-dimensional hyperspace, and

since transformations of the relevant tensorial quantities from one

coordinate system to another might be quite complicated, I suggest

that we standardise the orientation of these icosahedra in a similar

way that the orientation of a cube is typically taken to have

coordinate axes as the four-fold axes. I recommend that we use the

fact that the tetrahedral group is a subgroup of the icosahedral

group and that the two inequivalent three dimensional representations

of the icosahedral group both subduce the same three dimensional

vector representation of the tetrahedral group. A simple standard

orientation of a tetrahedron is the one in which its two-fold axes

coincide with the coordinate axes. The two icosahedra are then

oriented so that they contain the tetrahedron in the way shown in the

enclosed figure. In this way, we can, for example, construct relevant

icosahedral tensors by building from the tetrahedral tensors which are

particularly simple in the given orientation.