SOME REFLEXIONS AND QUESTIONS ABOUT APERIODIC CRYSTALS

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SOME REFLEXIONS AND QUESTIONS ABOUT APERIODIC CRYSTALS

L. Michel

To cite this version:

L. Michel. SOME REFLEXIONS AND QUESTIONS ABOUT APERIODIC CRYSTALS. Journal de

Physique Colloques, 1986, 47 (C3), pp.C3-495-C3-498. �10.1051/jphyscol:1986351�. �jpa-00225763�

SOME REFLEXIONS AND QUESTIONS ABOUT APERIODIC CRYSTALS

L . MICHEL

I n s t i t u t d e s H a u t e s E t u d e s S c i e n t i f i q u e s , 35, Route d e C h a r t r e s , F-91440 B u r e s - s u r - Y v e t t e

I e n j o y e d tremendously t h i s workshop ! Denis G r a t i a s and I thank c o r d i a l l y a l l t h e p a r t i c i p a n t s t o have made it. I l e a r n e d a l o t ; mainly a b o u t e x p e r i m e n t a l f a c t s and methods. How c l e v e r a r e e x p e r i m e n t a l i s t s 1 I sympathized very much w i t h t h e t h e o r i s t s : i t i s amusing t o p l a y w i t h models ! I was f a s c i n a t e d by t h e works of m a t h e m a t i c i a n s h e r e . Thanks t o t h e e x c e l l e n t l o g i s t i c of Les Houches " C e n t r e de P h y s i q u e " , I c o u l d b e v e r y r e l a x e d from t h e r e s p o n s a b i l i t i e s I had; s o I had a l o t of t i m e :or t h i n k i n g and d i s c u s s i n g . A s a r e s u l t , many q u e s t i o n s a r i s e t h a t I c a n n o t answer s o I w i l l s h a r e them w i t h you.

A s you know, i n t h e X I X c e n t u r y , c r y s t a l l o g r a p h y was i n t h e f o r e - Lrmt of s t u d i e s on symmetry i n N a t u r e ^: t h e 32 p o i n t g r o u p s were d e t e r - , mined ( F r a n k e n s t e i n , H e s s e l ) b e f o r e t h e word g r o u p s was i n t r o d u c e d by G a l o i s (1830) and t h e 230 s p a c e g r o u p s were c a t a l o g e d i n d e p e n d e n t l y by a m i n e r a l o g i s t Fedorov and a mathematician S c h 6 n f l i e s i n 1892. ( I t i s t r u e t h a t s p a c e g r o u p s were n o t a p p l i e d f o r t h e s t u d y 02 c r y s t a l s be.:ore t h e t w e n t i e s ) . I n t h e l a s t q u a r t e r of t h e XIXth c e n t u r y , symmetry a r g u - . ments were u s e d more and more by p h y s i c i s t s f o r e x p l a i n i n g o r p r e d i c t i n g p h y s i c a l p r o p e r t i e s of m a t t e r ( e . g . , J a c q u e s and P i e r r e C u r i e p r e d i c t i o n of p i e z o e l e c t r i c i t y ) and P. C u r i e p a p e r on t h e symmetries 02 p h y s i c a l s t a t e s [ 11 i s a c l a s s i c s t i l l v e r y much worth r e a d i n g . I n t h e XXth c e n t u r y , t h e symmetry of p h y s i c a l laws was a c e n t r a l s u b j e c t i n p h y s i c s . A f t e r s p e c i a l r e l a t i v i t y i t l e a d E i n s t e i n t o Eonrollate i t s g e n e r a l r e l a - t i v i t y t h e o r y . I t a l s o g u i d e d D i r a c f o r t h e d i s c o v e r y oZ i t s famous e q u a t i o n ; and a f t e r t h r e e y e a r s of e x t r a o r d i n a r y i n t e l l e c t u a l s t r u g g l e , D i r a c p r e d i c t e d from it t h e e x i s t e n c e of a new t y p e of symmetry i n p h y s i c s : t h a t of c h a r g e c o n j u g a t i o n (between m a t t e r and a n t i m a t t e r ) . F i f t e e n y e a r s ago emerged a new s t e p i n t h e u n i f i c a t i o n of p h y s i c s : QED (quantum e l e c t r o d y n a m i c s ) gauge group U (1

c o u l d be e n l a r g e d t o U(2) t o u n i f y weak and e l e c t r o m a g n e t i c i n t e r a c t i o n and t o U(3) ? o r chromodynamics and e l e c t r o m a g n e t i s m ; s o now, p h y s i c i s t s a r e a t work f o r d i s c o v e r i n g t h e g r a n d u n i f i e d t h e o r y (and even t h e supersymmetric t h e o r y which w i l l a l s o i n c l u d e g r a v i t a t i o n ) . Hence p h y s i c a l laws have a huge and b e a u t i f u l symmetry which i s r e a l i z e d nowhere i n t h e p h y s i c a l s t a t e s , b u t which may have b e e n t h e symmetry of t h e u n i v e r s e a t t h e b i g bang.

Tremendous e f f o r t s a r e made by p h y s i c i s t s f o r o b s e r v i n g p r e d i c t e d e Z F e c t s : p r o t o n decay ( l i f e t i m e o f 1 0 ~ ~ ' ~ y e a r s ) , t f H o o f t -Polyakov monopole w i t h mass i n t h e r a n g e 10-8 - ^{grams !} I t w i l l t a k e s o many y e a r s and s o much money f o r t h e s e and o t h e r e x p e r i m e n t s t o b e completed t h a t i t i s v e r y r e f r e s h i n g t o look simply a t t h e i n t r i g u i n g symmetry of a phase o f some A1-Mn a l l o y s .

I t i s a s t r a n g e paradox t h a t p h y s i c i s t s were a b l e t o p r e d i c t from symmetry arguments t h e e x i s t e n c e of a n t i m a t t e r , of t h e N' , ^{Z 0} bosons b u t n o t t h a t of t h e i c o s a h e d r a l q u a s i - c r y s t a l phase. On t h e c o n t r a r y it was a much b e l i e v e d dogma t h a t such c r y s t a l s c o u l d n o t e x i s t s ! Thanks t o D. Schechtman and h i s c o l l a b o r a t o r s t o have observed them !

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986351

C3-496 JOURNAL DE PHYSIQUE

I t must b e s a i d t h a t t h e o r i c i a n s a r e j u s t b e g i n n i n g t o e x p l a i n t h e e x i s t e n c e of c r y s t a l s from f i r s t p h y s i c a l p r i n c i p l e s ( s t a t i s t i c a l mechanics of t h e i n t e r a c t i n g atoms of t h e c r y s t a l ) and works of t h i s k i n d a r e n o t v e r y w e l l known ( s e e e . g . / 2 / , / 3 / , / 4 / and t h e i r b i b l i o - g r a p h y ) . I t i s t r u e t h a t , s i m i l a r t o t h e g e o m e t r i c a l d e d u c t i o n of t h e s p a c e g r o u p c l a s s i f i c a t i o n , t h e r e e x i s t s a g e n e r a l scheme f o r p r e d i c - t i n g t h e symmetry group of t h e mesomorphic p h a s e s / 5 / : t h e y a r e t h e c l o s e d co-compact ( i . e . t h e s e t of t h e i r c o s e t s i s compact) subgroups 02 t h e E u c l i d e a n g r o u p E ( 3 ) . These g r o u p s form a n i n f i n i t e ( b u t compact) s e t ; s o t h e p r e d i c t i v e v a l u e of t h e scheme i s weak ( w h i l e o n l y 8 s p a c e g r o u p s a r e n o t y e t i l l u s t r a t e d by a c t u a l c r y s t a l s ) . However, on such g e n e r a l grounds, one c a n a l s o c l a s s i f y t h e t o p o l o g i c a l l y s t a b l e symmetry d e f e c t s of t h o s e p h a s e s / 6 / and t h i s h a s been q u i t e u s e f u l f o r r e c o g n i z i n g r e c e n t l y new p h a s e s of l i q u i d c r y s t a l s . The same g e n e r a l scheme (decomposition of E u c l i d e a n i n v a r i a n t s t a t e s i n t o e x t r e m a l s t a t e s of t h e C*-algebra of l o c a l o b s e r v a b l e s ) p r e d i c t a l s o a g r e a t v a r i e t y of e q u i l i b r i u m p h a s e s w i t h o u t symmetrygroup c o r r e s p o n d i n g t o t h e e r g o d i c a c t i o n of E ( 3 ) . T h e i r c l a s s i f i c a t i o n h a s n o t been done : s u r e l y , i t must c o n t a i n t h e incommensurate and a p e r i o d i c c r y s t a l p h a s e s .

To my knowledge, t h e f i r s t p r e d i c t i o n of an a p e r i o d i c ( i n one d i r e c t i o n ) c r y s t a l s t a t e was made i n d e p e n d e n t l y by J . V i l l a i n / 7 / and

Yoshimori / 8 / i n 1959. T h a t was h e l i m a g n e t i s m , which was observed by Herpin e t a 1 . / 9 / . I t i s r e a l l y a p e r i o d i c M a u s e t h e r a t i o of t h e h e l i x p e r i o d t o t h e d i s t a n c e between neighbour r e t i c u l a r p l a n e s c o n t a i n i n g t h e s p i n s i s a c o n t i n u o u s f u n c t i o n of t e m p e r a t u r e ( i n g e n e r a l t h e r e i s no l o c k i n g ) . The n e x t o b s e r v a t i o n of a n incommensurate s t r u c t u r e by D e W o l f e t a l . / ? 0 / i s w e l l known h e r e and b o t h J a n n e r and J a n s s e n had t h e o p p o r t u n i t y t o c o n v i n c e u s once more t h a t t h e b e s t way t o d e s c r i b e t h e s e s t r u c t u r e s i s t o u s e q u a s i - p e r i o d i c f u n c t i o n s ; t h e y a r e r e s t r i c t i o n s of 4 , 5 o r 6 p e r i o d f u n c t i o n s . The l a t t e r c a n be c o n s i d e r e d a s t h e d e s c r i p - t i o n o: a c r y s t a l ( a s i m p l e and e l e g a n t a u x i l i a r y t h i n k i n g t o o l ) i n 4 , 5,6 dimensions.

The f i r s t time I d i s c u s s e d w i t h Denis G r a t i a s a b o u t t h e i c o s a h e d r a l phase i t was p e r f e c t l y c l e a r i n h i s mind t h a t it had a l s o t o be d e s c r i b e d by q u a s i p e r i o d i c Lunctions i n 3 dimension, r e s t r i c t i o n of 6-period Zunctionswith p e r i o d s i n r a t i o I l l , 1 ,

( 1 +&) /2) . T h i s i s a u n i q u e f i x e d p o i n t i n 6 d i m e n s i o n a l s p a c e , w h i l e t h e r e p r e s e n t a t i v e p o i n t of a modulated c r y s t a l depends on t e m p e r a t u r e , p r e s s u r e , e x t e r n a l i i e l d s , e t c . . . We h e a r d a l s o v e r y c o n v i n c i n g argument, e . g . by P. _Bak, t h a t t h i s 1s t h e good frame Eor d e s c r i b i n g t h i s new phase. Of c o u r s e , a s we were e x p l a i n e d by s e v e r a l c o l l e a g u e s , Penrose p a t t e r n s , a n d t h e i r e x t e n s i o n s t o t h r e e d i m e n s i o n a l s p a c e

Mackay; Kramer and N e r i ) were a l r e a d y a v a l a i b l e and g i v e u s a v e r y c o n c r e t e model t o p l a y w i t h . They w i l l h e l p t o e x p l a i n t h e s t r u c t u r e of t h e d i f f e r e n t a l l o y s . There i s however a d i f f i c u l t y n o t y e t met by c r y s t a l l o g r a p h e r s : t h e e x i s t i n g s o f t w a r e f o r c r y s t a l a n a l y s i s d o e s n o t work. How t o d e s c r i b e most economically t h e p o s i t i o n o f t h e atoms ? I b e l i e v e t h a t i t w i l l b e by g i v i n g t h e "motive" 02 t h e s i x d i m e n s i o n a l c r y s t a l , and t o check t h e p r e d i c t i o n s from t h i s s t r u c t u r e w i h e v e r y p o s s i b l e e x p e r i m e n t a l a n a l y - - s i s . I do hope t h a t t h e s i r n p l e t 1% , r 3 h y p e r c u b i c s t r u c t u r e i n

6--dimension w i l l work. I f i t w e r e n o t t h e c a s e , one may go h i g h e r i n dimension ( 1 0 and 15 were a l r e a d y s u g g e s t e d ) . The t r o u b l e i s t h a t t h e c h o i c e may be t o o l a r g e . i n d e e d what i s needed i s t o know t h e

n-dimensional l a t t i c e s whose holohedry group G c o n t a i n s Yh , t h e

120--element symmetry group 02 t h e dodecahedron o r i c o s a h e d r o n , w i t h

t h e c o n d i t i o n t h a t t h e c o r r e s p o n d i n g n 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

Y h by i n t e g e r m a t r i c e s must c o n t a i n , when e x t e n d e d t o r e a l m a t r i c e s ,

i t s 3-dimensional v e c t o r r e p r e s e n t a t i o n . However a f i n i t e g r o u p r e p r e -

s e n t a t i o n by i n t e g e r m a t r i c e s might b e r e d u c i b l e ( i . e . i t l e a v e s a

subspace i n v a r i a n t ) w i t h o u t b e i n g indecomposable i n t o a d i r e c t sum 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 . T h i s d o e s n o t o c c u r i n 2 o r 3 dimensions

i n t e g e r r e p r e s e n t a t i o n s i s r e d u c i b l e on R ( v e c t o r + o n e dimensional r e p r e s e n t a t i o n ) b u t indecomposable on t h e i n t e g e r s Z . ^{And Tor Y, t h e}

member of i n d e c o m p o s a b l e , . Z - i n e q u i v a l e n t r e p r e s e n t a t i o n s on t h e i n t e - g e r s i s i n E i n i t e ! T h i s i s a l s o t r u e :or Y i t i s d u e t o t h e f a c t t h a t i t s 2-Sylow g r o u p - ^Z

^{Z 2} i s n o t c y c l i c

/ 1 2 / . I t d o e s n o t s e e m t h a t t h e n o n - d e c o m p o s a ~ l e r e p r e s e n t a t i o n s of Y o r Yh have been y e t c l a s s i f i e d . However t h e r e i s n o r e a s o n t o worry a b o u t t h i s p r o b l e m a s l o n g a s t h e 6 - d i m e n s i o n a l s i m p l e h y p e r c u b l i c l a t t i c e works.

T h e r e i s a l s o t h e T p h a s e , q u a s i - c r y s t a l i n 2 d i m e n s i o n s p e r i o d i c i n o n e , n o t t o s p e a k of t h e p h a s e s known f o r s e v e r a l y e a r s , c r y s t a l i n two d i m e n s i o n s and q u a s i c r y s t a l i n one. There i s t h e r e f o r e a c h a l l e n g e t o t h e o r i s t s t o give a list of s u c h p o s s i b l e new p h a s e s , a n d t o experimen- t a l i s t s : f i n d them i f t h e y e x i s t .

I t was v e r y i n t e r e s t i n g t o see how Landau t h e o r y c a n accomodate t h e e x i s t e n c e oZ i c o s a h e d r a l and s i m i l a r p h a s e s . I would l i k e i n t h e w r i t t e n r z p o r t t o r e f e r t o some i m p r e s s i v e work t h a t I d i d n o t q u o t e i n t h e o r a l t a l k b u t which w e r e mentionned by J e a n T a y l o r : t h o s e of C. Radin / 1 3 / . H i s s t a t i s t i c a l mechanic models y i e l d b o t h c r y s t a l s a n d a p e r i o d i c c r y s t a l . T h e r e w e r e a l s o p r i v a t e d i s c u s s i o n s f o r t h e p o s s i - b i l i t y t o o b s e r v e what R u e l l e / 1 4 / c a l l s " t u r b u l e n t " c r y s t a l s ; a n d I d i d n o t e l e s s i g n o r a n c e a n d more s c e p t i c i s m a s u s u a l among t h e p a r t i c i p a n t s

W e h e a r d t h e g r e a t news t h a t t h e r e a r e now i c o s a h e d r a l p h a s e c r y s - t a l s w i t h a s i z e oT few m i l l i m e t e r s ( i n s t e a d of m i c r o n s ! ) . S o we w i l l soon l e a r n a l o t more, n o t o n l y a b o u t t h e i r s t r u c t u r e , b u t a l s o on t h e i r p h y s i c a l p r o p e r t i e s ; t h e r e i s much work a h e a d f o r t h e t h e o r i s t s t o e::plain them ! How d o t h e s e c r y s t a i s grow ? T h i s seems t o b e a com- p l e t e l y new problem, q u i t e d i f f e r e n t Zrom t h a t sf p e r i o d i c o r modulated c r y s t a l s . A d e t a i l e d s t u d y of t h e i r s h a p e w i l l p r o b a b l y g i v e s u s some h i l i t s .

To c o n c l u d e , I want t o t h a n k you a l l a g a i n t o b e h e r e , I t h i n k we d i d w e l l what w e had t o d o , a s I h e a r d i t e x p r e s s e d by N i e l s Bohr i n my y o u t h : "We t r i e d t o e x p l a i n t o e a c h o t h e r what we d o n o t y e t under-- s t z n d " .

REFERENCES

/ 1 / C u r i e , P., J o u r n a l d e P h y s i q u e T h g o r i q u e A p p l i q u k e , 3&me s 6 r i e , 3

(1894) 393.

/ 2 / Duneau,M., Katz,A., Ann. I n s t . H. P o i n c a r k 41 (1984) 269.

K a t z , A . , Duneau,M., J . S t a t . Phys. 37 (1984) 257.

/ 3 / V e n t e v o g e l , M. J . , N i j b o e r , B.R.A.,~hys=a

99 (1979) 569.

/ 4 / Radin, C., J . S t a t . Phys. 28 (1984) 109.

/ 5 / K a s t l e r , D . , L o u p i a s , G . , ~ z k o u t , M . , M i c h e l , L . , Commun Math. Phys.

2 7

T6/

/ 7 / /8/

/ 9 / / 10 Den

(1972) 195.

KlCman, M . , M i c h e l , L. , Phys. Rev. L e t t . 40 (1978) 1387.

V i l l a i n , J . , Phys. Chem. S o l i d s 11 ( 1 9 5 9 ) 3 0 3 . Yoshimori, A . , J . Phys. Soc. J ~ 14 ~ (1959) 807. G

H e r p i n , A . , M e r i e l , P . , J . Phys. Rad. 2 2 (1967) 337.

/ De Wolf, P.M., A c t a C r y s t . A 30 (19745-777, Van A a l s t , M.

H o l l a n d e r , J . , P e t e r s e , W . I . A . M . , De Wolf, P.M., Acta Crys

( 1

Thanks t o Andreas D r e s s f o r d i s c u s s i o n on t h i s p o i n t .

C3-498 JOURNAL DE PHYSIQUE

/11/ Brown, H., Eiilow, R., Neubiiser, J., Wondratschek, H., Zassenhauss, H., Crystallographic groups of four dimensional space, Willey J., and Sons, New-York 1978.

/12/ Curtis, C.W., Reiner, I., Methods of Representation Theory with Application t o Finite Groups and Orders, Vol. I, Theor. 33.6, Wiley, J.,

and Sons New-York 1981.

/13/ Radin, C., J. Math. Phys. 26 (1985) 1342; Phys. Lett. 114 A (1986)

381; Commun Math. Phys. to appear.