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SUPERSPACE EMBEDDING OF 1-DIMENSIONAL QUASICRYSTALS
A. Janner
To cite this version:
A. Janner. SUPERSPACE EMBEDDING OF 1-DIMENSIONAL QUASICRYSTALS. Journal de Physique Colloques, 1986, 47 (C3), pp.C3-95-C3-102. �10.1051/jphyscol:1986309�. �jpa-00225719�
JOURNAL DE PHYSIQUE
Colloque C3, s u p p l 6 m e n t a u n o 7, Tome 47, juillet 1986
SUPERSPACE EMBEDDING OF 1-DIMENSIONAL QUASICRYSTALS
A . JANNER
Instituut voor Theoretische Fysica, ~niversiteit Nijmegen, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands
Rdsum6 - On consid6re l e plongement de; q u a s i c r i s t a u x & une dimension dans un espace . Euclidien bidimensionnel . Les conditions pgur a b o u t i r 21 des r6seaux 1som5t-riques ( c a r r 6 e t hexagonal) f o n t a p p a r a i t r e deux p a r m & t r e s
qui c a r a c t 6 r i s e n t l a c l a s s 8 d'lsomorphisme l o c a l . Ces p6mes parami?tres d8termi'nent l a s t r u c t u r e ' des modules l i b c e s engendr6s par l e s compogantes e x t e r n e s e t - c e l l e s - i n t e r n e s d e s , vecteurs r 6 t i c u l a i r . e ~ de ,l'espape d i r e c t e t de c e l u i r6ciproque. La P y r r h o t i t e Fel-xS e s t t r a i t e e c m e exemple.
Abstract -
considered .'
hexagonal )
The embedding of 1D q u a s i c r y s t a l s i n a. 2D EuClidean space is The c o n d i t i o n s f o r g e t t i n g i s o m e t r i c l a t t i c e s (square and rameters c h a r a c t e r i z i n g t h e l o c a l isomorphism c l a s s . They a l s o determine th'e s t r u c t u r e of t h e f r e e modules generated by t h e i n t e r n a l and t h e e x t e r n a l components', r e s p e c t i v e l y , of t h e l a t t i c e v e c t o r s i n d l r e c t and i n r e c i p r o c a l space; ~ y r r h o t i t e Fel-xS is t r e a t e d a s .an example.
I - INTRODUCTION
The new c l a s s of c r y s t a l s t r u c t u r e s g e n e r a l l y i n d i c a t e d by-the name of q u a s i c r y s t a l s s h a r e s t h e fundamental property of a l l o t h e r c r y s t a l s known s o f a r of having F o u r i e r wave vectors e x p r e s s i b l e a s i n t e g r a l l i n e a r combination of a f i n i t e number of fundamental ones. When t h e s e b a s i s wave vectors a r e l i n e a r l y independent onL t h e r a t i o n a l s but l ~ i n e a r l y dependent on t h e r e l a t t l c e p e r i o d i c i t y is l o s t , t h e c r y s t a l i s a p e r i o d i c ( a t l e a s t p a r t i a l l y ) commonly 'called incommens~irate.
' Q u a s i c r y s t a l s belong t o t h i s c l a s s .
By bringing t h e fundamental wave v e c t o r s i n 1-to-1 correspondence with t h e b a s i s v e c t o r s of a l a t t i c e i n a higher dimensional Euclidean space ( c a l l e d s u p e r s p a c e ) , i t is p o s s i b l e t o embed t h e ' whole c r y s t a l s t r u c t u r e and t o recover space group symmetry even i n t h e incommensurate gase. I n present superspace appr'oach t h e o r i g i n a l s t r u c t u r e is r e l a t e d t o t h e embedded one by orthogonal projection_ i n
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986309
C3-96 JOURNAL DE PHYSIQUE
r e c i p r o c a l space and i n t e r s e c t i o n i n d i r e c t space /1/,/2/.
The c r y s t a l l o g r a p h i c symmetry of q u a s i c r y s t a l s has a l s o been recovered by embedding i n a higher dimensional Euclidean space following methods developed f o r t h e a l g e b r a i c c h a r a c t e r i z a t i o n of a p e r i o d i c t i l i n g of t h e Penrose t y p e /3/. One of t h o s e methods, t h e s o - c a l l e d d i r e c t p r o j e c t i o n approach, i s conceptually very s i m i l a r t o t h e superspace one, but involves p r o j e c t i o n and i n t e r s e c t i o n i n t h e r e c i p r o c a l as well a s i n t h e d i r e c t space. It has been shown by Janssen t h a t t h e two methods a r e a c t u a l l y equivalent /4/. I n t h e superspace approach, q u a s i c r y s t a l s appear a s modulated c r y s t a l s having discontinuous modulation l i n e s and a b a s i s s t r u c t u r e not n e c e s s a r i l y commensurate. P h y s i c a l l y speaking, t h i s can be expressed by s a y i n g t h a t t h e Bragg r e f l e c t i o n peaks of t h e d i f f r a c t i o n p a t t e r n of a q u a s i c r y s t a l do not s p l i t ( i n g e n e r a l ) i n t o a s u b s e t of 'main r e f l e c t i o n s ' on a r e c i p r o c a l l a t t i c e and one of a d d i t i o n a l s a t e l l i t e r e f l e c t i o n s a t incommensurate p o s i t i o n s . Accordingly, q u a s i c r y s t a l s r e p r e s e n t a new c l a s s of incommensurate c r y s t a l s t r u c t u r e s . This opens t h e p o s s i b i l i t y of point group symmetries which a r e c r y s t a l l o g r a p h i c i n a higher dimensional space only, a s it has been observed i n icosahedral quasi c r y s t a l s .
There is another new f e a t u r e , not shared by a l l q u a s i c r y s t a l s but t y p i c a l f o r t h o s e of the Penrose type: t h a t of being a s s o c i a t e d w i t h an hypercubic l a t t i c e , d e s p i t e t h e f a c t t h a t t h e d i f f r a c t i o n p a t t e r n i n space never has t h e f u l l hypercubic point group symmetry. The relevance of higher dimensional isometric l a t t i c e s has always been a b a s i c concern within t h e superspace approach because not o c c u r r i n g f o r incommensurate c r y s t a l s with main r e f l e c t i o n s on l a t t i c e p o s i t i o n s mapped o n t o themselves by symmetry transformations .
T h i s paper i s a preliminary i n v e s t l g a t i o n on t h e m e t r i c a l r e l a t i o n s one can a s s o c i a t e t o t h e embedding of an incommensurate c r y s t a l s t r u c t u r e . The c o n s i d e r a t i o n s are r e s t r i c t e d t o a 2D embedding of 1D q u a s i c r y s t a l s and t h e a t t e n t i o n i s focussed on l a t t i c e symmetry. The point group and t h e space group symmetry of the embedded s t r u c t u r e , a s well as the 2D and 3D q u a s i c r y s t a l c a s e r e p r e s e n t t h e n e x t s t e p s .
I1 - SUPERSPACE EMBEDDING
Considered is a 1D q u a s i c r y s t a l described by t h e d e n s i t y d i s t r i b u t i o n :
where n l , n2 a r e i n t e g e r s , a l , a 2 have an i r r a t i o n a l r a t i o and C(nln2) is a c h a r a c t e r i s t i c f u n c t i o n t a k i n g values 0 and 1 and ensuring a f i n i t e minimal d i s t a n c e between atomic p o s i t i o n s as well a s an i n f i n i t e c r y s t a l extension.
These c o n d i t i o n s can be r e a l i z e d by t h e s o - c a l l e d d i r e c t p r o j e c t i o n method, where p o i n t s of a 2D l a t t i c e w i t h i n a s t r i p of width D a r e p r o j e c t e d p e r p e n d i c u l a r l y t o t h e 1 D c r y s t a l l i n e . A l a t t i c e b a s i s a l s , a2s then p r o j e c t s onto a l and a
2' r e s p e c t i v e l y , and decomposes a c c o r d i n g t o :
with a i I t h e s o - c a l l e d i n t e r n a l components, and a . t h e e x t e r n a l ones. Choosing a p p r o p r i a t e i n i t i a l c o n d i t i o n s one t h e n has :
with 0 a s t e p f u n c t i o n a t t = 0. I n such a c a s e t h e F o u r i e r components of p ( r ) a r e given by:
Y *
with z , , z 2 i n t e g e r s and a l , a2 t h e ( e x t e r n a l ) space components of t h e r e c i p r o c a l b a s i s v e c t o r s a?s and a*
2s:
Y *
a:. = (a,, a ,I), a s a v = 6 , 1 ~ p p , v = 1 , 2 .
I n eq.(4) ? ( z l z 2 ) i s t h e c h a r a c t e r i s t i c f a c t o r given by:
From e q . ( 4 ) f o l l o w s t h a t s u c h a q u a s i c r y s t a l s a t i s f i e s t h e c o n d i t i o n s f o r Z-module c r y s t a l l o g r a p h y /5/.
Knowing t h e c r y s t a l s t r u c t u r e and its d i f f r a c t i o n p a t t e r n , t h e r e c i p r o c a l
Y *
f r e e Z-module M* = { a l , a 2 } f o l l o w s from t h e p o s i t i o n s of t h e Bragg peaks; t h e d i r e c t one M = [ a l , a 2 ] from t h e p o s s i b l e atomic p o s i t i o n s . Furthermore, from t h e s e t of i n t e n s i t i e s d e s c r i b e d by ? ( z l z 2 ) one g e t s ~ a and ~ a g ~ , y ~ and from t h e s e t of occupied atomic p o s i t i o n s given by C(nln2) a l s o alI/D and a21/D. Accordingly, a q u a s i c r y s t a l s t r u c t u r e determines i n a d d i t i o n t o M and MY a l s o t h e values of a *
u a v l f o r p , v = 1 , 2 , i .e. t h e two r e a l parameters o and p defined by:
JOURNAL DE PHYSIQUE
From d u a l i t y one g e t s t h e corresponding m e t r i c a l r e l a t i o n s i n d i r e c t space:
a l = pa2 and aZI = - aa 1 1
' (8)
These parameters a r e t h e same as t h o s e considered by Socolar and S t e i n h a r d t /6/. I n t h e i r paper q u a s i c r y s t a l s a r e c o n s i d e r e d having atomic p o s i t i o n s a t
w i t h n i n t e g e r , a , 6 and p r e a l , a i r r a t i o n a l and l....l t h e g r e a t e s t i n t e g e r f u n c t i o n . Denoting by 0 t h e a n g l e between a l s and t h e e x t e r n a l space d i r e c t i o n , and by q5 t h a t between a l s and a2s, t h e same p o s i t i o n s can be expressed by:
which i m p l i e s e q . ( 8 ) f o r a ? = a . The a d d i t i o n a l parameters a and 8 n o t c o n s i d e r e d above i n t h e m e t r i c a l r e l a t i o n s f o r t h e l a t t i c e embedding do not modify t h e l o c a l isomorphism c l a s s of t h e q u a s i c r y s t a l , which is f i x e d by a and p. A s explained i n / 6 / , a determines t h e r e l a t i v e frequency of t h e two s p a c i n g s a and a(1 + l / p ) between neighbouring atomic p o s i t i o n s i n t h e q u a s i c r y s t a l sequence. The o n l y a d d i t i o n a l parameter (up t o t h a t of a u n i t of l e n g t h expressed by a = l / a * ) a p p e a r i n g i n t h e embedding can be i n d i c a t e d by a l I = Xal. One t h u s a r r i v e s a t t h e f o l l o w i n g c a n o n i c a l p a r a m e t r i z a t i o n of t h e q u a s i c r y s t a l embedding:
a * = - a* cop, 11x1, a ; s = *(I
1 s l + o p l + o p , -1 1 , with aa* = 1. ( 1 1 )
I11 - TWO DIMENSIONAL BRAVAIS LATTICES
I t is not s e l f - e v i d e n t t h a t 2D Bravais l a t t i c e s can be r e l e v a n t f o r t h e embedding Of 1D q u a s i c r y s t a l s . Let us n e v e r t h e l e s s consider t h e c o n d i t i o n s imposed by t h e embedding i n t o one of t h e f i v e 2D B r a v a i s l a t t i c e s , d i s r e g a r d i n g i n t h i s e x p l o r a t o r y paper t h e q u e s t i o n of a r i t h m e t i c equivalence ( e x p r e s s i n g t h e freedom i n choosing t h e l a t t i c e b a s i s ) .
No c o n d i t i o n s a r e imposed by an Oblique l a t t i c e . Those f o r g e t t i n g a Rectangular l a t t i c e can be expressed by als.a2s = 0, implying
A s one can s e e from e q . ( 9 ) one can choose o , p > 0 without l o s s of g e n e r a l i t y . Therefore any I D q u a s i c r y s t a l a s above admits a r e c t a n g u l a r embedding. This i s , however, not t h e end of t h e s t o r y , a s i n s u p e r s p a c e embedding t h e d e s c r i p t i o n may i n f l u e n c e t h e s t r u c t u r a l r e l a t i o n s made e x p l i c i t / 7 / .
The c o n d i t i o n f o r g e t t i n g a Diamond l a t t i c e can be c a s t i n t o a:s = a:s l e a d i n g t o t h e r e l a t i o n :
implying t h a t u2 and p2 both have t o be e i t h e r l a r g e r o r s m a l l e r t h a n one. It appears t h a t , i f not a l r e a d y t r u e , t h a t c o n d i t i o n can be managed by changing t h e s i g n of p, i . e . i n t e r c h a n g i n g t h e r o l e of s h o r t and l o n g i n t e r v a l s . Therefore a diamond embedding is a l s o always p o s s i b l e .
One g e t s s t r u c t u r a l c o n d i t i o n s i n terms of o and p o n l y , f o r t h e embedding i n t o t h e two i s o m e t r i c B r a v a i s l a t t i c e s , t h e s q u a r e and t h e hexagonal one.
A Square l a t t i c e s a t i s f i e s both e q s . ( l 2 ) and (13) l e a d i n g t o t h e c o n d i t i o n :
2 2
P - 1 0 - 1 A
P u
imp1 ying
The parameter A i n e q . ( l 4 ) has been i n t r o d u c e d because i t appears i n t h e corresponding c h a r a c t e r i s t i c e q u a t i o n determining t h e i n f l a t i o n r u l e of t h e q u a s i c r y s t a l ( f o r A r a t i o n a l ) /8/. I n t h e c a s e of o = p t h e c h a r a c t e r i s t i c e q u a t i o n is given by:
which l e a d s t o t h e p a r a m e t r i z a t i o n :
The corresponding i n f l a t i o n r u l e i s Cn+, = c $ ~ - ~ . For A = 1 one o b t a i n s t h e well-known Fibonacci t i l i n g .
I n t h e c a s e of an Hexagonal l a t t i c e , one can c o n s i d e r it a s rhombic with an a n g l e 4 = ~ / 3 , i . e . a l s . a 2 s = %a:s = This i m p l i e s :
JOURNAL DE PHYSIQUE
with t h e parameter A a p p e a r i n g i n t h e corresponding c h a r a c t e r i s t i c e q u a t i o n
One then g e t s t h e s t r u c t u r a l c o n d i t i o n f o r hexagonal l a t t i c e embedding:
The s i m p l e s t s o l u t i o n i s o b t a i n e d f o r A = 1 g i v i n g t h e v a l u e s o = );( -1 + J? ) and p
= 20. Another s i m i l a r s o l u t i o n f o l l o w s f o r A = - y i e l d i n g o = ( 1 + J? ) and p =
$0.
The d i s c u s s i o n of t h e c a s e $ = 2n/3 i s q u i t e analogous and r e s u l t s i n a simple replacement of A by - A , and a change of t h e s i g n of o and of p i n e q . ( 2 0 ) . I V - MODULATED STRUCTURE
Q u a s i c r y s t a l s with atomic p o s i t i o n s a s i n e q . ( 9 ) can a l s o be e x p r e s s e d i n terms of a b a s i c s t r u c t u r e and a modulation:
with f r a c ( x ) t h e f r a c t i o n a l p a r t of x. Comparing e q s . ( l l ) and (21 ) one s e e s t h a t t h e
* *
r e c i p r o c a l wave v e c t o r s of t h e b a s i c s t r u c t u r e a r e generated by a l = a o p / ( l + o p ) ,
I * *
whereas t h e modulation wave v e c t o r s a r e generated by q = -a = a 2 . The Z-module M*
0 1
is t h e r e f o r e e x p r e s s i b l e i n terms of t h e same fundamental p e r i o d i c i t i e s a s i n t h e previous d e s c r i p t i o n , a s i t should be.
From e q . ( 2 1 ) i t follows furthermore t h a t t h e l a t t i c e d i s t a n c e i n t h e average s t r u c t u r e (which i s t h e b a s i c s t r u c t u r e a l s o ) i s given by:
Consider now a r a t i o n a l approximation f o r a , and denote, a s c o n v e n t i o n a l , t h e two s p a c i n g s between neighbouring p o s i t i o n s of t h e q u a s i c r y s t a l by:
a = S and a ( l + l / p ) = L .
Then f o r t h e u n i t c e l l i n t h e s u p e r s t r u c t u r e approximation one has:
n S S + n L L = (nS + nL) A, nS and nL i n t e g e r s .
One f i n d s indeed t h a t o e x p r e s s e s t h e r e l a t i v e frequency of S and L i n t e r v a l s
V - AN EXAMPLE: PYRRHOTITE
The c r y s t a l s t r u c t u r e of P y r r h o t i t e , Fel-xS is t h a t of a modulated c r y s t a l involving d i s p l a c i v e a s well as occupational modulation. The modulation wave vector q is o r i e n t e d a l o n g t h e hexagonal a x i s of t h e average s t r u c t u r e and r e l a t e d t o t h e compositional parameter x i n a simple way:
Such a compound i s denoted a s NC-Pyrrhotite. I n g e n e r a l , i t is thus an incommensurate c r y s t a l . Its (3,l)-dimensional superspace group symmetry has been determined, a s well as the c h a r a c t e r i s t i c s of t h e modulation waves /9/. Here t h e d i s p l a c i v e p a r t o n l y w i l l be considered. I t appears t h a t , within a good approximation, neighbouring Fe atoms along t h e c - d i r e c t i o n i n v o l v e two i n t e r v a l s only. D i s r e g a r d i n g t h e o t h e r two c r y s t a l l o g r a p h i c a x e s ( l y i n g i n t h e hexagonal p l a n e ) , P y r r h o t i t e can t h e r e f o r e be d e s c r i b e d a s a I D q u a s i c r y s t a l . I n r e f . /9/ t h e N = 5.5 c a s e is considered i n more d e t a i l ; within t h e u n i t c e l l t h e r e a r e 9 l o n g and 2 s h o r t i n t e r v a l s having r a t i o approximatively 1.0357. T h e r e f o r e , f o r e n s u r i n g t h e conventions adopted above ( o and p p o s i t i v e ) one has t o choose q = 9 c w / l l i n s t e a d of 2c*/11 a s i n e q . ( 2 6 ) . One t h e n g e t s t h e f o l l o w i n g s e t of parameters f o r t h e 5.5C P y r r h o t i t e :
n s = 2 . n L = 9 , o =11/9 and p - 28 . (27)
Embedding i n t o both t h e r e c t a n g u l a r o r t h e rhombic l a t t i c e is p o s s i b l e , but embedding i n t o t h e i s o m e t r i c ones, s q u a r e o r hexagonal, i s excluded. Indeed both t h e e q s . ( l 6 ) and ( 2 0 ) a r e not s a t i s f i e d . The same conclusion f o l l o w s from t h e d a t a one g e t s by choosing t h e q = 2c*/11 d e s c r i p t i o n .
V I - FINAL REMARKS
The s t a n d a r d embedding of a modulated s t r u c t u r e a s i n e q . ( 2 1 ) is given by:
JOURNAL DE PHYSIQUE
a = ( l / a a J , - b / a ) , bs = (0, b ) , with b*b = 1 ,
s o t h a t o n l y t h e r e c i p r o c a l 2-module M* = a;{l,o} appears. On t h e c o n t r a r y , t h e canonical q u a s i c r y s t a l embedding of eq. ( 1 1 ) , t a k e s t h e e q u i v a l e n t form:
which r e v e a l s an i n t e r t w i n n i n g of d i r e c t and r e c i p r o c a l s p a c e with r e s p e c t t o t h e i n t e r n a l and e x t e r n a l components. These g e n e r a t e t h e f o l l o w i n g f r e e modules:
showing t h e i n t e r t w i n n i n g .
ACKNOWLEDGEMENTS The very s t i m u l a t i n g d i s c u s s i o n s w i t h E. Springelkamp helped t o c l a r i f y t h e r o l e of d i f f r a c t i o n i n t e n s i t i e s i n t h e m e t r i c a l embedding's c o n d i t i o n s and a r e g r a t e f u l l y acknowledged.
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De B r u i j n , N.G., Proc. Kon. Ned. Ac. Wet. A84 (1981) 39.
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