VIBRATIONAL MODES IN A ONE DIMENSIONAL "QUASI-ALLOY" : THE MORSE CASE

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VIBRATIONAL MODES IN A ONE DIMENSIONAL

”QUASI-ALLOY” : THE MORSE CASE

F. Axel, J. Allouche, M. Kleman, M. Mendes-France, J. Peyriere

To cite this version:

F. Axel, J. Allouche, M. Kleman, M. Mendes-France, J. Peyriere. VIBRATIONAL MODES IN A ONE DIMENSIONAL ”QUASI-ALLOY” : THE MORSE CASE. Journal de Physique Colloques, 1986, 47 (C3), pp.C3-181-C3-186. �10.1051/jphyscol:1986318�. �jpa-00225729�

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abor oratoire d e P h y s i q u e d e s S o l i d e s , (UA 2 ) , Bâtiment 510.

U n i v e r s i t é P a r i s - S u d , F-91405 Orsay Cedex

a ab oratoire d e Mathématiques ( U A 226), U n i v e r s i t é de Bordeaux 1, 351, Cours de l a L i b e r a t i o n , F-33405 T a l e n c e . * * Equipe d ' A n a l y s e Harmonique (UA 757), B â t i m e n t 425, U n i v e r s i t é P a r i s - S u d , F-91405 Orsay Cedex

1 - INTRODUCTION

W e s t u d y t h e e f f e c t of d e t e r m i n i s t i c d i s o r d e r on t h e v i b r a t i o n a l d e n s i - t y o f s t a t e s and modes o f a one d i m e n s i o n a l e l a s t i c c h a i n . To t h i s end, we u s e a u t o m a t i c sequences and sequences g e n e r a t e d by a s u b s t i t u t i o n o p e r a t i n g on a two l e t t e r a l p h a b e t ( ( 0 , l ) o r ( a , b ) ) which have been i n v e s t i g a t e d and used by harmonic a n a l y s t s and number t h e o r e t i c i a n s /1, 2/. We g i v e two examples :

1) The F i b o n a c c i sequence g e n e r a t i n q a "ID P e n r o s e t i l i n g " ( w i t h o u t c o l o u r e d v e r t i c e s ) . The s u b s t i t u t i o n a i s d e f i n e d a s

a ( a ) = a b a (b) = abb

I t h a s non c o n s t a n t l e n g t h , t h e sequence g e n e r a t e d by r e p e a t e d l y a p p l y - i n g o i s q u a s i p e r i o d i c . Note t h a t t h e u s u a l 2 D P e n r o s e t i l i n g can be g e n e r a t e d by a s u b s t i t u t i o n o o p e r a t i n g on a l a r g e r a l p h a b e t .

2 ) The Thue-Morse sequence where a i s d e f i n e d by a ( a ) = ab

o ( b ) = ba and f o r i n s t a n c e o 4 ( a ) = abbabaabbaababba

i s net q u a s i - p e r i o d i c and c a l l e d au'tomatic b e c a u s e i t can a l s o b e gene- r a t e d by t r a v e l l i n g on t h e f o l l o w i n g 2-automaton s t a r t i n g £rom a : t h e f i f t h term of t h e s e q u e n c e , ( w h i c h h a s a z e r o t h t e r m l i s o b t a i n e d u s i n g t h e d e c o m p o s i t i o n i n b a s e 2 of 5 : 101. (The r e s u l t i s a ) . A l 1 t h e s e sequences a r e c o m p l e t e l y d e t e r m i n i s t i c ; t h e y h a v e z e r o entropyJ3/

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986318

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JOURNAL DE PHYSIQUE

Recently, the effect of quasi-periodic sequences on the properties of the spectrum of certain Schrodinger operators /4/ (for a review, see

/ 5 / ) andonthevibrational/6/ and electronic modes of discrete one

dimensional chains has been widely investiqated. We chose to concen- trate, instead, on the non quasi-periodic Morse sequence.

II

-

^THE"QUASI-ALLOY" MODEL

We study a chain of N = 2" masses and identical springs, with two different kinds of masses mo and ml.

The sequence of masses {m.) is such that the indices O and 1 are dis- tributed according, here, to ?Torse sequence. We coined the name of 3

"quasi-alloy" for this class of models by analogy with quasicrystal models where properties are diçtributed after Fibonacci or Fibonacci- like sequences. The u . beinq displacements, we look for time stationa-

1 ry solutions of

d 2 u .

(1) m 3 = K ( U ~ + ~ - u. - (u - u ) ) hence

j I j j-1

Let m ^{u 2} _po₌_{1, pl}_< 1, then one has

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When M h a s e i g e n v a l u e s o f modulus 1, p r o p a g a t i o n may o c c u r P

P and

( 7 ) c o s 8 = ¹Tr M ( w 2 ) P

where M (w2) i s a polynomial of d e g r e e p i n w 2 ( o r x) and 6 c h a r a c - t e r i z e s t h e r o t a t i o n of t h e wave f u n c t i o n p h a s e and p l a y s t h e r ô l e P o f t h e wave number i n t h e p e r i o d i c c a s e . One t h e n o b t a i n s t h e a n a l y t i c d i s p e r s i o n r e l a t i o n

I n one dimension, 6 , c o n v e n i e n t l y n o r m a l i z e d , i s a l s o t h e i n t e g r a t e d d e n s i t y of S t a t e s (IDS) / 9 / .

When Y h a s r e a l e i g e n v a l u e s , t h e phase i s blocked (qap), t h e

Ljapounov e x p o n e n t y i s non z e r o . I n t h e f i r s t c a s e P

I T ~

^Y ( < 2 , i n t h e second [ Tr M [ > 2 P

P

III

-

THE TRACE MAPPING THEOREM AND ITS APPLICATIONS

The fundamental r ô l e o f t h e t r a c e o f t h e mapping, which c o n t r o l s t h e b e h a v i o u r o f t h e phase 6 , t h e g a p s , t h e Ljapounov e x ~ o n e n t and t h e e s c a p e p r o p e r t i e s /Il/, h a s prompted two of u s t o i n v e s t i a a t e more g e n e r a l l y t h e p r o p e r t i e s o f the t r a c e of s u c h a m a t r i x p r o d u c t

f o r a c l a s s of s u b s t i t u t i o n s o ./IO/

Theorem : L e t o b e a s u b s t i t u t i o n on a two l e t t e r a l p h a b e t (a,b).

Then t h e r e e x i s t s a polynomial m a p @ : R 5 -+ R 5 w i t h i n t e a e r c o e f f i -

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JOURNAL DE PHYSIQUE

Fiq. 1 : The intesrated density of states 6 ( w 2 ) for a Morse elastic chain of 2' = 512 masses m and m l _O

, -

- - 0.8

mo

site number n

Fig. 2 : Normalized mode for x = 6.487 in a Morse elastic c h a h of Z7 = 128 sites with 1 ₌_0.5

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a c c u r a t e n u m e r i c a l c a l c u l a t i o n s . N b t e t h a t t h e theorem y i e l d s , f o r t h e F i b o n a c c i sequence t h e t r a c e mapping

( 1 0 ) t n + l - 2 t n t n - l - t n - 2 -

w i t h t h e q u a n t i t y

( I l ) 1 =

-

¹⁺^t; ⁺^t;-1 ⁺^t:4

-

2 t n tn-l tn-2

i n d e p e n d e n t of n. ( 1 0 ) and (11) have p r e v i o u s l y been found by s e v e r a l a u t h o r s / I l , 1'2, 13/.

The IDS f o r a v o r s e e l a s t i c c h a i n of 2' = 512 s i t e s d e r i v e d £rom ( 8 ) i s shown on F i q u r e 1. O b s e r v a t i o n o f a s u c c e s s i o n o f i t e r a t e s i n d i c a t e s t h a t gaps i n c r e a s e i n nuinber b u t a l s o s t a b i l i z e . The e x i s t e n c e o f a s e l f s i m i l a r s t r u c t u r e /14/ 1s a l r e a d y obvious a t t h i s s t a q e /7/.

IV

-

^-IODES

The nodes o f a Morse e l a s t i c c h a i n of 2 n s i t e s a r e s t u d i e s u s i n g t h e s y m e t r i c t r i d i a g o n a l d - - n a m i c a l m a t r i x deduced trom t h e 2" e q u a t i o n s ( ô ) w i t h f i x e d end boundary c o n d i t i o n s :

Having d e r i v e d 2" f r e q u e n c i e s w 2 one n u m e r i c a l l y c a l c u l a t e s t h e c o r r e s - 3

ponding modes. F i g u r e 2 shows an example of such a mode, l o c a l i z e d / l 5 A t h a t h a s a non t r i v i a l d e c r e a s e from t h e c e n t e r peak. Analoqous wave f u n c t i o n s have been d e s c r i b e d i n c e r t a i n q u a s i p e r i o d i c s i t u a t i o n s /16/.

The a u t h o r s t h a n k Louis I l i c h e l and Denis G r a t i a s f o r t h e i r k i n d h o s p i t a l i t y d u r i n g t h e workshop a t Les Houches. F. A. t h a n k s

P r o f e s s e u r J. F r i e d e l f o r h i s i n t e r e s t and f o r s t i m u l a t i n q d i s c u s s i o n s d u r i n g t h e course o f t h i s work.

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