INCOMMENSURATE STRUCTURE WITH NO AVERAGE LATTICE : AN EXAMPLE OF ONE DIMENSIONAL QUASI-CRYSTAL

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AVERAGE LATTICE : AN EXAMPLE OF ONE DIMENSIONAL QUASI-CRYSTAL

S. Aubry, C. Godreche

To cite this version:

S. Aubry, C. Godreche. INCOMMENSURATE STRUCTURE WITH NO AVERAGE LATTICE : AN EXAMPLE OF ONE DIMENSIONAL QUASI-CRYSTAL. Journal de Physique Colloques, 1986, 47 (C3), pp.C3-187-C3-196. �10.1051/jphyscol:1986319�. �jpa-00225730�

JOURNAL DE PHYSIQUE

Colloque C3, supplément au n o 7, Tome 47, juillet 1986

INCOMMENSURATE STRUCTURE WITH NO AVERAGE LATTICE : AN EXAMPLE OF ONE DIMENSIONAL QUASI-CRYSTAL

S. AUBRY* and C. GODRECHE* *

abor oratoire Léon Brillouin, CEN-Saclay.

F-91191 Gif-Sur-Yvette Cedex, France

* * Service de Physique du Solide et de Résonance Magnetique, CEN-Saclay, F-91191 Gif-sur-Yvette Cedex, France

Introduction and summary

Quasi-crystals are characterized by a discrete Fourier spectrum with a symmetry incompatlble with that of a crystal [Il. Unlike the standard incommensurate structures. they cannot be described as a crystal structure where the atomic positions are modulated as a periodic function of the space. These quasi-crystal structures can be considered as examples of "weakly peri.odic structures" the existence of which was conjectured i n an earlier paper 121.

In the one dimensional case, the projection method consists i n a projection of a strip of a two dimensional lattice on a line of arbitrary slope (F~Q. 2). However. Only standard one dimensional incommensurate structures with an average lattice periodically modulated i n space are thus obtained, for an irrational slope. In that sense such structures cannot be considered as one-dimensional quasi-crystals.

This talk is devoted to some considerations on the type of order that one finds in a one-dimensional quasi-lattice or i n an incommensurate structure with no average lattice. Details can be found i n Ref. 1141. The main results are the following : 1. I t i s possible t o find a simple structural mode1 describing a chain of atoms such that, i n the ground state, the fluctuation of the Positions of the atoms around their average lattice diverges logari thmical ly. The Fourier spectrum i s nevertheless discrete.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986319

2. This property i s related to a mathematical theorem due to Kesten i n Discrepancy Theory. We give simple arguments which illustrate the theorem in physical terms.

3. The model i s equivalent to a circle map with unusual mathematical properties : it i s ergodic, mixing and has zero entropy .

4. This map i s a natural generalization of the transformation obtained by the projection method i n one dimension.

These resul t s suggest extensions in higher dimension.

2. The model

We consider a chain of atoms j with coordinates u, such that the distance between two consecutive atoms i s 1 or 21,, , 1, being an arbitrary given number. This chain i s submitted to a periodic potential with period 2n, that for simpllcity we take

sinusoidal (Fig. 1). For a given concentration c of links 21, (not too large) we can find the ground state of the model.

We have to mlnimize the potential energy of the chain

@((u,))=L, coscu,, ( 1 )

on the space of configurations [uj) fulfilling the constraints : uj+,-u -1 or 21, _J- and c=c, given.

0 (21

-

Figure 1 :The model. Black dots are aroms, wnite ones vacancies.

This model describes a distribution of vacancies with fixed concentration in a periodic incommensurate potential (C=lO/2i1 is in general an irrational nurnber). I t i s a

particular case of the model studied i n Ref.[7] the purpose of which was the

understanding of the consequences of the non-convexity of the interactions i n the Frenkel-Kontorova model [2]. It can also be viewed as a simplified one-dimensional model corresponding t o systems i n two dimensions, such as the adsorbed monolayers of a mixture of two rare gas. Considering the ad-atoms as non interacting hard spheres which have two possible diameters, the problem of finding the ground-state of the close-compact monolayer i s similar and consists i n minimizing i t s energy i n the periodic potential of the substrate.

3. Generation of the ground state by a circle map T

The sites i are at the position v,=il,+ct where cx i s the location of the f i r s t site. In order t o have the smallest possible energy for the system, i t i s sufficient t o choose the location v, of the vacancies at the sites i which correspond t o the largest possible energy cos(vi). These sites are determined by a condition cos(v,)>cos(A/2). which defines a symmetric interval mod 2n, (called from now on the window) : W = ] -A/2, A / 2 [. Since the distribution of v, (mod 2111 i s u n i f o m on the interval [0,2n [, one has : A=2n c / ( 1 +cl. Therefore the sequence (u,) of atomic positions in the ground-state i s obtained by renumbering the sequence vi=ilo+vo after having withdrawn the t e m s vi which belong t o W (mod 2n). For instance, one would find :

vo VI v2 v5 v q v 5 vg vi ... v,, ...

U O U 1 y y ^U ',... U J . . . U . . . n

(uo=vo, u,=vl, u2=v3 ... 1 i f v2, vJ etc ... belong t o W mod 2n. Considering this interval W as an arc of the circle of length 2n, the mapping u'=T(u) of the circle ont0 itself defined as a piece-wise rotation by the following equations

u'= u +IO when u + 1, does not belong t o W (3-a) u'= u + 2 x 1, when u + 1, does belong t o W (3-b)

generates this sequence (u,). The f i r s t atom of the Chain i s at the arbitrary position u,=ct, which produces negligible boundary effects i n the l i m i t of an infinite system.

This sequence (uj) must f u l f i l l for al1 i the condition u,+,-ui= 1, or 2 l o t o be the ground-state of model Eq.( 1-2). This means that there exists no consecutive

vacancies, which implles

A< 1 and , A < 2n- 1, or equivalently c < 1, / ( 2 r - ^{1,) and c < (2n - ) 1, 1 1,}

4. Mathematical properties of the circle m a p T

Since i s irrational, the sequence {vi mod 2111 i s uniformly distributed on the circle.

Consequently the subsequence (uJ mod 2n) i s uniformly distributed on the complementary part W' of the window W on the circle.

This transformation T exchanges the order of three intervals, the union of which i s W' (see Ref.1141). The discrete dynamical system, the trajectorieç of which are defined by iterating the transformation T' on the initial point, has unusual properties. It i s erqodic w i t h respect t o the usual Lebesgue measure on the real axis. For most choice (in probability) of A and l,, T' i s also mixing as shown in ref, /IO]. The mixing property of a dynamical system i s usually associated w i t h strongly chaotic

trajectories and one usually expects i n that case a finite Kolmogorov entropy 1121. In fact i t can be shown that transformation f' has zero Kolmogorov entropy. This property can be related t o the fact that i t s unique Lyapounov number i s zero since

transformation T' i s measure-preserving (Pesin theorem (Ref.[l 11)). The behaviour of the trajectories i s not sufficiently chaotic t o produce a finite entropy. Transformation T' has also the property of being "minimal" [2]. This property means that the

trajectories for al1 initial point are dense on the whole set W' and implies that the ground state structure has the property of "weak periodicity" or equivalently of "local order at al1 scales".

Fluctuation of the ma0 T w i t h r e s ~ e c t t o a uniform averaae rotation maD.

Transformation T can also be written :

u'= T(u)= u+ ( 1 +x( u+l,)) 1, (4-a)

where ~ ( x ) i s the characteristic function of W, equal t o 1 when x belongs t o W and t o O when it does not. It i s also defined as

~(x)=lnt((x+A/2)/2n)-lnt([x-A/2)/2n) (4-b) where Int(x) denotes the integer part of x i.e. the largest integer G x.

The number of vacancies on the chain of s i tes IV,) i s '&=, ~(v,) and consequently, the position un of atom n i s equal to vN = N lo+vo where N f u l f i l l s :

n = N N &=,N~(vi) ( 5 )

Because the distribution of the sequence (vi mod 2111 on the circle i s uniform, the mean value of the characteristic function i s

limkw 1 /N &=, ^{x(vi) = A/2n}

A theorem due to Kesten [8,9] asserts that the fluctuation 6(N) = 1 /N

z=,

of the sum L,,~ x(vi) w i t h respect t o i t s expected value N A12n

i s bounded i n N i f and only i f the width A of W i s a multiple mod 2n of the rotation angle 1, that i s when A=r 1, mod 2n for some integer r. I f A does not f u l f i l l this condition, other theorerns (91 assert that S(N) diverges logarithmically as N grows.

5. Ground-state hull function

The ground-state of model ( 1 -2) appears as an incommensurate periodic modulation of the vacancy density. It i s similar to the kind of incommensurate structures which are formed for example i n certain alloys by antiphase boudaries [13]. By contrast, a standard displacive incommensurate structure i s described by a hull function which describes the modulation of the atomic positions from thelr average lattice site.

In one dimension, the simplest incommensurate structure (un) w i t h one atom per unit cell, i s given by :

un= uo + n 1 + g (q n) ( 6 )

where 1 i s the interatomic mean dlstance (l=<u,,, >) -LI, which determines the average lattice < un> = uo+ n 1, g(x) i s a 2n-periodic function which describes the modulation and q i s the wave-vector of this modulation. In order t o have a physical meaning, this function g(x) cannot be too pathologfcal. In the case of standard displacive

incommensurate structures, this function g(x) i s bounded and square summable i n order to have a discrete Fourier spectrum.

We show below that the ground-state of model ( t -2) cannot be described by a form ( 6 ) where g(x) i s a bounded function. The atomic mean distance i s a function of the

concentration c of bounds 2 lo and i s related t o A by A=2nc/( \ + c l : 1 = ( 1 -c) lo + c 2 1,

= 2n/(2n-A) l,,. The modulated average lattice, i f i t exlsts. i s defined by : <u,> = j 1 + u,,

. Suppose now that there exists a hull function g(x) such the ground-state be described by Eq.(61 for some wave-vector q. From Eq.(4) and (51, it cornes out that g(x) i s given for x=qn by : g(q n) = N 1, -nl = 1 6(N) where 6(N) i s the fluctuation

considered by the Kesten theorem. When A I r lo mod 2n, this fluctuation &(NI, and consequently the displacements of the atoms from their average position, are not bounded, which means that a bounded hull function g(x) does not exist. This result 1s

better understood by doing the analysis of the case where A=r 1, mod 2n = r 1, - ²ⁿ

/nt( r 1, /Zn) for some integer r. In that case, Eq.(4-b) gives : B(N) = &=,N x(vi) -N a 1211

= ' [ Frac[( i Io +vo-A12 ))/2n] - Frac [(itN) 1, +vo-A/2 ))/2II]

i=l

where Çrac(x) = x modulol. Thus : 6(N) = - [ Gr( N Io +vol - Gp(vo )]

The function G,(x) i s defined by Gr(x) = [ Frac[( x+ ilo-A/2)/2"]

and has period 2n. It i s a 'saw tooth" function, linear by part with a slope equal to r/2n and r discontinuities of amplitude - 1 per period Iocated at x,= (r/2-m) Io mod 2n (m= 1,2, ... r). Hence, when r+m, functions G(x) and 6(N), i f they exist, have non-bounded variations.

6. Fourier spectrum of the ground-state

Since the vacancy densi ty def ined by T i s periodical ly modulated, the Fourier spectrum of the ground state of mode1 ( 1 -2) i s discrete. The atomic density

~ ( r ) = Li 6 b u i ) =

xj

= [i - ~ ( r ) ] x & ^{8 ( r - j Io-a)}

i s the product of two periodic functions which have the periods 2n and Io respectively.

Consequently the structure factor S(K) = J exp(i ~ r ) p(r) dr

= ¹ l n L

,,

can be written as :

S(K)=N/n x 1 /N &, &,, a, exp [i(p+K)(m lo+uo)]

where a, are the Fourier coefficient of the 2n-periodic function 1 -x(x)=L, ,," a, exp(ipx1

def ined by

3 = 1 a n JO I -y(x)) exp(-ipx) dx

= - sin(pA/2)/pn for ptO

= I-A/ZII for P=O

Therefore the Fourier spectrum S(K) of the atomic density i s a sum of Dirac peaks

2 2

located at K,,, = -p+2qn/lo with intensities proportional to IS(I$,,W~ = (I+c) lad . ^As

i s the case for usual incommensurate structures, these vectors $, are obtained by

the rational combinations of two vectors of the reciprocal lattice :

K, ,= 2n/ l,, wave-vector of the one dimensional lattice i n the absence of any vacancies and K , B = l , wave Lector of the modulating 2n periodic sine potential.

Assuming that the ground-state lu,,) can also be described as a displacive

incommensurate structure, it i s possible t o prove that the width of the window W can be written i n the form D = r 1, mod 2n [14]. In other words there cannot exist any summable periodic hull function for the ground-state when A i s not a multiple of 1, (mod 2n) and that there exists one when A i s a multiple of 1,.

7. Connection with the projection method

Finally let us note that, i n the simple case A = 1, , the ground-state of Our mode1 i s an incommensurate structure isomorphic to the one obtained by the projection method [3-61. The Projection method in one dimension f i r s t consists in defining the strip of points of a two dimensional square lattice which are covered by al1 the translations of the unit square, parallel t o a given straight line D of slope t=tan(O). Next, the points contained in this strip are projected on the line D (Fig. 2). The dfstances between the consecutive points on the line are either c=cos(B) or s=sin(B).

It i s also poçsible to obtain the sequence of links (c, s) by a rotation on a circle. The length l + t of the circle i s equal t o the width of the section of the strip by the vertical axis. On this circle one defines a window of width unity (Fig. 2). The rotation w i t h a unit angle on the circle generates a sequence of points. By associating the links c t 0 the points which belong t o the window and the links s to the other points, one obtains the same sequence of links as w i t h the projection method. In conclusion, the sequence of links (c, sf i s generated by the transformation T defined i n Eq.(3) for 1, = A =

I /( 1 +tan 8 1 (except that now the length of the links are not double one from another).

One can describe more precisely the sequence of links denoted by (1,) as follows.

l i = c i f v , E W a n d l , = s i f v , L W

where v, = ⁱ 1, (Io= l / ( 1 +tN. Or : 1, = s( 1 -x(vil)+cx(vi) where 1 i s the characteristic function of W defined as :

~ ( x ) = Int(x)-lnt(x-1,).

1 i s given by :

Figure 2 : Equivalence between the projection method i n one dimension and a rotation on a circle. A i s the window.

1 = i i m N> ,(I/ &=; ^li

= s+(c-s)<x> = s+(c-SI/( 1 +t) = 1 /(s+c)

In the same manner one f inds the hull function for the sequence of links :

N N

g(N) = Li=, 1, - N 1 = z,=, ^[ s+(c-s)x(v~)] - ^N 1

= Ns + (c-s)lnt(N/( 1 +t)) - N 1 = (s-c) Frac(N/( 1 +t))

= (s-C) Frac(N c 1)

Therefore the structure generated by the projection method i n the one dimensional case i s a standard incornmensurate structure with a bounded periodic modulation. For that reason it is not the analogous of a quasi-crystal.

8. Concluding remarks

We studied here a model for which the ground-state i s obtained as an incommensurate modulation of a vacancy density (compositional modulation). For a discrete sequence of vacancy concentrations the same structure can also be described by a periodic crystal with an incommensurate displacive modulation. For other concentrations, corresponding t o the general case, the structure configuration i s generated by a one-dimensional map having unusual properties. The map i s ergodic, mixing and has zero Kolmogorov entropy. Thus, the type of order found i n the ground-state can be considered as intermediate between that of a standard incommensurate structure and that of a chaotic disordered structure.

The nature of the fluctuation of the position of the ith atom, defined by the distance between i t s actual position u, and i t s average position i l + u (where 1 i s the average distance between consecutive atoms), permits to distinguish between those degrees

of order. In the case of a standard incommensurate structure this fluctuation, described as a smooth function of the atomic position, i s bounded. For the model studied here, this fluctuation i s generally not bounded but i t s root mean square diverges proportionally t o Log(L1 as a function of the size L of the system. By

contrast, this root mean square diverges much faster and proportionnally to JL in the case of a chaotic distribution of atoms.

Although the model studied here i s very crude, it points out important aspects of the difference between the compositional incommensurate structures and the displacive

incommensurate structures. Extensions of this study t o two and three dimensions could help t o understand the format iqn of incommensurate ant iphase structures and of quasicrystals which both occur i n metallic alloys (e.g. AlTi and AlMn respectively).

Ref erences

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