Construction of average lattices for quasiperiodic structures by the section method

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Construction of average lattices for quasiperiodic

structures by the section method

C. Godrèche, Christophe Oguey

To cite this version:

Construction of

_average

lattices for

_{quasiperiodic}

structures

_by

the

section method

C. Godrèche

₍₁₎

and C.

_Oguey

₍₂₎

(1)

Service de

_Physique

du Solide et de Résonance

_Magnétique,

CEA,

Saclay,

F-91191 Gif-sur-Yvette _Cedex, France

(2)

Centre de

_{Physique Théorique,}

Ecole

_{Polytechnique,}

F-91128 Palaiseau _Cedex, France

(Reçu

le 21

_juillet

1989,

accepté

le 18

_septembre

₁₉₈₉₎

Résumé. 2014 La

construction d’un réseau moyen avec une modulation bornée pour des pavages

quasipériodiques

unidimensionnels est considérée du

_point

de vue de

l’espace

de dimension

supérieure R2.

Les pavages

quasipériodiques

sont:

_a)

_{le pavage}

_canonique

1D _{obtenu par}

exemple

par la méthode de coupe et

_projection,

_b)

_{des pavages}

_engendrés

_par un

_algorithme

du

cercle pour des valeurs

particulières

des

paramètres

définissant le modèle. Dans ce dernier cas, la construction comble un vide entre la méthode de coupe et

_projection

ou de section et le modèle de

_{l’algorithme}

du cercle et _apporte une autre _{preuve de l’ordre}

_{quasipériodique :}

nous

construisons des pavages 2D

périodiques appropriés

donnant les pavages 1D

quasipériodiques

par section. Cette

_approche

_{géométrique}

donne aussi une

_image

intuitive du mécanisme de la

disparition

du réseau moyen et de l’ordre

_{quasipériodique}

_{pour des valeurs}

_génériques

des

paramètres

du modèle. Les considérations données ici _peuvent servir de base à la construction de réseaux moyens avec modulation bornée, s’ils existent, pour des pavages de dimensions

supérieures.

Abstract. 2014

The construction of an _{average lattice with bounded} modulation, for one

dimensional

_{quasiperiodic tilings,}

is considered from the

_viewpoint

of the

_higher

dimensional space

R2.

The 1D

_{quasiperiodic tilings}

are :

_a)

the canonical 1D

_tiling

_{obtained e.g.,}

by

the cut and

project

method,

b)

tilings generated by

a _{circle map}

_algorithm,

for

_particular

values of the

parameters

defining

the model. In this last case, the construction

bridges

a _{gap between the} cut and

_project,

or _{section, methods, and the circle map} model, and

_provides

an alternative

_proof

of the

_{quasiperiodic ordering :}

we build suitable 2D

_periodic

_{tilings yielding}

the

_{quasiperiodic}

ones

by

section. This

_{geometrical approach gives}

also an intuitive

image

of the mechanism of the

disappearance

of the average latice, and of the

_{quasiperiodic ordering,}

for

_generic

values of the

parameters of the model. The considerations

_given

_{here may} serve as a basis for the construction

of average lattices with bounded modulation, if

they

exist, of

higher

dimensional

tilings.

Classification

Physics

Abstracts 61.50 - 02.40

1. Introduction.

One of the characterisations of order in one-dimensional incommensurate structures is obtained

_by

_considering

the

_question

of the existence of an _{average lattice. In} some sense, the

existence of an _{average lattice in} a 1D structure

_gives

a measure of its distance to

_crystalline

order,

and may

help

to

_classify

different

_aperiodic

structures, such as conventional

incommensurate structures or other more

_complex

_ones, with

_possibly

weaker order. Take a

distribution of atoms on a

_line,

for which it is

_possible

to define a mean interatomic distance or

inverse

_{density a ;}

if the deviation of the

_position

_{un of the n-th} atom with

_respect

to its average

position

na is

_bounded,

the structure is said to _possess an _average lattice

_[1].

The bounded deviation is called the modulation. For

instance,

un may be

given by

where

_{g (x )}

is a

_b-periodic

_function,

called the modulation

_{(or hull)}

function. In this case, the

Fourier

_spectrum

is discrete with resonant wavevectors at

_integer

combinations of 2

_{’TT la}

and 2

_’TT/b.

In other

words,

the structure is

_{quasiperiodic.}

_Examples

of 1D structures

which possess an _{average lattice with bounded modulation} are

_{given by :}

_a)

the

_ground

states

of the Frenkel-Kontorova model

_{[2], b)}

the so-called

_{« integrable}

cases » of a _{circle map}

model

_{[3, 4], c)}

structures obtained

_by

the cut and

_project

method

_[5-7]

from 2D to

_1D,

for suitable values of the width of the

_strip.

In all these cases, the modulation

_{g (x )}

is a

_periodic

function and the resultant structure is

_{quasiperiodic.}

On the other

_hand,

in recent

_works,

two

_examples

of structures _{without average lattice have}

been

_given.

These two

_examples

are built from the same

_{quasiperiodic}

_rule,

_involving

a circle

map which

generates

a

_binary

_{sequence of 0 and l’s. In the first}

_example

_{[1] («}

model 1 », or

« atoms and vacancies model

_»),

atoms and vacancies are associated to the

_binary

_sequence. In the second

_example

_{[3, 4] («}

model 2 », or « circle map

tiling

»),

tiles are associated to the sequence. For both

models,

in non

_integrable

cases,

i.e.,

for

_generic

values of the

_parameters

of the circle map

algorithm,

the structure does not _possess an _{average lattice.}

_Yet,

in the first

example,

the structure is

_{quasiperiodic,}

whereas in the second case, the

disappearance

of the average lattice has

strong

consequences on the Fourier

_{spectrum :}

the

_intensity

_spectrum

no

longer

contains any Dirac

peak

and it was claimed in references

_{[3, 4]}

that this structure, that may be named « almost

quasiperiodic »,

has a

_singular

continuous Fourier

_spectrum.

These results show the

_interplay

between two characteristic

_properties

of deterministic structures,

namely :

(1)

a bounded fluctuation of the atomic

_positions

with

respect

to their average

lattice ;

(2)

a

_{quasiperiodic}

Fourier

_spectrum.

The aim of this paper is to

_give

a

_{geometrical interpretation}

of this

_interplay

in a

_higher

dimensional space, here

R2,

from which the 1D structure

_{(or tiling)}

is recovered

_by

a section

method. In order to do so, a natural direction to

_explore

consists in

_trying

to cast the circle map

algorithm

in the section method framework. This seems a

_priori

possible,

at least for the

integrable

cases, since

they give

rise to

_{quasiperiodic}

structures. The section method

_gives

a

direct characterisation of the

_{quasiperiodic ordering :}

_{quasiperiodic}

_tilings

are sections of

periodic

ones,

just

as

_{quasiperiodic}

functions are sections of

_periodic

functions in several variables. The discrete nature of the Fourier

_spectrum

of

_{quasiperiodic}

_tilings

is a direct consequence of this fact.

More

_{precisely :}

i)

we first

show,

in section

2,

how to _construct,

_by

_section,

_{the average} lattice and the modulation for the canonical 1D

_tiling.

This is of

_importance,

since it is a first

_step

towards the

construction _{of the average}

lattices,

when

_they

exist,

of

_higher

dimensional

_tilings.

In

particular,

the construction

_given

here is

_easily

extendable to all codimension one _cases,

_{i. e. ,}

projection

from a N-dimensional _space to a

_{(N - 1 )-dimensional}

_{one ;}

ii)

in section

_3,

this construction is then

used,

together

with a renormalisation

_procedure,

2. From 1D to 2D and back.

_Average

lattice and modulation.

In this

_section,

we first recall model

2,

the _{circle map}

_tiling.

For

_particular

values of its

parameters,

this model is

_equivalent

to the

_projection

method from 2D to

_1D,

and

_generates

a

quasiperiodic tiling

which may, as

_well,

be viewed as a standard

_displacive

incommensurate

1D structure,

having

an _{average lattice with bounded modulation.}

We then show how to construct _{the average lattice and the} modulation of the

_position

of atoms, of this

particular tiling,

from the

_higher

_{dimensional space}

(R2.

2.1 MODEL

2,

A REMINDER. - This model

_{[3, 4],}

based on a _{circle map}

_algorithm,

_generates

a 1D atomic structure or a

_tiling

of the line. The structure is defined

_by

_putting

atoms on a

line,

the abscissa of the nth atom

_{being given by}

(uo

=

0).

The bond

lengths Àn

_are

given

by

the action of _{a «} window » function of width L1 on

the sequence

{nw

mod

_{1} ; w}

and L1 are

_given

numbers between 0 and 1 :

This is illustrated in

_figure

1.

_{X a (x )}

is a

_1-periodic

function defined

_by

Int

_(x)

and Frac

_(x)

are the

_integer

and fractional

_parts

_{of x,}

_{respectively.}

The

_binary

sequence

{Xn = XA (nw )}

is thus

quasiperiodic.

Fig.

1. - Generation of the

binary

sequence

{Xn}’

and of a

_{tiling, by}

a circle map. The sector

We have

The interatomic mean

_distance,

or inverse

_density,

of the model is

The modulation of the atomic abscissas with

_respect

to _{their average lattice}

_{(na )}

_{may be}

computed

as

8n

is the fluctuation of

_Sn,

the number of letters 1 after n

_steps,

with

_respect

to _{its average}

value na.

A

_theorem,

due to Kesten

_[8],

asserts that

_ân

is bounded in n, if and

only

if

where r and s are

_integers.

These cases are called

_{« integrable}

». If this condition is not

satisfied,

the fluctuation

_{Sn diverges}

with

_increasing

n. The nature of the

_{divergence depends}

on the arithmetical nature of w

_{[9, 10].}

This theorem

_implies

that there no

_longer

exists an

average lattice for the structure. As a _{consequence, it} was shown

_that,

for

_generic

non

integrable

cases, the Fourier

spectrum

of the structure is

_singular

continuous

_{[3, 4],}

which is the

_signature

of a weaker order than the

_{quasiperiodic}

one.

For

_integrable

_cases, it is

_possible

to

_compute

the fluctuation as

where

_yr(x)

is a

_{r-dependent 1-periodic}

_{function [1, 3],}

therefore a bounded one. The modulation _gn

_being

_periodic,

the structure is

_{quasiperiodic.}

Remark. Model 1 is built on the same

sequence {X n} .

Atoms and vacancies are

_put

on the . sites of a 1D lattice

according

to the rule : to 1

corresponds

a vacancy, to 0 an atom.

Though

this structure has no _average

_lattice,

it is

_{quasiperiodic [1].}

2.2 GEOMETRICAL APPROACH IN

R2.

- One way to look at the 1D structure from the

_point

of view of the

_higher

_{dimensional space}

_R2,

consists in

_associating

a broken line to the sequence

_{Xn},

by

replacing

1

_{(or fi)}

_by

_e1, and 0

_{(or Q2)}

_by

_E2, as shown in

_figure

2. si and E2 are the unit vectors of

Z2.

Hence,

a

_point

of the broken line is described

by

The average interatomic vector A

Fig.

2. 2013

Broken line in

_{Z2 corresponding}

to the 1D

_tiling.

where t denotes tan 0. The 1D

_tiling

is recovered

_{by projection}

of the broken line on

Ell.

Indeed,

since it is

_{always possible}

to rescale the

_lengths

_fi

and

_Î2

_{globally, only}

the ratio

Q2/Q1

is a

_significant

_parameter.

_Then,

if one restricts the ratio

_Q2/Q1

to t,

fi

may be identified

to c,

and Î2

to s, with c = _cos

6 ,

and s = sin 0. If _not, it is

possible

_to

project

the broken line

orthogonally

onto a line of

_slope

tan cp such

that,

now f,

1 should be identified to cos cp

and Î2

to sin cp.

The norm of A is

the mean interatomic distance. A has its

_extremity

on the

_diagonal

_Dl,

where,

for any

integer

n,

The nodes of the average lattice of the 1D

tiling,

in

_E~,

are the intersections of

Ep

with the

_{diagonals Dn.}

The modulation _gn has also an

_{interpretation}

in

R2

one _{may define} a vector

_Gn

=

This

_description

will be made more

_precise

_{below. P 1.}

_(Gn),

the

_projection

of

_Gn

onto

E_L

(1),

represents

the transversal extension of the broken line :

In

_particular,

in non

_integrable

cases, this

perpendicular

extension

_diverges.

Note that

equations

(2.9)-(2.15)

are true _{for any value of} Li.

Let us now restrict our

description

to the canonical case. This case is defined

_by

the choice

co = Li. One has

and

therefore,

This structure, which has an _{average lattice with}

_periodic

_modulation,

is the same

_{[1, 2]}

as

the

_{generated by}

the canonical

_projection

method from 2D to 1D. The

_{corresponding tiling}

is hereafter referred to as the canonical

_tiling.

This is illustrated

_{by figure}

3. The broken line is

entirely

contained in the

_{strip S = En}

+ _{C where C is the unit square}

The

_{correspondence}

between the

_parameters

of the two structures is

_{given by}

This

_geometrical

_{construction may also be described}

_by

a section method

_{[11, 12].}

Consider

the

_periodic

set of atomic surfaces

_(here

atomic

_segments),

built

_{by attaching}

an atomic

surface A to each

_{point §}

E

Z2.

The set of

_points

_{Un of the 1D} structure is obtained as the

intersection of

_{Ell by}

the set of atomic surfaces

In the canonical

case, A = - K

where

is the

_profile

of the

_strip.

The

_periodic

set _{of atomic surfaces may be}

_completed

into a 2D

_tiling

_[13].

Define the 2D tiles

These two

_{parallelograms}

_give

a

_partition

of a fundamental cell of

Z2.

The

_resulting

«

_{oblique » tiling}

_leads to the canonical

_{quasiperiodic}

_tiling

of

_E~

_by

section. The atomic

(1)

El

is the kemel

_{of pl.}

It

_specifies

the direction of the atomic segments. This direction is

_generated

by

(-

sin cp, cos

_{cp )}

and is held fixed for the various

examples

we will consider in _{this paper. This is} not the case for

_{E~ ,}

the

_physical

_{space, which will vary from} case to case. In

_particular,

_EL

is not

_necessarily

Fig.

3. -

Equivalence

of the

_projection

method from 2D to _{1D, with the circle map}

_algorithm,

when w =à.

surfaces described

_above,

are the boundaries of the

_{oblique tiling,}

transversal to

E~,

as shown in

_figure

4.

Let us note

_that,

since the

_points

_{of the average lattice} are the intersections of

EN by Dn,

the

_diagonals

are the atomic surfaces

_generating

_{the average lattice. In this}

Fig.

4.

_{- Oblique tiling}

and atomic surface, in the canonical case.

El’,

and the

_point

na of the average lattice obtained as a section of

_Dn

_(see

also

_Fig.

_2).

In other terms, the atomic

_surface,

seen

_along

a

_diagonal,

over a

_period

(here

one

_diagonal

of

the unit

_square),

_gives

a

_{representation}

of the hull of the

_{modulation, i.e.,}

of the modulation

function

_{g(x) = - (c - s)}

Frac

_(x).

_Indeed,

_figure

_{5b may be obtained from}

_figure

_5a,

_by

rescaling

AB to

_1,

and

_keeping

the direction of BC

_{perpendicular}

to

_AB,

without

_changing

is

length.

This

_{correspondence}

_{between the modulation function and the atomic surface may also be}

seen

_{by taking}

the converse

_point

of view : if the

_points

_{un of the} structure are

_given,

and if

one knows that

_they

come from a section

method,

it is

_possible

to

_get

the

_shape

of the atomic surface

_{by folding}

the

_plane

on a torus,

namely,

the unit square with

parallel edges

identified. The

_parametric

_equation

of the atomic surface A is

_{given by}

the hull of the set of

_points

satisfying

For

instance,

if u,, = _na,

since

_{a = 1 / (c + s ).}

_Hence,

the

_equation

in the torus of the atomic surface

_generating

the average lattice is

Fig.

5. -

Equivalence

between the atomic surface seen

_along

a

_diagonal,

and the modulation function,

for the canonical case.

simple

calculation

_yields

the

_equation

of the atomic surface in the torus, for the canonical

tiling

As mentioned in the

Introduction,

this

_{representation,}

_{in the extended space, of the} average lattice and of its

modulation,

is

_{easily generalised}

to _{any codimension} one case. We will now use the framework of this

section,

to extend the

_{geometrical description given}

here,

to the

_{general integrable}

cases of model 2.

3.

_Geometry

in

_higher

dimensional space of the

_{general integrable}

cases.

This section is devoted to the

_integrable

cases with finite fixed r =1= 1. As

_already

mentioned,

the

_{corresponding tilings}

are

_{quasiperiodic.}

It is thus natural to

_try

to describe these structures

in

_higher

dimensional _{space. In}

_particular,

we will build a set of atomic surfaces in

R2

that _generate,

_by

_section,

the 1D structure in

_Ell

_space.

reduces the

_integrable

cases to the canonical case

(to’, .d’ = Cù’),

for which the framework of

section 2

_applies.

In some _sense, this

_approach

_{fills in the gap between the}

_projection

method 2D --+ 1D and the circle map

algorithm

of model

_2,

at least for the

_integrable

cases.

We will end this section

_{by considering}

the first terms of the

_{sequence {rN}}

of values of r

leading,

when N - oo, to a non

_integrable

case. This will

_give

an intuitive

_geometrical

_image

of the mechanism of the

_{disappearance}

_{of the average lattice}

_(and

of

_{quasiperiodicity)}

in non

integrable

cases.

3.1

RENORMALISATION,

A REMINDER. - Renormalisation consists in

_changing

the scale at

which the

_{sequence {X n} ,}

and

_consequently

the structure that it

_generates,

are looked at. It is

an exact decimation

_{procedure acting}

_{on {X n} .}

We first need to define an intermediate

_labelling

_sequence on the unit circle. A _{sequence of} letters

_Ln

is associated to _{the sequence of numbers a n} = Frac

(nCù)

by

the rule :

Then

_{by choosing}

the

_{sequence {X n}}

is recovered.

A renormalization

_{operation lJl}

is a

_product

of a combination of four

elementary

transformations

S,

Tl,

T2, T3,

defined in the

table,

that

_changes

both the values of

(w , à )

and the

_{sequence {Ln} :}

In the last

_{equation, 31}

is a condensed notation for a substitution ag on the letters :

The order in which the

_{elementary operations}

enter in

3t,

i. e. ,

the order of a

_particular

sequence of renormalisation

operations,

is

uniquely

determined

by

the continuous fraction

expansion

of to and

_by

the

_w-expansion

of à

_[4].

For

_{example, S}

_always

_{maps £0 :::. 1/2} onto

Cd 1/2. In the

(w, A)

plane,

the iteration of the renormalisation transform Jt leads in

general,

to an

_aperiodic

orbit. For some values of

(co, A),

it may be a fixed

_point

or a

_cycle.

Let us

_give

the results of the

_application

of the renormalisation method to the cases of

interest for this paper.

a)

(co ,à

=

rw - s ).

Integrable

cases.

Table I . -

Definition of

the

_elementary

renormalisation

_{transformations}

The 1 D structure

_{corresponding}

to this case was studied

_analytically

in references

_{[3, 4].}

In

particular,

the Fourier

_spectrum

was shown to be

_singular

continuous. This

_example

is

_given

here,

since it

_corresponds

to the

_asymptotic

limit of the finite r cases considered in section

_3.2,

below. In

_particular,

the renormalisation

_operations

needed for the finite r cases and that of

the

_limiting

case

_{provided by}

this

_example,

are the same.

For these values of the

_parameters,

the renormalisation

_operation

leaves

_{(w,à )}

_{invariant, i. e,}

_{R(w,a) = (w , à).}

This fixed

_point

is

_repulsive,

i. e. ,

it is unstable

_{by 3t}

under a small

_{perturbation,}

_e.g.,

_along

the w = _const. direction :

Therefore,

the infinite sequence

{Ln}

is invariant

_by

a substitution o-R

{Ln}

is a fixed

_point

_{of U3t. In other terms, this sequence may be built}

_by

« inflation rules »,

From the n-th power of

M,

one finds that the number of letters in

_{Un (A )@ Un (B),}

o-nR(C)

are

_{F3n+l’ F 3n 2, F3n+2’ respectively,}

where

Fn

is the n-th Fibonacci number

defined

_by

Remark.

_Using

_{equations (3.5)-(3.6),}

the action of R on an

_integrable

case reads

c)

Approximation

of case

b)

_{by integrable}

cases.

The sequence of

integers

(rN’ SN)

which

_give

the best

_{approximants [}

is

_{given bey}

More

_generally

Hence,

the

_approach

to L1 = 1/2 is

given by

Remark. Let us note that

3t,

given by

equation (3.6),

satisfies

This

_property

is a _{consequence of the fact that}

(T-2, 1/2)

is a fixed

_{point, approached}

most

rapidly by

the best

_approximants

_[4].

_{Using equation}

_(3.11),

one

_gets

from which

_equation

_(3.12)

is obtained.

Since for any finite r, the 1D structure is

_{quasiperiodic, taking}

successive values of r means,

in some _sense, that the limit structure

_(no

_longer

_{quasiperiodic,}

since the Fourier transform does not _{contain any Dirac}

_peak)

is

_{approached by}

successive

_{quasiperiodic approximant}

structures, much in the same manner that one _may

_approach

a

_{quasiperiodic}

structure

_by

successive

_{periodic approximants}

_[14].

The

_geometry

in

1R2

of these structures is the

_subject

of section 3.2.

Remark. The canonical cases

_(w,

Li =

Cd)

_are

degenerate,

inasmuch _as the

labelling

sequence has

only

two letters. When w =

T - 2,

,

{X n}

is the Fibonacci sequence.

Using

the

same renormalisation

_technique,

one obtains the well-known inflation rules. Note that

(w

=

’T - 2, Li

= (0 )ils

_a fixed

point

of a renormalisation transform.

To conclude this

_section,

let us stress that a _{direct consequence of the renormalisation of}

the

_{sequence {X n}}

is the renormalisation of the 1 D

_tiling

itself. This is done

_{by replacing}

1 and O’s _{of the renormalised sequence}

_(x()

_by

renormalised tiles of

_{lengths e:2}

and

Qn,

respectively.

3.2 GEOMETRICAL APPROACH IN

R2.

- As seen

_above,

for

_integrable

cases, the renormalised sequence

{Xn}

gives

a renormalised

_tiling

with renormalised tiles. We want now to use the

renormalisation scheme

_directly

in 2D.

_We,

describe the

_general

method

first,

before

We first reduce an

_integrable

case

_{(lù, Li = rlù - s ) to a}

canonical one

(lù’, Li = lù’), by a

finite number of renormalisation

_steps.

We are thus in

_position

to use the framework of

section 2.2. We introduce renormalised coordinates

_{spanned by}

the vectors

₃₁

₁ and

{32,

defined as follows.

_Replace

the letters of the

_labelling

_sequence

_by

0 and l’s

_according

to

equation

(3.2),

then 1

_by

E1 and 0

by

E2. This

yields

the finite broken chains

:RE1

and

aE2-

131

and

_{13 2}

are the vectors

joining

the

_origin

to the end of the broken chain

RE1

(resp. :RE2).

These vectors

_generate

a sublattice of

BZ2

of

_Z2,

where B is the

matrix,

the columns of which are the

_components

of

₆

_{and 132.}

In these renormalised coordinates the 2D

geometry

is that of the canonical case. The renormalised

_{oblique tiling}

built in the coordinates of

_{13 1 13 2}

is invariant

_by

the lattice of translations

BZ2.

Since renormalisation is a

_decimation,

which consists in

_looking

at the structure at scale r,

points

are

_missing

in the 1D structure, or atomic surfaces in the

_{oblique tiling.}

In order to

_get

the

_{oblique tiling corresponding}

to scale

1,

with the initial tiles c, s in the

section,

one has to

partition

each renormalised tile

_{along Ell, according}

to the substitution induced

_by

cr 3t and

acting

on

_{13 l’}

_{8 2.}

This

_point

will be made clearer

_by

the

_examples

below. Four

_oblique

tiles are obtained that way.

_They

read

We illustrate the method

_{by taking}

the case

_(w

=

T - 2, L1 = 4 úJ - 1 ).

The renormalisation

transform to

_apply

is

_{given by equations}

_{(3.6), (3.8).}

_Following

the method described

_above,

the finite broken chains are

and the

_{vectors 8}

_i have the same

_expressions

with +

_signs

between the vectors.

The invariance lattice

BZ2

is

_{given by}

1 dent

(B)1,

the

« volume » of the unit

cell,

(the

index of the lattice in

Z2)

is

_equal

to four. In

_figure

_6,

the fundamental cell of the

_{oblique tiling}

is made of two

_oblique

_tiles,

the renormalised

tiles,

that may be

decomposed

into the smaller

_elementary

ones. The order in which each smaller tile appears inside the renormalised one is

_{given by}

the substitution eT 3t of

equation

(3.8).

From the

_{oblique tiling,}

one

_gets

the atomic surfaces. Note that

and

Therefore,

the

_{periodicity along}

a

_{diagonal Dn}

is

four,

in units of _{E2 - el}

_{(Fig. 6).}

_Again,

is

must be noticed that the atomic surface

_along

a

_{diagonal gives}

a

_{representation}

of the modulation function. In

_particular,

the atomic surface is made of three different

_segments

_(in

the

_E.L

_direction).

_{They correspond}

to the three

_segments

found in the function

Fig.

6. - 2D

geometry of the case

_(w

=

T-2,a

= 4 _{w -} 1

).

In bold : _a fundamental cell for the lattice 93 ; one of the

_periodic

atomic surfaces.

It is indeed known that Frac

_{(nw )}

divides the circle in three intervals

_{[8, 15].}

The order in which these

_segments

_{appear in the atomic surface is the} same as in the

_Y4(X)

_function,

namely a, b, c, b, where a : b : C = T - 4: T - 3: T- 2.

.Y4(x)

may be obtained

by

a deformation of

the atomic surface

_{(Fig. 7).}

Remark. Let us mention

_that,

as in section

_2,

_knowing

_un, we _{may write the}

_parametric

representation

of the atomic surfaces in the torus as

We end this section

_by

_studying

the

_asymptotic

behaviour of successive

_approximants

to the limit structure

_{generated by}

_(w

=

T -2,

d =

1/2 ).

The next best

_approximant

is

_(w

=

T-2, d

= 17 (ù -

6 ). Applying

twice the

renormalis-ation

_operation

_{equation (3.6),}

reduces this case to a canonical one. The

_geometrical

scheme in

IIBz

is similar to the

_previous

case.

_Figure

8 illustrâtes this case.

Let us consider now a

_{general (rN, sN)}

case.

_/31

_{(resp. 82) is}

obtained from

3tN

el

(resp.

Fig.

7. - The function

Fig.

8. - 2D

geometry of the case

where M is

_{given by equation}

_(3.9).

The « volume » of the unit cell

_{spanned by}

f3

1 and

f3 2,

is

given by

Generalising equation

(3.19),

one has

When N

increases,

the directions of

_{31

₁ and

_{32

get

progressively

nearer and nearer the

bisectrix,

since the number of vectors _El and E2

composing {31 and {32

are

_equal

in average.

Therefore,

the

_lengths

_{of /3i}

₁

_{and 16 2}

are well

_approximated

_by

F3N +2/..j2

and

F3 N + 1 / BF2,

respectively. Using

equation (3.24),

one deduces that the

_angle

between these

vectors _goes to zero as

T - 3 N.

°

4. Final remarks.

The

_description

of the

_integrable

cases of the circle _map

_tiling

in terms of a section

through

a

2D structure

_provides

a means to

_compute

the Fourier

spectrum

of the structure. _{This may be}

done

_along

the same lines as in references

_{[16], [17].}

The main difference with the canonical

tilings

is the presence of additional structural units in the renormalised tiles. In

_analogy

with standard

_{crystallography,}

the Fourier transform of the 2D

_tiling

is the

_product

of the lattice

(BZ2) *

reciprocal

to

BZ2 by

a

_geometric

form

factor ;

the form factor is the Fourier

transform of the set of atomic surfaces

where

In order to

_get

the Fourier

_spectrum

in the

_physical

1D space, we have to

_integrate

in the transversal direction. The result is a _pure

_point

Fourier

_transform,

with

_support

in the

projection

of

_{(BZ2 ) *}

as for a canonical

_tiling,

but with

_amplitudes

modulated

_by

the form factor.

It is worth

_drawing

the attention of the reader to the resemblances between

_aspects

of the work of

_Frenkel,

_Henley

and

_Siggia

_[18],

and some considerations

_given

here,

_{e.g., in}

section 2.

The existence of average lattices for 2D

quasiperiodic tilings

is a

_subject

of

_growing

interest

[19, 20].

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