Quasiperiodic models with microcrystalline structures

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Quasiperiodic models with microcrystalline structures

Michel Duneau

To cite this version:

Michel Duneau. Quasiperiodic models with microcrystalline structures. Journal de Physique I, EDP

Sciences, 1991, 1 (11), pp.1591-1601. �10.1051/jp1:1991227�. �jpa-00246438�

J.

Phys.

I France 1

(1991)

1591-1601 NOVEMBRE1991, PAGE 1591

Classification

Physics

Abstracts

61.50K 61.70N M.70K

Quasiperiodic models with microcrystalline structures

Michel Duneau

Centre de

Physique Th60rique,

Ecole

Polytechnique,

91128 Palaiseau Cedex, France

(Received17

May 1991, revbed 5

July

1991,

accepted

8 August 1991)

Rksumk, Nous _montrons comment la mdthode de coupe et

projection

pout ^dtre

adaptbe

I la construction de moddles

gkomktriques prksentant

une structure

microcrystalline.

La construction est illustrbe _par

l'exemple

du pavage

octogonal.

Premidrement, _une brisure

partielle

de la

symdtrie

_pennet de construire de _nouveaux pavages

prbsentant

deux bchelles

inddpendantes

_:

l'une est associbe I la taille des mailles et l'autre intervient dans les

parois

entre domaines.

L'ensemble des mailles _{a une} structure modulbe

simple.

Ensuite les mailles sont dbcordes selon le motif d'une structure

pbriodique approximante,

conduisant ainsi I _un moddle microcristallin

cohbrent. La transformbe de Fourier est calculbe exactement et montre des taches satellites

caractkristiques, dkpendantes

des

paramdtres

de

l'approximant, ajoutdes

au spectre

quasipkriodi-

que de rkfbrence.

Abstract. We show how the standard cut and

project

method _can be

adapted

to build

quasiperiodic geometrical

models

displaying

a

microcrystalline

structure. The

procedure

is illustrated with the

example

of the

octagonal tilings.

First, _a

partial

symmetry

breaking

allows to build _new

filings

with two

independent

scales _{: one} is associated to the size of the unit cells while the other occurs in walls between domains. The

corresponding

_array of unit cells has _a

simple

modulated structure. Then the unit cells _are decorated

according

to a

periodic approximant

structure, thus

providing

_a coherent

microcrystalline

model. The Fourier spectrum can be

computed

exactly and shows characteristic satellite spots, ^added to the

quasiperiodic

reference spectrum, which

depend

_on the

approximant

parameters.

I. Introducdon.

Since the

discovery

of icosahedral

quasicrystals

in temary

alloys

of Almnsi

[I]

a number of other

quasiperiodic

structures have been

prepared

_ans studied

by

Various methods and other

non

crystallographic symmetries

_were indicated

by

diffraction

experiments

and

high

resolution

microscopy (see

for instance

[2~4]).

^More

recently,

the

phase diagram

of the Alfecu

system

was

closely analysed [5]

in the

region

of the icosahedral

phase, showing

_a

biphased

state after

annealing.

Transitions between

quasicrystalline phases

and

microcrystal-

line

phases

have been

recently suggested

or observed

[6-1ii.

Recent observations of the AlLicu system ^showed a

complex

structure with _a network of translation walls

[12].

All these observations suggest that the

QC

states could be

closely related, by

_means of

displacive transformations,

to

microcrystalline

states with

special

kinds of defects and raise the

question

of the

possible

structural

relationships

between

quasicrystals

and

crystalline phases.

1592 JOURNAL DE PHYSIQUE I M 11

From _a theoretical

point

of

View, geometrical

models of

displacive

transformations between

QC

and

periodic crystals

have also been

proposed recently [13-17].

In this paper we propose a

simple construction,

based _on the cut and

project method,

which

provides

_a

geometrical

_setup for models of

microcrystalline

structures with _a

global quasiperiodic,

_or

periodic, long

_range

order. The skeleton of these models is obtained

by

_a standard CP method with

merely

_a different choice of the orientation of the

physical

_space in the

higher

dimensional space, or

by

the choice of different lattices and acceptance ^{domains. This}

usually implies

_a

partial

symmetry

breaking,

with respect to the

quasiperiodic

reference

model,

which results in _new

tilings showing

two different characteristic scales. The

large

scale dictates the size of the

typical

unit cells which _occur in domains

separated by

walls

involving

the smaller scale. At this

point

the ratio between the two scales _can be chosen

freely.

The skeleton thus obtained _can be shown to

correspond

to _a modulated

lattice,

the nodes of which _are attached to the unit cells.

It is different from the structures obtained in

[15]

which _seems to be

topologically

similar to the reference

tilings.

As in the usual CP method the

global

_average orientation of the

physical

space can be _set in order to have either _a

quasiperiodic skeleton,

_as assumed

above,

_{or a}

periodic

_one with

possibly large parameters corresponding

to some

approximant

structure.

For

particular

values of the two characteristic

scales,

^{the skeleton shows} a close connection with the reference

quasiperiodic

structure. These situations

correspond

to a characteristic ratio related to the inflation parameter ^{of the}

QC. Actually

a skeleton of empty unit cells _can be

exactly superimposed

on a

fully symmetric quasiperiodic

structure in the _sense that the

nodes of the skeleton fit with nodes of the reference structure. In this _case,

however,

the unit cells _cut in the

QC

show _a finite number of

different

decorations. In order _to obtain _a

microcrystalline

structure _we have to

prescribe

_a

unique

and constant decoration in all unit cells.

The Fourier spectrum ^{of these} structures can be

computed exactly.

Since

they

_are obtained

by

decoration of _a

quasiperiodic skeleton,

the Fourier transform

requires

two steps. ^The skeleton has _a nice modulated structure and tums out to be rather close to _a

periodic

_array of unit cells

(the approximant structure)

with _a network of defect walls

restoring quasiperiodici- ty.

^{On the other hand the unit cell}

yields

a structure factor which is close to the ideal Fourier

spectrum ^{since the unit cell} can be obtained

by

_a cut in the reference

QC. Finally,

the

resulting

_spectrum is carried

by

the reference

spectrum plus

satellite

spots.

^The

spectrum

can

be indexed with the _same basis but with fractional indices in the directions associated to the

approximant

structure. The characteristic denominator of the fractional indices is linked to the

parameters

^{of the}

approximant

and increases with the size of the unit cell.

The construction is

developed

in the _case of the

octagonal tiling

for

simplicity.

The _same ideas

apply

for 3 dimensional structures where the reference

QC

is icosahedral and where the

approximant

structures _can be either cubic _or rhombohedral.

2. Notations and

settings.

The

octagonal tiling

is obtained

by

the cut and

project

method

(CP method)

from

R~.

In order to have

edges

of unit

length

in the

physical plane

the

hypercubic

lattice is defined

as

Ao

=

/2f~.

_{The two}

_{orthogonal planes Eo}

_and

_E(

are the ranges of the

following orthogonal complementary projectors

:

0

II / _II /

0

II / _II /

~° 2

Ill Ill

₀

lI / _II /

₀

li° II

QUASIPERIODIC

MODELS WITH MICROCRYSTALLINE STRUCTURES 1593

0

-ill Ill

,

I o I

-ill -ill

~° 2

-ill -ill

₀

lll -ill

₀

The

projections _Lo

=

po(Ao)

and

L(

=

pi (Ao)

are dense 2f-modules

respectively generated by (ej,

,

e

~)

^and

(e(,

,

e

()

where _e~

=

po(/ _e~)

_and

_e)

=

pi (/ _e~).

_In orthonornlal coordinates

et = 1, 0

)

_e

=

(1,

0

~~

/~~~'~~

^~~

~~ ~' ~~

e~ =

(0,1) e(

=

(0,1)

Thus all vectors _e~ and

e)

^{have unit}

length.

The

octagonal tilings correspond

to _a cut

region

_So

=

Eoxwo

^where the acceptance domain

Wo is,

_up to a

translation,

the

projection

of the unit cube of

R~

_{in the}

perpendicular

space.

Wo

is _a

regular

_octagon

generated by

the four _vectors

(e(,...,

e

() (see Fig, I).

The

tilings

involve six different tiles

(two

_squares and four

rhombi)

which _are the

projections

of the six

possible

2 dimensional facets of the lattice Ao. ^The normalization of Ao

implies

that all tiles have all

edges equal

to I.

The Fourier spectrum ^{of the}

octagonal tiling

is carded

by

the

projection

_Lo* in

Eo

of the dual lattice

At

= 2L~ ₌

Ao.

Therefore

Lt

=

Lo

is

spanned by

the basis

/

2 2

(et,..,

_e

t)

where

e~* = e~ and all

Bragg peaks

_can be indexed with this basis.

2

The

octagonal tiling

shows _a

perfect

inflation

symmetry.

This is _a consequence of the existence of _a modular transfornlation

(')

J in

R~

which commutes with the action of the 8- fold

symmetry

_group. J has the

following expression

with respect ^{to the}

projectors

po and

pi

Ii

^{i o -i}

J

= =

(i

₊

/)po

+

(i /)pi

-i o i

As _a consequence the Z-modules

Lo, j

Lo* and

Lt'

_are invariant with respect to the

scaling

transfornlations of parameter + 2 _or I

/.

_In

_particular

_{the Fourier}

_spectrum

_is

carried

by

the scale invariant set of

points Lt'

Periodic

approximants

of the

octagonal filing

_can be

easily

obtained

using

the

underlying

inflation

symmetry [18].

The

approximant

of order k is associated to the

symmetric

matrix

J~

_{which reads}

:

O~+O~_j O~

0

-O~

J~

=

~~

°~ /~

~

/[ /

=

/

_i

_{)k pi}

+

(

i

/)kpi

O~ ^0~

~

O~~~~ O~+ (~_j

(')

A modular matrix has

integral

entries and determinant ±1.

1594 JOURNAL DE PHYSIQUE I li° II

e3 3

e4 e2

~~ lleill =

lle'ill

₌ ^~

,

2 ~4

Fig.

I.- The setup ^{of the standard}

octagonal filing

with the irrational

orthogonal planes

E and E'

yielding

8-fold symmetric

projections

of the cubic lattice

/ _{Z~ (top).}

_The

_{octagonal dungs (bottom}

left) _are obtained with windows

equal,

_up to a translation in perpendicular _space, to the octagon

spanned

by e(, e(, e],

e( (bottom right).

where the «Octonacci» _sequence

(O~)

is defined

by Oo=0, Oj= I, O~=2

and

°k

+1 ^" ~

°k

+

°k

-1.

3. Par6al symmetry

breaking

of the S-fold sgnmetry.

The

following

construction is devised to

give

_an ideal

geometrical

model of _a

quasiperiodic (or periodic)

_array of

large

unit cells that _we call the skeletons. This is obtained

by breaking

the 8- fold

symmetry

in order to have _a

filing

with two different scales _{: a}

large

scale will dictate the size of the unit cells and the smaller _one will be involved in the

grain

boundaries _or walls between domains. In the next section _we shall

give

_a

procedure

to decorate these unit cells with bases such _as the bases of

periodic approximant tilings.

In this _case the size of the unit cell will have to match the lattice parameters ^{of the}

approximant

lattices.

Let A

=

aY~

_{denote the} 4-dimensional

hypercubic

lattice with lattice

parameter

a m I and let _~ 0

satisfy a~

=

I _{+ ~.} The standard basis of A is

~).

Then there exist two

li° II

QUASIPERIODIC

MODELS WITH MICROCRYSTALLINE STRUCTURES 1595

orthogonal subspaces

E and E' ofdimension 2 in

R~,

with

corresponding orthogonal projections

p and

p' (see Fig. 2)

such that _:

p(~El)

~

(A, 0)

"

A~l p'(~~l)

"

(', 0)

"

~l

P(~E2)

"

~ _{(l, 1)}

" ~2

P'(~82)

"

~ _{(~ i,}

_~

"

~~i

P(aE3)

~

(°>

~

"

~~3 P'(~E3)

"

(°, 1)

"

e(

P(aE4)

~

~ _{(~ l, 1)}

= e4

P'(~E4)

"

~ _(~,

_~

₎

"

~e(

Thus A is the ratio of the two scales involved in the construction. If A is irrational with

respect

to

/,

a condition _we shall _assume in the

following,

the

projections

L

=

p(A)

and L'

=

p'(A)

_are dense Z-modules obtained _as all

integral

combinations of these vectors. For A

=

I _{we recover} the usual setup of the

octagonal tiling

with full 8-fold

symmetry.

A=aZ4

a2=1+l~~

~~

'

3 e3

e~i

e4 e~

l~ei

,

2

J~e4

Fig.

2. A convenient choice of the lattice parameter a and of the orientation of the

planes

E and E'

gives perpendicular projections showing

characteristic scales I and A.

The _new

projectors

_p and

p'

_are

given by

:

A~ ^All

⁰

^-All

_I All

_I

All

₀

~

a~

_o

All A~ All

-All

₀

All

and

~l ^-All

⁰

^All

,_I -All A~ -All

₀

~

a~

_o

-All

_I

-All

All

₀

-All A~

The

acceptance

^{domain W in E'}

is,

_up to _a

translation,

the

octagon spanned by

the vectors

p'(ae~ )

for I

=

I to 4. Notice that W has

only

_a four-fold symmetry. ^The

corresponding tiling

1596 JOURNAL DE PHYSIQUE I li° II

of E is associated to the cut

region

S

= E _x W and involves the six files

generated by

the four vectors p

(a

_e~

).

The symmetry

breaking

induced

by

A _~ l

implies

that the square tile

Tj~ generated by

Aej and Ae~ is

larger

than the others. The acceptance ^{domain of this tile dictates its}

frequency

in the

tiling.

One

easily

checks that this domain is the square

T(4 generated by Ae(

and

Ae(.

Therefore the

frequency

of

Tj~,

as a function of A, increases with its size. More

precisely,

the surface of the

octagon

W is ^~ ₊ 2

IA

+ I in such _{a way} that the relative

frequency

of

Tj~

^reads A~/(A^~ + 2

/

_A

+ I

).

The total relative surface covered

by

these tiles increases _{even more}

rapidly

with A since _a

simple

calculation

gives

A

~/(A ~+

_{2 A} ~

+

l).

As shown in the

example

of

figure 3,

the

large

tiles _are

frequently adjacent

to each other.

They

fornl _« domains which _are

separated by

boundaries made with the smaller tiles. Due to the construction in

hyperspace,

the

global

structure maintains _a strict

quasiperiodic

order in

the _sense that the Fourier transform of the structure is still _{a sum} of Dirac _measures which _can

be indexed

by

four

integers.

Therefore the arrangement ^of the domains _can be considered _as coherent.

Fig. 3. This tiling corresponds to the geometry ^of figure 2 showing two scales _: I and I ₊

/.

_The

large

_squares of

edge

I ₊

/

represent = 42.68 fb of the tiles and 85.36 fb of the surface.

They

occur in domains of several units.

The

example given

in

figure

3 _was

cojuted

^{with = I +}

^/

^{for which the relative}

frequency

of

Tj~

is

(3+2/)1(8+4 ₂₎

_{=0.4268 whereas this}

_frequency

_is

_only 11(2+2/)=0.2071

_{in the}

_{octagonal tiling.}

_The

corresponding

surface represents a

fraction

(17

+12

/)1(20+14 /)

= 0.8536 of the whole

tiling

instead of

I/4

in the

octagonal tiling.

An

important

feature of this construction is that the skeleton thus obtained is _a modulated square lattice. This _can be _{seen as} follows. The different

large

tiles _can be referred to

by

_a local

origin,

for instance the lower left

point

of the squares

Tj~ spanned by p(aei)

and- p

(ae~).

In the

perpendicular

_space the

corresponding points

fall inside _an acceptance ^domain

which

is,

_up to _a

translation,

the

complementary

_square

T(4 spanned by p'(ae~)

and

p'(ae4).

It has been

proved

elsewhere

[19]

^{that if} an

acceptance

^{domain tiles the}

perpendicular

_space

(with

respect to a sublattice of the

projection

L' of the

hyperlattice),

then

the structure associated to this domain has _an average lattice and

consequently

_can be

described _{as a} modulated lattice.

In the present case the acceptance domain of the local

origins, namely T(4

^tiles the

perpendicular

_space with

respect

to the 2D lattice

generated by p'(ae~)

and

p'(ae4)

which is

bt II

QUASIPERIODIC

MODELS WITH MICROCRYSTALLINE STRUCTURES 1597

the

projection

of the 2D lattice A~4

spanned by (ae~, ae4).

The skeleton _can be indexed

by

the

complementary

lattice

Aj~ spanned by (aej, ae~)

and the _average lattice is

given by

the

oblique projection

of

Aj~

on E

parallel

to the space

E~4 containing

A~4. ^A

simple

calculation shows that this average lattice is _a square lattice with basis

(a~/Aej, a~/Ae~).

The Fourier

spectrum

^{of the}

grain

skeleton _can

easily

be derived

analytically

since the construction is based _on the CP method. The dual lattice of A is A*

= a~ 2L~

=

a~~

_{A. The} spectrum ^is carried

by

the

projection

of A* on E which is the Z-module L*

= a~ ^~ L

spanned by (a~

^~

Aei, a~~e~, ^a~~ _Ae~, ^a~~ e4).

L* is the _sum of the dual average

lattice, spanned by

(a~~ Aej,

a~ ^~

Ae~),

^{and the} square lattice

spanned by (a~ ~e~, a~~e4)

^{which is}

responsible

for the satellites.

In the next section _we shall consider

particular

values of A for which the

large

tiles _can be identified with the

empty

^{unit cells of} an

approximant tiling:

this is achieved if

A =

(1+ /)~

_for

some even (2)

integer

k.

Figure

4 shows unit cells of the first two

approximant tilings.

In this _{case a more} convenient construction is

possible. Actually,

if J denotes the inflation matrix then

Aej

= po

J~(/

ej and

Ae~

₌ po

J~(/ _e~).

_This

_suggest

_to

consider the sublattice A

=

M~ Ao

of Ao ^associated to the

following integral

matrix

O~+O~_j

0 0 0

O~

I

O~

⁰

~"

0 0

O~+O~_j ^0'

-O~

0

O~

k=I k=2

Fig.

4. The unit cells of the first two

approximant,

k

=

I and k

=

2 of the

octagonal tiling

with

respectively

7 and 41nodes.

The first and third columns _are those of

J~

_{while columns 2 and 4}

are those of the

identity

matrix. The natural basis of A has therefore the

projections (Aei,

_e~,

Ae~, e4)

_on E and

(A e(, e(,

A

e(, e()

on E'.

(~) ^{For odd values of k the} construction

gives

rise to

overlappings

of _some tiles. This _can be handled by a

slight

modification of the

procedure (for

instance _ej and e~ are

changed

into -ej and

e~ in

physical space).

1598 JOURNAL DE

PHYSIQUE

I li° 11

The skeleton is obtained

by

the CP method with _{a square}

acceptance

^domain

spanned by e(

and

e(.

The Fourier transfornl is carried

by

the

projection po(A*)

of the dual lattice

A*

=

[(M~)~]~ At

of

A,

where the associated matrix is

given by

_:

I

-O~

0

O~

~~~~~~ ~"O~+~O~_j ~~~~) -~~

0 0 0

O~+O~_i

Consequently

the Fourier spectrum ^is

spanned by

the basis

et, et,

°k

+

°k

-1

~

et, et

Thus

po(A

^* contains the Fourier

spectrum

^{of the}

octagonal tiling

which

°k

+ _{k I}

is

spanned by

the basis

(et, et, et, et).

The _new

peaks

_are due _to the _presence of the

rational fraction and _can be understood _as satellites in the

et,

and

et

directions.

O~

+

O~

j

4. The

approximant octagonal filings.

Approximants

of the

octagonal tiling

have been studied in

[18]

where

they

were related to the inflation

mapping

J. An

explicit algorithm

was

given

to compute ^{the unit}

cell,

I-e- the

size,

the number of nodes and their

positions,

_{as a} function of the order of the

approximant.

More

precisely,

for _any

integer

k _m 0 _a rational

plane E~

is defined _as

J~ Ej~,

where J is the inflation

matrix and

Ej~

^is the 2-dimensional

plane

of

R~ spanned by

_ej and e~.

E~

and the

complementary orthogonal plane E( specify

the framework of the CP method where the

acceptance

^{domain W}

is,

_as

usual,

the

projection

of the unit cube _on

E( (the approximant

of order 0 is the

simple

_square lattice of unit

edge).

Due _to the rational orientation of

E~,

the

projection

of _{2L~ on}

E(

is _a 2-dimensional lattice

L(. Up

to _a modular transfornlation

(see [18]) L(

_can be identified to a 2-dimensional square lattice. The acceptance domain W is

an octagon

spanned by

the four lattice _vectors of coordinates

(O~

₊

O~_j, 0), (O~, O~), (0, O~

+

O~ j)

^and

(- O~, O~),

where

(O~)

is the Octonacci _sequence mentioned in

section 2.

The CP method

requires

that the

boundary

@W of the acceptance ^{domain W does} not

contain any

point

of

L(.

This is achieved

by

convenient relative translations t' between W and the lattice. Now _{on, we} shall _assume that the

octagon

^{W is fixed} and that the

origin

of the

plane

is _one of its vertices. The translated lattice is therefore

L(

₊ t' and the nodes

fatling

inside W

specify

the unit cell of the

periodic tiling

_{as a} function of t'. Now if

y denotes the unit _square of

L(,

_one

easily

checks that all

configurations

of the unit cell _are

obtained when t' _{runs over y.}

Moreover,

the

configurations

_are continuous w-r-t-

t'in the _sense that modifications of the unit cell

only

_occur when the

boundary

of the

octagon

meets the translated lattice

L(

+ t'. These

singular positions

result in

flip-flops

of nodes in the unit

cell,

_a well known

phenomenon

in

quasiperiodic tilings.

The

singular

translations t' _are

easily

identified _as those which

belong

to @W mod

L( (see Fig. 5).

This subset is the union of the

boundary

of _y with its two

diagonals.

The _square

y

splits

into 4

triangular

domains within which the unit cell is _a continuous and therefore constant function of t'. As a

conclusion,

_{we see} that four different unit cells _can result form the construction.

Finally,

these four domains _can be related

by symmetry operations

and

consequently,

the

corresponding

unit cells _are

exchanged by ar/2

successive rotations in such _a way that four different lattices related

by

_ar

/2

rotations _are obtained. In this _way the

global

^4-

li° 11 QUASIPERIODIC MODELS WITH MICROCRYSTALLINE STRUCTURES 1599

w

I

o

Fig.

5. The acceptance domain for k

=

I is _an octagon

spanned by

the four vectors

(1, 0), (1,

1),

(0,

1) and

(-I,I).

The _square lattice is shifted

by

t' in such _{a way that no}

point

falls _on

3W. This condition _means that t' does _{not meet} 3W mod

L(

which is the union of the

boundary

of y and its two

diagonals.

fold symmetry of the construction is recovered

although

_a

given approximant tiling

has _no

particular _symmetry.

Therefore

global phasons,

I.e. _a relative translations of the

hyperlattice

w-r-t- the cut

region

result

merely

in rotations of the lattice in the

physical

_space.

As

expected

from the

theory

of

phasons,

_we find that two lattices

corresponding

to

adjacent triangles only

differ

by ~periodic)

lines of

flip-flops.

5. The

complete nficrocrystalfine

structure.

A

complete

structure is obtained

by filling

each empty unit cell of the skeleton with _a basis of

an

approximant

structure. In the

following

_we shall _assume that the _same basis decorates all unit cells. This

assumption

is consistent with _our purpose which is _to get

microcrystalline

models.

However,

other structures could be obtained

by allowing

different decorations in dilTerent unit cells. These

possibilities

could be used for instance to include _some randomness in the structures. At the

opposite,

the reference

quasiperiodic tiling

_can be _{seen as a} skeleton of unit cells with

specific

decorations.

Finally,

a lot of internlediate _structures could be

obtained in

principle by

either _a random _{or a} deterministic

procedure.

Figure

6 represents a part ^of a

complete

structure associated to the second

approximant

of the

octagonal tiling.

The unit cells have _a constant decoration of 41nodes and it _can be observed that the small tiles _can be

given

_a constant decoration in such _a way that the overall

structure is _a

tiling.

In other words the tiles match of _course from unit cell _to unit cell but also

through

the

grain

boundaries.

In this

situation,

where the decoration of the unit cells is constant, we can

explicitly compute

^{the Fourier transform} of the whole structure. The

computation

is based _on the

decomposition

of the structure into _a basis and _a

(uasiperiodic

lattice of translation.

Actually,

the basis

gives

_a structure factor

S(q)

which _can be

computed explicitly,

_once the coordinates of the sites of the basis _are known.

However, S(q)

_can be _{seen to} be close to the Fourier transform of the

perfect

reference structure.

First,

the basis of _an

approximant

_can be identified _{as a}

part

^{of the}

quasiperiodic

reference _structure.

Consequently

the structure factor

can be recovered _as the exact Fourier transform of the infinite structure, smeared out

by

_a the Fourier transform of the cut function of the basis volume. This results in _{a sum} of smeared

Bragg peaks

centered _on the nodes of the

reciprocal quasilattice.

1600 JOURNAL DE

PHYSIQUE

I li° II

Fig.

6. A

complete microcrystalline

structure is obtained

by filling

the

quasiperiodic

skeleton with the unit cell of _an

approximant

structure.

D q§ ~ q~ a ^U

D

~ ~

_{q3 b ~ ~jj u ty P}

~ ~j'D

~ O

~

p

~

Jt~ ~

fjo ~

n

@ ~~j.

~ ~

~ _U

D 8

D P

b D q~ u ~ u ~ u

~ ~ ^O

El

u

e

_{o ~ ~}

u

f~

_~ a a a q~ _n

3

_n

~ ~ ~

~

G U _o

@

°

~q

_o P Q

b P _u 6~ n q _a q

~

j

© t£

%~ ~

~ij @

d

~ ~~j°

q3 ~

P j

~ Q

,

D

~ ~li

~ dl b QJ d' & q3

fl~ ~

_D

U O ~

~..

P u

Fig.

7. The Fourier transforrns of the

octagonal

tiling

(circles)

and of

the1nicrocrystalline

structure

(squares).

Satellites with rational shifts in the ej and e~ directions can be _seen.

On the other

hand,

the skeleton is obtained

by

_a standard CP method where the

acceptance

domain is the square

T(4

and where the

hyperlattice

is the sublattice of

/

_{2L~ associated to the}

matrix

M~,

where k is the order of the

approximant

unit cell. As

explained

in section 3 the Fourier

spectrum

^{of this} structure is carried

by

the

Z-module generated by

the basis

' ~*

~i

^' ~*

~/

°k

₊

°k-1

^{~ ~}

°k

+

°k-1

^~

bt 11

QUASIPERIODIC

MODELS WITH MICROCRYSTALLINE STRUCTURES 1601

Finally

the Fourier transform of the

quasiperiodic

_array of decorated unit cells is

given by

the

product

of the structure factor of the unit cell

by

the Fourier transform of the skeleton.

Figure

7 shows _a

superimposition

of the reference

spectrum

^{and the}

microcrystalline

_spectrum

in the _case of the second

approximant

k

=

2. Since

O~

+

O~

= 3 _{we see} that the Dirac

peaks

can be indexed w.r.t, the basis

(ef, et, et, e?)

with

integral

indices for the

et

and

et

vectors and rational indices with _a denominator

equal

to 3 for the

ef

and

et

vectors.

Acknowledgments.

The author is indebted to

P.Donnadieu, C.Oguey

and G.Coddens for _many fruitful

discussions.

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