Theory of the formation of quasicrystals

HAL Id: jpa-00212538

https://hal.archives-ouvertes.fr/jpa-00212538

Submitted on 1 Jan 1990

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Theory of the formation of quasicrystals

V.P. Dmitriev, Yu. M. Gufan, S.B. Rochal, R. Tolédano

To cite this version:

Short Communication

Theory

of the

formation

of

_{quasicrystals}

V.P. Dmitriev

_(1),

Yu.M. Gufan

_(1),

S.B. Rochal

₍₁₎

and R Tolédano

₍₂₎

(1)

Institute of

_Physics,

Rostov State

_University,

344104 Rostov on

_Don,

U.S.S.R.

(2)

Laboratoire des Transitions de

_Phases,

Université

_d’Amiens,

80000

_Amiens,

France

(Received

5 June 1990,

accepted in final form

11

_{July 1990)}

Résumé. 2014 Un modèle

_théorique

est

_proposé

_pour

_expliquer

la formation des

_alliages

quasicristal-lins. La

_phase

_{icosaédrique}

résulte dans ce

_modèle,

d’une transition reconstructive de _type

_{displacif à}

partir

d’une

_phase

mère cristalline. Le mécanisme de la transition met

_{en jeu}

deux

_paramètres

d’ordre distincts : un

_champ

de

_{déplacements qui}

transforme certains sous-réseaux cristallins en

_assemblages

icosaédriques,

et une onde de densité

_qui

brise de

_façon

incommensurable l’ordre translationnel de

la

_phase

mère. Les

_principaux

aspects de la théorie sont introduits à travers

_l’exemple

de AlMnSi.

Abstract. 2014 A theoretical model is

proposed

for the formation of

_{quasicrystalline alloys.}

The

icosa-hedral

_phase

is shown to result from a reconstructive transition of the

_displacive

_type from a _parent

crystalline

structure. Two order _parameters are involved in the transition mechanism: a

_displacement

field which transforms a number of sublattices into icosahedral clusters, and an icosahedral

_density

wave which breaks

_{incommensurately}

the initial translational order. The main features of the model

are introduced

_through

the illustrative

_example

of AlMnSi. Classification

Physics

Abstracts

61.50E - 02.40 - 63.20D

The

_{phenomenological}

models which have been

_proposed

to

_explain

the

_stability

of the icosa-hedral

_{(le) phases}

_[1-4]

refer to the ideas introduced

_by

Landau

_{[5] ,}

and to their extension

_by

Alexander and

_Mcîàgue

_{[6] ,}

for the

_description

of the

_{liquid-to-solid}

transition. The choice of the

_{isotropic liquid}

as a

_parent

_phase

for the Ic structure stems from the fact that

_initially

this

structure could

_only

be obtained in certain

_cooling

conditions,

below the range of

stability

of the

liquid phase.

However,

more recent

_experiments

_[7-9]

revealed that one could

_get

an Ic structure

under

_{as-equilibrium}

conditions,

and that a

_{quasicrystalline phase}

could coexist

_with,

or could be

obtained

from,

other

_crystalline

structures.

_Following

these

considerations,

the

_theory

_presented

in this

_paper

_{starts from a}

solid

_{crystalline phase}

as

_{the parent structure for}

the

Ic phase.

More

pre-cisely,

the Ic

_phase

is shown to result from a -reconstructive transition

_{of the}

_displacive

_type

_[10]

_front

a

_{crystal phase.}

The transition mechanism occurs in two

_steps:

a

_displacement

field which leads to

the formation of icosahedral atomic

clusters,

preserving

the initial

_unit-cell,

and an icosahedral

density

wave, which

destroys

the initial

_periodicity

and leads to icosahedral

_{quasiperiodicity.}

Let us introduce the essential features of our model

_through

the illustrative

_example

of

AIMnSi,

2400

and start from the cubic a-AIMnSi structure as described

_by

_Cooper

_{[11] .}

The 138 atoms

con-tained in one unit-cell of the ce

phase

(space-group Pm3) decompose

into 11 orbits

_which,

accord-ing

to the

_labelling

of reference

_[11],

can be shared into three

_groups:

_{AI(l), AI(2),}

and

_Al(7)

are

characterized

_by

a 6-fold

_{position; Al(3), AI(4), AI(8), AI(9), Mn(l),}

and

_Mn(2)

_possess

a 12-fold

degeneracy; Al(5)

and

_{AI(6) display}

a 24-fold

_degeneracy.

Fig.

1. - Icosahedra formed

by

the atoms

_belonging

to the

_AI(7)

_(Fig.

_la),

and

_{Mn(2) (Fig.lb)}

sublattices. White and black atoms _represent the cubic and icosahedral

_{positions, respectively.}

We first focus on the atoms of the

_AI(7)

orbit. From

_figure

la one can see that

_antiparallel

displacements

of each

_pair

of atoms, denoted

_{(1, 4), (2, 5),}

and

_{(3, 6)}

in

_figure

_la,

_along

the

respec-tive cubic directions

_{[100], [010],}

and

_[001],

_bring

the twelve atoms

_forming

the

_AI(7)

_orbit,

i.e. the three

_preceding

_{pairs plus}

the

_{translationally equivalent}

atoms,

toforni

an icosahedron for the spe-cific

_{displacements}

of

_magnitude

0.244

A

= 0.0194 _a, where a = 12.56

A

is the lattice

_parameter

of the cubic unit-cell

_[11].

The same

_displacement

field

_applied

to the atoms of the orbits

_Mn(l),

Mn(2), Al(3)

and

_Al(4)

also

_give

icosahedral clusters.

_Thus,

_{Al(3), Mn(l),}

and

_Al(7)

form three embedded icosahedra of

_increasing

_volume,

centered at the

_origin

of the

_unit-cell,

whereas

_Al(4)

and

_Mn(2)

form two embedded icosahedra centered at

_1/2,112,1/2

_{Fig.1b).

Table 1 summarizes the

_magnitude

of the

_{displacements corresponding}

to each of the

_preceding

orbits.

Besides,

one can show that under the same

_displacement

field the atoms

_pertaining

to the

_Al(1)

and

_AI(2)

orbits

give

octahedral clusters

_{(respectively}

centered at

_{1/2, 1/2, 1/2}

and

0,

_{0, 0) AI(9)}

and

_A1(11)

become

cuboctahedra

_(centered

at

_{1/2, 1/2, 1/2),}

whereas

_AI(5)

and

_AI(6)

remain

_{regular polyhedra}

with 24 vertices.

Let us now determine the

_symmetry

of the

_{order-parameter}

associated with the assumed

dis-placive

mechanism,

and show that the

_resulting

cubic structure with icosahedral clusters is

thermo-dynamically

stable. The basis function

_spanning

the

_{antiparallel displacements}

of the

_AI(7)

atoms,

can be

_written,

_using

the notation of

_figure

_{la: 0}

= _{xl - x4 + y2 - ys + z3 - z6.} It transforms as the

identity representation,

denoted rl, of the Pm3

space-group

at the center

_(k=0)

of the

_primitive

cu-bic Brillouin-zone. One can

_easily

_verify

that

_although

the atomic

_{displacements}

associated with the other orbits are different

_{(i.e. they}

are

_{spanned by}

différent basis

_functions),

_they

transform as the same irreducible

_{representation}

ri.

Accordingly,

the whole

displacement field

is associated

with a

_{one-component}

_{order parameter}

_’1], which has the

_symmetry

_of

71. The

corresponding

ther-modynamic potential

can thus be written

_{[12] : F( ’1])}

= F 0 +

a 2 +

b il 3 + -77 e 4.

°

2 3 4

Because of the

_magnitude

of the atomic

_{displacements}

which

_lead

to the icosahedral clusters

(see

Tab.

_I),

it should be

_unphysical

to

_put

forward a linear relation

_ship

between the

transi-Table 1. -

Magnitude of

the

_{displacements}

which lead to

_{the formation}

_{of icosahedral}

clusters.

Col-umns

₍₁₎

to

_{(5) give respectively: (1)}

the notation

_of

the sublattices

_{from reference}

_[11];

₍₂₎

the

crys-tallcgraphic positions

in the cubic

unit-cell ;

_{(3) experimental positions (from Ref .}

[11])

for

the cubic

clusters;

(4) calculated posirions for

the icosahedral

_clusters;

_{(5) conditions fulfilled by}

the icosahedral

coordinates. 0

is the

_golden

mean.

tions

_{[5] . Following}

the

_{generalization}

of the Landau

_theory

which has been

_recently

_proposed

for reconstructive transitions of the

_displacive

_type

_{[10] ,}

we will assume a transcendental

_dependence

of q

in functions of the

_{displacements}

_{[10] .}

One finds here:

where Ox =

xi - xj

represents

the distance between two

_antiparallel

atoms

_(e.g.

atoms 1 and 4 in

_{Fig. la) pertaining}

to the same

_{cluster. 0}

is the

_golden

mean. The form of the function

_(1),

which holds

_for

the whole set

_{of displacenlents,}

is

_represented

in

_figure

2. The

_points

indicated on

the curve of

_figure

2

_correspond

however to the

_{specific displacements}

associated with the

_Al(7)

atoms.

_Thus,

for the

_{specific displacements}

Ox =

a y 1

_{+ an,} and

a(o -

1)0-1

₊ an where n

is an

_integer,

the

_Al(7)

atoms form two sets of icosahedral clusters which can be deduced one

from another

_by

a

_shifting

of a , a , a

and a rotation of 90° around the

_[001]

cubic axis. For 2 2 2

the

_{specific displacements corresponding}

to such icosahedral

_domains,

_{the point group}

_symmetry

of the

_AI(7)

sublattice increases from m3 to

m35.

As indicated in

_figure

2 it is the space-group

symmetry

of the sublattice which increases to Pm3n and Pm3m for the

_{respective displacements}

a .

Ax =

(2n

+ 1)’2’

and na. For such virtual

_{displacements}

the

_system

_acquire

fourfold axes and its 2

2402

The

_{thermodynamic}

_stability

of the icosahedral clusters can be verified

_by

the introduction of

(1)

in the

_potential

_F,

and

_by

minimization of F with

_respect

to the

_{displacements}

_{[10] .}

One

gets

the

_équation

of

_{state: q}

_(a

+

_bq

+

crl2)

09

= 0. The stable solution _ri = 0 is obtained

a0x

for Ox -

_{a§- 1}

+ an, or Ox -

_{a(o -}

_1)0-1

-f- an, i.e. for the icosahedral clusters. This

re-suit is in

_agreement

with the

_spirit

of Landau’s

_theory

as a zéro value of the

_{order-parameter}

is associated with the

_{highest point-group}

_symmetry

realized

_by

the atomic clusters. The solution

n

_ -b -E-

(b2 -

4ac) ll2

/2c

_corresponds

to

_general

_temperature

_dependent

_{displacements}

Ox

(Pm3

symmetry

_{Y I‘Y} with cubic

_clusters

whereas

whereas

i9l7

= 0 is associated with the increased Pm3n and

â0x

Pm3m

_symmetries.

Fig.

2. - Periodic

_dependence

of the

_{order-parameter}

for the Pm3

_{(cubic clusters)}

to Pm3

_(icosahedral

clusters)

transition. The number of atoms _per

_{unit-cell, pertaining}

to the

_AI(7)

sublattice are indicated in brackets.

At this

_point

it should be stressed that the assumed

_displacive

field mechanism takes

_place

within the initial cubic

unit-cell,

and thus must be

_complemented

in order to loose the cubic

_periodicity

and stabilize a three-dimensional

_aperiodic

structure. A direct _way for

_expressing

the loss of cubic translational

_symmetry

is to

_find,

in the

_primitive

cubic

_{Brillouin-zone,}

a non-zero k-vector which

creates an icosahedral

_density

wave,

resulting

in an infinite

_{quasi-crystalline}

diffraction

_pattern

in the three-dimensional

_reciprocal

_space.

On that

_goal

let us consider the wave-vector:

with p

= 5 ;;tiJ ’" 0.09.

k,

is an incommensurate wave-vector

_lying

close to the center of

20

the first Brillouin-zone. Its star

_ki

is formed

_by

the six arms

_{ksi, 2}

_{= p}

± 21r

_0,

0 ,

_k3

4 =

1 2 =

M (

a

k3

4 =

a nd 1

In order to allow

_comparison

with

_experiment,

let us calculate the

_{displacements}

associated

in which

_only

the two first Fibonacci numbers 3 and 5 have been considered as an

_{approximation}

of the

_golden

mean

_0.

Minimization of the total

_density

_{wave 0}

=

_E1/;¡

with

_respect

_to

space

co-ordinates

_yields

the function which

_provides

the

_{displacements}

of the Mn atoms from their initial

position

in the cubic cell to their final

_position

in the icosahedral structure. The

_{corresponding}

diffraction

_pattern

is shown in

_figure

3a. It

_displays

_typical

fivefold

_symmetry

axes. An additional

verification of the

_consistency

of the assumed mechanism is

_given

in

_figure

3b where the calcu-lated icosahedral

_positions

are

_superposed

on the

_{high-resolution}

TEM

_micrograph

obtained

_by

Audier and

_Guyot

_{[13] .}

The

_squares

on the

_micrograph

denote the

_projections

of the vertices of the initial cubic lattice.

Fig.

3. -

(a)

Projection

on the

_{plane perpendicular}

to a five-fold axis of the calculated electronic diffraction

pattern

for the AIMnSi structure, when

_taking

into account the total

_displacive

mechanism assumed in our

model

_(formation

of icosahedral clusters

_plus

the incommensurate

_{density waves).}

Six hundred unit-cells have been used for the computer simulation.

_{(b) Superposition}

of the calculated electronic diffraction

pat-tern for the Mn

_quasilattice

on a

_{high-resolution}

TEM

_micrograph

obtained

_by

Audier and

_Guyot

_[13]

_along

the five-fold axis. The dots _represent the

_positions

of the calculated

_projections

in column

_{approximation.}

With the

_exception

of a small number

of defects,

denoted

_by

black dots a _very

_good

coincidence can be

ver-ified. The _squares indicate the

_projections

of the vertices of the initial cubic lattice. The size of the smaller icosahedra

_correspond

to the

_Mn(1)

sublattices.

In _summary, we have shown on the

_example

of AIMnSi that the formation of icosahedral

qua-sicrystal phases

could be understood as

_resulting

from the

_coupling

of two

_independent

mecha-nisms :

₁₎

a

_displacement

field which transforms

_groups

of atoms,

_initially

in cubic

_positions,

into icosahedral clusters within a cubic unit

_cell;

₂₎

an incommensurate

_propagation

of

_density

waves.

The fact that the two assumed mechanisms are

_{phenomenologically}

_independent

_suggests

that their

_experimental

realization in Nature should not be

_necessarily

correlated.

_Actually

there exist

a

_large

number of well-known structures which concretize

_only

the first

_mechanism,

i.e.

_crystal

structures in which near or ideal icosahedral clusters coexist with a three dimensional

2404

The

_effectivity

of the second mechanism assumed in our

_model,

the existence of an

incom-mensurate

_density

wave, received

_recently

two

_{independent possible}

confirmations in AlCuFe

alloys

[17,18]

namely 1)

the observation of soft

_phason

modes at the icosahedral to

_crystal

670° C

transition,

which is

_interpreted

_by

Bancel

_[17]

as an elastic

_instability,

in

_agreement

with our

prediction

that such an

_instability

should occur close to the center of the

_{Brillouin-zone;}

₂₎

the

existence in the

_{phase diagram}

of such

_alloys

of an intermediate

_phase

_interpreted

as an

incom-mensurate structure

_[18].

An additional illustration of the basic idea

_underlying

our

_model,

i.e. that the icosahedral

_phase

results from a solid state

_{transformation,}

was also

_suggested

in

AlCuFe

_alloys,

in which Dubois et al.

_[19]

showed that the icosahedral

_phase

transforms

re-versibly

at 747 °C into a

_closely

related rhombohedral modification

_{of symmetry}

R3m.

_Although

the

_preceding

results

_require

to be confirmed

_{experimentally they}

show indeed that the icosa-hedral

_phase

should be inserted in a more

_{complex phase diagram}

than the current

liquid-to-icosahedral model

_{initially proposed}

_[1-4].

In this

_respect,

let us

_emphasize

that the choice of the

Pm3

_group

as the

_{parent symmetry}

in our model is

_specific

to the case of AIMnSi. We have verified

that one could obtain icosahedral structures from a number of different initial

_{space-groups,}

i.e.

Im3, Fm3m, Pm3n,

P63/mcm,

or

R3m.

Even for AIMnSi the Im3

_{(Z = 138)}

or Im3m

_(Z

=

69)

structures, which can be deduced from the Pm3 structure

_by

small

_{displacements,}

could have been

used as

_parent

_phases.

The wave-vector

₍₂₎

has been selected as the more

_adapted

to the _{AIMnSi structure,} but there exist other wave-vectors,

adapted

to different structures, that _may create icosahedral

_density

waves.

The stars of such vectors must fulfil the two _{necessary conditions}

₁₎

to be formed

_by

at least six

independent

arms, and

₂₎

to allow construction of twelve vectors

_pointing

towards the vertices of

a

_regular

_icosahedron,

_by

combinations with the

_reciprocal

lattice basic vectors. For

_example,

the

wave-vector

_{k2 =}

(2J1.17:.,

_0,

2112 2a)

with ti 2 fulfil

the suficient

condition to

_possess

a twelve

a 2a _J.l2

arms star which

_span

a

_regular

icosahedron. The zone

_boundary

wave-vector

_k3

=

(tP, 0,

a a

the star of which has six arms also fulfils the

_preceding

condition.

The

_generality

of our

_approach

to

_{quasicrystals}

can also be found in the fact that it allows

to describe the

_variety

of one or two-dimensional

_{quasiperiodic}

structures that have been found

experimentally.

Thus,

we were able to show that

_providing

différent sets of atomic

_positions

and

stars k*

_decagonal

_[20],

_dodecagonal

_[21],

and

_octagonal

_[22]

2-d

_{quasicrystals}

could be obtained

as the result of a solid-state

transformation,

the incommensurate modulation

_being

restricted to

a

_plane.

7b

_complete

our model let us work out the theoretical

_{phase diagram corresponding}

to the

set of stable

_phases.

The

_{order-parameter}

associated with

_k*

has six

_components,

denoted

(j.

Observation of

_equal

diffraction intensities in

_{the.icosahedral}

_phase

_require

the

_equilibrium

con-ditions

(i = Ç for

all i. One can thus write the

_{order-parameter}

_expansion

under the effective

F, +

a 1

2 C

1

z

form:

_{F’(c) 0}

+ r c + r (4.

_Accordingly

the whole transition mechanism from the cubic to

()

2 4

, ,

the

_quasicrystal

_phase,

i.e. the formation of icosahedral

_{clusters plus}

the incommensurate

_density

wave can be acounted

_by

the two-order

_parameter

_{potential: F(r¡, ()}

=

F(r¡)

₊

F’(Ç)

₊

dr¡(2,

which

_corresponds

to the theoretical

_{phase diagrams represented}

in

_figure

4.

_Among

the various

Fig.

4. - Phase

diagram representing

the

_potential

_{F(q, ()}

for bd 0

_{(Fig. 4a),}

and bd > 0

_{(Fig. 4b)}

respectively. Solid,

dashed and mixed lines are first-order transition

lines,

second-order transition

lines,

and limit

_{of stability}

lines

_{respectively.}

AK is an isostructural transition line which ends at the critical

_point

K. The A and B multicritical

_points

are at the

_crossing

of a first and second-order transition lines.

Acknowledgements.

We thank M. Audier and R

_Guyot

for

_{kindly providing}

the

_{original micrograph}

used in

_figure

3b.

References

[1]

BAK

_{P., Phys.}

Rev. Lett. 54

₍₁₉₈₅₎

1517-1520.

[2]

MERMIN N.D. and TROIAN S.M.,

Phys.

Rev. Lett. 54

₍₁₉₈₅₎

1524-1527.

[3]

KALUGIN

_P.A.,

KITAEV A. Yu. and LEVITOV

_L.S.,

JETP Lett. 41

₍₁₉₈₅₎

145-149.

[4]

Ho

_T.L.,

_Phys.

Rev. Lett. 56

₍₁₉₈₆₎

468-471.

[5]

LANDAU

L.D.,

JETP 7

₍₁₉₃₇₎

627-633.

[6]

ALEXANDER S. and

MCTAGUE J.,

Phys.

Rev. Lett. 41

₍₁₉₇₈₎

702-705.

[7]

DUBOST

_B.,

LANG

_J.M.,

TANAKA

_M.,

SAINFORT P. and AUDIER _M., Nature 324

₍₁₉₈₆₎

48-50.

[8]

OHASHI W. and SPAEPEN

_F.,

Nature 330

₍₁₉₈₇₎

555-556.

[9]

SANYAL

_M.K.,

SAHNI V.C. and DEY

_G.K.,

Nature 328

₍₁₉₈₇₎

704-706.

[10]

DMITRIEV

_V.P.,

ROCHAL

S.B.,

GUFAN Yu.M. and

_{TOLEDANO P.,}

_Phys.

Rev. Lett. 60

₍₁₉₈₈₎

1958-1961.

[11]

COOPER

_{M., Acta}

_Cryst.

23

₍₁₉₆₇₎

1106-1107.

[12]

TOLEDANO J.C. and TOLEDANO

_P.,

The Landau

_Theory

of Phase

_Transitions,

_Chap.

4

_(World

_Scientific,

Singapore)

1987.

[13]

AUDIER M. and GUYOT

P.,

Philos.

_Mag.

B53

₍₁₉₈₆₎

L43-51.

[14]

PEARSON

W.B.,

A Handbook of Lattice

_Spacings

and Structures of Metals and

_{Alloys (Pergamon,}

Oxford)

1967.

[15]

FRANK F.C. and KASPER

_{J.S., Acta Cryst.}

11

₍₁₉₅₈₎

184-190.

[16]

GURYAN

_C.A.,

STEPHENS P.W., GOLDMAN A.I. and GAYLE _F.W.,

_Phys.

Rev. B37

₍₁₉₈₈₎

8495-8498.

[17]

BANCEL

_{P.A., Phys.}

Rev. Lett. 63

₍₁₉₈₉₎

2741-2744.

[18]

DENOYER

_F.,

HEGER

_G.,

LAMBERT _M., AUDIER M. and

_{GUYOT P., J. Phys.}

France 51

₍₁₉₉₀₎

651-660.

[19]

DUBOIS

_J.M.,

JANOT _Ch., DONG C., DE BOISSIEU M. and AUDIER

_{M., Preprint,}

to be

_published

in Phase Transitions

_(1990).

[20]

BENDERSKY _L.,

_Phys.

Rev. Lett. 55

₍₁₉₈₅₎

1461-1464.

[21]

CHEN

H.,

LI D.X. and KUO

_K.H.,

_Phys.

Rev. Lett. 60

₍₁₉₈₈₎

1645-1648.