Ferroelastic Phase Transition in Pb3(PO4)2 Studied by Computer Simulation

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Ferroelastic Phase Transition in Pb3(PO4)2 Studied by Computer Simulation

K. Parlinski, Y. Kawazoe

To cite this version:

K. Parlinski, Y. Kawazoe. Ferroelastic Phase Transition in Pb3(PO4)2 Studied by Computer Simu-

lation. Journal de Physique I, EDP Sciences, 1997, 7 (1), pp.153-175. �10.1051/jp1:1997131�. �jpa-

00247323�

J.

Phys.

I France 7

(1997)

153-175 JANUARY 1997, PAGE 153

Ferroelastic Phase Transition in Pb3(P04)2 Studied by Computer

Simulation

K. Parlinski

(~,~,*)

and Y. Kawazoe

(~)

(~) ^Institute for Materials

Research,

Tohoku

University,

Sendai 980-77, Japan (~) Laboratoire de

Physique

des Solides

(***),

Uuiversit6

Paris-Sud,

Bhtiment 510,

91405

Orsay

Cedex, France

(Received

2

July

1996, revised 6

September

1996,

accepted

27

September 1996)

PACS.63.70+h Statistical mechanics of lattice vibrations and

displacive phase

transitions PACS.63.20.-e Phonons in

crystal

lattices

PACS.77.84.Dy Niobates,

titanates, tantalates, PZT ceramics, etc.

Abstract. A model of lead

phosphate

which describes its rhombohedral-monoclinic im- proper ferroelastic

phase

transition is

proposed.

It contains _a reduced number of

degrees

of freedom but it is constructed

consistently

with symmetry

changes

at the phase transition. Po- tential parameters ^of the model

are derived from available

experimental

data. The

crystallites

of 25 _x 25 _x 25 and 121 x121 x 25 unit cells have been simulated by the

molecular-dynamics technique.

The results determine the

phase

transition at the L

point

of

reciprocal

_space, the order parameter, ^{and the} temperature ^{behavior of} monoclinic lattice parameters. ^{In the rhom-} bohedral

phase

the calculated

dynamical

structure factor shows inelastic

peaks

from which _a soft branch of uuderdamped

phonons

has been established. The model has been used to calculate _a diffuse

scattering

function which shows above Tc _a maximum at _an incommensurate _wave vector located

along

the L F line of the Brillouiu _zone. The mentioned line is

parallel

to the ternary symmetry ^{axis. On the basis of} the above results _{we were} able to vizualize the _nature of the

dynamical

monoclinic microdomains

persisting

in the

high-temperature

rhombohedral phase.

It has been shown that above Tc the fluctuations _can be treated _as temporary orientational monoclinic microdomains. Each type of microdomains

always

contains _an

irregular

_sequence of

antiphase

domains.

R4sum4. Uu modAle permettant ^de d6crire la transition de

phase ferro61astique

impropre,

rhombo6drique-monoclinique,

du

phosphate

de

plomb

est

propos6

ci-dessous. Il est construit h partir d'un uombre r6duit de

degr6s

de libert6, tout _en tenant compte des

changements

de

sym6trie caract6ristique ^de la transition de

phase.

Les

paramAtres

du modAle sont d6duits quau- titativement des r6sultats

exp6rimentaux disponibles.

La

technique

de dynamique moldculaire _a

permis

de simuler le comportement des cristallites comportant 25 x 25 _x 25 et 121 x 121 _x 25 mailles. Les rdsultats permettent de ddterminer la _nature de la transition de phase

(point

L de la _zone de

Brillouin),

le

paramAtre

d'ordre, la variation _avec la

tempdrature

des

paramAtres

cristallins de la structure

mouoclinique.

Dans le _cas de la

phase ternaire,

le calcul du facteur de

structure

dynamique

conduit h la ddtermination d'uue branche de

phonons

_mous sous-amortis.

Le modAle _a

6galement

6t6 utilis6 _pour obtenir la diffusion diffuse observable le

long

de la

ligue (*)

On leave from the Institute of Nuclear

Physics,

ul.

Radzikowskiego

152, 31-342

Cracow,

and

Academic

Computer

Centre,

Cyfronet,

Cracow, Poland Author for

correspondence (e-mail: [email protected])

(***) URA 2 CNRS

@

Les

#ditions

_de

_Physique

₁₉₉₇

L- F de la

zone de

Brillouin, ligne parallAle

h l'axe ternaire: cette diffusion

pr6sente

_un maximum d'intensitd _{pour une} valeur incommensurable du vecteur d'onde. Sur la base de _ces rdsultats, _nous

sommes parvenus h visualiser la nature des microdomaines

monocliniques

observables dans la

phase

temaire de haute

tempdrature.

Ces trois types de microdomaines

pr6sentent

des

sdquences irrdguliAres

de domaines

antiphases.

Lead

phosphate, Pb3(P04)2,

is _an

improper

ferroelastic material that

undergoes

a

phase

transition at about 453 K. Its

properties

are

relatively

well known froIn

X-ray

and neutron

diffractions,

diffuse

scattering

and inelastic neutron

scattering

studies and from the

optical

and electron transmission

microscopy.

Theoretical

investigations, however,

have been confined to the Landau

type theory

of

phase transition,

to the derivation of

siInple relationship

between the strain coInponents ^{in different}

crystalline phases,

and to the classification of doInain wall types.

In this work _we Inake _an

atteInpt

to relate the

microscopic

static and

dynamical properties

of lead

phosphate

to _a

single

model suitable for computer simulations.

The

high-temperature phase

of lead

phosphate belongs

to the rhombohedral

symmetry

with the space group

R$1n (D(~) [l,2],

and _one Inolecular unit per unit cell

(Z

=

I).

The

crystal

structure consists of chains of Pb atoIns and

P04

Inolecules. In each chain three

subsequent

Pb _{atoIns are}

separated by

two

P04

tetrahedra. The chains _are oriented

along

the three-fold

symmetry

^axis of the rhombohedral

system.

The structure contains three

types

of chains with

the _saIne chain structure, but shifted with

respect

to each other. At

low-teInperature

lead

phosphate

transforIns to Inonoclinic

C2/c (C(~)

_{space group} with two Inolecular units

(Z

=

2)

per unit cell.

The ferroelastic

properties

of lead

phosphate

should show _up in Ineasurelnents of the tem- perature

dependence

of the lattice parameters ^and

spontaneous

^strains. Dilatonietric _mea-

surements [3] showed discontinuities at the transition

teInperature

T~ ⁱⁿ dilatation

along

b and

c directions. The

X-ray

Ineasurelnents [4] estiInated the

changes

of lattice paraIneters as a

function of

teInperature.

^The

precise X-ray

diffraction

experiInent [5-7]

_gave the

teInperature dependence

of the _aM,

bM,

_cM and

fl

Inonoclinic lattice paraIneters below and above the

phase

transition and hence the variation of spontaneous ^strains en and _e13. The ferroelastic

phase

transition _was shown to be

slightly

of first order.

The Brillouin

scattering

Ineasurelnents [8] revealed the elastic constants in the

high-tempera-

ture

phase

in the interval of 457 523

K,

and

slight softening

of _cl

i and the

longitudinal

acoustic Inode has been observed. A number of

experiInental

studies have indicated that there exists

an intermediate teInperature interval froIn T~ to about 560

K,

in which the _average

symmetry

remains rhombohedral, but the local

symmetry

proves to be monoclinic and it is believed that this effect is

generated by

the nionodinic microdomains. The diffraction

X-ray scattering experinient

_[4] allowed to estimate the average size of the monoclinic microdomains to be of

mJ

501.

_The

neutron

scattering experiment performed

above T~ revealed _a

quasielastic

nature of the

supperlattice

reflection and confirmed _a

dominantly dynamic

character of the

monoclinic microdoInains with the characteristic relaxation tiIne of the order of 30 _ps at T~

[9j.

As _a consequence, in this interInediate

teInperature

_range, the actual

syInlnetry

ⁱⁿ Ralnan and infrared exaIninations is

expected

to be nionoclinic instead of rhombohedral.

Indeed,

_a

monoclinic

splitting

of Raman lines above T~ ^has been

reported

in references

[10-12].

^Froln

the full width _at half maximum of the 80

cm~~

Raman bands the life-tiIne of the Inonoclinic niicrodoniains _was estimated

again

to be of the order of _{30 ps}

[12j.

The infrared spectra

[13,14j

show siniilar behavior.

Moreover,

above T~ anoInalies related with

dynaInical

fluctuations in

Pb3(P04)2

have been observed in the

teInperature dependence

of

specific

heat [15] ^{and in the} excess"voluIne strain Ineasurelnents [7]. The

synchrotron X-ray diffuse-scattering experiInent

N°I FERROELASTIC PHASE TRANSITION IN

Pb3(P04)2

155

[16,17]

^shows that

just

above T~ one sees

pronounced

diffuse

scattering

streaks close to the L-

reciprocal

lattice

point

and

along

the L F direction

being parallel

to the ternary rhoInbohedral

crystallographic

axis. The Inaxilna of the diffuse _{streaks are}

placed

out of the

high-syInlnetry point,

but

they

do not lead to the forInation of _a

regular

incoInlnensurate structure. Previous

careful

X-ray

studies

perforIned

in the pure

crystal

of lead

phosphate

_[18] have not revealed any incoInlnensurate

phase.

In the

high-teInperature phase

the inelastic neutron

scattering

studies

[19]

^{have discovered}

a soft Inode at the

L-point

of the Brillouin _zone whose character is

coInpatible

with structural

changes

at the

phase

transition. The line

shape

of the soft Inode shows that it is

overdamped

between T~ and T~ + 100 K [9]. ^{In addition acoustic} and _some

optic phonon dispersion

_curves

have been measured in

principal

directions.

The group theoretical

analysis

of

symmetry

reduction of the rhombohedral-Inonodinic

phase

transition

predicts

three states of orientational Inonodinic doInains and tnro states of

antiphase

doInains within each orientationallnonoclinic doInain. These six doniains _Tray forIn coherent stress free doInain walls

along

either aInirror

crystallographic plane (doInain

wall type

W),

_or

along

_a

specific

orientation

(type W') _[20]. Variety

of these doInain walls has been observed

by optical [21-24]

and transInission electron

[25, 26] Inicroscopy.

^{The effective thickness of the}

W wall of around 10 unit cells has been deterInined

by X-ray

diffraction Inethod

[27].

The ferroelastic

phase

transition has been

theoretically

studied in terIns of Landau

theory [28].

^The

symmetry changes

involve

doubling

of the unit

cell,

hence, the lead

phosphate

is

a ferroleastic material of the

improper type.

^{The order} paraIneter transforms

according

to _a three-dimensional irreducible

representation

_[29]

R31n -

(k

=

L, Tl~'4.

j

# 0,

_q2

= ~3 "

0)

_-

C2/c (I)

which

corresponds

to the L

point

of the Brillouin _zone. It _can be also induced

by

another distinct irreducible

representations T~~.~) [28] leaving

the

ambiguity

_on the actual

symmetry

of the order

parameter.

The model used below exhibits

symmetry changes given by equation (I

The

free-energy expansion

consists of _{even terIns}

only, therefore,

six-order terIns have to be

kept

in order to describe the first order

phase

transition. The

expansion

^has been used to introduce the

syInlnetrized "flip

Inode" order

paraIneter [30]

^{and to describe the} nature of the Inonoclinic microdomains mentioned above. This

approach

involves the basic ideas of the

three-diInensional renorInalization of the Potts Inodel.

The aiIn of this work is to

perforIn

_a

coInputer

siInulation and to

study

consequences of the ferroelastic rhoInbohedral-Inonoclinic

phase

transition in lead

phosphate.

But _even

using

the

present-dayInost powerful supercoInputers,

it is not

possible

to simulate

crystallite

which contains all 13 atoms in the unit cell and which would be

sufficiently large

to

study phenoInena occuring

_on the intermediate scale where any

interesting object spreads

out at least several

lattice constants.

We propose to solve this

probleni by

the method of the reduction of

degrees

of freedom

(RDF).

It _means that the unit cell of the studied

crystal

^is

simplified

to a model unit

cell,

which _possesses

only

_very ^few but essential

degrees

of freedoni. The

computer

^time

nornially

needed to calculate forces between _many

neighbors

is then

considerably

reduced, and it is used to calculate effects _{over many} lattice constants.

Generally,

the model built in the frame of the method of RDF consists of elastic and

displacive subsystems.

The elastic

subsystem

is

proposed

_{as a} Bravais lattice. The

displacive subsystem

takes _care of the

changes

of the

crystal through

the

phase

transition. The two

subsysteIns

interact with linear

and/or

bilinear terIns

adjusted

to the

crystal syInlnetry.

In _{our case} we

require

that the Inodel _preserves the

following

features of lead

phosphate: ii)

it exhibits the _saIne

syInlnetry changes

at the

phase

transition;

in other words the soft mode

symmetries

of the model and of real

crystal

_are the _same;

(it)

the

soft mode

frequency

of both

systenis

_are

similar; (iii)

^{the elastic}

^properties

^remain

^similar;

(iv)

at

teInperature

^{0 K the}

degree

of deforniation of unit cells should be

comparable; (v)

the transition

teInperatures

of both

systeIns

are the _saIne. These conditions _are used _to derive the nuInerical values of the niodel

potential paraIneters.

The unit cell of the Inodel contains five

degrees

of freedoIn

only.

This reduction of the

coInplexity

allows to increase the size of siInulated

crystallite reInarkably

and _to derive _a nuInber of essential characteristics such _as the

teInperature

^{behaviour of the order}

paraIneter

and the lattice

paraIneters,

^the diffuse

scattering,

the low energy

phonon

Triodes

including

the

temperature dependence

of the soft Inode, and Inonodinic fluctuations in the rhoInbohedral

phase.

We show that above T~ lead

phosphate

has _a

tendency

to form _a

temporary

sequence of

antiphase

monoclinic domains which to

large

extent _are

kept

within _one orientational domain due to

strong coupling

to the elastic deformation. Studies of

annealing

_processes which lead to formation of doInain

patterns

can be

equally

well carried _on with the aim of the _saIne

Inodel

[31].

1. Model of Lead

Phosphate

The rhoInbohedral unit cell with _aR

" 7.48

I

_and

aR = 43.4°

[1, 2],

can be

conveniently represented by

the

hexagonal

_one with

paraIneters

ah "

2aRsin(oR/2)

=

5.531

and _ch

=

aR(9

-12

sin~(aR/2))~/~

=

20.301

_and with three1nolecular units. In Cartesian coordinates the basic vectors of the Inonoclinic unit cell inserted into the

hexagonal

lattice read: _{aM =}

(0,1/vi, _{2c)ah, bM}

"

(1, 0, 0)ah

and _cM

=

(0, vi, _0)ah

where _c

=

Ii /(4 sin~(oR/2)) -1/3)~/2,

the face is centered at

(bM

+

cM)/2,

and the Inonoclinic

angle fl'

_occurs between _aM and _cM unit cell

edges.

There _are three orientationallnonoclinic doInains I ₌

1,

2 and 3 and in the

right-hand setting

of lattice vectors the Inonoclinic

angles

are

fl', fl

and

fl', respectively,

where

fl'

= 180°

fl.

The rhoInbohedral

phase expressed

in terIns of monoclinic unit cell

paraIneters

is

a#~

= 13.896

Ji, _b#~

= 5.53

I, _c#~

= 9.58

I

_and

_fl'~~)

= 76.72°

= 180° -103.28°. At _room

temperature

^the rhombohedral unit cell is deforIned _to the monoclinic _one with

parameters

aM = 13.816

I,

_bM

= 5.692

I,

_cM

=

9.4291,

and

fl'

= 77.64°

= 180° 102.36°

[1].

The 39

degrees

of freedoIn of the rhoInbohedral unit cell of lead

phosphate

_can be reduced to 5

degrees

^of freedoIn

by

RDF. We have selected _a Inodel unit cell of the _saIne size _as the real _one, but

containing

two

objects only.

The first _one,

placed

at the _corner of the unit cell

Ii, j, k),

could be considered _{as a} coInbined

P04 P04 coInplex.

Its

position

is

specified by

the Cartesian coordinates

(X~,j,k, f,j,k, Zi,j,k).

The unit cell could be then identified with the elastic _cage. The second

object,

located at the surface

(generally inside)

elastic cage could be treated _{as a}

coInplex

of 3Pb atoIns with two coordinates ri,j,k "

(x~,j,~, yi,j,;)

counted froIn _a fixed

point

of the unit cell

(i, j, k).

It has been proven [4] that for

describing

the rhoInbohedral-

1nonoclinic

phase

transition it is sufficient _to consider the

displaceInents

of lead in the _xv

plane only. Figure

I shows the location of 3Pb and

2(P04) objects

in the

hexagonal

frame.

There,

we have

placed

also the basic vectors of the non-conventional rhombohedral unit cell _aR,

bR,

_cR used in the simulations. Their components are: aR =

(1,

0,

0)ah, bR

₌

(-1/2, vi12, _0)ah

_and

cR "

(0, -1/vi,c)ah.

_This choice makes the three surfaces

(aR,bR), (aR,cR)

and

(bR,cR)

not

equivalent

but it allows to build simulation

crystallites

in the forIn of _a thin

plate,

which is convenient when

siInulating

domain

patterns.

The

low-teInperaturelnonoclinic

unit cell

corresponding

to orientational doInain

= I is also indicated in

Figure

I.

Front the selected 5

degrees

of freedoni _{one can} construct the

phonon representation

of triodes at the _wave vector k

=

L,

which is

rphonon

₌

^2T2

+

3T~.

_{This list contains the} irreducible

representation T2, equation (I),

which drives the rhombohedral monoclinic

phase

transition

[32].

N°I FERROELASTIC PHASE TRANSITION IN

Pb3(P04)2

₁₅₇

' ,

'

, '

, ,

, ,

aM

,

CR

X

bM

Fig.

I. The structure of the model of lead

phosphate crystal. Open

and hatched circles denote

2(P04)

and 3Pb objects,

respectively.

The

(aR,

bR,

cR)

and

(aM,

bM,

cM)

vectors indicate the geomet- rical

relationships

between the rhombohedral and domain

= I of monoclinic unit cells.

respectively.

The

potential

_energy of the model is written _as

v

=

~ _(i[lit~

₊

iiimt

₊

ii,iP~

+

iaiiar)

,

(2)

1,j,~ j

where

~~~~

"

~

_{r~,j,k 4~(~)}

'r~(I),j(I),I.(I),

^(~~

l

velast

~

jj

_B

_(Ri

j k-in) J(")ik(")

~"~

^~

' ~~~

~~l~i

j~

p

(R

'

'(

k( ^~

~ ~~~~

_{~~~~ ~~~~}

~~ ~~~

i,j,k

~

" ~'~' '~ ~ '~ ~ ~ ~

m

~ ~

(~~~~~

"

~il

_(X~,j,k +

Y~,j,k)~

+G2 (vl,j,; isvfj,~x],j,~

+

isvl,j,txl,j,~ xl,j,~l

,

(6)

F L Z T F F _L F Z 3.0

, , , , , ,

,

, , , 1 , ,

, , ,

2.5

i i , i i

i , , ,

i i , , i

i , i , , i

, i , , ,

i _

~

' i

,

~ , , ,

,

$

_i,o

, , ,

z I,

LU _i , ,, i ,

~ ,

O °'5

, , ,

Lu i , , , ,

~

, ,

~ 0

, , , , i i ,

i , i , i , , i ,

, i i i , , , ,

05

, , , , , , , ,

, , , , , , ,

, , , , , , , ,

, , , 1 , 1

WAVE VECTOR

Fig.

2. Harmonic

dispersion

_curves of the model calculated

along

several lines of the rhombohedral Brillouin _zone and for the force constants at T

= 450 K. The

w~(k)

_< 0 values _are represented by negative

uJ(k).

First three

high-symmetry points

L F L _on the left-hand side

are located

parallel

to ternary symmetry ^axis.

and

-4mlrn)

=

(xl

₊

vl) _CoS128n,m)

₊

_{2Tnyn sin(28n,m) 17)}

R~,j,~,~,,j,,~,

=

jixi,j,t _xi,,j,,~,

_{)2 +}

(n,j.~ r,,j,,~,

_{)~ +}

iz~,j.~ zi,,j,,~, _)2j

^~/~

18)

r~,j,k "

(I~,j,k,

_Yi,j,k

). (9)

Here, 4l(1)

denotes _a two-dimensional matrix of force constants,

giving

^rise to

optic phonon

branches. The

4l(1)

has been set up

according

to the symmetry of the model. The summation

and _{n run over}

corresponding neighboring shells,

which _are

specified

below. Summation _over

m is limited to those

displacive objects 3Pb,

which interact

directly

with _a

given pair

of elastic

objects 2(P04)

taken into _account in the

(~(~(~ potential.

In order to indicate the Inethod of

findini'the

_{above set of} paraIneters we discuss first the harmonic terms of the

potential

_energy, I.e.

equations (3, 4). Differentiating them,

_one finds the

dynamical

matrix whose

eigenvalues

and

eigenvectors give

the

dispersion

_curves

w~(k),

and

polarization

vectors,

respectively.

The _Inasses which _enter the

dynaInical

Inatrix have been taken _as

fif~iast

= 811.5 _a.m.u. and fi~ldispi = 621.5 _a.1n.u., the total _Inass of unit cell and _mass of three lead atoms,

respectively.

In

Figure

2 harmonic

dispersion

_curves

uJ(k)

are

displayed

for _a nuInber of lines in the rhoInbohedral Brillouin _zone,

Figure

3. A

negative

soft

Inode, responsible

for the

phase transition,

occurs at the L

point.

The

dispersion

_curves have been derived under the

following assuInptions.

The elastic forces _are liInited to two central forces with first

(al

_"

5.53141)

and second

(a2

_"

7.48001)

_nearest

_neighbors

_{and with}

coefficients

Bi

= 9000

a.m.u.(THz)~

and

82

= 12000

a.1n.u.(THz)~, respectively.

The values of

Bi

and

82

are chosen _so that

they approxiInately reproduce

the

slope

of the acoustic modes Treasured

by

inelastic neutron

scattering _[19].

The

slopes

_are

equal

to sL "

5.14,

_{ST "} 2.63

N°1 FERROELASTIC PHASE TRANSITION IN

Pb3(P04)2

159

~

* , ' '

,

'

~' ' ,

'

Fig.

3. The Brillouin _zone of the rhombohedral unit cell.

and _sL

"

4.63,

_ST

" 2.96 THz

I,

_for

longitudinal

and transversal acoustic1nodes

along

r Z

and r L

directions, respectively.

The

displacive potential requires

Inore

paraIneters,

since _we wish that it will be able to forIn _an incoInmensurate modulation _as well. At T

= 0 K the soft

mode is assumed to be

equal

to its

experimental

value

w2(kL)

= -0.31

(THz)2,

obtained

by

linear

extrapolation

of its

high-teInperature experimental dependence

to T

= 0 K

[19].

^{In this}

assuInption

_we

siInplify

the soft Inode

picture disregarding

the relaxation _processes involved in

it,

but there _{seeIns to no} other way ^to estiInate the values of the model

parameters.

From

the T

= 0 soft niode _we find that the local force _constant

4xx(0; 0)

= 8500

a.m.u.(THz)~

is

positive, assuring

that the systeIn ^{is of}

displacive _type.

Two other force constants which

join

first _nearest

neighbors

_{in xv}

plane

_are 4l~~

lo;1, 0, 0)

= 1500

a.1n.u.(THz)~

and 4l~~

lo;1, 0, 0)

=

4200

a.1n.u.(THz)2,

Second nearest

neighbors

located

along

_cR lattice _{vector are}

interacting

with

4l~~(0; 0, 0,1)

= -1000

a.m.u.(THz)2

and

4l~~(0; 0, 0,1)

= -1500

a.m.u.(THz)2.

In order to enhance the incommensurate

type

of fluctuations _we have included also interactions with next nearest

neighbors

_4l~~

(0; 0, 0,

2 and

4lyy (0; 0, 0,

²

),

at the distance twice _a lattice constant cR. These force _{constants are} assumed to be temperature

dependent, Figure

4. Below T~ lead

phosphate produces

_a

typical

domain

pattern

which is

possible

to be

reproduced by

simulation

only

when

4lxx(0;0,0,2)

and

4l~~(0;0,0,2)

_are

negative.

In contrary, to simulate at

high-

temperatures ^the effect of monoclinic microdomains built _up from

antiphase domains,

the

same force _{constants must} be

positive. Taking

these facts into account we propose that

~4l~~(0;

0,

0, 2)

₌

4l~~(0: 0,

0,

2)

₌

Ao

₊

BOT~

+

CnT~ (10)

with

Ao

= -800

a.m.u(THz)~, Bn

= 5.76 _x

10~~ a.m.u.(THz)~K~~

and

Cn

= -7.74 _x

10~~ a.m.u.(THz)~K~~

Such

temperature

variation could be c~used

by

other

degrees

^{of fi.ee-}

dom of lead

phosphate

not taken

explicitly

into account in _our model.

Special

care has been

paid

to the

optic dispersion

_curves

along

the L F line,

parallel

to the ternary

axis,

since it is

expected

to be

responsible

for the existence of

dynamic

monoclinic

microdomains

occuring

in lead

phosphate _[16].

We have noticed that the curvature of the soft mode at L

point along

L F line is

quite

sensitive _to the value of

4l~x(0:0,0,2)

and

4lyy(0;0,0,2)

force constants. This

part

of

dispersion

_curves is shown in

Figure

5 and its

temperature dependence

follows from the force constants,

equation (10). Figure

5c indicates

clearly

that above T~ fluctuation of incommensurate nature could _appear with energy similar to the energy of fluctuation at the

L-point,

since the

dispersion

_curve is flat. Below

T~,

the

0 200 400 600 800

TEMPERATURE [K]

Fig.

4.-Assumed temperature

dependence

of the second nearest

neighbor

force constant

~~~(0;

0, 0,

2).

soft mode

dispersion

is

steeper

and incommensurate fluctuations will be less

probable.

The curvatures of

uJ(k) along

^directions

perpendicular

to the

ternary

axis is

positive

and

change

very little with temperature.

The anharmonic

potential, equation (6),

stabilizes the

system,

contributes to the

amplitude

of static

displacements

and defines the transition

temperature.

Notice that it does not contain

a fourth order term, which is allowed

by symmetry

and will be

isotropic.

It could be

neglected

because the

experimental

order parameter

exponent

is

fl

₌

1/4,

which is the

typical

value for

a tricritical

region _[18, 33],

when the fourth order _term vanishes. As _a matter of fact _we have been

trying

to

generate

a domain

pattern during annealing using only

fourth order anharmonic

term,

but the

crystal

rather

quickly

went to a

single

domain.

Indeed,

there is _no barrier for

rotation of static

displacements

of 3Pb

objects,

thus _no barrier

prevents

annihilation of the domain walls.

Therefore,

_one must _use the six order local anharmonic

potential.

Its

form, equation (6),

could

produce

in xv

plane

six minima which

correspond

to six

low-symmetry

domains of lead

phosphate.

If _{we set} the ratio _a

=

Vm/Vb

=

@,

_where

_~£

_{and Vb}

are the values of the

potential, equation (6),

_at

thy

^{minimum and at the}

^barrier, respectively,

then the ratio _{e =}

G2/Gi

_"

(a~ 1) /(a2

₊

₁₎

=

3/4.

The choice of

Gi

_" 92000

a.m.u.(THz)21~~

and

G2

" 69000

a.m.u.(THz)~l~~

leads _to the

phase

transition

temperature

at T~ ₌ 470 K.

The

symmetry

^{of the}

phase

transition allows for the

coupling

of elastic and

displacive degrees

of freedom in the

linear-quadratic

forIn. The VC°~P~ _term is

responsible

for the deforniation of the elastic cage caused

by

the soft mode. We limit the

coupling

interactions to nearest and next nearest elastic

objects, similarly

to

Bi

and

82

parameters. We then _assume that each

pair

of

2(P04) objects

interact with two nearest 3Pb

objects,

those which _are in the

closest distance from the line which connects the _two

2(P04) objects.

Front the structure of the

model,

_one finds then the

angles

^{for the}

coupling

function

~4~n(rn), equation ii),

to be

8n,~n

=

7r(n

+

))/3

for the six _nearest

neighbors

in the xv

)lane

numbered antidockwise _n

=

1, 2,. 6;

_{m =}

1,

2. The

8n_~n

_are

equal

for the two nearest

displacive

3Pb

objects (m

=

1, 2).

Siniilarly,

the six second nearest

neighbors

located

along

_cR lattice constant and

equivalent

directions,

are characterized

by phases 8n,m

=

27r(n

+

()/3

and

8n,~n

are

equal

for _{two nearest}

N°I FERROELASTIC PHASE TRANSITION IN

Pb3(P04)2

161

L F L

a T=300K

~

z 1

~ (

-

~ _'

~

w (

~ Q

W (

~ '

~ _'

'

' ' ' l I I I

-

z

~ (

~

z w ,

~ (_,

Q w

x (,

~ , ,

, , ,

,

p

I

j

~ ,

>

^j

fi

~ < l

#

1

~

' 1

For the ferroelastic materials the

boundary

conditions _are the crucial

point

^of simulations.

To avoid additional external forces induced

by

the ferroelastic deformation _we have used free

boundary

conditions. To minimize the surface

problem

for data

analysis

_we have removed _one

layer

of unit cells froIn each

crystal

surface.

2.

Computer

Simulation Results

The

present

^Inodel has been studied

by1nolecular-dynaInics technique.

The calculations have been

perforIned

_on the Hitachi

Supercolnputer S-3800/380

with 24

Gflops.

The siInulated

crystallite

had _a

shape of1nultiplicated

rhoInbohedral unit cell of either 25 _x 25 _x 25

=

15625,

_or

121 x121 _x 25

= 366025 unit cells. The

larger crystallite

has been used to

study

the Inonoclinic Inicrodolnains. The Newton

equations

of1notion have been solved

by

a

simple

difference scheme with time step ^At ₌ ^0.025 _ps. ^The average kinetic energy per

degree

of freedom

was taken _as the

system temperature

^{T. Canonical ensemble} with temperature ^constant was

used. As referred above, the elastic and

displacive subsystems

have been

coupled through

the

linear-quadratic

terms

only.

Such

couplings

do not facilitate transfer of _energy between the

subsysteIns. Therefore,

to

keep

the

teInperatures

of two

subsysteIns equal,

_we have stabilized their teInperatures to the _saIne value

by scaling

the velocities

independently.

2. I. ORDER PARAMETER AND LATTICE PARAMETERS. The order

paraIneter

^{of the1nono-}

clinic-rhoInbohedral

phase

transition has three

coInponents. They correspond

to three _arms of the three-diInensional irreducible star of the

point

L of the rhoInbohedral Brillouin _zone. The

particle configuration

_can then be

analysed

in terIns of local order

paraIneter, coInponents

of which _are defined _as

~))

~ ⁼

(xi,j,k

_{cos 01 + yz,j,k sin 01) cos}

27r(k/~ R)~

_~

). (11

The _suIn

~~~~ =

~j~)~j~ (12)

~

i,J,k ^{' '}

where N is the nuInber of unit

cells,

deterInines the

global

order

paraIneter

^of coInponent I.

Here,

=

1,2,3

^nuInbers the orientational doInains of the Inonoclinic

phase,

and 01

=

0,

02 _"

27r/3,

03 _"

47r/3

describe the

projection

of

displaceInents

of 3Pb

objects

_on the

arias of the irreducible _star L. The coordinates of the _arias in Cartesian

systeIn

are

k/~

=

(0, -1/vi, _~~)(llah), k/~

=

(), fi, ))(llah)

and

k/~

=

(-l/2, fi, ))(llah)

and

R),~,~

specifies

the

position

^of

(i,

_j,

k)

site in the non-deforIned rhombohedral lattice. The _states

~(i)

_~

_{~, ~j2)}

~

~, ~(3)

~ ~

~(21 ~

~, ~(2)

~

~, ~(3)

~

[

~J~~J = 0, ^J~~~

=

0,

^J~~~ >

o, j13)

correspond

to three orientational

domains, respectively.

Within each orientational doInain _one finds _two

antiphase

doInains defined

respectively by

the

sign

of _{a non-zero} coInponent I-e-

~(~) ^> 0 _or ~(~) ^{< 0.}

The

temperature dependence

of the order paraIneter

~(~),

_as resulted froIn _our

computer simulation,

is shown in

Figure

6. The transition

temperature,

where

~(~) vanishes,

is T~ =

470 K. The data of

Figure

6 have been collected

during heating

the

crystallite

from _a

single

domain 1= 3. Similar transition

temperature

was obtained

during heating

_runs which started

N°1 FERROELASTIC PHASE TR~iNSITION IN

Pb3(P04)2

163

~~

~~'.

-.

~ ~o

LU~ ._.

LU .

~ _.

<

~

<

~ .

~

i~

[

.

0

[ K ]

Fig.

6. Calculated temperature dependence of the order parameter ~l~~ as simulated in the crys- tallite of 25 _x 25 _x 25 unit cells.

Heating

rate _was

dT/dt

= 0A

K/ps.

froIn

reInaining

orientational doInains. Slower rates will still

slightly

diIninish the transition

teInperature T~,

while simulation of

larger

_system tends to increase

T~.

The

shape

of the order

parameter

curve

suggests

that the

phase

transition is of first order.

Indeed,

_maps of the order

parameter

distributions

show, Figure 7,

that the

phase

transition starts at the

crystallite

_corner

by forming

nuclei of _a

high-teInperature phase

and

proceeds by Inoving

the interface between

the two

phases

_so that it decreases the voluIne of the Inonoclinic

phase.

The

global potential

_{energy per} Inodel unit cell is

represented

in

Figure

8. It increases

Inonotonically exhibiting only

_a sInall

juInp

in the

vicinity

of the transition

teInperature.

The

slope

of the

potential

_energy in the rhoInbohedral

phase

is sInaller than in the Inonoclinic _one.

The lattice constants of lead

phosphate

_across the

phase

transition have

already

been _Trea- sured

[4, 5,

_7] ^{and it} has been found that _aM and _cM lattice constants and

angle fl

of the

Inonodinic unit

cell,

_are

increasing

_{as a} function of

teInperature,

while bM decreases.

During

the

heating

_run of the simulations _we have recorded the average values of the monoclinic lat- tice parameters ^{of the model for the domain} = 3 and the lattice

parameters

are shown in

Figures

^{9 and lo. These} _{are very} ^similar to the

experimental parameters. Figure

10 shows

as well _two

angles,

which

ought

to remain 90° _across the

phase

transition

and,

_{as one sees,}

they stay

constant with accuracy better than +0.2°. There

is, however,

another effect

quite

characteristic for the ferroelastic Inaterials. At low

teInperatures

the

system perforIns large

elastic vibrations

corresponding

to the _resonance

frequencies

of the whole

crystallite.

This

causes fluctuations of the lattice constants and lattice

angles

in the intermediate

teInperature

range as seen in

Figures

9 and 10.

2.2. DIFFUSE SCATTERING.

Systematic X-ray

studies of lead

phosphate

_(4j showed that

above T~ there _are

relatively

intense

spots

^located at the

position

of the

superlattice peaks.

This

X-ray pattern

could be

explained by

the existence of three sets of

equivalent

Inonoclinic

Inicrodomains,

such that the

ternary

symInetry ^{of the} mean structure is restored. Recent

precise synchrotron X-ray

diffuse

scattering

measureInent Trade above T~ indicates that there

are diffuse spots ^{which do} not concentrate around the L

point,

but _are

placed along

lines

parallel

to the

ternary

axis which connects the

high-symmetry reciprocal

lattice

points

^{L and} F

[16,17].

Our model is able to

reproduce

the diffuse

scattering anticipated by

the

X-ray synchrotron

diffuse

scattering.

For that _we had to _assume that above T~ the soft mode has _a shallow miniInum at the incoInmensurate _wave vector

along

L F

line,

but not

deep enough

to forIn _a

regular

incoInlnensurate

phase.

As _we shall _see in Section

2.4,

this Ininilnuln at incoInlnensurate _wave vector is also

responsible

for the Inonodiniclnicrodolnains which _appear above

T~.

We suppose that the anoInalous diffuse

scattering

is caused

by

the

displaceInents

of lead atoIns. The diffuse

scattering

function has been calculated

according

to the definition

F(k)

=

iP*ik, °)Pik,°)), (14)

where the

dynaInical

variable is

siInplified

to

p(k, t)

=

~j _exp[27rik R)~~~(t)]exp[- f

ei(R)[~~ R~en))], (15)

~

i,j,k l=1

'

Here, (..

denotes the tiIne average, and

R)~~~ (t)

₌

iaR

+

jbR

+

kcR

+ s + ri,j,k

It) (16)

defines the

position

^of the

ii, j, k) displacive object,

and _s

=

0.5(aR

₊

cR)

is the location vector of 3Pb

object

in the undeforIned rhoInbohedral unit cell. To Ininilnize the influence of

crystallite

surfaces _we have introduced the _space

daInping

factor

exp[- £)~~ _{ei(R)[(~ R~en))]}

in which

Rcen

describes the center of

crystallite, R)[(~

^{is the}

^{ii, j,} k)

site in the undeforIned rhoInbohedral

lattice,

and _et

"

e'/(Li/2)~ depends

_on the

daInping

constant

e',

and the linear

size of the simulated

crystallite Li along

its ₌

1, 2,

³

edge.

The e'

= 0.8 has been used.

Figure

11 shows the

F(k)

function for several

teInperatures

above

T~,

and

along

the line froIn L to F

point.

In this

geoInetry

^the transfer

polarization

vectors of the soft Inode _are

quite parallel

to the

scattering

vector k. The

F(k)

has been

averaged

_over 750 _ps. Just above

Tc

the diffuse

scattering

distribution is

peaked

at the L

point.

In the

temperature

_range between 525 and 600 K the maximuIn of

F(k)

shifts towards the incoInmensurate _wave vector at about

kin

= 0.025

i~~,

which

corresponds

to average fluctuation size of

mJ 40

I.

_{At still}

higher teInperature,

the incoInlnensurate Ininimum of the soft mode

disappears

and the diffuse

spot

at the incommensurate _{wave vector} merges ^to the diffuse

peak

at L.

2.3. SOFT MODE. The soft mode of lead

phosphate

has been measured

by

the inelastic

neutron

scattering technique

_[4] above and below transition

point.

Much above

Tc

it _can be

seen as a separate

peak,

which closer to

Tc

_merges to the central

component.

Its forIn _can be fitted to the

daInped

oscillator forInula with _square

frequency

w~

increasing linearly

_{as a}

function of

temperature starting

from _zero close to

Tc.

The width of the soft mode

peak,

which is of order of 0.33

THz, changes only

little in the whole

temperature

^interval.

The

phonon

excitations of the model could be calculated

by

the

dynaInical

structure factor

S(k,w)

Sik,uJ)

=

j~ dtjp*jk,0)pik,t))expj27riuJtjexpj-it/~)~j, _iii)

where

p(k, t)

is the

dynaInical variable, equation (15). Here,

_we have introduced the Gaussian

daInping

factor

exp[- it _/+~)~],

which is

required

while

calculating

the

long-tiIne

correlations and

N°1 FERROELASTIC PHASE TRANSITION IN

Pb3(P04

₎₂ 165

(a)

(b)

(C)

(d)

Fig.

7. Maps of the domain distribution drawn during

heating

large simulated

crystallite

of 65 x

65 x 35 unit cells. The _maps represent ^the

plane perpendicular

to ternary crystal axis and show monoclinic-rhombohedral

phase

transition mechanism. The color pairs: ^dark blue

light

blue, dark green

light

_green,

yellow

red denote the three orientational domains I

= 1, 2, 3,

respectively,

and two colors within each pair ^{define the}

antiphase

domains. Red color denotes then

single

monoclinic

domain. The color of _a

pixel

_on the _maps has been chosen with the aim of

equations (11, 13). Heating

rate _was

dT/dt

= DA

K/ps.

The

(a)

to

(d)

_maps have been recorded at temperatures 415, 443, 470 and 486 K,

respectively.

For this _run the transition temperature _was Tc ₌ 495 K.

d

~ _~'

~ .

it

o z

~ o

I

I

$

0 100 200 300 400 500 600 700 800 900

TEMPERATURE K

Fig.

8. Calculated temperature

dependence

of the potential _{energy per} unit cell of the model _as simulated in the

crystallite

of 25 _x 25 _x 25 unit cells.

Heating

rate _was

dT/dt

= DA

K/ps.

which

plays

the _saIne role _{as a} Gaussian resolution function in the instruInent which _Treasures

S(k, w).

The

experiInental frequency resolution, equation (17),

is then Am

=

[2/(7ri)](In 2)~/~.

Figure

12 shows the calculated

S(,k,uJ) representing

^the

phonon peaks belonging

to the soft branch. The _curves of

Figure

12

correspond

to r L line and _are labelled

by

the _wave vector k =

(2,-2q/vi,q/(3c))(llah),

where _q

=

n/12

and _n

=

1,2,.

6. At

(2,0,0) r-point

the

displaceInents

^{of 3Pb}

objects

_are allnost

parallel

to the

scattering

_wave vector k. The tiIne

dependent

correlation function

(p*(k,0)p(k.t))

has been calculated for tmax

= 20 _ps and is

averaged

over 730 ps.

Dalnping

factor

of1

= 12.5 _ps has been

used,

which results in energy resolution of Am

= 0.043 THz. As _seen from

Figure

12 the soft mode

peak

is not

overdaInped.

This feature follows froIn

entirely displacive

character of the Inodel. On the other hand the widths of

peaks

reInain finite

indicating

that Triodes _are far not harmonic.

The _mean

frequency

of the soft

phonons

has been collected in

Figure

13 for the _wave vector contour r-L-F-r

corresponding

to

points kr

"

(2,0, 0)(llah), kL

"

(?, -1/v5, 1/(6c))(llah)

and

kF

"

(2, -1/v5. -1/(3c))(llah)

and for three

teInperatures

^above

T~.

Most

softening

is

observed at L and F

points.

The

temperature dependence

of the _square

frequency

of the soft mode is shown in

Figure

14.

Within accuracy of the simulations the

points

form _a

straight

line which reaches _zero

frequency

at T

= 438 K, _a little less than T~

= 470

K,

^{obtained from the} order

paraIneter teInperature dependence.

That difference _can

again

be _an indication that the

phase

transition is of first order

type.

2.4. ORIGIN _OF DYNAMICAL IiONOCLINIC MICRODOMAINS IN RHOMBOHEDRAL PHASE.

Up

to _{now we} have

reproduced

the

phase transition,

the order

_and

lattice parameters, ^the diffuse

scattering

and soft Inode. These results _prove that the used model and the

input

^data

form _a consistent _set.

Now,

_we shall show that within the _same

model,

and without any additional parameters ^and

assumptions

_{we can} demonstrate how the monoclinic microdomains look like.

Experimentally,

the existence of the

dynamical

monoclinic microdomains has been

proved

_on

many

occasions,

but _one does not

really

know the _reasons

why they

_appear. One may ask what

N°1 FERROELASTIC PHASE TRANSITION IN

Pb3(P0412

167

13.9

W.,~~

,

(b)

- ~ ~~ _~ b

o< ~.

~

m

~'~ ^'

~W

~

^-~

~

,~

~'~

O ~

O w

~

~ ~

.

~~

~~

O 9,4

.

~ fi

i~

_~ -

8

~

9.2

~~W

fl

9,1

~

_.,.

5

_~'0

8.9

0 100 200 300 400 500 600 700 800 900

TEMPERATURE K

Fig.

9. Calculated temperature

dependence

of the monoclinic lattice constants _aM, bM, _cM of domain

= 3 _as simulated in the

crystallite

of 25 x 25 _x 25 unit cells.

Heating

rate,was

dT/dt

= 0A

K/ps.

is the forIn of

particle configurations

within _a microdomain?

ivhy

do the microdomains _re- main

dynamical?

The diffraction

X-ray scattering experiment

_[4] showed that the

superlattice

reflections

corresponding

to the monoclinic

phase,

which should vanish in the rhombohedral

I _(~)

if f

₉₁

<

X

~

~

"~~.~~../~*~.*~~-~.

d

W Z

<

W O

~ 5

-

b

.*#~.

<

$

~

~°~

o

z

~

<

~ ~

~ ₉₉ w

5 '.~~

98 92

I (C)

@3~

@

~ 91

<

~

~

l

90 -~."wW..::.'~...

W

~

~9 Z

<

W

j

j$

0 100 200 300 400 500 600 700 800 900

TEMPERATURE [Kl

Fig.

lo. Calculated temperature dependence ^{of all}

angles

of the monoclinic unit cell of domain f

= 3 _as simulated in the

crystallite

of 25 _x 25 _x 25 unit cells.

(b)

shows the

fl

= 180°

fl' angle

since for the

setting

of domain ₌ 3 the monoclinic

angle

is

fl'

< 90°.

Heating

rate _was

dT/dt

= DA

hips.

phase,

_possess above

Tc

non-zero intensities. The

synchrotron X-ray

diffuse

scattering

_exper- iments

[16,17]

^indicate a

pronounced

diffuse

scattering

streak close _to the L

reciprocal

lattice

point

and

along

the L F direction

being parallel

to the ternary rhombohedral

symmetry

^axis.