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 Simulation. Journal de Physique I, EDP Sciences, 1997, 7 (1), pp.153-175.
<10.1051/jp1:1997131>. <jpa-00247323>
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Phys.
J.
I
France
(1997)
7
Ferroelastic
Simulation
153-175
Phase
K.
Parlinski
(~)
Institute
(~)
Laboratoire
Transition
(~,~,*) and Y.
for
(Received
PACS.63.70+h
revised
of
mechanics
Phonons
PACS.77.84.Dy
153
by Computer
Studied
(~)
September
6
Statistical
PACS.63.20.-e
PAGE
Japan
980-77,
Sendai
Paris-Sud,
Bhtiment
510,
France
July 1996,
2
Kawazoe
Research, Tohoku University,
Physique des Solides (***), Uuiversit6
Orsay Cedex,
91405
Pb3(P04)2
in
Materials
de
1997,
JANUARY
crystal
in
Niobates,
1996,
lattice
accepted
September 1996)
27
vibrations
displacive
and
phase
transitions
lattices
titanates,
tantalates,
PZT
ceramics,
etc.
phosphate which
describes
its
rhombohedral-monoclinic
imis proposed.
It
contains
reduced
number
of degrees of
a
freedom
but it is
constructed
consistently with symmetry changes at the phase
transition.
Potential
of the
model
derived
from
available
experimental data. The crystallites
parameters
are
of 25 x 25 x 25 and 121
unit cells have
x121
been
simulated
by the
molecular-dynamics
x 25
technique. The results
determine
the phase
transition
the
at the L point of reciprocal
space,
Abstract.
A
model
order
and
parameter,
phase
bohedral
the
diffuse
located
along
of
symmetry
dynamical
has
It
On
the
shown
been
dynamical
phonons has been
which
basis
monoclinic
antiphase
of
Brillouiu
the
that
above
Each
type of
monoclinic
factor
established.
zone.
The
we
the
fluctuations
microdomains
can
at
has
peaks
able
been
parallel
line is
vizualize
to
high-temperature
be
treated
always
used
as
contains
the
In
from
a
to
the
nature
temporary
irregular
a
vector
wave
the
soft
calculate
to
rhombohedral
an
rhom-
which
incommensurate
an
mentioned
were
parameters.
inelastic
model
maximum
a
in
lattice
shows
The
results
persisting
Tc the
above
of
structure
above Tc
shows
microdomains
microdomains.
monoclinic
behavior
temperature
F line of the
the L
axis.
transition
calculated
the
uuderdamped
function
scattering
branch
lead
of
phase
ferroelastic
proper
ternary
of
the
phase.
orientational
sequence
of
domains.
de d6crire la
transition
de phase ferro61astique
impropre,
phosphate de plomb est propos6
Il est
ci-dessous.
construit
de degr6s de libert6,
h partir
d'un
uombre
r6duit
des
changements
de
tenant
tout
compte
en
sym6trie caract6ristique de la transition de phase. Les paramAtres du modAle sont d6duits
quautitativement
r6sultats
exp6rimentaux disponibles. La technique de dynamique
moldculaire
des
a
permis de simuler le comportement des
cristallites
25 x 25 x 25 et 121 x 121 x 25
comportant
mailles.
ddterminer
la
de la
de phase (point L
Les
rdsultats
transition
permettent de
nature
de la zone
de Brillouin), le paramAtre
d'ordre, la variation
la
tempdrature des paramAtres
avec
de la
mouoclinique. Dans le cas de la phase ternaire, le calcul du facteur de
cristallins
structure
dynamique conduit h la
d'uue
sous-amortis.
ddtermination
branche
de phonons
structure
mous
diffuse
observable
le long de la ligue
Le modAle a 6galement 6t6 utilis6
obtenir la
diffusion
pour
R4sum4.
Uu
modAle
rhombo6drique-monoclinique,
(*)
On
Author
for
(***) URA
@
Les
2
du
Physics, ul.
Radzikowskiego
from
the
Institute
of
Nuclear
Computer Centre, Cyfronet, Cracow, Poland
correspondence (e-mail: [email protected])
leave
Academic
permettant
CNRS
#ditions
de
Physique
1997
152,
31-342
Cracow,
and
JOURNAL
154
L- F de la
Brillouin, ligne parallAle
de
zone
d'intensitd
pour
phase
irrdguliAres
la
du
Ces
pr6sente
diffusion
cette
d'onde.
Sur la base
de
maximum
un
rdsultats,
ces
nous
observables
dans la
monocliniques
pr6sentent des sdquences
microdomaines
microdomaines
trois
N°1
I
ternaire:
vecteur
des
nature
tempdrature.
haute
types de
antiphases.
domaines
de
h l'axe
incommensurable
visualiser
h
parvenus
de
temaire
sommes
valeur
une
PHYSIQUE
DE
Pb3(P04)2,
ferroelastic
material
that undergoes a phase
improper
is an
relatively
well
known
froIn
X-ray and
properties
neutron
at
are
diffractions, diffuse scattering and inelastic
scattering studies and from the optical and
neutron
electron
microscopy.
Theoretical
investigations, however, have been confined to
transmission
the Landau type theory of phase transition, to the
derivation of siInple relationship
between the
classification
of doInain wall types.
strain
coInponents in different crystalline phases, and to the
In this work we Inake an
atteInpt to relate the microscopic static and dynamical properties of
phosphate,
Lead
transition
about
453
Its
K.
phosphate to a single model suitable for computer
simulations.
high-temperature phase of lead phosphate belongs to the
rhombohedral
R$1n
[l,2],
Inolecular
(D(~)
and
cell (Z
unit
unit
space
group
one
per
lead
The
the
Pb
atoIns
axis
saIne
unit
per
respect
C2/c (C(~)
showed
[3]
directions.
discontinuities
X-ray
The
of
of the
bM,
aM,
transition
and
hence
the
transition
was
shown
to
The
At
with
three
along
with
low-teInperature
lead
Inolecular
two
three-fold
the
types of chains
three
other.
group
(Z
units
=
2)
has
slightly
been
of 457
observed.
A
intermediate
teInperature
rhombohedral,
but
remains
effect is
this
experinient
501.
of first
generated by
[4] allowed to
The
neutron
below
and
en
revealed
elastic
the
number
slight softening of
of experiInental
studies
interval
froIn T~ to
local
estimate
about
symmetry
nionodinic
the
the
scattering
tem-
Dilatonietric
mea-
and
above
e13.
the
ferroelastic
The
phase
phase
order.
[8]
523 K, and
the
of the
Ineasurelnents
paraIneters
strains
spontaneous
in
up
strains.
estiInated
Ineasurelnents
interval
in the
show
spontaneous
teInperature
dilatation
along b and
T~ in
the changes of lattice
paraIneters
as
a
experiInent [5-7] gave the teInperature
transition
[4]
of
variation
be
should
and
precise X-ray diffraction
and fl Inonoclinic
lattice
cM
scattering
Brillouin
phase
the
at
Ineasurelnents
The
teInperature.
dependence
Inode
are
=
chain
oriented
contains
each
to
space
properties of lead phosphate
dependence of the lattice
parameters
function
that
with
each
In
chains
structure
ferroelastic
surements
an
The
The
system.
Inonoclinic
to
Inolecules.
crystal
subsequent
The
cell.
The
ture
tetrahedra.
shifted
but
P04
and
atoIns
P04
two
structure,
transforIns
perature
c
of Pb
rhombohedral
of the
chain
phosphate
chains
separated by
are
symmetry
the
of
consists
structure
with
symmetry
I).
to
proves
size
of
and
i
the
which
in
monoclinic
that
monoclinic
there
average
and it is
exists
symmetry
believed
X-ray scattering
diffraction
The
the
the
indicated
have
be
microdomains.
average
cl
K,
560
the high-temperalongitudinal acoustic
in
constants
microdomains
to
be
performed above T~ revealed a quasielastic
of the
supperlattice
reflection
confirmed a dominantly dynamic
nature
and
character
of the
monoclinic
microdoInains
with the
characteristic
relaxation
tiIne of the order of 30 ps at T~ [9j.
of
mJ
As
a
and
in this
consequence,
exaIninations
infrared
monoclinic
the full
splitting
width
niicrodoniains
show
siniilar
Pb3(P04)2
excess"voluIne
at
of
half
behavior.
have
interInediate
is
Raman
of the
maximum
again
Moreover,
been
strain
observed
in
Ineasurelnents
be
to
above
lines
to
nionoclinic
the
range,
instead
of
rhombohedral.
been
reported
in
references
teInperature
expected
estimated
was
experiment
80
T~ has
cm~~
be of the
above
the
[7].
T~
Raman
the
syInlnetry
life-tiIne
dependence
synchrotron
in
Ralnan
Indeed,
[10-12].
of the
of 30 ps [12j. The infrared
anoInalies
related
with dynaInical
order
teInperature
The
bands
actual
a
Froln
Inonoclinic
spectra [13,14j
fluctuations
in
of specific heat [15] and in the
X-ray diffuse-scattering experiInent
N°I
FERROELASTIC
[16,17] shows that just
reciprocal lattice point
crystallographic axis.
and
point,
they do
but
X-ray
above T~
The
the
to
studies
pronounced diffuse scattering streaks close to the Ldirection being parallel to the ternary
rhoInbohedral
of the diffuse
placed
streaks
of
the
high-syInlnetry
out
are
forInation of a regular
incoInlnensurate
Previous
structure.
the pure crystal of lead phosphate [18] have not
revealed
between
T~ and T~ + 100 K [9]. In addition
been
measured
directions.
in principal
have
The
predicts
doInains
within
free
stress
three
along a specific
by optical [21-24]
wall of
W
along
walls
and
cells
and
[19] have
coInpatible
is
shows
of the
structural
it is
that
phase
rhombohedral-Inonodinic
doInains
These
and
doniains
six
doInain
these
of antiphase
states
tnro
forIn
Tray
crystallographic plane (doInain
of
discovered
with
overdamped
optic phonon dispersion curves
Inode
some
doInain.
wall
been
has
walls
coherent
W),
type
or
observed
thickness
[25, 26] Inicroscopy. The effective
by X-ray diffraction
Inethod [27].
electron
been
has
soft
Inonodinic
aInirror
transInission
unit
10
studies
character
of the
(type W') [20]. Variety
orientation
around
either
whose
reduction
orientational
orientationallnonoclinic
each
doInain
of
states
scattering
neutron
acoustic
analysis of symmetry
theoretical
group
transition
155
F
perforIned in
incoInlnensurate
phase.
any
In the high-teInperature phase the
inelastic
soft
the
Inode
L-point
of
the
Brillouin
at
a
zone
changes at the phase transition.
The line shape
careful
Pb3(P04)2
IN
sees
the L
Inaxilna
lead
not
one
along
TRANSITION
PHASE
of the
deterInined
phase
has been
theoretically studied in terIns of Landau theory
transition
changes involve doubling of the unit cell, hence, the lead phosphate is
ferroleastic
of the improper type.
transforms
according to a
material
The order
paraIneter
three-dimensional
irreducible
representation [29]
ferroelastic
The
[28].
a
The
symmetry
R31n
corresponds
which
distinct
The
parameter.
free-energy expansion
kept
in
introduce
the
describe
to
Inonoclinic
aiIn
of
work
the
present-dayInost
lattice
all 13
We
atoms
is
even
q2
=
~3
Inode"
order
of the
perforIn
to
powerful
in the
unit
cell
and
be
can
(I)
also
by
induced
Potts
Inodel.
which
any
expansion
and
[30]
approach
This
phase
where
C2/c
-
The
paraIneter
above.
supercoInputers,
scale
0)
It
transition.
coInputer
a
"
another
ambiguity on the actual symmetry
exhibits
symmetry changes given by equation (I
therefore, six-order
have to be
only,
terIns
terIns
phase
order
mentioned
intermediate
the
on
first
the
below
rhoInbohedral-Inonoclinic
ferroelastic
contains
of
renorInalization
this
the
occuring
used
microdomains
three-diInensional
The
model
consists
syInlnetrized "flip
the
# 0,
j
the L point of the
Brillouin
zone.
representations T~~.~) [28] leaving the
The
order
L, Tl~'4.
=
to
irreducible
order
of the
(k
-
siInulation
transition
in
to
involves
and
lead
to
the
study
phosphate.
used
been
has
describe
the
basic
nature
of
of the
ideas
consequences
But
to
of
using
even
possible to simulate crystallite which
would be sufficiently large to study phenoInena
interesting object spreads out at least several
it is
not
constants.
propose
to
solve
this
probleni by
the
method
of
the
reduction
of
degrees of
freedom
cell,
a
nornially
freedoni.
The
time
of
degrees
computer
possesses
very
and it is
reduced,
neighbors is then considerably
calculate
forces
between
needed
to
many
frame
the
model
built
Generally,
the
in
lattice
effects
used to
calculate
constants.
many
over
The elastic
subsystem is
subsystems.
of elastic and displacive
of the
method of RDF
consists
proposed as a Bravais lattice. The displacive subsystem takes care of the changes of the crystal
bilinear
with
linear and/or
terIns
interact
The two subsysteIns
through the phase transition.
adjusted to the crystal syInlnetry. In our case we require that the Inodel preserves the following
transition;
features of lead phosphate: ii) it exhibits the saIne
syInlnetry changes at the phase
words the soft mode symmetries of the model and of real crystal are the same; (it) the
in other
(RDF).
which
It
means
that
only
the
unit
few
cell of the
but
essential
studied
crystal
is
simplified
to
model
unit
PHYSIQUE
DE
JOURNAL
156
N°1
I
frequency of both systenis are similar; (iii) the elastic properties remain similar;
of unit cells should be comparable; (v) the
teInperature 0 K the degree of deforniation
used to derive the
These
conditions
transition
teInperatures of both systeIns are the saIne.
are
potential paraIneters.
nuInerical
values of the niodel
soft
mode
(iv)
at
The
and
the
dependence
phase. We show that
antiphase monoclinic
due
to
to
formation
of
represented by
aR(9
-12
the
cell
unit
low
equally well
be
fluctuations
(0,1/vi, 2c)ah,
face is
within
order
the
derive
a
paraIneter
on
rhoInbohedral
the
in
temporary
a
domain
which
processes
aim of the
the
with
of
sequence
orientational
one
annealing
carried
bM
centered
"
at
aR
7.48
"
with
one
=
Inonoclinic
(1, 0, 0)ah and
(bM + cM)/2,
I
and
paraIneters
and
with
unit
cell
cM
the
43.4°
=
"
inserted
into
lead
saIne
=
the
=
rhombohedral
unit
cell
is
deforIned
and
ch
=
coordinates
=
c
=
doInains
fl', fl
are
=
I
=
and
1, 2 and 3 and in the
fl', respectively,
monoclinic
of
the
to
=
Cartesian
the
=
=
conveniently
5.531
hexagonal lattice read: aM
Ii /(4 sin~(oR/2)) -1/3)~/2,
between aM and cM
angle fl' occurs
orientationallnonoclinic
edges. There are three
right-hand setting of lattice vectors the Inonoclinic angles
fl'
rhoInbohedral
phase expressed in terIns
180°
fl. The
is a#~
5.53 I, c#~
9.58 I and fl'~~)
13.896 Ji, b#~
temperature
be
can
In
units.
where
Inonoclinic
[1, 2],
2aRsin(oR/2)
three1nolecular
(0, vi, 0)ah
=
and
aR
ah
cell
unit
form
to
of
to
phonon Triodes including the
energy
Studies
deformation.
20.301
of the
vectors
with
hexagonal
the
sin~(aR/2))~/~
basic
the
the
Inonodinic
of
reduction
and
of the
behaviour
phosphate has a tendency
kept
extent
to large
are
can
only. This
reInarkably
freedoIn
crystallite
Phosphate
Lead
of
rhoInbohedral
The
of
teInperature
and
Inode,
soft
patterns
the
as
elastic
the
to
siInulated
scattering,
diffuse
of the
doInain
degrees
of
[31].
Model
1.
the
five
such
above T~ lead
which
domains
coupling
strong
size
characteristics
paraIneters,
lattice
contains
the
increase
to
essential
temperature
Inodel
Inodel
the
allows
of
nuInber
of
cell
unit
coInplexity
76.72°
=
unit
180°
monoclinic
cell
-103.28°.
one
with
where
paraIneters
At
room
parameters
I, bM
9.4291, and fl'
77.64°
102.36° [1].
5.692 I, cM
180°
rhoInbohedral
The 39 degrees of freedoIn of the
unit cell of lead phosphate can be reduced
by RDF. We have selected a Inodel unit cell of the saIne size as the
to 5 degrees of freedoIn
real one, but containing two objects only. The first one, placed at the
of the unit cell
corner
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
The second object,
located at the surface (generally inside) elastic
cage.
cage could be
treated
with two
coordinates
atoIns
(x~,j,~, yi,j,;) counted froIn a
as a coInplex of 3Pb
ri,j,k
fixed point of the unit cell (i, j, k). It has been
rhoInbohedralproven [4] that for describing the
1nonoclinic
phase
transition
it is sufficient
consider the displaceInents of lead in the xv plane
to
only. Figure I shows the location of 3Pb and 2(P04) objects in the hexagonal frame. There,
have placed also the basic
of the
rhombohedral
non-conventional
unit cell aR, bR, cR
vectors
we
(-1/2, vi12, 0)ah and
used in the
simulations.
Their
(1, 0, 0)ah, bR
components
are:
aR
(0, -1/vi,c)ah.
choice makes the three
This
surfaces (aR,bR), (aR,cR) and (bR,cR)
cR
equivalent
but
allows
build
it
simulation
crystallites in the forIn of a thin plate, which
not
to
when
is
convenient
siInulating domain
The
low-teInperaturelnonoclinic
unit cell
patterns.
corresponding to orientational
doInain
also
indicated in Figure I.
I is
Front the
selected 5 degrees of freedoni
the phonon
representation of
construct
one
can
2T2 + 3T~.
triodes at the
k
L, which is rphonon
This list
the
contains
vector
wave
irreducible
representation T2, equation (I), which drives the
rhombohedral
phase
monoclinic
aM
=
13.816
=
=
=
=
"
=
=
"
=
=
transition
[32].
=
FERROELASTIC
N°I
PHASE
TRANSITION
'
IN
Pb3(P04)2
157
'
,
'
,
,
,
,
,
aM
,
CR
X
bM
Fig.
rical
The
I.
2(P04)
and
relationships
The
potential
the
of
structure
between
v
The
and
rhombohedral
the
of the
energy
lead
phosphate crystal. Open
(aR, bR, cR) and (aM, bM, cM)
of
model
respectively.
objects,
3Pb
model
is
domain
written
=
and
monoclinic
I of
circles
indicate
unit
cells.
the
denote
geomet-
respectively.
as
~ (i[lit~ + iiimt + ii,iP~ + iaiiar)
=
hatched
vectors
1,j,~
(2)
,
j
where
~~~~
~
"
r~,j,k
4~(~)
(~~
'r~(I),j(I),I.(I),
l
jj
velast
~~l~i
~
(R(Ri '(k-in)
B
'
j~ p
i,j,k
"
~'~'
'~
~
~"~
J(")ik(")
j
'~
~
k(
"
~
~
~
~~~~
~
~
~
~~~~ ~~~~
~~
~~~
m
~il (X~,j,k
+G2
~~~
'
~
(~~~~~
~
+
(vl,j,;
Y~,j,k)~
isvfj,~x],j,~
+
isvl,j,txl,j,~
(6)
xl,j,~l
,
JOURNAL
158
T
Z
L
F
PHYSIQUE
DE
N°1
I
F
F
F
L
3.0
,
,
,
,
,
,
,
,
,
,
Z
,
1
,
,
,
,
2.5
,
i
i
i
,
i
i
,
,
i
,
i
i
i
,
,
i
,
i
,
i
,
,
,
,
i
i
~
_
'
,
~
,
,
,
,
$
i,o
,
,
,
I,
z
,
LU
~
,
°'5
O
,
i
,
,
i
,
Lu
,
,
i
~
,
,
,
,
,
,
~
0
,
i
i
,
,
,
i
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1
,
1
,
,
i
,
i
,
,
i
i
i
,
,
,
,
,
,
,
,
i
05
,
,
,
VECTOR
WAVE
Fig.
ternary
for
and
zone
negative uJ(k).
to
dispersion
Harmonic
2.
Brillouin
First
three
symmetry
axis.
model
of the
curves
force
the
constants
at
high-symmetry
T
points
along
calculated
K.
450
=
L
F
L
several
w~(k)
The
0
lines
rhombohedral
of the
values
left-hand
the
on
<
represented
are
side
by
parallel
located
are
and
(xl
-4mlrn)
=
R~,j,~,~,,j,,~,
=
r~,j,k
"
Here, 4l(1)
denotes
vl) CoS128n,m)
+
jixi,j,t
(I~,j,k,
xi,,j,,~,
)2 +
(n,j.~
r,,j,,~,
17)
)~ +
iz~,j.~
zi,,j,,~, )2j
~/~
18)
(9)
Yi,j,k ).
two-dimensional
a
2Tnyn sin(28n,m)
+
of
matrix
force
giving
constants,
rise
phonon
optic
to
summation
4l(1) has been set up according to the symmetry of the model. The
and n run
corresponding
neighboring
shells,
which
specified
below.
Summation
over
are
over
limited to those displacive objects 3Pb, which
directly with a given pair of elastic
interact
m is
objects 2(P04) taken into account in the (~(~(~ potential.
branches.
The
order
In
harmonic
the
and
the
indicate
to
potential
of the
terms
of
Inethod
findini'the
energy,
and
taken
mass
displayed
Inode,
been
forces
fif~iast
as
of
three
for
a
nuInber
responsible
derived
with
Bi
of Bi and 82
Treasured
by
for
(al
=
are
"
9000
chosen
inelastic
of
paraIneters
we
first
discuss
the
811.5
and fi~ldispi
of unit cell
621.5
the total
a.m.u.
a.1n.u.,
Inass
respectively. In Figure 2 harmonic
dispersion
uJ(k) are
atoms,
curves
of lines in the
rhoInbohedral
Brillouin
Figure 3. A negative soft
zone,
=
phase
the
the
under
first
coefficients
=
lead
set
(3, 4). Differentiating them, one finds
w~(k),
eigenvectors give the dispersion
curves
which
have
the dynaInical
Inatrix
enter
Inasses
dynamical
matrix
whose eigenvalues and
polarization
respectively.
The
vectors,
been
above
equations
I.e.
transition,
following
5.53141)
and
a.m.u.(THz)~
so
that
neutron
and
The
(a2
second
82
the
at
occurs
assuInptions.
=
L
elastic
"
12000
point.
forces
The
are
dispersion
liInited
to
have
curves
central
two
7.48001) nearest neighbors and
a.1n.u.(THz)~, respectively. The
they approxiInately reproduce the slope of the acoustic
scattering [19]. The slopes are equal to sL
5.14, ST
"
with
values
modes
"
2.63
N°1
FERROELASTIC
PHASE
TRANSITION
IN
Pb3(P04)2
159
~
*
,
'
'
~'
Fig.
The
3.
and
sL
and
r
wish
"
Brillouin
'
,
'
'
rhombohedral
the
of
zone
,
unit
cell.
4.63, ST
2.96 THz I, for longitudinal and
transversal
directions, respectively. The displacive potential requires
it will be
that
able
forIn
to
incoInmensurate
an
modulation
along
acoustic1nodes
"
L
paraIneters,
Inore
well.
as
At
T
=
Z
r
since
0 K
we
soft
the
assumed
experimental value w2(kL)
-0.31 (THz)2,
obtained
by
to be equal to its
extrapolation of its high-teInperature experimental dependence to T
0 K [19]. In this
assuInption we siInplify the soft Inode picture disregarding the
relaxation
involved
processes
mode
is
=
linear
=
it, but
in
the
T
=
there
seeIns
0 soft
niode
assuring
positive,
other
to
no
we
find
that
estiInate
to
way
that
the
local
the
force
displacive
is of
values
model
of the
4xx(0; 0)
constant
=
From
parameters.
a.m.u.(THz)~
8500
is
join
first
neighbors in xv plane are 4l~~ lo;1, 0, 0)
1500 a.1n.u.(THz)~ and 4l~~ lo;1, 0, 0)
nearest
4200 a.1n.u.(THz)2,
Second
neighbors located along cR lattice
nearest
interacting
vector
are
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
the
systeIn
Two
type.
force
other
which
constants
=
=
=
the
enhance
to
next
=
incommensurate
neighbors 4l~~ (0; 0, 0,
nearest
fluctuations
type of
included
have
we
and 4lyy (0; 0, 0, 2 ), at
assumed to be
temperature
also
distance
the
2
with
interactions
twice
a
lattice
constant
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 highthe effect of
monoclinic
built
from
microdomains
antiphase domains, the
temperatures
up
force
that
into
constants
must be positive. Taking these facts
account
same
propose
we
force
These
cR.
constants
are
~4l~~(0; 0, 0, 2)
with
10~~
=
4l~~(0: 0, 0, 2)
Ao
-800 a.m.u(THz)~, Bn
a.m.u.(THz)~K~~ Such temperature
=
=
phosphate
5.76
taken explicitly
paid
to the optic
care
the ternary axis, since it is expected to be
microdomains
occuring in lead phosphate
soft
mode at L point along L
F line is
dom
of lead
Special
4lyy(0;0,0,2)
temperature
clearly
to
the
that
energy
has
force
not
been
constants.
dependence
follows
This
from
part
the
x
could
of
We
CnT~
+
our
have
sensitive
constants,
the L
existence
noticed
to
curves
Cn
-7.74
=
degrees
other
x
of fi.ee-
model.
along
the
(10)
and
by
c~used
be
curves
for
dispersion
force
BOT~
in
account
dispersion
responsible
[16].
quite
+
a.m.u.(THz)~K~~
10~~
variation
into
Ao
=
the
is
that
value
line,
F
dynamic
of
the
in
curvature
Figure
5
equation (10). Figure 5c
of
could
above T~
fluctuation
incommensurate
nature
of
fluctuation
L-point, since the dispersion
at the
appear
curve
is
to
monoclinic
of
4l~x(0:0,0,2)
of
shown
parallel
and
the
and
its
indicates
with
energy
similar
flat.
Below
T~, the
JOURNAL
160
DE
PHYSIQUE
[K]
TEMPERATURE
Fig.
4.-Assumed
~~~(0;
soft
dispersion
mode
of
a
fourth
the
second
neighbor
nearest
force
constant
order
the
tricritical
and
fluctuations
incommensurate
perpendicular
directions
to
ternary
the
will
be
axis
is
The
less probable.
positive and change
temperature.
equation (6),
potential,
displacements
because
a
with
anharmonic
static
steeper
is
uJ(k) along
of
little
The
of
2).
0, 0,
curvatures
very
dependence
temperature
800
600
400
200
0
N°1
I
term,
and
which
defines
is
amplitude
the
to
it
does
not
contain
symmetry and will be isotropic. It could be neglected
1/4, which is the typical value for
exponent is fl
parameter
allowed
experimental order
region [18, 33], when
contributes
the system,
Notice that
temperature.
stabilizes
transition
the
by
=
the
fourth
order
term
vanishes.
As
a
matter
of fact
we
have
during annealing using only fourth order
anharmonic
a
Indeed, there is no barrier for
term, but the crystal rather quickly went to a single domain.
of the
of static
displacements of 3Pb objects, thus no barrier
annihilation
rotation
prevents
Therefore, one must use the six order local
domain
walls.
anharmonic
potential.
Its form,
equation (6), could produce in xv plane six minima which correspond to six low-symmetry
@, where ~£ and Vb are the
If we set the ratio a
domains
of lead phosphate.
Vm/Vb
values of the potential, equation (6), at thy minimum
and at the barrier, respectively, then the
G2/Gi
3/4. The choice of Gi
ratio e
(a~
92000 a.m.u.(THz)21~~ and
1) /(a2 + 1)
a.m.u.(THz)~l~~
leads to the phase
G2
transition
470 K.
69000
temperature
at T~
The symmetry of the phase
allows for the coupling of elastic and displacive degrees
transition
of freedom in the
linear-quadratic
The VC°~P~ term is responsible for the
deforniation
forIn.
mode.
of the
elastic
caused by the soft
We limit
the coupling
interactions
to
nearest
cage
and next
elastic objects, similarly to Bi and 82
We then
that
nearest
parameters.
assume
each pair of 2(P04) objects
with two
interact
3Pb objects, those
which
in the
nearest
are
from the line which
closest
distance
the two 2(P04) objects.
Front the
of
connects
structure
the model, one finds then the angles for the coupling function ~4~n(rn), equation ii), to be
antidockwise
8n,~n
7r(n + ))/3 for the six nearest neighbors in the xv )lane numbered
n
displacive 3Pb objects (m
1, 2. The 8n_~n are equal for the two
1, 2).
1, 2,.
6; m
nearest
neighbors located along cR lattice
equivalent
Siniilarly, the six second
and
nearest
constant
characterized
by phases 8n,m
directions, are
27r(n + ()/3 and 8n,~n are equal for two nearest
trying
been
to
generate
domain
pattern
=
"
=
=
"
=
=
"
=
=
=
=
=
FERROELASTIC
N°I
PHASE
TRANSITION
L
IN
Pb3(P04)2
F
a
L
T=300K
~
z
1
~
-
(
~
~
'
(
w
~
Q
W
(
'
~
~
'
'
'
'
'
l
I
I
I
z
~
~
(
z
w
~
,
(
,
Q
w
(
x
,
~
,
,
,
,
,
,
p
I
j
~
,
>
j
fi
~
<
#
~
1
'
1
l
161
JOURNAL
162
the
For
ferroelastic
avoid
To
boundary
layer of unit
conditions.
cells
Computer
2.
external
To
froIn
boundary
by
the
materials
additional
forces
the
crystal
Simulation
I
conditions
induced
minimize
each
PHYSIQUE
DE
N°1
the
are
ferroelastic
the
point of
crucial
deformation
problem for data analysis
surface
simulations.
have
we
have
we
used
free
removed
one
surface.
Results
by1nolecular-dynaInics technique. The calculations have
Supercolnputer S-3800/380 with 24 Gflops. The siInulated
been perforIned on the
Hitachi
of1nultiplicated
crystallite had a shape
rhoInbohedral
unit cell of either 25 x 25 x 25
15625, or
larger
crystallite
366025 unit cells. The
has been used to study the
Inonoclinic
121 x121 x 25
Inicrodolnains.
The
Newton
of1notion
difference
equations
have
been
solved by a simple
scheme
with
At
freedom
time
The
kinetic
degree
of
0.025 ps.
step
average
energy
per
present
The
Inodel
been
has
studied
=
=
=
taken
was
As
used.
the
system
referred
above,
as
temperature
the
Canonical
T.
elastic
ensemble
displacive
and
linear-quadratic
only. Such couplings do not
terms
subsysteIns. Therefore, to keep the teInperatures of
their
value by scaling the
teInperatures
to the
saIne
ORDER
2. I.
PARAMETER
the
three-diInensional
irreducible
configuration
particle
which
defined
are
in
L of the
paraIneter
rhoInbohedral
of local
terIns
order
They correspond
order
to
the
have
stabilized
of
the1nono-
three
Brillouin
paraIneter,
the
between
energy
subsysteIns equal, we
velocities
independently.
The
was
of
arms
The
zone.
of
coInponents
as
~))
The
analysed
of
transfer
constant
coupled through
been
two
coInponents.
point
of the
star
be
then
can
three
has
transition
temperature
have
facilitate
PARAMETERS.
LATTICE
AND
phase
clinic-rhoInbohedral
with
subsystems
(xi,j,k
=
~
sin 01)
01 + yz,j,k
cos
R)~ ~ ).
27r(k/~
cos
(11
suIn
~~~~
=
~j~)~j~
~
'
(12)
'
i,J,k
N
where
0, 02
arias
nuInber
the
is
Here,
I.
1,2,3
=
27r/3, 03
"
of the
"
of
the
47r/3
describe
irreducible
position of (i,
the
L.
star
(0, -1/vi, ~~)(llah), k/~
specifies
cells,
unit
nuInbers
=
j,
The
~(i)
correspond
finds
three
to
antiphase
two
~(~) > 0 or ~(~) < 0.
The
temperature
simulation,
470
K.
domain
The
is
~(21
~
~J~~J
=
dependence
~, ~(3)
~
~, ~(3)
J~~~
=
0,
J~~~
defined
of
of
Similar
transition
in
0,
~
domains, respectively.
respectively by
orientational
Figure 6.
Figure 6 have
shown
data
1= 3.
~, ~j2)
~, ~(2)
the
order
paraIneter
The
transition
been
collected
temperature
was
paraIneter
~
of
Cartesian
=
systeIn
(-l/2,
fi, ))(llah)
rhombohedral
lattice.
=
coInponent
phase, and 01
of 3Pb objects on the
Inonoclinic
in
arias
k/~
and
order
the
displaceInents
of
non-deforIned
the
~
doInains
of
of the
coordinates
in
site
global
projection
the
(), fi, ))(llah)
k)
the
doInains
deterInines
orientational
k/~
are
=
and
The
R),~,~
states
~
[
~
>
j13)
o,
Within
the
each
sign of
~(~),
temperature,
as
a
orientational
non-zero
doInain
coInponent
one
I-e-
froIn our
computer
~(~) vanishes, is T~
resulted
where
during heating the crystallite
obtained during heating runs
from
which
a
=
single
started
N°1
FERROELASTIC
PHASE
TR~iNSITION
Pb3(P04)2
IN
163
~~'.
~~
-.
~
~o
LU
.
~
.
LU
.
~
.
<
~
<
.
~
~
i~
[
.
0
[
Fig.
Calculated
6.
tallite
of 25
froIn
reInaining
25
x
x
dependence
temperature
unit
25
]
K
Heating
cells.
rate
doInains.
orientational
of
Slower
parameter
will
tends
~l~~
simulated
as
in
the
crys-
K/ps.
0A
=
rates
of larger system
simulation
T~, while
transition
suggests that the phase
curve
teInperature
parameter
parameter
order
the
dT/dt
was
still
slightly
is of first
diIninish
T~.
increase
to
order.
The
Indeed,
the
transition
shape of the
order
of the
order
maps
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
Inodel
unit cell is represented in Figure 8.
It
increases
energy
per
Inonotonically exhibiting only a sInall juInp in the vicinity of the transition
The
teInperature.
slope of the potential energy in the
rhoInbohedral
phase is sInaller than in the Inonoclinic
one.
The lattice
of lead phosphate
the phase
have already been
transition
constants
across
Treasured [4, 5, 7] and it has been
found
that aM and cM
lattice
and angle fl of the
constants
Inonodinic
while bM decreases.
During
unit cell, are increasing as a function of teInperature,
of
the heating
the
simulations
have
recorded
the
values
of
the
monoclinic
lataverage
run
we
well
as
of
parameters
tice
Figures
and
9
they stay
for
characteristic
ought
corresponding
of the
as
in
seen
2.2.
DIFFUSE
above
T~
there
Figures
to
lattice
9 and
relatively
=
and
3
lattice
the
are
which
do
constants
intense
not
shown
in
and
lattice
angles
in
intermediate
the
teInperature
10.
spots
X-ray
located
of lead
studies
the
at
concentrate
around
the
L
phosphate (4j showed that
of the
superlattice peaks.
Inonoclinic
sets of equivalent
position
X-ray pattern could be explained by the existence of three
Inicrodomains, such that the ternary
symInetry of the
mean
precise synchrotron X-ray diffuse scattering
Trade
measureInent
spots
are
the
to
This
diffuse
parameters
experimental
Figure 10 shows
parameters.
phase
and,
remain
90°
the
transition
across
as
one
sees,
better
than
+0.2°.
effect quite
There is, however,
another
Inaterials.
At low
the system perforIns large
teInperatures
frequencies of the whole crystallite.
the
This
resonance
Systematic
SCATTERING.
are
to
accuracy
vibrations
domain
similar
ferroelastic
fluctuations
elastic
range
the
the
very
are
which
with
constant
for
model
These
angles,
two
causes
the
lo.
point,
is
structure
above
but
T~
are
restored.
indicates
Recent
that
there
placed along lines
parallel
the
to
[16,17].
F
DE
PHYSIQUE
the
high-symmetry reciprocal lattice points L and
diffuse
scattering anticipated by the X-ray
JOURNAL
164
synchrotron
shallow
model
the
at
regular
a
incoInlnensurate
which
able
is
connects
reproduce
to
scattering.
diffuse
miniInum
forIn
to
axis
ternary
Our
For
that
had
we
incoInmensurate
is also
vector
to
N°1
that
assume
along
vector
wave
phase. As we
responsible for
incoInlnensurate
wave
the
I
shall
above
L
soft
has
Ininilnuln
Inonodiniclnicrodolnains
the
mode
a
deep enough
not
2.4, this
Section
in
see
T~ the
line, but
F
which
at
appear
above T~.
We
the
that
suppose
function
has
F(k)
dynaInical
the
where
variable
p(k, t)
=
is
been
siInplified
denotes
the
the
position
object
of 3Pb
vector
crystallite
surfaces
(14)
f
R)~~~(t)]exp[-
R~en))],
ei(R)[~~
(15)
'
tiIne
of
the
in
the
we
lead
l=1
and
average,
R)~~~ (t)
defines
of
to
i,j,k
Here, (..
by the displaceInents
according to the definition
caused
is
calculated
iP*ik, °)Pik,°)),
=
~j exp[27rik
~
scattering
diffuse
anoInalous
scattering
diffuse
The
atoIns.
iaR
=
jbR
+
+
kcR
ii, j, k) displacive object,
undeforIned
introduced
have
rhoInbohedral
the
+
and
It)
ri,j,k
s
factor
(16)
0.5(aR
=
cell.
unit
daInping
space
+
s
To
+
cR)
Ininilnize
is
the
location
influence
the
of
exp[- £)~~ ei(R)[(~ R~en))]
ii, j, k) site in the undeforIned
crystallite, R)[(~ is the
e'/(Li/2)~ depends on the daInping constant e', and the linear
rhoInbohedral
lattice, and et
simulated
crystallite Li along its
size of the
1, 2, 3 edge. The e'
0.8 has been used.
Figure 11 shows the F(k) function for several
above
teInperatures
T~, and along the line
froIn L to F point. In this geoInetry the transfer
polarization
of
the soft Inode are quite
vectors
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
between
temperature
range
which
in
Rcen
describes
the
center
of
"
=
525
and
600 K the
i~~,
kin
0.025
teInperature,
=
the
at
maximuIn
the
of
F(k)
incoInlnensurate
incommensurate
shifts
corresponds
which
vector
wave
to
Ininimum
merges
=
towards
average
the
of the
soft
the
diffuse
to
incoInmensurate
fluctuation
mode
peak
size
wave
of
mJ
disappears
at
I.
40
and
vector
At
the
at
still
diffuse
about
higher
spot
L.
The soft mode of lead phosphate has been
measured
by the inelastic
technique [4] above and below
transition
point. Much above Tc it can be
Its forIn
separate peak, which closer to Tc merges to the central
component.
as
a
seen
can
be fitted to the daInped
forInula
with
oscillator
frequency w~ increasing linearly as a
square
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
interval.
temperature
The phonon
excitations
of the model
could be
calculated by the dynaInical
factor
structure
2.3.
SOFT
neutron
MODE.
scattering
S(k,w)
Sik,uJ)
where
p(k, t)
is the
daInping factor
=
j~ dtjp*jk,0)pik,t))expj27riuJtjexpj-it/~)~j,
dynaInical variable,
exp[- it /+~)~],
which is
iii)
equation (15). Here, we have introduced the Gaussian
required while calculating the long-tiIne correlations and
FERROELASTIC
N°1
PHASE
TRANSITION
IN
Pb3(P04 )2
165
(a)
(b)
(C)
(d)
heating large simulated
crystallite of 65 x
perpendicular to ternary crystal axis and show
mechanism.
The
color pairs:
dark
blue
light blue, dark
monoclinic-rhombohedral
phase
transition
denote
three
orientational
domains
I
respectively,
light
yellow
red
the
1,
and
2,
3,
green
green,
Red color
domains.
denotes
then single
monoclinic
colors
within
each pair define the antiphase
two
domain.
The color of a pixel on the maps has been chosen with the aim of equations (11, 13). Heating
Fig.
65
7.
x
35
Maps
unit
of
the
cells.
distribution
domain
The
represent
maps
drawn
the
during
plane
=
rate
was
and
486
dT/dt
=
DA
K/ps.
K, respectively.
For
The
this
(a)
to
run
the
(d)
maps
transition
have
been
temperature
recorded
was
temperatures
at
Tc
=
495
K.
415, 443,
470
JOURNAL
166
PHYSIQUE
DE
N°1
d
~
~'
~
o
.
it
z
I
o
~
I
$
0
300
200
100
400
Fig.
Calculated
8.
simulated
in
of 25
plays the saIne role
The experiInental
which
dependence
temperature
crystallite
the
as
25
x
x
25
Gaussian
a
of
the
cells.
unit
resolution
600
500
TEMPERATURE
800
700
900
K
potential
Heating
energy
rate
function
was
in
the
cell
unit
per
dT/dt
=
the
of
model
as
K/ps.
DA
which
instruInent
Treasures
[2/(7ri)](In 2)~/~.
frequency resolution, equation (17), is then Am
calculated S(,k,uJ) representing the phonon peaks belonging to the soft
12 shows the
of Figure 12 correspond to r
labelled by the
The
L line and are
branch.
vector
curves
wave
(2,-2q/vi,q/(3c))(llah), where q
n/12 and n
1,2,.
k
6. At (2,0,0) r-point the
k.
The tiIne
displaceInents of 3Pb objects are allnost parallel to the scattering wave
vector
(p*(k,0)p(k.t)) has been calculated for tmax
function
correlation
dependent
20 ps and is
averaged over 730 ps. Dalnping factor of1
12.5 ps has been used, which
results in energy
of Am
resolution
0.043 THz. As seen from Figure 12 the soft mode peak is not overdaInped.
follows froIn entirely displacive
This
feature
character
of the Inodel.
hand the
On the other
finite indicating that Triodes are far not
widths of peaks reInain
harmonic.
The
frequency of the soft phonons has been collected in Figure 13 for the wave vector
mean
(2,0, 0)(llah), kL
(?, -1/v5, 1/(6c))(llah)
r-L-F-r
corresponding to points kr
contour
-1/v5.
-1/(3c))(llah)
and kF
and for three
above T~. Most softening is
(2,
teInperatures
S(k, w).
Figure
=
=
=
=
=
=
=
"
"
"
observed
at
The
Within
T
at
=
order
points.
F
dependence
of the
frequency of
square
the
soft
mode
is
shown
in
Figure
14.
points form a straight line which reaches zero frequency
little less than T~
from the order
470 K, obtained
paraIneter
teInperature
a
difference
That
indication
again be an
that
the phase
transition
is of first
can
of the
accuracy
438 K,
dependence.
simulations
the
=
type.
ORIGIN
2.4.
Up
and
L
temperature
to
now
form
the
and
we
MICRODOMAINS
phase
These
Inode.
Now,
set.
parameters
shall
assumptions
transition,
results
show
we
the
that
prove
that
within
can
demonstrate
RHOMBOHEDRAL
IN
order _and
the
used
the
same
how
the
lattice
model
model,
the
input
data
and
the
and
monoclinic
PHASE.
parameters,
without
any
microdomains
like.
Experimentally,
many
reproduced
and soft
consistent
a
additional
look
have
scattering
diffuse
IiONOCLINIC
DYNAMICAL
OF
we
occasions,
one
does
dynamical
really know the
of the
existence
the
but
not
monoclinic
reasons
microdomains
why they
appear.
has
One
been
may
proved
ask
on
what
FERROELASTIC
N°1
PHASE
TRANSITION
IN
Pb3(P0412
167
13.9
(b)
W.,~~
~
-
o<
,
~~
~.
b
~
'
~'~
~
~W
m
~
-~
,~
~
~'~
O
~
O
w
~
~ ~
.
~~
.
O
~~
9,4
~
fi
i~
~
8
~
-
9.2
~~W
9,1
fl
~
.,.
5
~'0
8.9
0
100
300
200
500
400
TEMPERATURE
Fig.
=
is
9.
3
as
Calculated
simulated
in
temperature
the
crystallite
dependence
of 25
x
25
of the
x
25
600
monoclinic
unit
700
cells.
lattice
Heating
900
bM,
constants
aM,
rate,was
dT/dt
microdomain?
forIn of particle configurations
within a
ivhy do
dynamical? The diffraction X-ray scattering experiment [4] showed
reflections
corresponding to the
monoclinic
phase, which should vanish
the
main
800
K
the
cM
0A
of
domain
K/ps.
microdomains
that
in
=
the
the
re-
superlattice
rhombohedral
JOURNAL
168
I
PHYSIQUE
DE
N°1
I
(~)
if
f
91
<
X
~~
"~~.~~../~*~.*~~-~.
W
d
Z
<
W
O
~
5
b
-
.*#~.
<
~
$
~°~
o
~
z
<
~
~
w
~
99
5
'.~~
98
92
I
(C)
@
3~
@
~
91
<
~
~
l
-~."wW..::.'~...............
90
W
~
~9
Z
<
j
W
j$
0
100
200
300
500
400
Fig.
f
=
for
Calculated
lo.
3
as
the
phase,
simulated
setting
of
possess
[16,17]
point and along
iments
in
temperature
the
crystallite
domain
=
above Tc
indicate a
the L
F
3 the
dependence
of 25
x
25
monoclinic
of all
25
unit
angle
is
x
700
600
800
900
[Kl
TEMPERATURE
angles
cells.
fl'
<
of
(b)
90°.
the
monoclinic
shows
Heating
the
fl
rate
unit
cell
of
domain
fl' angle since
dT/dt
DA hips.
180°
=
was
=
The synchrotron X-ray diffuse
scattering
intensities.
non-zero
pronounced diffuse scattering streak close to the L reciprocal
rhombohedral
direction being parallel to the ternary
symmetry
exper-
lattice
axis.
FERROELASTIC
N°1
PHASE
TRANSITION
IN
Pb3(P04)2
169
L
F
(
-T=475K
-T=525K
-T=600K
-T=700K
-T=800K
~
$
E
0.02
0
O-M
Fig.
Calculated
11.
point of
crystal. The
diffuse
rhombohedral
to F
wave
given
is
vector
scattering
in
zone.
i~~
units.
L-F
0. lo
VECTOR
F(k)
function
The
Brillouin
0.08
0.06
WAVE
above Tc and
line is
parallel
along
the
vector
wave
axis
ternary
to
of the
line
from
L
rhombohedral
electron
microscopy is,
of the pheunderstanding
character.
Better
of course,
because of their dynamical
not possible
findings
with
experimental
the
computer modelling.
could be achieved by combining
nomena
real
If filled
with
cells.
consisted
of121
atoms
simulated
crystallite
121 x 25 unit
The
x
After
careful
equilibration
of
4.7
millions
at
than
this crystallite would
contain
atoms.
more
continued
during next 150 ps. The corresponding software
simulations
T
550 K the
were
and to analyse them in terms of six domains
fluctuations
allowed us to observe directly the
phase transition.
rhombohedral-monoclinic
created by the
direct
A
observations
of the
symmetry
reduction
monoclinic
microdomains
by
transmission
=
Due
to
monoclinic
domains.
six
the
domain
lead
Two
states
orientational
phosphate.
other
are
states
states
denoted
and
Rim
from
Three
within
states
each
to
C2/c
are
space
related
orientational
group,
by
120°
domain
six
domain
rotation,
differ by
11, 12 21, 22, 31 and 32, where
as Di
translational
denotes
the
the subscript
states
forming
translational
the first
in
the
orientational
shift.
The
index
distiiTguishes
Two
translational
=
states.
appear
orientational
walls
between
The domain
domains, i.e. ii and 12 are named antiphase domains.
electron
microscopy [34j.
observed by high-resolution
domains
have been directly
and antiphase
of these
domains.
Using
fluctuations
convenient
The analysis of
is
terms
to be performed in
corrected by the phase factor follo~ving from
equations iii. 13) and from the displacements
constructed
domain
index
of the order
the
parameter at L point, we have
vector
a local
wave
of the six
Dj ii, j. k). The Dj ii, j, k) array forms a volume map in which each pixel represents
one
PHYSIQUE
DE
JOURNAL
170
N°1
I
j/
(a)
T
)
=475
K
-q=1/12
-q-2/12
-q=3/12
-q=4/12
-qms/12
-q=1/2
e
j~
~
o
(L)
<
~~
w
i
#
$
l
~
~
§
£
li
fi
(b)
T=525K
e
-q=1/12
-q=2/12
-q=3/12
-q=4/12
-q=5/12
-q=1/2
*
8
<
O.25
0
O.5O
O.75
FREQUENCY
Fig.
dynamical
factor
structure
475 K, and 16) 525 K.
(2, -2q/3~/~,q/(3c))(llah) expressed in Cartesian
Calculated
12.
two
at
domains.
We
shows'selected
crystallite
the
whole
antiphase
an
(a)
temperatures:
shall
deformation
map.
the
use
maps
has
However,
incommensurate
a
in
six
terms
S(k,~)
The
curves
1.50
showing the phonons
correspond to the
coordinate
pixel values
of
1.25
THz
domain
at
the
wave
soft
vector
mode
k
=
system.
are
indices
shown
as
six
Di(I, j, k).
colours.
No
Figure
correction
15
for
Figure 15e the monoclinic
domain 3 covers
almost
which
visualizes
the
Figure 15a, shown in convention
pattern of antiphase domains of 31 and 32 types, which resembles
phase. These domains persist some time, but later they shrink
been
the
domains, displays
irregular
that
convention
fluctuations
of
1.00
(L)
same
made.
map,
In
FERROELASTIC
N°1
r
PHASE
L
TRANSITION
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
.
,
,
,
.
,
(
>
Z
UJ
~
,
Q
.
,
uj
fi0.
'
(
,
.
,
.
,
,
.
1
.
.
.
171
r
,
,
N
Pb3(P04)2
F
,
x
~
IN
,
.
,
,
,
,
.
-T=600K
,
.JOURNAL
112
DE
PHYSIQUE
N°1
I
~j
~
~P
@
~
(a)
Fig.
along
(e)
(b)
~~
(C)
ig)
(d)
~)
15.
The
maps
show
I.
the
evolution
of
monoclinic
microdomains
at
T
=
550
K.
Their
sizes
crystallite, and consist of
are
The
121 x 121 unit cells.
ternary axis is perpendicular to the maps. Maps (a, e), lb, f), (c, g) and id,
d) are drawn in terms of
h) have been taken at time t
0. 45,105
and 150 ps, respectively. Maps (a
(11) blue dark, (12) blue light, (21) green dark, (22)
domains:
monoclinic
and antiphase
orientational
(a-d), respectively,
Maps (e-f) represent the same
events
maps
as
green light, (31) yello>v, (32) red.
(1) blue, (2) green, (3) red.
domains:
Inonoclinic
domains
only. Color
convention
of
but in terms
of
the
edges
664
They
=
represent
the
middle
cross-section
of
the
FERROELASTIC
N°1
PHASE
TRANSITION
Pb3(P04)2
IN
173
and a region of another
monoclinic
domain 2 is created, Figure 15f. Its internal
shows
structure
again antiphase domains 21 and 22. Our video recording of the maps confirms that the process
of alternating
monoclinic
domains is continuing.
From the present
simulhtions
it
temporary
I
be
that
of
deduced
the
microdomains
of
size
the
order
of
is
300
and
their
relaxation
can
time
about
is
40
ps.
understanding of the origin of the monoclinic
microdomains
requires to
lowing questions: ii) why each monoclinic
microdomain
antiphase
contains
an
and iii) why such large
fluctuations
of monoclinic
domains
persist
at all.
may
The
question
first
Figures
at
we
and
2
of fact
matter
recall
5
This
two
for
reasons
rarely
and
this.
crystallite,
is of the
crystallite.
of the
for
as
a
40
be
can
very
at
energy.
the1v-ave
forms of
have
domains,
and
the
to
moreover,
Once
cost
pattern,
the
answer
looking
Indeed,
L is
vector
irregular
they do
as
a
incommensuinvolve
not
from
of
monoclinic
a
There
microdomains.
energy,
unit cell
oscillations
like
a
Second,
area.
usual
dispersion phonon
deforms
temporatly the
internal
involve
domain
appears
curves,
and
simulated
favor
which
stresses
the
the
body.
elastic
acoustic
course,
are
therefore, they
lot of
of103 K per
value
deformations
the
monoclinic
walls
macroscopic
estimated
dynamic
of the
size
oscillation,
Each
ps.
range
domain.
monoclinic
lead
model
our
little
mode
could
domain
whole, performs
long
Such
large
orientational
oscillations
of 20
order
fluctuations
the
to
the
cost
To
energy.
related
First
period of this
The
walls
slope of the soft
the
the
little
cost
Estimates
occur.
simulated
that
means
question is
second
The
that
domain
consisting of pairs of antiphase
deformation
lattice
antiphase
the
notices
one
flat.
modulations,
rate
that
the fol-
answer
domain
corresponding
one
pattern
coupling
for the
characteristic
crystal transforms to another
accompanied
The
dynamical
character
monoclinic
domain
order
reduce
the
in
stresses.
to
of the
monoclinic
microdomains
introduced
is equivalent in
to the "flip mode"
some
sense
reference [30j. In the present
simulations
have shown, however, that
the
monoclinic
in
we
microdomains
antiphase domains of smaller sizes, which resemble
are always consisting of many
phase.
These con>figurations lower the
irregular
incommensurate
energy of fluctuations,
an
diffuse scattering and exploit the flatness of the soft mode
around the
produce the mentioned
lattice point.
L reciprocal
of
antiphase
domains
crystals.
ferroelastic
Final
3.
it
elastic
due
elastic-displacive
strong
to
oscillation
continues,
and
the
of
the
Remarks
results
The
follows
The
of
the
confirm
simulation
the
applicability
RDF
method
reproduce
the
for
finding
the
phase transition,
diffuse
lattice
scattering, the
the
parameters,
parameter,
monoclinic
microdomains.
All that has been done using the
soft mode
behavior
above Tc, and
simplified model in which only 5 degrees of freedom from 39 of the real unit cell, have been
considered.
The
essential
point, however, was that these 5 degree of freedom do transform
representation
irreducible
with the
representation of the Rim space group as the active
same
rhombohedral-monoclinic
of the
phase transition, and that the elastic cage was able to mimic,
approximately, the elastic properties of lead phosphate.
at least
There is only local
anharmonicity.
The model potential
energy of lead phosphate has little
coupling between the neighbors. Hence, the
anharmonic
anharmonic
potential and even no
In principle,
the
really be well simulated.
mode
relaxation
damping and
cannot
processes
additional
couplings. In practice, however, one
with
anharmonic
model is easy to supplement
and this is not an easy
anharmonic
would
then face the problem of finding the
parameters,
main
accompanying the phase
dependence of the order
temperature
features
task.
Moreover,
of
limited
the
one
would
computer
be
time.
forced
to
Too
weak
transition.
reduce
We
the
size
anharmonicity
could
of the
involves
simulated
some
crystallite
consequences
because
in
the
JOURNAL
174
The
results.
approaching
little
soft
mode
the
phase
damped,
Enhancing
of
they
show
the
Also
larger
processes
sizes
(rw
would
one
I
function
elastic
the
large-size
drive
can
much
relaxation
monoclinic
S(k,uJ) scattering
in
transition.
therefore,
microdomains
our
peak
PHYSIQUE
DE
N°1
does
monoclinic
I)
300
damp
becomq overdamped
whole crystallite
not
oscillations
of
microdomains.
than
deformations
elastic
As
experimental
the
on
the
consequence
(rw 50
a
I).
ones
and
are
diminish
the
size
microdomains.
Acknowledgments
Special
thanks
phosphati,
due
are
for
many
for
to
Lambert
M.
fruitful
for
discussions
showing
suggestion
the
useful
and
undertake
to
The
remarks.
the
of lead
simulation
authors
grateful
are
to
B.
scattering prior the publication.
The
us
authors
would also like to thank T. Aihara, K. Esfarjani, P. Launois, K. Ohno and M. Sluiter
for
discussions
and i~. Maruyama for
help in handling the computer
One
extensive
systems.
of us (K.P.) would like to express his thanks to the staff of the Laboratory of Materials
Design
by qomputer Simulation,
for
Institute
Materials
Research, Tohoku University and of the staff
of
Laboratoire
de Physique des Solides,
Paris-Sud, for their great hospitality and
UniversitA
during his stays.
The visiting
assistance
research
professorship of the Hitachi
Corporation
is gratefully acknowledged.
This work
partially
supported
State
by
the
Committee
of
was
Hennion
al.
et
Scientific
the
(KBN), grant
Research
results
No 2
diffuse
of
P03B
145
10.
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