Relaxation Dynamis Near Ferroeletri
Phase Transitions and the Central-Peak Phenomenon
V.B. Kokshenev
Departamento deFsia,ICEx,UniversidadeFederalde MinasGerais,
CaixaPostal702,CEP30123-970,BeloHorizonte,MG,Brazil
eletroni address: valerysia.ufmg.br
Revisedformreeivedon12Otober,2000
The instability of the paraeletri phase in ferroeletris, driven by thermal utuations, is
dis-ussedonthebasisofthequantumthree-dimensionalspin-1/2transverse-eldIsingmodel(TIM)
withintheframeworkoftheGreenfuntionmethod.Thetwo-stepritialdynamisoftheTIMis
analyzedthroughthe ferroeletriorder-parameterutuationspetraobservedabove theritial
temperatureT. Thespetraprolesare givennearTinexpliitform. Theslow-downexponents
=
5
4 and
s =
1
4
are dedued for the slow and fast parts of strutural relaxation and are
omparedwiththoseknownfromtheliterature.
I Introdution
AfterElliot[1℄andStinhombe[2℄ithasbeen
repeat-edly reognized that the pseudo-spin Transverse Ising
Model(TIM) [3℄,
H = N
X
f S
xf 1
2 X
ff 0
J
ff 0
S
zf S
zf 0
; (1)
is generi to desribe the ritial dynamis observed
in ferroeletris and antiferroeletris near strutural
phase transitions that ourat aritial temperature
T
. The transverse eld provides spei dynamis
near T
: This is due to short-range thermal
utua-tions ofthe loalorder parameterthat anbetreated
in termsofompatorrelationregions(orlusters)of
the loal polarization. Critial dynamial eets are
diretly observed in the ferroeletri
order-parameter-utuation spetra both below and above T
and are
knownastheentralpeak (CP)phenomenon.
The main qualitative features of the CP are as
follows. The diusive-type entral-mode (Re!
= 0,
Im!
6= 0) appears in the dynamial spetra, in
ad-dition to the renormalized soft mode !
s
related to
the side-band peaks. Near T
theentral-mode
inten-sity I(!;T) divergesbut thesoft mode remainsnite
(Im!
s
6=0). The CP phenomenon appears to be
as-soiated with all strutural phase transitions [4℄, and
has been intensively studied by a great variety of
ex-perimental tehniques, namely, by neutron, light and
Mossbauer sattering, eletron and nulear
paramag-neti resonanes, dieletri dispersion and ultrasound
theobservedCP phenomena basedon theritial
dy-namis of the 3-dimensional (3D) TIM (1) developed
throughtheGreenfuntion method.
II CP Mirosopial
Desrip-tion
We disuss the CP phenomenon in termsof the
well-known phenomenologialdesriptionproposed by
sev-eral authors [5℄. This desription introduesthe
aux-iliary parametersÆ (oupling strength) and 1
(Debye
relaxation time) through the dynamial spetral
pro-les,
I(!;T)_ 1
! ImG
0 (!;T);
and
G
0
(!;T)_[! 2
0 !
2
i! Æ
2
i!
℄ 1
: (2)
Theauxiliary parametersÆ and of unknown nature
weresuggestedtodesribetheinterationofatrial
un-damped softmode !
0
with someunspeieddegreeof
freedom, alled arelaxation mode. Theauxiliary
pa-rametersaninturnbederivedfromtheobserved
spe-traandgivenintermsofthefrequenies!
s and!
and
theirwidths,respetively,
s and
. Thus,asT !T
,
theCPbehaviorisgivenby
! 2
s !!
2
1 =!
2
0 +Æ
2
;
!
! 2
0
! 2
1
: (3)
G
ff 0
(t)= i(t)<[S
zf (t)S zf 0 (0)℄> T (4)
with < S
yf (t) > T =< S zf (t) > T
= 0, and
<S
xf (t)>
T
=(T)= 1
2 tanh(
2T
). TheFourier
trans-formG
q
(!;T)hasbeendevelopeduptotheforthorder
and onsistently redued by appliation of the
sym-metrized Tyablikov deoupling sheme. The losed
hain-equation systemwasfound[6℄inexpliitform,
G
q
(!;T)= (T) [! 2 ! 2 q (T) q (!;T)℄
1
with ! 2
q
(T)= [ J
q
(T)℄; (5)
whereJ
q
standsfortheFourier-site3Dtransformednearest-neighborexhangeenergyandthepolarization
\oper-ator"is
q
(!;T)= ! 2 <S 2 z > T N X q 0 J 2 q 0 ! 2 ! 2 q 0 (T) 2 <S 2 x > 2 T N J q X q 0 J q 0 ! 2 ! 2 q 0 (T) : (6)
Forq!0andT !T +
,Eqs. (5)and(6)providethefollowingCPphenomenologialauxiliaryparameters,
! 0 ! ( T T 1) 1=2
;Æ! and ! ( T T 1) 1=4 : (7)
aswelltheobservableparameters,
! s ! ; s ! ( T T 1) 1=4 and ! ( T T 1) 5=4 : (8) d
The ritialregime is dened by the onditionsÆ >>
>> !
0
: Distint mirosopi approahes to the CP
problem, basedontheTIM (1)suggestadierent
de-sriptionofauxiliaryparametersÆand(fordisussion
seeRefs. 5,7).
TheanalysisoftheCPphenomenonisgiveninthe
TIMrigid-lattieapproximationbutthereis the
prob-lem of thestability of thesolutions(8) againstlattie
vibrationsharateristiof real rystals. Thestability
ofthethermal-utuationmehanismwasinvestigated
[8℄ by a omplete qualitative analyses, inluding
lat-tie distortions, spin-phonon interations and phonon
unharmoniity,arefullyaountedthroughtheETIM,
anextended versionofthe zero-orderlattie-vibration
TIM. The ETIM was introdued in the seond-order
lattie-vibration approximation,that,in away,
gener-alizesthewellknown(seee.g. Ref.3)rst-order
lattie-vibrationKobayashimodel. TheETIMisproposed [8℄
asarediblemodel forrealrystalsthatexpose
stru-turalorder-disordertransition.
III Summary
Wehavedisussedtherelaxationphenomenonobserved
near the strutural transition in three-dimensional
hydrogen-bonded ferroeletris above the transition
temperatureT
. Theanalysis is giventhroughthe
in-ofthe3Dspin-1/2TIMbytheGreen-funtionmethod.
Theobservedentral-modeintensity (2,3)isdesribed
mirosopially near T
by Eqs. (7) and (8). The
mean-eld dynamidesription,givenbythetrialsoft
modeand!
0
(T), withRe!
0
_(T T
)
1=2
andIm!
0
= 0, is explained by aounting for the short-range
order-parameter orrelations and is given in terms of
the divergent relaxation sale (T). The latter, in
turn,isgivenbythediusive-type,entral-mode!
(T)
(Re!
= 0and
= Im! 1
_ (T T
)
) and the
overdamped soft-mode !
s
(T) (Re!
s v and s = Im! 1 s
_ (T T
)
s
) order-parameter exitations.
As follows from Eq.(8) the slow-down exponents are,
respetively, = 5 4 and s = 1 4
. One of these
results an be ompared with the numerial estimate
=1:26elaboratedinRef.9.
The instability of the ferroeletri phase below T
was additionally analyzed on the basis of the
lassi-alIsing modelbytheThompsonmethod[10,11℄. The
following slow-down exponents, namely, 0 = 5 4 and 0 s = 5 16
,werededuedintheaseofthereal3Dspae.
We see that theIsing-model ritialdynamis,
assoi-atedwithT
-ritialdynamisinhydrogen-bonded
fer-roeletris[1-5℄, is driven by thermal order-parameter
utuations(intrinsiCPmehanism)anddoesnot
de-pendonthequantummodelorrelationsharateristi
SaBarreto, andPauloRobertodaSilva,fornumerous
helpful disussions,andto RonaldDikmanfora
riti-alreadingofthemanusript. Theworkwassupported
bytheBrazilianageniesCNPqandFAPEMIG.
Referenes
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