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Local dynamics around structural order-disorder phase transitions
B. Toudic, H. Cailleau, J. Gallier, R. Lechner
To cite this version:
B. Toudic, H. Cailleau, J. Gallier, R. Lechner. Local dynamics around structural order-disorder phase transitions. Journal de Physique I, EDP Sciences, 1992, 2 (6), pp.829-844. �10.1051/jp1:1992182�.
�jpa-00246604�
Classification
Physics
Abstracts64.60H 61.12E 76.60E
Local dynamics around structural order-disorder phase
transitions
B. Toudic
(I),
H. Cailleau(I),
J. Gallier(I)
and R. E. Lechner(2)
(1)
Groupe
Matihre Condens£e et Mat£riaux (*), Universit£ de Rennes I, F-35042 Rennes Cedex, France(2) Halm-Meitner Institut, Glienicker Strasse100, D-1000 Berlin 39,
Germany
(Received 21 November 1991,
accepted
infinal form
28 February 1992)Rdsum4. La
dynamique
locale autour d'une bansition dephase
structurale de type orate- d£sordre est £tudi£e enpadiculier
enanalysant
l'influence des fluctuations collectivespr£kansi-
tionnelles sur la
dynamique
individuelle. Nous utilisons [es r£sultatsexp£rimentaux
de r£sonance magn£tique nucl£aire du proton et de diffusion incoh£rente de neutrons du p-terph£nyle. Lacombinaison des deux
techniques
donne unedescription plus complme
de ladynamique
localecritique
etnon-critique
dans de telssyst~mes
et confirme notrepremihre interpr£tation
des r£sultats R-M-N-, dirt£rente de celle d'autres auteurs.Abstract. Local
dynamics
around an order-disorder structuralphase
transition is studied inparticular by analyzing
the influence ofpretransitional
collective fluctuations on individual dynamics. We use experimental results from proton nuclear magnetic resonance and incoherentneutron scattering on
p-terphenyl.
The combination of bothtechniques yields
a morecomplete
description
of the local critical and non-criticaldynamics
in such systems and confirms our firstinterpretation
of the N-M-R- results, different from that of other authors.1. Introduction.
The collective
aspect
ofpretransitional
fluctuations around structuralphase
transitions has beenextensively
studied. Inparallel,
much attention wasgiven
to the way these criticalfluctuations
modify
localproperties
such as individual relaxationalfunctions, phonon density
of states,
Debye-Waller factor,
heatcapacity,
etc. The clearestexperimental
evidence of localcritical
dynamics usually
comes from resonancetechniques.
In this paper we present ananalysis
of a molecularcrystal
around its order-disorderphase
transitiondiscussing
inparallel
Nuclear
Magnetic
Resonance and Incoherent NeutronScattering
results. The combined useof both
techniques actually
appears necessary to elucidate both the local critical and non- criticaldynamics
in such systems.(*) URA au CNRS 040804.
This paper is divided in five
pans.
First we recall the formalism used to describe criticaldynamics
around an order-disorderphase
transition. In the second part webriefly
review theexperimental techniques.
In thefollowing
two pans localdynamics
isanalysed
in the absence of criticalphenomena
in the disordered and in the orderedphase, respectively.
The last partdeals with local critical
dynamics
in theneighbourhood
of thephase
transition.Experimental techniques
thatanalyse
localdynamics probe
an autocorrelation function of the local variables,(t)
g(t)
=
)~( (s,(t)s,(o))
=
(s(t)s(0)) (I)
In the
simplest
case ofdynamical
disorder in a double-wellpotential,
the local variable s,(t)
may be defined as apseudo-spin
variable(± I).
The time average of this critical variableij
can bedirectly
related to the order parameter ~~ =
~
s,e'~~'" (2)
N
~'~~~
where q~ is the critical wavevector
defining
the staticperiodicity
of the orderedphase.
Pretransitional effects around such
phase
transitions are associated with the formation of short range ordered clusters of finite lifetime. In bothphases they correspond
to collectivefluctuations of the local order parameter around its mean value. A
description
of thespatial
and time variation of these fluctuations is
possible by introducing
the Fourier componentss~(t)
of the local variables,(t)
=
~ s~(t) e'~'~ (3)
q
The relaxational collective fluctuations are
govemed by
:(s~ (t)
s_~
(0))
= s~
~)
e °~~j4)
where
([s~[~)
is the mean squareamplitude
and T~ the relaxation time of the criticalfluctuation of wavevector q. On
approaching
thetransition,
the mean squareamplitude
becomes
larger
andlarger (divergence
of the staticsusceptibility,
I-e- of the correlationlengths f
which characterize thespatial
range of the critical fluctuations atq~)
and the latter livelonger
andlonger (critical slowing
down of thefluctuations,
I-e-divergence
of the correlation time T~ which characterizes the lifetime of the critical fluctuations atq~).
Coherentneutron
scattering
appears to be the best suitedtechnique
for thestudy
of the collective aspectof the
pretransitional phenomena
around structuralphase
transitionsmeasuring
thepair
correlation function (s~
(t)
s~
(0)) [1-3].
Let us note that the number of directobservations, by
this
technique,
of criticalslowing-down
of the fluctuations around order-disorderphase
transitions remains small[4-6].
The
general expression
of the autocorrelation function may begiven by
a summation over the Brillouin zone of the collective fluctuations of wavevector q :g(t)
=
) z (s, (ti
s,
(o))
=
) z (sq(t)
s-
~(o)j (5)
~~~~
q
The
spectral density
of the autocorrelation functionG(w),
Fourier transform in time ofg(t)
is normalised tounity
:ldw
G(w
=g(0)
=
(6)
Up
to now, almost all the studies of local criticaldynamics
have been carried outby using
resonance methods such as
N-M-R-, N-Q-R-,
E-P-R-[7-10].
Thesetechniques
are sensitive toslow fluctuations and thus have been very fruitful in
observing
andanalysing
thesepretransitional
local criticaldynamics. However,
theproblem
remainsquite complex
becausein such cases the
spectral density
of the fluctuations has a verycomplicated
energyshape.
Resonance methods
only
allow theanalysis
of a very limited number ofpoints (the
resonancefrequencies)
of thisfunction,
whereas its observation as a whole would be necessary.By
itsprinciple
incoherent neutronscattering
allows the observation of thecomplete spectral
autocorrelation function
[11-15].
Several conditions must be fulfilled however to obtain this result and up to now thistechnique
was seldomapplied
foranalysing
local criticaldynamics.
The most elaborate work
conceming
an order-disorder transition has been done onammonium
chloride,
with adescription
of thesingle particle dynamics
in bothphases [16-18].
Unfortunately,
as we will seelater,
thesuperposition
of differenttypes
of motions makes theproblem complicated,
and mayperhaps
have hidden the effect of thepretransitional
fluctuations. We have therefore retained a
compound
where no such aproblem
exists : p-terphenyl.
Thiscompound
isparticularly
suited for thestudy
of local criticaldynamics
for several other reasons :the
geometry
of the molecular motion issimple,
it exhibits an almost continuous structural
phase
transition withlarge pretransitional phenomena, previously
studiedby
coherent neutronscattering,
total or
selectively
deuteratedcompounds
areeasily
obtained aslarge single crystals
whichpermits
all types ofexperiments
to be made.p-terphenyl
is made of threephenyl rings
connectedby simple
C-C bonds. In the gaseous and theliquid
state, therepulsion
of orthohydrogen
atoms induces anon-planar
conformation of the molecule. In the
crystal X-ray
and neutron diffractionexperiments
have revealed adoubly peaked probability density
function for thephenyl rings,
associated to adouble-well
potential
for the conformations ofp-terphenyl [19].
These nonplanar
confor- mations may be describedby
the torsionalangles
of thephenyl rings
relative to the meanplanar
conformation. Theseangles
are about ± 13° for the centralring
ofp-terphenyl
and ± 5°for the extemal
rings.
In the lowtemperature phase,
the molecules stabilize in one of the twonon-planar
conformations and the critical wavevector is(1/2, 1/2, 0) [19-20].
Ourprevious
results
conceming
localdynamics
in thiscompound
have been described in references[21-24].
2.
Experimental techniques.
The two
techniques
which will be verybriefly
reviewed here are Incoherent NeutronScattering (I.N.S.)
and NuclearMagnetic
Resonance(N.M.R.).
Both areextensively presented
in books or review articles[7-12, 25-26].
2.I INCOHERENT NEUTRON SCATTERING. It is well known that incoherent
scattering
ofneutrons from
hydrogen
nuclei results fromspin
incoherence between the neutron and thescatterer. The
experimental separation
of coherent from incoherent neutronscattering
inmolecular
compounds
with ahigh
concentration ofhydrogen
atoms is not trivial butsomewhat facilitated due to the very
large
incoherentscattering
cross-section of the protons.The
description
of the « individualdynamics
» of the atom isperformed by introducing
the Van Hove autocorrelationfunction, g~(r', t),
Fourier transform in space of the intermediatescattering function1,~~(Q, t).
The total incoherentintensity
is the sum of the intensities of the individual incoherent scatterers. The intermediate incoherent neutronscattering
function reads1(Q, t)
=
'
L (e'~'~~~'e~'~'~'~°~) (?)
,
ii
Let us now consider the
simple
case of a relaxational motion of the atoms in a double-wellpotential,
whose minima for each atom are definedby
the vectors a and + a. The localpseudo-spin
variable may here be defined as :s, = + I if r, = +a
and
(8)
s~ = I if r, = a.
The intermediate incoherent
scattering
function for this relaxational process is then :Ij~(Q, t)
=
I ii
+ cos(2
aQ )i
+I
ii
cos(2
aQ )i I z (s, (t)
s,(0)j (9)
2 2 N
,
The first
time-independent,
term, is the well-known Elastic Incoherent Structure Factor(E.I.S.F.)
Ao(Q).
All the informationconceming
thedynamics
is included in thepseudo-spin
variable autocorrelation function
g(t)
definedby equation (I).
Thecomplete expression
of the incoherentscattering
function is~'~~~~~
~ ~~~
~
~~
~~~
(~~C(Q,
W +sjjj'(Q,
~
~j
~~~~The first term is the detailed balance factor, the second term is the
Debye-Waller
factor.S,$~(Q, w)
is the time Fourier transform of the reorientational intermediatescattering
function1$~(Q, t).
The last termcorresponds
to inelastic processes and isdirectly
related to thescatterer-weighted phonon density
of states. In the case of an order-disorderphase
transition the local
dynamics mainly
affects the reorientationalscattering
functionS$~(Q,
w
).
Let us mention here that in the
displacive
case, not further treated in this paper,Ao(Q)
andconsequently 1$~(Q, t)
isQ-independent
andalways equal
I. Local criticaldynamics,
which in thedisplacive
case results from asoftening
of aphonon
branch, willmodify S]]['(Q, w)
andconsequently
the associatedDebye-Waller
Factor.2.2 NUCLEAR MAGNETIC RESONANCE. In this paper we will
only
consider nuclearmagnetic
resonance of nucleihaving spin
I=
1/2 because this
technique
is the closest to incoherent neutronscattering.
In molecularsolids, dipolar
interactionsyield
the mainbroadening
of the resonance lines. Van Vleck[27]
has shown that in therigid
lattice the second moment of this line isdirectly
related to thegeometrical
arrangement of thespins
I,j
in the
crystal
M)
=
~
y~h~l(I
+I)~z~ (
~~°~~
~')
~
(11)
2N
jr
'= J» <J
b,~ is the
angle
between the vector r,~ and the direction of theapplied magnetic
field.Through
the timedependence
of theinteractions,
the motions narrow the resonance line when their characteristicfrequencies
are of the same order ofmagnitude
as thefrequency strengths
of thedipolar
interaction. The reduction of the second moment of the line isAM~
=M) f
with :3 N i N
~~
2 N"~~~~
~~ ~ ~~
i l~ l~~ ~°~~
~iJ') (rip (~~~~~~l~ (12)
The information
conceming
thespectral
function of the thermal fluctuations of thecrystal
is obtainedby measuring
thespin-lattice
relaxation rateT/
This value isproportional
to the transitionprobability W~~
between the statesk,
m of the system of thespins, resulting
from the interaction with the lattice :W~~
=~ ldt e~'~~~
~~~G~~ (t) (13)
h
G~~(t)
is the autocorrelation function of theperturbing
timedependent part
of thedipolar
interaction Hamiltonian
3C(t)
:G~~(t)
=
(3C~~(t)3C~~(0))
=
[3C~~[~ j(t) (14)
J(w ),
time Fourier transform ofj (t),
is thespectral density
for the involved autocorrelation function of the variablescoupling
the nuclearspins
and the lattice.In conclusion of this
experimental presentation,
one sees theanalogy
between thedynamical spectral
autocorrelation function G(w )
defined for I-N-S- andJ(w )
defined forN-M-R--
However,
one has tokeep
in mind that G(w probes
the localdynamics
of agiven
nucleus whereas
J(w ) probes
the localdynamics
of vectorsconnecting
agiven
nucleus with theinteracting
nucleiinteracting
with it.3. Local
dynamics
in the disorderedphase
far aboveTc.
In the
high temperature phase,
the mean value(~)
of the order parameter is zero.Furthermore,
the fluctuations of the order parameter around this mean value are also assumed to benegligible.
This is the case around a first orderphase
transition or around a second orderphase
transition far above the transitiontemperature Tc.
Then the motion ofone molecule in its double-well
potential
is not correlated with the motion of itsneighbours.
The
description
of such nondispersive dynamics
may be done in real space. In the case illustrated infigure
I of a relaxational process between twoequivalent
sites with a meanresidence time TR on each
site,
the calculation of theeigenvalues
of theequation
of motiongives
the well knownexpression
for thedynamical part
of the autocorrelation functiong(t)
:_u
g(t)
= e ~~
(15)
The incoherent
scattering
function is thencomposed
of an elasticpeak
and aquasielastic
lineof Lorentzian
shape
with a half width at half maximum2/T~,
Slc(Q,
W
)
"
A0(Q)
8 (W)
+ii Ao(Q)i
G(W ) (16)
with
G(w )
= £i
(w
=
1 ~/(R
~,
(17)
r~ «
(2/T~)
+ wT» ~~
a) b) c)
+ g
ill
-
e-t/r~
~
i
R
o
0 + t
p-
Zp+
Fig.
I. Random relaxational motion in a double-wellpotential
with mean residence timer~ in a well. Schematic
representation
a) of the double well potential, b) of the time evolution of the conformation of the molecule and c) of the autocorrelation function.The I-N-S-
experimental study
ofp-terphenyl
has beenperformed
at the Institut LaueLangevin combining
the time offlight
spectrometer INS and thehigh
resolutionbackscattering
spectrometer IN13. In order toamplify
the decrease inQ
of theEISF,
aselectively
deuteratedp-terphenyl C~DS-C~H4-C~DS
wassynthetised.
Diffraction and coherent neutronscattering
studies have shown that this
compound
presents a similarphase
transition asCjgHj4
orCjgDj4.
Theonly
difference observed(isotopic effect)
concems the transitiontemperature (Tc
= 193.3 K for
CjgHj4, Tc
= 185.3 K for
C~DS-C~H~-C~DS
andTc
= 178 K for
C181~14).
Figure
2 presents the I-N-S- results measured on thespectrometer
INS above 245 K, thatis,
far above the transitiontemperature
of theselectively
deuteratedp-terphenyl.
At thesetemperatures, the random reorientational model
gives
a verygood description
of these data.From this
analysis,
the reorientation rateI/T~
was found to follow an Arrhenius law. The activation energy, associated to theheight
of the barrier of the double-wellpotential
isE~
= I.I kcal/mole
[24].
At room temperature the reorientation rateI/T~ equals
84 GHz in thiscompound.
The same kind of information may be extracted from a proton Nuclear
Magnetic
Resonance
Study.
This wasperformed using
a Bruker SXPspectrometer operating
at theLarmor resonance
frequency
v~=
90 MHz. The
spin-lattice
relaxation times have been measuredby
the saturation 90°-pulse
method. The second momentM~
of the resonance linewas derived from the free induction
decay
be a least squares fit. All measurements wereperformed
onpowdered samples.
Far aboveTc,
the random reorientational motion of themolecule induces the same kind of motion for the
proton-proton
vectors which carry thedipolar
interaction :j~~~ ~- or ~
In the case of
p-terphenyl
T~ '= Tj ' for a
pair
of protons of the same molecule whereas T~ '=
2. Tj for
pairs
of protons of different molecules.Considering
the well known B-P-P- model, thespin-lattice
relaxation rate issimply expressed
as a function of thespectral density
J(w)
at the Larmor resonancefrequency
w~.Tj
' =y2 AM~, jJ(w~)
+ 4J(2 w~) j. (19)
2(50K
lN5
2707K
U~
8
295.5K3172K
-050 0 050 00 150 200
hw lmevl
Fig.
2. Fit of the incoherentscattering
spectra(points)
with the model of random relaxationaljump
motion in a double-wellpotential.
These results were obtained on INS onpowdered p-terphenyl C6D5- C~H4-C6D5
at ascattering angle
4=
96°
(Qo
" 1.5A~
~).
In the fast motion limit defined
by
I/T » w~, thespin-lattice
relaxation rateTj
isdirectly proportional
to T and thus follows the same Arrhenius law :Tj
'=
~~
y~ AM~
T
(20)
Such a behaviour is observed in
p-terphenyl
in thehigh
temperaturephase
above 240K,
as illustratedby
thestraight
line infigure
3. The activation energy isE~
= 1.05kcal/mole,
in verygood
agreement with I-N-S- results. The difference between the second moment of theresonance line in the
completely
orderedphase
and in the disorderedphase
isAM~
=
2.2
G~
(Fig. 6). Using
thisvalue,
the characteristic rate IIT is found to beequal
to 160GHz,
at roomtemperature.
This value is in rathergood
agreement with that foundby
incoherent neutronscattering, keeping
in mind the differencespointed
out above. It should be mentioned here that thisinterpretation [21]
is different from theinterpretation given by
other authors[28,
9-l0].
As we will see later, these authors assume that criticaldynamics
dominates thespin-
lattice relaxation process in the whole
high temperature phase. By analogy
withmagnetic
systems, where indeed individual Arrhenius processes do not exist, these authors did notconsider the molecular
jumps.
350 250 200 ~
~)
150 125
Ti<si
~18 ~14
' .
w
Tc
~~i3
Fig.
asured
Tj
4.
The
figure
mean
to
the
n esidence~~
=(21)
P- T-
T<< Tc
a) b) c)
g
Iii
_
~ ~-,i~
t
n2r
o.
o t
~
=p~-p_
#0
Fig.
4. The low temperaturephase, neglecting
the fluctuations of the order parameter : schematicrepresentation
a) of the double-wellpotential
whose asymmetry isgovemed by
the order parameter ~,b) of the individual
dynamics
of the molecule in thispotential
and c) of theresulting
autocorrelation function g(t).The
asymmetry
of the localpotential,
characterisedby
the energy difference AV between the twominima,
is then asimple
function of the order parameter[29]
:AV(~)
=
kT, ln ~~ ~
'~~. (22)
(1
~1)Neglecting
the fluctuations of the order parameter ~ around its mean non-zerovalue,
one canagain
calculate thedynamical
autocorrelation functiong(t)
in real space :g(t)
= 7~
~ +
(i
7~2~ e ~'~'(23)
with
T'~ '
=
Tj
+ TI(24)
This function is still
exponential
but it nolonger
tends to zero at infinite time(Fig. 4c).
In the low temperaturephase,
the reorientational incoherentscattering
function is then :s(c(Q, ~°)
"
IA 0(Q)
+Ii A0(Q)1'
'~~) '8(~°)
++
ID -Ao(Q)I li ~2j £T'-i(w) (25)
This result was first
developed
to describe the II to HIphase
transition ofNH4CI [16, 17].
There the
problem
isexperimentally
morecomplex
because there are two types ofstatistically independent
motions for theNH4
tetrahedron : the fourfold and the threefoldjumps.
At thistransition,
theordering
process concems the first motiononly
while the second one,essentially
notaffected,
continues toyield
aquasi-elastic scattering
inphase
III. The mainadvantage
inp-terphenyl
is that theordering phenomenon
isdirectly
associated with aunique
reorientational motion.
Actually
the use of asingle crystal
of theselectively
deuteratedcompound C~DS-C~H4-C~DS
even reduces theproblem
to thesimplest
case treated in part 2. I of oneproton
in a double-wellpotential
; in this case thetemperature dependence
of thepurely
elastic incoherentscattering
isdirectly given by
formulae 10 and25,
as shown at differentQ
values infigure
5. Thequasi-elastic
line measured in the lowtemperature phase
is found to be Lorentzian with anintegral
which decreasesquickly
uponcooling
down. However8.0
~
~ 'o
'
~
'', '
'~
~ ~,
a
~
a .
~ ~
~
i,
~ .O
',
~ A'
"
~~
O loo
Fig. 5.
Semi-logarithmic
plot of the elastic incoherentscattering
intensity measured on the spectrometer IN13 at the ILLusing
a single crystal ofselectively
deuteratedp-terphenyl
C~D~-C~H4- C6D~. ((.) Q~= I. 19
A ', (D)
Qo = 1.79A
', (A) Q~= 3. 76
A
withQofla
*.) Thestraight
lines in the high temperature phase take the temperature
dependence
of theDebye-Waller
factor into account. BelowTc,
the lines are meant as aguide
to the eye.this
component
remainsquite
broad as its width isessentially
dominatedby
the inverse of themean residence time Tj ' in the non-favorable well
[29-30].
The relaxational process may evenbecome faster in the low
temperature
orderedphase
than in the disorderedphase
as found inp-terphenyl using
formulae 22 and 25. Tendegrees
belowTc,
the reorientational rate I/T'in thiscompound
is found to be of the order of 30 GHz.A similar calculation in nuclear
magnetic
resonanceyields
the same kind of result for the second moment of the resonance line :M~(t)
=
T
+7~
it) [Ml ii (26)
The temperature
dependence
of the orderparameter
is determinedthrough
neutron diffractionexperiments
from the intensities ofsuperlattice
reflections in the low temperaturephase [20],
and it may be describedby
a power law~~ ((To T~/To)~fl
withp
0.14.Note that this small value of
p probably
indicates aweakly
first orderphase
transition within amean field
theory [31].
Indeed themetastability
limitTC
of the disorderedphase
is foundsome 0.I K below the transition temperature
To.
Theresulting
values ofM~(T~
areplotted
in full line infigure 6, showing
agood
agreement with theexperimental
data.Quite clearly
the increase of the second moment in the lowtemperature phase
occurs because of theordering
process of this
phase
andalthough
we are still in the fast motion limit(I/T'
»v~).
Thegeneral expression
of thespin-lattice
relaxation rate is then :Tj1
=
y2AM~(1
~2~
iJ(w~)
+ 4J(2 w~)1 (27)
By analogy
with theanalysis
of the incoherent neutronscattering
results in the lowtemperature phase,
the N-M-R- process may beinterpreted
in terms of aspectral
function whoseintegral
behaves as(1
~~.
However the behaviour of thespin-lattice
relaxation timeTi
isgovemed by
theorder-parameter
below the transition temperatureonly
when the contribution of the collective fluctuations in thisphase
becomesnegligible (Fig. 3).
Otherauthors
[28]
have taken the measurements betweenTc
andTc
5K,
andonly them,
todetermine the temperature
dependence
of the order parameterusing
thefollowing
formula :Tj
=
Tj/. (I
~
(28)
This
expression
is wrong since thespin-lattice
relaxation rate determined atTc, Tj/,
isgovemed by
the criticalpretransitional
fluctuations of the order parameter. Formula(28) corresponds
to formula(27),
ifTj (
is the value of thespin-lattice
relaxation rateextrapolated
at
Tc
withoutconsidering
the contribution of the critical fluctuation(Fig. 3).
Thisdescription
is based on the
assumption
that the localpotential
isactually
not much deformedby
thecollective fluctuations.
M~
jG2) ~18 ~14
12,O
ii,o
io,o
#
~~
~
Tc zo
~°°
5. Lucal critical
dynamics
(7'-7(.).
In the prcsencc
ol'pretmnsiiional phcnomcna,
around coniinuou; or almt>st coniinuou~pha,e
transitions the influence of'[he conl'orrnaiion of'a moleculc on ii;ncighbour,
becomc,important.
This local order may be describedby
alarger occupation probability
for one of thewells
during
the finite lifetime of the fluctuations in «clusters» characterizedby
thecorrelation
lengths (Fig. 7).
The autocorrelation function mustintegrate
all the,e collective fluctuations over the whole Brillouin zonesz
g(t) ~ ([s~[~) .e~"~~ (29)
q
T-Tc
~~
b)
q(r)
t
-r~ + +
= r
__
=÷-~ i
idusier
~,
d)
g t
I -t /~~
i
Sq ~ ~~~~l
q
,
iclusicr ~
i
Fig. 7. Schematic representation of the influence of the collective critical fluctuation, on the local properties a) the time evolution of the double-well potential for a given molecule, b) the ;patial fluctuation of the order parameter around its mean value, cl the dynamics of a molecule and d) the
resulting
autocorrelation function.The mean square
amplitude
of a collective relaxational mode of wavevecior q1;
~~~~~~~
q~~f~~
~~~~In [he ab;cncc ol'collective fluctuations
(T»Tc
andT«Tc),
thcrc I; inegligihlc
di;per,ion
in ihc correlation time; r~. On [he contrary, the collective t'luciuation, into(luce.i
,lr<)ng
di;per;it)n
in the;e correlation time, clo;e lo T~.(Fig.
8). Iii J nieJn l'iel(Ide,ciijiiioii
cl<),e ii) the critical wavcveclor q~. (), one ha;
T~~ T( /) II )
with
rj t ~ l'I'
I'~.i
(,1?)1-1 a)
T »Tcb)T~TC ~~ ~~~~
,'
' '
/ , , , / / / f J
q~ ~-i q
Fig.
8. Schematicrepresentation
of thedispersion
relation of the correlation times inreciprocal
space, a)
neglecting
the collective fluctuations (T MTc)
and b) in the presence of the collective fluctuations (T~ T
c).
From coherent neutron
scattering experiments performed
on thehigh
resolution spec-trometers IN10 and INll at the I-L-L- we found for deuterated
p-terphenyl
:Ti
m[0.25 (T
Tc)
GHz(33)
The
anisotropy
in the correlationlengths
weref~: f~. f~m3
: 8 :1[32]. Considering
anisotropic description,
the mean diffusion coefficient D=
f ~/Tc
was found to be of the order of 2xl~fA~
~LeV. The calculation of the critical part of thedynamical
autocorrelationfunction may been done
analytically
in theisotropic
case :g~~~(t) kT
~~
dq q~
~- ur~
0 q~ +
f
2(~
~~q~ defines the critical volume in the
reciprocal
space where thedispersion given by
formula(31)
is assumed valid. Let us note that this is the same calculation as thatperformed
todetermine the fluctuation correction for the
specific
heat[33].
For a second orderphase transition,
at T=
Tc(f~
= 0 andT£
=0),
g~~~(t) has thissimple expression
:g)~[ T~(t)
kT cl~ dq
e ~~~~ ~~(35)
For very
long
times(t»T~),
one canonly
consider fluctuation modes q«q~, I-e-q~ - tx~, and one has
gfi Tc(t) kTc ) (36)
corresponding
to thespectral
functionGi~f Tc(~°)
kTc
) (3?)
JOURNAL DE PHYSIQUEI T 2, N'6, JUNE 1992 33
At the transition temperature, the low energy part of the autocorrelation function
(w
«Tj~)
behaves as w~ ~'~. Even if itprobes
a local property, thepoint
at zero energy transferdiverges
for a second orderphase
transition. Theexperimental
evidence of the latter result has often beenreported by
N-M-R-through
thedivergence,
in the fast motionlimit,
of thespin-lattice
relaxation rateT/~
atTc [7, 9-10].
Thetemperature dependence
of thisdivergence
is muchdependent
of theanisotropy
of the correlations. If d characterizes thisspatial dimension,
one findsw
J(0) T/ (T
Tc)
~(38)
Thus, in
principle,
N-M-R- measurementsgive
thedimensionality
of the correlations.However the determination of the exponent
goveming T/
is not obvious at all because hereone has to consider both critical and non critical
dynamics. If,
like Guillon et al.[28],
one considers that the criticalphenomena
govem the N-M-R- process at all temperatures, the fluctuations are found to have a 2-dimensional character. Butif,
as shown in part3,
the criticalphenomena actually
dominate thespin-lattice
relaxation processonly
up to 30 K or~
~
<
~
~
= 0
~
m©
~ z
~ w
= 12 uev
= 24
uev
O
T-T< (K)
Fig.
9. Theoretical temperaturedependence
of thespectral density
of the autocorrelation function atpoints
hw=
0, hw
= 12 iueV (v
m 6 GHz) and hw
= 24 iueV (v
m 12
GHz)
inp-terphenyl,
in the presence of collective fluctuations. Thephase
kansition is assumed to be continuous and is described in the frame of a mean fieldtheory.
40K above the transition
temperature,
the fluctuations are found to haveessentially
a 3- dimensional character[17, 19].
Theanalytical
calculation of local criticaldynamics considering
the actual
anisotropy
of the correlation isquite complicated.
The numericalsimulation, however,
shows that thespectral shape
of this critical part isessentially
not different from thatobtained in the
isotropic
case, its relativeweight being simply
increased[34].
The theoretical behaviour of the differentpoints
of the criticalpart
of the autocorrelation function is shown in theisotropic
case infigure
9. Theexperimental
evidence of this result for these threefrequencies
isgiven
in reference[24].
Thecomplete
observation of thespectral
autocorre- lation function has been obtainedby
incoherent neutronscattering [23]. Contrary
toN-M-R-,
theanalysis
of these datarequire
a realisticdescription
of thedispersion
of the collectivecorrelation times over the whole
reciprocal
space[23, 30].
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