Nagel Saling and Relaxation in the Kineti
Ising Model on an n-Isotopi Chain
L. L.Gonalves,
Departamento deFisia,UniversidadeFederaldoCeara,
Campusdo Pii,C.P.6030,
60451-970, Fortaleza,Ceara, Brazil
M. Lopez de Haro, y
Centrode InvestigaionenEnerga,UNAM,
Temixo,Morelos 62580,Mexio
and J. Tague~na-Martnez z
Centrode InvestigaionenEnerga,UNAM,
Temixo,Morelos 62580,Mexio
Reeivedon1September,2000
ThekinetiIsingmodelonann-isotopihainisonsideredintheframeworkofGlauberdynamis.
ThehainisomposedofN segmentswithnsites,eahoneoupiedbyadierent isotope. Due
to the isotopi mass dierene, the n spins in eah segment have dierent relaxation times in
theabsene ofinterations,andonsequentlythedynamisof thesystemis governedby multiple
relaxation mehanisms. Thesolution is obtainedin losed formfor arbitrary n, by reduing the
problem to a setof n oupledequations, and it is shownrigorously that the ritial exponent z
is equalto 2. Expliit results are obtainednumeriallyfor any temperatureand it isalso shown
thatthedynamisuseptibilitysatisesthenewsaling(Nagelsaling)proposedforglass-forming
liquids. This is in agreement with our reent results (L. L. Gonalves, M. Lopez de Haro, J.
Tague~na-Martnez and R. B.Stinhombe, Phys. Rev. Lett. 84, 1507 (2000)), whih relatethis
newsalingfuntiontomultiplerelaxationproesses.
Universal behavior in various physial model
sys-tems has been one of the lues to unovering
funda-mental regularities of nature. About ten years ago,
a saling hypothesis (Nagel saling) was proposed to
desribeexperimental work on dieletri relaxationin
glass-formingliquids[1℄. Thishypothesiswas meantto
replaethemoreusualnormalizedDebyesaling,where
thefrequenyissaledwiththeinverseofthe (single)
relaxationtimeandtherealandimaginarypartsofthe
dynamisuseptibilityarethendividedbytheirvalues
at zeroandone,respetively. However,andin spiteof
itsphenomenologialsuessandapparentubiquity,its
physialoriginisas yetnotquitewellunderstood.
Very reently, we put forward what we believe to
betherstbonademirosopimodelinwhihNagel
salingis shownto arise[2℄. Inthis previouswork,we
omputedthe magnetization, thedynami ritial
ex-ponentz andthefrequenyandwavevetordependent
suseptibilityofthekinetiIsingmodelonan
alternat-ing isotopi hain with Glauberdynamis[3℄. Our
re-sultsindiatedthatassoonasthetworelaxationtimes
ofthismodelbeamesubstantiallydierent,the
agree-mentwiththeNagelsalingimprovedsigniantly. The
questionremainedofwhether theinlusionofmore
re-laxation mehanisms would lead to the same sort of
results. Itisthisissuethatwemainlywanttoexamine
inthispaper. Henewepresentanextensionofthe
al-ternatingisotopihainin whihratherthan onlytwo
relaxationtimes,nrelaxationtimesareinvolved.
The model onsists of a linear hain with N
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ments, eah one ontaining n sites oupied by
iso-topesharaterizedbyndierentspinrelaxationtimes.
TheHamiltonianforthel th
segmentistheusualIsing
Hamiltoniangivenby
H l = J n X j=1 l;j l;j+1 ; (1) where l;j
is a stohasti (time-dependent) spin
vari-able assuming the values1 and J the oupling
on-stant. Note that if n = 2, this model redues to the
alternatingisotopihaintreatedin Ref. [2℄. The
on-gurationofthesegmentisspeiedbythesetofvalues
f l;1 ; l;2 ;::: l;n g l;n
at time t. This
ongura-tion evolves in time due to interations with a heat
bath. Weassume for thesegmentstheusual Glauber
dynamis sothat thetransitionprobabilitiesaregiven
by w li ( l;i )= 1 2 i 1 2 ( l;i 1 l;i + l;i l;i+1 ) ; (2)
where = tanh(2J=k
B T), k
B
being the Boltzmann
onstantandT theabsolutetemperature,and
i isthe
inverseoftherelaxationtime
i
ofspiniintheabsene
ofspininterations.
ThetimedependentprobabilityP l;n ;t fora
givenspinongurationsatises themasterequation
dP l;n ;t dt = n X i=1 w li ( l;i )P l;n ;t + n X i=1 w li ( l;i )P T
li l;n ;t ; (3) where T li l;n f l;1 ; l;2 ;::: l;i 1 ; l;i ; l;i+1 ;::: l;n
g. The dynamialpropertieswe areinterestedin, namely
themagnetization,thedynamiritialexponentandthesuseptibility,requiretheknowledgeofsomemomentsof
theprobabilityP
l;n
;t
. Hene,weintroduethefollowingexpetation valuesdened as:
q
li
(t)=h
li (t)i= X f l;n g l;i P l;n ;t ; (4)
wherethesumrunsoverallpossibleongurations. Usingthemasterequation(3)andthisdenitionofq
li
( t),one
aneasily derivetheresult
dq l;1 dt = 1 q l;1 2 (q l 1;n +q l;2 ) ; dq l;j dt = j q l;j 2 (q l;j 1 +q l;j+1 )
;(2jn 1)
dq l;n dt = n q l;n 2 (q l;n 1 +q l+1;1 ) : (5)
IntroduingnowtheFouriertransformqe
Q;l
(l=1;2;:::n)dened by
e q Q;l = 1 p N N X l=1 exp(iQld)q l;j (6)
whereQ= 2m
Nd
(m=0;1;2:::N),d=na,abeingalattieparameterandthevetor
Q givenby Q = 0 B B B B e q Q;1 : : : e q Q;n 1 C C C C A ; (7)
oneanderivethefollowingresult
d
Q
=M
Q Q
where thematrixM
Q
hasarathersuggestivestruturegivenby
M Q = 0 B B B B B B B B 1 1 2
0 : : 0
1 2 e iQd 2 2 2 2 2
0 : : 0
0 3 2 3 3 2
0 : 0
: : : : : : :
: : : : : : :
: : : 0
n 1 2 n 1 n 1 2 n 2 e iQd
0 : : :
n 2 n 1 C C C C C C C C A : (9)
ThesolutiontoEq. (8),whihyieldsthemagnetization,isstraightforward,namely
Q (t)=e
M
Q t
Q
(0): (10)
The relaxationproess of the wave-vetordependent magnetization is determined by the eigenvalues of M
Q
denoted as
lQ
(l = 1;2;:::n) and whih for n 5 haveto be omputed numerially. The same applies to the
dynamiritialexponentz,whihisobtainedbyimposingthesalingrelation
Q
z
f(Q)fortheritialmode,
1Q
;withthe(Q-dependent)relaxationtime
Q =
1
1Q
;intheregionT !0andQ!0where
1Q
!0,andthe
orrelationlength is exp(2J=k
B
T). Thisyields
ln 1 1Q
=C+ 2Jz
k
B T
; (11)
whereCisanirrelevantonstant. Byplottingln 1 10 vs. 2J k B T
andonsideringthelimitT !0,wehaveheked
analytiallyfor n=2[2℄ andnumeriallyfor n=3;4and 5andvarious valuesofthe 0
s that z=2,so that this
modelbelongstothesameuniversalitylassastheuniformIsinghain.
LetusnowintroduethespatialFouriertransformb
Q (t
0
;t 0
+t)of thetime-dependentorrelationdened by
b Q (t 0 ;t 0
+t)= 1 nN N X l=1 N X l 0 =1 n X i=1 n X j=1 e iQdl e iQdl0 h l;i ( t 0 ) l 0 ;j (t 0
+t)i: (12)
Then, thet 0
!1limitofthetemporalFouriertransformofb
Q (t
0
;t 0
+t);denoted by e
C
Q
(!);isgivenby
e
C
Q
(!)= lim
t 0 !1 1 2n n X i=1 n X j=1 Z 1 1 he Q;i ( t 0
)e
Q;j (t
0
+t)iexp( i!t)dt; (13)
with e
Q;m = 1 p N N P l=1 exp(iQld) l;m
. After somelengthyand tedious but not verydiÆult algebra, using eq. (4)
andtheabovedenitionsonemayderivetheresult
e C Q (!)= n X l=1 g lQ i! lQ (14) where g lQ = n X i=1 n X j=1 n X m=1 a ml a lj n he Q;i e Q;j i eq : (15)
Here,thestatiorrelationfuntionshe
Q;i e Q;l i eq
aregivenby
he Q;i e Q;l i eq = 8 > > > < > > > : 1 u 2n 1 2u n os(nQ)+u 2n
(i=l)
u jl ij e
iQn u n 1 e iQn u n +u jl ij e
iQn u n 1 e iQn u n jl ij
(i6=l)
while u=tanh( J
kBT ), a
ml
denotes the(m;l) elementof thematrixA formed withthe eigenvetorsofM
Q and
a
lj
the (l;j) elementof the matrix A 1
. Finally, by using the utuationdissipation theorem[4℄, the response
funtion S
Q
( !)turnsoutto begivenby
S
Q
(!) = 1
k
B T
0
1
n n
X
i=1 n
X
j=1 he
Q;i e
Q;l i
eq i!
e
C
Q (!)
1
A
= 1
k
B T
n
X
l=1
lQ g
lQ
i!
lQ
: (17)
d
It should be noted that the frequeny dependent
sus-eptibilityis(!)k
B TS
0
(!)(1 u)=(1+u). Thus,
(!)= u 1
u+1 n
X
l=1
l0 g
l0
i!
l0
(18)
If we set all the 0
s equal in Eq. (17), i.e., we
take the uniform hain, then of ourse the
result-ing suseptibility has the simple Debye form. In any
ase,its generalstrutureis alinearombination ofn
Debye-liketerms. In orderto investigate whether the
Nagel saling holds for this suseptibility, it is
onve-nientto reall that in theso-alledNagel plotthe
ab-sissais (1+W)log
10 ( !=!
p )=W
2
andthe ordinateis
log
10 (
00
(!)!
p
=!)=W. Here, 00
is the imaginary
part of the suseptibility, W is the full width at half
maximumof 00
, !
p
is thefrequenyorrespondingto
thepeak in 00
,and =(0)
1
is thestati
sus-eptibility. ForthesakeofillustrationinFigs. 1and2
wepresentNagelplotsfortheasesn=2(with
1 =1
and
2
=2), andn=3(with
1
=1and
2
=2and
3
= 3), respetively, and dierent values of the
re-dued (dimensionless)temperature T
k
B
T=2J. By
omparingthetwogures,weanseethatevenforthis
ase,wheretherelaxationtimesareverylose,thereis
a marked improvement of the saling in the low
tem-peraturelimitwiththeinreaseoftherelaxation
meh-anisms. Onthe other hand,theDebyesaling, shown
asinsetinthegures,presentsanopposite behaviour.
Thisresultgivesfurthersupporttothehypothesisthat
Nagel saling is related to multiple relaxation
meha-nisms,asdisussedin ourpreviouswork[2℄.
Oneofus(L.L.Gonalves)wantstothankthe
Cen-trodeInvestigaionenEnergia(UNAM)forthe
hospi-talityduringhisvisit,andtheBrazilianageniesCNPq
andFINEPforpartialnanialsupport. Theworkhas
alsoreeivedpartialsupportbyDGAPA-UNAMunder
Figure 1. Nagelplotfor n =2 (with1 =1 and2 =2)
and for T
=5;7;10;50;100and150. There isreasonable
agreement withthe salingform for this hoieexept for
low T
. Insert: 00
(!)= 00
(!
p
) vs. !=!
p
in order to test
Debye-likebehavior.
Figure 2. Thesame as Fig. 1but with n =3 (1 = 1 ,
2=2and3=3)andT
=5;7;10;50;100and150. The
improvementintheagreementwithNagelsalingisrather
notieable,whiletheoppositetrendisobservedwithrespet
Referenes
[1℄ P. K.Dixon, L. Wuand S.R. Nagel, B. D. Williams
andJ. P.Carini, Phys.Rev.Lett. 65, 1108 (1990); L.
Wu,P. K.Dixon, S.Nagel, B. D. Williams, and J. P.
Carini,J.Non-Cryst.Solids131-133,32(1991); D.L.
Lesley-Peleky and N. O.Birge, Phys.Rev. Lett. 72,
1232(1992);M.D.Ediger,C.A.AngellandS.R.Nagel,
J.Phys.Chem.100,13200-13212(1996).
[2℄ L. L. Gonalves, M. Lopez de Haro, J. Tague~
na-Mart
inez,andR.B.Stinhombe,Phys.Rev.Lett.84,
1507(2000).
[3℄ R.J.Glauber,J.Math.Phys.4,294(1963).
[4℄ M. Suzuki and R. Kubo, J. Phys. So. Jpn. 24, 51
