Rotational Dynamis of the
Magneti Moments in Ferrouids
Claudio Sherer and Trieste F. Rii
InstituteofPhysis,FederalUniversityofRioGrandedoSul
91501-970PortoAlegre,RS-Brazil
Reeivedon19January,2001
Wepresent aset ofsixnon-linearstohasti dierentialequations forthe sixvariableswhihare
relevantforthedynamialbehaviorofthemagnetimomentsinferrouids,namely,thethreeEuler
anglesofthemagnetipartile,thetwopolaranglesofthemagnetimomentrelativetothepartile
andthemodulusofthemagnetimoment. Theinterationbetweenthemagnetipartileandthe
solventuidismodeledbydissipativeandrandomnoisetorques,andsoistheinterationbetween
thepartileanditsmagnetimoment,treatedasanindependentphysialentity. Intheappropriate
limits,themodelsystemreduestotheasesofsuper-paramagnetiorofnon-super-paramagneti
(blokedmagnetimoments)partiles. Numerialresultsshowthatfornon-zeromomentofinertia
thepreessionofthemagnetimomentaroundthemagnetieldisaompaniedby\nutation". It
isalsoindiatedhowthedynamiomplexsuseptibility maybealulatedfromthe equationsof
motionandthenumerialresultsshowthatthenutationleadstoaseondresonanepeak.
I Introdution
The rotational dynamis of the magneti moments in
ferrouidsis the essential phenomenon to explain the
frequeny dependent magneti suseptibility of these
materials. Two distint rotational relaxation
meha-nisms may oexist in ferrouids: the Neel relaxation,
bywhihthe magnetimomentmoveswithrespetto
the mehanial partile, and the Brownian, or Debye
relaxation,orresponding to thepartile's rotation
in-side the uid. In mostexperimental situations oneof
these mehanisms is dominant, and this may be the
reason why there is not, up to now, that we know, a
satisfatory theory, suÆiently general to be applied
forallsituations,fromthepureNeeltothepure
Brow-nianrelaxation,passingbyallpossibleombinationsof
those mehanisms. In this respet the model of \two
spheres", by Fannin and Coey[14℄ should be
men-tionedasarsteort. Thenon-inertiallimit,i.e.,when
theontributionof themoments ofinertia of the
par-tile to the equations of motion is negligible, in
om-parisonto the other fores involved, hasbeen treated
byShliomisandStepanov[3℄,wheretheyintroduethe
egg model. There they ompare themagneti partile
with an egg, the yolk orresponding to the magneti
moment, and show that in the non-inertial limit and
for weak applied eld the equations of motion
deou-ple, soallowingone to simultaneouslyaount forthe
tile.
Thepurposeofthepresentpaperistopresenta
gen-eralsetofequationsofmotionfortheombinedsystem
ofmagneti momentplusmehanialpartile,inside a
uid. The main limitationof ourapproah isthat we
deal only with axially symmetri partiles, with easy
axis of magnetization parallel to the symmetry axis.
However,themagnetimomentisallowedtorotate
in-sidethepartile,aswellastohaveanosillating
modu-lus,andthepartileisallowedtorotatewithrespetto
thesolvent,whihisimmobilewithrespettothe
labo-ratory. Thesuspensionis onsideredsuÆientlydilute
forthe partile-partileinteration tobenegligible,so
that wedealonlywithsinglepartiledynamis.
In ontradistintion to most existing theories, our
approahinludes in the equationsof motionthe
par-tile's momentof inertia. Tonegletinertia may bea
goodapproximationformanyferrouids,beauseofthe
smallnessofthepartiles,butwearepresentinga
the-ory whih intends to be suÆientlygeneral to inlude
non-stable suspensions,forwhih thepartilesmaybe
onsiderablybigger. Intheaseofsuper-paramagneti
partiles,anaspetwhihdistinguishesourtheoryfrom
theusualapproahesisthattherotationofthe
poten-tial gradient on the magneti moment, aompanying
theBrownianrotationofthepartile,istakeninto
a-ount.
rota-auid(Langevin-typeequations),basedonthe
gener-alized Euler-Lagrangeequations. InsetionIIIwe
ob-tain, from theequations of setion II, in a onvenient
limit,theequationsofmotionforthemagnetimoment
,whihredue,intheaseofonstantmodulusof,
to the Gilbert's equation. In setion IV we arrive at
the set ofsix oupledequations, for thesix degreesof
freedom, thethree Euler angles of the partile's
rota-tions, the two polar angles of and its modulus. In
setionV weindiate,briey,howto alulate,by
nu-merialsimulationof theequationsof motion,the
dy-nami magneti suseptibility, aproedure whih was
morearefullyexplainedinapreviouspaper[22℄. Some
less general situations are onsidered in setion VIas
partiularasesandnumerialresultsaregiven.
II Rotational Dynamis of a
Par-tile in a Fluid
Consider apartileof axially symmetri shapein
sus-pension in a uid. The prinipal moments of inertia
will bedenoted byI
1 =I
2 andI
3
. Disregarding
trans-lationaldegreesoffreedom,itsLagrangianmaybe
writ-tenintermsoftheEulerangles,and (inthe
nota-tionofGoldstein[16℄),takenasgeneralizedoordinates,
as L= I 1 2 ( _ 2 + _ 2 sin 2 )+ I 3 2 ( _ + _ os) 2
V(;) (1)
whereV(;)issomeorientationdependentpotential.
Itannot dependon beauseof theaxial symmetry
ofthepartile.
Theinteration fores(torques)betweenthe
parti-leandtheuidareofthedissipativeandnoisetypes.
Therefore, they are not inluded in the Lagrangian,
but instead, we have to use the \generalized
Euler-Lagrange equations", with the orresponding torques,
representedbyQ
i
,attherighthandside:
d dt L q_ i L q i =Q i ; (2) whereq i
=;or .
We write thenon-onservativetorquesQ
i
as sums
ofdissipativeandnoiseterms,intheform
Q i = F q_ i + i
(t); (3)
whereF isthefollowingRayleighdissipationfuntion
[16℄, F= 1 2 ( _ 2 + _ 2 sin 2 )+ 1 2 0 ( _ + _ os) 2 ; (4) and i
(t) are the noise torques. The dissipation
on-stants and 0
maybe dierent beause 0
is
assoi-ated with the partile rotation around the symmetry
axis, while is assoiated with the rotations
perpen-diular to it. Substituting Eqs. (1), (3) and (4) into
Eq. (2)weobtainthefollowingsystemofequationsfor
thepartile'srotation:
I 1 ( _ 2
sinos)+I
3 _ ( _ + _
os)sin+ _ +V = ; (5a) I 1 ( sin 2 +2 _ _
sinos)+I
3 os d dt ( _ + _ os)+ I 3 ( _ + _ os) _
sin+ _ sin 2 +V = ; (5b) I 3 d dt ( _ + _
os)+ 0
( _
+ _
os)= : (5)
d
where V
=V= andV
=V=. The expression
( _
+ _
os)wasleft unbrokenwhereveritappearsin
the above equations beause it represents the
ompo-nentoftheangularveloityvetor!alongthe
symme-tryaxisandwemakeuseofthisfatin the
interpreta-tion of the dissipative torquesin termsof the
ompo-nentsof!,asfollows.
Let us dene the following four unit vetors: z,
metryaxis,a,perpendiulartotheplaneontaining
andz(z-plane)andb,perpendiulartothea-plane,
namely,
z=(0; 0; 1); (6a)
=(sinos; sinsin; os); (6b)
a= z
sin
b=a=( osos; ossin; sin): (6d)
Asanotationtobeusedthroughoutthiswork,
sub-sriptsz; ; a or bon avetorindiate itsorthogonal
projetiononthez, , aorbdiretions andsubsript
indiatesthevetor'sprojetionontheplane
perpen-diularto.
Thepartile'sangularveloityvetor!maybe
de-omposedintoasumoftwovetors,perpendiularand
parallelto,respetively,
!=!
+!
;
with
!
=_ =( _
sin _
sinosos;
_
os _
sinossin; _
sin 2
)
and
!
=
_
+ _
os:
Theorthogonalprojetionof!
onthez-axisis
!
z
=!
z= _
sin 2
;
and the orthogonal projetion of ! (or of !
) on the
diretionperpendiulartothez planeis
!
a
=!a=!
a= _
:
ThusweseethatthedissipativetorquespresentinEqs.
(5a),(5b)and(5)aregivenby!
a ,!
z
and!
,
respe-tively,timesthedissipationparametersor 0
.
Thenoisetorques willbetreatedalongthese same
lines. Westart bydening thenoisetorque vetorby
itsorthogonalomponents,
=
a a+
b b+
:
The noise beomes ompletely dened by stating
the statistisof itsthree omponents. The usual
pro-edureistoonsiderthemasstatistiallyindependent,
Gaussian white noise. This is, however, not a
nees-sary assumptionand weleaveit open forfuture
mod-eling. What we need now is to know how the three
omponentsomeinto Eqs. (5). Guidedby theabove
deomposition of the dissipativetorque, weare led to
identify
=
a ;
=
z
=
z=
b sin;
=
:
Before weproeed to dedue theequations of
mo-tion for the general ase of magneti partiles in
sus-pensionsweshow, in the nextsetion, how to obtain,
fromEqs. (5),theequationsofmotionforthespherial
III Equations of Motion for a
Magneti Moment
Themagnetimomentofamono-domainpartileis
relatedtoitsinternalangularmomentumSby=S,
whereisthegyro-magnetifator. Althoughthe
mod-ulusSofSistakenasonstantinmostworkson
super-paramagnetismandmagnetiuids,forverysmall
par-tilesitsosillationmaybesigniantandwepreferto
allow it to be time dependent. The modern
tehnol-ogy allows the preparation of samples with magneti
partileswhosediametersaresmallerthan20
A[17℄and
super-paramagneti lusters ontaining only 12
mag-netiatomshavealsobeenreported[18℄. Weanmodel
themagnetimomentbyarotatinghargedpartile,in
the limit ofzero momentsof inertia, I
1
! 0,I
3 ! 0,
and _
!1so thatI
3 _
=S. Beausein thenext
se-tionwewillworkwiththejointsystem,apartileand
itsutuatingmagnetimoment,wewritethe
general-izedoordinates,potentialenergy,dissipativeandnoise
torques,withanotationdistintfromthat
orrespond-ingtothepartile. Namely,wemakethefollowing
sub-stitutions: !#; !'; I
3 _
!S; V !W; !
;
0
! 0
and !T. We alsointrodue two
mod-iations in the equation orresponding to Eq. (5),
namely,wewriteS S
0
insteadofS inthedissipative
term andintrodue atorque W
s
, whose originwill be
explained below. In the said limit and with the new
notationthesystemofEqs. (5)beomes:
S'_sin#+ _
#+W
# =T
#
; (7a)
_
Sos# S _
#sin#+'_sin 2
#+W
' =T
'
; (7b)
_
S+ 0
(S S
0 )+W
s =T
s
: (7)
Here wehavewritten S S
0
, insteadofS, in the
dis-sipation termof Eq. (7) toaount forthe fat that
therelaxationoftheutuationsofSistowardsamost
probable(equilibrium) valueS
0
andnottowards0. It
mayappearstrangethat,eventhoughwehavederived
the equations of motion for S from the equations of
motionforasymmetripartile,in aonvenientlimit,
wehavenowto addaterm\adho" (S
0
),whih does
nothaveanequivalentinthepartile'sequations. This
is sobeause in lassialphysistheequilibrium
mag-netization is always zero. Non-zero equilibrium
mag-neti moments anonlyexist beauseof thequantum
W
s
wasintroduedbeausearystaleldmayhavean
eetive interation with , with origin in an orbital
ontribution to S[29℄, with a possible torque
ompo-nentparallel to S. There isnotanequivalenttermin
Eq. (5)beauseoftheassumedaxialsymmetryofthe
partile.
It is interesting to study the behaviorof Eqs. (7)
in the abseneof noise,T
i
=0and with W
s
=0. Eq.
(7) has then the trivial stationary solution S = S
0 .
Assuming this onstant value for S in Eqs. (7a)and
(7b) theyredue to
S
0 _
'sin#+ _
#+W
#
=0; (8a)
S
0 _
#sin#+'_sin 2
#+W
'
=0: (8b)
The onservative torques, W
#
and W
' , have,
usually,ontributionsfromtwodierentorigins,the
in-terationofSwitharystalline,anisotropyeldand/or
with amagnetield,whihan alsobeofseveral
dif-ferent origins. In the ase of magneti eld, H, the
potentialenergy is W = H. With alittle of
al-gebrai work one an show, in this ase, that the set
of Eqs. (8) is equivalent to the well known Gilbert's
equation[10℄,
d
dt
=
H
2
d
dt
; (9)
for = S and S = S
0
. This equation was used by
W. F.Brown[12℄ asastarting point forhis stohasti
theoryofsuper-paramagnetism,whereheassumedthe
magnetieldH toontainanoiseterm. Amore
gen-eraltheoryforsuper-paramagnetism,whihallowsalso
for osillations on the modulus = S of the
mag-neti moment, was worked out by Rii and Sherer
[20, 21,22℄,basedontheset ofEqs. (7). Forthis
rea-son we will not ontinue to explore the onsequenes
of Eqs. (7) in the present paper, turning, instead, to
the moregeneral approah, wherethe rotationsof the
mehanialpartilearetakenintoaount,in addition
to themotionofS relativeto thepartile.
IV Equations of Motion for a
Small Magneti Partile in
Suspension
In reent years several researhers[1, 4, 14, 23℄ have
drawn attention to the importane of the motion of
themagnetipartile,itsinertiaandvisousinteration
with the uid, to thedynami magneti suseptibility
of ferrouids. Atheoretialtreatmentofthisproblem,
whihisboth,morefundamentalandmoregeneralthan
those previously published, followsnaturally from the
Takentogether,thesystemsofEqs. (5)and(7)
on-tainallthedegreesoffreedomrelevanttotheproblem.
Tothepotentialenergyterms,V inEqs. (5)andW in
Eqs. (7), theinteration energybetweenthemagneti
momentand the partile, whih we will denote by U,
hastobeadded. Due tothe partile'ssymmetry, this
termanonlydepend onS and ontheangle between
S andthesymmetryaxis,. Itisonvenientto dene
another orthogonal set of unit vetors, related to the
diretion of the magneti moment, namely, s, in the
S diretion, u, perpendiular to the sz-plane and v,
perpendiularto thesu-plane,
s= S
S
=(sin#os'; sin#sin'; os#); (10a)
u= zs
sin#
=( sin';os';0); (10b)
v=s u=( os#os'; os#sin'; sin#): (10)
TheinterationenergyU anthenbewrittenasU(S;s
).Inpriniplethepartileaninteratalsowithother
elds, besides H, as is the ase if it has an eletri
dipole and aneletrield ispresent. Forthis reason
wekeepalsothepotentialenergyV(;)inthenewset
ofequations.
Thedissipativeinterationassoiatedwiththe
rota-tionofSrelativetothepartilewillbewritteninterms
of the relative angular veloity vetor. Sine only
ro-tations perpendiular to S an lead to a meaningful
interation torque with originon the relative motion,
wedenetherelativeangularveloity!
r as
!
r
=$ !
s ;
where
$=ss_
istheangularveloityofrotationofthemagneti
mo-mentwithrespettothelaboratoryand
!
s
=s!s=! (s!)s:
is the orthogonal projetion of the partile's angular
veloity!ontheplaneperpendiularto S. The
dissi-pativeinterationtorqueonthepartileisthen+!
r .
The plus sign is beause of the way we dened !
r ,
where the partile's angular veloity appears with a
minus sign. Guided by the interpretation of the
dis-sipative torque terms of Eqs. (5) in terms of angular
veloityomponents,asexplainedbellowthesaid
equa-tions,wewritedownimmediatelythedissipativetorque
termstobeaddedtotheleft-handsides(therefore,with
!
ra
= !
r a;
!
rz
= [!
r (!
r
)℄z= (!
rz !
r os;
!
r
= !
r :
d
Of ourse, all this salar produts, as well as those
whihfollow,inthenextequations,maybeeasily
writ-tenas funtions of thefour angles ; ; # and 'and
theirtimederivatives,byusingEqs. (6)and(10).
How-ever,beausesalarprodutsareveryeasilyhandledin
numerial proedures,we prefer to leavethem in this
form.
Clearly,thetorqueonthemagnetimoment,dueto
therelativemotion,isthe \reation"to thetorqueon
thepartile,i.e.,itisequalto !
r
,and,inplaeof _
#
and '_sin 2
# in Eqs. (7) we shall use (remembering
that!
rs =!
r )
!
ru =!
r u;
!
rz =!
r z:
Noterm omingfrom therelativeangular veloity!
r
hastobeaddedto Eq. (7)beause!
r
is
perpendiu-lartoS. However,thereistheterm 0
(S S
0
)already
presentin thatequation, withoriginin the(quantum)
utuations of S, and this term will be kept. Sine
angular momentum has to be onserved, its reation
ounterpart on the partile has to be added to Eqs.
(5). Calling
R=(S S
0 )s;
thetermstobeaddedtotheleft-handsidesofEqs. (5)
are
0
R
a
=
0
Ra= 0
(S S
0 )sa;
0
R
z
=
0
[R (R)℄z= 0
(S S
0
)[s (s)℄z;
0
R
=
0
R= 0
(S S
0 )s:
d
The noise torques of interation between the
par-tile and the magneti moment an be written down
alongthesamelinesofproedure asdoneforthenoise
torques ofthe uid on thepartile, at theend of
se-tionII.Weassumethreeorthogonal,independent,noise
torque vetors,alongthe unit vetorsdened with
re-spettothediretionofthemagnetimoment:
T =T
s s+T
u u+T
v
v: (11)
BeingT thetorqueonthemagnetimoment,thenthe
torqueon thepartileis T. Followingthesameline
ofreasoningasdonebefore,weidentifythe torquesin
Eqs. (7):
T
# =T
u ;
T
' =T
sz
=T
v sin#;
T
s =T
s :
Correspondingly,thefollowingtermshavetobeadded
totheright-hand-sidesof Eqs. (5):
T
= T
a
= T a;
T
= T
z
= [T (T )℄z;
T = T
= T :
Therefore, the state of the omposed system, the
partile andits magnetimoment, is desribedby the
6generalizedoordinates,; ; ;#; ' andS, whose
dynamialbehaviorisgovernedbythefollowingsetof
I 1 ( _ 2
sinos)+I
3 _ ( _ + _
os)sin+
_ ! ra + 0 R a +V +U = a T a ; (12a) I 1 ( sin 2 +2 _ _
sinos)+I
3 os d dt ( _ + _ os)+ I 3 ( _ + _ os) _
sin+ _ sin 2 ! rz 0 R z +V +U = b
sin T
z ; (12b) I 3 d dt ( _ + _
os)+ 0
( _
+ _
os) !
r 0 R = T ; (12)
S'_sin#+!
ru +W # +U # =+T u ; (12d) _
Sos# S _
#sin#+!
rz +W ' +U ' =+T sz ; (12e) _ S+ 0 (S S 0 )+U
S =T
s
: (12f)
d
This set of six equations is of very general
appli-abilityonmagneti suspensions. Itallowsfor alarge
varietyofmodeling: Thereare threeindependent
on-servativeinterationpotentials,V; U and W,four
dis-sipativeparameters,; 0
; ,and 0
,andalsothenoise
torques andT,whosestatistial propertiesareopen
for modeling. Partile-partileinteration was not
ex-pliitly takeninto aount.
V Dynami Suseptibility
Toalulate,fromthesetofEqs.(12)thedynami
mag-neti suseptibility, and therefore theabsorption lines
of magneti resonane,it is better to transformthem
into the usual form ofrst order dierentialLangevin
equations[19℄. Notingthattherstthreeequationsare
seond order,weintroduenewvariables,
_ = _ = _ + _
os= (13)
and transform Eqs.(12), so that, together with
Eqs.(13), we have a set of nine rst order equations.
Forexample,Eq.(12a)beomes
I
1 (_
2
sinos)+I
3
sin+
! ra 0 R a +V +U = a T a
WeusealsotheWienerproesses W
j
(t), whih are
relatedtothewhitenoiseomponents (orwhatever
appearsattherighthand sideof Eqs.(12))by
W j (t)= Z t 0 j (t 0 )dt 0 ;
andmaketheusual substitutions
j
(t)dt!dW
j (t) to
writethe setof stohastidierentialequationsin the
form
dX
i (t)=A
i
(X(t);t)dt+ X j B ij (X(t))dW j (t) (14) where X i
(t) are the dependent variables
; ; ; ; ; ; #; '; S and A
i
(X(t);t) and
B
ij
(X(t))areobtainedbyomparisonbetweenEq.(14)
and those from the set of rst order equations
men-tioned above, after the expressions for V;U and W
havebeenintrodued. Inthetypialaseof magneti
resonane, with a strong onstant magneti eld H
0
paralleltothez-axisandaperiodiweakeldF(t)
per-pendiulartoit,theA
i
(X(t);t)turnouttobewritten
intheform
A
i
(X(t);t)= X j ij (X)F j (t)+A
0
i
(X): (15)
Following theproedure of referene[22℄, theresponse
funtionsarethengivenby
ij (t)= X k h k j (x) k hX i (tjx)i
0 i
eq
: (16)
ThesymbolX
i
(tjx)meansthestohastivariableX
i
at time t, given that the \vetor" of stohasti
vari-ables X had the value x at the initial time t = 0,
hX
i (t j x)i
0
means average over many realizations of
startingfromthepointx,
k
meansderivativewith
re-spetto thek omponentof theinitial \point" xand
hi
eq
meansaverageovertheequilibriumdistribution
ofinitialpoints. Thisequilibrium averagemayinlude
average overthe distribution of partile's
harateris-tis,ifpolydispersityistobeonsidered. Forexample,
if thepartiles are all madeof the samematerial and
have the same shape, varying only in size, assuming
somegivendistribution of alinear dimension, r, then
theother partile's parametersshell besaled
aord-ingly. Forexample,
S
0 /r
3
; I /r 5
; /r 3
: (17)
Aswasshowninreferene[22℄,Eq. (16)maybe
eval-uated from numerial simulations of Eq. (14). From
the results obtained for
ij
(t) we an then alulate
thesuseptibility
ij
(!) bynumerialFourier-Laplae
transform.
VI Some Limit Cases and
Re-sults of the Simulation
Severalinterestinglimitsituationsarereadilyobtained
from Eqs.(12). The \super-paramagneti" limit, for
whihthepartile'soordinates,; ,and ,aretaken
as onstants, so that the system redues to the last
three equations, or, equivalently, to Eqs. (7), has
been treated in three previous papers by Rii and
Sherer[20, 21, 22℄. A further simpliation, in this
limit, whih is appropriate for most ases of
prati-al interest, follows by assuming S = S
0
=onstant.
In thisase theonlyrelevant equationsare Eq. (12d)
andEq. (12e), and,moreover,thetermin _
S also
van-ishes. ThenoisetorquesT
u andT
sz
maythenbe
writ-ten in terms of a stohasti magneti eld, rendering
our set of equations in a form equivalent to Brown's
generalization[12℄of Gilbert's equation, Eq. (9). This
ase has been treated by several authors, of whih a
veryinteresting aount is given in areent paper by
Garia-PalaiosandLazaro[15℄,wheremuhnumerial
workispresented.
The \bloked" limit (also alled \Brownian"
limit[24℄or\inertiallimit"[2℄),orrespondstothease
when the magneti moment is bloked along the
par-tile's symmetry diretion, i.e., # = and ' = .
Thismayhappenbeausethesampleiskeptbelowthe
\bloking temperature" T
B
[28℄ or beause the
mate-rialissohighlyanisotropithatthemagnetimoments
onlyexistsparalleltotheeasyaxis[18℄. Thepartileis
still immersed in a uid arrier, being able to rotate,
togetherwithitsmagnetimoment.
Intermsof thesetofEqs.(12),theblokedlimitis
obtainedbyassuminganinterationpotentialU ofthe
form U
0
Æ(s ),withU
0
!1,sothattheonlystates
energetiallypossibleare thosewith s=, i.e.,#=
and ' = . By summing Eq. (12a) with Eq. (12d)
andEq. (12b)withEq. (12e)theinterationtermsU
and U
#
as well as U
and U
'
anel out. The terms
ontaining !
ra , !
rz , R
a , R
z
, T
a
, and T
z
beome
identiallyzero,andR
beomes(S S
0
). Choosing
and to denotethe ommonpolar angles, thesystem
of equations, in the notation of the previous setion,
beomes:
I
1 (_
2
sinos)+I
3
sin++V
+Ssin+W
=
a
; (18a)
I
1 (_sin
2
+2sinos)+I
3
os_ I
3
sin+sin 2
+V
+
_
Sos Ssin+W
=
b
sin; (18b)
I
3 _ +
0
0
(S S
0 )=
T
; (18)
_
S+ 0
(S S
0 )=+T
: (18d)
d
This is still a rather general set of equations. A
rstobvioussimpliationours,in mostasesof
in-terest,whenS =S
0
. Then also _
S=0andT
=0and
the systemis redued to three equations. Muh work
has been done in this ase, mainly in the ontext of
eletridipolar moleules, forwhih S =S
0
=0. For
example, MConnell[25℄,Coeyet al.[26℄, and Gaiduk
and MConnell[27℄ desribe dieletri relaxation and
dynamisofpolarmoleulesin greatdetail.
As asimple illustrationwe will assume aonstant
modulus for the magneti moment, i.e., S = S
H = S
0
sH, where H =H
0
z is aonstant
eld. Theinterationenergybetweenthemagneti
mo-mentandtheeldisthen
W = H= S
0 H
0 os:
With this simpliations, the systemof Eqs. (18)
be-omes
I
1 (_
2
sinos)+I
3
sin++S
0
sin+S
0 H
0 sin=
a
(19a)
I
1 (_sin
2
+2 sinos)+I
3
os_ I
3
sin+sin 2
S
0
sin=
b
sin (19b)
I
3 _ +
0
=
: (19)
Wewillonsiderinthissimpleillustrationonlythelimitofveryweaknoise. ThenEq.(19)hastheapproximate
stationarysolution'0. We,therefore,negletin Eqs.(19a)and(19b),whihbeome
I
1 (_
2
sinos)++S
0
sin+S
0 H
0 sin=
a
; (20a)
I
1
(_sin+2 os)+sin S
0 =
b
; (20b)
d
whih, together with the denition of and in
Eqs.(13), formaset of fourrst orderLangevin
equa-tions.
The simulations, of whih wepresent someresults
in thegures,havebeendevelopedalongthefollowing
proedure: An ensembleof 500partileswereinitially
put ontheequator,i.e.,=0andwith ===0
and the equations of motion, with a dierent
realiza-tionofthenoiseforeahpartile,weresolvedforsome
time,longenoughfortheensembletoaquirea
station-arydistribution. Alongthisperiodtheaveragevalueof
S
z
=os()(wehooseS
0
=1)wasalulated. The
re-sultisshowninFig.1,forseveralvaluesofthe\moment
of inertia", I=I
, where I is I
1 and I
is an arbitrary
referene value. We see that for very low I, hS
z i
in-reases monotoniallyto itsstationaryvalue(whihis
very lose to 1 beausethe noiseis very low), like in
the super-paramagneti ase, when the magneti
mo-ment preesses around the magneti eld following a
notbehavemonotonially,butshowsosilations,a
phe-nomenonknownas\nutation".
Inordertoalulatetheresponsefuntions,thenew
stateofeahpartile wastakenastheinitial statefor
500independentrealizationsofthenoise. Average
val-ues of these 250000 realizations have been alulated
along the time, aording to Eq. (16). By numerial
Fourier-Laplaetransformoftheresponsefuntionswe
obtainthedynami suseptibilities,
ij
(!). The
imag-inary part of the diagonal omponent, Im
xx (!), is
diretly related to the magneti resonane line. We
showit,forseveralvaluesofthemomentofinertia,in
Fig.2. Forsmall I ( super-paramagnetilimit) there
isonlyoneresonanepeak( !isphysiallyequivalent
to+!). ForbiggervaluesofI e seondpeakbeomes
apparent,whihorrespondstothephenomenonof
nu-tation,seeninFig.1. Wenotethatthefrequenyunits,
inFig.2,aredierentforthedierentplots,and,
Figure1. AveragevalueofS
z
(t)forfourdierentvaluesofthemomentofinertia.
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