Larmor precession and Debye relaxation of single-domain magnetic nanoparticles

0.3 MB·English

by  Zs. Janosfalvi

Larmor precession and Debye relaxation of single-domain magnetic nanoparticles
Zs. Jánosfalvi1, J. Hakl1 and P.F. de Châtel2,1
1Institute of Nuclear Research, P.O.Box 51, H-4001 Debrecen, Hungary
2Institute of Metal Research, CAS, Shenyang 110016, P.R. China
The numerous phenomenological equations used in the study of the behaviour of single-domain
magnetic nanoparticles are described and some issues clarified by meansof qualitativecomparison.
Toenablea quantitativeapplication of themodel based on theDebye(exponential) relaxation and
2 the torque driving the Larmor precession, we present analytical solutions for the steady states in
1 presenceoflinearlyandcircularlypolarizedacmagneticfields. Thepowerlossiscalculatedforboth
0 casesandtheresultscomparedwiththepredictionsoftheLandau-Lifshitz-Gilbert equation. Using
2 the exact analytical solutions, we can confirm the insight that underlies Rosensweig’s introduction
of the "chord" susceptibility for an approximate calculation of the losses. We also find that this
n
approximation provides satisfactory numerical accuracy up to magnetic fields that would generate
a
3/4 of thesaturation magnetization if applied constantly.
J

5
PACSnumbers: 47.65.Cb,75.30.Cr,75.75.Jn
2

]
l I. INTRODUCTION tionintheabsenceofastrongstaticmagneticfield,which
l
a give, in addition to the specific losses, some insight into
h Enduring interest in colloidaldispersions of ferro- and the underlying physical processes.
-
s ferrimagneticnanoparticlesisfuelledbytheirmedicalap-
e plications. Colloids of small particles of iron oxide are
m used in magnetic resonance imaging (MRI)1 as contrast II. EQUATIONS OF MOTION AND
SIMPLIFICATIONS
. agentsandinhyperthermiatreatmentasheat-generating
t
a media.2 Inbothcases,thenanoparticlesaresubjectedto
m atime-dependentmagneticfield,butthefrequenciesand The behavior of single-domain ferro- or ferrimagnetic
- magnetic field intensities are quite different. In the case nanoparticles in an external magnetic field has much in
d of MRI they are determined by the resonance frequency common with that of atomic or nuclear magnetic mo-
n
ofproton,whichis63MHzinafieldcommonlyused,1.5 ments. The torque on a magnetic moment µ,
o
c Tesla. In hyperthermia, medical considerations require
[ frequencies of the order 105 Hz and magnetic fields of
T =µ B (1)
1 10−2 Tesla. The frequency of ferrimagnetic resonance in ×
magnetiteishigherthan1GHzatzeroexternalfield,3 so
v determines the equation of motion of the angular mo-
6 that hyperthermia takes place at frequencies way below ment, dL/dt = T. The gyromagnetic relation, µ = γL
3 resonance, irrespective of the applied field.
enables a closed equation for µ which, applied to the
2 The theoretical background of the processes taking
magnetic moment of unit volume, provides the equation
5 placeintheseapplicationsiscolorful. Variousapproaches
of motion of the magnetization,
.
1 are being pursued with little attention for attempts to
0 compare and evaluate the available methods. Berger et
2 al.4 have undertaken to compare eight phenomenologi- dM/dt=γM B. (2)
1 cal equations of motion for magnetic moments of ferro- ×
:
v or ferrimagnetic nanoparticles dispersed in a nonmag- Here γ is the gyromagnetic ratio. The magnetization
i netic medium. They found that the calculated line- of materials we have in mind in this work is due to the
X
shapes in the narrow-linewidth limit, pertaining to high spin of electrons, accordingly γ = 1.76 1011 Am2/Js.
r − ×
temperatures, are of the Lorentzian form, irrespective The vector product in eq.(2) implies that any change
a
of the equation of motion used, but in the case of of the magnetization is perpendicular to M, that is, the
broadresonancelines significantvariationsareobserved. modulus of M remains constant. Also, if B is constant,
Low-temperature superparamagnetic resonance spectra d(M B)/dt = 0, that is, the angle between the two
·
proved to enable a distinction between different equa- vectors is constant. The only motion satisfying these
tions of motion, ofwhich only the Landau-Lifshitz equa- conditions is a precession of M around B. The angular
tion provided a satisfactory fit. velocity of the magnetization in this Larmor precession
The purpose of the work reported here is a theoreti- isωL =γM B/M⊥,whereM⊥ istheprojectionofM
×
cal comparison of two equations of motion, the Landau onthe plane perpendicular to B. The Larmorfrequency
-Lifshitz-Gilbert equation and the modified Bloch equa- is defined as a positive quantity, ω = γ B. To reduce
L
tion (in the nomenclature of Berger et al.4). Our focus the potential energy, U = M B, the|c|ontribution to
− ·
isonhyperthermia,ratherthanmagneticresonance. We dM/dtduetorelaxationmusthaveacomponentparallel
are seeking analytical solutions to the equations of mo- with B. Landau and Lifshitz5 have chosen a "damping
2
tteormM", w(hMich evBid)en=tl(yMachiBev)eMs thisM, b2eBin.gTphroepLoartnidoanua-l 1.0-Π4 0 Π4 Π2 34Π Π
× × · −
Lifshitz equation of motion is then a
0.5
b
dM
=γ[M B+αM−1M (M B)]. (3) Ms
dt × × × 0.0
(cid:144)
L
Ht
The coefficient α goes under the name of "the dimen- M
-0.5
sionless damping coefficient", which is something of a
misnomer, because addition of the "damping term" evi-
dently enhances the motion of M: -1.0
-Π 0 Π Π 3Π Π
4 4 2 4
dM
=γ M B (1+α2)1/2. (4) t
dt | × |
(cid:12) (cid:12)
Gilbert’s a(cid:12)(cid:12)(cid:12)pproa(cid:12)(cid:12)(cid:12)ch6 is closer to the notion of friction, rFiIvGed. 1fr:om(Ctohloersoonlulitnioen) TofimLLeGdeepqeunadteionnce(7o)f (Mcu(rtv)/eM(aS))iswditeh-
as it subtracts from the Larmor torque a torque propor- condition αω˜L =1. For comparison an (1−exp−x) type ex-
tionalto dM/dt. Thisisreminiscentoffrictioninlinear ponential relaxation is also given (curve (b)). Both saturate
motion, w| here a|force opposite to dr/dt is introduced to thesame MS butthe path is different.
| |
intotheequationofmotion. Theanalogyallowsaderiva-
tion of the Gilbert equation,
tothisfactor,Gilbert’sdampingtermreduces themotion
of M:
dM
=γ[M B ηµ M dM/dt], (5)
0
dt × − ×
dM
by adding a Rayleigh dissipation function to the La- =γ M B (1+α2)−1/2. (8)
dt | × |
grangian which describes the Larmor precession of the (cid:12) (cid:12)
(cid:12) (cid:12)
magnetic moment. (cid:12) (cid:12)
Strictlysp(cid:12)eakin(cid:12)g,theeffectoftheLandau-Lifshitzand
AcomparisonoftheLandau-LifshitzandGilbertequa-
Gilbert damping coefficients cannot be described with a
tions reveals that dM/dt (i) is perpendicular to M in
relaxation time, because the relaxation they stand for
both equations and (ii) in the former it consists of two is not exponential. If B is a constant field, pointing
mutually perpendicular terms while in the latter this is
in the z direction, the solution of the LLG equation is
notthecase. Observation(ii)impliesthatdecomposition M = Mtanh(αω˜ t), with ω˜ = ω /(1+α2)−1/2. If
z L L L
ofthedampingtermintheGilbertequationintocompo-
the magnetization is aligned perpendicular to B at the
nents parallel and perpendicular to that of the Landau-
outset, switching into the energetically favourable state
Lifshitz equation will offer a direct comparison of the
proceeds as given in Fig.1, curve (a). For comparison,
two. In fact, the Gilbert equation itself provides a de-
curve (b) shows the path of exponential relaxation with
composition into components which delivers the desired
the same initial state. The similarity may suggest the
transformation. Multiplying both sides of eq.(5) by M
definition of
and taking (i) in account yield
1 (1+α2)1/2
dM τ = = (9)
M × dt =γ[M ×(M ×B)+ηµ0M2dM/dt]. (6) LLG αω˜L αγB
Substituting this result in the last term of the Gilbert as the relaxation time of the LLG dynamics. However,
equation and rearranging terms lead to the Landau- its dependence on B would make τLLG an awkward pa-
Lifshitz-Gilbert (LLG) equation, rameterwhenthe magneticfieldistime-dependent. Fur-
thermore, a constant value for the product τ ω˜ , is
LLG L
counter-intuitive as different values of the ratio of the
dM =γ(1+α2)−1[M B αM−1M (M B)], (7) relaxation time and the Larmor precession time are ex-
dt × − × × pected to lead to qualitatively different behavior.
Shliomis8 hassuggestedthatunderwell-definedcondi-
where α = γηµ M. Clearly, for α 1 the Landau-
0 ≪ tions the equation of exponential relaxation,
Lifshitz equation is a good approximation, but for the
general case the (1+α2)−1 factor is essential to elimi-
natethenon-physicalimplicationsoftheLandau-Lifshitz dM M M
equationpointedoutbyKikuchi7andGilbert.6 Also,due = − eq, (10)
dt − τ
3
shouldsufficetodescribethebehaviorofacolloidofmag- Demonstrating’chord’-susceptibility
netic nanoparticles. Here, M is the average magnetiza- MHHoL(cid:144)Ms=71%
tion of the particles, -Ho 0 Ho
MHHoL MHHoL
Exact
µ HM V
M =M 0 d eˆ , (11)
eq S H
L kT
(cid:18) (cid:19)
"Chord"
V is the particle volume, eˆ =H/H is the unite vector
H L
pointing along H and MS is the saturation magnetiza- Hqt 0 0
e
tionofthecolloidM =φM ,whereφisthevolumefrac- M
S d
tion and M is the magnetization of the single-domain
d
magnetic nanoparticle.
The Langevin function, (x) = coth(x) 1/x, gives
L −
the magnitude of the magnetization in thermal equilib-
rium. Note that M must be an ensemble average and
eq -MHHoL -MHHoL
consequently so is M. In this respect, in the context of -Ho 0 Ho
superparamagnetic resonance or hyperthermia , eq.(10)
HHtL
is more expedient than the LLG equation. In the latter
case, having found the possible solutions of the equation
FIG.2: Color online. Thevariation of exact equilibrium and
of motion, one has to face the issue of the appropriate ’chord’-susceptibilitybasedmagnetizationsatfieldamplitude
weighted averageof the associated energy losses. of H0 =50 kA/m.
Clearly, the parameter τ in eq.(10) is a proper relax-
ation time. In a stationary field, where M is also con-
eq
stant, the solution of this equation for the component of equilibrium magnetization is defined in terms of the to-
M along H is tal effectivemagneticfield,whichincludesthemicrowave
field as well as the internal fields (demagnetizing field
and crystalline anisotropic field). The latter are rea-
(M(t) M )=(M(0) M )exp( t/τ). (12) sonable, internal fields are often included in the Bloch-
eq eq
− − −
Bloembergen equation as well. However, the legitimacy
In reference 9, Shliomis names eq.(10) the Debye re- of a time-dependent field in the same place implies the
laxation equation. Yet it is often reduced from what is assumption that the response of the magnetization to a
generally called the Shliomis relaxation equation, rapidly changing field can be described in the same way
as in the case of a stationary field.
Recently, Cantillon-Murphy et al.16 have published a
dM M M
+M( v) ω M = − eq. (13) thorough analysis of the implications of the Shliomis
dt ∇· − × − τ
relaxation equation for the relaxation process and en-
DevoidofthepossibilityofLarmorprecession,theDe- ergy dissipation, in superparamagnetic colloids contain-
bye relaxation equation, eq.(10), can be looked upon ing magnetite nanoparticles, under the conditions of
as a truncated and simplified version of the Bloch- magnetic resonance imaging (MRI) and hyperthermia
Bloembergenequation. Thelatterdoesincludethegyro- treatment of cancer patients.2 They justify the neglect
magnetic torque, µ γM H. The equation of motion, ofv andω, ineq.(13)so thatin factthe calculationsare
0
which Bloch has set up fo×r the interpretation of nuclear based on the Debye relaxation equation, eq.(10). Apart
magnetic resonance experiments14 and Bloembergen ap- from the approximation inherent in the exclusion of
pliedtoferromagneticresonance15wasspecificallyaimed the gyromagnetic torque they also applied Rosensweig’s
at the configuration of such experiments. The z axis is chord susceptibility,17
aligned along the static magnetic field, which provides
the axis of the Larmor precession, and enables the defi- M µ H M V
S 0 0 d
nition of two distinct relaxationtimes: τ1 for the z com- χch = H L kT . (14)
ponentofthe magnetizationvectorandτ foritscompo- 0 (cid:18) (cid:19)
2
nents in the xy plane. As the Debye relaxationequation instead of the Langevin function appearing in eq.(11).
is not meant to be limited to ferrofluids in a dominant The use of the chord susceptibility in the present con-
static magnetic field, the simplification to a single relax- text is illustrated in Fig.2. As the alternating applied
ation time seems inevitable. field is increasing towards its maximum at H , eq.(11)
0
Usually, the equilibrium magnetization in the Bloch- requiresthe equilibrium magnetization to follow the con-
Bloembergenequationisassumedtobegeneratedbythe cave curve denoted by , whereas the approximate equi-
L
static fields. The effect of the much weaker microwave librium magnetization given by the chord susceptibil-
field on the equilibrium magnetization is neglected. In ity increases along the straight line between M (0) at
eq
whatBergeret al.4 callthe modifiedBlochequation,the M (H ). The equation of that line is
eq 0
4
Ho=7kA(cid:144)m Mmax.
0 T(cid:144)4 T(cid:144)2 3T(cid:144)4 T Mž7(cid:144)8T
0 100 200 300 400 500
Dm 400 400 440300 Lmax& R max ô ô ô ô440300
(cid:144)A 200 200 ôôô L
(cid:144)D@kAm,Mk-2000 HHLtL& R -0200 @(cid:144)DMkAm230000 ôòôòôòôòôòôòòòò R ò ò ò ò230000
@H 3 a 100 ôò 100
-400 -400 ôò 4 a
ôò
0 0
0 T(cid:144)4 T(cid:144)2 3T(cid:144)4 T
t
0 100 200 300 400 500
Ho=50kA(cid:144)m Ho@kA(cid:144)mD
0 T(cid:144)4 T(cid:144)2 3T(cid:144)4 T
Mmax.
400 400 Mž7(cid:144)8T
Dm L 0 20 40 60 80 100
@(cid:144)MkA 200 HRHtL 200 380 Lmax&R maxô ô ôL ô380
Dm, 0 0 300 ô ô ò ò ò ò300
@(cid:144)HkA-200 3 b -200 (cid:144)DkAm200 ôò ôò ò ò R 200
@
-400 -400 M ôò
0 T(cid:144)4 T(cid:144)2 3T(cid:144)4 T 100 ôò 100
t ôò
4 b
ôò
Ho=100kA(cid:144)m 0 0
0 T(cid:144)4 T(cid:144)2 3T(cid:144)4 T 0 20 40 60 80 100
Ho@kA(cid:144)mD
400 400
Dm R L FIG.4: (Coloronline)Equilibriummagnetization(L)andap-
(cid:144)A 200 200 proximateequilibrium magnetization calculated fort=7T/8
k HHtL
@ andt=T,plottedagainsttheamplitudeH0oftheoscillating
M
0 0 magnetic field. At t = T the two are identical by definition
D,
m of χch, but at t = 7T/8 the relative difference between the
(cid:144)A-200 -200 (L) curve (down triangles) and the (R) curve (up triangles)
k
@H 3 c increases with H0, reaching 20% at 100 kA/m.
-400 -400
0 T(cid:144)4 T(cid:144)2 3T(cid:144)4 T
t
H µ H M V
Ho=500kA(cid:144)m M =M 0 0 d eˆ , (15)
eq S H
0 T(cid:144)4 T(cid:144)2 3T(cid:144)4 T H0L(cid:18) kT (cid:19)
HHtL
400 L 400 hence the definition of the chord susceptibility given by
Dm R eq.(14).
(cid:144)
A 200 200 Equation (10) implies that in a field H(t) =
k
@M H0cos(2πt/T)the relaxationpulls themagnetizationto-
0 0
D, wards the equilibrium magnetization, which is oscillat-
m
(cid:144) ing in time. Figure 3 shows this equilibrium magnetiza-
A-200 -200
k tion determined by the Langevin function ( ) and the
@H 3 d L
Rosensweigchordsusceptibility(R). Bydefinitionofthe
-400 -400
latter, the two curves meet at H = 0 and H = H . In
0
0 T(cid:144)4 T(cid:144)2 3T(cid:144)4 T the limit of strong external field (µ H M V/kT 1),
0 0 d
≫
t Fig.3(d), the curve is flipping between the extrema,
L
while the R curve has sinusoidal characteristic. The
FIG. 3: (Color online) The external magnetic field depen- magnetization does not exceed the saturation magneti-
dence of the behavior of equilibrium and approximate equi- zation M , even for field H > M . In the weak field
S 0 S
librium magnetizations for a single H cycle. The equilib-
riummagnetizations definedwith Rosensweig’s susceptibility
and Langevin function are denoted by R (dotdashed) and
L (dashed) respectively. At weak fields the two curves are
identical Fig.3(a). Increasing field strenght only the extrema
remain identical.
5
limit, Fig.3(a), where hyperthermia is applied, there is the solution, eq.(12), found in Section II for the De-
no difference. Figures 3(b) and 3(c) show the transition bye relaxation equation. For a rotating magnetic field,
between the two behaviors. It is clear that when the H = H cos(ωt);H = H sin(ωt), the coupled equa-
x 0 y 0
field reaches a value where M /M > 0.8 (Figs. 3(c) tions to be solved are as follows:
eq S
and 3(d)), the magnetization calculated with the chord
susceptibility is substantially deviates from the correct
function.
dM 1
The dependence of the discrepancy between the two x = µ γM H sin(ωt) (M M (H )cos(ωt));
0 z 0 x eq 0
dt − − τ −
curvesonthe amplitude ofthe ac fieldis shownin Fig.4.
dM 1
In the low-field limit there is no discrepancy, because y = µ γM H cos(ωt) (M M (H )sin(ωt));
0 z 0 y eq 0
χ = χ and M = χH, but it is visible at 20 kA/m dt − τ −
ch eq
and increases with increasing field. Beyond H0 = 100 dMz = µ γ(M H sin(ωt) M H cos(ωt)) Mz. (17)
0 x 0 y 0
kA/m the relative difference remains constant at about dt − − τ
20%, which is too large to use the approximation in cal-
culations of the magnetization. In Session IV we show
To handle the entanglement of the three components
that this does not disqualify the chord susceptibility in
of M, it will prove to be convenient to apply the series
calculation of the energy loss. of transformations that enabled us in a previous work18
InsectionsIIIandIVwepresentanalyticalsolutionsto
to solve this set of equations for free precession, i.e. in
the Debye relaxationequation enrichedwith the Larmor
the absence of relaxation (1/τ = 0). Three transforma-
torque,
tions create a coordinate system, which rotates as dic-
tated by free Larmor precession in the rotating field.
dM M M First,arotationaroundthe z axisby anangleωτ makes
=µ γM H − eq, (16)
0 the xy plane follow the magnetic field, then the z axis
dt × − τ
is turned by an angle Θ into the direction of the to-
with Meq as given in eq.(11). We will use these results tal angular velocity Ω = ω+ωL, and finally a rotation
to assess the effects of the torque on the steady-state so- around this new, z′ axis by an angle Ωτ drives the x′y′
lutions under linear and circular polarization of the ac plane to rotate together with the magnetization vector.
magnetic field. As we take into account the curvature of The description is reminiscent of an Euler transforma-
the Langevin function, we can also discuss the implica- tion and indeed the product of the three matrices repre-
tions of the chord susceptibility. senting the rotations listed above, is of the form of the
canonical Euler transformation19 with the replacements
α Ωt,β Θ and γ ωt. As the axis of the
III. CIRCULAR POLARIZATION La↔rmorprec↔essi−onis the ma↔gne−tic field, which rotatesin
the planeperpendicularto the z axis,ω is perpendicular
Inthissectionwegivetheanalyticalsolutionofeq.(16) to ωL and Ω = ω2+ω2. Also, it follows from this
L
for a rotating magnetic field. The first term, represent- configuration that sinΘ = ω /Ω and cosΘ = ω/Ω. In
L
p
ing the Larmor torque, spoils the separation of Carte- what follows, the transformation matrix will be used in
sian components, which has enabled the derivation of the form
ωcosωtcosΩt+ΩsinωtsinΩt ωsinωtcosΩt ΩcosωtsinΩt ω cosΩt
1 − − L
O = ωcosωtsinΩt ΩsinωtcosΩt ωsinωtsinΩt+ΩcosωtcosΩt ω sinΩt (18)
L
Ω − − 
ω cosωt ω sinωt ω
L L
 
To find the derivative of the transformed magnetiza- equations for the transformed magnetization:
tion vector M′, we need the derivative of the matrix O:
M (H )cos(ωt)
dM′ dOM dO dM dM′ eq 0
= = M +O . (19) τ = OM +O Meq(H0)sin(ωt) . (20)
dt dt dt dt dt −  0 
Substituting here dM from eq.(16), we find that the  
dt The first term on the right-hand side gives trivially
contribution of the Larmor torque to OdM cancels
dt M′,thesecondonecanbefoundapplyingeq.(18),tofind
dO
M, as it should, leaving the following differential thedifferentialequationsforthethreecomponentsofthe
dt
6
transformed magnetization vector. Ultimately, the ana- is
lyticalsolutionforthe transformedmagnetizationvector
ω cosδ ω cos(Ωt δ)
M′(t) = M′(0) M (H ) exp( t/τ)+M (H ) − ;
x " x − eq 0 Ω 1+(Ωτ)2# − eq 0 Ω 1+(Ωτ)2
ω p sinδ ω psin(Ωt δ)
M′(t) = M′(0)+M (H ) exp( t/τ)+M (H ) − ;
y " y eq 0 Ω 1+(Ωτ)2# − eq 0 Ω 1+(Ωτ)2
ω
M′(t) = M′(0) M (H ) Lpexp( t/τ). p (21)
z z − eq 0 Ω −
h i
Here δ is defined by sinδ = Ωτ/ 1+(Ωτ)2, cosδ = E~AHΤΩL
1/ 1+(Ωτ)2. Since the exponentpially decaying terms 0.50 æ n=1
areofnointerestforapplicationsontimescalesexceeding
p æ
τ by several orders of magnitude (t/τ 1) and we seek
≫ 0.30 æ n=2
thesteady-statesolutioninthelaboratoryframe,weshall æ
æ
drop the exponential terms. The inverse transformation 0.20 n=3
is easily carried out with the transposed of matrix (18). A
0.15
A compact form of the final result is
0.10
0.5ΩΤ
ωcos(ωt)+Ω2τsin(ωt)
ω 1
M(t)=M (H ) ωsin(ωt) Ω2τcos(ωt) .
eq 0 Ω21+(Ωτ)2  − 
ω 0.1 0.5 1 2
L
−
 (22) ΩΤ
Note that |M(t)| = Meq(H0) 1− ωΩL22 √1+1(Ωτ)2, indi- FIG. 5: (Color online) The dependence of energy E ∼ A =
catingthattheinterplayofLarmqorprecessionandrelax- A(ωτ)on ωτ for differentωL/ω ratios as n=1+ωL/ω. The
ation pushes the magnitude of the magnetization, which curvesreach maximum of0.5n−1 at 1/n. Largest energydis-
istime-independent,belowthethermalequilibriumvalue sipation is achieved at ωL ≪ ω. The maximum is shifted
towardslowerfrequenciesasnisincreased,providedτ iscon-
in a static field H . The energy loss per cycle is easily
0 stant.
calculated,
relaxation and Larmor precession suppresses the energy
0+2π/ω dH
E = µ M dt loss, in the case of circular polarization.
0
− Z0 · dt The specific absorption rate (SAR) defined as energy
ω Ωτ losspersecondandperkg,isthemeasureofenergylosses
= 2πµ M (H )H . (23)
0 eq 0 0Ω1+(Ωτ)2 relevant to applications:
This Debye-type dependence on Ωτ is familiar in low-
fieldmagneticresonance20 andwasshownbyGarstens21 . Eω µ0Meq(H0)H0ω2 Ωτ
SAR= = . (24)
to be exact in the limit of vanishing static field. 2πρ ρ Ω 1+(Ωτ)2
The basic functional dependence of E on H , ω and τ
0
is dictated by the term H0ωτ . Noting, that ω2 H2, The function describing the dependence on ωτ and n
1+Ω2τ2 L ∼ 0 is now
it has maximum either as a function of H , ω, τ or
0
products ωτ and ω τ. To facilitate comparison with
L
Garstens’20 result for frequency dependence of E, we se- 1 ω2τ2
lect the dimensionless parameters ωτ and ω /ω and de- B = , (25)
L τ 1+n2(ωτ)2
finen= ω2/ω2+1andA= ω Ωτ . Figure5shows
L Ω1+(Ωτ)2
A(whichisproportionaltoE)asafunctionofωτ atvar- which grow monotonically with ωτ, saturating at B =
p
ious value of n. The energy loss is seen to increase with 1 1 . Figure 6 shows that the saturation takes place at
τ n2
ωτ up to a maximum at 1/n, the value of A at the max- aboutωτ =1,meaningthatincreasingthefrequencywill
ima being 1/(2n). It is remarkable how the interplay of not basically further improve the SAR.
7
SAR~BHΤΩL IV. LINEAR POLARIZATION
1.00
n=1
To find the equation of motion for the magneti-
n=2
0.50
zation subjected to a magnetic field oscillating along
n=3
the z axis, the following vectors have to be in-
0.20
serted into eq.(16): H(t) = (0,0,H cos(ωt)),M
0
B0.10 H = µω0|Lγ|(Mycos(ωt),−Mxcos(ωt),0) and Meq(H) ×=
0.05 0,0,φMdL µ0MkdTVH0 cos(ωt) , where ωL = µ0|γ|H0.
T(cid:16)he matrix o(cid:16)f the transformati(cid:17)o(cid:17)nthat will eliminate the
0.02 Larmor term of eq.(16) depends on the sign of the gyro-
magnetic ratio. For both cases we can write the matrix
0.01
0.1 0.2 0.5 1.0 2.0 5.0 as
ΩΤ
sing cosg 0
FIG.6: (Coloronline) Thedependenceofspecificabsorption ±
rate SAR ∼ B = B(ωτ) on ωτ for different ωL/ω ratios as O± =−cosg ±sing 0, (26)
n = 1+ωL/ω. B saturates at ωL ≫ 1, however most of 0 0 1
change takes place up to ωτ =1. Largest specific absorption  
rate is achieved at ωL ≪ω (n=1). where the upper sign equals that of γ and g(t) =
(ω /ω)sin(ωt). The equation of motion for M′(t) =
L
O M(t) is
±
dM′(t) dO dM dO 1
= ±M +O = ±M µ γ O (M H) O (M M (H)). (27)
0 eq
dt dt ± dt dt ± | | ± × − τ ± −
Takingintoaccountthatdg/dt=ω cos(ωt)(irrespec-
L
tive of the sign of ω ), the first two terms in eq.(27)
L
cancel each other. The equation of motion for M′(t) in M(t) = ±My(0)sing+Mx(0)cosg exp( t/τ)
terms of the new coordinate system is easy to find, as My(0)cosg Mx(0)sing −
(cid:18) ∓ (cid:19)
Oco±mMpon=enMt,w′,hbicyhdiseefivniidtieonnt,lyannodtMaffeeqc(tHed)bhyaOs on.lWy haazt = M⊥(0) cosisn[g[g∓ϕϕ(0(0)])] exp(−t/τ), (30)
± (cid:18)∓ ∓ (cid:19)
remains in the transformed frame is the set of equations
of motion where ϕ(0) is the azimuth angle of M(0) and M⊥(0) =
M2(0)+M2(0). We can determine the time depen-
x y
dM′ 1 dqence of the angular velocity:
x = M′;
dt −τ x
dMy′ = 1M′; ϕ = tan−1 My =tan−1 tan ωL sin(ωt) ϕ(0)
dt −τ y Mx ∓ ω ∓
dM′ 1 µ M VH cos(ωt) ωL (cid:16) (cid:16) (cid:17)(cid:17)
z = M′ M 0 d 0 (.28) = sin(ωt) ϕ(0);
dt −τ z− SL kT ∓ ω ∓
(cid:20) (cid:18) (cid:19)(cid:21) dϕ
= ω cos(ωt), (31)
L
The solution for the first two equations must be the dt ∓
familiar exponential relaxation to zero,
withtheuppersignforγ >0andthelowersignforγ <0.
Equation (31) describes a Larmor precession whose fre-
quency is followingthe time dependence of the magnetic
Mx′,y(t)=Mx′,y(0)exp(−t/τ). (29) field (note that ωL is defined as the Larmor frequency
at H = H , so that ω cos(ωt) gives the Larmor fre-
0 L
Since O separates the xy plane and the z axis, the quency at H = H(t)). Whether the time dependence
±
transformationbacktothelaboratoryframecanbedone of the precession frequency is observable will depend on
in two dimensions. As g vanishes at t = 0, we have the magnitude of the projection of M on the xy plane,
M (0)= M′(0) and M (0)=M′(0), whence which, according to eq.(30), decays exponentially,
x − y y x
8
µ H M V/kT 1 one can use the susceptibility
0 0 d
≪
instead of the Langevin function, i.e., the first term in
M2(t)+M2(t)=M (0)exp( t/τ). (32) the Taylor series of instead of the function . In fact,
x y ⊥ − L L
the integrals can be done for higher order terms also
q
If the relaxation time is very short (like in hyperther- (Ref. 22, p.228). As the Taylorseries of cothx necessary
mia, where ωτ 10−4, the magnetization will be fully here(Ref.22,p.42)isonlyvalidfor x <π,theresulting
aligned along th≈e z axis before the magnetic field under- expression for the time dependenc|e |of the equilibrium
goes a substantial change, let alone a change of sign. Of magnetization in terms of odd powers of cos(ωt),
course, to observe the time dependence given in eq.(30),
we also need a Larmor frequency larger than the fre-
quency of H, that is, ωL > ω. This conditions is met in (ζcos(ωt))= ∞ 22m B (ζcos(ωt))2m−1, (34)
2m
hyperthermia, at H =200 kA/m the Larmor frequency L (2m)!
0 m=1
is about 40 GHz, whereas ω is of the order of 100 kHz. X
The z component,being separatedfromthoseperpen- where the B are Bernoulli numbers as defined and
2m
dicular to it, need not be put in a rotating frame. The listedinRef.22,p.1040andp.1045,respectively. Substi-
searchfor an analytical solution to the last equation un- tuting the series (34) into eq.(33),
der (28) leads to a single (though not simple) integral if
wesubstitute intoeq.(28)thedM /dtfunctionextracted
z
from M ∞ 22m
et/τM = S B ζ2m−1 et/τcos2m−1ωtdt.
z 2m
τ (2m)!
m=1 Z
X
d(et/τM ) 1 dM (35)
z = et/τMz+et/τ z . (33) Now is straightforward to find a solution to the equa-
dt τ dt
tion of motion for M which is analytical, albeit in the
z
To avoid the difficult integral, for ζ = form of an infinite series :
∞ m−1
2M 1 (1/τ)cos((2k+1)ωt)+(2k+1)ωsin((2k+1)ωt)
Mz = S B ζ2m−1 (36)
τ m 2m (m 1 k)!(m+k)![(1/τ)2+(2k+1)ω2]
m=1 k=0 − −
X X
There are three features of this solution that can be
revealed without evaluating the series: (i) the exponen-
tial factors have disappeared, meaning that the function ∞
1 cos(ωt)+ωτsin(ωt)
describes a steady state, (ii) keeping only the m = 1 Meff =2M B ζ2m−1 .
z S (m!)2 2m 1+(ωτ)2
termintheseriesthewell-knownresultforthefrequency- m=1
X
dependent Curie susceptibility [Ref. 11, eqs. (2) and (38)
(10)], Mz =χ0H0[cos(ωt)+ωτsin(ωt)]/[1+(ωτ)2] with Looking back at Fig.3(d), it becomes now clear that
abruptchangesofthemagnetizationtriggeredbytheflip-
ping of the equilibrium magnetization will not influence
µ M2Vφ
χ = 0 d (37) the calculated energy loss, because they are represented
0
3kT
by high-frequency terms in M .
z
emerges and (iii) the double summation is essentially a The effective magnetizationof eq. (38) may be looked
Fourier series, of which only the k = 0 sine terms are upon as a fictitious magnetization, which, if realized,
needed for the calculation of losses, because M , mul- would give the correct losses without reproducing the
z
tiplied by dH(t)/dt = H ωsin(ωt), will be integrated correct fictitious hysteresis loop. Instead, the shape of
0
−
over a cycle. the hysteresis loop and its frequency dependence are ex-
Keeping only the k = 0 terms in the second sum in actly the same as those one gets at low fields, where
eq.(36) amounts to eliminating the higher harmonics in the static magnetization is proportional to the magnetic
thetimedependenceofthemagnetization,whicharegen- field. Equation(38)confirmsthentheinsightthatunder-
eratedduetothesubstantialnonlinearityoftheLangevin lies Rosensweig’sintroductionof the chordsusceptibility
17
function beyond x 0.5. It follows then that for any [Ref. eq.(2)] and enables the calculation of the "field
≈
valueofH asusceptibilitycanbe found,whichwillpro- dependence"ofafictitioussusceptibility. Infact,thesus-
0
vide the exact energy loss on the assumption that the ceptibility relevant to the energy loss, implicitly defined
magetization is byMeff =χ H [cos(ωt)+ωτsin(ωt)]/[1+(ωτ)2]with
z loss 0
9
0.45 is not field dependent (that would contradict the defi-
nition of the susceptibility as a limes at H 0), but
(cid:144)Ζ 0.40 modd amplitude dependent, through ζ, which is pr→oportional
L
LHΖ 0.35 Χ0 toH0. NotethatB2 =1/6,sothatχloss =χ0 forζ 1.
r ≪
o
yl 0.30
a
T Inpractice,thesummationineq.(39)willbelimitedto
,Ζ 0.25
HL(cid:144)Ζ 0.20 meven ΧL naufimnbiteersinwdietxhmmmisaxa.cTonhceerrna,pbidutinFcirge.a7ssehoofwtshtehBatertnaokuinllgi
L
m = 17, the convergence is reliable up tp ζ = 2.8.
0.15 max
The converged values of (ζ)/ζ represent the amplitude
0 1 2 3 4 SLARžΩ =Μ ÈΓÈH
L 0 0
Ζ 1M 10M 100M 1G
Ω=Ω
1 ΩΤ=1 L
FIG. 7: (Color online) Convergence of the Taylor series ex-
pansion of L(ζ)/ζ representingthesummingterm in eq.(39).
0.01
Thin constant line is related to mmax = 1 (χ0), while thin Dg
(cid:144)
curves are superposed functions up to mmax = 17. Red W 10-4
dashed line shows the exact χL = L(ζ)/ζ. Arrows point to- @k
wards the direction of increasing m. AR 10-6
S
EžΩ =Μ ÈΓÈH 10-8
L 0 0
1M 10M 100M 1G
104 10-10
Ω=Ω 1M 10M 100M 1G
D L
L
cle 100 ΩΤ=1 Ω@HzD
y
c
3
m 1 FIG. 9: (Color online) Specific absorption rate as a function
H
(cid:144)J of frequency ω, parameterized by field strength H0. Thick
@
s lines correspond to linear polarized cases, while dashed ones
s 0.01
o
l tocircularpolarizedcases. Auxiliarythinlineisdrawntoex-
y
g plore behavior around resonance ωτ =1. Maximum reached
ner10-4 for each circular polarized case when condition ω > ωL is
E
fulfilled (Thin, dashed auxiliary lines).
10-6
1M 10M 100M 1G
Ω@HzD
FIG.8: (Color online) Energy loss asafunction offrequency
ω, parameterized by field strength H0. Thick lines corre-
spondtolinear polarized cases, while dashedonestocircular dependence of of the fictitious susceptibility χ . Note
loss
polarized cases. Auxiliary thin line is drawn to explore be-
that while the deviation of χ from χ is increasing
chord 0
havioraroundresonanceωτ =1. Thereislocalmaximumfor
substantially, the difference between the exact χ and
eachcircularpolarizedcasewhenconditionω=ωLisfulfilled loss
χ is small and remains constant at about 5% even
(Thin, dashed auxiliary lines). chord
for 2 < ζ < 2.8. Hence for ζ > 1 calculating the energy
lossusing χ is not reliable,but close enoughresults can
0
be obtained using the χ susceptibility.
chord
µ H M2Vφ ∞ 1
χ =2 0 0 d B ζ2(m−1), (39) Calculating the energy loss per cycle,
loss kT (m!)2 2m
m=1
X
t0+2π/ω dH t0+2π/ω dH τω
E = µ M dt= µ Meff zdt=πµ H χ (ζ) . (40)
− 0 · dt − 0 z · dt 0 0 loss 1+(τω)2
Zt0 Zt0
Comparing the ωτ dependence of E in both polar- ized cases, prompt can be realized that frequency de-
10
pendence of energy dissipation in circularly and linearly kA/m with r = 5 nm nanoparticles. For small driving
polarized cases can be equivalent by a factor of 2, if fieldamplitudes acommonenergylosslocalmaximumis
n = ω /ω+1 = 1 and the linear approximation of observed. By increasing the driving field amplitude, the
L
M is taken for both polarized cases. tent shape formof the energy loss curve is not changing.
eq
p
At high driving field amplitudes (ζ 1) the energy Forthelinearlypolarizedfielditisproportionallyshifted
≫
absorption per cycle can be estimated as follows. For to higher energy loss values, while for the circularly po-
ω 1/τ eq.(28) implies that M lags behind the tar- larized case the energy absorption maximum is shifted
z
≪
getM (ζcos(ωt)) value by time τ, whichyields for the towards higher frequencies. Notably, in the latter case
S
L
energy absorption E 4µ M H ωτ. For ω 1/τ ac- at the given set of parameters the energy loss at lower
0 S 0
≈ ≫
cording to eq.(28) the driving magnetic field causes only frequencies just started to decrease with increasing field
saw-tooth curve like oscillation of magnetization around strength (see eq.(23)).
the zero value with amplitude MSπ/2ωτ. The energy Figure 9 shows the same data in practical units of
≈
absorption per cycle in this case is E 4µ0MSH0/ωτ. kWg−1 as defined for SAR. The presented results con-
≈
cerning the shift of the energy dissipation maximum
andhigh frequency behaviour arein excellent agreement
V. SUMMARY AND CONCLUSIONS withtheresultsderivedfromtheLandau-Lifshitz-Gilbert
equation18.
Analytical solutions of Bloch-Bloembergen equation Contrarythe resultsobtainedforlowfrequency region
for both polarized cases are presented. They are in ex- showdifferentbehaviour. Thisdifferenceisoutsetbythe
cellent agreement with the numerical solutions. possibility of changing the magnitude of magnetization.
Differencesintime-dependenceofmagnetizationusing For example this produces resonance like E(H ) depen-
0
Rosensweig’s"chord"-susceptibilityandLangevinfunc- dence in Bloch-Bloembergen case, while the lack of this
tion areemphasizedin Figs.3 and4, where H0 is param- possibility results in H0 independent E(H0) behaviour
eterised. Formulaeforenergydissipationandspecificab- intheLandau-Lifshitz-Gilbertframeforcircularlypolar-
sorption rate (SAR) are also provided. Short analysis is ized field. The difference in results for linearly polarized
givenforstrongandweakfieldlimit. Itcanbeconcluded field also originates from this possibility.
that in hyperthermia region Rosensweig’s suggestion for At low frequency and relatively high driving field am-
"chord"-susceptibilityisreasonable. Linearapproxima- plitudes, linearly polarized field would yield more dissi-
tion of Langevin function, (x) x/3, can also be used pationcomparedtothecircularcasewhentherelaxation
L ≈
providedthevalueofargumentxissmallenough. While processescharacterizedby the Bloch-Bloembergenequa-
x=0.4means 1%difference in linearapproximation,at tion. Our final conclusion is alike as drawn from analyz-
x=1.2 already deviates with 10%. ing Landau-Lifshitz-Gilbert equation: when the relax-
The frequency dependence of energy dissipation and ationprocessis characterizedby the Bloch-Bloembergen
SAR is also clarified, see Figs. 5 and 6. Possible maxi- equation,the technicalcomplicationsofgeneratingacir-
mumofenergydissipationdecreaseswithincreasingratio cularly polarized field in difficult geometry need not be
ωL/ω and shifted to lower ω when relaxation time τ is considered.
taken constant. SAR in the high ωτ limit shows satu-
ration and its value is decreasing with increasing ratio
ω /ω. Equivalence of linear and circular polarized field
L
Acknowledgement
can be stated with a factor of two, if the Larmor fre-
quency is negligiblesmallto externalfieldfrequency and
linear approximation of Langevin function is taken for One of the author, Zs.J., wishes to express special
expressing equilibrium magnetization. In Fig.8 the re- thankstoI.Nándoriandtocolleaguesinthelabfortheir
sulting energy loss per cycle is depicted as a function of valuablecriticaladvicesandcontinuoussupport. Theau-
frequencyforbothpolarizedcasesasaparameterofdriv- thorsacknowledgesupportfromtheHungarianScientific
ing field amplitude H where τ =10−9 s andM =446 Research Fund (OTKA) No.101329.
0 0 S
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