ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 300 (2006) 397–406
www.elsevier.com/locate/jmmm
Thermal and particle size distribution effects on the
ferromagnetic resonance in magnetic fluids
C.N. Marin
Faculty of Physics, West University of Timisoara, Bd. V.Parvan, no. 4, Timisoara 300223, Romania
Received 24 January 2005; received in revised form 29 March 2005
Available online 21 June 2005
Abstract
Thermal and particle size distribution effects on the ferromagnetic resonance of magnetic fluids were theoretically
investigated, assuming negligible interparticle interactions and neglecting the viscosity of the carrier liquid. The model is
based on the usual approach for the ferromagnetic resonance description of single-domain magnetic particle systems,
which was amended in order to take into account the finite particle size effect, the particle size distribution and the
orientation mobility of the particles within the magnetic fluid. Under these circumstances the shape of the resonance
line, the resonance field and the line width are found to be strongly affected by the temperature and by the particle size
distribution of magnetic fluids.
r 2005 Elsevier B.V. All rights reserved.
PACS: 76.50.+g; 61.46+w; 75.50 Mm
Keywords: Ferro-ferrimagnetic resonance; Nanoparticles; Magnetic fluids
1. Introduction
Magnetic fluids are stable colloidal systems
consisting of single-domain ferro-ferrimagnetic
particles coated with a surfactant and dispersed
in a carrier liquid [1]. These remarkable magnetic
systems have attracted the interest of the scientific
community because of applicative reasons as well
as from a fundamental research point of view. As
Tel.: +40722331695; fax: +40256282026.
E-mail address: [email protected].
their structure and particle concentration can be
easily controlled depending on the production
process, temperature or presence of external fields
[1], the magnetic fluids are suitable systems for the
study of the properties of nanometric size particles
and in researches regarding disordered systems.
The ferromagnetic resonance measurements are
a powerful tool in investigations concerning both
the macroscopic properties of a magnetic fluid and
the properties of individual single-domain particles
within the magnetic fluid. The analysis of the
colloidal stability of magnetic fluids [2,3], the
0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.jmmm.2005.05.026
ARTICLE IN PRESS
398
C.N. Marin / Journal of Magnetism and Magnetic Materials 300 (2006) 397–406
experimental determination of the effective anisotropy constant of particles within the fluid [4,5]
and determination of the particle size distribution
within magnetic fluids [6,7] are examples in which
the ferromagnetic resonance measurements can be
successfully used. Therefore development of the
theoretical description of the ferromagnetic resonance phenomenon in magnetic fluids is of interest
both from a fundamental point of view and for
different applications.
In Ref. [8] are presented the first experimental
results concerning the effect of particle sizes on the
ferromagnetic resonance behaviour of single-domain particle systems. In the above-mentioned
reference, the ferromagnetic resonance measurements revealed that by increasing the mean
diameter of magnetite particles from 3.3 to
3.6 nm one obtains an increase of the ferromagnetic resonance line width from 350 to 475 G.
Furthermore, the ferromagnetic resonance lines in
Ref. [8] are asymmetric.
In case of magnetic fluids, one of the first
experimental results regarding the effect of the
particle size distribution on the ferromagnetic
resonance line was reported by Sharma and
Waldner [9]. They showed that at room temperature
the ferromagnetic resonance line of some magnetic
fluids has a two line pattern, consisting of a narrow
line superposed to a broad line. Sharma and
Waldner [9] assumed that the broad line is due to
the large particles within the magnetic fluid and the
narrow line is due to the small particles, which are in
superparamagnetic state. Similar experimental results in which the ferromagnetic resonance line of
magnetic fluids consisted of a narrow line superposed to a broad line were reported in Ref. [10–12].
The influence of the temperature on the
ferromagnetic resonance behaviour of magnetic
fluids was investigated by different authors
[12–15]. The experimental results have shown that
by increasing the temperature, the ferromagnetic
resonance line becomes narrower and the resonance field shifts towards larger values.
A correct qualitative explanation to the abovementioned experimental results was supplied by
the well-known model of Raikher et al. (RSS
model) [16–18]. The RSS model is an effective-field
model, which is based on the solution of the
Focker–Planck equation under condition of strong
polarizing field, but this model is limited to
systems in which the particles have the same
magnetic diameter.
The aim of this article is not to revise, but to
present an alternative theoretical approach to the
RSS model and to go further by taking into
account the particle size distribution within the
fluid. This theoretical model is based on the usual
theory of the ferromagnetic resonance phenomenon in single-domain particle systems, which was
amended in order to take into account the effect of
finite particle sizes, the particle size distribution
and the orientation distribution of anisotropy axes
of particles within the fluid. By means of this
model a theoretical investigation of the effect of
which the temperature, the mean particle diameter,
m and the standard deviation of the particle size
distribution, s have on the resonant behaviour of
magnetic fluids is performed.
2. Theoretical considerations
The usual approach of the theoretical description of the ferromagnetic resonance phenomenon
in single-domain particle systems is based on the
analysis of the free magnetic energy of a representative particle within the system [19]. Following the
theoretical approach as in Ref. [20], one obtains the
expressions of the resonance condition (Eq. (1)), of
the line width (Eq. (2)) and of the contribution at
the complex susceptibility of the system, corresponding to the representative particle (Eq. (3)).
ggð1 þ a2R Þ1=2
ðF yy F jj F 2jy Þ1=2 ,
M sin y0
F jj
ggaR
F yy þ 2
DoR ¼
,
M
sin y0
o0;R ¼
w00R ¼
g2 g2 ð1 þ a2R Þ
ðo20;R o2 Þ2 þ o2 ðDoR Þ2
s2 F jj
F yj
2
oDoR l F yy þ 2 þ 2ls
sin y0
sin y0
2
2
oaR Mðo0 o Þ 2
ðl þ s2 Þ .
ggð1 þ a2R Þ
(1)
(2)
ð3Þ
ARTICLE IN PRESS
C.N. Marin / Journal of Magnetism and Magnetic Materials 300 (2006) 397–406
In the above equations g is the spectroscopic
splitting factor; g is the gyromagnetic electronic
ratio (g ¼ 8:79 106 G1 s1 ); aR is the damping
parameter from the Landau-Lifshitz equation,
corresponding to the representative particle; o is
the pulsation of the microwave field; M is the
magnetization of the representative particle; y and
~ F yy ,
j are the azimuth and the polar angles of M;
F jj and F yj are the second derivatives of the
magnetic free energy of the representative particle,
divided by the volume of the particle. F yy , F jj and
F yj are calculated in the equilibrium position of
the magnetization of the representative particle, in
which the angular coordinates are denoted (y0 , j0 ).
The parameters l and s from Eq. (3) are given by
Eqs. (4) and (5), where d and l are the azimuth and
the polar angles of the microwave field:
l ¼ sin d sinðj0 lÞ,
(4)
s ¼ cos y0 sin d cosðj0 lÞ cos d sin y0 .
(5)
In the following considerations uniaxial anisotropy and a spherical shape for the representative
particle is assumed. Neglecting the interparticle
interactions, the density of the magnetic free energy
of the representative particle (which is the magnetic
free energy of the particle divided by its volume) is
given by relation (6):
F ¼ MB0 ð~
e ~
eB Þ Kð~
e ~
eA Þ2 .
(6)
In Eq. (6), B0 is the induction of the polarizing
field, K is the uniaxial anisotropy constant, ~
e, ~
eB and
~
eA are the unit vectors defining the direction of
~ of the polarizing field and of the
magnetization, M,
anisotropy axis, respectively.
In the following considerations, the polarizing
field is assumed parallel to the oy-axis (yB ¼ p=2
and jB ¼ p=2). In case of a small anisotropy by
comparison with the polarizing field (i.e. 2K/
M5B0), the resonance condition and the line
width of the representative particle will have
respectively the following expressions:
o0;R ¼ ggð1 þ a2R Þ1=2
K
B0 þ ð3 sin2 yA sin2 jA 1Þ ,
M
ð7Þ
DoR ¼ 2ggaR
399
K
2
2
B0 þ ð3 sin yA sin jA 1Þ .
M
(8)
The first amendment to the usual theory of the
ferromagnetic resonance of magnetic fluids consists of taking into account the orientation
distribution of the uniaxial axes of particles within
magnetic fluid, in the presence of a polarizing
magnetic field. As it was shown in Refs. [16,17], if
~0 is parallel to the
the polarizing magnetic field, B
oy-axis, then the orientation distribution function
of the uniaxial axes of particles within the
magnetic fluid is given by expression:
Z Z
1 2p p
MV B0
f ðyA ; jA ; B0 Þ ¼
sin y sin j
exp
Z 0
kT
0
KV
þ
½cos y cos yA þ sin y sin jA
kT
cosðj jA Þ2 sin y dy dj.
ð9Þ
In Eq. (9), V is the magnetic volume of one
particle, k is the Boltzmann’s constant, T is the
temperature of the system, yA and jA are
the azimuth and the polar angles of ~
eA and Z is
the partition function, having the following
expression:
kT
MV B0
Z ¼ 16p2
sinh
MV B0
kT
Z 1
KV 2
y dy:
exp
ð10Þ
kT
0
As can be observed from Eq. (9), the Brown
rotation of particles within the magnetic fluid is
neglected. This approximation is correct because
the frequency range within which the ferromagnetic resonance in magnetic fluids is observed is
108–1010 Hz and the characteristic frequency of the
Brown rotation of particles is ranged within
103–105 Hz. Hereby, the Brown rotation of particles within magnetic fluids can be assumed frozen
up relative to much faster process of the ferromagnetic resonance.
The existence of the orientation distribution of
the uniaxial axes of particles within magnetic
fluids, in the presence of a polarizing magnetic
field has been experimentally proven in Ref. [14].
ARTICLE IN PRESS
400
C.N. Marin / Journal of Magnetism and Magnetic Materials 300 (2006) 397–406
Due to the orientation distribution of the anisotropy axes of particles within the magnetic fluid,
the imaginary part of the complex magnetic
susceptibility of a magnetic fluid having particles
with the same magnetic diameter, D, is given by
the relation (11).
Z 2p Z p
w00 ðB0 ; DÞ ¼ Fm ðDÞ
f ðyA ; jA ; B0 ; DÞ
0
0
w00R ðyA ; jA ; B0 ; DÞ sin yA dyA djA .
ð11Þ
In relation (11), Fm ðDÞ is the volume fraction of
the particles within the magnetic fluid. If the
particle size distribution is taken into account,
then the susceptibility must be averaged over the
particle size distribution, g(D). Thus the imaginary
part of the complex magnetic susceptibility of a
magnetic fluid is given by relation (12):
Z
w00 ðB0 Þ ¼ w00 ðB0 ; DÞgðDÞ dD:
(12)
In Ref. [21], de Biasi and Devezas demonstrated
for the first time that in the case of nanometric size
particles, due to the thermal fluctuations, the
effective values of the anisotropy constant and of
the magnetization of each particle (as measured in
strong magnetic field) are smaller than the
corresponding values of the bulk material. As it
was shown in Ref. [21], the uniaxial anisotropy
constant, K and the magnetization of a particle,
M, have the following expressions in the presence
of a strong magnetic field:
K ¼ K m ½1 3x1 cothðxÞ þ 3x2 ,
(13)
1
M ¼ M S cothðxÞ .
x
(14)
In relations (13) and (14), Km is the uniaxial
anisotropy constant of the bulk material of
particles, MS is the saturation magnetization of
the bulk material of particles and x ¼ VM S B0 =kT.
Both Km and MS depend on temperature obeying
functions, which are specific to the material of
particles. In order to keep generality, here we
shall neglect the dependence on temperature of
Km and MS, assuming constant values both for
the uniaxial anisotropy constant and for the
saturation magnetization. However, the temperature range where the magnetic fluids are in fluid
state does not exceed the range 250–450 K and
within this range of temperature, the most of the
magnetic materials used in magnetic fluids are not
substantially changing their values of Km and MS.
The second amendment to the usual theory of
the ferromagnetic resonance of magnetic fluids is
the taking into account of the dependence on
particle sizes and on temperature of the anisotropy
constant and of the magnetization of the particles,
in accordance with relations (13) and (14). Thus, in
all above equations the anisotropy constant and
the magnetization of the particles are replaced
with their thermal averages, as given by relations
(13) and (14).
3. Results and discussions
In the experimental arrangement which is
characteristic to the ESR spectrometers, the
investigated sample is placed within a resonance
cavity. The frequency of the microwave field is
kept constant and the polarizing field is slowly
changed within a fixed range and during a settled
time interval. The ESR signal is proportional with
the imaginary part of the complex magnetic
susceptibility of the sample w00 . In the following
considerations the dependence on the polarizing
field of the imaginary part of the complex
magnetic susceptibility will designate the absorption line and its derivative will designate the
resonance line.
In order to investigate theoretically the thermal
and particle size distribution effects on the
ferromagnetic resonance of magnetic fluids, in
the first instance, using Eq. (11), the imaginary
part of the complex magnetic susceptibility,
w00 ðB0 ; DÞ was numerically computed for different
magnetic diameters of particles and at different
temperatures. The polarizing field was assumed
parallel to the oy-axis and the microwave field, h~
was assumed parallel to the ox-axis. The constants used in computations were M S ¼ 480 G,
K m ¼ 1:1 105 erg=cm3 , aR ¼ 0:03, T ¼ 300 K,
g ¼ 2 and the frequency of the microwave field,
f 0 ¼ 9060 MHz.
ARTICLE IN PRESS
C.N. Marin / Journal of Magnetism and Magnetic Materials 300 (2006) 397–406
3.1. System with particles having the same
magnetic diameter
In Fig. 1 are plotted some dependencies on the
polarizing field of the imaginary part of complex
magnetic susceptibility, w00 ðB0 ; DÞ. In each of these
dependencies the particles are assumed as having
the same magnetic diameter, D, and the volume
fraction of particles, Fm ðDÞ was considered as
being equal to unity. As it can be observed from
Fig. 1, the shape, the amplitude, the position of the
maximum and the width of w00 ðB0 ; DÞ depend both
on temperature and on the particle sizes.
The orientation of anisotropy axes generally
differ from one particle to another and the
dependence w00 ðB0 ; DÞ is an average of the contribution of all particles within the magnetic fluid.
In case of the particles which have the anisotropy axes parallel to the polarizing field, the
maximum of the absorption line, w00 ðB0 ; DÞ corresponds to the polarizing field denoted BII. In case
of the particles which have the anisotropy axes
perpendicular to the polarizing field, the maximum
of the absorption line, w00 ðB0 ; DÞ corresponds to
the polarizing field denoted B? . As results from
Eq. (7), B? BII ¼ 3K
M . Depending on the difference B? BII , the dependence w00 ðB0 ; DÞ can
2.0
1.5
401
exhibit: (a) one maximum if B? BII is small,
(b) one maximum and one shoulder or (c) two
maximums if B? BII is large.
Regarding the width of the dependence
w00 ðB0 ; DÞ, it is extrinsic due to the orientation
distribution of anisotropy axes of the particles,
being determined by dispersion of the resonance
field of individual particles, Bres. Based on the Van
Vleck’s method of moments [22], Eq. (15) gives the
line width, where Bres results from Eq. (7):
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
DB0 ¼ 2 dispðBres Þ ¼ 2ðhB2res i hBres i2 Þ1=2 . (15)
As a conclusion, the shape of the dependence
w00 ðB0 ; DÞ, the resonance field and the line width
depend on the ratio K/M and on the orientation
distribution function of the anisotropy axes of
particles, f ðyA ; jA ; B0 Þ. On the other hand, both
K/M and f ðyA ; jA ; B0 Þ depend on the ratio V/T,
where V is the magnetic volume of one particle and
T is the temperature of the system.
Based on the dependencies w00 ðB0 ; DÞ, in Fig. 2
are plotted the dependencies on the ratio V/T of
the magnetic field corresponding to the maximum
of w00 ðB0 ; DÞ, denoted Bmax and of the peak to peak
00
ðB0 ;DÞ
line width of the derivative dw dB
, denoted DB0 .
0
As one can be observe from Fig. 2, there are
intervals of V/T for which the dependence
DB0 ðV =TÞ increases with increasing V/T as well
as intervals of V/T for which DB0 ðV =TÞ decreases
with increasing V/T. From Eqs. (13) and (14) can
6
1.0
Bmax
Bmax [kG]
χ″
3.4
5
4
0.5
3
1
2.8
3.0
3.2
B0 [kG]
3.4
3.6
∆B0 [kG]
2.8
3.0
2
0.0
2.6
∆B0
3.2
0.5
0.0
Fig. 1. Absorption lines at different temperatures, for different
magnetic diameters of particles, as follows: (1) – D ¼ 3 nm;
T ¼ 400 K; (2) – D ¼ 3 nm; T ¼ 250 K; (3) – D ¼ 6 nm;
T ¼ 400 K; (4) – D ¼ 6 nm; T ¼ 250 K; (5) – D ¼ 16 nm;
T ¼ 400 K; (6) – D ¼ 16 nm; T ¼ 250 K.
0.3
1
10
100 200
V / T [cm3 / K] x 1022
Fig. 2. The dependencies of Bmax and of DB0 on the ratio
between the magnetic volume of a particle and temperature, V/T.
ARTICLE IN PRESS
402
C.N. Marin / Journal of Magnetism and Magnetic Materials 300 (2006) 397–406
be seen that by decreasing the ratio V/T, the ratio
K/M decreases, which (from Eqs. (15)) leads to a
decrease of the line width DB0 . On the other hand,
as results from Eq. (9), a decrease of the ratio V/T
leads to a broadening of the orientation distribution of the anisotropy axes of particles within the
system. Consequently, the line width DB0 is a
result of the competition between the two abovementioned mechanisms of extrinsic broadening.
Thus for some intervals of the parameter V/T, the
dependence DB0 ðV =TÞ can increase with increasing V/T, but for other intervals of the parameter
V/T, the dependence DB0 ðV =TÞ can decrease with
increasing V/T.
Regarding Bmax, one can observe from Fig. 2
that are intervals of V/T for which Bmax ðV =TÞ
increases with increasing V/T and intervals of V/T
for which Bmax ðV =TÞ decreases with increasing
V/T. Moreover, around the value of V =T ¼ 20 1022 cm3 K1 the dependence w00 ðB0 ; DÞ has two
maximums. This behaviour is a result of two
causes. The first is the dependence of the ratio
K/M on the ratio V/T and the second is the
dependence of the orientation distribution of the
anisotropy axes of particles on the ratio V/T. For
small values of V/T, the orientation distribution
function of the anisotropy axes of particles,
f ðyA ; jA ; B0 Þ is approximately constant and
BII ffi B? , resulting in narrow and symmetric
absorption lines. Increasing V/T, the orientation
distribution function of the anisotropy axes of
particles, f ðyA ; jA ; B0 Þ is narrowing and the
difference B? BII is increasing due to the
increase of the ratio K/M. This leads to asymmetric absorption lines having two maximums or
one maximum and one shoulder, depending on the
number of particles which have the anisotropy axis
parallel to the polarizing field. Also, because BII
decreases with increasing of V/T and B? increases
with increasing of V/T, depending on the number
of particles which have the anisotropy axis parallel
to the polarizing field, Bmax can increase or can
decrease with increasing V/T. In case of large
values of V/T, the majority of particles have the
anisotropy axis parallel to the polarizing field.
Consequently, Bmax is essentially determined by
BII, resulting in a decrease with increasing V/T
(see Fig. 2).
3.2. System with particles which obey a particle size
distribution
In order to take into account the particle size
distribution, the imaginary part of complex
magnetic susceptibility of the magnetic fluid was
numerically computed as the average:
w00 ðB0 Þ ¼
14
X
pi w00 ðB0 ; Di Þ.
(16)
i¼1
In relation (16), pi is the fraction of particles which
have the magnetic diameter Di and w00 ðB0 ; Di Þ is the
contribution at the imaginary part of susceptibility
due to the particles which have the magnetic
diameter Di (as computed with Eq. (11)). In the
present computations the index i ¼ 1 corresponds
to a magnetic diameter of particles, D1 ¼ 3 nm, the
index i ¼ 2 corresponds to a magnetic diameter of
particles, D2 ¼ 4 nm and so on up to the index
i ¼ 14, which corresponds to a magnetic diameter
of particles, D14 ¼ 16 nm. In order to compute pi is
assumed that the magnetic diameter of the
particles obeys a Gaussian distribution, which
was normalized to unity over the range
2.5–16.5 nm. This means that the particles taken
into account are only in the above-mentioned
interval. The normalization is necessary because in
real magnetic fluid samples, the magnetic diameter
of particles has not all the values between zero and
infinity, but it ranges in a certain interval depending
P14from a sample to another. Therefore, the sum
i¼1 pi ¼ 1, where
Z
Di þ0:5
pi ¼
Di 0:5
w m2 1
pffiffiffiffiffiffi
exp 0:5
dw
s
2psN
(17)
and
Z
D14 þ0:5
N¼
D1 0:5
u m2 exp 0:5
du.
s
(18)
In relations (16)–(18) the magnetic diameter of
particles are in nanometres and the parameters m
and s are respectively the mean value of the
magnetic diameter of particles and the standard
deviation of the particle size distribution.
ARTICLE IN PRESS
C.N. Marin / Journal of Magnetism and Magnetic Materials 300 (2006) 397–406
3.2.1. The effect of the mean value of particle
diameters, m on the resonant behaviour of magnetic
fluids
Based on the numerical computations, w00 ðB0 ; DÞ
for different values of m, the resonance lines are
obtained using Eq. (16). In Fig. 3 are presented
dw00
four dependencies of the derivative dB
on the
0
polarizing field (also called the resonance lines)
which were computed for magnetic fluids having
the same value of s ¼ 0:5 nm, but different values
of m, at the temperature T ¼ 300 K. One can
observe from Fig. 3 that the shape of the
resonance line strongly depends on m. With
increasing m the resonance line changes from a
simple and symmetric line to a composite and
asymmetric one. By further increasing m, the
amplitude of the resonance line corresponding to
high magnetic field values decreases and the
amplitude of the resonance line corresponding to
small magnetic field values increases. This behaviour allows us to assert that the small field region
of the resonance line is due to the large size
particles within the magnetic fluid and the high
field region of the resonance line is due to the small
size particles.
The resonance field can be determined in two
ways: (a) as the value of the polarizing field, B(0),
at which the resonance line becomes zero and (b)
as the average, BPP ¼ ðB1 þ B2 Þ=2, where B1 is
the polarizing field corresponding to the first
5.00x10-4
4
dχ″
dB0
2.50x10-4
1
3
0.00
2
-2.50x10-4
1 4
2
3
-5.00x10-4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
B0 [kG]
Fig. 3. Resonance lines of magnetic fluids having particles with
the same standard deviation of the particle size distribution,
s ¼ 0:5 nm, but different mean values of the magnetic diameter
of particles, m, at the temperature T ¼ 300 K.
403
maximum and B2 is the polarizing field corresponding to the last minimum of the resonance
line. Because in the case of composite and
asymmetric lines, B(0) can have more than one
value, for this situations it is useful to compute the
resonance line as BPP ¼ ðB1 þ B2 Þ=2. In the
following considerations, the resonance field is
computed as the average, BPP ¼ ðB1 þ B2 Þ=2. As
presented in Fig. 3, the resonance field, BPP
decreases with increasing m.
The resonance line width, DB was computed as
DB ¼ B2 2B1 , where B1 is the polarizing field
corresponding to the first maximum and B2 is the
polarizing field corresponding to the last minimum
of the resonance line (see Fig. 3). One can observe
from Fig. 3 that by increasing m, the resonance line
width increases.
An explanation of the effect, which m has on the
resonant behaviour of magnetic fluids is based on
the results presented in Figs. 1 and 2. From these
figures one can observe that by increasing the
size of particles, the dependence w00 ðB0 ; DÞ is
broadening, shifts towards the low magnetic field
region and its amplitude is increasing. Thus,
increasing the mean magnetic diameter of particles, the resonance line of magnetic fluids is
changing its shape from a simple, narrow and
symmetric line to a composite, broad and asymmetric one and shifts towards small magnetic field
region.
3.2.2. The effect of the standard deviation of
particle size distribution, s, on the resonant
behaviour of magnetic fluids
In order to investigate theoretically the effect of
s on the resonant behaviour of magnetic fluids, the
resonance lines were computed for magnetic fluids
having the same mean values of m ¼ 9 nm, but
different values of s, at temperature T ¼ 300 K. In
Fig. 4 are presented three such resonance lines and
one can be observed that the resonance lines are
composite and asymmetric. As presented in Fig. 4,
the resonance field, BPP decreases with increasing
s, the resonance line shifting thus towards
smaller values of the polarizing field. In addition,
from Fig. 4, one can observe that the peak-topeak resonance line width, DB increases with
increasing s.
ARTICLE IN PRESS
C.N. Marin / Journal of Magnetism and Magnetic Materials 300 (2006) 397–406
dχ″
dB0
0.0
-2.0x10-4
-4.0x10-4
2.6
2.8
3.0
3.2
B0 [kG]
3.4
3.6
3.8
Fig. 4. Resonance lines of magnetic fluids having the same
mean value of the magnetic diameter of particles, m ¼ 9 nm, but
different standard deviations, s, at the temperature T ¼ 300 K.
An explanation of the numerical results obtained in this section, regarding the effect of which
s has on the resonant behaviour of magnetic fluids
is based on the results presented in Figs. 1 and 2.
From these figures, one can see that the large
particles give a larger contribution to the susceptibility of the magnetic fluid than the small
particles. The large particles have a dependence
w00 ðB0 ; DÞ broader and with a higher amplitude
than the small particles. Also, from Figs. 1 and 2,
one can observe that by increasing the size of
particles, the dependence w00 ðB0 ; DÞ shifts towards a
lower magnetic field region. Thus, increasing the
number of large particles in the magnetic fluid, by
increasing s, the resonance line of the magnetic
fluid is broadened and is shifted towards a lower
magnetic field region.
3.2.3. The effect of the temperature on the resonant
behaviour of magnetic fluids
Theoretical investigations of the thermal effect
on the ferromagnetic resonance of magnetic fluids,
is based as in previous sections on the numerical
computations of the imaginary part of the complex
magnetic susceptibility, w00 ðB0 ; DÞ, for different
magnetic diameters of particles and different
temperatures. The resonance lines are obtained
using Eq. (16).
In Fig. 5 are plotted some ferromagnetic
resonance lines, at different temperatures, for
magnetic fluids having the same values of s, but
different mean magnetic diameters of particles, m.
One can observe that the ferromagnetic resonance
line becomes more symmetric by increasing the
temperature and by decreasing m.
The dependence on temperature of the resonance field is plotted in Fig. 6 for a magnetic fluid
having particles with m ¼ 5 nm, as well as for a
magnetic fluid having particles with m ¼ 9 nm. In
the case of the magnetic fluid having particles with
m ¼ 5 nm, the resonance field decreases with
increasing temperature, whilst in the case of the
magnetic fluid having m ¼ 9 nm, the resonance
field increases with increasing the temperature.
The majority of the reported experimental
results show that the resonance field of magnetic
fluids increases by increasing the temperature
[12–14], being in qualitative agreement with the
theoretical dependence obtained for the magnetic
fluid having particles with m ¼ 9 nm (see Fig. 6).
Currently, the only experimental results in which
the resonance field decreases with increasing
temperature are reported in Ref. [23]. However,
complete experimental results regarding the particle size distribution effect on the resonant behaviour of magnetic fluids at different temperatures
are lacking. It is true that many researchers
investigated the temperature effect on the resonant
behaviour of magnetic fluids, but the results are
not accompanied by a necessary characterization
of the samples, i.e. a colloidal stabilization analysis
2.0x10-4
1: T = 250 K
2: T = 300 K
3: T = 450 K
1
2
3
0.0
dχ″
2.0x10-4
dB0
404
3 2 1
3
2
-2.0x10-4
1
µ = 5 nm ; σ = 1nm
µ = 9 nm ; σ = 1nm
-4.0x10-4
1
2
2.6
2.8
3.0
3.2
3.4
3
3.6
3.8
B0 [kG]
Fig. 5. Resonance lines of magnetic fluids having the same
standard deviation of the particle size distribution, but different
mean value of the magnetic diameter of particles, at different
temperatures.
ARTICLE IN PRESS
C.N. Marin / Journal of Magnetism and Magnetic Materials 300 (2006) 397–406
3.30
∆B µ = 5nm σ = 1nm
∆B µ = 9nm σ = 1nm
Bpp [kG]
Bpp
Bpp
3.25
3.20
∆B [kG]
3.15
0.6
0.4
0.2
250
300
350
T [K]
400
450
Fig. 6. The dependence on temperature of the resonance field
and of the resonance line width for two magnetic fluids having
different mean magnetic diameter of particles.
apart from the particle size distribution. Hereby,
we cannot be sure that the reported experimental
results are not influenced by the particle agglomerations. As is known from theoretical models, a
decrease in temperature favours the occurrence of
particle agglomerations within magnetic fluids (see
for example Refs. [24–26]). Also, as it is demonstrated in Ref. [27], increasing the degree of
particle agglomerations leads to a decrease of the
resonance field. Thus, the increase of the resonance
field by increasing the temperature, as experimentally reported in different articles, can be a result
of particle agglomerations.
An explanation of the numerical results obtained in this section, regarding the effect of which
the temperature has on the resonance field of
magnetic fluids is based on the results presented in
Figs. 1 and 2. From these figures one can see that
in case of systems for which V =To10 1022 cm3 K1 (i.e. small particles and/or high
temperature), increasing the temperature leads to
a decrease of the magnetic field corresponding to
the maximum of the absorption line (which is
equal to the resonance field). In our case,
for particles having the magnetic diameter,
D ¼ 5 nm, at T ¼ 250 K, one obtains V =T ¼
2:6 1022 cm3 K1 . Therefore for the magnetic
fluid with m ¼ 5 nm and s ¼ 1 nm, the resonance
field of the majority of individual particles
405
decreases with increasing temperature (see Fig. 2),
resulting in a corresponding decrease of the
resonance field of the magnetic fluid. One can
also observe from Fig. 2, that for V =T414
1022 cm3 K1 , Bmax increases with increasing
temperature, as was obtained in the case of the
magnetic fluid with m ¼ 9 nm and s ¼ 1 nm (see
Fig. 6). In this case, for particles having the
magnetic diameter, D ¼ 9 nm, at T ¼ 250 K, one
obtains V =T ¼ 15:2 1022 cm3 K1 . Therefore
for the magnetic fluid with m ¼ 9 nm and
s ¼ 1 nm, the resonance field of the majority of
individual particles increases with increasing temperature, resulting in a corresponding increase of the
resonance field of the magnetic fluid (see Fig. 2).
The dependence on temperature of the resonance
line width is plotted in Fig. 6 for the magnetic fluids
having particles with m ¼ 5 nm, and m ¼ 9 nm,
respectively. One can observe that the line width
decreases with increasing temperature.
In the case of magnetic fluids the broadening of
the ferromagnetic resonance line is extrinsic, originating in the polydispersity of magnetic fluids. As
can be observed from Fig. 2,00 the peak-to-peak line
ðB0 ;DÞ
width of the derivative dw dB
decreases with
0
increasing temperature, for particles for which
V =To67 1022 cm3 K1 . In case of magnetite
particles, the value V =T ¼ 67 1022 cm3 K1 at
T ¼ 250 K corresponds to a magnetic diameter of
particles, D ¼ 15 nm. Therefore, because both in the
case of the magnetic fluid with m ¼ 5 nm and s ¼
1 nm and in the case of the magnetic fluid with m ¼
9 nm and s ¼ 1 nm, the particles fulfil the condition
V =To67 1022 cm3 K1 , the line width decreases
with increasing temperature, as shown in Fig. 6. The
magnetic diameter of particles within magnetic
fluids are smaller than 15 nm (fulfilling the condition
V =To67 1022 cm3 K1 ), hereby in all reported
experimental results [12–15] the line width decreases
with increasing temperature, in qualitative agreement with the theoretical results obtained by means
of the present model.
4. Conclusions
The thermal and the particle size distribution
effects on the ferromagnetic resonance of diluted
ARTICLE IN PRESS
406
C.N. Marin / Journal of Magnetism and Magnetic Materials 300 (2006) 397–406
magnetic fluids were theoretically investigated,
neglecting possible effects due to the viscosity of
the carrier liquid. The model originates from the
usual theory of the ferromagnetic resonance of
single-domain particle systems, which was modified in order to take into account the orientation
mobility of the particles within the fluid and the
finite size of magnetic particles.
Under these circumstances the shape of the
resonance line, the resonance field and the line
width of magnetic fluids are found to be strongly
affected by the temperature and by the particle size
distribution.
References
[1] J.-C. Bacri, R. Perzynski, in: B. Berkovski, V. Bashtovoy
(Eds.), Magnetic Fluids and Applications Handbook,
Begell House Inc., New York, 1996.
[2] P.C. Fannin, C.N. Marin, V. Socoliuc, G.M. Istrătucă,
A.T. Giannitsis, J. Phys. D: Appl. Phys. 36 (2003) 1227.
[3] P.C. Fannin, C.N. Marin, V. Socoliuc, G.M. Istrătucă, J.
Magn. Magn. Mater. 284 (2004) 104.
[4] I. Mălăescu, J. Optoelectron. Adv. Mater. 5 (2003) 234.
[5] P.C. Fannin, I. Mălăescu, C.N. Marin, J. Magn. Magn.
Mater. 289 (2005) 162.
[6] W.S.D. Folly, R.S. de Biasi, Brazilian J. Phys. 31 (2001)
398.
[7] P.C. Fannin, B.K.P. Scaife, A.T. Giannitsis, S.W. Charles,
J. Phys. D: Appl. Phys. 35 (2002) 1305.
[8] S.M. Aharoni, M.H. Litt, J. Appl. Phys. 42 (1971) 352.
[9] V.K. Sharma, F. Waldner, J. Appl. Phys. 48 (1977) 4298.
[10] M.D. Sastry, Y. Babu, P.S. Goyal, R.V. Mehta, R.V.
Upadhyay, D. Srinivas, J. Magn. Magn. Mater. 149 (1995)
64.
[11] I. Hrianca, I. Mălăescu, F. Claici, C.N. Marin, J. Magn.
Magn. Mater. 201 (1999) 126.
[12] P.C. Morais, G.R.R. Gonc- alves, A.F. Bakuzis, K. Skeff
Neto, F. Pelegrini, J. Magn. Magn. Mater. 225 (2001)
84.
[13] J.M. Patel, S.P. Vaidya, R.V. Mehta, J. Magn. Magn.
Mater. 65 (1987) 273.
[14] F. Gazeau, J.-C. Bacri, F. Gendron, R. Perzynski, Yu.L.
Raikher, V.I. Stepanov, E. Dubois, J. Magn. Magn.
Mater. 186 (1998) 175.
[15] P.C. Morais, M.C.F.L. Lara, A.L. Tronconi, F.A.
Tourinho, A.R. Pereira, F. Pelegrini, J. Appl. Phys. 79
(1996) 7931.
[16] Yu.L. Raikher, M.I. Shliomis, Adv. Chem. Phys. 87 (1994)
595 (Chapter 8).
[17] Yu.L. Raikher, V.I. Stepanov, Phys. Rev. B 50 (1994)
6250.
[18] Yu.L. Raikher, V.I. Stepanov, J. Magn. Magn. Mater. 149
(1995) 34.
[19] E.P. Valstyn, J.P. Hanton, A.H. Morrish, Phys. Rev. 128
(1962) 2078.
[20] U. Netzelmann, J. Appl. Phys. 68 (1990) 1800.
[21] R.S. de Biasi, T.C. Devezas, J. Appl. Phys. 49 (1978)
2466.
[22] J.H. Van Vleck, Phys. Rev. 74 (1948) 1168.
[23] G.R.R. Gonc- alves, A.R. Pereira, A.F. Bakuzis, K. Skeff
Neto, F. Pelegrini, P.C. Morais, J. Magn. Magn. Mater.
225 (2001) 84.
[24] P.C. de Gennes, P.A. Pincus, Phys. Cond. Matt. 11 (1970)
189.
[25] P.C. Jordan, Mol. Phys. 25 (1973) 961.
[26] A.Yu. Zubarev, L.Yu. Iskakova, JETP 80 (1995) 857.
[27] C.N. Marin, J. Magn. Magn. Mater. 250 (2002) 197.