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A harmonic oscillator model of the complex ac susceptibility of a magnetic fluid
Journal of Magnetism and Magnetic Materials 149 (1995) 29-33
i Journal of magnetic , ~
ELSEVIER
malerlals
A harmonic oscillator model of the complex ac susceptibility of a magnetic fluid P.C. F a n n i n a,* S . W . Charles b T. Relihan a a Department of Microelectronics and Electrical Engineering, Trinity College, Dublin 2, Ireland b Department of Chemistry, University College of North Wales, Bangor LL57 2UW, UK Abstract Plots of the real and imaginary components, X'(to) and X"(co), of the complex ac susceptibility of a distribution of magnetic particles (magnetic fluid) have been generated using a classical harmonic oscillator model. To achieve a better fit between experimental data and the model, the model was extended to include F distribution functions to represent the distribution of the particle sizes and their magnetic properties, After-effect functions of the magnetic fluid are also determined.
1. Introduction In terms of the classical oscillator model [1], the complex, frequency-dependent magnetic susceptibility, X(co), where X ( c o ) = X ' ( c o ) - i x " ( c o ) , can be written in the form X(co) = Xo/( 1 - co2/co2 + icoa/coo2),
(1)
with X'(co) = Xo[( 1 - co2/co~)/(( 1 - co2/cooZ)2
where /xo is the permeability of free space, 7 is the gyro magnetic ratio and H A is the anisotropy field within the particle. By means of appropriate processing of the complex susceptibility data, information on the magnetisation decay or after-effect function, b(t), of the sample can be determined. This can be realised by means of the fluctuation dissipation theorem [2] which relates the normalised aftereffect function, b(t)/b(O), and the inverse Fourier transform (IFT) of (X"(co)/co) in the following manner, with
b(t)/b(O) (2/'rrXo) R e £ ~ ( X " ( c o ) / c o )
and
x"(co) =
exp(icot) do),
(5)
where Re denotes the real part.
[(
-
voo ¢ +
(3) where Xo is the susceptibility value at to = 0, w o is the resonant frequency and a is the damping coefficient of the classical oscillator having the dimension of s - ~. The existence of resonance is indicated by the real part of the susceptibility going negative and, in the case of a uniaxial particle, is characterised by the precession of the magnetic moment, rap, about its axis of easy magnetisation with an angular frequency coo = THa/Xo,
* Corresponding author. Fax: + 353-1-677 2442.
(4)
2. Experimental Measurements of the real and imaginary components of the complex susceptibility, over the frequency range 300 kHz to 3 GHz, were obtained for a colloidal dispersion of manganese ferrite in isopar-m and are shown in Fig. 1. The sample had an approximate average particle size of 9 nm and a M s of 0.03 T. These data were measured as a function of log(f) Hz, however in order to determine its time domain equivalent it was necessary to remeasure the data in 1 MHz steps over the approximate frequency range 1 MHz to 1 GHz [3]. These data are represented in Fig. 2(a) plotted against log(co/coo). Theoretical complex susceptibility components, X'(co) and X"(co), were generated by Eqs. (2) and (3), and then fitted to the experimental data as shown in Fig. 2(b).
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(95)00323-1
30
P.C Fannin et al. /Journal of Magnetism and Magnetic Materials 149 (1995) 29-33
0.2
0.15
5MHz
o.1
IJ
0.05 ~-
Y.,(~)
L ......... "
°i
.0.05 I 1o5
65MH~~<.
I0 6
10 7
IO s
!0~o
10 9
tog{f Hz) Fig. 1. Plot of X'(to) and X"(to) against log(f) Hz.
3. Results and discussion
For the experimental data shown in Fig. 1, X'(to) goes negative at a frequency of approximately 65 MHz, indicating an apparent resonance at a frequency lower than the theoretical estimates [4], whilst the peak in X"(oJ) occurs at a frequency, fmax, of approximately 35 MHz. In Fig. 2, two curves are plotted: (a), of the experimen-
tal data for X'(to) and X'(to), and (b), generated by Eqs. (2) and (3) using the parameter tOo/(2"rr) = 65 MHz and a damping parameter of a = 7 x 108 rad s-1. Not surprisingly, the fit is not good since no regard has been paid to the presence of a distribution of particle sizes and effective magnetic anisotropy constant. The latter is a combination of magnetocrystalline and shape anisotropies. In Fig. 3, again two curves are plotted: (a), of the
"''"
"~ /~Go) 0.8
b
i:,:"""'",,
o.6
0.4
~;C~I
o.~ o
2
i
i1
i
i
to g Fig. 2. Plot of X'(W) and X"(t°) against log(to/to0) for (a) experimental and (b) theoretical data.
P.C. Fannin et al. /Journal of Magnetism and Magnetic Materials 149 (1995) 29-33
31
1 "'''--
.
,
!
II
i
-0.2 .2
-1.5
-I
-0.5
0
0.5
1
log Fig. 3. Plot of X'(~o) and X"(co) against log(oJ/~o o) for (a) experimental and (b) theoretical data generated with F distribution of ~o~.
experimental data, and (b), generated as before but with the inclusion of a F distribution function of to o as given by ~ ( t o o ) = (too/to~) (~- i) exp( - t o o / t o ~ ) / ( to~ F ( / 3 ) ) .
(6)
t
Comparison of the curves shows a much more satisfactory fit than that of Fig. 2, using the fitting parameters (to~/2"n-)= 70 MHz and / 3 = 4 . 2 5 . Ten values of (too/2'tr) in the range 70-700 MHz were used. Low-temperature studies of the particles used in this study [5] indicate that the particles possess uniaxial
a
....
b 0.6
w
),2_~,)
0.4
0.2 / ....°..--'"
o
°2-2
-;.5
J
-~
-d.s
;
i
o.~
tog( Fig. 4. Plot of X'(co) and X"(co) against log(co/co 0) for (a) experimental and (b) theoretical data generated with F distribution of damping parameter o~.
32
P.C. Fannin et al. /Journal of Magnetism and Magnetic Materials 149 (1995) 29-33 1 0.9
a
b(t) 0.,
b
(o) "7 0.6
0.5
°"I i 0'3I 0.2 0.I 0
0
°'~,
0.5
i
J
i
1,5
2
2.5
3.5
4
4.5
fS.
5 xlO-S
Fig. 5. Plot of decay functions against t (s) corresponding to the data of Fig. 2.
anisotropy. This probably arises predominantly from a shape anisotropy rather than the magnetocrystalline anisotropy, which should be cubic. Thus if we are justified in treating too as a Larmor precession then one can write it in terms of Eq. (4). Thus the F distribution function of too arises from a distribution of HA, which in turn arises from a distribution of shape. However, no panicle volume distribution is considered in this analysis. Even though the fit of
the experimental and theoretical analyses is good, this seems difficult to justify since it implies that the damping parameter a is a constant independent of panicle size. In view of this, a fit between the experimental data and a generated curve using Eqs. (2) and (3) with a F distribution function of a was investigated. The damping parameter a can be considered to depend on the thermally induced rotational mobility of the magnetic moment either
0.9-
o~1 0
I 0.5
a
.
3
3.5
"~-.... 1
1.5
2
2.5
t
4
4.5
S.
Fig. 6. Plot of decay functions against t (s) corresponding to the data of Fig. 3.
xlO-'
P.C. Fannin et al. /Journal of Magnetism and Magnetic Materials 149 (1995) 29-33 by a Brownian rotation of panicles within a liquid of viscosity "1/ or by reorientation of the magnetic moment within the panicles. As was pointed out by Raikher and Shliomis [6], in order that it may be possible to speak of precession at all, it is obviously necessary that its period be small in comparison with the rotational diffusion time. To get a feel for the appropriate range of values of a, the damping parameter, the effective relaxation times, zeff, for a distribution of particles in a liquid of known viscosity can be generated from a knowledge of the particle size distribution and a constant effective anisotropy constant [7,8]. Fig. 4 compares the experimental data and the data generated using a F distribution of a. As in the case of a F distribution of too, the two curves compare favourably using the fitting parameters /3 = 4.25, tOo/27r = 65 MHz and a ' = 2.7 × 10 s rad s -1. Again, ten values of a in the range 2.7 × 108-2.7 × 109 rad s -1 were used. These latter values of a are similar in magnitude to values of 1/~-,fe from data described previously [7], but do not cover quite such a wide range of values. At present, the authors are in the process of extending the calculations to a wide range of values of a , to see if an even better fit between the experimental and theoretical data can be achieved. Figs. 5 and 6 compare the corresponding decay functions based on the experimental data and that generated with and without F distributions. The decay functions of Figs. 5(a) and (b) correspond to the processed data of Figs. 2(a) and (b), respectively; the obvious discrepancy between the two decay functions is quite apparent. In contrast, the improvement resulting from the inclusion of a /" distribution function is demonstrated by Figs. 6(a) and (b) with both decay functions being almost identical up to a time constant time. These results are very satisfactory and lead one to hope that when the theoretical treatment is further refined, very good agreement can be achieved.
33
4. Conclusions Plots of the real and imaginary components, X'(to) and ×"(to), of the complex ac susceptibility of a distribution of magnetic particles have been generated using a classical harmonic oscillator model. The plots obtained were fitted to experimental data for a magnetic fluid, measured over the approximate frequency range 1 MHz to 1 GHz, by adjusting the damping parameter a and resonant angular frequency too of the system. To achieve a better fit between the experimental data and the model, the model was extended to include F distribution functions to represent the distribution of the panicle sizes and their magnetic properties. These plots are presented and compared. By processing the experimental data of the ac complex susceptibility, after-effect functions of the magnetic fluid are determined and compared with the corresponding functions generated by the classical harmonic oscillator model. The improvement resulting from the inclusion of F distribution function is clearly demonstrated, with both decay functions being closely similar up to a time constant time. References [1] E.P. Gross, Phys. Rev. 97 (1955) 395. [2] B.K.P. Scaife, Principles of Dielectrics (London, Oxford Science Publications, 1989). [3] P.C. Fannin, S.W. Charles and T. Relihan, J. Phys. D: Appl. Phys. 27 (1994) 189. [4] P.C. Fannin, B.K.P. Scaife and S.W. Charles, Meas. Sci. Technology 3 (1992) 1014. [5] R.V. Upadhyay, K.J. Davies, S. Wells and S.W. Charles, J. Magn. Magn. Mater. 132 (1994) 249. [6] Y.L. Raikher and M.I. Shliomis, Sov. Phys. JETP 40 (1975) 526. [7] P.C. Fannin and S.W. Charles, J. Phys. D: Appl. Phys. 22 (1989) 187. [8] P.C. Fannin, J. Magn. Magn. Mater. 136 (1994) 49.
i Journal of magnetic , ~
ELSEVIER
malerlals
A harmonic oscillator model of the complex ac susceptibility of a magnetic fluid P.C. F a n n i n a,* S . W . Charles b T. Relihan a a Department of Microelectronics and Electrical Engineering, Trinity College, Dublin 2, Ireland b Department of Chemistry, University College of North Wales, Bangor LL57 2UW, UK Abstract Plots of the real and imaginary components, X'(to) and X"(co), of the complex ac susceptibility of a distribution of magnetic particles (magnetic fluid) have been generated using a classical harmonic oscillator model. To achieve a better fit between experimental data and the model, the model was extended to include F distribution functions to represent the distribution of the particle sizes and their magnetic properties, After-effect functions of the magnetic fluid are also determined.
1. Introduction In terms of the classical oscillator model [1], the complex, frequency-dependent magnetic susceptibility, X(co), where X ( c o ) = X ' ( c o ) - i x " ( c o ) , can be written in the form X(co) = Xo/( 1 - co2/co2 + icoa/coo2),
(1)
with X'(co) = Xo[( 1 - co2/co~)/(( 1 - co2/cooZ)2
where /xo is the permeability of free space, 7 is the gyro magnetic ratio and H A is the anisotropy field within the particle. By means of appropriate processing of the complex susceptibility data, information on the magnetisation decay or after-effect function, b(t), of the sample can be determined. This can be realised by means of the fluctuation dissipation theorem [2] which relates the normalised aftereffect function, b(t)/b(O), and the inverse Fourier transform (IFT) of (X"(co)/co) in the following manner, with
b(t)/b(O) (2/'rrXo) R e £ ~ ( X " ( c o ) / c o )
and
x"(co) =
exp(icot) do),
(5)
where Re denotes the real part.
[(
-
voo ¢ +
(3) where Xo is the susceptibility value at to = 0, w o is the resonant frequency and a is the damping coefficient of the classical oscillator having the dimension of s - ~. The existence of resonance is indicated by the real part of the susceptibility going negative and, in the case of a uniaxial particle, is characterised by the precession of the magnetic moment, rap, about its axis of easy magnetisation with an angular frequency coo = THa/Xo,
* Corresponding author. Fax: + 353-1-677 2442.
(4)
2. Experimental Measurements of the real and imaginary components of the complex susceptibility, over the frequency range 300 kHz to 3 GHz, were obtained for a colloidal dispersion of manganese ferrite in isopar-m and are shown in Fig. 1. The sample had an approximate average particle size of 9 nm and a M s of 0.03 T. These data were measured as a function of log(f) Hz, however in order to determine its time domain equivalent it was necessary to remeasure the data in 1 MHz steps over the approximate frequency range 1 MHz to 1 GHz [3]. These data are represented in Fig. 2(a) plotted against log(co/coo). Theoretical complex susceptibility components, X'(co) and X"(co), were generated by Eqs. (2) and (3), and then fitted to the experimental data as shown in Fig. 2(b).
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(95)00323-1
30
P.C Fannin et al. /Journal of Magnetism and Magnetic Materials 149 (1995) 29-33
0.2
0.15
5MHz
o.1
IJ
0.05 ~-
Y.,(~)
L ......... "
°i
.0.05 I 1o5
65MH~~<.
I0 6
10 7
IO s
!0~o
10 9
tog{f Hz) Fig. 1. Plot of X'(to) and X"(to) against log(f) Hz.
3. Results and discussion
For the experimental data shown in Fig. 1, X'(to) goes negative at a frequency of approximately 65 MHz, indicating an apparent resonance at a frequency lower than the theoretical estimates [4], whilst the peak in X"(oJ) occurs at a frequency, fmax, of approximately 35 MHz. In Fig. 2, two curves are plotted: (a), of the experimen-
tal data for X'(to) and X'(to), and (b), generated by Eqs. (2) and (3) using the parameter tOo/(2"rr) = 65 MHz and a damping parameter of a = 7 x 108 rad s-1. Not surprisingly, the fit is not good since no regard has been paid to the presence of a distribution of particle sizes and effective magnetic anisotropy constant. The latter is a combination of magnetocrystalline and shape anisotropies. In Fig. 3, again two curves are plotted: (a), of the
"''"
"~ /~Go) 0.8
b
i:,:"""'",,
o.6
0.4
~;C~I
o.~ o
2
i
i1
i
i
to g Fig. 2. Plot of X'(W) and X"(t°) against log(to/to0) for (a) experimental and (b) theoretical data.
P.C. Fannin et al. /Journal of Magnetism and Magnetic Materials 149 (1995) 29-33
31
1 "'''--
.
,
!
II
i
-0.2 .2
-1.5
-I
-0.5
0
0.5
1
log Fig. 3. Plot of X'(~o) and X"(co) against log(oJ/~o o) for (a) experimental and (b) theoretical data generated with F distribution of ~o~.
experimental data, and (b), generated as before but with the inclusion of a F distribution function of to o as given by ~ ( t o o ) = (too/to~) (~- i) exp( - t o o / t o ~ ) / ( to~ F ( / 3 ) ) .
(6)
t
Comparison of the curves shows a much more satisfactory fit than that of Fig. 2, using the fitting parameters (to~/2"n-)= 70 MHz and / 3 = 4 . 2 5 . Ten values of (too/2'tr) in the range 70-700 MHz were used. Low-temperature studies of the particles used in this study [5] indicate that the particles possess uniaxial
a
....
b 0.6
w
),2_~,)
0.4
0.2 / ....°..--'"
o
°2-2
-;.5
J
-~
-d.s
;
i
o.~
tog( Fig. 4. Plot of X'(co) and X"(co) against log(co/co 0) for (a) experimental and (b) theoretical data generated with F distribution of damping parameter o~.
32
P.C. Fannin et al. /Journal of Magnetism and Magnetic Materials 149 (1995) 29-33 1 0.9
a
b(t) 0.,
b
(o) "7 0.6
0.5
°"I i 0'3I 0.2 0.I 0
0
°'~,
0.5
i
J
i
1,5
2
2.5
3.5
4
4.5
fS.
5 xlO-S
Fig. 5. Plot of decay functions against t (s) corresponding to the data of Fig. 2.
anisotropy. This probably arises predominantly from a shape anisotropy rather than the magnetocrystalline anisotropy, which should be cubic. Thus if we are justified in treating too as a Larmor precession then one can write it in terms of Eq. (4). Thus the F distribution function of too arises from a distribution of HA, which in turn arises from a distribution of shape. However, no panicle volume distribution is considered in this analysis. Even though the fit of
the experimental and theoretical analyses is good, this seems difficult to justify since it implies that the damping parameter a is a constant independent of panicle size. In view of this, a fit between the experimental data and a generated curve using Eqs. (2) and (3) with a F distribution function of a was investigated. The damping parameter a can be considered to depend on the thermally induced rotational mobility of the magnetic moment either
0.9-
o~1 0
I 0.5
a
.
3
3.5
"~-.... 1
1.5
2
2.5
t
4
4.5
S.
Fig. 6. Plot of decay functions against t (s) corresponding to the data of Fig. 3.
xlO-'
P.C. Fannin et al. /Journal of Magnetism and Magnetic Materials 149 (1995) 29-33 by a Brownian rotation of panicles within a liquid of viscosity "1/ or by reorientation of the magnetic moment within the panicles. As was pointed out by Raikher and Shliomis [6], in order that it may be possible to speak of precession at all, it is obviously necessary that its period be small in comparison with the rotational diffusion time. To get a feel for the appropriate range of values of a, the damping parameter, the effective relaxation times, zeff, for a distribution of particles in a liquid of known viscosity can be generated from a knowledge of the particle size distribution and a constant effective anisotropy constant [7,8]. Fig. 4 compares the experimental data and the data generated using a F distribution of a. As in the case of a F distribution of too, the two curves compare favourably using the fitting parameters /3 = 4.25, tOo/27r = 65 MHz and a ' = 2.7 × 10 s rad s -1. Again, ten values of a in the range 2.7 × 108-2.7 × 109 rad s -1 were used. These latter values of a are similar in magnitude to values of 1/~-,fe from data described previously [7], but do not cover quite such a wide range of values. At present, the authors are in the process of extending the calculations to a wide range of values of a , to see if an even better fit between the experimental and theoretical data can be achieved. Figs. 5 and 6 compare the corresponding decay functions based on the experimental data and that generated with and without F distributions. The decay functions of Figs. 5(a) and (b) correspond to the processed data of Figs. 2(a) and (b), respectively; the obvious discrepancy between the two decay functions is quite apparent. In contrast, the improvement resulting from the inclusion of a /" distribution function is demonstrated by Figs. 6(a) and (b) with both decay functions being almost identical up to a time constant time. These results are very satisfactory and lead one to hope that when the theoretical treatment is further refined, very good agreement can be achieved.
33
4. Conclusions Plots of the real and imaginary components, X'(to) and ×"(to), of the complex ac susceptibility of a distribution of magnetic particles have been generated using a classical harmonic oscillator model. The plots obtained were fitted to experimental data for a magnetic fluid, measured over the approximate frequency range 1 MHz to 1 GHz, by adjusting the damping parameter a and resonant angular frequency too of the system. To achieve a better fit between the experimental data and the model, the model was extended to include F distribution functions to represent the distribution of the panicle sizes and their magnetic properties. These plots are presented and compared. By processing the experimental data of the ac complex susceptibility, after-effect functions of the magnetic fluid are determined and compared with the corresponding functions generated by the classical harmonic oscillator model. The improvement resulting from the inclusion of F distribution function is clearly demonstrated, with both decay functions being closely similar up to a time constant time. References [1] E.P. Gross, Phys. Rev. 97 (1955) 395. [2] B.K.P. Scaife, Principles of Dielectrics (London, Oxford Science Publications, 1989). [3] P.C. Fannin, S.W. Charles and T. Relihan, J. Phys. D: Appl. Phys. 27 (1994) 189. [4] P.C. Fannin, B.K.P. Scaife and S.W. Charles, Meas. Sci. Technology 3 (1992) 1014. [5] R.V. Upadhyay, K.J. Davies, S. Wells and S.W. Charles, J. Magn. Magn. Mater. 132 (1994) 249. [6] Y.L. Raikher and M.I. Shliomis, Sov. Phys. JETP 40 (1975) 526. [7] P.C. Fannin and S.W. Charles, J. Phys. D: Appl. Phys. 22 (1989) 187. [8] P.C. Fannin, J. Magn. Magn. Mater. 136 (1994) 49.