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The wavelength and concentration dependence of the magneto-dielectric anisotropy effect in magnetic fluids determined from magneto-optical measurements
Journal of Magnetism and Magnetic Materials 184 (1998) 375—386
The wavelength and concentration dependence of the magneto-dielectric anisotropy effect in magnetic fluids determined from magneto-optical measurements N.A. Yusuf*, A. Ramadan, H. Abu-Safia Department of Physics, Yarmouk University, Irbid, Jordan Received 25 August 1997; received in revised form 10 November 1997
Abstract The concentration and wavelength dependence of the magneto-dielectric anisotropy effect in magnetic fluids has been investigated. Our results show that the field-induced change in the dielectric constants for light polarized in the field direction increases with the applied field while that for light polarized perpendicular to the field decreases with the field. Furthermore, our results show that this field-induced change is linear with concentration. The results also show that the field-induced change, for light polarized in the field direction, increases with frequency till it reaches a peak at a given frequency and then decreases with further increase in frequency; while for light polarized perpendicular to the field, it decreases with frequency till it reaches a minimum at a given frequency and then increases with further increase in frequency. Moreover, our results yield a value of two for the magneto-dielectric anisotropy factor. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Anisotropy — magneto-dielectric; Magneto-dielectric behavior; Dielectric constant; Magneto-optics; Anisotropy — optical; Birefringence; Dichroism; Magnetic fluids
1. Introduction Magnetic fluids are colloidal suspension of ultra-fine ferro or ferrimagnetic particles dispersed in a suitable liquid carrier. The dielectric behavior of magnetic fluids changes with the application of
* Corresponding author. Tel.: #962 2 271115; fax: #962 2 274725.
an external magnetic field and with the relative orientation of the electric and magnetic fields. This effect is known as magneto-dielectric anisotropy effect. Magneto-dielectric effects in magnetic fluids have been investigated by many workers [1—13] both experimentally and theoretically. The experimental investigations were based on impedance measurement techniques where the magnetic fluid is placed in a capacitor. Measurements of the impedance parameters such as the modulus and
0304-8853/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 1 1 5 8 - X
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phase are carried out using a bridge or an RLC meter [1—8]. It is known that impedance measurement techniques suffer some serious disadvantages such as electrode effects, parasitic impedances, skin depth and accuracy-related problems. Very recently, Yusuf et al. [9] have determined the magneto-dielectric effect from magneto-optical measurements where the disadvantages suffered by conventional impedance measurement techniques are avoided. On the theoretical side, Monte Carlo simulations were used to calculate the magneto-dielectric anisotropy effect [10—13]. In most of the previous works the magnetodielectric anisotropy factor which is defined as e (H, u)!e (0, u) 0 g(H, u)" , e (0, u)!e (H, e) 0 M was either 2 or 1. Experimentally, Cotae [3] and Espurz et al. [7] have obtained the value of 1 for g(H, u) while Mailfert [6] has obtained a value of 2. Experimental results of Espurz et al. [7] show that the value of g(H, u) is 1 at low fields and 1.23 at high fields. Our previous results [9] on the magneto-dielectric anisotropy effect calculated from magneto-optical measurements yield a value of 1 for g(H, u). It is important to note that in our previous work the sample was considered as a two dimensional one through the use of n #n "2n , M 0 and k #k "2k where n and k are the real and , M 0 imaginary parts of the index of refraction of the sample, respectively. Fannin et al. [8] have obtained experimentally the value 2 for g(H, u) for a sample of a magnetic fluid with a thickness of 10 lm and particle size of &11 nm. Theoretically, Chantrell [10] and Mailfert et al. [6] have calculated the magneto-dielectric anisotropy effect in magnetic fluids and both obtained a value of 2 for g(H, u). It is worth mentioning that the samples considered in their works were three dimensional samples. Shobaki et al. [13] have numerically calculated the dielectric constant, e par, allel to the applied field and e in a direction M perpendicular to the field for two cases: for a two dimensional sample and a three dimensional sample, and showed that g(H, u) is 1 for the two dimensional case and is two for the three
dimensional case. It is worth mentioning that calculations of the optical anisotropy in magnetic fluids have been carried out using a similar procedure [23]. In this work the concentration and wavelength dependence of the magneto-dielectric anisotropy effect are investigated. The magneto-dielectric anisotropy effect is determined from magneto-optical measurements in a way similar to that reported in Ref. [9] but with the sample considered three dimensional through the use of n #2n "3n and , M 0 k #2k "3k . , M 0 2. Theoretical background The dielectric constant of a magnetic fluid in the absence of an external magnetic field exhibits no anisotropy due to the random orientation of the particles. Therefore, the dielectric constant seen by light with different states of polarization is the same. However, when a magnetic field is applied, orientation of particles and field-induced chain formation in the field direction take place leading to two different average lengths l in the field direc, tion and l perpendicular to the field; consequently, M the dielectric constant will exhibit some degree of anisotropy. The orientation function /(p, q) for a uniaxial single-domain particle suspended in a nonmagnetic liquid carrier under the application of an external magnetic field was treated by Hartmann and Mende [14,15] and Scholten [16,17] and is given by /(p, q)"m(q) f (p),
(1)
where m(q) represents the coupling between the magnetic moment of the particle and its easy axis and is usually expressed as
C
D
3 q1@2exp(q)!I(q) m(q)" !1, 2 4q I(q)
(2)
where
P
I(q)"
q1@2 exp(x2) dx 0
(3)
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and q"K»/k ¹. The function f (p) is given by B f (p)"[1!(3/p)¸(p)], (4) where ¸(P) is the Langevin function, and p" (k M »H/k ¹). 0 4B B In general, the dielectric constant of the magnetic fluid is complex and is written as e*"e@!ieA,
(5)
where e@ and eA are the real and imaginary parts of the dielectric constant, respectively. Furthermore, the index of refraction of the fluid is also complex and is written as n*"n!ik
(6)
where n and k are the real index of refraction and the extinction coefficient, respectively. Conventionally, the dielectric constant and the index of refraction are related by e*"n*2"n2!k2!2ink.
(7)
When an external magnetic field is applied, the dielectric constant e* becomes anisotropic thus exhibiting two different behaviors for light polarized parallel to the magnetic field e* and perpendicular , to the magnetic field e*. Similarly, the index of M refraction will exhibit two indices n* and n*. In this M , case Eq. (7) will be modified such that it incorporates this induced anisotropy. The modified equation will read: e*"e@ !ieA "n*2"n2 !k2 !2in k , , , , , , , , and
(8)
(9) e*"e@ !ieA "n*2"n2 !k2 !2in k . M M M M M M M M By equating the real and imaginary parts in Eq. (8) one gets: (10) e@ "n2 !k2 , , , , (11) eA "2n k . , , , Similarly, by equating the real and imaginary parts in Eq. (9) one gets e@ "n2 !k2 , M M M eA "2n k . M M M
(12) (13)
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Since the sample is a three dimensional one the refractive indices and the extinction coefficients both parallel and perpendicular to the magnetic field are related through the following relations: n #2n "3n , (14) , M 0 n !n "*n, (15) , M k #2k "3k , (16) , M 0 k !k "*k, (17) , M where n and k are the real index of refraction and 0 0 the extinction coefficient of the sample under zero field, respectively; and *n is the birefringence and *k is proportional to dichroism. Solving the sets of Eqs. (14) and (15) and Eqs. (16) and (17) simultaneously the refractive indices and the extinction coefficients both parallel and perpendicular to the magnetic field are obtained in terms of n , k , *n and *k as 0 0 n "n #2*n, (18) , 0 3 n "n !1*n, (19) M 0 3 k "k #2*k, (20) , 0 3 k "k !1*k. (21) M 0 3 Substituting Eqs. (14)—(17) in Eqs. (10)—(13) the values of the dielectric constant are obtained as e@ "(n #2*n)2!(k #2*k)2, (22) , 0 3 0 3 (23) eA "2(n #2*n)(k #2*k), , 0 3 0 3 e@ "(n !1*n)2!(k !1*k)2, (24) M 0 3 0 3 (25) eA "2(n !1*n)(k !1*k). M 0 3 0 3 The last equations, Eqs. (22)—(25) are the fundamental equations in calculating the dielectric constants (e* and e*) of the magnetic fluid and hence in M , determining the magneto-dielectric anisotropy effect. They show that magneto-dielectric effects may be calculated from optically measured physical quantities such as birefringence *n, dichroism (*A), the real refractive index n and the extinction coef0 ficient k (under zero magnetic field). It is important 0 to note that these physical quantities can be measured with excellent accuracies, e.g. values of *n and *k&10~6 can be easily measured [16—22].
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The frequency dependence of the index of refraction or the dielectric constant of matter (dispersion) is explained in terms of the distortion of the internal charge distribution under the influence of an applied external electric field. This distortion results in induced electric dipole moments leading to an induced electric polarization. Considering the charge carriers to be elastically bound, the electric field of the electromagnetic wave provides the driving force of the harmonic oscillator. Following standard procedures for solving the equation of a forced damped harmonic oscillator, the dielectric constant is given by q2N e"e # , 0 m(u2!u2#icu) 0
(26)
where e is the dielectric constant of free space, 0 q the charge of the charge carrier, N the number of charge carriers per unit volume, m the mass of the charge carrier, u the natural frequency of the 0 oscillator, u the frequency of the electromagnetic wave and c a damping parameter. For an elongated particle, the charge carriers can oscillate along the easy direction much easier than along other directions and with larger amplitudes leading to more electric polarization in the easy direction. Therefore, the contribution to the dielectric constant of charge carriers oscillating along the easy axis of the particle is the dominant one. Therefore, we suggest that the number of charge carriers contributing to the dielectric constant is direction dependant when a magnetic field is applied. This leads to the anisotropy in the dielectric constant. Consequently, the dielectric constant is given by q2N x e "e # , x 0 m(u2!u2#icu) 0
(26a)
q2N y e "e # , y 0 m(u2!u2#icu) 0
(27)
q2N z e "e # , z 0 m(u2!u2#icu) 0
(28)
where e , e and e are the dielectric constants seen x y z by the electromagnetic waves polarized in the x, y, and z directions, respectively; and N , N and x y
N are the number of charge carriers per unit volz ume oscillating in the x, y, and z directions, respectively. In the absence of the magnetic field N "N "N "1N, where N is the density of x y z 3 charge carriers. Therefore e , e , and e are all equal x y z to e (dielectric constant at zero magnetic field) and ;% given by q2N e "e # . (29) ;% 0 3m(u2!u2#icu) 0 When a magnetic field H is applied in the zdirection, the particles tend to align in the field direction with a probability P(H, ¹). This probability is equal to [(1#2/(p, q))/3] thus insuring that P(H, ¹)"1 at zero field or very high temperature 3 and is 1 at very high fields or low temperatures. The densities of the charge carriers are given by N "NP(H, ¹), (30) z (31) N "N "1N(1!P(H, ¹)). x y 2 Substituting the values of N and N in Eqs. (26) x z and (28) yields the following values for e and e : , M q2N P(H, ¹) e "e # , (32) , 0 m(u2!u2#icu) 0 q2N(1!P(H, ¹)) e "e # . (33) M 0 2m(u2!u2#icu) 0 Separating the real and imaginary parts in Eqs. (29), (32) and (33) yields: e@ "e #f (u)NP(H, ¹), , 0 1 e@ "e #1 f (u)N(1!P(H, ¹)), M 0 2 1 eA "f (u)NP(H, ¹), , 2 eA "1 f (u)N(1!P(H, ¹)), M 2 2 e@ "e #1 f (u)N, ;% 0 3 1 eA "1 f (u)N, ;% 3 2 where f (u) is given by 1 (u2!u2)q2 0 f (u)" 1 mM(u2!u2)2#(cu)2N 0
(34) (35) (36) (37) (38) (39)
(40)
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and f (u) is given by 2 cuq2 . f (u)" 2 mM(u2!u2)2#(cu)2N 0
(41)
It is important to note that these results presented in Eqs. (35)—(39) are in agreement with the assumption that the dielectric constant is proportional to the average projection length of the particles which has been previously used to calculate and explain the magneto-dielectric anisotropy effect in magnetic fluids [9,13]. The field-induced change in the real dielectric constants *e@ and *e@ and that in the imaginary M , dielectric constants *eA and *eA are then given by M , *e@ "f (u)N(P(H, ¹)!1), 3 , 1
(42)
*e@ "1 f (u)N(1!P(H, ¹)), M 2 1 3
(43)
*eA "f (u)N(P(H, ¹)!1), , 2 3
(44)
*eA "1 f (u)N(1!P(H, ¹)). M 2 2 3
(45)
These last equations describe the behavior of the magneto-dielectric anisotropy effect with field, concentration and frequency.
3. Experimental 3.1. Birefringence Measurements of the birefringence were undertaken using an optical arrangement consisting of a conventional quartz lamp, narrow band interference filters, a polarizer, sample cell, an analyzer, a compensator and a photomultiplier. The use of the compensator allows for a direct measurement of the relative retardation, d, of the components of the transmitted light through the sample with their planes of polarization parallel and perpendicular to the applied magnetic field, even when the sample exhibits dichroism. In this experiment, the plane of polarization was set at 45° with the magnetic field. With the field off, the reading of the compensator, l , was adjusted to give a minimum photo0 multiplier output; then with the field on the reading
379
l was readjusted to give a minimum once more. B From the difference (l !l ) and the calibration of B 0 the compensator the relative retardation d was obtained. 3.2. Dichroism Measurements of dichroism were undertaken using a set-up similar to that used for the birefringence experiment but with the compensator and analyzer removed. The transmission of light with planes of polarization parallel and perpendicular to the applied magnetic field I and I were measured. , M The values of the parallel and perpendicular dichroism *A and *A , are defined as the change in , M the absorbance of the sample as a result of applying a magnetic field and are given by *A "!ln(I /I ), , , 0
(46)
*A "!ln(I /I ), M M 0
(47)
where I is the transmitted intensity at zero field. 0 The dichroism *A is then given by *A"*A !*A . , M
(48)
The sample was contained in a standard 1 mm thick rectangular cell which in turn was placed between the poles of an electromagnet that is capable of producing fields up to 10 kOe. The samples used in this work are Fe O particle 3 4 magnetic fluids with Isopar-M as a liquid carrier. The average volume of the particles and the standard deviation were determined using scanning electron microscopy; their values are 75 and 2 A_ , respectively. The volumic fraction, e, of the samples ranged from 0.548 to 5.173]10~3, and the wavelengths used in this study ranged from 372.5 to 694.3 nm. The real part of the refractive index n of the 0 samples under zero magnetic field (for all wavelengths used) is determined by measuring the Brewster angle. Moreover, the extinction coefficient k of the samples (for all wavelengths used) is deter0 mined by measuring the transmission of light through cells of different optical paths filled with the sample. The cells are standard cells made of the same glass having the same thickness.
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4. Results The birefringence *n and dichroism *A, for different concentrations and at a wavelength j" 633 nm, versus magnetic field are presented in Fig. 1a and Fig. 1b, respectively. The results in the figures, similar to previously reported results, show that both *n and *A for a given concentration increases rapidly with the field for low fields then less rapidly for intermediate fields and tends to saturate at high fields. The birefringence *n and dichroism *A, for different wavelengths and e"0.548]10~3, versus magnetic field are presented in Fig. 2a and Fig. 2b, respectively. The results in the figures show that both *n and *A, at a given field, increase with wavelength reaching a peak at a given j then decreases with further increase with the wavelength. From the results in Fig. 2a and Fig. 2b, the birefringence *n and dichroism *A at measuring fields 500 and 2500 Oe are plotted versus the wavelength j in Fig. 3a and Fig. 3b, respectively. The results in the figures show that *n peaks at j"514 nm while *A peaks at j"488 nm. Furthermore, the results show that the position of the peak does not change with the applied magnetic field even though the width of the peak increases with the field. The real part of the refractive index n and the 0 imaginary part i.e., the extinction coefficient k , for 0 the samples, at zero field are plotted versus concentration in Fig. 4a and Fig. 4b, respectively. The results in the figures show that both n and 0 k increase almost linearly with concentration. This 0 is not too surprising since the range of concentration is not large. Furthermore, although the k line 0 practically passes through the origin reflecting the fact that Isopar-M has a zero extinction coefficient, the n line has an intercept corresponding to the 0 refractive index of Isopar-M. The measured real refractive index n and the 0 extinction coefficient k of the sample with e" 0 0.548]10~3 are plotted versus j in Fig. 5a and Fig. 5b, respectively. The results in the figures show that n decreases monotonically with j in the used 0 range of wavelengths, while they show that the extinction coefficient increases with j for low wavelengths exhibiting a peak at j"488 nm then decreases for higher wavelengths.
Fig. 1. Birefringence *n (a) and dichroism *A (b) for different concentrations versus magnetic field measured at ¹"300 K and j"63 nm.
N.A. Yusuf et al. / Journal of Magnetism and Magnetic Materials 184 (1998) 375—386
381
Fig. 2. Birefringence *n (a) and dichroism *A (b) for different wavelengths versus magnetic field measured at ¹"300 K. Fig. 3. Birefringence *n (a) and dichroism *A (b) at a measuring field of 500 Oe versus the wavelength j.
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Fig. 4. Real part of the index of refraction n (a); the extinction 0 coefficient k (b) versus concentration. 0
Fig. 5. Real part of the index of refraction n (a); the extinction 0 coefficient k (b) versus wavelength. 0
N.A. Yusuf et al. / Journal of Magnetism and Magnetic Materials 184 (1998) 375—386
Using Eqs. (22)—(25) and the measured values of *n, *A, n and k , the real and imaginary parts of 0 0 the dielectric constants for light polarized parallel and perpendicular to the magnetic field (e@ , eA ) and , , (e@ , eA ); and those at zero field (e@ , eA ) were cal;% ;% M M culated. The field-induced change in the real dielectric constants (*e@ "e@ !e@ ) and (*e@ "e@ !e@ ); ;% M M ;% , , and that in the imaginary dielectric constants (*eA "eA !eA ) and (*eA "eA !eA ), for samples ;% M M ;% , , with different concentration and at j"633 nm, are plotted versus field in Fig. 6a and Fig. 6b, respectively. The results show that the changes in the real and imaginary parts of the dielectric constant for light polarized in the field direction increase with the field while those for light polarized normal to the field decrease with the field. These results are in good agreement with previously reported results [1—13]. Furthermore, the results show that the value of the magneto-dielectric anisotropy factor at any given field and for any given concentration, in agreement with previous works [6,8,10,13], is two. The field-induced changes in the real dielectric constants *e@ and *e@ and that in the imaginary M , dielectric constants *eA and *eA measured at M , j"633 nm are plotted versus concentration in Fig. 7a and Fig. 7b, respectively. The results in the figures show that this change with concentration is linear in the studied range, and also show that the magneto-dielectric anisotropy factor at any given field and for any given concentration is again two. The field-induced changes in the real dielectric constants *e@ and *e@ and those in the imaginary M , dielectric constants *eA and *eA for the sample M , with e"0.548]10~3 and at selected measuring fields are plotted versus wavelength j in Fig. 8a and Fig. 8b, respectively. The results in the figures show that both *e@ and *eA increase with j till each , , reaches a maximum at a given wavelength j then .!9 decreases for longer wavelengths. However, *e@ M and *eA both decreases with j till each reaches M a minimum at a given wavelength j then in.*/ creases for longer wavelengths. Furthermore, the wavelengths at which the maximum in *e@ and the , minimum in *e@ (real part) occur are the same; and M those wavelengths corresponding to the maximum in *eA minimum in *eA (imaginary part) are also M , the same though different from those corresponding to the real part. For the real part the maximum
383
Fig. 6. Field-induced changes *e@ and *e@ (a); and *eA and *eA , M , M (b) at j"633 nm versus magnetic field.
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Fig. 7. Field-induced changes *e@ and *e@ (a); and *eA and *eA , M , M (b) at different fields and j"633 nm versus concentration.
Fig. 8. Field-induced changes *e@ and *e@ (a); and *eA and *eA , M , M (b) versus wavelength j at selected fields.
N.A. Yusuf et al. / Journal of Magnetism and Magnetic Materials 184 (1998) 375—386
in *e@ and the minimum in *e@ occur at j" M , 514 nm while for the imaginary part they occur at j"488 nm.
5. Discussion The behavior of birefringence and dichroism with field and concentration have been fully explained previously [20—22]. Therefore, we only discuss the magneto-dielectric effect presented in this work. The magneto-dielectric anisotropy effect is mainly attributed to the orientation of non-spherical particles and to the field-induced chain formation in the field direction. For a given frequency u and at a given temperature ¹, the probability P(H, ¹) increases with the field and as can be seen from Eqs. (42) and (44) the field-induced change in the dielectric constants for light polarized parallel to the field *e@ and *eA , , increases with the field. However, the field-induced change in the constants for light polarized perpendicular to the field *e@ and *eA decreases with the M M field. Furthermore, using Eqs. (42) and (43) and Eqs. (44) and (45) one can see that the magnetodielectric anisotropy factor g(H, u) equals 2. Looking at our results presented in Fig. 6a and Fig. 6b one finds out an excellent agreement between the results and the theory. Again for a given frequency u and a given applied field and at a given temperature, the functions f (u) and f (u) and the probability P(H, ¹) are 1 2 constants. Therefore, the field-induced changes in the dielectric constant are proportional to the density of charge carriers N with positive slopes for the parallel components and negative slopes for the perpendicular components. Furthermore, the ratio of the slopes for the two cases is (!2). Noting that N is proportional to the number of particles per unit volume in the fluid, then the field-induced changes in the dielectric constant are proportional to the concentration of the sample. Our results in Fig. 7a and Fig. 7b show a behavior consistent with the prediction of the theory. From Eqs. (42) and (43) one sees that both *e@ , and *e@ have the same dependence on the freM quency of the electromagnetic waves and thus should exhibit their optimal conditions, i.e. the
385
maximum in *e@ and the minimum in *e@ at the M , same frequency. Similarly *eA and *eA have the M , same frequency dependence and therefore the maximam in *eA and the minimum *eA occur at M , the same frequency. However, since the functional dependence on frequency for the parallel and perpendicular components of the dielectric constant are not the same, their optimal conditions occur at two different frequencies. Our results presented in Fig. 8a and Fig. 8b show a behavior similar to that predicted by the theory.
6. Conclusions The magneto-dielectric anisotropy effect in magnetic fluids has been determined from magnetooptical measurements. The field, concentration and wavelength dependence of the effect have been investigated. Our results showed that the field-induced change in both the real and imaginary parts of the dielectric constant for light polarized parallel to the field increase with the field while those for light polarized perpendicular to the field decrease with the field. Furthermore, our results show that the field-induced changes vary linearly with the concentration of the sample with a positive slope for the parallel components and negative slope for the perpendicular components. Moreover, our results show that the field-induced change in the parallel component increases with frequency reaching a maximum at a given frequency then it decreases with further increase of frequency; while the change in the perpendicular component decreases with frequency reaching a minimum and then increases with further increase of frequency. Our results have been explained in terms of the theory of dispersion of electromagnetic waves in matter with the assumption that the density of charge carriers oscillating in a given direction is proportional to the number of particles oriented in that direction.
References [1] O. Derriche, L. Jorat, G. Noyel, J. Monin, J. Magn. Magn. Mater. 102 (1991) 155. [2] A. Colteu, J. Magn. Magn. Mater. 39 (1983) 88.
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[3] A. Cotae, J. Magn. Magn. Mater. 39 (1983) 85. [4] P. Kopcansky, J. Cernak, P. Macko, D. Spisak, K. Marton, J. Phys. D 22 (1989) 1410. [5] B.Z. Kaplan, D.M. Jacobson, Nature 259 (1976) 654. [6] A.J. Mailfert, B. Nahounou, IEEE Trans. Magn. 16 (1980) 254. [7] A. Espurz, J.M. Alameda, A. Espurz-Nieto, J. Phys. D 22 (1989) 1174. [8] P.C. Fannin, B.K. Scaife, S.W. Charles, J. Magn. Magn. Mater. 122 (1993) 168. [9] N.A. Yusuf, J. Shobaki, H. Abu-Safia, I. Abu-Aljarayesh, J. Magn. Magn. Mater. 149 (1995) 373. [10] R.W. Chantrell, J. Magn. Magn. Mater. 45 (1984) 100. [11] R.W. Chantrell, J. Popplewell, S.W. Charles, IEEE Trans. Magn. 14 (1987) 975. [12] N.Y. Ayoub, A. Bradburay, R.W. Chantrell, J. Popplewell, J. Magn. Magn. Mater. 65 (1987) 185.
[13] J. Shobaki, S. Musameh, F. Rawwagah, N.A. Yusuf, Phys. Rev. B 54 (1996) 13063. [14] U. Hartmann, H.H. Mende, J. Magn. Magn. Mater. 45 (1984) 100. [15] U. Hartmann, H.H. Mende, Phys. Stat. Sol. 82 (1984) 481. [16] P.C. Scholten, IEEE Trans. Magn. 16 (1980) 221. [17] P.C. Scholten, J. Magn. Magn. Mater. 39 (1983) 99. [18] H.W. Davies, J.P. Llewellyn, J. Phys. D 13 (1980) 2327. [19] H.W. Davies, J.P. Llewellyn, J. Phys. D 12 (1979) 311. [20] H. Abu-Safia, I. Abu-Aljarayesh, S. Mahmood, N.A. Yusuf, J. Magn. Magn. Mater. 87 (1990) 333. [21] H. Abu-Safia, I. Abu-Aljarayesh, H.M. El-Ghanem, N.A. Yusuf, J. Magn. Magn. Mater. 103 (1992) 19. [22] N.A. Yusuf, D.A. Manasrah, M.A. Abdallah, H. Abu-Safia, I. Abu-Aljarayesh, J. Magn. Magn. Mater. 138 (1994) 173. [23] J. Shobaki, F. Rawwagah, I. Abu-Aljarayesh, N.A. Yusuf, J. Magn. Magn. Mater. 159 (1996) 367.
The wavelength and concentration dependence of the magneto-dielectric anisotropy effect in magnetic fluids determined from magneto-optical measurements N.A. Yusuf*, A. Ramadan, H. Abu-Safia Department of Physics, Yarmouk University, Irbid, Jordan Received 25 August 1997; received in revised form 10 November 1997
Abstract The concentration and wavelength dependence of the magneto-dielectric anisotropy effect in magnetic fluids has been investigated. Our results show that the field-induced change in the dielectric constants for light polarized in the field direction increases with the applied field while that for light polarized perpendicular to the field decreases with the field. Furthermore, our results show that this field-induced change is linear with concentration. The results also show that the field-induced change, for light polarized in the field direction, increases with frequency till it reaches a peak at a given frequency and then decreases with further increase in frequency; while for light polarized perpendicular to the field, it decreases with frequency till it reaches a minimum at a given frequency and then increases with further increase in frequency. Moreover, our results yield a value of two for the magneto-dielectric anisotropy factor. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Anisotropy — magneto-dielectric; Magneto-dielectric behavior; Dielectric constant; Magneto-optics; Anisotropy — optical; Birefringence; Dichroism; Magnetic fluids
1. Introduction Magnetic fluids are colloidal suspension of ultra-fine ferro or ferrimagnetic particles dispersed in a suitable liquid carrier. The dielectric behavior of magnetic fluids changes with the application of
* Corresponding author. Tel.: #962 2 271115; fax: #962 2 274725.
an external magnetic field and with the relative orientation of the electric and magnetic fields. This effect is known as magneto-dielectric anisotropy effect. Magneto-dielectric effects in magnetic fluids have been investigated by many workers [1—13] both experimentally and theoretically. The experimental investigations were based on impedance measurement techniques where the magnetic fluid is placed in a capacitor. Measurements of the impedance parameters such as the modulus and
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phase are carried out using a bridge or an RLC meter [1—8]. It is known that impedance measurement techniques suffer some serious disadvantages such as electrode effects, parasitic impedances, skin depth and accuracy-related problems. Very recently, Yusuf et al. [9] have determined the magneto-dielectric effect from magneto-optical measurements where the disadvantages suffered by conventional impedance measurement techniques are avoided. On the theoretical side, Monte Carlo simulations were used to calculate the magneto-dielectric anisotropy effect [10—13]. In most of the previous works the magnetodielectric anisotropy factor which is defined as e (H, u)!e (0, u) 0 g(H, u)" , e (0, u)!e (H, e) 0 M was either 2 or 1. Experimentally, Cotae [3] and Espurz et al. [7] have obtained the value of 1 for g(H, u) while Mailfert [6] has obtained a value of 2. Experimental results of Espurz et al. [7] show that the value of g(H, u) is 1 at low fields and 1.23 at high fields. Our previous results [9] on the magneto-dielectric anisotropy effect calculated from magneto-optical measurements yield a value of 1 for g(H, u). It is important to note that in our previous work the sample was considered as a two dimensional one through the use of n #n "2n , M 0 and k #k "2k where n and k are the real and , M 0 imaginary parts of the index of refraction of the sample, respectively. Fannin et al. [8] have obtained experimentally the value 2 for g(H, u) for a sample of a magnetic fluid with a thickness of 10 lm and particle size of &11 nm. Theoretically, Chantrell [10] and Mailfert et al. [6] have calculated the magneto-dielectric anisotropy effect in magnetic fluids and both obtained a value of 2 for g(H, u). It is worth mentioning that the samples considered in their works were three dimensional samples. Shobaki et al. [13] have numerically calculated the dielectric constant, e par, allel to the applied field and e in a direction M perpendicular to the field for two cases: for a two dimensional sample and a three dimensional sample, and showed that g(H, u) is 1 for the two dimensional case and is two for the three
dimensional case. It is worth mentioning that calculations of the optical anisotropy in magnetic fluids have been carried out using a similar procedure [23]. In this work the concentration and wavelength dependence of the magneto-dielectric anisotropy effect are investigated. The magneto-dielectric anisotropy effect is determined from magneto-optical measurements in a way similar to that reported in Ref. [9] but with the sample considered three dimensional through the use of n #2n "3n and , M 0 k #2k "3k . , M 0 2. Theoretical background The dielectric constant of a magnetic fluid in the absence of an external magnetic field exhibits no anisotropy due to the random orientation of the particles. Therefore, the dielectric constant seen by light with different states of polarization is the same. However, when a magnetic field is applied, orientation of particles and field-induced chain formation in the field direction take place leading to two different average lengths l in the field direc, tion and l perpendicular to the field; consequently, M the dielectric constant will exhibit some degree of anisotropy. The orientation function /(p, q) for a uniaxial single-domain particle suspended in a nonmagnetic liquid carrier under the application of an external magnetic field was treated by Hartmann and Mende [14,15] and Scholten [16,17] and is given by /(p, q)"m(q) f (p),
(1)
where m(q) represents the coupling between the magnetic moment of the particle and its easy axis and is usually expressed as
C
D
3 q1@2exp(q)!I(q) m(q)" !1, 2 4q I(q)
(2)
where
P
I(q)"
q1@2 exp(x2) dx 0
(3)
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and q"K»/k ¹. The function f (p) is given by B f (p)"[1!(3/p)¸(p)], (4) where ¸(P) is the Langevin function, and p" (k M »H/k ¹). 0 4B B In general, the dielectric constant of the magnetic fluid is complex and is written as e*"e@!ieA,
(5)
where e@ and eA are the real and imaginary parts of the dielectric constant, respectively. Furthermore, the index of refraction of the fluid is also complex and is written as n*"n!ik
(6)
where n and k are the real index of refraction and the extinction coefficient, respectively. Conventionally, the dielectric constant and the index of refraction are related by e*"n*2"n2!k2!2ink.
(7)
When an external magnetic field is applied, the dielectric constant e* becomes anisotropic thus exhibiting two different behaviors for light polarized parallel to the magnetic field e* and perpendicular , to the magnetic field e*. Similarly, the index of M refraction will exhibit two indices n* and n*. In this M , case Eq. (7) will be modified such that it incorporates this induced anisotropy. The modified equation will read: e*"e@ !ieA "n*2"n2 !k2 !2in k , , , , , , , , and
(8)
(9) e*"e@ !ieA "n*2"n2 !k2 !2in k . M M M M M M M M By equating the real and imaginary parts in Eq. (8) one gets: (10) e@ "n2 !k2 , , , , (11) eA "2n k . , , , Similarly, by equating the real and imaginary parts in Eq. (9) one gets e@ "n2 !k2 , M M M eA "2n k . M M M
(12) (13)
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Since the sample is a three dimensional one the refractive indices and the extinction coefficients both parallel and perpendicular to the magnetic field are related through the following relations: n #2n "3n , (14) , M 0 n !n "*n, (15) , M k #2k "3k , (16) , M 0 k !k "*k, (17) , M where n and k are the real index of refraction and 0 0 the extinction coefficient of the sample under zero field, respectively; and *n is the birefringence and *k is proportional to dichroism. Solving the sets of Eqs. (14) and (15) and Eqs. (16) and (17) simultaneously the refractive indices and the extinction coefficients both parallel and perpendicular to the magnetic field are obtained in terms of n , k , *n and *k as 0 0 n "n #2*n, (18) , 0 3 n "n !1*n, (19) M 0 3 k "k #2*k, (20) , 0 3 k "k !1*k. (21) M 0 3 Substituting Eqs. (14)—(17) in Eqs. (10)—(13) the values of the dielectric constant are obtained as e@ "(n #2*n)2!(k #2*k)2, (22) , 0 3 0 3 (23) eA "2(n #2*n)(k #2*k), , 0 3 0 3 e@ "(n !1*n)2!(k !1*k)2, (24) M 0 3 0 3 (25) eA "2(n !1*n)(k !1*k). M 0 3 0 3 The last equations, Eqs. (22)—(25) are the fundamental equations in calculating the dielectric constants (e* and e*) of the magnetic fluid and hence in M , determining the magneto-dielectric anisotropy effect. They show that magneto-dielectric effects may be calculated from optically measured physical quantities such as birefringence *n, dichroism (*A), the real refractive index n and the extinction coef0 ficient k (under zero magnetic field). It is important 0 to note that these physical quantities can be measured with excellent accuracies, e.g. values of *n and *k&10~6 can be easily measured [16—22].
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The frequency dependence of the index of refraction or the dielectric constant of matter (dispersion) is explained in terms of the distortion of the internal charge distribution under the influence of an applied external electric field. This distortion results in induced electric dipole moments leading to an induced electric polarization. Considering the charge carriers to be elastically bound, the electric field of the electromagnetic wave provides the driving force of the harmonic oscillator. Following standard procedures for solving the equation of a forced damped harmonic oscillator, the dielectric constant is given by q2N e"e # , 0 m(u2!u2#icu) 0
(26)
where e is the dielectric constant of free space, 0 q the charge of the charge carrier, N the number of charge carriers per unit volume, m the mass of the charge carrier, u the natural frequency of the 0 oscillator, u the frequency of the electromagnetic wave and c a damping parameter. For an elongated particle, the charge carriers can oscillate along the easy direction much easier than along other directions and with larger amplitudes leading to more electric polarization in the easy direction. Therefore, the contribution to the dielectric constant of charge carriers oscillating along the easy axis of the particle is the dominant one. Therefore, we suggest that the number of charge carriers contributing to the dielectric constant is direction dependant when a magnetic field is applied. This leads to the anisotropy in the dielectric constant. Consequently, the dielectric constant is given by q2N x e "e # , x 0 m(u2!u2#icu) 0
(26a)
q2N y e "e # , y 0 m(u2!u2#icu) 0
(27)
q2N z e "e # , z 0 m(u2!u2#icu) 0
(28)
where e , e and e are the dielectric constants seen x y z by the electromagnetic waves polarized in the x, y, and z directions, respectively; and N , N and x y
N are the number of charge carriers per unit volz ume oscillating in the x, y, and z directions, respectively. In the absence of the magnetic field N "N "N "1N, where N is the density of x y z 3 charge carriers. Therefore e , e , and e are all equal x y z to e (dielectric constant at zero magnetic field) and ;% given by q2N e "e # . (29) ;% 0 3m(u2!u2#icu) 0 When a magnetic field H is applied in the zdirection, the particles tend to align in the field direction with a probability P(H, ¹). This probability is equal to [(1#2/(p, q))/3] thus insuring that P(H, ¹)"1 at zero field or very high temperature 3 and is 1 at very high fields or low temperatures. The densities of the charge carriers are given by N "NP(H, ¹), (30) z (31) N "N "1N(1!P(H, ¹)). x y 2 Substituting the values of N and N in Eqs. (26) x z and (28) yields the following values for e and e : , M q2N P(H, ¹) e "e # , (32) , 0 m(u2!u2#icu) 0 q2N(1!P(H, ¹)) e "e # . (33) M 0 2m(u2!u2#icu) 0 Separating the real and imaginary parts in Eqs. (29), (32) and (33) yields: e@ "e #f (u)NP(H, ¹), , 0 1 e@ "e #1 f (u)N(1!P(H, ¹)), M 0 2 1 eA "f (u)NP(H, ¹), , 2 eA "1 f (u)N(1!P(H, ¹)), M 2 2 e@ "e #1 f (u)N, ;% 0 3 1 eA "1 f (u)N, ;% 3 2 where f (u) is given by 1 (u2!u2)q2 0 f (u)" 1 mM(u2!u2)2#(cu)2N 0
(34) (35) (36) (37) (38) (39)
(40)
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and f (u) is given by 2 cuq2 . f (u)" 2 mM(u2!u2)2#(cu)2N 0
(41)
It is important to note that these results presented in Eqs. (35)—(39) are in agreement with the assumption that the dielectric constant is proportional to the average projection length of the particles which has been previously used to calculate and explain the magneto-dielectric anisotropy effect in magnetic fluids [9,13]. The field-induced change in the real dielectric constants *e@ and *e@ and that in the imaginary M , dielectric constants *eA and *eA are then given by M , *e@ "f (u)N(P(H, ¹)!1), 3 , 1
(42)
*e@ "1 f (u)N(1!P(H, ¹)), M 2 1 3
(43)
*eA "f (u)N(P(H, ¹)!1), , 2 3
(44)
*eA "1 f (u)N(1!P(H, ¹)). M 2 2 3
(45)
These last equations describe the behavior of the magneto-dielectric anisotropy effect with field, concentration and frequency.
3. Experimental 3.1. Birefringence Measurements of the birefringence were undertaken using an optical arrangement consisting of a conventional quartz lamp, narrow band interference filters, a polarizer, sample cell, an analyzer, a compensator and a photomultiplier. The use of the compensator allows for a direct measurement of the relative retardation, d, of the components of the transmitted light through the sample with their planes of polarization parallel and perpendicular to the applied magnetic field, even when the sample exhibits dichroism. In this experiment, the plane of polarization was set at 45° with the magnetic field. With the field off, the reading of the compensator, l , was adjusted to give a minimum photo0 multiplier output; then with the field on the reading
379
l was readjusted to give a minimum once more. B From the difference (l !l ) and the calibration of B 0 the compensator the relative retardation d was obtained. 3.2. Dichroism Measurements of dichroism were undertaken using a set-up similar to that used for the birefringence experiment but with the compensator and analyzer removed. The transmission of light with planes of polarization parallel and perpendicular to the applied magnetic field I and I were measured. , M The values of the parallel and perpendicular dichroism *A and *A , are defined as the change in , M the absorbance of the sample as a result of applying a magnetic field and are given by *A "!ln(I /I ), , , 0
(46)
*A "!ln(I /I ), M M 0
(47)
where I is the transmitted intensity at zero field. 0 The dichroism *A is then given by *A"*A !*A . , M
(48)
The sample was contained in a standard 1 mm thick rectangular cell which in turn was placed between the poles of an electromagnet that is capable of producing fields up to 10 kOe. The samples used in this work are Fe O particle 3 4 magnetic fluids with Isopar-M as a liquid carrier. The average volume of the particles and the standard deviation were determined using scanning electron microscopy; their values are 75 and 2 A_ , respectively. The volumic fraction, e, of the samples ranged from 0.548 to 5.173]10~3, and the wavelengths used in this study ranged from 372.5 to 694.3 nm. The real part of the refractive index n of the 0 samples under zero magnetic field (for all wavelengths used) is determined by measuring the Brewster angle. Moreover, the extinction coefficient k of the samples (for all wavelengths used) is deter0 mined by measuring the transmission of light through cells of different optical paths filled with the sample. The cells are standard cells made of the same glass having the same thickness.
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4. Results The birefringence *n and dichroism *A, for different concentrations and at a wavelength j" 633 nm, versus magnetic field are presented in Fig. 1a and Fig. 1b, respectively. The results in the figures, similar to previously reported results, show that both *n and *A for a given concentration increases rapidly with the field for low fields then less rapidly for intermediate fields and tends to saturate at high fields. The birefringence *n and dichroism *A, for different wavelengths and e"0.548]10~3, versus magnetic field are presented in Fig. 2a and Fig. 2b, respectively. The results in the figures show that both *n and *A, at a given field, increase with wavelength reaching a peak at a given j then decreases with further increase with the wavelength. From the results in Fig. 2a and Fig. 2b, the birefringence *n and dichroism *A at measuring fields 500 and 2500 Oe are plotted versus the wavelength j in Fig. 3a and Fig. 3b, respectively. The results in the figures show that *n peaks at j"514 nm while *A peaks at j"488 nm. Furthermore, the results show that the position of the peak does not change with the applied magnetic field even though the width of the peak increases with the field. The real part of the refractive index n and the 0 imaginary part i.e., the extinction coefficient k , for 0 the samples, at zero field are plotted versus concentration in Fig. 4a and Fig. 4b, respectively. The results in the figures show that both n and 0 k increase almost linearly with concentration. This 0 is not too surprising since the range of concentration is not large. Furthermore, although the k line 0 practically passes through the origin reflecting the fact that Isopar-M has a zero extinction coefficient, the n line has an intercept corresponding to the 0 refractive index of Isopar-M. The measured real refractive index n and the 0 extinction coefficient k of the sample with e" 0 0.548]10~3 are plotted versus j in Fig. 5a and Fig. 5b, respectively. The results in the figures show that n decreases monotonically with j in the used 0 range of wavelengths, while they show that the extinction coefficient increases with j for low wavelengths exhibiting a peak at j"488 nm then decreases for higher wavelengths.
Fig. 1. Birefringence *n (a) and dichroism *A (b) for different concentrations versus magnetic field measured at ¹"300 K and j"63 nm.
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381
Fig. 2. Birefringence *n (a) and dichroism *A (b) for different wavelengths versus magnetic field measured at ¹"300 K. Fig. 3. Birefringence *n (a) and dichroism *A (b) at a measuring field of 500 Oe versus the wavelength j.
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Fig. 4. Real part of the index of refraction n (a); the extinction 0 coefficient k (b) versus concentration. 0
Fig. 5. Real part of the index of refraction n (a); the extinction 0 coefficient k (b) versus wavelength. 0
N.A. Yusuf et al. / Journal of Magnetism and Magnetic Materials 184 (1998) 375—386
Using Eqs. (22)—(25) and the measured values of *n, *A, n and k , the real and imaginary parts of 0 0 the dielectric constants for light polarized parallel and perpendicular to the magnetic field (e@ , eA ) and , , (e@ , eA ); and those at zero field (e@ , eA ) were cal;% ;% M M culated. The field-induced change in the real dielectric constants (*e@ "e@ !e@ ) and (*e@ "e@ !e@ ); ;% M M ;% , , and that in the imaginary dielectric constants (*eA "eA !eA ) and (*eA "eA !eA ), for samples ;% M M ;% , , with different concentration and at j"633 nm, are plotted versus field in Fig. 6a and Fig. 6b, respectively. The results show that the changes in the real and imaginary parts of the dielectric constant for light polarized in the field direction increase with the field while those for light polarized normal to the field decrease with the field. These results are in good agreement with previously reported results [1—13]. Furthermore, the results show that the value of the magneto-dielectric anisotropy factor at any given field and for any given concentration, in agreement with previous works [6,8,10,13], is two. The field-induced changes in the real dielectric constants *e@ and *e@ and that in the imaginary M , dielectric constants *eA and *eA measured at M , j"633 nm are plotted versus concentration in Fig. 7a and Fig. 7b, respectively. The results in the figures show that this change with concentration is linear in the studied range, and also show that the magneto-dielectric anisotropy factor at any given field and for any given concentration is again two. The field-induced changes in the real dielectric constants *e@ and *e@ and those in the imaginary M , dielectric constants *eA and *eA for the sample M , with e"0.548]10~3 and at selected measuring fields are plotted versus wavelength j in Fig. 8a and Fig. 8b, respectively. The results in the figures show that both *e@ and *eA increase with j till each , , reaches a maximum at a given wavelength j then .!9 decreases for longer wavelengths. However, *e@ M and *eA both decreases with j till each reaches M a minimum at a given wavelength j then in.*/ creases for longer wavelengths. Furthermore, the wavelengths at which the maximum in *e@ and the , minimum in *e@ (real part) occur are the same; and M those wavelengths corresponding to the maximum in *eA minimum in *eA (imaginary part) are also M , the same though different from those corresponding to the real part. For the real part the maximum
383
Fig. 6. Field-induced changes *e@ and *e@ (a); and *eA and *eA , M , M (b) at j"633 nm versus magnetic field.
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Fig. 7. Field-induced changes *e@ and *e@ (a); and *eA and *eA , M , M (b) at different fields and j"633 nm versus concentration.
Fig. 8. Field-induced changes *e@ and *e@ (a); and *eA and *eA , M , M (b) versus wavelength j at selected fields.
N.A. Yusuf et al. / Journal of Magnetism and Magnetic Materials 184 (1998) 375—386
in *e@ and the minimum in *e@ occur at j" M , 514 nm while for the imaginary part they occur at j"488 nm.
5. Discussion The behavior of birefringence and dichroism with field and concentration have been fully explained previously [20—22]. Therefore, we only discuss the magneto-dielectric effect presented in this work. The magneto-dielectric anisotropy effect is mainly attributed to the orientation of non-spherical particles and to the field-induced chain formation in the field direction. For a given frequency u and at a given temperature ¹, the probability P(H, ¹) increases with the field and as can be seen from Eqs. (42) and (44) the field-induced change in the dielectric constants for light polarized parallel to the field *e@ and *eA , , increases with the field. However, the field-induced change in the constants for light polarized perpendicular to the field *e@ and *eA decreases with the M M field. Furthermore, using Eqs. (42) and (43) and Eqs. (44) and (45) one can see that the magnetodielectric anisotropy factor g(H, u) equals 2. Looking at our results presented in Fig. 6a and Fig. 6b one finds out an excellent agreement between the results and the theory. Again for a given frequency u and a given applied field and at a given temperature, the functions f (u) and f (u) and the probability P(H, ¹) are 1 2 constants. Therefore, the field-induced changes in the dielectric constant are proportional to the density of charge carriers N with positive slopes for the parallel components and negative slopes for the perpendicular components. Furthermore, the ratio of the slopes for the two cases is (!2). Noting that N is proportional to the number of particles per unit volume in the fluid, then the field-induced changes in the dielectric constant are proportional to the concentration of the sample. Our results in Fig. 7a and Fig. 7b show a behavior consistent with the prediction of the theory. From Eqs. (42) and (43) one sees that both *e@ , and *e@ have the same dependence on the freM quency of the electromagnetic waves and thus should exhibit their optimal conditions, i.e. the
385
maximum in *e@ and the minimum in *e@ at the M , same frequency. Similarly *eA and *eA have the M , same frequency dependence and therefore the maximam in *eA and the minimum *eA occur at M , the same frequency. However, since the functional dependence on frequency for the parallel and perpendicular components of the dielectric constant are not the same, their optimal conditions occur at two different frequencies. Our results presented in Fig. 8a and Fig. 8b show a behavior similar to that predicted by the theory.
6. Conclusions The magneto-dielectric anisotropy effect in magnetic fluids has been determined from magnetooptical measurements. The field, concentration and wavelength dependence of the effect have been investigated. Our results showed that the field-induced change in both the real and imaginary parts of the dielectric constant for light polarized parallel to the field increase with the field while those for light polarized perpendicular to the field decrease with the field. Furthermore, our results show that the field-induced changes vary linearly with the concentration of the sample with a positive slope for the parallel components and negative slope for the perpendicular components. Moreover, our results show that the field-induced change in the parallel component increases with frequency reaching a maximum at a given frequency then it decreases with further increase of frequency; while the change in the perpendicular component decreases with frequency reaching a minimum and then increases with further increase of frequency. Our results have been explained in terms of the theory of dispersion of electromagnetic waves in matter with the assumption that the density of charge carriers oscillating in a given direction is proportional to the number of particles oriented in that direction.
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