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A 3-D moving vectorial Preisach-type hysteresis model
Journal of Magnetism and Magnetic Materials 207 (1999) 188}192
A 3-D moving vectorial Preisach-type hysteresis model C. Michelakis, D. Samaras*, G. Litsardakis Department of Electrical and Computer Engineering, Aristotle University, GR-54006 Thessaloniki, Greece Received 2 February 1999; received in revised form 30 June 1999
Abstract A moving vectorial model of hysteresis for media consisting of uniaxial interacting particles is proposed. A 3D distribution of the particles' easy axes is considered, described by two independent symmetrical angular distributions. Interaction "elds following the magnetization direction are used. An identi"cation procedure is proposed. Comparison with experimental major loops and dm curves shows a very good agreement with data taken from both longitudinal and tranverse directions. ( 1999 Elsevier Science B.V. All rights reserved. PACS: 75.60.Ej; 75.50.S Keywords: Hysteresis modeling
1. Introduction An e$cient simulation of magnetic recording, especially in modern recording technologies, requires a vectorial model of magnetic hysteresis. As the scalar Preisach model proves insu$cient to describe a vectorial behavior, Oti and Della Torre [1] introduced the idea of combining this model and the vectorial Stoner}Wohlfarth model in order to develop a vectorial 3D model for non-identical interacting particles. Building a vectorial model requires, besides the vectorial expression of the applied "eld, a consideration about the vector of the local interaction "eld. Cramer [2] and more
* Corresponding author. Tel.: #30-31-996384; fax: #30-31996302. E-mail address: [email protected] (D. Samaras)
recently Charap [3,4] included a magnetizationdependent mean interaction "eld and a #uctuation term along the easy axis of each particle to develop moving 3D models. In these models a "xed distribution of the interaction "eld #uctuations, independent of the magnetization direction and centered at the medium's easy axes is used. In the present model the interaction "eld vectors follow the magnetization direction. The distribution of the particles's easy axis is described by two independent angular distributions. The applied "eld is considered on the symmetry plane of one of the two distributions. Moreover, in order to include the observation that the medium shows a much bigger coercivity on the transverse direction [5], a truncated Stoner}Wohlfarth astroid [6] is used in this work. A systematic identi"cation procedure leading to a good agreement of the model in a limited number of trials with metal particle medium data is proposed.
0304-8853/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 5 1 8 - 1
C. Michelakis et al. / Journal of Magnetism and Magnetic Materials 207 (1999) 188}192
2. Description of the model The classical Preisach model describes perfectly oriented media subjected to an applied "eld acting along the direction of orientation. The medium is considered as an assembly of single domain particles with uniaxial anisotropy, interacting with each other. In order to add vectorial properties, the two state scalar switching operator of the Preisach model is replaced by a vector operator, describing both rotation and switching, according to the Stoner}Wohlfarth model. In this way, a vectorial model for media consisting of single domain, interacting, non-identical, uniaxial particles is built [1]. A three-dimensional distribution of the directions of the particles's easy axes with respect to the medium easy axis is considered, described by two independent symmetrical angular distributions, one in the plane of the medium and the other perpendicular to it. Each particle's axis is characterized by a set of two angles, the azimuthal (h) and the polar one (/) (Fig. 1). The interaction "elds are assumed to be of the form: H "aM#h . The "rst term is a mean "eld i i contribution, proportional to the global magneti-
189
zation M, in accordance with the molecular "eld approximation. The proportionality factor a is thus analogous to the usually used moving parameter [7]. The second term accounts for the interaction "eld #uctuations from the mean value [8]. In the present work the term h is assumed collinear to the i global magnetization vector. The interaction "eld H is thus described by its mean value aM and i a standard deviation p i. In general p i is magnetiH H zation dependant [9] but, as "rst approximation, it is kept constant. The applied "eld H is considered 0 to lie on the symmetry plane of one of the two angular distributions. A more general con"guration has not been thought as worthwhile, since the con"guration considered is of su$cient generality for the application of the model to magnetic recording simulation. The implementation of the model is based on the calculation of the magnetization of oriented Preisach planes, each one de"ned by a set of azimuthal and polar angles h, u. The response of each particle f(H, h, u) is vectorial, calculated by means of a truncated SW astroid. The SW astroid equation is
A B A B
H 2@3 H 2@3 x # y "1, H H % )
Fig. 1. Con"guration of directions involved in the model. E: medium's easy axis, h, u: angles determining the particles' easy axis e , H : applied "eld, J , J: magnetization of particle and i 0 i medium, respectively.
where H , H , are the applied "eld components and x y H , H are the particle anisotropies along its easy % ) and hard axes, respectively. In the truncated astroid H is modi"ed by the truncation factor q(q'0): % H@ "H q. % % The astroid is subjected to the e!ective "eld, M(H) "f(H #aM#h ). The e!ective "eld calh,r 0 i culation is performed on each particle. The term aM depends explicitly on the global magnetization vector while the h depends on the magnetization i direction. In this sense, H is a feedback "eld and i calls for an iterative procedure. The starting value of the magnetization direction, needed for the "rst iteration, is taken as the one of the &most probable' particle, i.e. the one corresponding to the mean values of the Preisach and angular distributions. The contributions of the di!erent Preisach planes, weighed by the density function o(h, u), are "nally integrated for all the h, u sets.
190
C. Michelakis et al. / Journal of Magnetism and Magnetic Materials 207 (1999) 188}192
The calculated magnetization value for a given "eld is as follows:
P P P P p@2 p@2
g(0, u) ~p@2 H4!5 Hk ] f(H, 0, u)o(H , h ) dh k i i ~H4!5 ~H4!5 ]dH d0 du. k Density function o(H , H ) is taken as a product k i of the Gaussian distribution of the anisotropy "elds, with a mean value of SH T and a standard k deviation p k, and the Gausian distribution of the H interaction "elds having a zero mean value and a standard deviation p i. Density function g(h, u) is H taken as a product of the Gaussian distribution of the azimuthal angle h having a standard deviation of p and the Gaussian distribution of the h polar angle u having a standard deviation of p : g(h, u)"g (h) g (u) cos(u). The "eld H used for ( 1 2 the calculation of the particle response f(H, h, u) is the total "eld acting on each particle including the applied one, the mean interaction "eld aM and the interaction "eld #uctuation h . The moving parai meter a is the feedback tensor; as a "rst approach the same value of a has been used for every "eld direction. The integral is solved numerically. For the calculation, the double Everett-type integral M(H)"
0
P P H4!5
Hk
f(H, 0, u)o(H , h ) dh dH k i i k ~H4!5 ~H4!5 representing the classical Preisach model (including however a vectorial operator) is calculated at "rst. It is then followed by a double integration for all h and u sets. The steps used for the numerical calculation are equal. A 30]30 discretization for the double Everett integral proves enough for major loop calculation accuracy better than 1%. In order to achieve the same accuracy for the calculation of dm curves, a "ner discretization is necessary. The model proves very stable numerically, showing a robust behavior even at usual points of &instability' such as the cases where magnetization reaches zero and the determination of the magnetization direction is uncertain. In this case, the solu-
tion given is that J is calculated for a very small value (nearly zero), large enough however for the model to converge in a few iterations.
3. Identi5cation method and results The model parameters used are: the mean anisotropy "eld SH T, the standard deviation of the ank isotropy "elds p k, the standard deviation of the H interaction "elds p i, the standard deviation of the H particles' easy axes on the plane of the medium p , h the standard deviation of the particles' easy axes on the plane perpendicular to the one of the medium p , the moving parameter a and the truncation ( factor q. For the identi"cation procedure major loops as well as dm curves are used. A value for the mean anisotropy "eld SH T as well as for the angular k distributions can be derived by major loop data. Data of dm curves can be used to specify the interaction "eld parameters. As a matter of fact, dm curves, directly related to Henkel plots, are very sensitive to interaction e!ects. A direct correspondence of the remanence values with the angular distribution characteristics cannot be done due to the dependence of the remanence values on the moving parameter a as well. In general, the relation between curve characteristics and model parameters is complex; however, a careful examination of the variation of the calculated values of the curve characteristics can lead to some rules of the thumb. f A rise in the angular distributions' standard deviations leads to a decrease in the longitudinal remanence value and an increase in the transverse remanence, while an increase of the moving parameter a leads to an increase of both remanence values. f The value of the coercivity is primarily dependent on the mean anisotropy "eld SH T. Addik tionally, the increase in the deviations of the angular distributions leads to a decrease in the longitudinal coercivity and an increase in the transverse coercivity value. f SH mainly depends on the p k. H f The transverse major loops at "elds higher than the coercivity are highly depended on q.
C. Michelakis et al. / Journal of Magnetism and Magnetic Materials 207 (1999) 188}192
191
f The maximum of the positive peak in the dm curve is primarily dependent on the moving parameter a and secondarily on p i. The same H parameters in inverse priority de"ne the maximum of the negative peak. The identi"cation procedure uses the remanence coercivity values of both longitudinal and transverse major loops and SH, maximum and minimum of the longitudinal dm curve. The identi"cation procedure is performed as follows: f Values for p and p are determined so that the h ( experimental and simulated Mr values * both longitudinal and transverse * match, for a starting value of the moving parameter equal to zero. f Given the longitudinal H value and the para# meters already speci"ed a SH T value is deterk mined. f The SH is used for the determination of the p k value. H f After longitudinal major loop has been matched, q is used to improve "t on the transverse loop. f For these values of p , p SH T and p k and for h (, k H a starting value of p i equal to p k dm curves are H H traced. The positive peak is matched varying a. The negative peak is matched varying p i. H f Given the a value, repeat the procedure for a more accurate determination of the parameters. The model has been compared to experimental data. Major loops and dm curves have been traced using an Oxford Instruments MagLab VSM, having a sensitivity of 10~5 emu. The sample used was in the form of a 6 mm diameter disk cut from a commercial metal particle 8 mm video tape. The parameters speci"ed are given in Table 1. In Figs. 2 and 3 experimental and simulated major loops and dm curves are shown. A very good
Table 1 Determined model parameters SH T"3100 Oe k
p "323 h p "263 (
p k"900 Oe H p i"240 Oe H
a"380 q"0.5
Fig. 2. Longitudinal and transverse major loops.
Fig. 3. Longitudinal and transverse delta M curves.
agreement for the longitudinal ones is achieved. For the transverse loops, in which the agreement is less satisfactory, we have previously employed two di!erent SH T values [5], while in this work we k have introduced the truncation factor q. The results of the proposed model are compared with those of a model where the interaction "eld #uctuations are taken along the particle's easy axis. The two models coincide for longitudinal loops, since the magnetization in this case lies on the medium's easy axis. On the contrary, they exhibit considerable di!erences for transverse loops. The lower the anisotropy (soft and mid-soft materials) and the smaller the standard deviations for the polar and azimuthal angle distributions, the bigger
192
C. Michelakis et al. / Journal of Magnetism and Magnetic Materials 207 (1999) 188}192
along the medium's easy axis is a good approximation for highly anisotropic particles and external "elds applied close to the medium's easy axis, while a magnetization-dependent #uctuation "eld is expected to be a more realistic approach of a low anisotropy material in "elds far from the easy axis. The ability of the model to simulate the behavior of soft materials, as the ones used in technology of high currents is worth a closer examination.
References Fig. 4. Transverse major loop comparison: present work (solid line) * model with interaction #uctuations along particles' easy axes (dotted line).
the di!erences observed. A comparison of transverse major loops computed according to the two models is given in Fig. 4 for the following parameter values: SH T"2000 Oe, p k"100 Oe, p i" k H H 500 Oe, p "53, p "53, q"0.5, a"150. Di!erh ( ences as high as 15% are observed near the knee of the curve. A "xed interaction "eld distribution
[1] J. Oti, E. Della Torre, J. Appl. Phys. 67 (1990) 5364. [2] H.A.J. Cramer, J. Magn. Magn. Mater. 88 (1990) 194. [3] F. Ossart, R. Davidson, S.H. Charap, IEEE Trans. Magn. 31 (1995) 1785. [4] R.J. Davidson, S.H. Charap, IEEE Trans. Magn. 32 (1996) 4198. [5] C. Michelakis, D. Samaras, G. Litsardakis, J. Magn. Magn. Mater. 196}197 (1999) 599. [6] I.A. Beardsley, J. Appl. Phys. 53 (1982) 2582. [7] E. Della Torre, IEEE Trans. Audio AU-14 (1966) 86. [8] X.D. Che, H.N. Bertram, J. Magn. Magn. Mater. 116 (1992) 121. [9] F. Vajda, E. Della Torre, IEEE Trans. Magn. 32 (1996) 1112.
A 3-D moving vectorial Preisach-type hysteresis model C. Michelakis, D. Samaras*, G. Litsardakis Department of Electrical and Computer Engineering, Aristotle University, GR-54006 Thessaloniki, Greece Received 2 February 1999; received in revised form 30 June 1999
Abstract A moving vectorial model of hysteresis for media consisting of uniaxial interacting particles is proposed. A 3D distribution of the particles' easy axes is considered, described by two independent symmetrical angular distributions. Interaction "elds following the magnetization direction are used. An identi"cation procedure is proposed. Comparison with experimental major loops and dm curves shows a very good agreement with data taken from both longitudinal and tranverse directions. ( 1999 Elsevier Science B.V. All rights reserved. PACS: 75.60.Ej; 75.50.S Keywords: Hysteresis modeling
1. Introduction An e$cient simulation of magnetic recording, especially in modern recording technologies, requires a vectorial model of magnetic hysteresis. As the scalar Preisach model proves insu$cient to describe a vectorial behavior, Oti and Della Torre [1] introduced the idea of combining this model and the vectorial Stoner}Wohlfarth model in order to develop a vectorial 3D model for non-identical interacting particles. Building a vectorial model requires, besides the vectorial expression of the applied "eld, a consideration about the vector of the local interaction "eld. Cramer [2] and more
* Corresponding author. Tel.: #30-31-996384; fax: #30-31996302. E-mail address: [email protected] (D. Samaras)
recently Charap [3,4] included a magnetizationdependent mean interaction "eld and a #uctuation term along the easy axis of each particle to develop moving 3D models. In these models a "xed distribution of the interaction "eld #uctuations, independent of the magnetization direction and centered at the medium's easy axes is used. In the present model the interaction "eld vectors follow the magnetization direction. The distribution of the particles's easy axis is described by two independent angular distributions. The applied "eld is considered on the symmetry plane of one of the two distributions. Moreover, in order to include the observation that the medium shows a much bigger coercivity on the transverse direction [5], a truncated Stoner}Wohlfarth astroid [6] is used in this work. A systematic identi"cation procedure leading to a good agreement of the model in a limited number of trials with metal particle medium data is proposed.
0304-8853/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 5 1 8 - 1
C. Michelakis et al. / Journal of Magnetism and Magnetic Materials 207 (1999) 188}192
2. Description of the model The classical Preisach model describes perfectly oriented media subjected to an applied "eld acting along the direction of orientation. The medium is considered as an assembly of single domain particles with uniaxial anisotropy, interacting with each other. In order to add vectorial properties, the two state scalar switching operator of the Preisach model is replaced by a vector operator, describing both rotation and switching, according to the Stoner}Wohlfarth model. In this way, a vectorial model for media consisting of single domain, interacting, non-identical, uniaxial particles is built [1]. A three-dimensional distribution of the directions of the particles's easy axes with respect to the medium easy axis is considered, described by two independent symmetrical angular distributions, one in the plane of the medium and the other perpendicular to it. Each particle's axis is characterized by a set of two angles, the azimuthal (h) and the polar one (/) (Fig. 1). The interaction "elds are assumed to be of the form: H "aM#h . The "rst term is a mean "eld i i contribution, proportional to the global magneti-
189
zation M, in accordance with the molecular "eld approximation. The proportionality factor a is thus analogous to the usually used moving parameter [7]. The second term accounts for the interaction "eld #uctuations from the mean value [8]. In the present work the term h is assumed collinear to the i global magnetization vector. The interaction "eld H is thus described by its mean value aM and i a standard deviation p i. In general p i is magnetiH H zation dependant [9] but, as "rst approximation, it is kept constant. The applied "eld H is considered 0 to lie on the symmetry plane of one of the two angular distributions. A more general con"guration has not been thought as worthwhile, since the con"guration considered is of su$cient generality for the application of the model to magnetic recording simulation. The implementation of the model is based on the calculation of the magnetization of oriented Preisach planes, each one de"ned by a set of azimuthal and polar angles h, u. The response of each particle f(H, h, u) is vectorial, calculated by means of a truncated SW astroid. The SW astroid equation is
A B A B
H 2@3 H 2@3 x # y "1, H H % )
Fig. 1. Con"guration of directions involved in the model. E: medium's easy axis, h, u: angles determining the particles' easy axis e , H : applied "eld, J , J: magnetization of particle and i 0 i medium, respectively.
where H , H , are the applied "eld components and x y H , H are the particle anisotropies along its easy % ) and hard axes, respectively. In the truncated astroid H is modi"ed by the truncation factor q(q'0): % H@ "H q. % % The astroid is subjected to the e!ective "eld, M(H) "f(H #aM#h ). The e!ective "eld calh,r 0 i culation is performed on each particle. The term aM depends explicitly on the global magnetization vector while the h depends on the magnetization i direction. In this sense, H is a feedback "eld and i calls for an iterative procedure. The starting value of the magnetization direction, needed for the "rst iteration, is taken as the one of the &most probable' particle, i.e. the one corresponding to the mean values of the Preisach and angular distributions. The contributions of the di!erent Preisach planes, weighed by the density function o(h, u), are "nally integrated for all the h, u sets.
190
C. Michelakis et al. / Journal of Magnetism and Magnetic Materials 207 (1999) 188}192
The calculated magnetization value for a given "eld is as follows:
P P P P p@2 p@2
g(0, u) ~p@2 H4!5 Hk ] f(H, 0, u)o(H , h ) dh k i i ~H4!5 ~H4!5 ]dH d0 du. k Density function o(H , H ) is taken as a product k i of the Gaussian distribution of the anisotropy "elds, with a mean value of SH T and a standard k deviation p k, and the Gausian distribution of the H interaction "elds having a zero mean value and a standard deviation p i. Density function g(h, u) is H taken as a product of the Gaussian distribution of the azimuthal angle h having a standard deviation of p and the Gaussian distribution of the h polar angle u having a standard deviation of p : g(h, u)"g (h) g (u) cos(u). The "eld H used for ( 1 2 the calculation of the particle response f(H, h, u) is the total "eld acting on each particle including the applied one, the mean interaction "eld aM and the interaction "eld #uctuation h . The moving parai meter a is the feedback tensor; as a "rst approach the same value of a has been used for every "eld direction. The integral is solved numerically. For the calculation, the double Everett-type integral M(H)"
0
P P H4!5
Hk
f(H, 0, u)o(H , h ) dh dH k i i k ~H4!5 ~H4!5 representing the classical Preisach model (including however a vectorial operator) is calculated at "rst. It is then followed by a double integration for all h and u sets. The steps used for the numerical calculation are equal. A 30]30 discretization for the double Everett integral proves enough for major loop calculation accuracy better than 1%. In order to achieve the same accuracy for the calculation of dm curves, a "ner discretization is necessary. The model proves very stable numerically, showing a robust behavior even at usual points of &instability' such as the cases where magnetization reaches zero and the determination of the magnetization direction is uncertain. In this case, the solu-
tion given is that J is calculated for a very small value (nearly zero), large enough however for the model to converge in a few iterations.
3. Identi5cation method and results The model parameters used are: the mean anisotropy "eld SH T, the standard deviation of the ank isotropy "elds p k, the standard deviation of the H interaction "elds p i, the standard deviation of the H particles' easy axes on the plane of the medium p , h the standard deviation of the particles' easy axes on the plane perpendicular to the one of the medium p , the moving parameter a and the truncation ( factor q. For the identi"cation procedure major loops as well as dm curves are used. A value for the mean anisotropy "eld SH T as well as for the angular k distributions can be derived by major loop data. Data of dm curves can be used to specify the interaction "eld parameters. As a matter of fact, dm curves, directly related to Henkel plots, are very sensitive to interaction e!ects. A direct correspondence of the remanence values with the angular distribution characteristics cannot be done due to the dependence of the remanence values on the moving parameter a as well. In general, the relation between curve characteristics and model parameters is complex; however, a careful examination of the variation of the calculated values of the curve characteristics can lead to some rules of the thumb. f A rise in the angular distributions' standard deviations leads to a decrease in the longitudinal remanence value and an increase in the transverse remanence, while an increase of the moving parameter a leads to an increase of both remanence values. f The value of the coercivity is primarily dependent on the mean anisotropy "eld SH T. Addik tionally, the increase in the deviations of the angular distributions leads to a decrease in the longitudinal coercivity and an increase in the transverse coercivity value. f SH mainly depends on the p k. H f The transverse major loops at "elds higher than the coercivity are highly depended on q.
C. Michelakis et al. / Journal of Magnetism and Magnetic Materials 207 (1999) 188}192
191
f The maximum of the positive peak in the dm curve is primarily dependent on the moving parameter a and secondarily on p i. The same H parameters in inverse priority de"ne the maximum of the negative peak. The identi"cation procedure uses the remanence coercivity values of both longitudinal and transverse major loops and SH, maximum and minimum of the longitudinal dm curve. The identi"cation procedure is performed as follows: f Values for p and p are determined so that the h ( experimental and simulated Mr values * both longitudinal and transverse * match, for a starting value of the moving parameter equal to zero. f Given the longitudinal H value and the para# meters already speci"ed a SH T value is deterk mined. f The SH is used for the determination of the p k value. H f After longitudinal major loop has been matched, q is used to improve "t on the transverse loop. f For these values of p , p SH T and p k and for h (, k H a starting value of p i equal to p k dm curves are H H traced. The positive peak is matched varying a. The negative peak is matched varying p i. H f Given the a value, repeat the procedure for a more accurate determination of the parameters. The model has been compared to experimental data. Major loops and dm curves have been traced using an Oxford Instruments MagLab VSM, having a sensitivity of 10~5 emu. The sample used was in the form of a 6 mm diameter disk cut from a commercial metal particle 8 mm video tape. The parameters speci"ed are given in Table 1. In Figs. 2 and 3 experimental and simulated major loops and dm curves are shown. A very good
Table 1 Determined model parameters SH T"3100 Oe k
p "323 h p "263 (
p k"900 Oe H p i"240 Oe H
a"380 q"0.5
Fig. 2. Longitudinal and transverse major loops.
Fig. 3. Longitudinal and transverse delta M curves.
agreement for the longitudinal ones is achieved. For the transverse loops, in which the agreement is less satisfactory, we have previously employed two di!erent SH T values [5], while in this work we k have introduced the truncation factor q. The results of the proposed model are compared with those of a model where the interaction "eld #uctuations are taken along the particle's easy axis. The two models coincide for longitudinal loops, since the magnetization in this case lies on the medium's easy axis. On the contrary, they exhibit considerable di!erences for transverse loops. The lower the anisotropy (soft and mid-soft materials) and the smaller the standard deviations for the polar and azimuthal angle distributions, the bigger
192
C. Michelakis et al. / Journal of Magnetism and Magnetic Materials 207 (1999) 188}192
along the medium's easy axis is a good approximation for highly anisotropic particles and external "elds applied close to the medium's easy axis, while a magnetization-dependent #uctuation "eld is expected to be a more realistic approach of a low anisotropy material in "elds far from the easy axis. The ability of the model to simulate the behavior of soft materials, as the ones used in technology of high currents is worth a closer examination.
References Fig. 4. Transverse major loop comparison: present work (solid line) * model with interaction #uctuations along particles' easy axes (dotted line).
the di!erences observed. A comparison of transverse major loops computed according to the two models is given in Fig. 4 for the following parameter values: SH T"2000 Oe, p k"100 Oe, p i" k H H 500 Oe, p "53, p "53, q"0.5, a"150. Di!erh ( ences as high as 15% are observed near the knee of the curve. A "xed interaction "eld distribution
[1] J. Oti, E. Della Torre, J. Appl. Phys. 67 (1990) 5364. [2] H.A.J. Cramer, J. Magn. Magn. Mater. 88 (1990) 194. [3] F. Ossart, R. Davidson, S.H. Charap, IEEE Trans. Magn. 31 (1995) 1785. [4] R.J. Davidson, S.H. Charap, IEEE Trans. Magn. 32 (1996) 4198. [5] C. Michelakis, D. Samaras, G. Litsardakis, J. Magn. Magn. Mater. 196}197 (1999) 599. [6] I.A. Beardsley, J. Appl. Phys. 53 (1982) 2582. [7] E. Della Torre, IEEE Trans. Audio AU-14 (1966) 86. [8] X.D. Che, H.N. Bertram, J. Magn. Magn. Mater. 116 (1992) 121. [9] F. Vajda, E. Della Torre, IEEE Trans. Magn. 32 (1996) 1112.