- Email: [email protected]
Experimental study and theoretical interpretation of hysteresis loops and Henkel plots in soft magnetic materials
~
journalof magnetism
,~ ELSEVIER
and
magnetic materials
Journal of Magnetism and Magnetic Materials 133 (1994) 111-114
Experimental study and theoretical interpretation of hysteresis loops and Henkel plots in soft magnetic materials V. Basso a,,, M. Lo Bue b, A. Magni a G. Ummarino a, G. Bertotti
a
a Istituto Elettrotecnico Nazionale Galileo Ferraris and GNSM-INFM, 1-10125 Torino, Italy b Dipartimento di Fisica, Politecnico di Torino, 1-10129 Torino, Italy
Abstract
In this paper we discuss an application of the moving Preisach model (MPM) and the zero-temperature Sherrington-Kirkpatrick spin-glass model (SKM) to the interpretation of hysteresis phenomena in soft materials, with particular attention to the analogies and differences between the models. Both models are compared with hysteresis measurements performed on strips of amorphous F e - B - S i alloy. We describe experimental data using the same distribution of local coercivities and interactions fields in both of them. The results confirm that the MPM is actually able to reproduce the hysteresis behavior of soft magnetic materials and can be useful to discuss the complicated behavior of the SKM, where we explicitly know the form of interactions.
I. Introduction
The interpretation of hysteresis phenomena in soft magnetic materials requires the detailed treatment of several nontrivial aspects, namely the difficulty to connect the stochastic nature of domain wall (DW) Barkhausen jumps with macroscopic hysteresis properties, the presence of important reversible magnetization contributions, the dominant role of long-range magnetostatic fields. Several authors have stressed the fact that Barkhausen jumps are equivalent to local switching of some small volume in the material, so that models based on sets of Ising-spin-like bistable elements s i = + 1 should be able to provide a good physical description [1]. This conclusion is confirmed by N6el's model of DW motion in the Rayleigh region [2], which was shown to be equivalent to the Preisach model, one of the simplest and best known models based on bistable interacting elements. The main drawback of the Preisach model is that the statistical distribution of the interactions among the s i elements must
* Corresponding author. Tel. +39 (11) 3919753; fax +39 (ll) 650761; e-mail: [email protected].
be given a priori, although on physical grounds it would be expected to be state-dependent. In this respect, a major improvement is represented by the socalled moving Preisach model (MPM) [3], where statedependent mean-field effects are introduced through the relation H = H a + k l between the effective field H acting on each PM element si, the applied field H a and the system magnetization I. A more detailed treatment of this aspect is better worked out, however, through models where the coupling between the switching elements is explicitly given. As shown in Ref. [4], an interesting approach of this kind is provided by the zero-temperature Sherrington-Kirkpatrick spinglass model (SKM), where the collective s i behavior is controlled by the set of equations: Hi = E j J i j s j + Ha, s i = sgn(H/), where H i is the local field acting on si, Jij are random interaction constants, and the second equation describes the local stability condition for each s i. This model, possibly supplemented with some description of the local coercive field acting on each si, contains the same basic concepts of the Preisach model, but expressed in much more detailed form. In this paper, we discuss the application of the MPM and SKM to the interpretation of hysteresis phenomena in soft materials, with particular attention to the analogies and differences between the MPM
0304-8853/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00055-V
112
V. Basso et at/Journal of Magnetism and Magnetic Materials 133 (1994) 111-114
and SKM as physical hysteresis models. To this end, hysteresis measurements were performed on strips of amorphous F e - B - S i alloys. We have found that both the models give a good description of experimental data and that, interestingly enough, this is obtained by using the same distribution of local coercivities and interaction fields in both of them. This confirms that the MPM is actually able to reproduce the hysteresis behavior of soft magnetic materials and can be useful to discuss the complicated behavior of the model system (SKM) where, although in a highly idealized and somewhat unrealistic form, we explicitly know the form of interactions.
2. The MPM and SKM models
As discussed in previous papers [5], the MPM can provide a good physical framework for the description of soft magnetic materials, once a proper characterization of reversible magnetization contributions is introduced. Reversible processes can be described, in the simplest form, by adding to the PM switching field distribution p(a,[3) an extra distribution of zero-width PM elementary loops, according to the relation:
l(t)
=
(Ii
(1Hm
~'u Ot
-- A l r e v ) J _ H m d O L j _ H m d [ 3 p ( o ~
, [3)
x 6 [ a , [3; H ( t ) ] + A l r e v F [ H ( t ) ] ,
(1)
where I(t) is the magnetization at time t, I S is the saturation magnetization, H m is some peak field associated with the problem, ~b(a, [3; [H(t)])= +1 describes the state of each elementary PM loop, dependent on the whole past history [H(t)] of the magnetic field, p ( a , [3) is the PM switching field distribution, Alrev is the total contribution due to reversible processes, and F ( H ) is proportional to the integral of the reversible susceptibility. Mean-field effects are described by assuming that the local field H experienced by each PM elementary loop is H / H ~ = H a / H c + k I / I s = h a + ki, where the strength of the mean-field contribution, proportional to the magnetization I, is measured by the dimensionless parameter k, H¢ is some characteristic field giving the field scale of the problem, and small letters are used, here as well as in the following, to represent dimensionless quantities. By making a specific choice for the functions p(oe, [3) and F ( H ) , a complete description of hysteresis properties is obtained. According to previous studies [5,6], amorphous materials are well described by taking F ( H ) equal to the integral of a stretched exponential function: 1 fHU)/Hr~'dx exp( -Ix I C), F[H(t)] F ( 1 + l / c ) "o
(2)
and the switching field distribution p(ce, [3) equal to a log-normal Gaussian distribution of the form: p ( h c , hu) = Y ( h c ) g ( h u ) 1
1
[
4-fro'cotu hc exp
×exp
-
(loghc) 2 ] ff~r-~
j,
J
(3)
where h c = ( a - [ 3 ) / 2 H c and h u = (a + [3)/2H c represent, in units of He, the width and the shift of each elementary PM loop. Notice that the MPM amounts to changing h u into h u - ki in Eq. (3). The basic concept underlying MPM is the existence of bistable contributions, say s i = ___1, whose mutual interactions give rise, together with the presence of some local coercive field he, to the PM distribution p(a, [3). Instead of considering some arbitrary a priori choice for p ( a , [3), we can try a more detailed description of interactions, in view of a subsequent analysis of the consequences on the macroscopic hysteresis properties and of the possible analogies with the PM description. A rather interesting model of this kind is the generalization of the zero-temperature SKM considered in Ref. [4], where the collective behavior of N Ising-like spins s~ is described by a set of random spin-spin coupling constants {J~j} and by a statistical distribution {hT} of local spin coercive fields. The zerotemperature behavior of the system is fully controlled by the local fields h i experienced by each spin s~, according to the equations: h i = E J i j s j + s i h c + h a = h u + s i h c + h a, J Si =
sgn(hi),
(4)
where h i is the (dimensionless) local field acting on si, and the equation s i = sgn(hi) describes the local stability condition for each s i. Note that the local spin coercive field h7 and the interaction field h~ play the same role as the PM coordinates h c and - h u previously introduced. Any configuration satisfying Eqs. (4) is stable against single spin-flipping. However, this stability is destroyed when h a changes in time. The spins are forced towards instability (i.e., s i --g sgn(hi)) and progressively flip, thus producing magnetization changes [4]. This dynamics gives rise to realistic hysteresis loops and also to a realistic description of the Barkhausen effect, arising from spin-spin coupling. The random constants {Ji~} are infinite-range spin-spin coupling constants, in the sense that Ju is different
V. Basso et al. /Journal of Magnetism and Magnetic Materials 133 (1994) 111-114 from zero for each spin couple, and are generated independently for each spin couple from a Gaussian distribution P ( ~ j ) ct e x p [ - ( J i j - h o / N ) Z / ( 2 o ' 2 / N ) ] . This implies that when the system is saturated (i.e. s i = + 1 (or - 1) for each i), also the distribution of the interaction field h~ will be Gaussian, P(h~) ct e x p [ - ( h ~ - h 0 ) 2 / 2 ~ ] . This distribution plays the role of the function g(h~) of Eq. (3). In addition, the presence of the constant h 0 shifts the whole distribution just in the same way as the term ki in the MPM. By comparing the saturation h~ distributions in the two models it is found that h 0 = ( 1 - Alrev/Is)k. In a similar way, the local spin coercive field distribution P ( h c) is expected to coincide with the function f(h~) of Eq. (3). In this way, the MPM and SKM are fully determined and can be directly compared and used to reproduce experimental data. To this end, we have to make a choice for H~, cry, % and k and we have to sum to both models the reversible magnetization contribution described by Eq. (2). This analysis is of interest from two rather different viewpoints. On the one hand, we are interested in testing the ability of the MPM to reproduce the description of interactions implied by the SKM. This is a case in fact where, although for a highly idealized model system, we explicitly know the form of interactions and we can check to what extent the MPM is able to reproduce the basic consequences of such interactions on the hysteresis phenomenology. On the other hand, we are interested in checking the ability of both models to reproduce experimental data, in order to study their value and possible applications as physical descriptions of hysteresis phenomena.
3. Experimental results and discussion Experiments were performed on samples (length 300 ram, width 5 mm, thickness 15 Ixm) of annealed Fe7sB13Si 9 amorphous ribbons (350°C in longitudinal field of 2000 A / m ) . The virgin magnetization curve, the saturation and minor hysteresis loops, the loop areas and various first-order reversal curves were measured by a computer-controlled low-drift fluxmetric setup. The specimen was placed in a 702 turns, 0.284 m long coil. The signal was detected by a narrow 400 turns coil placed in the middle of the strip. The specimen and the coils were placed in a double mumetal box and all measurement were performed in a shielded room in order to avoid electromagnetic interference. The sample was magnetized by triangular primary current waveform of variable frequency generated by a Krohn-Hite 5920 arbitrary function generator amplified with a Kepco bipolar operational amplifier. All the magnetization curves were performed with a constant low rate d H / d t = 2.28 A / m s , and analyzed with an HP 3562A dynamic signal analyzer.
i
113 I
1.0
0.5
~" 0.0
< i is
-0.5 -1.0 I
-10
-5
0 5 10 H (A/m) Fig. 1. Hysteresis loops. (a) Dotted line: experimental loop measured with constant rate d H / d t =2.28 A/ms on annealed Fe-B-Si amorphous ribbons; (b) solid line: MPM loop; (c) dashed line: SKM loop simulated with N = 1600 spins. Parameters (from Eqs. (l)-(3)): % = 0.5; o-~= 7, k = 5, H e = 1.6 A/m, I~ = 1.56 T, Alrcv = 0.1431s, //rev = 45.6 A/m, c = 0.7.
Fig. 1 shows a typical measured hysteresis loop compared with the two model predictions. The choice of parameter values follows a previous analysis [8] and implies a positive value of k. We obtain a good description of experimental data with the MPM, while the SKM simulated loop shows a definite discrepancy. The large coercive field obtained with the SKM partly depends on the presence of random spin-spin interactions. In Ref. [4], it was found that the SKM loop coercive field is proportional to o"u but inversely proportional to some power of the number of spins N considered in the computer simulations. We believe that the difference of coercivities in the MPM and SKM should disappear in the limit N ~ ~. We also found that the introduction of h 0 in the SKM model has actually the same effect of the mean-field parameter k in the MPM: the shift of the Jij distribution is able to change the loop shape, but not the loop area. This can also be seen in the agreement of loop vertices. In order to investigate the role of interactions, Henkel plots have been measured and simulated with the MPM and SKM. In Fig. 2, the I r and I d remanences, obtained by applying and removing fields H and - H , respectively, to the demagnetized and the saturation remanent states I~ are plotted. The agreement of the MPM and SKM Henkel plots with experimental data confirms that the internal disorder of the material gives a demagnetizing-like contribution (i.e. the Henkel plot is below the non-interaction line I d / I ~ = 1 -- 2 I J l ~ ) and is well described by the wide interaction field distributions g(h u) and P(Jis). As shown in a previous paper [7], for the MPM a very large value of ~ru gives a demagnetizing-like effect, while a positive
114
V. Basso et al. / Journal of Magnetism and Magnetic Materials 133 (1994) 111-114 |,
0
~
, " " 0.(
,
I 0.2
i
~i
'
I
'
I
H e n k e l plot. This fact confirms that, in this case, the M P M gives a good description of r a n d o m s p i n - s p i n interactions of the S K M in spite of the differences in the hysteresis loop. In future work we will investigate the r e l a t i o n b e t w e e n the models using a n a r r o w interaction field distribution and a larger n u m b e r of spins, in o r d e r to minimize the additional energy losses present in the SKM and not in the M P M . In this case a more detailed comparison, of b o t h hysteresis curves a n d H e n k e l plots, can be w o r k e d out.
'
~ 0.4
0.6
0.8
l-.0
Ir /Ico Fig. 2. Henkel plots. Symbols: experimental data on annealed F e - B - S i amorphous ribbons; solid line: MPM prediction; broken line: SKM simulation. Straight lines I~ = 1 ~ - 2 1 r and I d =I r are the upper and lower limits allowed by the classical Preisach model.
value of the m e a n - f i e l d p a r a m e t e r k gives a magnetizing-like effect. In this case the d e m a g n e t i z i n g effect is d o m i n a n t . It is very interesting to note t h a t also in the S K M the s p r e a d of the r a n d o m interactions distribution gives a demagnetizing-like c o n t r i b u t i o n to the
References
[1] G. Bertotti and M. Pasquale, J. Appl. Phys. 67 (1990) 5255. [2] L. N~el, Cabiers de Physique 12 (1942) 1. [3] E. Della Torre, IEEE Trans. Audio 14 (1966) 86. [4] G. Bertotti and M. Pasquale, J. Appl. Phys. 69 (1991) 5066. [5] G. Bertotti, F. Fiorillo and M. Pasquale, IEEE Trans. Magn. 29 (1993). [6] C. Appino and F. Fiorillo, J. Magn. Magn. Mater. 133 (1994) 107 (this volume). [7] V. Basso and G. Bertotti, IEEE Trans. Magn. 30 (1994) to appear. [8] G. Bertotti, V. Basso and M. Pasquale, EMMA-93.
journalof magnetism
,~ ELSEVIER
and
magnetic materials
Journal of Magnetism and Magnetic Materials 133 (1994) 111-114
Experimental study and theoretical interpretation of hysteresis loops and Henkel plots in soft magnetic materials V. Basso a,,, M. Lo Bue b, A. Magni a G. Ummarino a, G. Bertotti
a
a Istituto Elettrotecnico Nazionale Galileo Ferraris and GNSM-INFM, 1-10125 Torino, Italy b Dipartimento di Fisica, Politecnico di Torino, 1-10129 Torino, Italy
Abstract
In this paper we discuss an application of the moving Preisach model (MPM) and the zero-temperature Sherrington-Kirkpatrick spin-glass model (SKM) to the interpretation of hysteresis phenomena in soft materials, with particular attention to the analogies and differences between the models. Both models are compared with hysteresis measurements performed on strips of amorphous F e - B - S i alloy. We describe experimental data using the same distribution of local coercivities and interactions fields in both of them. The results confirm that the MPM is actually able to reproduce the hysteresis behavior of soft magnetic materials and can be useful to discuss the complicated behavior of the SKM, where we explicitly know the form of interactions.
I. Introduction
The interpretation of hysteresis phenomena in soft magnetic materials requires the detailed treatment of several nontrivial aspects, namely the difficulty to connect the stochastic nature of domain wall (DW) Barkhausen jumps with macroscopic hysteresis properties, the presence of important reversible magnetization contributions, the dominant role of long-range magnetostatic fields. Several authors have stressed the fact that Barkhausen jumps are equivalent to local switching of some small volume in the material, so that models based on sets of Ising-spin-like bistable elements s i = + 1 should be able to provide a good physical description [1]. This conclusion is confirmed by N6el's model of DW motion in the Rayleigh region [2], which was shown to be equivalent to the Preisach model, one of the simplest and best known models based on bistable interacting elements. The main drawback of the Preisach model is that the statistical distribution of the interactions among the s i elements must
* Corresponding author. Tel. +39 (11) 3919753; fax +39 (ll) 650761; e-mail: [email protected].
be given a priori, although on physical grounds it would be expected to be state-dependent. In this respect, a major improvement is represented by the socalled moving Preisach model (MPM) [3], where statedependent mean-field effects are introduced through the relation H = H a + k l between the effective field H acting on each PM element si, the applied field H a and the system magnetization I. A more detailed treatment of this aspect is better worked out, however, through models where the coupling between the switching elements is explicitly given. As shown in Ref. [4], an interesting approach of this kind is provided by the zero-temperature Sherrington-Kirkpatrick spinglass model (SKM), where the collective s i behavior is controlled by the set of equations: Hi = E j J i j s j + Ha, s i = sgn(H/), where H i is the local field acting on si, Jij are random interaction constants, and the second equation describes the local stability condition for each s i. This model, possibly supplemented with some description of the local coercive field acting on each si, contains the same basic concepts of the Preisach model, but expressed in much more detailed form. In this paper, we discuss the application of the MPM and SKM to the interpretation of hysteresis phenomena in soft materials, with particular attention to the analogies and differences between the MPM
0304-8853/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00055-V
112
V. Basso et at/Journal of Magnetism and Magnetic Materials 133 (1994) 111-114
and SKM as physical hysteresis models. To this end, hysteresis measurements were performed on strips of amorphous F e - B - S i alloys. We have found that both the models give a good description of experimental data and that, interestingly enough, this is obtained by using the same distribution of local coercivities and interaction fields in both of them. This confirms that the MPM is actually able to reproduce the hysteresis behavior of soft magnetic materials and can be useful to discuss the complicated behavior of the model system (SKM) where, although in a highly idealized and somewhat unrealistic form, we explicitly know the form of interactions.
2. The MPM and SKM models
As discussed in previous papers [5], the MPM can provide a good physical framework for the description of soft magnetic materials, once a proper characterization of reversible magnetization contributions is introduced. Reversible processes can be described, in the simplest form, by adding to the PM switching field distribution p(a,[3) an extra distribution of zero-width PM elementary loops, according to the relation:
l(t)
=
(Ii
(1Hm
~'u Ot
-- A l r e v ) J _ H m d O L j _ H m d [ 3 p ( o ~
, [3)
x 6 [ a , [3; H ( t ) ] + A l r e v F [ H ( t ) ] ,
(1)
where I(t) is the magnetization at time t, I S is the saturation magnetization, H m is some peak field associated with the problem, ~b(a, [3; [H(t)])= +1 describes the state of each elementary PM loop, dependent on the whole past history [H(t)] of the magnetic field, p ( a , [3) is the PM switching field distribution, Alrev is the total contribution due to reversible processes, and F ( H ) is proportional to the integral of the reversible susceptibility. Mean-field effects are described by assuming that the local field H experienced by each PM elementary loop is H / H ~ = H a / H c + k I / I s = h a + ki, where the strength of the mean-field contribution, proportional to the magnetization I, is measured by the dimensionless parameter k, H¢ is some characteristic field giving the field scale of the problem, and small letters are used, here as well as in the following, to represent dimensionless quantities. By making a specific choice for the functions p(oe, [3) and F ( H ) , a complete description of hysteresis properties is obtained. According to previous studies [5,6], amorphous materials are well described by taking F ( H ) equal to the integral of a stretched exponential function: 1 fHU)/Hr~'dx exp( -Ix I C), F[H(t)] F ( 1 + l / c ) "o
(2)
and the switching field distribution p(ce, [3) equal to a log-normal Gaussian distribution of the form: p ( h c , hu) = Y ( h c ) g ( h u ) 1
1
[
4-fro'cotu hc exp
×exp
-
(loghc) 2 ] ff~r-~
j,
J
(3)
where h c = ( a - [ 3 ) / 2 H c and h u = (a + [3)/2H c represent, in units of He, the width and the shift of each elementary PM loop. Notice that the MPM amounts to changing h u into h u - ki in Eq. (3). The basic concept underlying MPM is the existence of bistable contributions, say s i = ___1, whose mutual interactions give rise, together with the presence of some local coercive field he, to the PM distribution p(a, [3). Instead of considering some arbitrary a priori choice for p ( a , [3), we can try a more detailed description of interactions, in view of a subsequent analysis of the consequences on the macroscopic hysteresis properties and of the possible analogies with the PM description. A rather interesting model of this kind is the generalization of the zero-temperature SKM considered in Ref. [4], where the collective behavior of N Ising-like spins s~ is described by a set of random spin-spin coupling constants {J~j} and by a statistical distribution {hT} of local spin coercive fields. The zerotemperature behavior of the system is fully controlled by the local fields h i experienced by each spin s~, according to the equations: h i = E J i j s j + s i h c + h a = h u + s i h c + h a, J Si =
sgn(hi),
(4)
where h i is the (dimensionless) local field acting on si, and the equation s i = sgn(hi) describes the local stability condition for each s i. Note that the local spin coercive field h7 and the interaction field h~ play the same role as the PM coordinates h c and - h u previously introduced. Any configuration satisfying Eqs. (4) is stable against single spin-flipping. However, this stability is destroyed when h a changes in time. The spins are forced towards instability (i.e., s i --g sgn(hi)) and progressively flip, thus producing magnetization changes [4]. This dynamics gives rise to realistic hysteresis loops and also to a realistic description of the Barkhausen effect, arising from spin-spin coupling. The random constants {Ji~} are infinite-range spin-spin coupling constants, in the sense that Ju is different
V. Basso et al. /Journal of Magnetism and Magnetic Materials 133 (1994) 111-114 from zero for each spin couple, and are generated independently for each spin couple from a Gaussian distribution P ( ~ j ) ct e x p [ - ( J i j - h o / N ) Z / ( 2 o ' 2 / N ) ] . This implies that when the system is saturated (i.e. s i = + 1 (or - 1) for each i), also the distribution of the interaction field h~ will be Gaussian, P(h~) ct e x p [ - ( h ~ - h 0 ) 2 / 2 ~ ] . This distribution plays the role of the function g(h~) of Eq. (3). In addition, the presence of the constant h 0 shifts the whole distribution just in the same way as the term ki in the MPM. By comparing the saturation h~ distributions in the two models it is found that h 0 = ( 1 - Alrev/Is)k. In a similar way, the local spin coercive field distribution P ( h c) is expected to coincide with the function f(h~) of Eq. (3). In this way, the MPM and SKM are fully determined and can be directly compared and used to reproduce experimental data. To this end, we have to make a choice for H~, cry, % and k and we have to sum to both models the reversible magnetization contribution described by Eq. (2). This analysis is of interest from two rather different viewpoints. On the one hand, we are interested in testing the ability of the MPM to reproduce the description of interactions implied by the SKM. This is a case in fact where, although for a highly idealized model system, we explicitly know the form of interactions and we can check to what extent the MPM is able to reproduce the basic consequences of such interactions on the hysteresis phenomenology. On the other hand, we are interested in checking the ability of both models to reproduce experimental data, in order to study their value and possible applications as physical descriptions of hysteresis phenomena.
3. Experimental results and discussion Experiments were performed on samples (length 300 ram, width 5 mm, thickness 15 Ixm) of annealed Fe7sB13Si 9 amorphous ribbons (350°C in longitudinal field of 2000 A / m ) . The virgin magnetization curve, the saturation and minor hysteresis loops, the loop areas and various first-order reversal curves were measured by a computer-controlled low-drift fluxmetric setup. The specimen was placed in a 702 turns, 0.284 m long coil. The signal was detected by a narrow 400 turns coil placed in the middle of the strip. The specimen and the coils were placed in a double mumetal box and all measurement were performed in a shielded room in order to avoid electromagnetic interference. The sample was magnetized by triangular primary current waveform of variable frequency generated by a Krohn-Hite 5920 arbitrary function generator amplified with a Kepco bipolar operational amplifier. All the magnetization curves were performed with a constant low rate d H / d t = 2.28 A / m s , and analyzed with an HP 3562A dynamic signal analyzer.
i
113 I
1.0
0.5
~" 0.0
< i is
-0.5 -1.0 I
-10
-5
0 5 10 H (A/m) Fig. 1. Hysteresis loops. (a) Dotted line: experimental loop measured with constant rate d H / d t =2.28 A/ms on annealed Fe-B-Si amorphous ribbons; (b) solid line: MPM loop; (c) dashed line: SKM loop simulated with N = 1600 spins. Parameters (from Eqs. (l)-(3)): % = 0.5; o-~= 7, k = 5, H e = 1.6 A/m, I~ = 1.56 T, Alrcv = 0.1431s, //rev = 45.6 A/m, c = 0.7.
Fig. 1 shows a typical measured hysteresis loop compared with the two model predictions. The choice of parameter values follows a previous analysis [8] and implies a positive value of k. We obtain a good description of experimental data with the MPM, while the SKM simulated loop shows a definite discrepancy. The large coercive field obtained with the SKM partly depends on the presence of random spin-spin interactions. In Ref. [4], it was found that the SKM loop coercive field is proportional to o"u but inversely proportional to some power of the number of spins N considered in the computer simulations. We believe that the difference of coercivities in the MPM and SKM should disappear in the limit N ~ ~. We also found that the introduction of h 0 in the SKM model has actually the same effect of the mean-field parameter k in the MPM: the shift of the Jij distribution is able to change the loop shape, but not the loop area. This can also be seen in the agreement of loop vertices. In order to investigate the role of interactions, Henkel plots have been measured and simulated with the MPM and SKM. In Fig. 2, the I r and I d remanences, obtained by applying and removing fields H and - H , respectively, to the demagnetized and the saturation remanent states I~ are plotted. The agreement of the MPM and SKM Henkel plots with experimental data confirms that the internal disorder of the material gives a demagnetizing-like contribution (i.e. the Henkel plot is below the non-interaction line I d / I ~ = 1 -- 2 I J l ~ ) and is well described by the wide interaction field distributions g(h u) and P(Jis). As shown in a previous paper [7], for the MPM a very large value of ~ru gives a demagnetizing-like effect, while a positive
114
V. Basso et al. / Journal of Magnetism and Magnetic Materials 133 (1994) 111-114 |,
0
~
, " " 0.(
,
I 0.2
i
~i
'
I
'
I
H e n k e l plot. This fact confirms that, in this case, the M P M gives a good description of r a n d o m s p i n - s p i n interactions of the S K M in spite of the differences in the hysteresis loop. In future work we will investigate the r e l a t i o n b e t w e e n the models using a n a r r o w interaction field distribution and a larger n u m b e r of spins, in o r d e r to minimize the additional energy losses present in the SKM and not in the M P M . In this case a more detailed comparison, of b o t h hysteresis curves a n d H e n k e l plots, can be w o r k e d out.
'
~ 0.4
0.6
0.8
l-.0
Ir /Ico Fig. 2. Henkel plots. Symbols: experimental data on annealed F e - B - S i amorphous ribbons; solid line: MPM prediction; broken line: SKM simulation. Straight lines I~ = 1 ~ - 2 1 r and I d =I r are the upper and lower limits allowed by the classical Preisach model.
value of the m e a n - f i e l d p a r a m e t e r k gives a magnetizing-like effect. In this case the d e m a g n e t i z i n g effect is d o m i n a n t . It is very interesting to note t h a t also in the S K M the s p r e a d of the r a n d o m interactions distribution gives a demagnetizing-like c o n t r i b u t i o n to the
References
[1] G. Bertotti and M. Pasquale, J. Appl. Phys. 67 (1990) 5255. [2] L. N~el, Cabiers de Physique 12 (1942) 1. [3] E. Della Torre, IEEE Trans. Audio 14 (1966) 86. [4] G. Bertotti and M. Pasquale, J. Appl. Phys. 69 (1991) 5066. [5] G. Bertotti, F. Fiorillo and M. Pasquale, IEEE Trans. Magn. 29 (1993). [6] C. Appino and F. Fiorillo, J. Magn. Magn. Mater. 133 (1994) 107 (this volume). [7] V. Basso and G. Bertotti, IEEE Trans. Magn. 30 (1994) to appear. [8] G. Bertotti, V. Basso and M. Pasquale, EMMA-93.