192 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 1, JANUARY 2011

Reversible and Irreversible Components Evaluation in Hysteretic ProcessesUsing First and Second-Order Magnetization Curves

Ilie Bodale, Laurentiu Stoleriu, and Alexandru Stancu

Department of Physics, Alexandru Ioan Cuza University, Iasi 700506, Romania

This paper presents a method to separate the reversible and irreversible parts of magnetization in real ferromagnetic systems. Thebasic hypothesis is that the first-order reversal curve (FORC) diagram includes effects of both the reversible and irreversible componentswhile the remanent second-order reversal curve ������� diagram includes essentially only the irreversible component. This meansthat the difference of the two diagrams describes only the reversible part. The experimental and simulated results were compared andthe differences are discussed. Various scalar Preisach-type models have been used to simulate the FORC, ����� and the differencebetween the two distributions.

Index Terms—FORC diagram, Preisach model, remanent SORC diagram, reversible and irreversible magnetization.

I. INTRODUCTION

I N RECENT years, researchers have focused their atten-tion on increasing the recording density in magnetic storage

media. A key to achieve good technological performances is tohave a good knowledge of both irreversible switches and re-versible behavior of magnetization within the magnetic mate-rial. Any magnetization procedure implies both processes andall the attempts to evaluate separately the two parts of magne-tization had, so far, only limited success [1]. The origins of theirreversible magnetization processes are usually associated withthe dissipation of energy and with the switching of the magneticmoment between two equilibrium positions [2]. On the otherhand, the reversible processes are associated with the rotationof the magnetic moment [3] or with domain wall motions [4] inthe absence of switches. The irreversible component depends onthe actual applied field and on the history of the field extremes,while the usual assumption is that the reversible component de-pends only on the actual value of the applied field.

A first deconvolution technique of magnetization into re-versible and irreversible components has been introduced byMayergoyz [5] by using the nonzero initial slopes of magne-tization curves in the reversal points to evaluate the reversibleprocesses in the Preisach model [6].

In subsequent studies of Della Torre and coworkers it hasalso been proven that the reversible component is correlatedwith the irreversible distribution. To include this effect in thePreisach-type models, a number of solutions have been pro-posed (DOK model [7], VD1-2 models [8], CMH model [9]).The reversible component in these models is linked with theirreversible component. The most complex model in this se-ries is the complete moving hysteresis (CMH) model, which isusing a distribution of nonrectangular but similar in shape hys-terons, (with the same squareness as the major hysteresis loop).Our micromagnetic study has shown that the squareness of the

Manuscript received July 02, 2010; revised August 09, 2010; acceptedSeptember 20, 2010. Date of publication October 04, 2010; date of currentversion December 27, 2010. Corresponding author: A. Stancu (e-mail: alstancu@uaic.ro).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2010.2083679

pseudo-particles (equivalent of the nonrectangular hysterons) isnot uniformly distributed in the Preisach plane [10].

In a recent paper, Winklhofer and coworkers [11] have usedfirst-order reversal curve (FORC) and second-order reversalcurve (SORC) techniques to separately quantify reversible andirreversible magnetization changes within the major hysteresisloop. The susceptibility on the SORCs in the reversal points iscalculated and shows a clear state dependency of the reversiblepart. The irreversible susceptibility calculated as the first deriva-tive of the FORCs show also a bimodal shape and dependenceon the magnetic state of the sample. However, the use of theSORCs was limited to the evaluation of these susceptibilities.

The effect of the connection between the reversible and ir-reversible parts as a result of mean field interactions and statedependent variable variance of the statistical distribution of in-teractions was discussed by us in a previous paper [12]. We haveobserved that the reversible part of magnetization, calculated asthe susceptibility in the reversal point on the FORCs, is clearlyasymmetric for most ferromagnetic samples, especially for theparticulate media. This was correlated with the shape of the re-versible contribution in the Stoner-Wohlfarth (SW) model forsingle domain particles [13]. Moreover, as a direct consequenceof the physical source of the reversible ridge, the reversible partis fundamentally bimodal and state dependent.

This result has indicated a straightforward method to includethe reversible part in the Preisach model for patterned media[14], model which is already using a bimodal state dependentdistribution of the irreversible part.

In this paper, we discuss a deconvolution method for re-versible and irreversible components which is also based onFORC and SORC measurements. If one makes a selection ofpoints from the measured sets of SORC data—the value of themoment in the remanent state on the second-order magnetiza-tion curve—one obtains a set of data very similar to the FORCset. One identifies this set as “remanent SORC”because, in fact, this set is covering the remanent hysteresisloop as the FORC is covering the surface of the in-field majorhysteresis loop (MHL). If one takes into account that all the

moments are measured in the absence of an appliedfield, one can make the hypothesis that the reversible influ-ence on these data is negligible. Consequently, the differencebetween the FORC and data in each state will give acomprehensive image of the reversible magnetization processesin correlation with the irreversible magnetization processes and

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BODALE et al.: REVERSIBLE AND IRREVERSIBLE COMPONENTS EVALUATION IN HYSTERETIC PROCESSES 193

their interdependence. In this paper, we analyze the ability ofthe main Preisach-type models to reproduce the typical featuresobserved on the diagrams.

In the next section, one presents the main concepts used inthe classical Preisach model and then one defines the FORC and

measurements that are used as experimental techniquesin this analysis. In the third section, one presents the typicalfeatures observed on the diagrams and howthe models can reproduce these features.

II. CLASSICAL PREISACH MODEL. FORC AND

MAGNETIZATION PROCESSES

The classical Preisach model (CPM) [6] essentially deals withthe magnetization processes of a system of interacting SW par-ticles [13] with the easy axes aligned on the applied field direc-tion. The magnetic moment of each particle has a single point as-sociated in the Preisach plane whose coordinates arethe particle’s switching fields. As the particles are not identicaland their number is extremely large, a continuous distribution ofthe particle’s moments as a function of the switching fields canbe defined (Preisach distribution). In many cases, especially forferromagnetic materials, the Preisach distribution can be ratheraccurately decomposed in two statistically independent distribu-tions, of coercive and interaction fields. Thesefields are directly related to the switching fields by the relations

(1)

For simplification, a 45 rotated coordinate system is oftenused and the coercivity and interaction fields in this system aregiven by .

As the two distributions are statistically independent, thePreisach distribution is given by

(2)

In CPM, the total magnetic moment is related only to irre-versible processes and is calculated as the difference betweenthe magnetic moments of the particles in “up” and “down”

states in the Preisach plane

(3)

While in order to calculate the total magnetic moment using(3) we have to know the Preisach distribution of the measuredsample, it is of paramount importance to have a reliable methodto identify this distribution. Mayergoyz proposed in [1] an iden-tification technique for a material which is correctly describedby the CPM, based on a set of FORCs.

Pike and coworkers [15] have later launched the idea to usethis identification technique as a purely experimental method.In this case, one obtains a distribution which is not the Preisachdistribution (the only exception is when the hysteretic processesare completely described by the CPM which are the systemsobeying the wiping-out and congruency properties [1]).

Fig. 1. The FORC and ���� magnetizations.

The FORC distribution for non-CPM systems is only anapproximation of the Preisach distribution of rectangular (ir-reversible) hysterons. Mean field interaction, the complexityof the reversible magnetization processes, and the couplingsbetween the reversible and irreversible processes are the maincauses of the differences between the Preisach and FORCdistributions. Usually the FORC distribution is seen as an eval-uation of the irreversible part of magnetization. The contourplot of the FORC distribution is named FORC diagram.

The reversible part is taken into account as a singular distri-bution on the first bisector of the Preisach plane which is calcu-lated as the susceptibility on the FORCs exactly in the point ofreversal. The points on the first bisector are associated to degen-erated reversible hysterons (step functions) with no coercivity.

Effectively, from a set of experimental FORCs one can cal-culate two FORC distributions, one representing the irreversibleprocesses, which has nonzero values in the Preisach plane, anda singular ridge on the first bisector of the same plane associatedto the reversible processes in the sample. However, the physicalreality is much more complicated than that.

To observe this fact, we discuss the second-order reversalcurves which are measured between a point on a FORC andthe point where the FORC started on the major-loop [16]. TheSORCs may cover with magnetization curves the surface be-tween the MHL and a FORC (see Fig. 1). The introduction ofthe SORCs in this manner is not easy to use and to study experi-mentally. The number of experimental data is increasing signifi-cantly and this huge amount of data is really hard to be analyzed.Therefore, we are introducing a simpler measurement which isalready familiar in many laboratories [17]. From each SORC,we propose to use only the intersection with the zero field axis(remanent data). In this way, starting from a FORC measure-ment, we replace the value measured in-field on FORC with theremanent value (we measure after we apply a typical FORC se-quence of field followed by a zero field). This type of data setis obtained from second-order reversal curves and contains onlyremanent measurements. We shall call it measurement.As we do not apply an external field during the moment mea-surement one expects that the variations of the magnetic mo-ment measured on such a process will be essentially due to irre-

194 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 1, JANUARY 2011

Fig. 2. A SORC set measured for � (a) and ���� s constructed as remanent magnetizations from experimental SORCs (b).

versible processes. One can easily show in CPM that the FORCand diagrams should be identical, even if a reversiblepart is associated to the first bisector. This fact observed in realsamples it would confirm that the FORC distribution associatedto the Preisach plane is a representation of irreversible processesonly. However, measurements made on real ferromagnetic par-ticulate samples show a clear difference between the FORC and

distributions. The possible origin of this difference willbe discussed in the following sections.

The FORCs are measured starting form one branch of theMHL by changing the sign of variation of the external field (Fig.1) [16]. Therefore, the magnetic moment on a certain FORC willbe characterized by both the value of the applied field when thereversal took place (called the first reversal field, ) and theactual value of the field .

Using the same strategy, a point on a curve [18]can be obtained by starting from a first-order reversal curveand measuring the remanent moment corresponding to eachpoint, thus the value of the magnetic moment on a second-ordermagnetization curve will be a function of the first reversal field

—associated with the FORC curve—as well as the lastvalue of the field applied before measuring the remanence,(called second reversal field ),(Fig. 2).

Based on the hypothesis that there is a negligible contribu-tion of the reversible processes to the total magnetization if theexternal applied field is zero [19], one may consider that a setof FORCs contains information about both irreversible and re-versible magnetization components while a set of s de-scribes only the irreversible magnetization because it is mea-sured in the absence of an external applied filed.

Taking that into account, one can define

(4)

which will be used in the evaluation of reversible component.

III. EXPERIMENTAL DETAILS

Fig. 3 shows two sets of FORC and SORC experimentaldata measured on a commercial magnetic recording mediausing a vibrating sample magnetometer system (Dual system

Fig. 3. The FORC (red lines—online) and ���� curves (black lines—on-line) measured for commercial magnetic recording media.

AGM/VSM MicroMag 2900/3900 from Princeton Measure-ment Co.). The moment is calculated using(4) and Fig. 4 shows the experimental results.

The FORC distribution is defined as the mixed second deriva-tive of the set of first order reversal curves [20], [21]:

(5)

and in a similar way, the distribution is the mixedsecond derivative of the set of remanent SORCs:

(6)

The diagram (Fig. 6) clearly shows only the regioncorresponding to the distribution of the irreversible component

.The diagram of the (Fig. 7) was obtained

as the mixed second derivative (6) of the experimental data setsin Fig. 4 (see also [18]). In addition to the classical reversibledistribution , the other three regions become visible: onenegative region near the reversible distribution and a pair

BODALE et al.: REVERSIBLE AND IRREVERSIBLE COMPONENTS EVALUATION IN HYSTERETIC PROCESSES 195

Fig. 4. The difference �� � between � and �measured for commercial magnetic recording media.

Fig. 5. The FORC diagram for a commercial magnetic recording media.

of regions, one positive and the other negative situ-ated along the second bisector of the Preisach plane, in the sameregion with the irreversible distribution.

IV. PREISACH TYPE MODELS SIMULATIONS

In order to study the features observed in the difference be-tween the FORC and diagrams, we have simulated theexperimental data with three models.

1) The CPM [6] was extended to the generalized Preisachmodel (GPM) [1], [22] by introducing a purely reversiblecomponent based on the degenerated (completely re-versible) hysteresis loops. The reversible part is associatedwith the first bisector of the Preisach plane and the re-versible magnetization is calculated following the samerules for the distribution as in the CPM through integrationof the reversible distribution :

(7)

Fig. 6. The ���� diagram for a commercial magnetic recording media.

The most important parameter in calculating the irre-versible and reversible weights of magnetization is thesquareness defined as the normalized remanent mag-netization . The magnetization of thesample is the sum of the two magnetizations:

(8)

2) The CPM hypothesis that the interaction field acting on acertain particle is independent of magnetization is usuallynot true for the real systems. The moving Preisach model(MPM) [23] considers that the interaction field, , is asum between the statistical interaction field, as describedby CPM (and associated with the demagnetization stateof the sample ), and a mean field interaction termproportional to the total magnetic moment of the sample

(9)

The proportionality term is called the “moving”parameter. The generalized moving Preisach model(GMPM) takes into account the reversible distribution inthe same manner as in GPM [24].

3) The Preisach model for patterned media (PM2) [14] ap-proach started from a micromagnetic analysis of the in-teraction field distribution and contains as particular casesboth moving and variable variance models [25]. PM2 isbased on the hypothesis that the interaction field distribu-tion is a superposition of two distributions with the sameabsolute value of the mean field but with opposite signs.The maximum values of the two interaction distributionsare dependent on the total magnetic moment of the sample[14], [26]:

(10)

To separate the reversible distribution from irreversible distri-bution, we have used the generalized PM2 (GPM2) type modelwhere the total magnetization is a sum of the two magneti-zation components (11), like in GPM:

(11)

196 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 1, JANUARY 2011

Fig. 7. The ��������� diagram for a commercial magnetic recordingmedia.

Symmetrical and asymmetrical reversible part can beadded to PM2 model. The symmetrical reversible component

was considered for all mentioned models as

(12)

An asymmetrical reversible distribution with a nonzero av-erage value of the distribution can be included in GPM2 only(noted GPM2-asymmetric) because the asymmetric part cannotbe introduced as a static function but just as magnetization pro-cesses dependent function. In our approach the reversible distri-bution depend on the current value of total magnetic moment,like in PM2 [14], and is given by

(13)

We have identified the parameters of the irreversible Preisachdistributions from experimental data with a log-normal functionfor coercive field distribution and a Gaussian function for inter-action field distribution. The asymmetric reversible distributionhas been identified using a modified Gauss function, which is aproduct between a Gaussian and an error function [12].

As we expect, one obtains identical FORC and dia-grams in the simulations when no reversible component is takeninto account.

Case 1. GPM: Using the GPM, one can separate the two com-ponents of the magnetization using the three diagrams (FORC,

, and ). In this case, each distributionis well separated but the experimental shape of and the threeregions in Fig. 7 are not explained (Fig. 8).

Simulations with an asymmetrical reversible functions inGPM and GMPM models have not been able to correctlydescribe the FORC and SORC’s curves and, consequently, onlysymmetrical functions can be used in these models.

Case 2. GMPM: In order to understand the origin of the fea-tures observed in experimental diagrams, we have constructedthe diagrams using a simulation with moving term—GMPM(Fig. 9). Indeed, the three additional reversible regions (

Fig. 8. The ��������� diagrams in the GPM.

Fig. 9. The��������� diagram in the GMPM. In this case, the additionthe three reversible regions �� �� � and� ) appear but the shape of � don’tfit the experimental date.

and ) are obtained but the shape of differs from the onein Fig. 7.

Case 3. GPM2-Asymmetric: In GPM2-asymmetric, we haveused for the reversible function a state dependent exponentiallymodified Gauss function (13) which explains the experimentalshape of (Fig. 10), its maximum position as well as three

and regions.If in a GPM model, one tries to decouple the two components

by changing with in (13):

(14)

one obtains an asymmetric reversible distribution but the threeadditional regions ( and ) cannot be observed any-more, thus one can conclude that these regions are associatedto the coupling between the irreversible and the reversiblecomponents.

In simulations made for systems with very strong interactions(also called “three quadrant media” by Della Torre [27], [28])

BODALE et al.: REVERSIBLE AND IRREVERSIBLE COMPONENTS EVALUATION IN HYSTERETIC PROCESSES 197

Fig. 10. The ��������� diagram in the GPM2-asymmetry.

we have observed that the described deconvolution procedure isnot essentially affected.

V. CONCLUSION

We present a new identification method for the irreversibleand reversible components based on the FORC, , and

diagrams. The experimental data obtainedon a particulate ferromagnetic medium were compared with thesimulations made with three Preisach-type models. The first andthe second are the well-known GPM and the GMPM and cannotexplain the experimentally observed features.

Only the GMPM model can reproduce the specific shape ofthe diagram in the Preisach plane whichis essentially linked to the correlation between reversible andirreversible magnetization processes. Unfortunately, the asym-metric irreversible ridge cannot be included in these models andconsequently they cannot correctly reproduce the reversible partof the process.

However, all the details of the experimentaldiagram can be reproduced only by

the GPM2-asymmetric model that seems to include the majorelements of the reversible processes in an interacting system offerromagnetic particles.

ACKNOWLEDGMENT

This work was supported by the CNCSIS Romania grantsIDEI FASTSWITCH 1994 and PNII RU-TE 185/2010.

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