- Email: [email protected]
Separation of magnetic phases in alloys
ARTICLE IN PRESS Physica B 403 (2008) 3137– 3140
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Separation of magnetic phases in alloys J. Takacs a,, I. Me´sza´ros b a b
Department of Engineering Science, University of Oxford, 5. Pound Close, Yarnton, Oxon OX5 1QG, Oxford, UK Department of Materials Science and Technology, Budapest University of Technology and Economics, Budapest, Hungary
a r t i c l e in fo
abstract
Article history: Received 8 March 2008 Accepted 26 March 2008
In this paper we present a study of the separation of phases in multi-phase alloys. The proposed technique is based on the hyperbolic model of magnetization. By using this model it is possible to decompose the magnetic phases of alloys and determine their magnetic properties separately. Experimental verification was carried out on a transformer-like setup, constructed from layered samples representing the various magnetic phases. The samples were constructed from elements of strongly different magnetic properties. The results given by the model are in an excellent agreement with the experimental results, giving justification for the proposed method of decomposition. The proposed method is the first step towards the recognition and the separation of magnetic constituencies of different magnetic properties in an alloy by analytical means. & 2008 Elsevier B.V. All rights reserved.
Keywords: Hysteresis Modelling Magnetic measurements Phase separations
1. Introduction Magnetic alloys are used in a very wide range of applications with properties tailored to one specific need. The behaviour of the constituent parts of these alloys and the changes due to external influences like heat [1,2] radiation, ageing [2,3] or plastic deformation [3,4] long occupied the mind of researchers. Although a number of papers have been published on the optical [2,3,5] EDS, XRD, USM [2,6,7] and other methods [8,9] for determining the constituent parts of these alloys, so far no approach has been made, to our knowledge, to separate these components analytically. In this paper we propose a way to analyse alloys with strictly controlled composition as the first step towards wider practical applications. In this approach we used the hyperbolic model for the analysis, which gives a relatively simple but effective mathematical description of magnetic, sigmoidshaped hysteresis loops and normal magnetization curves in closed mathematical forms. This choice was made after a careful consideration of the suitability of various models. This particular model, first published in 2001, is well documented in the literature [10,11] and only a brief summary of it will be given later in the paper. For more details and other applications the reader is referred to the literature. The model formulates the hysteretic loops and the shape of the anhysteretic curve as well the commutation curve. By using the parameters of the modelled major hysteresis loop, in a simple case, the magnetic components of the alloy can be recognized, described separately and the
Corresponding author. Tel./fax: +44 1865 373391.
E-mail addresses: [email protected], [email protected] (J. Takacs). 0921-4526/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2008.03.023
changes during processing or ageing can be followed and analysed independently [12]. The model enables us to decompose a measured compound magnetization curve and to separate and identify the elementary magnetization curves in multi-phase alloys. In case of phase transformation due to temperature [2,9,13], ageing [3] or radiation this analytical method allows the observer to model the stepby-step changes in the magnetic properties of the material under investigation.
2. Experiment A specially designed permeameter-type magnetic analyzer, developed at the Department of Materials Science and Engineering of BUTE, was used for measuring the hysteresis curves. The applied measuring yoke contains a robust U- shaped laminated Fe–Si iron core with a magnetizing coil. The excitation current was sinusoidal produced by a digital function generator and a power amplifier, used in voltage regulated, current generator mode. The pick-up coil was around the middle of the specimen. The permeameter was under the control of a computer in which a 16 bit National Instruments input–output data acquisition card accomplished the measurements. The applied maximum excitation field strength was 2100 A/m. In each case 200 minor hysteresis loops were measured. Each hysteresis loops were recorded by measuring their coordinates at 1000 points. The magnetic-excitation field was increased in steps with 5 s delay between the steps and the data acquisition in order to ensure the sample’s perfect magnetic accommodation, free of the effects of magnetic transients. All the
ARTICLE IN PRESS ´ros / Physica B 403 (2008) 3137–3140 J. Takacs, I. Me´sza
3138
sigmoid-type loops of the individual components, by using Maxwell’s superposition principle. This supposition was put to the test by using the hyperbolic model. This model facilitates the linear superposition of the individual sigmoid loops and indeed provides the way to separate the simple phases and/or the different parallel magnetic processes acting in an alloy during magnetization. The model itself is well described in the literature and here only a brief summary of the formulation will be given [10,11]. In an alloy, the contribution of the individual phases to the combined hysteresis loop can be described by the following mathematical equations in normalized form:
measurements were carried out by using sinusoidal excitation at a frequency of 5 Hz. Due to the relatively low excitation frequency and the small thickness of the samples (0.35–0.5 mm), the completed magnetic measurements should be regarded as pseudo-static. The effect of eddy-current on the magnetization curves were negligibly small, well within the measuring error. The permeameter enabled us to obtain all the practical magnetic parameters such as the saturation induction, remanence, coercivity, relative permeability and hysteretic losses, directly from the measured hysteresis loops. For testing the model for the magnetic phase decomposition, special-layered test objects were constructed out of six, selected elementary samples (identified as S4, S6, S8, S11, S13 and S17), made of structural steels and a permalloy. All the selected samples had simple sigmoid-like hysteresis loops. The coercivity values and geometrical details of each layer samples are summarized in Table 1. The magnetization curves of each of the component material were measured separately for reference purposes. Following that, a number of layered test samples were built with the combination of the different laminae. The phases were represented in the experiment by the laminated material of known character, composed into different combinations. The volume ratios of the components, present in the experimental samples were calculated from the geometry and the number of layers. After that, the details of the measured hysteresis curves passed on for analysis with the identity and the properties of the layered samples concealed. Then each of the major loops were modelled and decomposed to their constituent parts. The magnetic properties and the hysteresis loops of all component parts were recorded and the volume ratio calculated.
yþn ¼
n X ðAk f þk þ bn Þ
(1a)
k¼1
yn ¼
n X ðAk f k bn Þ
(1b)
k¼1
f þk ¼ tanh½ak ðx a0k Þ
(2a)
f k ¼ tanh½ak ðx þ a0k Þ
(2b)
bn ¼
n 1X A ðf f þk Þ 2 k¼1 k k
for x ¼ xm
(3)
The model is characterized by the practical parameters used in magnetism. Here y+n and yn are the normalized ascending and descending magnetization functions, respectively, x is the field excitation, a0k is the coercivity of the kth process. Ak is the amplitude of the components present, ak is the sheering factor and bn is the integration constant [14], while xm represents the maximum field excitation. The index k refers to the individual component phases and n is the number of total magnetic components involved. For most of the magnetic materials, used in practical application n equals 3. Ferro-magnetic materials, specially made or selected for purity, have only single sigmoidtype character. In the elementary samples, used in this experiment, the existence of one irreversible phase and a very small content of nearly reversible phase were found, which made up their hysteretic properties (see Tables 1 and 2).
3. Model description The analytical approach described here was based on the initial assumption that, when an alloy contains two or more magnetic metallurgical phases they are not interacting magnetically, therefore their magnetization curves can be linearly superimposed. From this, it has followed, that in a simple case the hysteresis loop of an alloy can be composed by linear superposition of the Table 1 Measured-coercivity values and cross sections of the elementary samples Sample name
Description
Measured coercive field (A/m)
Cross section (mm2)
S4 S6 S8 S11 S13 S17
Low carbon steel (AISI 1010) annealed Medium carbon steel (AISI 1050) cold rolled High carbon steel (AISI 1074) cold rolled Permalloy (Fe-76%Ni) cold rolled Medium carbon steel (AISI 1050) annealed High carbon steel (AISI 1080) normalized
266 308 555 172 336 513
15.28 15.28 20.13 9.7 6.35 7.98
(2 6 5) (3 0 1) (5 6 7) (1 6 5) (3 2 1) (5 0 3)
Table 2 Experimental and calculated coercivity (Hc) and remanence (Br) values for the combinations of the elementary samples shown Alloy
Hc A/m Hc1 Hc1 A/m Hc2 Hc2 A/m
S6+S17
S4+S17
S6+S8
S11+S13
2xS11+S13
Exp.
Calc.
Exp.
Calc.
Exp.
Calc.
Exp.
Calc.
Exp.
Calc.
365 308 513
368 305 503
322 266 513
333 289 510
400 308 555
400 302 550
236 172 336
240 180 328
208 172 336
214 181 397
Br T
1.51
1.5
1.4
1.37
1.5
1.5
1.2
1.1
1
0.95
A1/A2
1.91
2.1
1.91
1.89
1.31
1.65
1.53
1.66
3.1
3.05
ARTICLE IN PRESS ´ros / Physica B 403 (2008) 3137–3140 J. Takacs, I. Me´sza
3139
4. Test results In this blind test, out of the six selected samples, the prepared layered structures were the following combinations of the elementary samples: S6+S17, S4+S17, S6+S8, S11+S13 and 2xS11+S13. Not until all the relevant parameters were calculated from the fitted loops and the two sets, of data (experimental and calculated) were compared was the test sample identified. The fitted hysteresis loops provided the coercivity and remanence values and also allowed us to determine the relative volume of the elementary components in the alloy and the relative magnitudes of the hysteresis loops of the component phases. The measured and calculated coercivity of the combined (Hc) and the component hysteresis loops ðHc1 ; Hc2 Þ with the associated remanence (Br) as well as the volume ratio (A1/A2) of the two major components are tabulated in Table 2. Fig. 1 depicts the measured and fitted-combined hysteresis loops of the experimental sample S6+S8. Fig. 2 shows the calculated constituent loops with the combined loop. These are shown as a representative set, out of the five combined sets measured in the experiment. The modelled
Fig. 1. Measured (broken line) and calculated (solid line) hysteresis loops of combined sample S6+S8.
Fig. 3. Measured (broken line) and calculated (solid line) reference hysteresis loops; (a) elementary component S6, (b) elementary component S8 and (c) near reversible component. Fig. 2. The calculated combined and constituent loops.
ARTICLE IN PRESS 3140
´ros / Physica B 403 (2008) 3137–3140 J. Takacs, I. Me´sza
loops have been calculated by using Eqs. (1a),(1b),(2a),(2b) and (3) with parameters of the following normalized and physical values: A1 ¼ 0.825 ¼ 0.891 T, A2 ¼ 0.5 ¼ 0.54 T, A3 ¼ 0.325 ¼ 0.4 T, a1 ¼ 0.83, a2 ¼ 0.55, a3 ¼ 0.042, a01 ¼ 3 ¼ 302 A/m, a02 ¼ 5.5 ¼ 553 A/m, a03 ¼ 6 ¼ 603 A/m. Figs. 3a and b show the reference hysteresis loops of the elementary samples S6 and S8 fitted with the loops calculated with the same ak, a0k and xm parameters. The agreement between the elementary reference and the calculated loops is unexpectedly good as one can see from Fig. 3. An identical underlying component (A3, a3, a03) present in all samples, shown in Fig. 3c, having a near reversible character, is not listed in Table 1. Its presence shows up in the slight inclination of the hysteresis loop (see Fig. 2) near saturation. Its shearing coefficient is an order of magnitude smaller than that of the other two. Due to this, its effect on the characteristic parameters of the test sample is negligibly small relative to that of the other two larger strongly hysteretic components.
5. Discussion and conclusions The blind test, carried out on a number of well-controlled samples, demonstrated that the proposed method by using the hyperbolic model could be used for separation of the magnetic phases in alloys. The initial assumption, that in a non-interacting case the elementary components can be identified and their magnetic properties correctly calculated is well proven. The experiment has also proved that all the practical magnetic parameters such as the coercivity, remanence, the volume ratio and the hysteresis loops of the individual contributory elements, can be calculated with very good accuracy. Although the accuracy
achieved so far seems acceptable in most practical applications, further experiments are in progress to improve it and also to use this method for decomposition of other elementary magnetic materials with various combinations. Further investigations will cover composites with more than one magnetic process, deviating from the sigmoid character. This method, described here, could become an important tool in practical tests, particularly in investigating phase transitions as well as in theoretical investigations in magnetism. References [1] S. Chatterjee, S. Giri, S. Majumdar, S.K. De, J. Magn. Magn. Mater. 320 (2008) 617. [2] K.H. Lo, J.K.L. Lai, C.H. Shek, D.J. Li, Mater Sci. Eng. A 452–453 (2007) 149. [3] K.H. Lo, J.K.L. Lai, C.H. Shek, D.J. Li, Mater Sci. Eng. A 452–453 (2007) 78. [4] Y.D. Zhang, C. Esling, M.L. Gong, G. Vincent, X. Zhao, L. Zuo, Scri. Mater. 54 (2006) 1897. [5] S.P. Sagar, B.R. Kumar, G. Dobman, D.K. Bhattacharya, NDT E Int. 38 (2005) 674. [6] S. Groudeva-Zotova, H. Karl, A. Savan, J. Feydt, B. Wehner, T. Walther, N. Zotov, B. Stritzker, A. Ludwig, Thin Solid Films 495 (2006) 169. [7] S.K. Putatunda, S. Unni, G. Lawes, Mater. Sci. Eng. A 406 (2005) 254. [8] A. Senas, J.I. Espeso, J.R. Fernandez, J.G. Soldevilla, J.C.G. Sal, J.R. Carvajal, R. Ibarra, Physica B 276–278 (2000) 614. [9] J.S. Blasquez, V. Franco, C.F. Conde, J. Ferenc, T. Kulik, Mater. Lett. 62 (2008) 780. [10] J. Takacs, Mathematics of Hysteresis Phenomena, Wiley-VCH Verlag, Weinheim, 2003. [11] J. Takacs, COMPEL 20 (4) (2001) 1002. [12] L.K. Varga, Gy. Kova´cs, J. Taka´cs, Anhysteretic and biased first magnetization curves for Finemet type toroidal samples, J. Magn. Magn. Mater (2008) in press, doi:10.1016/j.jmmm.2008.04.135. [13] J. Taka´cs, Gy. Kova´cs and L.K. Varga. Decomposition of the hysteresis loops of nanocrystalline alloys below and above the decoupling temeperature. J. Magn. Magn. Mater (2008) in press. [14] J. Taka´cs, Physica B 372 (1–2) (2006) 57.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Separation of magnetic phases in alloys J. Takacs a,, I. Me´sza´ros b a b
Department of Engineering Science, University of Oxford, 5. Pound Close, Yarnton, Oxon OX5 1QG, Oxford, UK Department of Materials Science and Technology, Budapest University of Technology and Economics, Budapest, Hungary
a r t i c l e in fo
abstract
Article history: Received 8 March 2008 Accepted 26 March 2008
In this paper we present a study of the separation of phases in multi-phase alloys. The proposed technique is based on the hyperbolic model of magnetization. By using this model it is possible to decompose the magnetic phases of alloys and determine their magnetic properties separately. Experimental verification was carried out on a transformer-like setup, constructed from layered samples representing the various magnetic phases. The samples were constructed from elements of strongly different magnetic properties. The results given by the model are in an excellent agreement with the experimental results, giving justification for the proposed method of decomposition. The proposed method is the first step towards the recognition and the separation of magnetic constituencies of different magnetic properties in an alloy by analytical means. & 2008 Elsevier B.V. All rights reserved.
Keywords: Hysteresis Modelling Magnetic measurements Phase separations
1. Introduction Magnetic alloys are used in a very wide range of applications with properties tailored to one specific need. The behaviour of the constituent parts of these alloys and the changes due to external influences like heat [1,2] radiation, ageing [2,3] or plastic deformation [3,4] long occupied the mind of researchers. Although a number of papers have been published on the optical [2,3,5] EDS, XRD, USM [2,6,7] and other methods [8,9] for determining the constituent parts of these alloys, so far no approach has been made, to our knowledge, to separate these components analytically. In this paper we propose a way to analyse alloys with strictly controlled composition as the first step towards wider practical applications. In this approach we used the hyperbolic model for the analysis, which gives a relatively simple but effective mathematical description of magnetic, sigmoidshaped hysteresis loops and normal magnetization curves in closed mathematical forms. This choice was made after a careful consideration of the suitability of various models. This particular model, first published in 2001, is well documented in the literature [10,11] and only a brief summary of it will be given later in the paper. For more details and other applications the reader is referred to the literature. The model formulates the hysteretic loops and the shape of the anhysteretic curve as well the commutation curve. By using the parameters of the modelled major hysteresis loop, in a simple case, the magnetic components of the alloy can be recognized, described separately and the
Corresponding author. Tel./fax: +44 1865 373391.
E-mail addresses: [email protected], [email protected] (J. Takacs). 0921-4526/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2008.03.023
changes during processing or ageing can be followed and analysed independently [12]. The model enables us to decompose a measured compound magnetization curve and to separate and identify the elementary magnetization curves in multi-phase alloys. In case of phase transformation due to temperature [2,9,13], ageing [3] or radiation this analytical method allows the observer to model the stepby-step changes in the magnetic properties of the material under investigation.
2. Experiment A specially designed permeameter-type magnetic analyzer, developed at the Department of Materials Science and Engineering of BUTE, was used for measuring the hysteresis curves. The applied measuring yoke contains a robust U- shaped laminated Fe–Si iron core with a magnetizing coil. The excitation current was sinusoidal produced by a digital function generator and a power amplifier, used in voltage regulated, current generator mode. The pick-up coil was around the middle of the specimen. The permeameter was under the control of a computer in which a 16 bit National Instruments input–output data acquisition card accomplished the measurements. The applied maximum excitation field strength was 2100 A/m. In each case 200 minor hysteresis loops were measured. Each hysteresis loops were recorded by measuring their coordinates at 1000 points. The magnetic-excitation field was increased in steps with 5 s delay between the steps and the data acquisition in order to ensure the sample’s perfect magnetic accommodation, free of the effects of magnetic transients. All the
ARTICLE IN PRESS ´ros / Physica B 403 (2008) 3137–3140 J. Takacs, I. Me´sza
3138
sigmoid-type loops of the individual components, by using Maxwell’s superposition principle. This supposition was put to the test by using the hyperbolic model. This model facilitates the linear superposition of the individual sigmoid loops and indeed provides the way to separate the simple phases and/or the different parallel magnetic processes acting in an alloy during magnetization. The model itself is well described in the literature and here only a brief summary of the formulation will be given [10,11]. In an alloy, the contribution of the individual phases to the combined hysteresis loop can be described by the following mathematical equations in normalized form:
measurements were carried out by using sinusoidal excitation at a frequency of 5 Hz. Due to the relatively low excitation frequency and the small thickness of the samples (0.35–0.5 mm), the completed magnetic measurements should be regarded as pseudo-static. The effect of eddy-current on the magnetization curves were negligibly small, well within the measuring error. The permeameter enabled us to obtain all the practical magnetic parameters such as the saturation induction, remanence, coercivity, relative permeability and hysteretic losses, directly from the measured hysteresis loops. For testing the model for the magnetic phase decomposition, special-layered test objects were constructed out of six, selected elementary samples (identified as S4, S6, S8, S11, S13 and S17), made of structural steels and a permalloy. All the selected samples had simple sigmoid-like hysteresis loops. The coercivity values and geometrical details of each layer samples are summarized in Table 1. The magnetization curves of each of the component material were measured separately for reference purposes. Following that, a number of layered test samples were built with the combination of the different laminae. The phases were represented in the experiment by the laminated material of known character, composed into different combinations. The volume ratios of the components, present in the experimental samples were calculated from the geometry and the number of layers. After that, the details of the measured hysteresis curves passed on for analysis with the identity and the properties of the layered samples concealed. Then each of the major loops were modelled and decomposed to their constituent parts. The magnetic properties and the hysteresis loops of all component parts were recorded and the volume ratio calculated.
yþn ¼
n X ðAk f þk þ bn Þ
(1a)
k¼1
yn ¼
n X ðAk f k bn Þ
(1b)
k¼1
f þk ¼ tanh½ak ðx a0k Þ
(2a)
f k ¼ tanh½ak ðx þ a0k Þ
(2b)
bn ¼
n 1X A ðf f þk Þ 2 k¼1 k k
for x ¼ xm
(3)
The model is characterized by the practical parameters used in magnetism. Here y+n and yn are the normalized ascending and descending magnetization functions, respectively, x is the field excitation, a0k is the coercivity of the kth process. Ak is the amplitude of the components present, ak is the sheering factor and bn is the integration constant [14], while xm represents the maximum field excitation. The index k refers to the individual component phases and n is the number of total magnetic components involved. For most of the magnetic materials, used in practical application n equals 3. Ferro-magnetic materials, specially made or selected for purity, have only single sigmoidtype character. In the elementary samples, used in this experiment, the existence of one irreversible phase and a very small content of nearly reversible phase were found, which made up their hysteretic properties (see Tables 1 and 2).
3. Model description The analytical approach described here was based on the initial assumption that, when an alloy contains two or more magnetic metallurgical phases they are not interacting magnetically, therefore their magnetization curves can be linearly superimposed. From this, it has followed, that in a simple case the hysteresis loop of an alloy can be composed by linear superposition of the Table 1 Measured-coercivity values and cross sections of the elementary samples Sample name
Description
Measured coercive field (A/m)
Cross section (mm2)
S4 S6 S8 S11 S13 S17
Low carbon steel (AISI 1010) annealed Medium carbon steel (AISI 1050) cold rolled High carbon steel (AISI 1074) cold rolled Permalloy (Fe-76%Ni) cold rolled Medium carbon steel (AISI 1050) annealed High carbon steel (AISI 1080) normalized
266 308 555 172 336 513
15.28 15.28 20.13 9.7 6.35 7.98
(2 6 5) (3 0 1) (5 6 7) (1 6 5) (3 2 1) (5 0 3)
Table 2 Experimental and calculated coercivity (Hc) and remanence (Br) values for the combinations of the elementary samples shown Alloy
Hc A/m Hc1 Hc1 A/m Hc2 Hc2 A/m
S6+S17
S4+S17
S6+S8
S11+S13
2xS11+S13
Exp.
Calc.
Exp.
Calc.
Exp.
Calc.
Exp.
Calc.
Exp.
Calc.
365 308 513
368 305 503
322 266 513
333 289 510
400 308 555
400 302 550
236 172 336
240 180 328
208 172 336
214 181 397
Br T
1.51
1.5
1.4
1.37
1.5
1.5
1.2
1.1
1
0.95
A1/A2
1.91
2.1
1.91
1.89
1.31
1.65
1.53
1.66
3.1
3.05
ARTICLE IN PRESS ´ros / Physica B 403 (2008) 3137–3140 J. Takacs, I. Me´sza
3139
4. Test results In this blind test, out of the six selected samples, the prepared layered structures were the following combinations of the elementary samples: S6+S17, S4+S17, S6+S8, S11+S13 and 2xS11+S13. Not until all the relevant parameters were calculated from the fitted loops and the two sets, of data (experimental and calculated) were compared was the test sample identified. The fitted hysteresis loops provided the coercivity and remanence values and also allowed us to determine the relative volume of the elementary components in the alloy and the relative magnitudes of the hysteresis loops of the component phases. The measured and calculated coercivity of the combined (Hc) and the component hysteresis loops ðHc1 ; Hc2 Þ with the associated remanence (Br) as well as the volume ratio (A1/A2) of the two major components are tabulated in Table 2. Fig. 1 depicts the measured and fitted-combined hysteresis loops of the experimental sample S6+S8. Fig. 2 shows the calculated constituent loops with the combined loop. These are shown as a representative set, out of the five combined sets measured in the experiment. The modelled
Fig. 1. Measured (broken line) and calculated (solid line) hysteresis loops of combined sample S6+S8.
Fig. 3. Measured (broken line) and calculated (solid line) reference hysteresis loops; (a) elementary component S6, (b) elementary component S8 and (c) near reversible component. Fig. 2. The calculated combined and constituent loops.
ARTICLE IN PRESS 3140
´ros / Physica B 403 (2008) 3137–3140 J. Takacs, I. Me´sza
loops have been calculated by using Eqs. (1a),(1b),(2a),(2b) and (3) with parameters of the following normalized and physical values: A1 ¼ 0.825 ¼ 0.891 T, A2 ¼ 0.5 ¼ 0.54 T, A3 ¼ 0.325 ¼ 0.4 T, a1 ¼ 0.83, a2 ¼ 0.55, a3 ¼ 0.042, a01 ¼ 3 ¼ 302 A/m, a02 ¼ 5.5 ¼ 553 A/m, a03 ¼ 6 ¼ 603 A/m. Figs. 3a and b show the reference hysteresis loops of the elementary samples S6 and S8 fitted with the loops calculated with the same ak, a0k and xm parameters. The agreement between the elementary reference and the calculated loops is unexpectedly good as one can see from Fig. 3. An identical underlying component (A3, a3, a03) present in all samples, shown in Fig. 3c, having a near reversible character, is not listed in Table 1. Its presence shows up in the slight inclination of the hysteresis loop (see Fig. 2) near saturation. Its shearing coefficient is an order of magnitude smaller than that of the other two. Due to this, its effect on the characteristic parameters of the test sample is negligibly small relative to that of the other two larger strongly hysteretic components.
5. Discussion and conclusions The blind test, carried out on a number of well-controlled samples, demonstrated that the proposed method by using the hyperbolic model could be used for separation of the magnetic phases in alloys. The initial assumption, that in a non-interacting case the elementary components can be identified and their magnetic properties correctly calculated is well proven. The experiment has also proved that all the practical magnetic parameters such as the coercivity, remanence, the volume ratio and the hysteresis loops of the individual contributory elements, can be calculated with very good accuracy. Although the accuracy
achieved so far seems acceptable in most practical applications, further experiments are in progress to improve it and also to use this method for decomposition of other elementary magnetic materials with various combinations. Further investigations will cover composites with more than one magnetic process, deviating from the sigmoid character. This method, described here, could become an important tool in practical tests, particularly in investigating phase transitions as well as in theoretical investigations in magnetism. References [1] S. Chatterjee, S. Giri, S. Majumdar, S.K. De, J. Magn. Magn. Mater. 320 (2008) 617. [2] K.H. Lo, J.K.L. Lai, C.H. Shek, D.J. Li, Mater Sci. Eng. A 452–453 (2007) 149. [3] K.H. Lo, J.K.L. Lai, C.H. Shek, D.J. Li, Mater Sci. Eng. A 452–453 (2007) 78. [4] Y.D. Zhang, C. Esling, M.L. Gong, G. Vincent, X. Zhao, L. Zuo, Scri. Mater. 54 (2006) 1897. [5] S.P. Sagar, B.R. Kumar, G. Dobman, D.K. Bhattacharya, NDT E Int. 38 (2005) 674. [6] S. Groudeva-Zotova, H. Karl, A. Savan, J. Feydt, B. Wehner, T. Walther, N. Zotov, B. Stritzker, A. Ludwig, Thin Solid Films 495 (2006) 169. [7] S.K. Putatunda, S. Unni, G. Lawes, Mater. Sci. Eng. A 406 (2005) 254. [8] A. Senas, J.I. Espeso, J.R. Fernandez, J.G. Soldevilla, J.C.G. Sal, J.R. Carvajal, R. Ibarra, Physica B 276–278 (2000) 614. [9] J.S. Blasquez, V. Franco, C.F. Conde, J. Ferenc, T. Kulik, Mater. Lett. 62 (2008) 780. [10] J. Takacs, Mathematics of Hysteresis Phenomena, Wiley-VCH Verlag, Weinheim, 2003. [11] J. Takacs, COMPEL 20 (4) (2001) 1002. [12] L.K. Varga, Gy. Kova´cs, J. Taka´cs, Anhysteretic and biased first magnetization curves for Finemet type toroidal samples, J. Magn. Magn. Mater (2008) in press, doi:10.1016/j.jmmm.2008.04.135. [13] J. Taka´cs, Gy. Kova´cs and L.K. Varga. Decomposition of the hysteresis loops of nanocrystalline alloys below and above the decoupling temeperature. J. Magn. Magn. Mater (2008) in press. [14] J. Taka´cs, Physica B 372 (1–2) (2006) 57.