ELSEVIER Journal of Magnetism and Magnetic Materials 171 (1997) 94-102 Journal of magnetism ~i and magnetic materials Tensor magnetostriction loops for a steel cube Y. Yu", L. Clapham a, D.L. Atherton a'*, T.W. Krause b aDepartment of Physics, Queen's University, Kingston, Ont., Canada K7L 3N6 bAECL Research, Chalk River Laboratories, Chalk River, Ont., Canada KOJ 1JO Received 2 January 1997 Abstract Tensor magnetostriction loops were measured for a 25 mm steel cube. The volume magnetostriction of the cube was calculated for the magnetic field applied in three orthogonal directions parallel to the edge directions of the cube. The tensor magnetostriction loops are interpreted in terms of domain wall motion and magnetization vector rotation, as well as the existence of a magnetic easy axis in the sample, which is evidenced by the asymmetry of the tensor magnetostriction loops. The relative magnetic easy axis determined from the tensor magnetostriction loops is consistent with that determined from the tensor magnetic hysteresis loops measured previously. A new parameter termed magnetic Poisson's ratio is introduced to describe the relative dimensional changes in the longitudinal and transverse directions with respect to the applied field directions. PACS: 75.30.Gw; 75.60.Ej Keywords: Magnetostriction; Magnetic Poisson's ratio; Easy axis; Magnetization rotation; Domain wall motion 1. Introduction It is well known that magnetization processes in a ferromagnet usually consist of domain wall motion and magnetization rotation [-1]. Although the concept of domain wall motion and magneti- zation rotation is not new, the study of these pro- cesses and related phenomena continues to be a major research topic in magnetics. Recently, ten- sor magnetic hysteresis loops have been proposed *Corresponding author. Tel.: + 1-613-545-2701; fax: + 1- 613-545-6463. and used in the study of magnetization processes in two typical materials: materials whose dominant magnetization processes are both domain wall motion and magnetization rotation [-2] and those whose primary magnetization process is magneti- zation rotation [3]. The magnetic hysteresis loop tensor measurement contains abundant informa- tion about magnetization processes and is helpful in understanding and modeling magnetic hyster- esis. Tensor magnetic hysteresis loops pose a new challenge for modeling magnetic hysteresis. A suc- cessful magnetic hysteresis model should explain not only the loops of the diagonal elements of the tensor, but also the off-diagonal ones. Recently, 0304-8853/97/$17.00 ,~, 1997 Elsevier Science B.V. All rights reserved PII S0304-88 53(97)00057-7
Transcript
ELSEVIER Journal of Magnetism and Magnetic Materials 171 (1997) 94-102
Journal of magnetism
~ i and magnetic materials
Tensor magnetostriction loops for a steel cube
Y. Yu", L. Clapham a, D.L. Atherton a'*, T.W. K r a u s e b
a Department of Physics, Queen's University, Kingston, Ont., Canada K7L 3N6 b AECL Research, Chalk River Laboratories, Chalk River, Ont., Canada KOJ 1JO
Received 2 January 1997
Abstract
Tensor magnetostriction loops were measured for a 25 mm steel cube. The volume magnetostriction of the cube was calculated for the magnetic field applied in three orthogonal directions parallel to the edge directions of the cube. The tensor magnetostriction loops are interpreted in terms of domain wall motion and magnetization vector rotation, as well as the existence of a magnetic easy axis in the sample, which is evidenced by the asymmetry of the tensor magnetostriction loops. The relative magnetic easy axis determined from the tensor magnetostriction loops is consistent with that determined from the tensor magnetic hysteresis loops measured previously. A new parameter termed magnetic Poisson's ratio is introduced to describe the relative dimensional changes in the longitudinal and transverse directions with respect to the applied field directions.
It is well known that magnetization processes in a ferromagnet usually consist of domain wall motion and magnetization rotation [-1]. Although the concept of domain wall motion and magneti- zation rotation is not new, the study of these pro- cesses and related phenomena continues to be a major research topic in magnetics. Recently, ten- sor magnetic hysteresis loops have been proposed
and used in the study of magnetization processes in two typical materials: materials whose dominant magnetization processes are both domain wall motion and magnetization rotation [-2] and those whose primary magnetization process is magneti- zation rotation [3]. The magnetic hysteresis loop tensor measurement contains abundant informa- tion about magnetization processes and is helpful in understanding and modeling magnetic hyster- esis. Tensor magnetic hysteresis loops pose a new challenge for modeling magnetic hysteresis. A suc- cessful magnetic hysteresis model should explain not only the loops of the diagonal elements of the tensor, but also the off-diagonal ones. Recently,
0304-8853/97/$17.00 ,~, 1997 Elsevier Science B.V. All rights reserved PII S0304-88 53(97)00057-7
Y. Yu et al. / Journal of Magnetism and Magnetic Materials" 171 (1997) 94-102 95
a hysteresis model based on the Atherton-Beattie model [4] was developed to explain successfully the tensor hysteresis loops resulting from magneti- zation rotations [3].
Similar to magnetic hysteresis loops, magneto- striction loops also exhibit hysteresis [5-9]. Mag- netic hysteresis and magnetostrictive hysteresis are intrinsically coupled through their dependence on magnetization [7, 9]. A magnetic hysteresis loop may reflect every possible magnetization process, including 90 and 180 ° domain wall motions and magnetization rotations. The 90 and 180 ° domain wall motions cannot be easily distinguished from a magnetic hysteresis loop. In contrast, a magneto- striction loop can separate the effects of 90 ° domain wall motions from the 180 ° domain wall motions at low and intermediate fields. Thus, the measurement of magnetostriction loops can help to understand 90 ~) domain wall motion at low and intermediate fields and magnetization rotation at high field. The magnetostrictive behavior for single crystal mater- ials is very well understood [1, 10, 11]. However, magnetostrictive properties of polycrystalline ma- terials are complicated and have not been investi- gated in complete detail. The Sablik-Jiles model [9] describes both magnetic and magnetostrictive hysteresis as due to domain wall motion and mag- netoelastic effect without taking into account the contribution from domain magnetization vector rotation since the model was developed based part- ly on the Jiles Atherton model [12] and Jiles- Atherton-Sablik model [9].
We present here tensor magnetostriction loops for a polycrystalline steel cube whose tensor mag- netic hysteresis loops were measured previously [2]. Our study is intended to shed light on under- standing magnetization processes and related mag- netic phenomena observed in bulk materials through measurements of the magnetostriction loop tensor.
to the computer. The 350 f~ sample gauge was mounted as one arm of a Wheatstone bridge with the other three arms consisting of 350f~ strain gauges mounted on a piece of steel. The excitation voltage for the Wheatstone bridge is 3.0 V. The magnetostriction signal from the bridge was ampli- fied by a DC amplifier with a gain of 1000 before it was fed into a 486 personal computer.
To minimize the compressive force applied to the sample by the variable-gap pole pieces upon the application of a magnetic field, the two pole pieces were fixed tightly by cotter pins. The sample used in the present study was a 0.10 wt%C mild steel cube of dimensions 25 x 25 x 25 mm. Microstructural examination revealed a typical a w e plus pearlite microstructure with a relatively large grain size (~ 100 jam), consistent with very slow cooling after rolling at temperatures higher than 800°C. The large, equiaxed grain structure suggested that little or no crystallographic texture would be expected in the sample.
Strain gauges were mounted on the center of three orthogonal faces of the cube in order to measure the magnetostriction loop tensor. Only one gauge was mounted each time before mag- netostriction was measured. Some of the magnetos- triction data were measured twice with remounted strain gauges. The sample was demagnetized in the tensor diagonal directions before magnetostriction measurements were performed in those directions and in the corresponding off-diagonal directions. The magnetostriction data were measured from the demagnetized state, for which the magnetostriction is assumed to be zero.
3. Results and discussion
3.1. The linear magnetostriction versus magnetic field and the sample easy axis
2. Experiments
The experimental apparatus is similar to the one used for measuring tensor magnetic hysteresis loops [2] except that the magnetostriction signal now replaces the flux density B as the analog input
To assist the description of tensor magnetostric- tion loops, the cube sample and the associated coordinate system are shown in Fig. 1. Fig. 2 shows the tensor magnetostriction loops for the steel cube as a function of magnetic field. The loops are char- acteristic of a polycrystalline sample represented by the initial rapid increase of magnetostriction, 2,
96 Y. Yu et al. /Journal of Magnetism and Magnetic Materials 171 (199D 94 102
Z J~xY , /
k~z i ;7 Y
Fig. 1. Schematic diagram showing the 25 mm cube arid the xyz coordinate system. The diagram also shows the magnetostric- tion components, 2xx, )oxY, and 2xz, when a magnetic field, H~, is applied in the x direction.
with the applied field until a peak value is reached,
then followed by the gradual decrease of magnetos- triction unti l the curve levels off at high field. The rapid initial increase in 2 is due to the 90 ° domain wall motion. As the magnet ic field increases to a range in which magnet iza t ion vectors move to- wards the nearest easy axes with respect to the applied field direction, the bulk magnetos t r ic t ion reaches a maximum. A further increase in the mag- netic field rotates the magnet iza t ion vectors away from those nearest easy axes toward the applied field direction, leading to a decrease in magnetos- triction. The above processes could be illustrated in
Fig. 3, showing the expected magnetostrict ion chan- ges for a closure doma in structure when the mag- netic field increases from low to high. Fig. 3
describes the magnetostr ict ive behavior of mate- rials with positive magnetos t r ic t ion in the direction
30 ' I '
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Fig. 2. Tensor magnetostriction loops as a function of magnetic field for a steel cube. Labels xX, xY, and xZ represent magnetostriction measured in the x, y, and z directions for magnetic field applied in the x direction, respectively, and so on. For each subfigure, the horizontal axis is the magnetic field in kA/m; the vertical axis is magnetostriction in microstrain (10-6).
E Yu et al./Journal of Magnetism and Magnetic Materials 171 (1997) 94-102 97
A H B Directionof net ~ . elongation
D o m a i n ~ . . ~ . ~ " / ~ Direct . . . . fnet
W a l l ¢ ~ v - - . / / contraction Motion
D Direction of net C ontraetion
ioT;?tlo° ., °, Direction of net elongation
Fig. 3. A simplified two-dimensional description of magneto- striction changes during the magnetization processes starting from the demagnetized state for a closure domain structure. This model describes a material with positive magnetostriction, such as the steel cube studied here. Stage A: demagnetized state; Stage B: initial stage of domain wall motion; Stage C: nearing the final stage of domain wall motion with major magnetization vector in the nearest easy axis; Stage D: saturated state reached through magnetization rotation from stage C.
of magnetization, as is the case in our cube sample. Here the closure domain structure can also be modeled as an interaction region of approximately the average grain size of the sample [13]. The measured magnetostriction is the superposition of magnetostrictions of such individual grains or in- teraction regions with a particular orientation dis- tribution.
The magnetostriction loop tensor is asymmetric, indicating that the polycrystalline sample is mag- netically anisotropic and that there may exist at least one easy axis. The overall bulk magnetic easy axis of the polycrystalline sample can be deter- mined by analysis of the relative magnitudes of the tensor magnetostriction loops. Magnetically, the steel cube can be thought of as consisting of interac- tion regions [13], each of which has an effective easy axis. The orientation distribution of the easy axes for individual interaction regions determines the bulk easy axis of the sample. Since both 90 ° and
180 ° domains may exist in an interaction region, the easy axis of the interaction region can also be defined as the direction in which a majority of the 180 ° domains are aligned [14], as shown in Fig. 3. If a magnetic field is applied close to or in the easy axis direction of such a domain structure, less mag- netostriction will be expected than if the field is applied in a direction perpendicular to the easy axis, since the sample volume swept by 90 ° domain walls in the former case is less than that swept by 90 ° domain walls in the latter case. Similar argu- ments have been used previously for the correlation between preferred domain orientation and mag- netostriction data [-11, 15]. Since the cube sample can be assumed to consist of a distribution of such oriented interaction regions, larger magnetostric- tion will result when a magnetic field is applied in or close to the hard axis. By examining the tensor magnetostriction loops of Fig. 2, we found that the y and z directions are relatively easy axes with the y direction slightly magnetically harder than the z direction. The x direction is the relatively hard axis although the relative anisotropies in the x, y, and z directions are not very strong, as can be seen from the degree of asymmetry of the tensor mag- netostriction loops.
For elastically and magnetically isotropic mate- rials, the elongation in one direction is always ac- companied by corresponding contractions in the transverse directions and vice versa [,16, 17]. This statement may not always be true for magnetically anisotropic materials, as is the case in our sample as seen in Fig. 2. An elongation in one direction is accompanied by a contraction in the transverse directions when the magnetic field is in the x or z direction. However, when magnetized in the y di- rection, the sample increases its length in both the applied field direction and one of the transverse directions, x, except at certain low-field ranges where the sample contracts slightly in the x direc- tion. Again, this phenomenon may be related to the magnetic anisotropy of the sample. In particular, the asymmetry between 90 ° versus 180 ° domain wall motion and the amount of domain vector rotation required to fully magnetize the sample will define the amount and form of the observed mag- netostriction that takes place when the sample is magnetized along any particular direction.
98 Y. Yu et al. / Journal of Magnetism and Magnetic Materials 171 (1997) 94-102
3.2. Magnetostr ict ion versus magnetizat ion
Fig. 4 shows the tensor magnetostriction loops for the steel cube as a function of the diagonal component of the magnetization vector measured previously [2]. Unlike a magnetic hysteresis loop, the magnetostriction loops shown here are actually double loops because the magnetostriction is inde- pendent of the sense of the magnetization 1-15]. In some materials, even quadruple loops are observed [-8]. On the one hand, the plot of magnetostriction, 2, versus magnetization, M, is helpful for under- standing the coupling between magnetostriction and magnetization when stress is present I-5, 9]. Alternatively, 2 versus M is also important for the interpretation of magnetization processes in a poly- crystalline sample, particularly when tensor mag- netostriction loops are considered.
An appreciable amount of magnetostrictive hys- teresis is observed at low and intermediate magnet- ization levels where irreversible domain wall motion is involved. As the magnetostriction reaches its
maximum, the magnetization vectors within the sample lie predominantly along the nearest easy axes to the applied field direction. Further increase in the applied field rotates these magnetization vectors towards the applied field direction. These rotation processes are primarily reversible in na- ture since little hysteresis is observed at high mag- netization. The magnetostrictive behavior in an isotropic system during the rotation processes can be best described by an approximate mathematical relation [11, 15]:
where 2 and M are measured along the applied field direction, Ms is the saturation magnetization and a and b are constants. When magnetostriction is measured perpendicular to the applied field direc- tion, the magnetostriction may also be given by Eq. (1). Fig. 5 shows magnetostrictions plotted against the square of the diagonal component of magnetization vector. The magnetostriction loops
3 0 ' I ' 10 I
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Fig. 4. Tensor magnetostriction loops as functions of diagonal magnetization components. Refer to Fig. 2 for the meaning of the indices, xX, xY, and xZ, etc. For each subfigure, the horizontal axis is the magnetization component in the corresponding diagonal direction, in units of A/m; the vertical axis is the magnetostriction in microstrain (10 6).
E Yu et al./Journal of Magnetism and Magnetic Materials 171 (1997) 94 102 99
30
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10
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3.0E+12 O.OE+O 3.0E+12
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Fig. 5. Tensor magnetostriction loops as functions of the square of the diagonal magnetization components. Refer to Fig. 2 for the meaning of the indices, xX, x Y, and xZ, etc. For each subfigure, the horizontal axis is the square of the magnetization component in the corresponding diagonal direction, in units of AZ/mZ; the vertical axis is the magnetostriction in rnicrostrain (10 6).
are now banana-shaped with the stem representing the high magnetization region. These magnetostric- tion versus M 2 curves follow approximately Eq. (1) at high-magnetization, verifying the contribution of magnetization rotation processes to the magnetos- triction in the high-magnetization region. The mi- nor deviation from Eq. (1) is probably caused by the contribution from the small magnetic aniso- tropy of the cube sample, as discussed above.
3.3. Volume magnetostriction
The volume magnetostriction of the sample can be calculated from the tensor magnetostriction loops. Given the magnetostriction tensor elements 2xX, 2xy, and 2~z, etc., the volume magnetostriction in the i direction may be given by [1]
(BY) = 2 i x + 2 1 y + 2 i z , (2) T where the index i stands for the magnetic field direction, x, y, or z. Volume magnetostrictions are
plotted as functions of magnetic field and corre- sponding magnetization component in the x, y, and z directions, as shown in Fig. 6.
Fig. 6 shows that the volume of the sample de- creases for the field applied in the x and z directions while the volume of the sample increases for field applied in the y direction, except at certain low magnetization levels, in which the sample con- tracts. The changes in the volume of the sample depend on the material and its corresponding mag- netization processes. The volume of the sample changes in such a way that the sample tends toward both mechanical and magnetic equilibrium [1, 9, 16].
Significant hysteresis is apparent in the volume magnetostriction, particularly when the magnetic field is applied in the x direction. Since the changes in the volume magnetostriction are related to the magnetization processes, all of the irreversible pro- cesses, such as irreversible 90 ° domain wall motion and irreversible magnetization rotation, contribute to the observed hysteresis in volume magnetostric- tion. The orientation dependence of hysteresis in
100 Y. gu et al. // Journal of Magnetism and Magnetic Materials 171 (1997) 94-102
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Fig. 6. Volume magnetostriction, 6v/v, in microstrain (10 6) for magnetic field applied in the x, y, and z directions. Subfigures (a), (b), and (c) are plots of 3v/v versus magnetic field in the x, y, and z directions; subfigures (d), (e) and (ft are plots of 8v/v versus diagonal magnetization components in the x, y, and z directions, respectively. The volume magnetostriction is assumed to be zero when the sample is in its demagnetized state.
vo lume magne tos t r i c t ion is ind ica ted in Fig. 6. A s imilar p h e n o m e n o n has a l r eady been r epor t ed for tensor magnet ic hysteresis loops for the same steel cube s tudied here and for a ferrite pe rmanen t magne t [2, 3].
3.4. Magne t ic Poisson 's ratio definedJ?om the tensor magnetostr ict ion loops
An equivalent Poisson ' s rat io, which m a y be te rmed the magnet ic Poisson ' s ra t io (MPR), may be defined from the tensor magne tos t r i c t ion loops. Two M P R s can be defined for magnet ic field ap- plied in any of the x, y, and z direct ions. Given tensor magne tos t r i c t ion loops, a total of six M P R s can be defined as vxy, Vxz, Vyx, vyz, vzx, vzy, where
vxy s tands for the M P R in the y d i rec t ion with magnet ic field in the x direct ion, and so on. Figs. 7 and 8 show the respective magnet ic Poisson ' s ra t ios as funct ions of magne t ic field and co r respond ing magne t i za t ion components .
The M P R as defined, describes the relat ive chan- ges in longi tud ina l and t ransverse magnetos t r ic - t ions in fe r romagnet ic mate r ia l s under the influence of appl ied magne t ic fields ins tead of stress. The M P R is not a constant , but depends on magnet ic field or magnet iza t ion , as seen in Fig. 7 or Fig. 8. M P R s are much greater than the elastic Poisson ' s rat ios, which are between 0.2~0.5. M P R s for the sample s tudied show typica l ly relat ively large values at low magne t i za t ion c o m p a r e d to those at high magnet iza t ion , indica t ing the r ap id change of
8 . 0 I i ; 8.0
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i I
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-20.0 I I I -20.0 , I I -60 -30 0 30 60 -60 -30 0 30 60
Fig. 7. Magnetic Poisson's ratios as a function of magnetic field, H (kA/m), for H in the x, y, and z directions. The magnetic Poisson's ratio, %., represents the ratio of magnetostriction in the y direction to the magnetostriction in the x direction which is also the applied field direction, and so on.
K Yu et al./Journal of Magnetism and Magnetic Materials 171 (1997) 94-102 101
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0.0
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0 , 0 - -
' -20.0 2.0E+6 -2.0E+6
, I i
O.OE+O 2.0E+6
Fig. 8. Magnetic Poisson's ratios as functions of diagonal mag- netization component, M (kA/m), for H in the x, y, and z direc- tions. The magnetic Poisson's ratios, Vxy and vx=, etc., have the same meaning as those in Fig. 7.
the magnetization state at low magnetization. The sign of the MPR is either positive or nega- tive, depending on how the magnetostriction be- haves at certain magnetization levels. The MPR also shows hysteresis due to its dependence on the magnetization. Therefore, magnetic Poisson's ratio is another physical quantity which can be used to describe the behavior of ferromagnetic materials.
3.5. Correlation between tensor magnetostriction and tensor magnetic hysteresis loops
Both tensor magnetic hysteresis loops and mag- netostriction loops describe the magnetization properties in ferromagnetic materials. The mag- netostriction loops reflect the changes in the
amount of volume swept by 90 ° domain walls and the amount of domain magnetization rotation, while magnetic hysteresis loops include effects of all possible magnetization processes, including 90 and 180 ° domain wall motion and magnetization vector rotation. One essential difference between the two tensor loops is that the off-diagonal elements in the magnetostriction tensor are of the same order of magnitude as the diagonal elements, in contrast to the tensor magnetic hysteresis loops in which the off-diagonal elements are about an order of magni- tude smaller than the corresponding diagonal ele- ments [2]. This is explained by the fact that the dimensional change of a grain or an interaction region in one direction is always accompanied by a comparable change in the transverse directions. This argument also holds for a polycrystalline sample where the macroscopic deformation of the sample is the average of individual grains.
Both tensor magnetic hysteresis loops and tensor magnetostriction loops give consistent results with regards, for example, to the magnetic easy axis. In principle, we can determine the relative easy axis of the cube sample by comparing the diagonal mag- netic hysteresis loops. However, this method is not used because the difficulty in locating the Hall sensor close to the surface of the sample leads to possible uncertainty in the magnetic field when the field is applied in different directions of the cube sample. Instead, the tensor magnetic hysteresis loops can be utilized in the determination of rela- tive sample easy axis. The study of magnetization rotation processes using tensor magnetic hysteresis loops [3] has indicated that the relative easy axis direction can be defined as the one in which the corresponding off-diagonal element is larger than all other off-diagonal elements at all magnetization levels provided that the applied field directions do not happen to be the easy or hard axis directions. This definition is also suitable for a sample in which magnetization processes consist of both domain wall motion and magnetization rotation, as is the case of the steel cube studied here. When tensor magnetic hysteresis loops are measured for the sample, larger off-diagonal loops will be expected in the direction of the relative magnetic easy axis than in the non-easy axis directions. By reviewing the tensor magnetic hysteresis loops [2], we find
102 E Yu et al. /Journal of Magnetism and Magnetic Materials 171 (1997) 94 102
that the relative easy, medium, and hard axes are in the z, y, and x directions, respectively. This is in agreement with that obtained from the tensor mag- netostriction loops, as discussed before.
in explaining the magnetostrictive behavior at vari- ous levels of magnetization.
Acknowledgements
4. Conclusions
Our understanding of magnetization processes and associated magnetic hysteresis has been en- hanced by analysis of the tensor magnetostrictive hysteresis loops. The magnetostriction at low mag- netization originates from the motion of 90 ° do- main walls. The magnetostrictive behavior at high magnetization can be characterized primarily as magnetization vector rotation. The magnetostric- tion for diagonal and off-diagonal components due to domain magnetization rotation can be described approximately by 2 = a + b ( M / M s ) 2, where M and Ms are the magnetization prior to saturation and the saturation magnetization of the sample in the diagonal directions, respectively, and a and b are constants. Magnetostriction also exhibits hysteresis because of its dependence on the magnetization. Volume magnetostriction and magnetic Poisson's ratio were determined from the magnetostriction loop tensor, all of which are a function of magneti- zation. The measurements of tensor magnetostric- tion loops have demonstrated that there is the need for magnetic and magnetostrictive hysteresis mod- els that can explain the full tensor behavior of magnetic and magnetostrictive loops, involving both domain wall motion and magnetization vec- tor rotation.
The asymmetry of the magnetostriction loops' tensor indicates the existence of magnetic aniso- tropy of the sample. The relative magnetic easy axis determined by the tensor magnetostriction loops and that determined by the tensor magnetic hyster- esis loops are consistent. The concept of interaction region is used in the definition of easy axis as well as
The authors would like to thank Pat Weyman for help in the magnetostriction measurements and Jonathan Makar for valuable discussion. The re- search is supported by Natural Sciences and Engine- ering Research Council of Canada, Pipetronix, and the Province of Ontario.
References
[1] S. Chikazumi, Physics of Magnetism, Wiley, New York, 1964.
[2] Y. Yu, T.W. Krause, P. Weyman, D.L. Atherton, J. Magn. Magn. Mater. 166 (1997) 290.
