ELSEVIER Journal of Magnetism and Magnetic Materials 166 (1997) 290-296
Tensor magnetic hysteresis loop measurements of a steel cube
Y. Yu, T.W. Krause, P. Weyman, D.L. Atherton *
Department of Physics, Queel~ ' s Uniuersity, Kingston, 01~tario, Canada K7L 3N6
Received 4 March 1996; revised 12 August 1996
Abstract
The change in magnetic moment of a 25.4 mm steel cube sample was measured in three orthogonal directions with fields up to magnetic saturation applied in each of the three cube directions. An analysis of the diagonal components of the resulting 3 x 3 tensor indicated the relative magnetic easy axis amongst the three directions measured. Additional information on the magnetization processes was obtained from an analysis of the off-diagonal changes in the magnetic moment tensor. Hysteresis was also observed in these off-diagonal loops. A net magnetization vector of magnitude M was considered. Deviations of magnetization components from linearity along the diagonal directions, 2x Mii , were related to the off-diagonal components, Mij and M~K, by the simple relation ~xM~r = +_(M~ + M~K)/2M, where i is the field direction x, y, or z and I, J, and K cyclic coordinates X, Y, and Z indicating the magnetization components measured by coils X, Y, and Z, respectively. Changes in the off-diagonal components were associated with various field dependent magnetization processes such as domain wall motion and domain vector rotation. The asymmetry of the magnetic hysteresis loop tensor is associated with either a texture of the material or the existence of internal stresses.
Magnetic hysteresis loops are useful for charac- terizing the behavior of magnetic materials [1,2]. Many important bulk magnetic parameters, such as coercivity, remanence, and saturation magnetization, can be determined from these loops. Historically, magnetic hysteresis loops plot magnetic induction (B) or magnetization ( M ) components aligned with the applied field direction. For convenience, we refer
* Corresponding author. Fax: + 1-613-545-6463.
tO magnetization loops measured parallel to the ap- plied field direction as the diagonal elements of the magnetization tensor and those perpendicular to the applied field direction as the off-diagonal elements.
Very little attention has yet been paid to the components of the off-diagonal changes in magneti- zation. It is shown here that the measurement of the off-diagonal elements may be used to recover the loss of magnetization along the diagonal elements of the magnetization tensor. This is performed by iden- tifying a net magnetization vector along which the maximum magnetization of the sample takes place. Magnetization processes are considered that may generate off-diagonal contributions to the magnetiza- tion tensor. Results are also considered in terms of
Y. Yu et al . / JounTal of Magnetism and Magnetic Materials 166 (1997) 290-29d 291
the relative magnetic easy axis of the sample as identified from the diagonal hysteresis loops.
2. Experimental technique
A 25.4 mm steel cube sample was cut from a steel plate. Perpendicular grooves, 2.0 mm wide by 0.15 mm deep, were cut centered on each surface of the sample with each groove parallel to one side of the cube. 20-turn coils, X, Y, and Z, were wound in each groove of the sample, using the right-hand rule, with the area vectors aligned in the +x , +y , and + z directions, respectively. Fig. 1 is a schematic dia- gram of the cube, showing the coil windings and definitions of the x, y, and z directions. Care was taken to obtain exact alignment of the coils so that the effect of misalignment was minimized.
The sample was placed between the pole pieces of a variable gap electromagnet. The pole pieces were moved against the sample to minimize air gaps and shape effects. Demagnetization effects were there- fore negligible for this type of measurement tech-
nique. Magnetic induction, B, was measured using a Walker MF-3A fluxmeter. The fluxmeter was ther- mally insulated in order to reduce thermal drift. The magnetic field H was measured using a Hall probe which was fastened close to the surface of the sam- ple and perpendicular to the applied field. The ana- log B and H signals were fed into a personal computer (PC) through an A / D converter. A real time data acquisition program, written previously, was used to collect data and to calibrate B and H [3]. The electromagnet was powered by a bipolar power supply controlled by the PC through a D / A converter.
The sample was demagnetized in the direction of the subsequently applied field before any loop mea- surements were made in that direction or in the off-diagonal directions. Since the loop measurements were normally started from a positive saturation field, the demagnetization process did not have sig- nificant effects on the loop data.
3. Results and discussion
a
Coil Z/
x
/ oiIY kA A / /
Fig. 1. Coordinatmsystem for a steel cube sampIe. Coil windings are within the grooves (not shown) and consistent with the right-hand rule with respect to the + x, + y, and + z directions. Note that actual number of turns of each coil is 20 and adjacent turns are side by side without overlap.
3. l. Magnetic tmiformity of the sample
Surface magnetic Barkhausen noise (MBN) mea- surements performed on a similar cube sample cut from the same plate indicated the surface magnetic easy axis of the sample in the plane of the plate [4]. This was correlated with the relative magnetic easy axis of the sample obtained from bulk hysteresis loop measurements performed in the three diagonal directions [4]. Since surface MBN measurements are localized to a 0.02-0.20 mm skin depth and reflect the area sampled by a 9 mm diameter pickup coil [4,5], these measurements established the general uniformity of the plate, from which the sample re- ported on here, was removed. Only the relative magnetic easy axis of the cube sample is determined by hysteresis loop measurement in the three orthogo- nal directions since, ideally, the angular dependence of magnetic hysteresis loops needs to be measured in order to locate the easy axis of the sample. This is similar to using MBN measurements to determine the easy axis [6,7].
292 E Yu et a l . / Journal of Magnetism and Magnetic Materials 166 (t997) 290-296
3.2. Diagonal and off-diagonal BH hysteresis loops
Magnetic hysteresis loops for the steel cube sam- ple are shown in Fig. 2. For comparison, all the diagonal loops are presented with the same B scale and all the off-diagonal ones use a B × 20 scale. Measurements were repeated several times with re- wound coils indicating repeatable results. AI1 the diagonal loops reach the same near-saturation flux density B of 2.00 T. The off-diagonal loops are not of the same shape as the diagonal elements, negating the possibility of misaligned coil magnetization con- tributions. The peak magnetic flux density for the off-diagonal loops is about an order of magnitude smaller than that for the diagonal loops due to the fact that the applied field is parallel to the plane of these off-diagonal coils. The off-diagonal compo- nents of the magnetization vector are therefore not negligible, particularly when the sample is not satu- rated. As the sample approaches saturation, all the off-diagonal components tend to zero.
We replot the diagonal loops of the sample in Fig. 3. Obviously, the z direction is the relative hard axis in the sample and the x and y directions are rela- tively easy axes with the x direction slightly harder than the y direction.
A detailed analysis of the off-diagonal compo- nents shown in Fig. 2 indicates that the maxima and minima usually occur just before the sample begins to saturate. Since domain wall motions are associated primarily with the low to intermediate field magneti- zation processes [8], domain wall motions may be identified with the changes in magnetization occur- ring between the maxima and minima in the off-di- agonal loops. The magnetization processes beyond the maxima and minima of the off-diagonal loops may be associated primarily with magnetization rota- tions since they occur during the approach to satura- tion [8]. Therefore, the maxima and minima of the off-diagonal loops indicate a transition of magnetiza- tion mechanisms between domain wall motions and magnetization rotations as the sample approaches
Fig. 2. Nine B-H hysteresis loops for the steel cube. Diagonal and off-diagonal Ioops are arranged as diagonal and off-diagonal elements in the matrix. Figs. xX, xY, and xZ are for coils X, Y, and Z, respectively, with H in the x direction, and so on. In each figure, the horizontal axis is the magnetic field H (kA/m) and the vertical axis the magnetic flux density B(T). Note: for off-diagonal loops B is multiplied by 20.
E Yu et al./Journal of Magnetism and Magnetic Materials 166 (7997) 290-296 293
-2 ' - I 0 -5 0 5 10
H (k.A./m)
Fig. 3. Diagonal B - H hysteresis loops for the cube.
saturation. As the magnetic field in a diagonal direc- tion is reduced from the nearly saturated state, mag- netization vectors rotate towards the nearest magne-
tocrystalline easy axis, contributing to the corre- sponding off-diagonal components of the total mag- netization vector. The off-diagonal components reach the maxima when the majority of the magnetization vectors complete their rotations. As the magnetic field is further reduced, more magnetic domains begin to nucleate. The magnetization processes are then followed by domain wall motions and the mag- nitudes of the off-diagonal loops start to decrease.
The relative magnitudes and directions of the off-diagonal loops are different when the fields are applied in the different orthogonal directions. In other words, the magnetic hysteresis loop tensor is asymmetric. It is interesting to note that off-diagonal loops are out of phase with respect to each other for field applied in the y direction and are in phase with each other for field in the x and z direction. We have taken into account the coil winding directions to adjust the B senses for both diagonal and off-di-
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Fig. 4. Plots of magnetization components, M.,. x, Mxy, and M.,.z, etc., as functions of the amplitude of total effective magnetization M for the sample, xX represents magnetization component measured by coil X with applied field in the x direction and xY magnetization component measured by coil g with field in the x direction, and so on. In each figure, the horizontal axis is the amplitude ( A / m ) of total magnetization vector and the vertical axis the magnetization component ( A / m ) in the corresponding x, y, or z direction. Note: for off-diagonal loops the magnetization component is multiplied by 20.
294 Y Yu et aI. / Jou~vTal of Magnetism and Magnetic Materials 166 (I997) 290-296
agonat loops so that a positive value indicates a component in +x, +y, or +z direction and a negative value a component in - x , - y , or - z direction. This assists us in a description of the total effective magnetization vector in the .ryz coordinate system shown in Fig. 1.
The asymmetry of the tensor loops are presumed to be caused by a texture of the material or the existence of internal stresses which, in turn, deter- mine the easy axis of the sample and the distribution of grain easy axis directions. The apparent correla- tion of the relative amplitudes of the off-diagonal loops with the relative easy axes of the sample is not yet well understood. However, this intermediate field behavior of the off-diagonal loops is not inconsistent with the interpretation that the entire sample has an intrinsic easy axis since easy axis is defined as the direction in which the magnetization is attained at all levels with ease, or more specifically, in which both the knee and /Zma ~ emerge at the lowest field strengths [9].
3.3. Magnetization components versus total magneti- zation
In Fig. 4, the magnetization components of the sample in the x, y, and z directions are plotted as functions of the magnitude of the total magnetization vector M. Since the signs of the magnetization com- ponents have been adjusted by taking into account the senses of coil winding, Fig. 4 may therefore be related to the total effective magnetization vector in the sample volume.
One advantage of these magnetization plots is that magnetization processes and associated hysteresis effects are more easily distinguished. With field applied in the x direction, more hysteresis effects are observed in the z direction than in the y direction; with field in the z direction, hysteresis is prominent in both the x and y directions. This suggests that hysteresis effects for the off-diagonal elements may be field orientation dependent. Fig. 4 also shows that hysteresis effects are reduced at high fields or mag- netizations, which may be associated with more re- versible magnetization vector rotation processes [8]. The hysteresis observed at lower field or magnetiza- tion may be related to the irreversible motion of 180 ° and/or 90 ° domain walls [1,2] contributing to the
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Fig. 5. Deviation of diagonal magnetization components from the straight line determined by the origin and either positive or negative end point in Fig. 4 as functions of the magnitude of the total magnetization. Both axes are in units of A / m and the deviations are multiplied by 200. The exact and estimated data are calculated from Eqs. (4) and (6), respectively. Figs. (a), (b), and (c) are for magnetic field in the x, y, and z directions, respec- tively.
magnetization components in the corresponding di- rections.
Although the diagonal components of the total effective magnetization vector are dominant over the corresponding off-diagonal ones, they vary nonlin- early with the total effective magnetization due to the non-negligible off-diagonal components when the sample is not yet saturated. Fig. 5a-5c show the deviation of diagonal magnetization components from the straight line determined by the origin and either
II. Yu et at./Journal of Magnetism and Magnetic Materials 166 (1997) 290-296 295
positive or negative end point in Fig. 4. A positive value in Fig. 5a-5c indicates a point below the straight line determined by the origin and positive near-saturation point. Likewise, a negative value in- dicates a point above the straight line determined by the origin and negative near saturation point. The maximum deviation from the straight line occurs when the corresponding off-diagonal magnetization components reach a maximum or minimum. The maxima and minima in Fig. 4 are thus the positions of maximum deviation of the sampte magnetization from the field direction. As the sample approaches saturation, the diagonal components vary linearly with total magnetization. This is expected since all domain magnetization directions are now aligned with the field direction [8].
It can be seen from Figs. 4 and 5 that the devia- tions from linearity of the diagonal magnetization components are about an order of magnitude smaller than corresponding off-diagonal components and are about two orders of magnitude smaller than the diagonal magnetization components. This can be de- rived mathematically. The total effective magnetiza- tion vector M for magnetization along a direction i may be expressed as
M = M i I + M i : + M i K , (1)
where Mir is the diagonal magnetization vector component, Mij and MiK the off-diagonal compo- nents, i the field direction x, y, or z, and I, J, and K cyclically refer to the coils X, Y, and Z, which measure the respective magnetization components in the x, y, and z directions. The magnitude of the total magnetization vector may be written as
M = tMi~ + Mi2s + Mi~ . (2)
The diagonal component, Mil , may then be ex- pressed as
Mir = +__ ~/M 2 - M?: - Mi~ . (3)
where the + and - signs indicate magnetization components in the + i and - i directions, respec- tively. The deviation from linearity can be expressed a s
A~l i i = + ( M - M i I )
= 4 - ( M - t M 2 - M i S - M i % ) . (4)
Since M >> Mi: and Mix, Eq. (3), to first order, can be rewritten as
Mi1~ 4- M 2M ' (5)
A Miz can then be written as
A Mii ~ 4- Mid + Mi~ 2M ' (6)
where + and - signs are for Mil > 0 and Mil < 0, respectively. A Mir is the second order quantity since Mij and Mix are the first order quantities compared to M. Fig. 5 shows both the exact and estimated deviations from linearity with field in the x, y and z directions, using Eqs. (4) and (6), respectively. It is seen that the estimated curve agrees vet5, well with the exact curve.
An analysis of the angle of the net magnetization vector with respect to the applied field may also be performed. In this case the maximum deviation of the net magnetization vector from the field direction where AMiz is also a maximum occurs at 7.0 °, 6.0 °, and 3.2 ° for the field applied along the x, y, and z directions, respectively. The relative magnitudes of these angles are in agreement with the relative mag- nitudes of the A Miz from the total sample magni- tude. The respective size of the deviations corre- sponds to the degree of offset of the net magnetiza- tion vector from the applied field direction.
4. Conclusions
Tensor magnetic hysteresis loops have been mea- sured for a steel cube. The diagonal elements en- abled the relative magnetic easy axis of the steel cube sample to be deduced.
Our study showed that the off-diagonal loops are not negligible for the steel cube studied. The off-di- agonal loops are useful for measuring magnetization components as well as understanding the magnetiza- tion processes in bulk magnetic materials. The off- diagonal components are correlated with various magnetization processes such as domain wall motion and magnetization rotation. The asymmetry of the
296 Y. Yu et al. / Journal of Magnetism and Magnetic Materials 166 (1997) 290-296
loop tensor is attributed to texture in the material or the existence of internal stresses.
The magnetization components in the x, y, and z directions vary with the magnitude of the total effec- tive magnetization. The diagonal magnetization com- ponents vary nonlinearly with the total magnetization before the sample begins to saturate. This is ex- plained by magnetization contributions to off-diago- nal components. Compared to total magnetization, the deviation of the diagonal magnetization compo- nents from linearity is of second order and the off-diagonal magnetization components are of first order; the deviation from linearity is described by the approximate expression 5 M i I ~ +(Mid + M i ~ ) / 2 M , where M is the magnitude of the total magnetization vector, Mz] and Mix the off-diagonal magnetization components, i the magnetic field di- rection x, y, or z, and I, J, and K cyclic coordi- nates X, Y, and Z indicating the magnetization components measured by coils X, 17, and Z, respec- tively. The total magnetization of the sample may be recovered at high field by domain vector rotation. These magnetization processes are dominant along certain directions and define the direction of the net magnetization vector.
Acknowledgements
The authors would like to thank J. Thompson for cutting cube samples. This work was supported by Natural Sciences and Engineering Research Council of Canada, the Province of Ontario, and Pipetronix Ltd.
