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Interactions among magnetic dipoles induced in feeble magnetic substances under high magnetic fields
ARTICLE IN PRESS
Physica B 346–347 (2004) 272–276
Interactions among magnetic dipoles induced in feeble magnetic substances under high magnetic fields Tomohiro Takayamaa,*, Yasuhiro Ikezoea, Hiromichi Uetakea, Noriyuki Hirotaa,b, Koichi Kitazawaa,b a
Department of Advaced Materials Science, University of Tokyo, 5-1-5-402 Kashiwanoha, Kashiwa, Chiba 277-8561, Japan b Japan Science and Technology Corporation, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan
Abstract Interactions among induced magnetic dipoles were observed in the systems of feeble magnetic substances. Usually, such interactions are too small to be observed, and have been neglected so far. However, we focused on the interactions, and succeeded in observing the interactions through elaborate experiments. Furthermore, by applying this interactions to many-particle systems, some peculiar alignments were obtained. These phenomena would be of use as a control of materials structures, and bring new applications of magnetic fields in various processing. r 2004 Elsevier B.V. All rights reserved. PACS: 75.90.+w; 81.90.+c Keywords: Magnetic dipole interaction; Feeble magnetic substance; High magnetic field; Magneto-Archimedes effect
1. Introduction Recently, effects of magnetic fields on para- and diamagnetic, namely, feeble magnetic substances are investigated energetically, and various phenomena have been found such as magnetic levitations [1,2], Moses effect [3] and so on. These effects are mainly based on magnetic forces, and magnetic forces can be expressed as interactions between feeble magnetic substances and gradient fields. On the contrary, interactions among feeble magnetic substances under magnetic fields have been neglected so far. In ferromagnetic substances, *Corresponding author. Tel.: +81-4-7136-3767; fax:+81-47136-3792. E-mail address: [email protected] (T. Takayama).
such interactions through their magnetic dipoles can be observed clearly [4,5], and the energy of the interactions of two magnetic dipoles is expressed as m0 ma mb 3ðma rÞðmb rÞ U¼ ½J; ð1Þ r5 4p r3 where m0 is the permeability of vacuum, ma and mb are the magnetic dipoles, and r and r are the vector between two dipoles and its distance, respectively. On the other hand, in feeble magnetic substances, magnetic dipoles are induced only under magnetic fields, and their values are extremely small. Therefore, the energy of the interactions becomes too small, and they are not usually observed. However, through elaborate experiments using high magnetic fields of several teslas, we confirmed
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.01.064
ARTICLE IN PRESS T. Takayama et al. / Physica B 346–347 (2004) 272–276
that such interactions can be observed visually even in feeble magnetic substances. Then, it is expected that alignments or structures of feeble magnetic substances can be controlled utilizing the interactions, and such applications would be useful, for example, in material processing. This paper reports on the observations of the induced magnetic dipole interactions and basic researches on the control of structures of feeble magnetic substances.
2. Induced dipole interactions between two samples In this section, we report on the experiments of the observations of interactions between two magnetic dipoles. The experimental set-up is shown in Fig. 1. In this study, we used a cryocooler cooled superconducting magnet with a 100 mm f room-temperature bore. The magnet was placed vertically and the direction of fields was parallel to that of gravity. The samples used in this experiment were palladium (paramagnetic, volume magnetic susceptibility w=7.78 104 [in SI units]) and gold (diamagnetic, w=3.45 105 [in SI units]). Both samples had cylindrical shapes of 1.0 mm in diameters and 5.0 mm in height.
273
These samples were held side by side with some spacing (B1.0 mm) in the bore of the magnet with polyester fibers, and the area where magnetic fields are almost flat in the horizontal direction was selected as the samples position to avoid effects of magnetic forces in the horizontal direction. This experiment was performed in three pairs of the samples such as palladium–palladium, palladium– gold and gold–gold. From this configuration, magnetic fields were increased gradually, and the distance between samples was observed. Fig. 2 shows the result of the case of the palladium– palladium pair. The upper figure shows the initial state, and the lower one shows the state with a magnetic field of 6 T. In this figure, the magnetic field was applied parallel to this space and directed from lower side to upper side. From this result, the distance between samples was increased about 0.4 mm with applying magnetic field of 6 T. This was because the samples repelled with each other through magnetic dipoles induced in them. However, no interaction was observed in the cases of the palladium–gold and gold–gold pair. This seems because the absolute value of the volume magnetic susceptibility of gold is only 101 as large
0T
Polyester fiber
1.4 mm
Samples s.c. magnet
1.0 mm
6T
1.8 mm Field Fiel 1.0 mm Fig. 1. Experimental set-up for the observations of interactions between two induced magnetic dipoles.
Fig. 2. Repulsive interaction of palladium cylinders under a magnetic field of 6 T.
ARTICLE IN PRESS 274
T. Takayama et al. / Physica B 346–347 (2004) 272–276
as that of palladium, and the magnetic dipoles induced in the gold samples were too small. Then, quantitative analysis was performed on the result of the palladium–palladium pair. From the experimental result, the force derived from the dipole interaction was estimated to be 8.5 107 N. On the other hand, this value of the force can be also given from calculations. The magnetic field around the samples was distorted spatially by induced magnetic dipoles, and its distribution was calculated by computer. From the result, the value of magnetic forces acting in the horizontal direction, that is, the value of magnetic forces derived from dipole interactions was obtained. This value was found to be 2.4 107 N. From these results, it could be possible to say that the experimental result and the calculate value were in substantial agreement. Therefore, it was confirmed that the observed phenomenon was caused by the induced magnetic dipole interaction. Next, the enhancement of dipole interactions was considered. In the previous results, existence of the environmental surroundings was neglected. Then, by considering the environmental surroundings, Eq. (1) can be modified as follows: m Dma Dmb 3ðDma rÞðDmb rÞ U¼ 0 ½J; ð2Þ r5 4p r3 where Dm represents the difference of magnetic dipoles between samples and surroundings. According to this equation, the energy of interactions can be enlarged by selecting environmental surroundings properly, and then the interactions can be enhanced. This means that magneto-Archimedes effect, enhancement of magnetic forces by environmental surroundings [2], also appears in magnetic dipole interactions. The magneto-Archimedes effect was examined experimentally in the gold–gold pair. By considering magneto-Archimedes effects, interactions between diamagnetic samples can be enhanced in a paramagnetic medium. In the experiment, manganese dichloride aqueous solution was selected as a paramagnetic medium. The concentration of used solution was 40 wt%, and its susceptibility is 7.99 104 [in SI units]. By using this solution as a surrounding medium, the same procedure was performed. The result of this case is shown in Fig. 3. The upper
0T
1.0 mm
1.0 mm
3T
1.3 mm
1.0 mm
Fig. 3. Repulsive interaction of gold cylinders in 40 wt% MnCl2 aqueous solution under a magnetic field of 3 T.
figure shows the initial state, and the lower one shows the state with a magnetic field of 3 T. In this case, magnetic dipoles directed opposite to the applied field were induced in the gold samples, and repulsive interaction between the samples appeared. In addition, the interaction was enhanced by the surrounding MnCl2aq in this case. Then, the samples were observed to go away about 0.3 mm, which could not be observed without using MnCl2aq. Therefore, it was confirmed that induced magnetic dipole interactions can be enhanced by selecting environmental surroundings properly, that is, by considering the magnetoArchimedes effect.
3. Alignments of feeble magnetic particles utilizing induced magnetic dipole interactions Subsequently, the experiments using many feeble magnetic particles were carried out. It has been known that magnetic dipole interactions lead dispersed particles to some ordered alignments [4,5]. However, only systems containing ferromagnetic substances have been considered so far, and such applications have been restricted in a few
ARTICLE IN PRESS T. Takayama et al. / Physica B 346–347 (2004) 272–276
materials. The applications of this effect to the systems of feeble magnetic substances would make it possible to control structures of various materials, which would be of use in material processing. Therefore, the experiments to examine such possible applications were performed. In this section, alignments parallel and perpendicular to fields were considered. First, alignments parallel to magnetic fields were observed. The experimental set-up for this case is shown in Fig. 4. The sample particles used in this experiment were glass beads (B0.8 mm f), which are diamagnetic and have volume magnetic susceptibility of 1.8 105 [in SI units]. Manganese dichloride aqueous solution of 40 wt% was used as a medium in consideration of magnetoArchimedes effect. In this experiment, the same magnet as used in previous section was set horizontally. The glass beads and MnCl2aq were put in a glass cell (B70 25 5 mm3), and the cell was inserted to the bore of the magnet. One of the cell edges was fixed at the center of the fields, and the glass beads were gathered at that side in the initial condition. From this configuration, magnetic fields were increased gradually. Then, magnetic forces acted on the glass beads, and they moved to the other side of the cell to go away from the center of the fields. These processes were
275
observed from the bottom side of the cell with a CCD camera. The result of this experiment is shown in Fig. 5. In this figure, the magnetic field was applied parallel to this space directed from left to right, and its intensity was 2.5 T. As seen in this figure, the glass beads formed chain-like alignments parallel to the applied field during moving away from the center of the fields. These alignments were derived from the attractive interactions among magnetic dipoles induced in the glass beads. From this result, alignments of feeble magnetic particles can be controlled by magnetic fields in one-dimensional direction parallel to the fields. Next, alignments perpendicular to fields were observed. The experimental set-up for this case is shown in Fig. 6. In this experiment, the magnet was placed vertically, and a petri dish, where the
Fig. 5. Chain-like alignments of glass beads in 40 wt% MnCl2aq. The magnetic field was directed from left to right, and its intensity was 2.5 T at the field center.
S. C. Magnet MnCl2 aq
Gold balls
B Glass beads CCD cam camera ra S. C. Magnet
Mirror MnCl 2 aq Center of the magnetic fields
Fig. 4. Experimental set-up for observation of dipole interactions in many-particle systems (1).
Magnetic Field Fig. 6. Experimental set-up for observation of dipole interactions in many-particle systems (2).
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sample particles and surrounding medium were put, was placed in its bore. The sample particles used in this case were gold balls of 1.0 mm in diameters, and 40 wt% MnCl2aq was used as a surrounding medium. The vertical position of the gold balls was 149 mm above the center of fields, where magnetic fields are only slightly larger (B0.2%) at the wall side than at the middle of the bore. From this configuration, magnetic fields were applied and the two-dimensional alignments of gold balls were observed from above. To observe the forces derived from dipole interactions in the horizontal direction effectively, the intensity of magnetic fields was adjusted and the apparent weights of the gold balls were set to be zero utilizing vertical magnetic forces. That is, magneto-Archimedes levitation was applied [2]. Fig. 7
shows the magneto-Archimedes levitation state of gold balls observed from above. In the levitation condition, the gold balls gathered to the middle of the bore because of the slight radial magnetic forces. However, the gathering balls were not packed closely, and formed triangle-lattice alignments with some spacing. This formation of the lattices was caused by the repulsive interactions among each magnetic dipole induced in the gold balls. From this result, alignments perpendicular to fields can be controlled by repulsive interactions among induced magnetic dipoles. Therefore, the formations of ordered alignments can be obtained not only in the systems containing ferromagnetic substances, but also in the systems of feeble magnetic substances by controlling experimental conditions properly.
4. Summary In this study, interactions among magnetic dipoles induced in feeble magnetic substances were observed. Usually, such interactions are too small to be observed and have been neglected so far. However, by controlling experimental conditions carefully, we succeeded in observing the subtle interactions. Furthermore, by applying the interactions to the many-particle systems, some ordered alignments were obtained such as chain-like alignments and triangle-lattice alignments. These results show that structures of systems of feeble magnetic substances can be controlled by this interactions. Therefore, these phenomena suggest new applications of magnetic fields to various fields such as material processing.
References
Fig. 7. Triangle-lattice alignments of gold balls in 40 wt% MnCl2aq. The direction of magnetic field was perpendicular to this space, and its intensity was 4.9 T. The lower figure is a close-up view of the upper one.
[1] E. Beaugnon, R. Tournier, Nature 349 (1991) 470. [2] Y. Ikezoe, N. Hirota, J. Nakagawa, K. Kitazawa, Nature 393 (1998) 749. [3] N. Hirota, T. Homma, H. Sugawara, K. Kitazawa, M. Iwasaka, S. Ueno, H. Yokoi, Y. Kakudate, S. Fujiwara, K. Kawamura, Jpn. J. Appl. Phys. 34 (1996) L991. [4] P.G. de Gennes, P.A. Pincus, Phys. Kondens. Materie 11 (1970) 189. [5] A.T. Skjeltorp, Phys. Rev. Lett. 51 (1983) 2306.
Physica B 346–347 (2004) 272–276
Interactions among magnetic dipoles induced in feeble magnetic substances under high magnetic fields Tomohiro Takayamaa,*, Yasuhiro Ikezoea, Hiromichi Uetakea, Noriyuki Hirotaa,b, Koichi Kitazawaa,b a
Department of Advaced Materials Science, University of Tokyo, 5-1-5-402 Kashiwanoha, Kashiwa, Chiba 277-8561, Japan b Japan Science and Technology Corporation, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan
Abstract Interactions among induced magnetic dipoles were observed in the systems of feeble magnetic substances. Usually, such interactions are too small to be observed, and have been neglected so far. However, we focused on the interactions, and succeeded in observing the interactions through elaborate experiments. Furthermore, by applying this interactions to many-particle systems, some peculiar alignments were obtained. These phenomena would be of use as a control of materials structures, and bring new applications of magnetic fields in various processing. r 2004 Elsevier B.V. All rights reserved. PACS: 75.90.+w; 81.90.+c Keywords: Magnetic dipole interaction; Feeble magnetic substance; High magnetic field; Magneto-Archimedes effect
1. Introduction Recently, effects of magnetic fields on para- and diamagnetic, namely, feeble magnetic substances are investigated energetically, and various phenomena have been found such as magnetic levitations [1,2], Moses effect [3] and so on. These effects are mainly based on magnetic forces, and magnetic forces can be expressed as interactions between feeble magnetic substances and gradient fields. On the contrary, interactions among feeble magnetic substances under magnetic fields have been neglected so far. In ferromagnetic substances, *Corresponding author. Tel.: +81-4-7136-3767; fax:+81-47136-3792. E-mail address: [email protected] (T. Takayama).
such interactions through their magnetic dipoles can be observed clearly [4,5], and the energy of the interactions of two magnetic dipoles is expressed as m0 ma mb 3ðma rÞðmb rÞ U¼ ½J; ð1Þ r5 4p r3 where m0 is the permeability of vacuum, ma and mb are the magnetic dipoles, and r and r are the vector between two dipoles and its distance, respectively. On the other hand, in feeble magnetic substances, magnetic dipoles are induced only under magnetic fields, and their values are extremely small. Therefore, the energy of the interactions becomes too small, and they are not usually observed. However, through elaborate experiments using high magnetic fields of several teslas, we confirmed
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.01.064
ARTICLE IN PRESS T. Takayama et al. / Physica B 346–347 (2004) 272–276
that such interactions can be observed visually even in feeble magnetic substances. Then, it is expected that alignments or structures of feeble magnetic substances can be controlled utilizing the interactions, and such applications would be useful, for example, in material processing. This paper reports on the observations of the induced magnetic dipole interactions and basic researches on the control of structures of feeble magnetic substances.
2. Induced dipole interactions between two samples In this section, we report on the experiments of the observations of interactions between two magnetic dipoles. The experimental set-up is shown in Fig. 1. In this study, we used a cryocooler cooled superconducting magnet with a 100 mm f room-temperature bore. The magnet was placed vertically and the direction of fields was parallel to that of gravity. The samples used in this experiment were palladium (paramagnetic, volume magnetic susceptibility w=7.78 104 [in SI units]) and gold (diamagnetic, w=3.45 105 [in SI units]). Both samples had cylindrical shapes of 1.0 mm in diameters and 5.0 mm in height.
273
These samples were held side by side with some spacing (B1.0 mm) in the bore of the magnet with polyester fibers, and the area where magnetic fields are almost flat in the horizontal direction was selected as the samples position to avoid effects of magnetic forces in the horizontal direction. This experiment was performed in three pairs of the samples such as palladium–palladium, palladium– gold and gold–gold. From this configuration, magnetic fields were increased gradually, and the distance between samples was observed. Fig. 2 shows the result of the case of the palladium– palladium pair. The upper figure shows the initial state, and the lower one shows the state with a magnetic field of 6 T. In this figure, the magnetic field was applied parallel to this space and directed from lower side to upper side. From this result, the distance between samples was increased about 0.4 mm with applying magnetic field of 6 T. This was because the samples repelled with each other through magnetic dipoles induced in them. However, no interaction was observed in the cases of the palladium–gold and gold–gold pair. This seems because the absolute value of the volume magnetic susceptibility of gold is only 101 as large
0T
Polyester fiber
1.4 mm
Samples s.c. magnet
1.0 mm
6T
1.8 mm Field Fiel 1.0 mm Fig. 1. Experimental set-up for the observations of interactions between two induced magnetic dipoles.
Fig. 2. Repulsive interaction of palladium cylinders under a magnetic field of 6 T.
ARTICLE IN PRESS 274
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as that of palladium, and the magnetic dipoles induced in the gold samples were too small. Then, quantitative analysis was performed on the result of the palladium–palladium pair. From the experimental result, the force derived from the dipole interaction was estimated to be 8.5 107 N. On the other hand, this value of the force can be also given from calculations. The magnetic field around the samples was distorted spatially by induced magnetic dipoles, and its distribution was calculated by computer. From the result, the value of magnetic forces acting in the horizontal direction, that is, the value of magnetic forces derived from dipole interactions was obtained. This value was found to be 2.4 107 N. From these results, it could be possible to say that the experimental result and the calculate value were in substantial agreement. Therefore, it was confirmed that the observed phenomenon was caused by the induced magnetic dipole interaction. Next, the enhancement of dipole interactions was considered. In the previous results, existence of the environmental surroundings was neglected. Then, by considering the environmental surroundings, Eq. (1) can be modified as follows: m Dma Dmb 3ðDma rÞðDmb rÞ U¼ 0 ½J; ð2Þ r5 4p r3 where Dm represents the difference of magnetic dipoles between samples and surroundings. According to this equation, the energy of interactions can be enlarged by selecting environmental surroundings properly, and then the interactions can be enhanced. This means that magneto-Archimedes effect, enhancement of magnetic forces by environmental surroundings [2], also appears in magnetic dipole interactions. The magneto-Archimedes effect was examined experimentally in the gold–gold pair. By considering magneto-Archimedes effects, interactions between diamagnetic samples can be enhanced in a paramagnetic medium. In the experiment, manganese dichloride aqueous solution was selected as a paramagnetic medium. The concentration of used solution was 40 wt%, and its susceptibility is 7.99 104 [in SI units]. By using this solution as a surrounding medium, the same procedure was performed. The result of this case is shown in Fig. 3. The upper
0T
1.0 mm
1.0 mm
3T
1.3 mm
1.0 mm
Fig. 3. Repulsive interaction of gold cylinders in 40 wt% MnCl2 aqueous solution under a magnetic field of 3 T.
figure shows the initial state, and the lower one shows the state with a magnetic field of 3 T. In this case, magnetic dipoles directed opposite to the applied field were induced in the gold samples, and repulsive interaction between the samples appeared. In addition, the interaction was enhanced by the surrounding MnCl2aq in this case. Then, the samples were observed to go away about 0.3 mm, which could not be observed without using MnCl2aq. Therefore, it was confirmed that induced magnetic dipole interactions can be enhanced by selecting environmental surroundings properly, that is, by considering the magnetoArchimedes effect.
3. Alignments of feeble magnetic particles utilizing induced magnetic dipole interactions Subsequently, the experiments using many feeble magnetic particles were carried out. It has been known that magnetic dipole interactions lead dispersed particles to some ordered alignments [4,5]. However, only systems containing ferromagnetic substances have been considered so far, and such applications have been restricted in a few
ARTICLE IN PRESS T. Takayama et al. / Physica B 346–347 (2004) 272–276
materials. The applications of this effect to the systems of feeble magnetic substances would make it possible to control structures of various materials, which would be of use in material processing. Therefore, the experiments to examine such possible applications were performed. In this section, alignments parallel and perpendicular to fields were considered. First, alignments parallel to magnetic fields were observed. The experimental set-up for this case is shown in Fig. 4. The sample particles used in this experiment were glass beads (B0.8 mm f), which are diamagnetic and have volume magnetic susceptibility of 1.8 105 [in SI units]. Manganese dichloride aqueous solution of 40 wt% was used as a medium in consideration of magnetoArchimedes effect. In this experiment, the same magnet as used in previous section was set horizontally. The glass beads and MnCl2aq were put in a glass cell (B70 25 5 mm3), and the cell was inserted to the bore of the magnet. One of the cell edges was fixed at the center of the fields, and the glass beads were gathered at that side in the initial condition. From this configuration, magnetic fields were increased gradually. Then, magnetic forces acted on the glass beads, and they moved to the other side of the cell to go away from the center of the fields. These processes were
275
observed from the bottom side of the cell with a CCD camera. The result of this experiment is shown in Fig. 5. In this figure, the magnetic field was applied parallel to this space directed from left to right, and its intensity was 2.5 T. As seen in this figure, the glass beads formed chain-like alignments parallel to the applied field during moving away from the center of the fields. These alignments were derived from the attractive interactions among magnetic dipoles induced in the glass beads. From this result, alignments of feeble magnetic particles can be controlled by magnetic fields in one-dimensional direction parallel to the fields. Next, alignments perpendicular to fields were observed. The experimental set-up for this case is shown in Fig. 6. In this experiment, the magnet was placed vertically, and a petri dish, where the
Fig. 5. Chain-like alignments of glass beads in 40 wt% MnCl2aq. The magnetic field was directed from left to right, and its intensity was 2.5 T at the field center.
S. C. Magnet MnCl2 aq
Gold balls
B Glass beads CCD cam camera ra S. C. Magnet
Mirror MnCl 2 aq Center of the magnetic fields
Fig. 4. Experimental set-up for observation of dipole interactions in many-particle systems (1).
Magnetic Field Fig. 6. Experimental set-up for observation of dipole interactions in many-particle systems (2).
ARTICLE IN PRESS 276
T. Takayama et al. / Physica B 346–347 (2004) 272–276
sample particles and surrounding medium were put, was placed in its bore. The sample particles used in this case were gold balls of 1.0 mm in diameters, and 40 wt% MnCl2aq was used as a surrounding medium. The vertical position of the gold balls was 149 mm above the center of fields, where magnetic fields are only slightly larger (B0.2%) at the wall side than at the middle of the bore. From this configuration, magnetic fields were applied and the two-dimensional alignments of gold balls were observed from above. To observe the forces derived from dipole interactions in the horizontal direction effectively, the intensity of magnetic fields was adjusted and the apparent weights of the gold balls were set to be zero utilizing vertical magnetic forces. That is, magneto-Archimedes levitation was applied [2]. Fig. 7
shows the magneto-Archimedes levitation state of gold balls observed from above. In the levitation condition, the gold balls gathered to the middle of the bore because of the slight radial magnetic forces. However, the gathering balls were not packed closely, and formed triangle-lattice alignments with some spacing. This formation of the lattices was caused by the repulsive interactions among each magnetic dipole induced in the gold balls. From this result, alignments perpendicular to fields can be controlled by repulsive interactions among induced magnetic dipoles. Therefore, the formations of ordered alignments can be obtained not only in the systems containing ferromagnetic substances, but also in the systems of feeble magnetic substances by controlling experimental conditions properly.
4. Summary In this study, interactions among magnetic dipoles induced in feeble magnetic substances were observed. Usually, such interactions are too small to be observed and have been neglected so far. However, by controlling experimental conditions carefully, we succeeded in observing the subtle interactions. Furthermore, by applying the interactions to the many-particle systems, some ordered alignments were obtained such as chain-like alignments and triangle-lattice alignments. These results show that structures of systems of feeble magnetic substances can be controlled by this interactions. Therefore, these phenomena suggest new applications of magnetic fields to various fields such as material processing.
References
Fig. 7. Triangle-lattice alignments of gold balls in 40 wt% MnCl2aq. The direction of magnetic field was perpendicular to this space, and its intensity was 4.9 T. The lower figure is a close-up view of the upper one.
[1] E. Beaugnon, R. Tournier, Nature 349 (1991) 470. [2] Y. Ikezoe, N. Hirota, J. Nakagawa, K. Kitazawa, Nature 393 (1998) 749. [3] N. Hirota, T. Homma, H. Sugawara, K. Kitazawa, M. Iwasaka, S. Ueno, H. Yokoi, Y. Kakudate, S. Fujiwara, K. Kawamura, Jpn. J. Appl. Phys. 34 (1996) L991. [4] P.G. de Gennes, P.A. Pincus, Phys. Kondens. Materie 11 (1970) 189. [5] A.T. Skjeltorp, Phys. Rev. Lett. 51 (1983) 2306.