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Large gyromagnetic effect in soft magnetic amorphous ribbons and wires
Journal of Magnetism and Magnetic Materials 215}216 (2000) 413} 415
Large gyromagnetic e!ect in soft magnetic amorphous ribbons and wires Horia Chiriac*, Tibor-Adrian OD vaH ri, Cay tay lin Sandrino Marinescu National Institute of R&D for Technical Physics, 47 Mangeron Blvd., 6600 Iasi 3, Romania
Abstract We report results on the Einstein}de Haas rotation of Fe Si B amorphous ribbons and wires, a phenomenon called large gyromagnetic e!ect. The laws of this e!ect are "rstly inferred from the experimental data, and then a comprehensive model that explains these laws is presented. 2000 Elsevier Science B.V. All rights reserved. Keywords: Gyromagnetic e!ect; Magnetic standing wave; Soft magnetic materials
We have recently reported that an Fe Si B amorphous wire with a certain diameter U and length L, subjected to an axial alternating magnetic "eld H(t)"H sin 2plt, where H is the "eld amplitude,
and l its frequency, starts rotating around its axis at given values of H and m [1]. We called this e!ect the
large gyromagnetic ewect (LGE). The aim of this paper is to report for the "rst time the appearance of the LGE in Fe Si B amorphous ribbons, to present the laws of the LGE in both wires and ribbons with this composition, based on the experimental investigations performed on such samples, as well as the &magnetic standing wave' model that explains the characteristics and appearance conditions of the LGE. We prepared Fe Si B amorphous wires with U"120 lm by in-rotating-water spinning, and 20 lm thick and 0.565 mm wide amorphous ribbons with the same composition by chill block melt spinning, in order to obtain samples with the same cross section areas (S"11.3;10\ m). For the experiments we have chosen samples with 15 mm)¸)120 mm. The ribbonand wire-shaped samples were vertically placed * one at a time * in a 10 mm long magnetizing coil with 4 mm diameter and 200 turns/mm. They were positioned on
* Corresponding author. Tel.: #40-32-130680; fax: #40-32231132. E-mail address: [email protected] (H. Chiriac).
a quartz bearing to reduce friction and to center the wire within the coil. H(t) was generated with a current supplied by a programmable function generator. The maximum value of H was 300 A m\, and its frequency
l varied from 1 to 60 kHz. We measured the values of H and l at which the LGE occurs in both ribbons and
wires, for di!erent sample lengths ¸. We observed that both ribbon- and wire-shaped samples having a given length, ¸, and subjected to a "eld with a given amplitude H , start rotating in a certain direc tion when the "eld frequency reaches a certain value l !*l/2. If the "eld frequency is slightly increased, the sample rotation is maintained in the same direction, until the frequency reaches the value l , at which the rotation stops, but the sample oscillates. Further increase of the "eld frequency determines again a sample rotation, but in the opposite direction, the rotation being maintained until the frequency reaches l #*l/2. When this value is exceeded, the rotation disappears, and also does any oscillation. Thus, the LGE occurs within a speci"c frequency range *l. The entire process reappears at multiples of l : l "2l , l "3l , etc. This is the xrst law of the LGE: for a given sample length and amplitude of the axial alternating "eld, the e!ect appears, irrespective of the samples shape * ribbon or wire * in a certain frequency range *l centered around a well-de"ned fundamental frequency l , and it reappears in frequency ranges centered around multiples of this frequency, the superior harmonics l "2l , m "3m , etc. The frequen cies l , l , l , etc. and the width of the frequency range in
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 1 7 4 - 8
414
H. Chiriac et al. / Journal of Magnetism and Magnetic Materials 215}216 (2000) 413}415
Fig. 1. Dependence of the frequencies at which LGE occurs on the inverse of the sample length for H "200 A m\ (solid
lines), and on the "eld amplitude, in 25 mm long samples (dashed lines), for Fe Si B amorphous ribbons with 20 lm in thickness and 0.565 mm in width.
which the LGE appears are the same for ribbon- and wire-shaped samples having the same length and cross section area. The appearance of the LGE at discrete values of l indicates the presence of a low-frequency resonance phenomenon, and consequently that a speci"c wave can be associated to this process. The frequencies l , l , and l are plotted against the inverse value of the sample length 1/¸ for ribbons in Fig. 1 (solid lines), at H "200 A m\. It is meaningful
to emphasize that these dependencies are identical for wires with the same value of S [1]. From these results, we can infer the second law of the LGE: the frequencies of the applied "eld at which LGE occurs are directly proportional to the inverse value of the sample length: l J1/¸, where i"0, 1, 2, etc. The linear dependence of l on 1/¸ indicates that the wave involved in the LGE has a standing character. Another experimental fact is that the LGE does not appear irrespective of frequency, if H is smaller than
a certain threshold value, almost equal to the switching "eld, H* at which a large Barkhausen e!ect appears in the Fe Si B amorphous wires, and to the coercive force H of the Fe Si B amorphous ribbons. This ! fact constitutes the third law of the LGE: the "eld amplitude must be larger than a certain threshold value in order for the LGE to occur: H *H* for wires and
H *H for ribbons.
! The dependencies of l , m , and l on H for
Fe Si B amorphous ribbons with ¸"const. (25 mm) are also illustrated in Fig. 1 (dashed lines). The values of H in this experiment were chosen to be
larger than both H for ribbons (16 A m\) and H* ! for wires (30 A m\). These dependencies are identical
for wires with the same S [1], and they point to the fourth law of the LGE: the "eld frequencies at which LGE occurs are directly proportional to its amplitude: l JH , i"0, 1, 2, etc. G
The speci"c domain structure of highly positive magnetostrictive amorphous wires such as Fe Si B ones, originating in the minimization of the large magnetoelastic energy determined by the coupling between magnetostriction and internal stresses induced during preparation, mainly consists of an axially magnetized inner core and a radially magnetized outer shell, the volume of the inner core being &70% of the entire wire volume [2]. This domain structure favors the appearance of a large Barkhausen e!ect, that consists of a 1803 domain wall propagation determined by a "eld larger than a certain threshold value, called the switching "eld H*. Since the changes in the directions of the magnetic moments from the inner core occur with "nite velocity, we can consider the wall as a disturbance that propagates within a "nite medium, and we can associate a magnetic planar wave to its movement. When the applied "eld is a high-frequency alternating one, there is a nucleation and propagation of new 1803 domain walls at each wire end, each time the "eld reverses, and consequently, new magnetic planar waves are generated. These waves interfere, and therefore it is plausible to expect the appearance of a magnetic standing wave in certain conditions related to the correlation between l and ¸. In the case of Fe Si B amorphous ribbons, which are also prepared by rapid solidi"cation, the magnetoelastic energy minimization leads to the formation of a maze-like domain structure that consists of multiple domains with in-plane axial anisotropy, separated by 1803 domain walls. This structure is somewhat equivalent to that of the wires subjected to an alternating axial "eld: the 1803 domain walls which are propagating within the ribbon subjected to H(t) can also be treated as planar waves which interfere, resulting in given circumstances in a standing wave. Both cases (ribbon and wire) can be explained in a single manner, irrespective of the sample's shape, starting from a wave associated to the movement of a 1803 domain wall. Such a wave is well described by h(z, t) * the angle of the magnetic moment located at the coordinate z on the axial direction, with respect to the sample's axis, at the instant of time t; h3[0, p], if the sample is subjected to H(t). h changes in this range with a "nite velocity * the phase velocity of the magnetic wave * which is actually the domain wall velocity < . "5 h(z, t) is given by h(z, t)"h e SRe\?> IX,
(1)
where the maximum amplitude h is p/2, and k"2p/j , k being the wavelength of the &domain "5 "5
H. Chiriac et al. / Journal of Magnetism and Magnetic Materials 215}216 (2000) 413}415
Fig. 2. Frequency dependence of Re(h), for a 60 mm long FeSiB amorphous ribbon or wire-shaped sample.
wall' wave, j "< /m. The energy losses in the "5 "5 sample are described by the damping factor a, which for simplicity, is assumed to be constant in time. The wave that results from the interference of n such waves (generated by nucleation and propagation in the case of wires and pre-existent in the case of ribbons), can be expressed after several transformations as h(t)"h [Re(h)#j Im(h)]e SR, in which the real, Re(h), and imaginary parts, Im(h), respectively, are given by sinh(2a¸) Re(h)" , (2) cosh(2a¸)!cos(4pl¸/< ) "5 sin(4pl¸/< ) "5 Im(h)" , (3) cosh(2a¸)!cos(4pl¸/< ) "5 in which we considered the explicit frequency dependencies. The resulting wave is a standing one only when Re(h) is maximum, i.e. when the conditions for interference maxima are accomplished. According to Eq. (2), Re(h) is maximum when ¸"nj /2, being equal to coth (a¸). "5 Fig. 2 illustrates the frequency dependence of Re(h), for ¸"60 mm, in which we used < "580 m s\, value "5 determined by the Sixtus}Tonks technique for an Fe Si B amorphous sample. The "rst maximum (l "4.84 kHz) corresponds to n"1, the wavelength of the standing wave in this particular case being 120 mm (2¸). This means that we obtain a standing wave with a wavelength of 120 mm in a 60 mm long sample, when the "eld frequency is 4.84 kHz. The conditions for the appearance of the standing wave are also ful"lled for n"2, 3,2, at l "2l "9.68 kHz, l "3l " 14.52 kHz, etc. These values are close to the experimentally determined ones for the ribbon and wire with the same S and ¸: l"4.76, l"9.66, and l"13.92 kHz.
415
When the standing wave appears, the magnetic moments perform a complex 3-D motion, whose projection in the samples transverse cross section is a spiral, and in its longitudinal cross section an oscillation, due to the simultaneous precession and forced oscillation around the z-axis, both determined by H(t). The maximum amplitudes of the forced oscillations performed by the magnetic moments depend on their position on the z-axis, ranging from }n/2 to n/2, their positions changing in time according to Re(h) e xR. All the magnetic moments relax toward the "eld direction simultaneously, and then they proceed simultaneously in the opposite direction, as the "eld changes its direction. When the magnetic moments relax toward the "eld direction, their angular momentum changes determining a rotation of the sample to conserve the angular momentum of the system. As the magnetic moments move toward the "eld direction, which is also the easy axis of magnetization, the magnetic energy decreases, and it is transferred into kinetic energy for rotation. The decrease in the magnetic energy depends on the magnitude of the axial magnetoelastic anisotropy, being directly proportional to the magnetostriction constant. Thus, the angular momentum transfer between the system of magnetic moments and the sample as a whole is related to the magnitude of the magnetostriction constant. When the magnetic moments are moving away from the z-axis, the spin ensemble receives energy from H(t), and therefore, there is no kinetic energy released for the sample rotation, but it still rotates due to inertial reasons. The imaginary part of h(t), Im(h), determines a temporal phase di!erence, u, between H and h, given by: u"arctan+Im(h)/Re(h),.
(4)
For a 60 mm long sample, u is }p/4 at m , l , l , etc., showing that the spin oscillations lag behind the "eld in phase by }p/4. These results explain the observed change in the direction of the sample rotation. When u increases over }n/4 (as the frequency increases over l , but still in the *l frequency range), the equilibrium positions around which the magnetic moments perform precession and oscillation change from upwards to downwards, and consequently, the sample will rotate in the opposite direction. Summarizing, the model o!ers a good qualitative explanation of the phenomenon, as well as a good quantitative agreement between the experimental results and the theoretically predicted ones.
References [1] H. Chiriac, C.S. Marinescu, T.-A. OD vaH ri, IEEE Trans. Magn. 33 (1997) 3349. [2] H. Chiriac, T.-A. OD vaH ri, M. VaH zquez, A. Hernando, J. Magn. Magn. Mater. 177 (1998) 205.
Large gyromagnetic e!ect in soft magnetic amorphous ribbons and wires Horia Chiriac*, Tibor-Adrian OD vaH ri, Cay tay lin Sandrino Marinescu National Institute of R&D for Technical Physics, 47 Mangeron Blvd., 6600 Iasi 3, Romania
Abstract We report results on the Einstein}de Haas rotation of Fe Si B amorphous ribbons and wires, a phenomenon called large gyromagnetic e!ect. The laws of this e!ect are "rstly inferred from the experimental data, and then a comprehensive model that explains these laws is presented. 2000 Elsevier Science B.V. All rights reserved. Keywords: Gyromagnetic e!ect; Magnetic standing wave; Soft magnetic materials
We have recently reported that an Fe Si B amorphous wire with a certain diameter U and length L, subjected to an axial alternating magnetic "eld H(t)"H sin 2plt, where H is the "eld amplitude,
and l its frequency, starts rotating around its axis at given values of H and m [1]. We called this e!ect the
large gyromagnetic ewect (LGE). The aim of this paper is to report for the "rst time the appearance of the LGE in Fe Si B amorphous ribbons, to present the laws of the LGE in both wires and ribbons with this composition, based on the experimental investigations performed on such samples, as well as the &magnetic standing wave' model that explains the characteristics and appearance conditions of the LGE. We prepared Fe Si B amorphous wires with U"120 lm by in-rotating-water spinning, and 20 lm thick and 0.565 mm wide amorphous ribbons with the same composition by chill block melt spinning, in order to obtain samples with the same cross section areas (S"11.3;10\ m). For the experiments we have chosen samples with 15 mm)¸)120 mm. The ribbonand wire-shaped samples were vertically placed * one at a time * in a 10 mm long magnetizing coil with 4 mm diameter and 200 turns/mm. They were positioned on
* Corresponding author. Tel.: #40-32-130680; fax: #40-32231132. E-mail address: [email protected] (H. Chiriac).
a quartz bearing to reduce friction and to center the wire within the coil. H(t) was generated with a current supplied by a programmable function generator. The maximum value of H was 300 A m\, and its frequency
l varied from 1 to 60 kHz. We measured the values of H and l at which the LGE occurs in both ribbons and
wires, for di!erent sample lengths ¸. We observed that both ribbon- and wire-shaped samples having a given length, ¸, and subjected to a "eld with a given amplitude H , start rotating in a certain direc tion when the "eld frequency reaches a certain value l !*l/2. If the "eld frequency is slightly increased, the sample rotation is maintained in the same direction, until the frequency reaches the value l , at which the rotation stops, but the sample oscillates. Further increase of the "eld frequency determines again a sample rotation, but in the opposite direction, the rotation being maintained until the frequency reaches l #*l/2. When this value is exceeded, the rotation disappears, and also does any oscillation. Thus, the LGE occurs within a speci"c frequency range *l. The entire process reappears at multiples of l : l "2l , l "3l , etc. This is the xrst law of the LGE: for a given sample length and amplitude of the axial alternating "eld, the e!ect appears, irrespective of the samples shape * ribbon or wire * in a certain frequency range *l centered around a well-de"ned fundamental frequency l , and it reappears in frequency ranges centered around multiples of this frequency, the superior harmonics l "2l , m "3m , etc. The frequen cies l , l , l , etc. and the width of the frequency range in
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 1 7 4 - 8
414
H. Chiriac et al. / Journal of Magnetism and Magnetic Materials 215}216 (2000) 413}415
Fig. 1. Dependence of the frequencies at which LGE occurs on the inverse of the sample length for H "200 A m\ (solid
lines), and on the "eld amplitude, in 25 mm long samples (dashed lines), for Fe Si B amorphous ribbons with 20 lm in thickness and 0.565 mm in width.
which the LGE appears are the same for ribbon- and wire-shaped samples having the same length and cross section area. The appearance of the LGE at discrete values of l indicates the presence of a low-frequency resonance phenomenon, and consequently that a speci"c wave can be associated to this process. The frequencies l , l , and l are plotted against the inverse value of the sample length 1/¸ for ribbons in Fig. 1 (solid lines), at H "200 A m\. It is meaningful
to emphasize that these dependencies are identical for wires with the same value of S [1]. From these results, we can infer the second law of the LGE: the frequencies of the applied "eld at which LGE occurs are directly proportional to the inverse value of the sample length: l J1/¸, where i"0, 1, 2, etc. The linear dependence of l on 1/¸ indicates that the wave involved in the LGE has a standing character. Another experimental fact is that the LGE does not appear irrespective of frequency, if H is smaller than
a certain threshold value, almost equal to the switching "eld, H* at which a large Barkhausen e!ect appears in the Fe Si B amorphous wires, and to the coercive force H of the Fe Si B amorphous ribbons. This ! fact constitutes the third law of the LGE: the "eld amplitude must be larger than a certain threshold value in order for the LGE to occur: H *H* for wires and
H *H for ribbons.
! The dependencies of l , m , and l on H for
Fe Si B amorphous ribbons with ¸"const. (25 mm) are also illustrated in Fig. 1 (dashed lines). The values of H in this experiment were chosen to be
larger than both H for ribbons (16 A m\) and H* ! for wires (30 A m\). These dependencies are identical
for wires with the same S [1], and they point to the fourth law of the LGE: the "eld frequencies at which LGE occurs are directly proportional to its amplitude: l JH , i"0, 1, 2, etc. G
The speci"c domain structure of highly positive magnetostrictive amorphous wires such as Fe Si B ones, originating in the minimization of the large magnetoelastic energy determined by the coupling between magnetostriction and internal stresses induced during preparation, mainly consists of an axially magnetized inner core and a radially magnetized outer shell, the volume of the inner core being &70% of the entire wire volume [2]. This domain structure favors the appearance of a large Barkhausen e!ect, that consists of a 1803 domain wall propagation determined by a "eld larger than a certain threshold value, called the switching "eld H*. Since the changes in the directions of the magnetic moments from the inner core occur with "nite velocity, we can consider the wall as a disturbance that propagates within a "nite medium, and we can associate a magnetic planar wave to its movement. When the applied "eld is a high-frequency alternating one, there is a nucleation and propagation of new 1803 domain walls at each wire end, each time the "eld reverses, and consequently, new magnetic planar waves are generated. These waves interfere, and therefore it is plausible to expect the appearance of a magnetic standing wave in certain conditions related to the correlation between l and ¸. In the case of Fe Si B amorphous ribbons, which are also prepared by rapid solidi"cation, the magnetoelastic energy minimization leads to the formation of a maze-like domain structure that consists of multiple domains with in-plane axial anisotropy, separated by 1803 domain walls. This structure is somewhat equivalent to that of the wires subjected to an alternating axial "eld: the 1803 domain walls which are propagating within the ribbon subjected to H(t) can also be treated as planar waves which interfere, resulting in given circumstances in a standing wave. Both cases (ribbon and wire) can be explained in a single manner, irrespective of the sample's shape, starting from a wave associated to the movement of a 1803 domain wall. Such a wave is well described by h(z, t) * the angle of the magnetic moment located at the coordinate z on the axial direction, with respect to the sample's axis, at the instant of time t; h3[0, p], if the sample is subjected to H(t). h changes in this range with a "nite velocity * the phase velocity of the magnetic wave * which is actually the domain wall velocity < . "5 h(z, t) is given by h(z, t)"h e SRe\?> IX,
(1)
where the maximum amplitude h is p/2, and k"2p/j , k being the wavelength of the &domain "5 "5
H. Chiriac et al. / Journal of Magnetism and Magnetic Materials 215}216 (2000) 413}415
Fig. 2. Frequency dependence of Re(h), for a 60 mm long FeSiB amorphous ribbon or wire-shaped sample.
wall' wave, j "< /m. The energy losses in the "5 "5 sample are described by the damping factor a, which for simplicity, is assumed to be constant in time. The wave that results from the interference of n such waves (generated by nucleation and propagation in the case of wires and pre-existent in the case of ribbons), can be expressed after several transformations as h(t)"h [Re(h)#j Im(h)]e SR, in which the real, Re(h), and imaginary parts, Im(h), respectively, are given by sinh(2a¸) Re(h)" , (2) cosh(2a¸)!cos(4pl¸/< ) "5 sin(4pl¸/< ) "5 Im(h)" , (3) cosh(2a¸)!cos(4pl¸/< ) "5 in which we considered the explicit frequency dependencies. The resulting wave is a standing one only when Re(h) is maximum, i.e. when the conditions for interference maxima are accomplished. According to Eq. (2), Re(h) is maximum when ¸"nj /2, being equal to coth (a¸). "5 Fig. 2 illustrates the frequency dependence of Re(h), for ¸"60 mm, in which we used < "580 m s\, value "5 determined by the Sixtus}Tonks technique for an Fe Si B amorphous sample. The "rst maximum (l "4.84 kHz) corresponds to n"1, the wavelength of the standing wave in this particular case being 120 mm (2¸). This means that we obtain a standing wave with a wavelength of 120 mm in a 60 mm long sample, when the "eld frequency is 4.84 kHz. The conditions for the appearance of the standing wave are also ful"lled for n"2, 3,2, at l "2l "9.68 kHz, l "3l " 14.52 kHz, etc. These values are close to the experimentally determined ones for the ribbon and wire with the same S and ¸: l"4.76, l"9.66, and l"13.92 kHz.
415
When the standing wave appears, the magnetic moments perform a complex 3-D motion, whose projection in the samples transverse cross section is a spiral, and in its longitudinal cross section an oscillation, due to the simultaneous precession and forced oscillation around the z-axis, both determined by H(t). The maximum amplitudes of the forced oscillations performed by the magnetic moments depend on their position on the z-axis, ranging from }n/2 to n/2, their positions changing in time according to Re(h) e xR. All the magnetic moments relax toward the "eld direction simultaneously, and then they proceed simultaneously in the opposite direction, as the "eld changes its direction. When the magnetic moments relax toward the "eld direction, their angular momentum changes determining a rotation of the sample to conserve the angular momentum of the system. As the magnetic moments move toward the "eld direction, which is also the easy axis of magnetization, the magnetic energy decreases, and it is transferred into kinetic energy for rotation. The decrease in the magnetic energy depends on the magnitude of the axial magnetoelastic anisotropy, being directly proportional to the magnetostriction constant. Thus, the angular momentum transfer between the system of magnetic moments and the sample as a whole is related to the magnitude of the magnetostriction constant. When the magnetic moments are moving away from the z-axis, the spin ensemble receives energy from H(t), and therefore, there is no kinetic energy released for the sample rotation, but it still rotates due to inertial reasons. The imaginary part of h(t), Im(h), determines a temporal phase di!erence, u, between H and h, given by: u"arctan+Im(h)/Re(h),.
(4)
For a 60 mm long sample, u is }p/4 at m , l , l , etc., showing that the spin oscillations lag behind the "eld in phase by }p/4. These results explain the observed change in the direction of the sample rotation. When u increases over }n/4 (as the frequency increases over l , but still in the *l frequency range), the equilibrium positions around which the magnetic moments perform precession and oscillation change from upwards to downwards, and consequently, the sample will rotate in the opposite direction. Summarizing, the model o!ers a good qualitative explanation of the phenomenon, as well as a good quantitative agreement between the experimental results and the theoretically predicted ones.
References [1] H. Chiriac, C.S. Marinescu, T.-A. OD vaH ri, IEEE Trans. Magn. 33 (1997) 3349. [2] H. Chiriac, T.-A. OD vaH ri, M. VaH zquez, A. Hernando, J. Magn. Magn. Mater. 177 (1998) 205.