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Magnetic anisotropy and spin reorientation in Nd2Fe14B
Journal of Magnetism and Magnetic Materials 212 (2000) 195}200
Magnetic anisotropy and spin reorientation in Nd Fe B 2 14 Sherif Yehia!,*, Samy H. Aly" !Department of Physics, Faculty of Science Helwan, University of Helwan, Cairo, Egypt "Department of Physics, Faculty of Science at Demiatta, University of Mansoura, Egypt Received 2 June 1999; received in revised form 21 October 1999
Abstract We present calculations of the magnetic properties of Nd Fe B system using the laws of statistical thermodynamics 2 14 in the mean "eld approximation. The energy of the system depends strongly on temperature via the temperature dependence of the "ve anisotropy constants of the tetragonal crystal. The classical partition function is evaluated and used to calculate the dependence of the magnetic properties on "eld and temperature. The calculated magnetization and magnetic heat capacity in the temperature range from 77 K to room temperature support the in-plane anisotropy and spin reorientation phenomena in Nd Fe B. Calculations of the torque and energy dependence on magnetic "eld 2 14 and/or temperature are also reported. The "ndings of our method are in good agreement with both the experimental and theoretical work on this important material. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 75.30.GW; 75.30.Sg; 75.50.Ww Keywords: Rare-earth transition metal borides; Anisotropy; Magnetic heat capacity; Torque; Spin reorientation; Mean "eld approximation
1. Introduction The magnetocrystalline anisotropy, magnetization process and spin reorientation in R Fe B 2 14 crystals have been studied extensively [1}4]. The anisotropy coe$cients of some members of this system (e.g. Nd Fe B) are known to be strongly 2 14 temperature dependent or even change sign in a wide temperature range [3,5]. In addition, basal plane anisotropy has been shown to exist [3] up to ¹&250 K which emphasizes the important role of the in-plane anisotropy constants in the magnetization process of this important class of compounds. * Corresponding author. E-mail address: [email protected] (S. Yehia)
The interesting phenomenon of spin reorientation from the easy c-axis to an easy cone has been reported by a number of authors. Torque measurements [6,7], stepwise decrease in the demagnetization curve [7,8], the angular dependence of the total energy in (1 0 0) and (1 1 0) planes [7,9] and magnetization as a function of temperature [1,4,10], all support the presence of a spin reorientation at a temperature about 125$10 K. Further studies on heat capacity [10], DSC [11] and magnetostriction and thermal expansion in Nd Fe B 2 14 [12] con"rm the spin reorientation detected by magnetic measurements. In a recent work [13], we reported on a simple method in which the magnetic properties of a rareearth transition metal system may be derived from
0304-8853/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 7 7 2 - 6
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S. Yehia, S.H. Aly / Journal of Magnetism and Magnetic Materials 212 (2000) 195}200
the classical partition function in the mean "eld approximation. In this work we apply our method to Nd Fe B and report on the magnetization 2 14 process in the temperature range from 77 K up to room temperature. A study on the magnetic heat capacity, torque and energy is also included.
3. Results and discussion Examples of the calculated magnetization curves in the [0 0 1], [1 0 0] and [1 1 0] directions in magnetic "elds up to 250 kOe are shown in Figs. 1}4 for ¹"77, 125, 176 and 270 K, respectively. The simulated curves agree fairly well with the experimental data for a single crystal [3]. The anisotropy
2. Model and analysis Our starting point is the total energy of the tetragonal system, which is the sum of the anisotropy and Zeeman energies: E(¹, h, /, H)"K sin2 h#K sin4 h 1 2 #K sin4 h cos 4/#K sin6 h 3 4 #K sin6 h cos 4/!H ) M , (1) 5 S where h is the angle the magnetization vector makes with the c-axis and / is the azimuthal angle of the magnetization vector measured from the [1 1 0] direction in the basal plane. The partition function of the system is given by
PP
1 2p p Z(¹, H)" e~EVb sin h dh d/, (2) 4p 0 0 where < is the volume of the magnetic spherical particle, taken to be 10~19 cm3 throughout this work. The partition function depends on the temperature via the temperature dependence of the "ve anisotropy constants and of course on b. We have used the tabulated data of K(¹) reported by Bolzoni et al. [3] to "nd the best "ts of each K(¹) in the temperature range 77}293 K and thus to evaluate Z(¹, H). The magnetic and thermomagnetic properties of the system are well known to be derivable from the partition function [13]. The torque in the [1 0 0] and [1 1 0] directions is calculated at di!erent temperatures using the relation ¸"![K #2(K $K ) sin2 h 1 2 3 #3(K $K ) sin4 h] sin 2h, (3) 4 5 where the # and ! signs stand for the [1 1 0] and [1 0 0] directions, respectively. Mathematica software is used throughout this work to perform symbolic and numeric analyses.
Fig. 1. Magnetization versus magnetic "eld at ¹"77 K for [0 0 1], [1 0 0] and [1 1 0] directions. The points are experimental results of Ref. [3], and the solid line represents the computer simulation.
Fig. 2. Magnetization versus magnetic "eld at ¹"125 K for [0 0 1], [1 0 0] and [1 1 0] directions. The points are experimental results of Ref. [3], and the solid line represents the computer simulation.
S. Yehia, S.H. Aly / Journal of Magnetism and Magnetic Materials 212 (2000) 195}200
197
Fig. 3. Magnetization versus magnetic "eld at ¹"176 K for [0 0 1], [1 0 0] and [1 1 0] directions. The points are experimental results of Ref. [3], and the solid line represents the computer simulation.
Fig. 5. Magnetization versus temperature in the temperature range 77}293 K in a magnetic "eld H"10 kOe directed along the [0 0 1] direction.
Fig. 4. Magnetization versus magnetic "eld at ¹"270 K for [0 0 1], [1 0 0] and [1 1 0] directions. The points are experimental results of Ref. [3], and the solid line represents the computer simulation.
within the basal plane decreases with increasing the temperature until the [1 0 0] and [1 1 0] directions become isotropic above &250 K. At certain critical values of the magnetic "eld a jump to saturation occurs in the [1 0 0] direction. Below a critical "eld, the [1 0 0] direction is magnetically harder than the [1 1 0] direction, along which the magnetization su!ers no discontinuity until saturation. In Figs. 5 and 6 is shown the magnetization dependence on
temperature in the [0 0 1] direction (Fig. 5) and in the basal plane (Fig. 6). A spin reorientation is shown in Fig. 5 at ¹"136 K above which the magnetization drops steeply reaching &168 emu/g in a "eld of 10 kOe at 292 K. This temperature agrees well with both the calculated and experimentally measured values reported earlier (e.g. Refs. [1}4,10]). The spin reorientation temperature has not shown "eld dependence in our simulated M versus ¹ relations at 1 and 5 kOe (not shown). Fig. 6 shows the behavior of the magnetization dependence on temperature along the [1 0 0] and [1 1 0] directions in 1 kOe "eld. The in-plane anisotropy is evident at ¹)¹ and disappears for SR ¹'¹ . The relative magnetic hardness of the SR [1 0 0] direction increases as the temperature decreases. The dependence of the magnetic heat capacity on temperature in the 77}290 K range is shown in Fig. 7 for H"10 kOe. A jump in C followed V by a sharp drop takes place in the temperature range 120}140 K, the range within which the spin
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S. Yehia, S.H. Aly / Journal of Magnetism and Magnetic Materials 212 (2000) 195}200
Fig. 6. Magnetization versus temperature in the temperature range 77}293 K in a magnetic "eld H"1 kOe directed along either [1 0 0] or [1 1 0] direction.
reorientation takes place. This range agrees well with the experimental ¹ (135 K) and the value SR 115 K calculated by taking crystal "eld and exchange interaction into account [10]. The data extracted from torque measurements in certain crystallographic directions are known to shed light on the spin reorientation phenomenon [6,7]. We have used Eq. (3) to simulate the torque along the [1 0 0] and [1 1 0] directions at di!erent temperatures. Figs. 8 and 9 show combined plots of the torque ¸ versus the magnetization angle h, at di!erent temperatures below and above ¹ , along the SR [1 0 0] and [1 1 0] directions, respectively. From these two "gures it is evident that the cone angle decreases as the temperature increases. We have also plotted, for convenience, the torque curves along [1 0 0] and [1 1 0] at the same temperature. At high temperatures (&270 K) the curves are identical. Anisotropy starts to show up clearly at &124 K and lower temperatures (Fig. 10). It is also
Fig. 7. Magnetic heat capacity dependence on temperature in the temperature range 77}293 K in a magnetic "eld H" 10 kOe.
shown from the torque plots that the transition from easy cone to easy axis, at a given temperature )¹ , commences at a smaller h in the case of SR # [1 0 0] as compared to the [1 1 0] case. The variation of the energy with the magnetization angle provides additional evidence for the di!erence in the magnetization process along di!erent crystallographic directions [7,9]. We have simulated the E(h) dependence for di!erent "elds and temperatures in the case of the magnetic "eld parallel to the [1 0 0] direction. In the [1 0 0] direction, two equivalent energy minima (one at 903"1.57 rad, and another at &0.8 rad) separated by a local maximum appear in the E(h) plot at ¹"124 K for H"143 kOe (Fig. 11). Changing the "eld to values other than the critical value of 143 kOe resulted in shifting the minimum at 903 to higher or lower energies relative to the other minimum. We have plotted E(h) in the angular range from 0 to p in
S. Yehia, S.H. Aly / Journal of Magnetism and Magnetic Materials 212 (2000) 195}200
Fig. 8. Torque variation with magnetization angle h in the [1 0 0] direction at temperatures ¹"78, 104, 124, 148, 271 and 293 K.
order to clearly show the local minimum in energy at p/2, the plot however is symmetric about its bisecting energy axis.
4. Conclusions The classical partition function of a magnetic system whose energy is the sum of the magnetocrystalline anisotropy and Zeeman energies is used in deriving and calculating the magnetic properties of Nd Fe B in the temperature range from 77 K 2 14 to room temperature.
c Fig. 10. Torque variation with magnetization angle h at ¹"78 K in the [1 0 0] and [1 1 0] directions.
199
Fig. 9. Torque variation with magnetization angle h in the [1 1 0] direction at temperatures ¹"78, 104, 124, 148, 271 and 293 K.
200
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References [1] Y.B. Kim, M.J. Kim, Jin Han-min, T.K. Kim, J. Magn. Magn. Mater. 191 (1999) 133. [2] D. Givord, H.S. Li, R. Perrier, dela Batchie, Solid State Commun. 51 (1984) 857. [3] F. Bolzoni, O. Moze, L. Pareti, J. Appl. Phys. 62 (1987) 615. [4] A.T. Pedziwiatr, W.E. Wallace, J. Magn. Magn. Mater. 65 (1987) 139. [5] K.D. Durst, H. KronmuK ller, J. Magn. Magn. Mater. 59 (1986) 86. [6] O. Yamada, H. Tokuhara, F. Ono, M. Sagawa, Y. Matsuura, J. Magn. Magn. Mater. 54 (1986) 585. [7] Y. Otani, H. Miyajima, S. Chikazumi, J. Appl. Phys. 61 (1987) 3436. [8] Y. Otani, H. Miyajima, S. Chikazumi, S. Hirosawa, M. Sagawa, J. Magn. Magn. Mater. 60 (1986) 168. [9] R. Verhoef, Q. Wang, J.J.M. Franse, Physica B 177 (1992) 211. [10] Y. Takano, Y. Satoh, K. Sekizawa, J. Phys. Colloque C 8 (1988) 561. [11] C.D. Fuerst, J.F. Herbst, E.A. Alson, J. Magn. Magn. Mater. 54 (1986) 567. [12] M.R. Ibarra, P.A. Algarabel, A. Alberdi, J. Bartolome', A. del Moral, J. Appl. Phys. 61 (1987) 3451. [13] Samy H. Aly, Sherif Yehia, J. Magn. Magn. Mater. 202 (1999) 565. Fig. 11. Energy as a function of magnetization angle h at ¹"124 K in "elds H"50, 100, 120, 143 and 160 kOe in the [1 0 0] direction.
The results of our method agree well with the experimental and theoretical work on this material. Further studies on the torque and energy support the "ndings of our method.
Magnetic anisotropy and spin reorientation in Nd Fe B 2 14 Sherif Yehia!,*, Samy H. Aly" !Department of Physics, Faculty of Science Helwan, University of Helwan, Cairo, Egypt "Department of Physics, Faculty of Science at Demiatta, University of Mansoura, Egypt Received 2 June 1999; received in revised form 21 October 1999
Abstract We present calculations of the magnetic properties of Nd Fe B system using the laws of statistical thermodynamics 2 14 in the mean "eld approximation. The energy of the system depends strongly on temperature via the temperature dependence of the "ve anisotropy constants of the tetragonal crystal. The classical partition function is evaluated and used to calculate the dependence of the magnetic properties on "eld and temperature. The calculated magnetization and magnetic heat capacity in the temperature range from 77 K to room temperature support the in-plane anisotropy and spin reorientation phenomena in Nd Fe B. Calculations of the torque and energy dependence on magnetic "eld 2 14 and/or temperature are also reported. The "ndings of our method are in good agreement with both the experimental and theoretical work on this important material. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 75.30.GW; 75.30.Sg; 75.50.Ww Keywords: Rare-earth transition metal borides; Anisotropy; Magnetic heat capacity; Torque; Spin reorientation; Mean "eld approximation
1. Introduction The magnetocrystalline anisotropy, magnetization process and spin reorientation in R Fe B 2 14 crystals have been studied extensively [1}4]. The anisotropy coe$cients of some members of this system (e.g. Nd Fe B) are known to be strongly 2 14 temperature dependent or even change sign in a wide temperature range [3,5]. In addition, basal plane anisotropy has been shown to exist [3] up to ¹&250 K which emphasizes the important role of the in-plane anisotropy constants in the magnetization process of this important class of compounds. * Corresponding author. E-mail address: [email protected] (S. Yehia)
The interesting phenomenon of spin reorientation from the easy c-axis to an easy cone has been reported by a number of authors. Torque measurements [6,7], stepwise decrease in the demagnetization curve [7,8], the angular dependence of the total energy in (1 0 0) and (1 1 0) planes [7,9] and magnetization as a function of temperature [1,4,10], all support the presence of a spin reorientation at a temperature about 125$10 K. Further studies on heat capacity [10], DSC [11] and magnetostriction and thermal expansion in Nd Fe B 2 14 [12] con"rm the spin reorientation detected by magnetic measurements. In a recent work [13], we reported on a simple method in which the magnetic properties of a rareearth transition metal system may be derived from
0304-8853/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 7 7 2 - 6
196
S. Yehia, S.H. Aly / Journal of Magnetism and Magnetic Materials 212 (2000) 195}200
the classical partition function in the mean "eld approximation. In this work we apply our method to Nd Fe B and report on the magnetization 2 14 process in the temperature range from 77 K up to room temperature. A study on the magnetic heat capacity, torque and energy is also included.
3. Results and discussion Examples of the calculated magnetization curves in the [0 0 1], [1 0 0] and [1 1 0] directions in magnetic "elds up to 250 kOe are shown in Figs. 1}4 for ¹"77, 125, 176 and 270 K, respectively. The simulated curves agree fairly well with the experimental data for a single crystal [3]. The anisotropy
2. Model and analysis Our starting point is the total energy of the tetragonal system, which is the sum of the anisotropy and Zeeman energies: E(¹, h, /, H)"K sin2 h#K sin4 h 1 2 #K sin4 h cos 4/#K sin6 h 3 4 #K sin6 h cos 4/!H ) M , (1) 5 S where h is the angle the magnetization vector makes with the c-axis and / is the azimuthal angle of the magnetization vector measured from the [1 1 0] direction in the basal plane. The partition function of the system is given by
PP
1 2p p Z(¹, H)" e~EVb sin h dh d/, (2) 4p 0 0 where < is the volume of the magnetic spherical particle, taken to be 10~19 cm3 throughout this work. The partition function depends on the temperature via the temperature dependence of the "ve anisotropy constants and of course on b. We have used the tabulated data of K(¹) reported by Bolzoni et al. [3] to "nd the best "ts of each K(¹) in the temperature range 77}293 K and thus to evaluate Z(¹, H). The magnetic and thermomagnetic properties of the system are well known to be derivable from the partition function [13]. The torque in the [1 0 0] and [1 1 0] directions is calculated at di!erent temperatures using the relation ¸"![K #2(K $K ) sin2 h 1 2 3 #3(K $K ) sin4 h] sin 2h, (3) 4 5 where the # and ! signs stand for the [1 1 0] and [1 0 0] directions, respectively. Mathematica software is used throughout this work to perform symbolic and numeric analyses.
Fig. 1. Magnetization versus magnetic "eld at ¹"77 K for [0 0 1], [1 0 0] and [1 1 0] directions. The points are experimental results of Ref. [3], and the solid line represents the computer simulation.
Fig. 2. Magnetization versus magnetic "eld at ¹"125 K for [0 0 1], [1 0 0] and [1 1 0] directions. The points are experimental results of Ref. [3], and the solid line represents the computer simulation.
S. Yehia, S.H. Aly / Journal of Magnetism and Magnetic Materials 212 (2000) 195}200
197
Fig. 3. Magnetization versus magnetic "eld at ¹"176 K for [0 0 1], [1 0 0] and [1 1 0] directions. The points are experimental results of Ref. [3], and the solid line represents the computer simulation.
Fig. 5. Magnetization versus temperature in the temperature range 77}293 K in a magnetic "eld H"10 kOe directed along the [0 0 1] direction.
Fig. 4. Magnetization versus magnetic "eld at ¹"270 K for [0 0 1], [1 0 0] and [1 1 0] directions. The points are experimental results of Ref. [3], and the solid line represents the computer simulation.
within the basal plane decreases with increasing the temperature until the [1 0 0] and [1 1 0] directions become isotropic above &250 K. At certain critical values of the magnetic "eld a jump to saturation occurs in the [1 0 0] direction. Below a critical "eld, the [1 0 0] direction is magnetically harder than the [1 1 0] direction, along which the magnetization su!ers no discontinuity until saturation. In Figs. 5 and 6 is shown the magnetization dependence on
temperature in the [0 0 1] direction (Fig. 5) and in the basal plane (Fig. 6). A spin reorientation is shown in Fig. 5 at ¹"136 K above which the magnetization drops steeply reaching &168 emu/g in a "eld of 10 kOe at 292 K. This temperature agrees well with both the calculated and experimentally measured values reported earlier (e.g. Refs. [1}4,10]). The spin reorientation temperature has not shown "eld dependence in our simulated M versus ¹ relations at 1 and 5 kOe (not shown). Fig. 6 shows the behavior of the magnetization dependence on temperature along the [1 0 0] and [1 1 0] directions in 1 kOe "eld. The in-plane anisotropy is evident at ¹)¹ and disappears for SR ¹'¹ . The relative magnetic hardness of the SR [1 0 0] direction increases as the temperature decreases. The dependence of the magnetic heat capacity on temperature in the 77}290 K range is shown in Fig. 7 for H"10 kOe. A jump in C followed V by a sharp drop takes place in the temperature range 120}140 K, the range within which the spin
198
S. Yehia, S.H. Aly / Journal of Magnetism and Magnetic Materials 212 (2000) 195}200
Fig. 6. Magnetization versus temperature in the temperature range 77}293 K in a magnetic "eld H"1 kOe directed along either [1 0 0] or [1 1 0] direction.
reorientation takes place. This range agrees well with the experimental ¹ (135 K) and the value SR 115 K calculated by taking crystal "eld and exchange interaction into account [10]. The data extracted from torque measurements in certain crystallographic directions are known to shed light on the spin reorientation phenomenon [6,7]. We have used Eq. (3) to simulate the torque along the [1 0 0] and [1 1 0] directions at di!erent temperatures. Figs. 8 and 9 show combined plots of the torque ¸ versus the magnetization angle h, at di!erent temperatures below and above ¹ , along the SR [1 0 0] and [1 1 0] directions, respectively. From these two "gures it is evident that the cone angle decreases as the temperature increases. We have also plotted, for convenience, the torque curves along [1 0 0] and [1 1 0] at the same temperature. At high temperatures (&270 K) the curves are identical. Anisotropy starts to show up clearly at &124 K and lower temperatures (Fig. 10). It is also
Fig. 7. Magnetic heat capacity dependence on temperature in the temperature range 77}293 K in a magnetic "eld H" 10 kOe.
shown from the torque plots that the transition from easy cone to easy axis, at a given temperature )¹ , commences at a smaller h in the case of SR # [1 0 0] as compared to the [1 1 0] case. The variation of the energy with the magnetization angle provides additional evidence for the di!erence in the magnetization process along di!erent crystallographic directions [7,9]. We have simulated the E(h) dependence for di!erent "elds and temperatures in the case of the magnetic "eld parallel to the [1 0 0] direction. In the [1 0 0] direction, two equivalent energy minima (one at 903"1.57 rad, and another at &0.8 rad) separated by a local maximum appear in the E(h) plot at ¹"124 K for H"143 kOe (Fig. 11). Changing the "eld to values other than the critical value of 143 kOe resulted in shifting the minimum at 903 to higher or lower energies relative to the other minimum. We have plotted E(h) in the angular range from 0 to p in
S. Yehia, S.H. Aly / Journal of Magnetism and Magnetic Materials 212 (2000) 195}200
Fig. 8. Torque variation with magnetization angle h in the [1 0 0] direction at temperatures ¹"78, 104, 124, 148, 271 and 293 K.
order to clearly show the local minimum in energy at p/2, the plot however is symmetric about its bisecting energy axis.
4. Conclusions The classical partition function of a magnetic system whose energy is the sum of the magnetocrystalline anisotropy and Zeeman energies is used in deriving and calculating the magnetic properties of Nd Fe B in the temperature range from 77 K 2 14 to room temperature.
c Fig. 10. Torque variation with magnetization angle h at ¹"78 K in the [1 0 0] and [1 1 0] directions.
199
Fig. 9. Torque variation with magnetization angle h in the [1 1 0] direction at temperatures ¹"78, 104, 124, 148, 271 and 293 K.
200
S. Yehia, S.H. Aly / Journal of Magnetism and Magnetic Materials 212 (2000) 195}200
References [1] Y.B. Kim, M.J. Kim, Jin Han-min, T.K. Kim, J. Magn. Magn. Mater. 191 (1999) 133. [2] D. Givord, H.S. Li, R. Perrier, dela Batchie, Solid State Commun. 51 (1984) 857. [3] F. Bolzoni, O. Moze, L. Pareti, J. Appl. Phys. 62 (1987) 615. [4] A.T. Pedziwiatr, W.E. Wallace, J. Magn. Magn. Mater. 65 (1987) 139. [5] K.D. Durst, H. KronmuK ller, J. Magn. Magn. Mater. 59 (1986) 86. [6] O. Yamada, H. Tokuhara, F. Ono, M. Sagawa, Y. Matsuura, J. Magn. Magn. Mater. 54 (1986) 585. [7] Y. Otani, H. Miyajima, S. Chikazumi, J. Appl. Phys. 61 (1987) 3436. [8] Y. Otani, H. Miyajima, S. Chikazumi, S. Hirosawa, M. Sagawa, J. Magn. Magn. Mater. 60 (1986) 168. [9] R. Verhoef, Q. Wang, J.J.M. Franse, Physica B 177 (1992) 211. [10] Y. Takano, Y. Satoh, K. Sekizawa, J. Phys. Colloque C 8 (1988) 561. [11] C.D. Fuerst, J.F. Herbst, E.A. Alson, J. Magn. Magn. Mater. 54 (1986) 567. [12] M.R. Ibarra, P.A. Algarabel, A. Alberdi, J. Bartolome', A. del Moral, J. Appl. Phys. 61 (1987) 3451. [13] Samy H. Aly, Sherif Yehia, J. Magn. Magn. Mater. 202 (1999) 565. Fig. 11. Energy as a function of magnetization angle h at ¹"124 K in "elds H"50, 100, 120, 143 and 160 kOe in the [1 0 0] direction.
The results of our method agree well with the experimental and theoretical work on this material. Further studies on the torque and energy support the "ndings of our method.