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Modeling proximity-coupling in multifilamentary wires by grained Bean model
Physica C 468 (2008) 1481–1484
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Modeling proximity-coupling in multifilamentary wires by grained Bean model T. Akune *, W. Yumoto, N. Sakamoto Department of Electrical Engineering and Information Technology, Kyushu Sangyo University, 2-3-1 Matsukadai, 813-8503 Fukuoka, Japan
a r t i c l e
i n f o
Article history: Available online 28 May 2008 PACS: 74.72 Keywords: Grained Bean model AC susceptibility Pinning penetration field
a b s t r a c t Proximity-currents between filaments in a multifilamentary wire show a close resemblance with the inter-grain current in a high-Tc superconductor. The critical current densities of the proximity-induced superconducting matrix Jcm can be estimated from measured twist-pitch dependence of magnetization and have been shown to follow the well-known scaling law of the pinning strength. In the grained Bean model, the filaments are immersed in the proximity-induced superconducting matrix. Difference of the superconducting characteristics of the filament, the matrix and the filament content factor give a variety of deformation on the AC susceptibility curves. The computed AC susceptibility curves of multifilamentary wires using the grained Bean model are favorably compared with the experimental results. Ó 2008 Published by Elsevier B.V.
1. Introduction
2. Experimental
It is necessary to enhance the inter-grain coupling in high-Tc superconductors for fulfilling the demand of high critical current densities and low losses for industrial AC applications. It has been reported [1] that the AC losses in the multifilamentary wires increased with decreasing interfilamentary spacing dN and were noticeable when dN became comparable to or smaller than the coherence length nN of the matrix. It is widely known that this effect results from an excess magnetization caused by interfilamentary proximity-coupling, and the proximity effect is reduced by replacing the Cu in the interfilamentary region by CuNi or by introducing a small percentage of Mn in the Cu between the filaments. In a preceding paper [2,3] we reported the agreement of both the upper critical field Bc2p obtained by applying the scaling law to Jcm data estimated from the twist-pitch ‘p dependence of the magnetization and Bc2p obtained from the peak temperatures Tp appearing in the imaginary parts of the AC susceptibilities of multifilamentary wires with Cu and Cu–0.5wt% Mn matrices. In addition the values of Bc1p were estimated from the temperature dependence of real parts of the AC susceptibilities. In this paper the relation of the proximity effect and the AC susceptibilities in the multifilamentary wire is discussed. Field distribution and magnetization in the multifilamentary wire are calculated by Bean model [4]. Fourier integration of magnetization is carried out numerically and gives rise to the AC susceptibilities. Measured results of NbTi multifilamentary wires with CuMn matrix are compared with the simulated results to give the matrix characteristics.
Sample specifications of multifilamentary wires are shown in Table 1. The matrix is Cu–0.5wt% Mn for series A, B and C samples. Samples were manufactured by double stacking and are composed of 24186 NbTi filaments embedded in the Cu–0.5wt% Mn matrix with the volume ratio being 1.58. The value of the interfilamentary spacing dN was changed by drawing the initial untwisted wire, and the drawn wire was twisted for each series. Therefore, the values of dN and df were calculated from the measured values of the wire diameter Dw. The edges of twisted wire were polished with sandpaper to separate filaments from each and then cleaned by ultrasonic agitation in acetone. After wire length dependence of the magnetization was measured, the sample length of L > 70 mm was determined to be sufficient for safely neglecting the effect of flux entry and exit along the wire axis. About five turns of each sample were wound on a straw with the diameter of 5.5 mm and fixed on it by adhesives. A commercial SQUID magnetometer was used to measure magnetic moments. The magnetization measurement was carried out in a temperature range 2–10 K, and the DC magnetic fields up to 1 T and the superposed AC magnetic fields of 0.05 mT were applied perpendicular to the wire axis. Low frequency of 10 Hz is used to avoid eddy-currents in the matrix. The superconducting transition temperatures Tc of wires are 8.5 K, which were measured from temperature dependence of magnetization under zero-field cooling and field cooling conditions.
* Corresponding author. Tel.: +81 92 673 5632; fax: +81 92 673 5091. E-mail address: [email protected] (T. Akune). 0921-4534/$ - see front matter Ó 2008 Published by Elsevier B.V. doi:10.1016/j.physc.2008.05.143
3. Grained Bean model [5,6] Proximity-induced superconducting matrix with a penetration field Bpm determines the magnetic fields at the filament surface
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T. Akune et al. / Physica C 468 (2008) 1481–1484
Table 1 Sample specifications
4. Proximity-coupling in multifilamentary wire
Sample
Matrix
Dw (mm)
df (lm)
dN (lm)
‘p (mm)
A B C
Cu–0.5wt% Mn Cu–0.5wt% Mn Cu–0.5wt% Mn
0.102 0.145 0.260
0.324 0.461 0.827
0.050 0.072 0.129
11.7 10.4 9.2
Mp
hBi ¼ Bo
Bpm þ 2
i¼1
ð2Þ
ð3Þ
Filament magnetization mi is easily computed following magnetization process of each filaments. In the case of uniform wire structure with constant filament size df ( =dfi for all i) and interfilamentary spacing dN ( =dNi for all i), n
hBi ¼ Bo
g X Bpm mi þ ff 2 i¼1
ð4Þ
where ff ( = df/(dN + df)) is the filament content factor. Then the magnetization M of the superconductor at the field Bo for each magnetization processes is given by
M ¼ hBi Bo
ð6Þ
J cm ¼ J 0
dfi mfi dNi þ dfi
l0 kp km Jcm ðBÞ‘p
ð1Þ
where Dw is a wire diameter, x is a position inside the filament from the each filament surface as shown in the inset. Average of the magnetic flux density hBi in the increasing period for both of the matrix and filaments with interfilamentary spacing dN is given by ng X
1
p2
where kp denotes the volume fraction of the superconducting matrix in the multifilamentary region and km is the volume fraction of the multifilamentary region in the wire. Thus Jcm values can be estimated from the slope of the ‘p dependence of the magnetization. Field dependence of Jcm estimated from the ‘p dependence of magnetization is approximately expressed by the scaling law of the pinning effect as [7]
with the penetration field Bpf as shown in Fig. 1. Field distribution Bm outside the ith filament at a filament position {i and field inside Bf of ith filament with the filament diameter dfi are given by using Bpm, Bpf and Bean model as [4]
2X i : outside the filament Bm ¼ Bo Bpm Dw x : inside the filament Bf ¼ Bm Bpf dfi
The proximity-effected magnetization Mp assisted by Jcm is expressed as [1]
ð5Þ
B Bc2p
c
B Bc2p
d ;
ð7Þ
where c and d are adjustable pinning parameters and Bc2p is the upper critical field of the weakly superconducting matrix. Temperature dependence of Bc2p is as follows:
Bc2p ¼ Bc2N expðK N dN Þ; 1/2
K 1 N
ð8Þ 1/2
where = b/T and b (K lm) in the low-temperature range of T < 4 K is 0.126, 0.172 and 0.282 for series A, B and C samples, respectively. In the high-temperature range of T > 4 K, b is 0.062 for series A and 0.086 for series B samples [7]. 5. AC susceptibilities of the grained Bean model The AC susceptibilities v0 and v00 under an AC field Ba cos xt are derived from Fourier integrals of the magnetization. The fundamental Fourier components v01 and v001 are denoted as v0 and v00 hereafter. Temperature dependence of v0 and v00 is obtained by introducing the temperature variation of the penetration fields of Bpf and Bpm. If the Bpf has a usual parabolic dependence of the form as
( Bpf ¼ Bpf ð0Þ 1
Fig. 1. Field distribution Bm outside the filament and distribution Bf inside the filament. Penetration fields are Bpm and Bpf for matrix and filament, respectively.
1
T T cf
2 )nf ;
ð9Þ
where over the temperature Tcf the pinning effect for the fluxoid motion disappears in the filament. Since the relation of the penetration field of Bpm and the critical current densities of the proximityinduced superconducting matrix Jcm is Bpm = l0DwJcm/2, the temperature variation of the penetration field of Bpm is given by Eq. (7) and (8). The AC susceptibilities are numerically computed and plotted in Fig. 2 where Bf = 4 mT, ff = 0.32, nf = 2 and dN = 0.047–0.051 lm. The first peak of v00 at higher temperature decreases and the additional peak shifts to lower temperature with increasing dN. At a fixed interfilamentary spacing of dN = 0.05 lm and ff = 0.32 the effect of nf is shown in Fig. 3. With increasing nf the imaginary peaks increase at the second peak and coexist. This type of double peaks is often observed in high-Tc materials [8]. The double peak structure in the v00 curve is specifically indicated to originate in the cooperative work of the filament and the matrix. Temperature dependence of the real and imaginary parts of AC susceptibility for sample C with twist-pitch ‘p = 9.2 mm at various DC magnetic fields BDC of 0.1–1.0 T is shown in Fig. 4a. With increasing DC magnetic field, the real part v0 becomes broad due to the entry of flux into the wire but the critical temperature does not change almost. The imaginary part v00 shows two peaks due to the filament and the proximity-induced superconducting matrix. For the higher DC magnetic fields, the peaks are well separated as the coupling peaks move to lower temperature. The computed curves of the grained Bean model are plotted in Fig. 4b. These
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T. Akune et al. / Physica C 468 (2008) 1481–1484
a
= 4 (mT) = 0.32 nf = 2
B pf
0.047
sample C
0.1
0.04
l p = 9.2 (mm) Ba = 0.05 (mT)
χ ''
χ ''
ff
f = 10 (Hz)
0.02
0.051 0
0
dN ( μ m) = 0.051
B DC (T) 0.051
0.1
χ'
χ'
0.050 0.049 0.048 0.047
0.047
0.4 0.6
—0.1
0.8 —1
1.0
0
1
0
T/T c
10
5
T (K) Fig. 2. AC susceptibilities v0 and v00 as a function of temperature T reduced by the critical temperature Tc of wire for interfilamentary spacing dN of 0.047–0.051 lm at a filament content of ff = 0.32 and nf = 2.
b
= 4 (mT) d N = 0.050 (mm) f f = 0.32
B pf
χ ''
0.2
nf
0.1
χ ''
0.04
6
0.02 2
= 6, 5, 4, 3, 2 0
B DC (T) 0.1
χ'
0 6
—0.1
0.4 0.6
χ'
0.8 —0.2
2
1.0 0
10
5
T (K) —1 0
Fig. 4. (a) Temperature dependence of the real and imaginary parts of AC susceptibility for sample C with twist-pitch ‘p = 9.2 mm at various fields of 0.1– 1.0 T. (b) Numerically computed AC susceptibilities as a function of temperature.
1
T/Tc
Fig. 3. Influence of nf on the temperature dependence of AC susceptibilities v0 and v00 at ff = 0.32 and dN = 0.050 lm.
the magnetization and that obtained from simulation using the grained Bean model, respectively. Jcm values estimated from AC susceptibilities are 4–14 times larger than those estimated from measured twist-pitch dependence of magnetization and their temperature dependence is weak.
curves are favorably compared with the experimental results in Fig. 4a. Fit parameters in Eq. (6)–(9) are shown in Table 2. Field dependence of Jcm is shown in Fig. 5. Symbols of open circle and solid circle present Jcm estimated from the ‘p dependence of
Table 2 Fit parameters BDC (T)
Bpf (mT)
ff
J0 (106 A/m2)
b
Bc2N (T)
c
d
Jcm (106 A/m2)
0.1 0.4 0.6 0.8 1.0
0.21 0.11 0.09 0.08 0.07
0.14 0.14 0.14 0.14 0.14
10 7 6 4 4
0.1 0.2 0.3 0.6 0.9
4 4 4 4 4
0.4 0.4 0.4 0.4 0.4
1 1 1 1 1
17 8.2 7.2 5.7 5.2
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T. Akune et al. / Physica C 468 (2008) 1481–1484
6. Conclusions
sample C
AC susceptibilities of NbTi multifilamentary wires with CuMn matrix were measured using a SQUID magnetometer at temperatures of 2–10 K, DC magnetic fields BDC up to 1 T. Texture of multifilamentary wire is simulated by the grained Bean model, where the superconducting regions are divided two parts; filaments and proximity-induced superconducting matrix. The critical current densities of the proximity-induced superconducting matrix Jcm were estimated from the susceptibilities numerically analyzed using the grained Bean model. These values are 4–14 times larger than those estimated from measured twist-pitch dependence of magnetization.
l p = 9.2 (mm)
10
8
Ba = 0.05 (mT)
2
Jcm (A/m )
f = 10 (Hz)
References
10
[1] N. Harada, Y. Mawatari, O. Miura, Y. Tanaka, K. Yamafuji, Cryogenics 31 (1991) 183. [2] T. Akune, R. Maeda, N. Sakamoto, K. Funaki, IEEE Trans. Appl. Supercond. 11 (2001) 2764. [3] T. Akune, R. Maeda, N. Sakamoto, M. Takeo, Physica B 284–288 (2000) 2095. [4] C.P. Bean, Phys. Rev. Lett. 8 (1962) 250. [5] M. Kubo, T. Akune, N. Sakamoto, Physica C 463–465 (2007) 478. [6] W. Yumoto, T. Akune, N. Sakamoto, Annual Report of RISS, vol. 4, 2007, p. 86 (in Japanese). [7] T. Akune, N. Sakamoto, Jpn. J. Appl. Phys. 34 (1995) 6041. [8] L. Fàbrega, A. Sin, A. Calleja, J. Fontcuberta, Phys. Rev. B 61 (2000) 9793.
6
—1
10
0
B DC(T)
10
Fig. 5. Dependence of Jcm on DC magnetic field. Symbols of open circle and solid circle present Jcm estimated from the twist-pitch ‘p dependence of the magnetization and that obtained from simulation using the grained Bean model, respectively.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Modeling proximity-coupling in multifilamentary wires by grained Bean model T. Akune *, W. Yumoto, N. Sakamoto Department of Electrical Engineering and Information Technology, Kyushu Sangyo University, 2-3-1 Matsukadai, 813-8503 Fukuoka, Japan
a r t i c l e
i n f o
Article history: Available online 28 May 2008 PACS: 74.72 Keywords: Grained Bean model AC susceptibility Pinning penetration field
a b s t r a c t Proximity-currents between filaments in a multifilamentary wire show a close resemblance with the inter-grain current in a high-Tc superconductor. The critical current densities of the proximity-induced superconducting matrix Jcm can be estimated from measured twist-pitch dependence of magnetization and have been shown to follow the well-known scaling law of the pinning strength. In the grained Bean model, the filaments are immersed in the proximity-induced superconducting matrix. Difference of the superconducting characteristics of the filament, the matrix and the filament content factor give a variety of deformation on the AC susceptibility curves. The computed AC susceptibility curves of multifilamentary wires using the grained Bean model are favorably compared with the experimental results. Ó 2008 Published by Elsevier B.V.
1. Introduction
2. Experimental
It is necessary to enhance the inter-grain coupling in high-Tc superconductors for fulfilling the demand of high critical current densities and low losses for industrial AC applications. It has been reported [1] that the AC losses in the multifilamentary wires increased with decreasing interfilamentary spacing dN and were noticeable when dN became comparable to or smaller than the coherence length nN of the matrix. It is widely known that this effect results from an excess magnetization caused by interfilamentary proximity-coupling, and the proximity effect is reduced by replacing the Cu in the interfilamentary region by CuNi or by introducing a small percentage of Mn in the Cu between the filaments. In a preceding paper [2,3] we reported the agreement of both the upper critical field Bc2p obtained by applying the scaling law to Jcm data estimated from the twist-pitch ‘p dependence of the magnetization and Bc2p obtained from the peak temperatures Tp appearing in the imaginary parts of the AC susceptibilities of multifilamentary wires with Cu and Cu–0.5wt% Mn matrices. In addition the values of Bc1p were estimated from the temperature dependence of real parts of the AC susceptibilities. In this paper the relation of the proximity effect and the AC susceptibilities in the multifilamentary wire is discussed. Field distribution and magnetization in the multifilamentary wire are calculated by Bean model [4]. Fourier integration of magnetization is carried out numerically and gives rise to the AC susceptibilities. Measured results of NbTi multifilamentary wires with CuMn matrix are compared with the simulated results to give the matrix characteristics.
Sample specifications of multifilamentary wires are shown in Table 1. The matrix is Cu–0.5wt% Mn for series A, B and C samples. Samples were manufactured by double stacking and are composed of 24186 NbTi filaments embedded in the Cu–0.5wt% Mn matrix with the volume ratio being 1.58. The value of the interfilamentary spacing dN was changed by drawing the initial untwisted wire, and the drawn wire was twisted for each series. Therefore, the values of dN and df were calculated from the measured values of the wire diameter Dw. The edges of twisted wire were polished with sandpaper to separate filaments from each and then cleaned by ultrasonic agitation in acetone. After wire length dependence of the magnetization was measured, the sample length of L > 70 mm was determined to be sufficient for safely neglecting the effect of flux entry and exit along the wire axis. About five turns of each sample were wound on a straw with the diameter of 5.5 mm and fixed on it by adhesives. A commercial SQUID magnetometer was used to measure magnetic moments. The magnetization measurement was carried out in a temperature range 2–10 K, and the DC magnetic fields up to 1 T and the superposed AC magnetic fields of 0.05 mT were applied perpendicular to the wire axis. Low frequency of 10 Hz is used to avoid eddy-currents in the matrix. The superconducting transition temperatures Tc of wires are 8.5 K, which were measured from temperature dependence of magnetization under zero-field cooling and field cooling conditions.
* Corresponding author. Tel.: +81 92 673 5632; fax: +81 92 673 5091. E-mail address: [email protected] (T. Akune). 0921-4534/$ - see front matter Ó 2008 Published by Elsevier B.V. doi:10.1016/j.physc.2008.05.143
3. Grained Bean model [5,6] Proximity-induced superconducting matrix with a penetration field Bpm determines the magnetic fields at the filament surface
1482
T. Akune et al. / Physica C 468 (2008) 1481–1484
Table 1 Sample specifications
4. Proximity-coupling in multifilamentary wire
Sample
Matrix
Dw (mm)
df (lm)
dN (lm)
‘p (mm)
A B C
Cu–0.5wt% Mn Cu–0.5wt% Mn Cu–0.5wt% Mn
0.102 0.145 0.260
0.324 0.461 0.827
0.050 0.072 0.129
11.7 10.4 9.2
Mp
hBi ¼ Bo
Bpm þ 2
i¼1
ð2Þ
ð3Þ
Filament magnetization mi is easily computed following magnetization process of each filaments. In the case of uniform wire structure with constant filament size df ( =dfi for all i) and interfilamentary spacing dN ( =dNi for all i), n
hBi ¼ Bo
g X Bpm mi þ ff 2 i¼1
ð4Þ
where ff ( = df/(dN + df)) is the filament content factor. Then the magnetization M of the superconductor at the field Bo for each magnetization processes is given by
M ¼ hBi Bo
ð6Þ
J cm ¼ J 0
dfi mfi dNi þ dfi
l0 kp km Jcm ðBÞ‘p
ð1Þ
where Dw is a wire diameter, x is a position inside the filament from the each filament surface as shown in the inset. Average of the magnetic flux density hBi in the increasing period for both of the matrix and filaments with interfilamentary spacing dN is given by ng X
1
p2
where kp denotes the volume fraction of the superconducting matrix in the multifilamentary region and km is the volume fraction of the multifilamentary region in the wire. Thus Jcm values can be estimated from the slope of the ‘p dependence of the magnetization. Field dependence of Jcm estimated from the ‘p dependence of magnetization is approximately expressed by the scaling law of the pinning effect as [7]
with the penetration field Bpf as shown in Fig. 1. Field distribution Bm outside the ith filament at a filament position {i and field inside Bf of ith filament with the filament diameter dfi are given by using Bpm, Bpf and Bean model as [4]
2X i : outside the filament Bm ¼ Bo Bpm Dw x : inside the filament Bf ¼ Bm Bpf dfi
The proximity-effected magnetization Mp assisted by Jcm is expressed as [1]
ð5Þ
B Bc2p
c
B Bc2p
d ;
ð7Þ
where c and d are adjustable pinning parameters and Bc2p is the upper critical field of the weakly superconducting matrix. Temperature dependence of Bc2p is as follows:
Bc2p ¼ Bc2N expðK N dN Þ; 1/2
K 1 N
ð8Þ 1/2
where = b/T and b (K lm) in the low-temperature range of T < 4 K is 0.126, 0.172 and 0.282 for series A, B and C samples, respectively. In the high-temperature range of T > 4 K, b is 0.062 for series A and 0.086 for series B samples [7]. 5. AC susceptibilities of the grained Bean model The AC susceptibilities v0 and v00 under an AC field Ba cos xt are derived from Fourier integrals of the magnetization. The fundamental Fourier components v01 and v001 are denoted as v0 and v00 hereafter. Temperature dependence of v0 and v00 is obtained by introducing the temperature variation of the penetration fields of Bpf and Bpm. If the Bpf has a usual parabolic dependence of the form as
( Bpf ¼ Bpf ð0Þ 1
Fig. 1. Field distribution Bm outside the filament and distribution Bf inside the filament. Penetration fields are Bpm and Bpf for matrix and filament, respectively.
1
T T cf
2 )nf ;
ð9Þ
where over the temperature Tcf the pinning effect for the fluxoid motion disappears in the filament. Since the relation of the penetration field of Bpm and the critical current densities of the proximityinduced superconducting matrix Jcm is Bpm = l0DwJcm/2, the temperature variation of the penetration field of Bpm is given by Eq. (7) and (8). The AC susceptibilities are numerically computed and plotted in Fig. 2 where Bf = 4 mT, ff = 0.32, nf = 2 and dN = 0.047–0.051 lm. The first peak of v00 at higher temperature decreases and the additional peak shifts to lower temperature with increasing dN. At a fixed interfilamentary spacing of dN = 0.05 lm and ff = 0.32 the effect of nf is shown in Fig. 3. With increasing nf the imaginary peaks increase at the second peak and coexist. This type of double peaks is often observed in high-Tc materials [8]. The double peak structure in the v00 curve is specifically indicated to originate in the cooperative work of the filament and the matrix. Temperature dependence of the real and imaginary parts of AC susceptibility for sample C with twist-pitch ‘p = 9.2 mm at various DC magnetic fields BDC of 0.1–1.0 T is shown in Fig. 4a. With increasing DC magnetic field, the real part v0 becomes broad due to the entry of flux into the wire but the critical temperature does not change almost. The imaginary part v00 shows two peaks due to the filament and the proximity-induced superconducting matrix. For the higher DC magnetic fields, the peaks are well separated as the coupling peaks move to lower temperature. The computed curves of the grained Bean model are plotted in Fig. 4b. These
1483
T. Akune et al. / Physica C 468 (2008) 1481–1484
a
= 4 (mT) = 0.32 nf = 2
B pf
0.047
sample C
0.1
0.04
l p = 9.2 (mm) Ba = 0.05 (mT)
χ ''
χ ''
ff
f = 10 (Hz)
0.02
0.051 0
0
dN ( μ m) = 0.051
B DC (T) 0.051
0.1
χ'
χ'
0.050 0.049 0.048 0.047
0.047
0.4 0.6
—0.1
0.8 —1
1.0
0
1
0
T/T c
10
5
T (K) Fig. 2. AC susceptibilities v0 and v00 as a function of temperature T reduced by the critical temperature Tc of wire for interfilamentary spacing dN of 0.047–0.051 lm at a filament content of ff = 0.32 and nf = 2.
b
= 4 (mT) d N = 0.050 (mm) f f = 0.32
B pf
χ ''
0.2
nf
0.1
χ ''
0.04
6
0.02 2
= 6, 5, 4, 3, 2 0
B DC (T) 0.1
χ'
0 6
—0.1
0.4 0.6
χ'
0.8 —0.2
2
1.0 0
10
5
T (K) —1 0
Fig. 4. (a) Temperature dependence of the real and imaginary parts of AC susceptibility for sample C with twist-pitch ‘p = 9.2 mm at various fields of 0.1– 1.0 T. (b) Numerically computed AC susceptibilities as a function of temperature.
1
T/Tc
Fig. 3. Influence of nf on the temperature dependence of AC susceptibilities v0 and v00 at ff = 0.32 and dN = 0.050 lm.
the magnetization and that obtained from simulation using the grained Bean model, respectively. Jcm values estimated from AC susceptibilities are 4–14 times larger than those estimated from measured twist-pitch dependence of magnetization and their temperature dependence is weak.
curves are favorably compared with the experimental results in Fig. 4a. Fit parameters in Eq. (6)–(9) are shown in Table 2. Field dependence of Jcm is shown in Fig. 5. Symbols of open circle and solid circle present Jcm estimated from the ‘p dependence of
Table 2 Fit parameters BDC (T)
Bpf (mT)
ff
J0 (106 A/m2)
b
Bc2N (T)
c
d
Jcm (106 A/m2)
0.1 0.4 0.6 0.8 1.0
0.21 0.11 0.09 0.08 0.07
0.14 0.14 0.14 0.14 0.14
10 7 6 4 4
0.1 0.2 0.3 0.6 0.9
4 4 4 4 4
0.4 0.4 0.4 0.4 0.4
1 1 1 1 1
17 8.2 7.2 5.7 5.2
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T. Akune et al. / Physica C 468 (2008) 1481–1484
6. Conclusions
sample C
AC susceptibilities of NbTi multifilamentary wires with CuMn matrix were measured using a SQUID magnetometer at temperatures of 2–10 K, DC magnetic fields BDC up to 1 T. Texture of multifilamentary wire is simulated by the grained Bean model, where the superconducting regions are divided two parts; filaments and proximity-induced superconducting matrix. The critical current densities of the proximity-induced superconducting matrix Jcm were estimated from the susceptibilities numerically analyzed using the grained Bean model. These values are 4–14 times larger than those estimated from measured twist-pitch dependence of magnetization.
l p = 9.2 (mm)
10
8
Ba = 0.05 (mT)
2
Jcm (A/m )
f = 10 (Hz)
References
10
[1] N. Harada, Y. Mawatari, O. Miura, Y. Tanaka, K. Yamafuji, Cryogenics 31 (1991) 183. [2] T. Akune, R. Maeda, N. Sakamoto, K. Funaki, IEEE Trans. Appl. Supercond. 11 (2001) 2764. [3] T. Akune, R. Maeda, N. Sakamoto, M. Takeo, Physica B 284–288 (2000) 2095. [4] C.P. Bean, Phys. Rev. Lett. 8 (1962) 250. [5] M. Kubo, T. Akune, N. Sakamoto, Physica C 463–465 (2007) 478. [6] W. Yumoto, T. Akune, N. Sakamoto, Annual Report of RISS, vol. 4, 2007, p. 86 (in Japanese). [7] T. Akune, N. Sakamoto, Jpn. J. Appl. Phys. 34 (1995) 6041. [8] L. Fàbrega, A. Sin, A. Calleja, J. Fontcuberta, Phys. Rev. B 61 (2000) 9793.
6
—1
10
0
B DC(T)
10
Fig. 5. Dependence of Jcm on DC magnetic field. Symbols of open circle and solid circle present Jcm estimated from the twist-pitch ‘p dependence of the magnetization and that obtained from simulation using the grained Bean model, respectively.