Materials Science and Engineering B 151 (2008) 95–100

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Materials Science and Engineering B
journal homepage: www.elsevier.com/locate/mseb

Possible explanation for trapped ﬁeld enhancement on REBaCuO bulk by
modiﬁed multi-pulse technique with stepwise cooling (MMPSC)
H. Fujishiro a,∗ , T. Hiyama a , T. Naito a , T. Tateiwa a , Y. Yanagi bab

Faculty of Engineering, Iwate University, 4-3-5 Ueda, Morioka 020-8551, Japan
Imra Material R&D Co. Ltd., 5-50 Hachiken-cho, Kariya 448-0021, Japana r t i c l e

i n f oArticle history:
Received 11 October 2007
Received in revised form 4 March 2008
Accepted 4 March 2008
Keywords:
Pulse ﬁeld magnetization
Bulk superconductor
Viscous force
Pinning force
MMPSC method
Temperature risea b s t r a c t
The time evolutions of the local ﬁelds BL (t) have been measured on the surface of the superconducting
bulk disk magnetized by a two-stage pulse-ﬁeld magnetizing technique, called a modiﬁed multi-pulse
technique combined with stepwise cooling (MMPSC), and the magnetic ﬂux movement and the ﬂux
trapping have been investigated. The optimum concaved (“M-shaped”) trapped ﬁeld proﬁle, which isa necessary condition at the ﬁrst stage to enhance the ﬁnal trapped ﬁeld BT , makes a larger magnetic
gradient (dB/dx) at the bulk periphery in the ascending stage of the applied magnetic pulse at the second
stage due to the large viscous force Fv . The magnetic ﬂuxes, which stay at the bulk periphery, start to ﬂow
to the center of the bulk, after the applied pulse ﬁeld reaches a maximum, at which the ﬂux velocity v
is nearly zero and then Fv decrease. As a result, a large number of the magnetic ﬂuxes are trapped at thebulk center. The effect of the “M-shaped” proﬁle at the ﬁrst stage in MMPSC on the enhancement of BT is
discussed.
© 2008 Elsevier B.V. All rights reserved.

1. Introduction
Recently, a pulse ﬁeld magnetization (PFM) for REBaCuO bulks(RE: rare earth element and Y) has been intensively investigatedfor practical applications, instead of a ﬁeld-cooled magnetization(FCM), because PFM is an inexpensive and mobile experimental
setup with no use of a superconducting magnet. The trapped ﬁeld
BTP by PFM is, however, lower than that by FCM at temperatures
lower than 50 K due to the large temperature rise by the dynamical motion of the magnetic ﬂuxes. At 77 K, several approaches havebeen performed and succeeded to enhance BTP by an iteratively magnetizing pulsed-ﬁeld method with reducing amplitude (IMRA) [1], amulti-pulse technique with stepwise cooling (MPSC) [2] and so on.
We have systematically studied the time and spatial dependencesof the temperature T(t,x), local ﬁeld BL (t) and the trapped ﬁeld BTPon the surface of cryocooled REBaCuO bulks 45 mm in diameter
during PFM for various starting temperatures Ts and applied ﬁelds
Bex [3–5]. To enhance BTP , the reduction in temperature rise T is
an indispensable issue. The lowering of Ts is also effective because
the critical current density Jc and resultant trapped ﬁeld BTP increase
with lowering Ts . However, in the PFM technique, the lowering ofTs as low as 10–20 K is not necessarily effective because of the large
heat generation due to the enhancement of pinning loss and the∗ Corresponding author. Tel.: +81 19 621 6363; fax: +81 19 621 6363.
E-mail address: fujishiro@iwate-u.ac.jp (H. Fujishiro).
0921-5107/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.mseb.2008.03.010

decrease in the heat capacity of the bulk [6]. Taking the obtained
experimental results into consideration, we proposed a new PFM
technique named as a modiﬁed MPSC (MMPSC), which consistedof the two-stage procedures [7]. In this technique, a small number of magnetic ﬂuxes should be trapped on the bulk periphery forthe reduction in T at a ﬁrst stage at T1 (<Tc ), and the optimum
pulse ﬁeld B2 should be applied at a second stage at T2 (<T1). The“M-shaped” trapped ﬁeld proﬁle, which means the lower trapped
ﬁeld at the bulk center than that at the bulk periphery, is also necessary at the ﬁrst stage to enhance BTP . Using the MMPSC method,
we have succeeded a highest ﬁeld trapping of BTP = 5.20 T on the45 mm GdBaCuO bulk, on which 3.6 T had been trapped by a single pulse application at 40 K [8]. BTP = 5.20 T is a record-high value
by PFM to date. The MMPSC method has been proved to be a universal and effective method to enhance BTP using other bulks suchas the 45 mm SmBaCuO disk with different pinning ability [9], therectangular-shaped GdBaCuO bulk (33 mm × 33 mm × 15 mm) [10]and the large GdBaCuO disk 65 mm in diameter [11]. In the previous study, we measured the time evolutions of the local ﬁelds BL (t)at three points on the bulk surface [8] and the two-dimensional
trapped ﬁeld proﬁle just above 0.5 mm on the bulk for various Bex
and Ts [12]. It has been found that the “M-shaped” proﬁle should be
experimentally optimized to enhance BTP . However, there has been
no reasonable explanation for the question, why the “M-shaped”
proﬁle is necessary at the ﬁrst stage in MMPSC.
In this paper, we measure the time evolutions of the local ﬁelds
BL (t) at several points on the bulk surface for various conditions in96

H. Fujishiro et al. / Materials Science and Engineering B 151 (2008) 95–100

Fig. 1. (a) The experimental setup around the bulk and the pulse coil. Three Hall sensors are adhered at #1, #2 and #3 on the surface of the bulk. (b) The experimental
sequence of the MMPSC method, where the estimated time dependence of the temperature is shown.

the MMPSC process, and analyze the magnetic ﬂux movement andtrapping. The meaning of the “M-shaped” proﬁle at the ﬁrst stagein MMPSC is discussed.

a scanning device, which has an x–y stage controller with a ﬂexible
bellows. The trapped ﬁeld proﬁles BT (0.5 mm) were also measured
0.5 mm above the bulk surface under the same MMPSC conditions,after removing the adhered Hall sensors.

2. Experimental
3. Results and discussion
The GdBaCuO bulk disk (Nippon Steel, Japan) used in thisstudy was 45 mm in diameter and 18 mm in thickness, which wasdifferent from that attained BT = 5.20 T [8]. Fig. 1(a) shows the experimental setup around the bulk and the magnetizing pulse coil. The
bulk was mounted on soft iron yoke cylinder 40 mm in diameterand 20 mm in thickness and tightly anchored onto the cold stageof a Gifford–McMahon (GM) cycle helium refrigerator. The temperature of the bulk was controlled over the range from 30 K to70 K. The magnetizing solenoid coil, which generated the pulse
ﬁeld up to Bex = 6.3 T with a rise time of 12 ms, was placed outside the vacuum chamber, in which the central axis of the resulting
ﬁeld coincides with that of the bulk. The trapped ﬁeld characteristics of the bulk by the single pulse application have been reported
elsewhere [12]. Fig. 1(b) shows the experimental sequence of the
MMPSC technique. Four magnetic pulses were applied at different
initial temperatures T1 and T2 on the bulk surface. At the ﬁrst stage,a pulse ﬁeld B1 was applied twice at T1 (=70 K or 30 K) in order torealize various shapes of the trapped ﬁeld proﬁle on the bulk. Hereafter, we refer to these two pulses as Nos. 1 and 2. At the second
stage, the bulk was cooled down to T2 (=30 K) and a higher pulse
ﬁeld B2 (=6.3 T) was applied twice (Nos. 3 and 4), after recovering
to T2. Three Hall sensors (F.W. Bell, model BHT 921) were adhered
in a line at the positions #2 (bulk center), #1 and #3 (12.5 mm distant from the bulk center) on the bulk surface as shown in Fig. 1(a).
For each pulse of the MMPSC method, the time evolutions of thelocal ﬁelds [BL (#1)(t), BL (#2)(t) and BL (#3)(t)] were monitored byusing a digital oscilloscope. The trapped ﬁeld proﬁle BT (2 mm) was
measured 2.0 mm above the bulk surface by scanning an axial-type
Hall sensor (F.W. Bell, BHA 921) inside the vacuum chamber using

3.1. Trapped ﬁeld proﬁles at the ﬁrst stage for various conditions
In the MMPSC method, a desired trapped ﬁeld proﬁle at the ﬁrst
stage can be controlled by choosing the conditions of T1 and B1,
and the subsequent trapped ﬁeld proﬁle of the second stage is realized by T2 and B2. Table 1 summarizes four cases (Case A–Case D)
and their conditions (T1, B1, T2 and B2) investigated in this study.
Four trapped ﬁeld proﬁles with different shape were realized at theﬁrst stage and, at the second stage, the magnetic ﬂux movement
and the trapping were measured after the pulse ﬁeld application of
B2 = 6.3 T at T2 = 30 K. In the Case D, the pulse ﬁelds were not appliedat the ﬁrst stage and the pulse ﬁeld of B2 = 5.7 T was applied twice
to the virgin state bulk at T2 = 30 K.
Fig. 2(a) shows the vertical sections of the BT (0.5 mm) proﬁlealong different directions and the trapped ﬁelds [BT (#1), BT (#2)
and BT (#3)] after applying pulse ﬁeld of B1 = 3.2 T at T1 = 70 K (No.
1 pulse of the Case A). The two-dimensional BT (0.5 mm) proﬁle isalso shown in the inset. The vertical sections are nearly symmetric,
which suggest that the trapped ﬁeld proﬁle can be estimated using
three Hall sensors at #1, #2 and #3. Fig. 2(b) shows the trapped ﬁeld
proﬁles from Case A to Case D at the end of the ﬁrst stage. The measured BT values at the positions from #1 to #3 and the estimatedones (BT = 0) at the bulk edge (X = 22.5 mm) were used. In the Cases
B and C, a “large M” and a “trapezoid” BT proﬁles were realized,
respectively. In the Case A, the M-shaped proﬁle was somewhat
smaller than that in the Case B, in which the maximum BT (#2) ashigh as 4.0 T was ﬁnally obtained. We abbreviate the BT proﬁle in
the Case A as “optimum M”.

Table 1
The conditions (T1, B1, T2 and B2) and the ﬁnal trapped ﬁeld BT (#2) from Case A to Case D in the MMPSC method

Case A (optimum M)
Case B (large M)
Case C (trapezoid)
Case D (without)

1st stage (T1 and B1) (No. 1 and No. 2)

2nd stage (T2 and B2) (No. 3 and No. 4)

BT (#2) (No. 4)

70 K, 3.2 T (No. 1)
70 K, 3.2 T (No. 2)
30 K, 6.2 T (No. 1)
30 K, 6.2 T (No. 2)
70 K, 3.9 T (No. 1)
30 K, 4.6 T (No. 2)–

30 K, 6.3 T

4.0 T

30 K, 6.3 T

1.5 T

30 K, 6.3 T

1.2 T

30 K, 5.7 T

1.2 T

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Fig. 2. (a) The example of vertical sections of the BT (0.5 mm) proﬁle along different directions and BT at #1, #2 and #3 after applying pulse ﬁeld of B1 = 3.2 T at T1 = 70 K. (b)The trapped ﬁeld proﬁle at the ﬁrst stage in MMPSC method for the cases from Case A to Case D.

3.2. The ﬂux movement and ﬂux trapping at the second stage
The ﬂux movement and the ﬂux trap are investigated on thebulk with different trapped ﬂux distribution from Case A to CaseD, after applying the No. 3 pulse of B2 = 6.3 T (=5.7 T in the Case D)
at T2 = 30 K. Fig. 3 shows the results for the Case A. In Fig. 3(a), thetime evolutions of the applied ﬁeld 0 Ha (t) and the local ﬁelds BL (t)at #1, #2 and #3 are shown. The applied ﬁeld smoothly increased
and took a maximum at t = 12 ms and then decreased. BL (#1)(t) andBL (#3)(t) started to rise at t = 6 ms with a time delay and took amaximum at t = 15 and 17 ms, respectively. BL (#2)(t) started to rise
at t = 13 ms with a long time delay, took a maximum at t = 17 ms andthen decreased. The magnitude of BL (#2)(t) is larger than that at #1
and #3 for t > 20 ms. Fig. 3(b) and (c) shows the time dependencesof the transient magnetic distribution for the ascending (t ≤ 12 ms)
and the descending (t ≥ 12 ms) stages of the No. 3 pulse ﬁeld, respectively, both of which were reconstructed using BL (t) at #1, #2 and#3 shown in Fig. 3(a). BL (#1)(t) and BL (#3)(t) hardly change fort ≤ 6 ms and the magnetic ﬂux distribution in the bulk showed akink at #1 and #3. This result cannot be explained using a Bean’s
critical-state model under zero-ﬁeld cooling, because the magnetic
gradient dBL /dx, which is proportional to a critical current density
Jc , should be constant along the radius direction and the ﬂux distribution should be expressed by a straight line. This result strongly
suggests that the viscous force Fv for the ﬂux movement is strong
early in the ascending stage. For t ≥ 6 ms, BL (#1)(t) and BL (#3)(t)gradually increase and the ﬂux distribution changes to a straightline. The BL (#2)(t) value slightly increases during the ascending

stage. For the descending stage as shown in Fig. 3(c), however,BL (#2)(t) sharply increased at t = 15 ms, just after the applied ﬁeld
0 Ha (t) took a maximum. BL (t) at #1 and #3 gradually decreased
with increasing time and a cone-shaped trapped ﬁeld distribution
with BTP = 4.0 T was ﬁnally obtained.
Fig. 4 shows the similar results for the Case B, where a large
M-shaped trapped-ﬁeld proﬁle was realized at the ﬁrst stage. In
Fig. 4(a), BL (#1)(t) and BL (#3)(t) started to rise at t = 6 ms and thentook a maximum of 4.5 T at t = 13 ms. The maximum value wassmaller, and the time, at which BL (t) took a maximum, was shorter
than that for the Case A. In Fig. 4(b) and (c), BL (#1)(t) and BL (#3)(t)smoothly increased and then decreased. As a result, the large Mshaped proﬁle was almost maintained. It should be noted thatthe magnetic gradient (dBL /dx) at the end of the ascending stage(t = 12 ms) was smaller than that for the Case A.
Fig. 5 shows the similar results for the Case C, where a trapezoidshaped trapped ﬁeld proﬁle was realized at the ﬁrst stage. In
Fig. 5(a), BL (#1)(t) and BL (#3)(t) keep the initial values up tot = 10 ms and then the magnetic ﬂuxes abruptly intrude into thebulk like a ﬂux jump. BL (#2)(t) was not enhanced even at thedescending stage, contrary to the Case A. In Fig. 5(b), Fv is strong
for t ≤ 10 ms and magnetic ﬂuxes cannot intrude into the bulk center. In the descending stage, BL (#1)(t) and BL (#3)(t) decreased withincreasing time and the BL (#2)(t) kept a initial value. Since the large
temperature rise might take place at the bulk periphery due to theﬂux jump, the ﬂux ﬂow did not take place to the bulk center but took
place to the bulk periphery. As a result, the BT (#2) value was notenhanced only with a slight deformation of the trapped ﬁeld proﬁle.

Fig. 3. (a) The time dependence of the applied ﬁeld 0 Ha (t) and the local ﬁelds BL (t) at #1, #2 and #3 for the No. 3 pulse in the Case A (optimum M). The time dependenceof the ﬂux movement in the bulk for (b) the ascending (t ≤ 12 ms) and (c) the descending (t ≥ 12 ms) stages of the No. 3 pulse.

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Fig. 4. (a) The time dependence of 0 Ha (t) and BL (t) at #1, #2 and #3 for the No. 3 pulse in the Case B (large M). The time dependence of the ﬂux movement in the bulk for(b) the ascending (t ≤ 12 ms) and (c) the descending (t ≥ 12 ms) stages of the No. 3 pulse.

Fig. 5. (a) The time dependence of the 0 Ha (t) and BL (t) at #1, #2 and #3 for the No. 3 pulse in the Case C (trapezoid). The time dependence of the ﬂux movement in the bulkfor (b) the ascending (t ≤ 12 ms) and (c) the descending (t ≥ 12 ms) stages of the No. 3 pulse.

The similar results for the Case D are indicated in Fig. 6, wherea ﬁrst magnetic pulse was applied to the virgin state bulk as a No.
3 pulse. In the previous paper, the temperature rise for the ﬁrst
pulse-ﬁeld application is larger, compared with that in the case,where a small number of the magnetic ﬂuxes are already trapped[5]. Then the maximum of BL (#1)(t) and BL (#3)(t) reaches 5 T and
BL (#2)(t) increases to 1.5 T due to the large temperature rise. In
Fig. 6(b) and (c), the magnetic ﬂux distribution is not a straight line
along the radius direction at the end of the ascending stage, andﬁnally shows the asymmetric M-shaped trapped ﬁeld proﬁle. The
BTP value at the bulk center cannot be enhanced. These results from
Figs. 3–6 suggest that the magnetic ﬂuxes enter to the bulk centerat the speciﬁed trapped ﬁeld proﬁle at the ﬁrst stage.
3.3. Possible explanation of BT enhancement in MMPSC
In the previous subsection, it was clariﬁed that the optimum“M-shaped” proﬁle such as “Case A” should be realized at the ﬁrst

Fig. 6. (a) The time dependence of 0 Ha (t) and BL (t) at #1, #2 and #3 in the virgin state bulk in the Case D (without). The time dependence of the transient magnetic
distribution in the bulk for (b) the ascending and (c) the descending (t ≥ 12 ms) stages of the No. 3 pulse.

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99

Fig. 7. The conceptual time dependence of the transient magnetic ﬁeld distribution for the optimum M-shaped proﬁle (Case A) ((a) and (b)), and for the large M-shaped
proﬁle (Case B) ((c) and (d)) for the ascending and the descending stages of the No. 3 pulse ﬁeld.

stage in MMPSC to enhance BTP . We qualitatively discuss about
the magnetic ﬂux movement and trapping after the No. 3 pulse
application, compared with the estimated results by a Bean’s
critical-state model.
The ﬂux motion in the bulk is determined by the force balancein the critical-state model:
FL = −(FP + Fv ) = Jc B + 

B
V,
0

(1)

where FL is the Lorentz force, FP the pinning force, Jc the criticalcurrent density, B the applied pulse ﬁeld,  the viscosity coefﬁcient,0 a ﬂuxoid quantum and v is the ﬂux velocity. Fv is proportional
to v, but FP is independent of v.
Fig. 7(a) and (b) shows the conceptual time dependence of thetransient magnetic ﬁeld distribution for the optimum M-shaped
proﬁle (Case A) for the ascending and the descending stages of theNo. 3 pulse ﬁeld, respectively, both of which were estimated by useof the results in Fig. 3(b) and (c). In these ﬁgures, thin and thicklines show the distribution for the ZFC method (Bean’s criticalstate model) and for the No. 3 pulse application in the MMPSC
method, respectively. In the MMPSC method, a large magnetic gradient (dB/dx) exists in the bulk periphery at the ascending stage,
which is fairly larger than that estimated by a Bean’s critical-statemodel due to the large viscous force Fv . When the applied pulse
ﬁeld reaches a maximum at t = 12 ms, a large number of magnetic
ﬂuxes, which stay at the bulk periphery, start to ﬂow to the centerof the bulk and then the trapped ﬁeld proﬁle changes from the “Mshaped” proﬁle to the “cone-shaped” one at the descending stage. At
this moment, the ﬂux velocity v is nearly zero and Fv decreases tozero as estimated from Eq. (1). The magnetic gradient ∂B(x,t)/∂x in
the bulk can be derived as follows [13]:x ∂B(x , t)∂B(x, t)
B(x, t)= 0 ± Jcx dx .
+|B|
0 |B|xx0
∂x
∂t

(2)

In the ZFC method, since the time dependence of the magnetic ﬁeld
∂B(x ,t)/∂t in the bulk is almost zero, ∂B(x,t)/∂x is given by 0 Jc . How-

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H. Fujishiro et al. / Materials Science and Engineering B 151 (2008) 95–100

ever, in PFM, the magnitude of ∂B(x,t)/∂x is markedly affected by thepresence of the second term in Eq. (2). For instance, ∂B(x ,t)/∂t is positive in the ascending stage and the second term of Eq. (2) becomes
positive. Hence, ∂B(x,t)/∂x is enhanced by an amount relative to 0 Jc
in the ZFC method.
Fig. 7(c) and (d) shows the similar conceptual view for the large
M-shaped proﬁle (Case B) for the ascending and the descending
stages of the No. 3 pulse ﬁeld. In this case, a large magnetic gradient(dB/dx) exists at the bulk periphery at the end of the ascending
stage (t = 12 ms), but the dB/dx value is smaller than that for the
optimum M-shaped proﬁle (Case A) due to the higher height at thebulk center of the large M-shaped proﬁle. Then the magnetic ﬂuxes,
which stay at the bulk periphery, do not ﬂow to the bulk center, but
ﬂow to out of the bulk at the descending stage, even though theﬂux velocity v is nearly zero and the viscous force Fv decreases.
Finally, we comment on the inﬂuence of the temperature riseon the ﬁnal trapped ﬁeld during PFM. In the previous study, it wasclariﬁed that the already trapped ﬂuxes in the bulk should reduce
the temperature rise due to the decrease of the pinning loss Qp[5]. In this study, the temperature rise seems to be the largest for
Case D (“without”) and is the smallest for Case B (“large M”), after
applying the No. 3 pulse. The temperature rise for the Case C (“trapezoid”) is estimated to be also large because of the ﬂux jump. In thissense, a larger number of the magnetic ﬂuxes should be trapped
and the BT (#2) value should be enhanced in the Case B. However,the BTP value was not enhanced for the Case B and the optimum “Mshaped” proﬁle such as Case A (“optimum M”) exists in MMPSC.
These results strongly suggest that a moderate temperature rise isnecessary to enhance the BTP value in the MMPSC method. The penetration of the ﬁeld into the bulk during PFM could have been caused
by a combination of heating due to the magnitude of the induced
screening currents which corresponds to the slope of the BL vs. x
curve, and a suppression of Jc by the local ﬁeld. It is necessary to
study the ﬂux dynamics using a theoretical treatment.
4. Summary
The time evolutions of the local ﬁelds BL (t) have been measured
on the surface of the superconducting bulk disk magnetized bya modiﬁed multi-pulse technique combined with stepwise cooling (MMPSC) under various trapped ﬁeld proﬁles at the ﬁrst stage,
and the magnetic ﬂux movement and the ﬂux trapping have been
investigated. The important experimental results and conclusions
obtained in this study are summarized as follows:(1) The trapped ﬁeld BTP by the MMPSC method strongly depends on
the trapped ﬁeld proﬁle at the ﬁrst stage. The “M-shaped” pro-ﬁle, which means the higher trapped ﬁeld at the bulk peripherythan that at the bulk center, is a necessary condition to enhance
BTP and should be optimized.
(2) The time dependences of the ﬂux distribution were reconstructed for the ascending and the descending stages of theapplied pulse ﬁeld using BL (t). These graphs enable us to understand the ﬂux motion and trapping in the bulk clearly.
(3) Under the optimum M-shaped proﬁle at the ﬁrst stage, the large
viscous force Fv is created at the end of ascending stage of theNo. 3 pulse, at which the temperature rise is relatively small.
When the applied pulse ﬁeld reaches a maximum, at which
the viscous force Fv decreases due to v = 0, a large number ofmagnetic ﬂuxes, staying at the bulk periphery, start to ﬂow to
the center of the bulk and then the trapped ﬁeld proﬁle changes
from the “M-shaped” to the “cone-shaped” one at the descending
stage.
(4) A moderate amount of temperature rise is necessary to enhance
the BTP value at the second stage, which is decided the relations
between the trapped ﬁeld proﬁle at the ﬁrst stage, T2 and B2.
Acknowledgements
The authors thank Professor T. Oka of Niigata University for thevaluable discussions. This work is supported in part by a Grant-inAid for Scientiﬁc Research from the Ministry of Education, Culture,
Sports, Science and Technology, Japan (No. 19560003).
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