Superconductor Science and Technology
Supercond. Sci. Technol. 29 (2016) 034006 (7pp)

doi:10.1088/0953-2048/29/3/034006

Flux jumps in high-Jc MgB2 bulks during
pulsed ﬁeld magnetization
H Fujishiro1, H Mochizuki1, T Naito1, M D Ainslie2 and G Giunchi3
1

Department of Materials Science and Engineering, Faculty of Engineering, Iwate University, 4-3-5 Ueda,
Morioka 020-8551, Japan2
Bulk Superconductivity Group, Department of Engineering, University of Cambridge, Trumpington
Street, Cambridge CB2 1PZ, UK
3
Materials Science Consultant, via Teodosio 8, I-20131 Milano, Italy
E-mail: fujishiro@iwate-u.ac.jp
Received 16 October 2015, revised 5 January 2016
Accepted for publication 6 January 2016
Published 2 February 2016
Abstract

Pulsed ﬁeld magnetization (PFM) of a high-Jc MgB2 bulk disk has been investigated at 20 K, inwhich ﬂux jumps frequently occur for high pulsed ﬁelds. Using a numerical simulation of the
PFM procedure, we estimated the time dependence of the local magnetic ﬁeld and temperatureduring PFM. We analyzed the electromagnetic and thermal instability of the high-Jc MgB2 bulk
to avoid ﬂux jumps using the time dependence of the critical thickness, dc(t), which shows theupper safety thickness to stabilize the superconductor magnetically, and the minimum
propagation zone length, lm(t), to obtain dynamical stability. The values of dc(t) and lm(t) change
along the thermally-stabilized direction with increasing temperature below the criticaltemperature, Tc. However, the ﬂux jump can be qualitatively understood by the local
temperature, T(t), which exceeds Tc in the bulk. Finally, possible solutions to avoid ﬂux jumps in
high-Jc MgB2 bulks are discussed.
Keywords: ﬂux jump, MgB2, pulsed ﬁeld magnetization, numerical simulation, critical
thickness, temperature rise
(Some ﬁgures may appear in colour only in the online journal)
1. Introduction

bulk 35 mm in diameter fabricated by a hot isostatic pressing(HIP) method [4]. A record-high trapped ﬁeld of Bz=5.4 T
has been attained at 12 K on a single MgB2 bulk 20 mm indiameter, which was fabricated by hot-pressing of ball-milled
Mg and B powders using ﬁne-grained boron powders [5].
Pulsed-ﬁeld magnetization (PFM) has also been investigated to magnetize such bulk superconductors. However, thetrapped ﬁeld Bz by PFM is generally lower than that by FCM
because of a large temperature rise caused by the dynamic
motion of the magnetic ﬂux. It is well known that multi-pulse
techniques are extremely effective in enhancing Bz for
REBaCuO bulks due to the reduction of this temperature rise[6, 7]. We have achieved the highest trapped ﬁeld of
Bz=5.20 T with a GdBaCuO bulk 45 mm in diameter at
30 K using a modiﬁed multi-pulse technique with stepwise
cooling [8], which is a record-high value for a REBaCuO
bulk magnetized by PFM to date. The PFM technique has
also been performed to MgB2 bulks fabricated by various

Bulk MgB2 superconductors have a promising potential as
quasi-permanent magnets, and have a number of attractive
properties, such as being rare-earth-free, light-weight andpresenting a homogeneous trapped ﬁeld distribution, which
are in clear contrast with REBaCuO bulk magnets. The problem of weak-links at grain boundaries can be ignored in
MgB2 polycrystalline bulks due to their long coherence
length, ξ [1]. These characteristics enable us to realize betterand larger polycrystalline MgB2 bulk magnets below thetransition temperature Tc=39 K, which are expected to be
applicable in magnetically levitated trains and wind power
generators using liquid H2 or cryocooler systems. Several
groups have already reported the trapped ﬁeld Bz for MgB2
bulks activated by ﬁeld-cooled magnetization (FCM) [2, 3].
We have attained Bz=3.6 T and 4.6 T at 14 K on a single
bulk and in a bulk pair, respectively, for a Ti-doped MgB20953-2048/16/034006+07$33.00

1© 2016 IOP Publishing Ltd Printed in the UK

Supercond. Sci. Technol. 29 (2016) 034006

H Fujishiro et al

methods [9, 10]. However, Bz(PFM)=0.80 T at 14 K is thehighest value for an MgB2 bulk fabricated by the HIP method,
for which Bz(FCM)=2.23 T was trapped at 16 K by FCM[11]. The numerical simulation is a valuable technique tounderstand the mechanism of the ﬂux intrusion and the ﬂux
trapping during PFM for REBaCuO and MgB2 bulks [12–14].
Considering the experimental results and the numericalsimulation for the REBaCuO system, the Bz(PFM) value for
the MgB2 bulk is expected to be enhanced by using a bulk
with a higher Bz(FCM) and/or a higher critical current density, Jc. However, ﬂux jumps took place frequently during
PFM in the MgB2 bulk and consequently the Bz(PFM) value
decreased for higher applied magnetic ﬁelds, which has been
scarcely observed for REBaCuO bulks at around 40 K during
PFM. Flux jumps are scarcely observed in low-Jc MgB2 bulks
and even in high-Jc MgB2 bulks for lower applied ﬁelds, Bex.
The ﬂux dynamics and heat generation in the MgB2 bulk ataround 20 K are in clear contrast with those in the REBaCuO
bulk at around 40 K because of the small speciﬁc heat, large
thermal conductivity, and a narrow temperature margin
against Tc for the MgB2 bulk. A concentric-circled MgB2
bulk composite disk is one of the possible solutions to avoid
ﬂux jumps by PFM, because many MgB2 thin rings are ﬁlled
in both surfaces of highly thermal conductive metal disk, andthe generated heat is removed quickly [15].
In this study, we analyze and discuss the electromagnetic
and thermal instability during PFM of a high-Jc MgB2 bulk
fabricated by the HIP method to avoid ﬂux jumps using
conventional relations, such as the critical thickness, dc, andthe minimum propagation zone (MPZ) length, lm.

Figure 1. Schematic view of the experimental setup for PFM.

temperature, T(t), was measured at the center of the bulk
surface using a thermometer (CernoxTM; Lakeshore), which
was adhered by GE7031TM varnish.
After the PFM experiments, the bulk was cut and small
pieces about 1×1×2 mm3 in size were prepared formagnetization curve M(H) measurements at 10, 20 and 30 K
under a magnetic ﬁeld up to μ0H=5 T using a commercial
SQUID magnetometer (MPMS-5 T; Quantum Design). The
Jc–B relationship was estimated from the M(H) hysteresis
loop using an extended Bean model, Jc=20ΔM/a(1−a/3b), where a and b are the dimensions of the plane of thesample perpendicular to the applied ﬁeld and ΔM is the widthof the M(H) loop. The temperature dependence of the thermal
conductivity, κ(T), was measured by a steady-state heat ﬂow
method.

2. Experimental procedure
A bulk MgB2 superconducting disk was fabricated by a HIP
method. The diameter and thickness is 38 mm and 7 mm,
respectively, and the relative mass density was as high as 93%
of the ideal mass density. The detailed fabrication process is
described elsewhere [4]. The bulk MgB2 disk was mounted ina stainless steel ring (38.1 mm I.D., and 56.0 mm O.D.) using
epoxy resin (Stycast 2850TM) and was tightly anchored onto
the cold stage of a Gifford–McMahon cycle helium refrigerator. The details of the PFM procedure and experimental
setup are described elsewhere [9, 11]. The initial temperature,Ts, of the bulk was set to 20 K. The solenoid copper magnetizing coil (99 mm I.D., 121 mm O.D., and 50 mm height),
which was submerged in liquid nitrogen, was placed outside
the vacuum chamber. Magnetic pulses, Bex, up to 2.2 T, witha rise time of 0.013 s and duration of 0.15 s were applied via apulse current in the coil. The time evolution of the local ﬁeld,
Bz(t), and the subsequent trapped ﬁeld, Bz, at the center of thebulk surface, which was deﬁned as the ﬁnal value of Bz(t),were monitored by a Hall sensor (BHA 921; F W Bell). Twodimensional trapped ﬁeld proﬁles of Bz were mapped stepwise
with a pitch of 1 mm by scanning the same Hall sensor using
an x–y stage controller. During PFM, the time evolution of the

3. Numerical analysis
Based on the experimental setup, a model for the numericalsimulation of PFM was constructed as shown in ﬁgure 1, in
which the physical phenomena occurring during the magnetization process are described using the fundamental electromagnetic and thermal equations in axisymmetric coordinates.
The details of the numerical simulation have been described
elsewhere [16, 17]. We set a spacing plate 2 mm in thickness
with a thermal conductivity of κcont=0.5 W m−1 K−1
between the bulk and the cold stage of the refrigerator, which
imaginarily represents both the cooling power of the refrigerator and the thermal contact. The power-n model (n=100)
was used to describe the nonlinear E–J characteristic of the
MgB2 bulk. The measured Jc(B) of the MgB2 bulk at 20 K

2

Supercond. Sci. Technol. 29 (2016) 034006

H Fujishiro et al30
20Bex=1.41 T(b)

Y (mm)

10
0
-10
-20
-30
-30 -20 -10

010

X (mm)

30
20

(c)

20

30

Bex=1.96 T

Y (mm)

10
0
-10
-20
-30
-30 -20 -10

010

X (mm)

20

30

0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
(T)

0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
(T)

Figure 2. (a) Applied pulsed-ﬁeld dependence of trapped ﬁeld, Bz, for the MgB2 bulk at Ts=20 K at the center of the bulk surface. The
dashed line shows the estimated Bz value, in the case that no ﬂux jump took place during PFM (see ﬁgure 3(b)). The trapped ﬁeld proﬁles for(b) Bex=1.41 T and (c) Bex=1.96 T, mapped on the bulk at 1 mm above the bulk surface, are shown.

was ﬁtted using the following equation3⎡ ⎛ B ⎞b ⎤⎧⎛ T ⎞2 ⎫ 2
Jc (B , T ) = a ⎨1 - ⎜ ⎟ ⎬ exp ⎢ - ⎜ ⎟ ⎥ ,⎢⎣ ⎝ B0 ⎠ ⎥⎦⎝ Tc ⎠ ⎭
⎩
⎪

⎪

⎪

⎪

(1 )

where Tc=39 K, and α=4.5×109 A m−2, B0=1.3 T andβ=1.7 are the ﬁtted parameters at 20 K. The magnetic ﬁeld
and temperature dependence of the critical current density Jc(B,T) was estimated according to equation (1). Iterative calculations were performed for analyzing the combined problem ofelectromagnetic ﬁelds and heat diffusion by the ﬁnite element
method using commercial software, Photo-Eddy, combinedwith Photo-Thermo (Photon Ltd, Japan). In the analysis ofthermal conduction, the temperature dependence of the thermal
conductivity, κ(T), and the speciﬁc heat, C(T), of the MgB2
and the stainless steel [18] were introduced.

4. Experimental results
Figure 2(a) shows the relationship between the trapped ﬁeld,
Bz, by PFM and the applied pulsed ﬁeld, Bex, for the MgB2
bulk at Ts=20 K. Similar to the previous experimental
results for REBaCuO and MgB2 bulks magnetized by PFM[6, 9], the Bz value increases, attains a maximum value, andthen decreases with further increasing Bex. However, for thebulk with a higher Jc, ﬂux jumps took place frequently forhigher applied ﬁelds.
Figures 3(a) and (b) depict the time evolution of the local
ﬁeld, Bz(t), at the center of the bulk surface and the applied
pulsed ﬁeld, Bex(t), for Bex=1.41 T and Bex=1.96 T,
respectively. For Bex=1.41 T, Bz(t) started to increase with aslight time delay, took a maximum at 0.016 s, and thendecreased smoothly with increasing time. However, for
Bex=1.96 T, Bz(t) saw a slight drop at t=0.05 s and a larger
drop of about 40% at t=0.07 s due to ﬂux jumps. Then, Bz(t)approaches to the ﬁnal trapped ﬁeld, Bz, due to ﬂux creep. The

Figure 3. Time evolution of the local ﬁeld, Bz(t), at the center of thebulk surface and the applied pulsed ﬁeld, Bex(t), for (a) Bex=1.41 T
and (b) Bex=1.96 T at 20 K. The magnetization curves, which were
calculated by ΔB=Bz(t)−Bex(t), are shown in the insets.
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Figure 4. Time dependence of the temperature, T(t), at the center ofthe bulk surface for each applied ﬁeld, Bex, during PFM. The
maximum temperature, Tmax, as a function of Bex, is also shown in
the inset.

estimated Bz(t) curve is also shown in the ﬁgure, which was
ﬁtted before the ﬂux jumps using an exponential curve. The
estimated Bz(t) introduces the ﬁnal trapped ﬁeld, Bz, which isshown in ﬁgure 2(a) as a dotted line. If any ﬂux jumps can be
avoided, the Bz value over 0.9 T can be realized as shown inﬁgure 2(a). The typical trapped ﬁeld proﬁles are shown inﬁgures 2(a) and (b). For Bex=1.41 T, the proﬁle is concave
and nearly concentric because of the lower Bex. On the other
hand, for Bex=1.96 T, the proﬁle is asymmetric and themagnetic ﬂux can escape from the bulk center due to ﬂux
jumps.
The insets of ﬁgure 3 show the magnetization curves,which were calculated by ΔB=Bz(t) – Bex(t). In theascending stage of Bex(t), the magnetic ﬂux can hardly
penetrate into the bulk center and the ΔB value is nearly
proportional to the negative applied ﬁeld Bex (ΔB=−Bex).
A wide hysteresis loop can be seen clearly with increasing
time, which mainly results from ﬂux penetration and trappingin the bulk disk. A ﬂux jump can be also conﬁrmed in thehysteresis curve. Flux jump phenomena in the magnetic
hysteresis measurement were frequently observed in the small
MgB2 samples at low temperature using a SQUID magnetometer [19, 20], where the temperature rise due to the ﬂux
jump quickly recovered and the magnetizing process started
again over a long time scale. On the other hand, the escaped
magnetic ﬂux did not recover in the present PFM processusing the large MgB2 bulk: the magnetization process wasﬁnished within 0.1 s, but the elevated temperature did notrecover to the initial temperature quickly.
Figure 4 presents the time dependence of the temperature,
T(t), during PFM on the side surface of the stainless steel ringfor each applied ﬁeld, Bex. The maximum temperature, Tmax, asa function of Bex, is also shown in the inset. The temperature, T(t), increases, takes a maximum at around t=2 s, and thendecreases with increasing time. Since the thermometer has aﬁnite size, the measured temperature shows an average temperature change of the bulk disk within the ring.

Figure 5. (a) Magnetic ﬁeld dependence of the critical current

density, Jc(B), of the MgB2 bulk under analysis at Ts=10, 20 and
30 K. (b) Temperature dependence of the measured thermal
conductivity, κ, and the speciﬁc heat, C, referred from [18] forbulk MgB2.

Figure 5(a) shows the magnetic ﬁeld dependence of themeasured critical current density, Jc(B), of the present MgB2
bulk at Ts=10, 20 and 30 K. Jc under zero ﬁeld increases
with decreasing temperature and Jc monotonically decreases
with increasing magnetic ﬁeld at each temperature. The
values of Jc at 20 K are 4.3×109 A m−2 at μ0Ha=0 T and3.1×108 A m−2 at μ0Ha=2 T, which are typically high Jc
values in an MgB2 bulks. Figure 5(b) shows the temperaturedependence of the measured thermal conductivity, κ(T), andthe speciﬁc heat, C(T), referred from [18] for bulk MgB2. The
κ(T) value was as high as 18 W mK−1 at 20 K due to thehigher mass density.

5. Numerical simulation results
Figure 6(a) depicts the results of the numerical simulation of
the trapped ﬁeld, Bz, at the center of the bulk surface, as afunction of the applied ﬁeld, Bex, at 20 K. The Bex versus Bz
proﬁle using the numerical simulation qualitatively reproduces the experimental results shown in ﬁgure 2(a). However,
the maximum Bz value is twice as large as that obtained in theexperiment, which may come from the overestimated κcont in
4

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H Fujishiro et al

Figure 7. Temperature and magnetic ﬁeld dependences of the critical
thickness dc for the present MgB2 bulk.

where Tc is the transition temperature and T is the operating
temperature. Jc(T, B), C(T) and γ are critical current density,
speciﬁc heat and mass density (2360 kg m−3) at T,
respectively.
The MPZ length, lm, is usually used to evaluate thedynamical instability under adiabatic conditions. If thesuperconductor is divided with the size below lm, and isthermally connected to highly thermal conductive metal, thesuperconductor seems to be thermally stable. lm can be
expressed using the following equationlm =Figure 6. Numerical simulation results of the trapped ﬁeld, Bz, at thecenter of the bulk surface, as a function of the applied ﬁeld, Bex, at
20 K. (b) Cross-sections of the trapped ﬁeld proﬁle, Bz(x), foreach Bex.

2k (Tc - T )
,
rJc 2

(3 )

where κ(T) is the thermal conductivity and ρ is the normal
state resistivity just above Tc.
Figure 7 shows the temperature dependence of the critical
thickness, dc, of the present MgB2 bulk for each magnetic
ﬁeld, which is calculated using the Jc(T, B) and C(T) values
shown in ﬁgure 5. These results suggest a maximum thicknessof 0.86 mm, when the present MgB2 bulk is carrying its critical current, above 20 K and above 1 T [22], which is about
one order of magnitude shorter than that for the GdBaCuO
bulk (dc=4.9 mm at 40 K and 1 T) [11]. This suggests that
the MgB2 bulk at 20 K is magnetically unstable, comparedwith the GdBaCuO bulk at 40 K.

the simulation compared to the actual κcont in the experiment.
Figure 6(b) presents cross-sections of the trapped ﬁeld proﬁle,Bz(x), for each Bex. The Bz(x) proﬁle changes from concave
for lower Bex to trapezoidal for higher Bex.

6. Discussion
6.1. Estimation of critical thickness using experimental results

6.2. Time dependence of critical thickness dc and MPZ lm using
the results of numerical simulation

Here we discuss the magnetic instability during PFM using
the conventional equations for superconducting wires [21]. In
the classical slab model, the critical thickness, dc, is usuallyused to evaluate magnetic instability. In order to avoid a ﬂux
jump or quench, a subdivision of the superconductor smaller
than dc is effective in minimizing ﬂux movement. That is, dc
describes an upper safety limit of the thickness and can be
written as follows
dc = 23gC (Tc - T ),m 0 Jc 2

In section 6.1, the critical thickness dc becomes longer withincreasing temperature, T, and magnetic ﬁeld, B, which
implies ﬂux jumps are less likely to happen during PFM.
However, ﬂux jumps often happen for higher applied ﬁelds,
Bex. Inhomogeneous heat generation takes place adiabaticallyover a short time period in the disk-shaped bulk. We supposethat the ﬂux jump mainly results from a local temperature rise
above Tc and calculate the time dependence of the critical
thickness, dc(t), using the T(t) estimated by the numericalsimulation at three typical points (center: (x, z)=(0, 0),

(2 )

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Figure 8. The positions for the calculation of the time dependence ofdc and lm.

Figure 10. Estimated time dependence of the temperature, T(t),critical current density, Jc(t), speciﬁc heat, C(t), and critical
thickness, dc(t), at the center position for Bex=1.6 T at 20 K. dc(t)was calculated using equation (2).

Figure 9. Estimated time dependence of the temperature, T(t), at thethree positions (center, middle, side) after applying a pulsed ﬁeld of
Bex=1.6 T at 20 K, which were obtained by numerical simulation.

middle: (8, 0) and side: (17, 0)) in the bulk disk, as shown inﬁgure 8.
Figure 9 presents the estimated time dependence of thetemperature, T(t), at these three positions after applying apulsed ﬁeld of Bex=1.6 T at 20 K, at which the trapped ﬁeld
takes a maximum, as shown in ﬁgure 6(a). The timedependence of Bex(t) is also shown on the right vertical axis.
The temperature rise starts to increase from the bulk periphery
due to the ﬂux penetration. At the side of the bulk, (x,y)=(17, 0), T(t) exceeds Tc=39 K at t=0.009 s anddecreases below Tc at t=0.04 s. The time, at which T(t)exceeds Tc and recovers to below Tc, is delayed for the middle
and the center positions of the bulk in this order.
Figure 10 shows the time dependences of the temperature, T(t), critical current density, Jc(t), speciﬁc heat, C(t),and critical thickness, dc(t), at the center position, (x, y)=(0,0), for Bex=1.6 T at 20 K. dc(t) was calculated using
equation (2). Jc(t) decreased and C(t) increased withincreasing time due to the temperature rise, both of which
contribute the increase of dc In order to decrease the value ofdc and to induce a ﬂux jump, T(t) must be increased andexceed Tc. In this case, the ﬂux jump should take place at thebulk center at 0.03t0.08 s. Even if the temperature, T

Figure 11. Time dependence of the estimated dc(t) at the threepositions for Bex=1.6 T at 20 K.

(t), recovers below Tc at t>0.08 s, the trapped ﬁeld cannot
be recover to the previous value because it takes much time torecover the initial temperature and the pulsed ﬁeld, Bex(t), hasdecreased signiﬁcantly by this time. In the PFM experiments
shown in ﬁgure 3, the ﬂux jumps almost always happen
during the descending stage of the magnetic pulse (i.e.,t>0.02 s).
Figure 11 shows the time dependence of the estimateddc(t) at the three positions for Bex=1.6 T at 20 K. The valueof dc(t) for the center position is also reused. If the ﬂux jump
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Acknowledgments
This work was supported by Open Partnership Joint Projectsof Japan Society for the Promotion of Science (JSPS) Bilateral Joint Research Projects, and JSPS KAKENHI grant
number 23560002 and 15K04646. Dr Mark Ainslie would
like to acknowledge the support of a Royal Academy ofEngineering Research Fellowship and a Royal SocietyInternational Exchanges Scheme grant, IE131084.

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Figure 12. Time dependence of the estimated lm(t) at the threepositions for Bex=1.6 T at 20 K. lm(t) was calculated using
equation (3).

is supposed to take place during the time when T(t)>Tc, theposition at which the ﬂux jump happens proceeds from theside to the center of the bulk with increasing time. Figure 12
presents the time dependence of the MPZ length, lm(t), at thethree positions for Bex=1.6 T at 20 K, which was calculatedusing equation (3). The value of lm(t) is about one order ofmagnitude smaller than dc(t), which suggests that the MgB2
bulk is thermally unstable, rather than magnetically. These
analyses suggest a possible approach to understand and avoid
this ﬂux jump behavior. The use of a longer applied pulsed
ﬁeld and a powerful and stable cooling apparatus for the bulk
is a valuable approach to avoid ﬂux jumps because it restricts
the temperature rise [16]. The use of a concentric-circled
MgB2 bulk composite disk, which consists of many MgB2
thin rings in the highly thermally conductive metal disk, isalso another potentially valuable solution to avoid ﬂux jumps
caused by PFM [15].

7. Conclusion
PFM of a high-Jc bulk MgB2 disk has been investigatedexperimentally, in which ﬂux jumps frequently occur for
higher pulsed ﬁelds, Bex. Using a numerical simulation of the
PFM procedure, we estimated the time dependence of thelocal magnetic ﬁeld and temperature during PFM, and analyzed the electromagnetic and thermal instability of the highJc MgB2 bulk to avoid the ﬂux jump using the timedependence of the critical thickness, dc(t), and the MPZ
length, lm(t). The values of dc(t) and lm(t) change along thethermally stabilized direction with increasing temperature
below the critical temperature, Tc. However, the ﬂux jump
behavior can be qualitatively understood by the local temperature, T(t), which exceeds Tc in the bulk. The use of a longer
applied pulsed ﬁeld and a powerful and stable coolingapparatus for the bulk is are valuable approaches to avoid ﬂux
jumps because it restricts the temperature rise.

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