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Numerical simulation of flux jump behavior in REBaCuO ring bulks withan inhomogeneous J c profile during pulsed-field magnetization
To cite this article: Tatsuya Hirano et al 2020 Supercond. Sci. Technol. 33 044003

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Superconductor Science and Technology
Supercond. Sci. Technol. 33 (2020) 044003 (11pp)

https://doi.org/10.1088/1361-6668/ab7297

Numerical simulation of ﬂux jump behavior
in REBaCuO ring bulks with an
inhomogeneous Jc proﬁle during pulsedﬁeld magnetization
Tatsuya Hirano1,3 , Hiroyuki Fujishiro1 , Tomoyuki Naito1
Mark D Ainslie2

and1

Department of Physical Science and Materials Engineering, Faculty of Science and 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, United Kingdom
E-mail: g0318132@iwate-u.ac.jp
Received 27 September 2019, revised 5 January 2020
Accepted for publication 3 February 2020
Published 17 February 2020
Abstract

We have investigated the electromagnetic and thermal properties of a REBaCuO ring bulk withan inhomogeneous critical current density, Jc, proﬁle during pulsed ﬁeld magnetization (PFM)
using a numerical simulation and compared those to a bulk with a homogeneous Jc proﬁle. A
notch was introduced in the bulk periphery, which was assumed as a crack existing in the actual
bulk material. A sudden ﬂux penetration (ﬂux jump) took place through the notch area and as aresult, a large temperature rise also took place around this notch. Consequently, the ﬁnal trapped
ﬁeld proﬁle was simulated to be a ‘C-shaped proﬁle’, which qualitatively reproduced ourprevious experimental results. The size and position dependences of the notch on the ﬂux
penetration behaviour were also simulated, in which a larger and outer notch promotes the ﬂux
jump phenomenon easily. On the other hand, in the homogeneous model, under the same
conditions, no ﬂux jump phenomenon was observed. These results suggest that the imperfectionin the bulk can be a possible starting point of the ﬂux jump. The electromagnetic and thermalhoop stresses were also simulated in the ring bulk during PFM, in which the electromagnetic
stress and the thermal stress were both observed to be lower than the fracture strength of the bulk
material. This provides good evidence that the experimentally observed ‘C-shaped proﬁle’results from the ﬂux jump rather than the fracture of the bulk.
Keywords: bulk superconductor, pulsed ﬁeld magnetization, ﬂux jump, mechanical stress,
numerical simulation, inhomogeneous Jc proﬁle(Some ﬁgures may appear in colour only in the online journal)
1. Introduction
3

REBaCuO (RE: rare earth element or Y) superconducting
bulks can be used as strong trapped ﬁeld magnets (TFMs),
which have generated magnetic ﬁelds of over 17 T to date
[1–3]. Such TFMs have been used for practical applications
such as a compact cryogen-free nuclear magnetic resonance(NMR) spectrometer, magnetic separation, magnetic drug

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Supercond. Sci. Technol. 33 (2020) 044003

T Hirano et al

Figure 1. Time evolution of the trapped ﬁeld, Bt(t), at the center ofthe ring bulk (r=z=0 mm) and the applied ﬁeld, Bex(t), for
Bex=3.10 T at 65 K. The inset shows the trapped ﬁeld proﬁlemapped 1.0 mm above the bulk surface, in which the contour of thering bulk is also shown. © 2016 IEEE. Reprinted, with permission,from [21].

Figure 2. 2D cross-section of the numerical model of the ring bulk

and magnetizing coil for pulsed-ﬁeld magnetization.

behaviour in the ring-shaped bulk was also observed experimentally during PFM [22, 23].
During the magnetizing process of REBaCuO superconducting bulks, a ﬂux jump sometimes occurs, whichusually causes a locally large temperature rise and reduces theﬁnal trapped ﬁeld, especially during FCM. However, a ﬂux
jump, which was named a ‘giant ﬂux leap’ by Weinstein et al[24], enables the magnetic ﬂux to penetrate the bulk moreeasily in the ascending process of PFM and can enhance theﬁnal trapped ﬁeld [24–27]. Recently several authors havereported the numerical simulation of the ﬂux jump phenomena during the PFM process [27–29]. However, the inﬂuenceof the ﬂux jump and the resultant large temperature rise on
mechanical fracture during PFM has not been well clariﬁed yet.
In this study, we reconsider the cause of the reported ‘Cshaped proﬁle’ and perform numerical simulation of the PFM
process for a realistic REBaCuO ring bulk. The ﬂux jump
phenomenon and mechanical stress behavior were simulated
for ring bulks with homogeneous and inhomogeneous Jc
proﬁles. We discuss the mechanical stresses in such a realistic
REBaCuO ring bulk during PFM to clarify the risk of fracture
for the bulk. This study contributes to the development ofring-shaped TFMs from the viewpoints of the electromagnetic, thermal and stress behaviors during PFM.

delivery systems and rotating machines [4–8]. Today, stateof-the-art bulk materials have potential to trap magnetic ﬁeldof over 20 T at low temperatures from the viewpoint ofmagnetic ﬁeld and temperature dependence of the criticalcurrent density, Jc(B, T). However, REBaCuO TFMs sufferfrom mechanical fracture of the brittle ceramic material due toa large Lorentz force, F=J×B, (J is the current density
and B is the magnetic ﬁeld), that is developed during ﬁeldcooled magnetization (FCM) in high applied magnetic ﬁelds[9–12]. We have investigated the mechanical properties of
disk- and ring-shaped REBaCuO bulks during FCM using
numerical simulations based on the ﬁnite element method(FEM), and proposed optimal reinforcement structures madeof metal to prevent mechanical fracture [13–17]. The
mechanical properties have also been investigated analyticallyfor bulks with an inﬁnite height during FCM and zero-ﬁeldcooled magnetization (ZFCM) [18–20]. However, looking
towards industrial applications, the pulsed ﬁeld magnetization(PFM) is the most practical technique to magnetize superconducting bulks because of the inexpensive and mobile
experimental setup in contrast to FCM/ZFCM.
In our previous study [21], we performed PFM experiments on a GdBaCuO ring bulk. Figure 1 shows the results oftime evolution of the trapped ﬁeld, Bt(t), at the center of thering bulk and the applied ﬁeld, Bex(t), for Bex=3.10 T at
65 K. The ﬁnal trapped ﬁeld of the center of the ring bulk wasa negative value after PFM and exhibited a ‘C-shaped’ trapped ﬁeld proﬁle, as shown in the inset of ﬁgure 1. We concluded, in this work, that these results may come from thebulk fracture: the ring bulk was reinforced only by an outer
metal ring, which was effective for the tensile stress during
the descending stage of PFM, but not effective for the compressive stress during the ascending stage. In addition, at thattime, we had not yet developed a simulation technique toinvestigate mechanical properties precisely. The ﬂux jump

2. Numerical simulation framework
Based on our experimental setup for PFM [21], we constructed a three-dimensional (3D) model for a REBaCuO ringbulk (60 mm in outer diameter (O.D.), 36 mm in inner diameter (I.D.) and 17 mm in height (H)), which was thesame size as the previous study. Figure 2 shows the twodimensional (2D) cross-section of the 3D numerical model.
The ring bulk was mounted in an Al alloy ring holder 5 mm inwidth (70 mm in O.D. and 60 mm in I.D.) with the same2

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Table 1. Numerical parameters for the Jc (B, T) characteristics inequation (1) at 65, 70, 75 and 80 K used in the numerical simulation.

T (K)
65
70
75
80

Jc1 (Am−2)91.0×10
7.9×108
6.1×108
4.3×108

BL (T)

Jc2 (Am−2)

0.46
0.40
0.32
0.21

8

5.9×10
3.8×108
2.2×108
1.1×108

Bmax (T)

k

2.4
2.2
1.8
0.65

1.1
1.3
1.8
2.8

height as the bulk. The Al alloy is a suitable material for thereinforcement of the ring bulk for NMR apparatus because it
is non-magnetic and has a higher mechanical strength, compared to pure aluminum. The bulk was cooled to Ts=65 K
from the bottom surface attached to the cold stage using a thinindium sheet and was magnetized by PFM using an outermagnetizing coil (120 mm in O.D., 100 mm in I.D. and
50 mm in H). Single magnetic pulses with a rise time of
13 ms, a duration of 200 ms, and amplitudes ranging fromBex=2.53 to 4.0 T, were applied to the bulk via a pulsed
current in the coil. During the PFM process (t=1∼300 ms),
the time evolution of the trapped magnetic ﬁeld, Bt(t), wasmonitored at the center of ring bulk (r=z=0 mm). Afterthe PFM process (t=7000 ms) at the steady state without
ﬂux ﬂow, the Bt distribution was also mapped 0 mm
(z=8.5 mm) or 1.0 mm (z=9.5 mm) above the bulk surface. Physical phenomena during PFM are described by thefundamental electromagnetic and thermal equations in [30].
The power-n law (n=20) was used to describe the highly
nonlinear E-J characteristic of the bulk material [31]. The
following magnetic ﬁeld and temperature dependence of critical current density, Jc(B, T), proposed by Jirsa et al [32], wasused in the simulation.
⎛
B ⎞B
Jc (B , T ) = Jc1(T ) exp ⎜ ⎟ + Jc2 (T )⎝ BL (T ) ⎠Bmax (T )⎡⎛⎞k (T ) ⎞ ⎤B
1 ⎛⎜⎜1 - ⎜⎟⎟ ⎥ ,´ exp ⎢⎟
⎝ Bmax (T ) ⎠⎠ ⎦⎥⎣⎢ k (T ) ⎝Figure 3. Magnetic ﬁeld and temperature dependences of the criticalcurrent density, Jc(B, T), between 60 K and 77 K used in thesimulation. The Jc(B, T) proﬁles of GdBaCuO bulk measured using
SQUID magnetometer are also shown in comparison [33].

Table 2. Mechanical parameters of the REBaCuO bulk and Al alloy
used in the numerical simulation.

REBaCuO bulk
Al alloy

E (GPa)ν

α0 (K−1)

100
78

0.33
0.34

5.20×10−6
1.48×10−5(E: Young’s modulus, ν: Poisson ratio, α0: thermal
expansion coefﬁcient).

REBaCuO bulk were assumed to be independent of temperature. The temperature-dependent speciﬁc heat, C, of the bulk
was used [34]. The simulation procedure of electromagnetic
and thermal properties are described elsewhere in detail [30].
The elastic behaviour of an isotropic material can be
explained by Hooke’s law, in which the stress is linearly
proportional to the strain [13]. The electromagnetic hoop
stress, sq EM, and the thermal hoop stress, sq therm, during PFM
were calculated for each case. Table 2 summarizes themechanical parameters (Young’s modulus, E, Poisson ratio,
ν, and thermal expansion coefﬁcient, α0) of the REBaCuO
bulk and the Al alloy used in the simulation.
In this study, homogeneous and inhomogeneous models
are constructed for the Jc distribution, as shown in ﬁgures 4(a)and (b). The same model was used for the electromagnetic,
thermal, and mechanical simulations, which was equallydivided into 36 elements along the circumferential (θ) direction. Along the z- and r-directions, the bulk region was
divided by a 0.5 mm mesh and the other region was divided
by a 1.0 mm mesh. The total number of mesh elements was
227 304. The computational time needed to solve all theelectromagnetic, thermal, and mechanical models for one
condition (Ts, Bex) was approximately about 6 h. In theinhomogeneous model, a notch (2 mm in width, 17 mm in
H and 1 element along the θ-direction) is placed at thebulk periphery (r=28–30 mm) as a non-superconducting
material with very low thermal conductivity (κ=1×10−8
Wm−1K−1) and speciﬁc heat (C=1×10−8 Jkg−1K−1). The(1 )where the parameters (Jc1, BL, Jc2, Bmax and k) for the Jc(B) at
each temperature are shown in table 1. Figure 3 shows theJc(B, T) proﬁles used in the simulation, compared with Jc(B,T) of the GdBaCuO bulk measured using SQUID magnetometer, which has typical characteristics in the recent bulk
material [33]. The similar experimental Jc(B, T) were also
reported [34]. The Jc(B, T) characteristics used in this simulation were slightly modiﬁed from the SQUID data [33] toreproduce the experimental Bt(t) proﬁle shown in ﬁgure 3.
The temperature dependences of Jc(B, T) were linearlyinterpolated to simulate the results during PFM, as describedin [35]. The REBaCuO ring bulk are assumed to be isotropic
and homogeneous for simplicity. The commercial software
package, Photo-Eddy, combined with Photo-Thermo andPhoto-Elas (Photon Ltd, Japan), was used for analyses ofelectromagnetic, thermal and mechanical properties. The
anisotropic thermal conductivities κab=20 Wm−1K−1 in theab-plane and κc=4 Wm−1K−1 along the c-axis of3

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Figure 4. Schematic view of the numerical models for the ring-shaped bulk: (a) homogeneous model (without notch), (b) inhomogeneousmodel (with notch) (see text).

Figure 5. Schematic view of the numerical models for the ring-shaped bulk with various notches: (a) inner notch, (b) center notch, (c) outernotch, (d) inner tiny notch, (e) center tiny notch and (f) outer tiny notch (see text).

notch’ is the same as that shown in ﬁgure 4. In addition, themodels with a tiny notch (1.0 mm in width, 1.0 mm in H and1 element along the θ-direction) at r=18, 23, 28 mm aredenoted as ‘inner tiny notch’, ‘center tiny notch’ and ‘outer
tiny notch’, respectively, as shown in ﬁgures 5(d)–(f).
Since a ﬂux jump is a thermomagnetic instability, thenumerical simulation of the ﬂux jump is sensitive to the meshsizes, the initial and boundary conditions, and so on. However, this study presents the qualitative tendency of the ﬂux
jump behavior based on the chosen mesh.

boundary condition between the notch region and bulk region(or outer air region) is not adiabatic, but heat conductive.
The size of the notch shown in ﬁgure 4 is much larger
than that of a realistic crack. To clarify the inﬂuence of notch
size and position on the ﬂux jump, the inhomogeneousmodels with various types of notches are also constructed.
The models with the notch (2.0 mm in width, 17 mm in H and1 element along the θ-direction) at r=18, 23, 28 mm aredenoted as ‘inner notch’, ‘center notch’ and ‘outer notch’,
respectively, as shown in ﬁgures 5(a)–(c). Note that the ‘outer4Supercond. Sci. Technol. 33 (2020) 044003

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Figure 6. (a) Experimental (© 2016 IEEE. Reprinted, with permission, from [21]) and (b) present simulation results of time evolution of thetrapped ﬁeld, Bt(t), at the center of the ring bulk (r=z=0 mm) and applied pulsed ﬁeld, Bex(t), for Bex=3.10 T at 65 K. The inset of eachﬁgure shows the trapped ﬁeld proﬁle mapped 1.0 mm above the bulk surface (z=9.5 mm). The contour of the ring bulk is shown in the inset
of each ﬁgure.

Figure 7. Time evolution of the trapped ﬁeld, Bt, at the center of the ring bulk (r=z=0 mm) for Bex=3.1∼4.0 T at Ts=65 K and applied
pulsed ﬁeld, Bex(t), for Bex=4.0 T using (a) the homogeneous model and (b) the inhomogeneous model.

magnetic ﬂux might penetrate into the actual bulk easily froma lower Jc region and the ﬁnal trapped ﬁeld becomes
lower [36].

3. Numerical simulation results and discussion
3.1. Verification of simulation results

To verify the accuracy of numerical simulation, the numericalresults are compared to the previous experimental results.
Figures 6(a) and (b), respectively, show the experimental
results by Mochizuki et al [21] and the present simulation
results using the homogenous model, in which the time
evolution of the trapped ﬁeld, Bt(t), at the center of the ringbulk (r=z=0 mm) and applied pulsed ﬁeld, Bex(t), for
Bex=2.53 T at 65 K are shown. The inset in each ﬁgureshows the trapped ﬁeld proﬁle mapped 1.0 mm above thebulk surface (z=9.5 mm), in which the contour of the ringbulk is also shown. The experimental Bt(t) behavior at thecenter of the ring bulk was reproduced by numerical simulation well, in which the ﬁtting parameters for equation (1) areadjusted to adequately reproduce the maximum Bt value
during PFM process. However, the Bt value at the surface ofthe bulk periphery in the simulation is about 2.5 times higher
than that in the experiment. The difference in the Bt value
may come from the ideally homogenous Jc(B, T) characteristics in the simulation model, assuming no position
dependence of Jc. Furthermore, in the experiment, the3.2. Comparison of simulation results between homogeneousand inhomogeneous models

Figure 7(a) shows time evolution of the trapped ﬁeld, Bt(t), at
the center of the ring bulk (r=z=0 mm) for Bex=
3.1∼4.0 T at Ts=65 K using the homogeneous model. Foran applied ﬁeld of Bex=3.1 T, Bt(t) took a peak value andthen decreased to small Bt value at t=300 ms, at which themagnetic ﬂux cannot penetrate well to the center of the bulk
due to its magnetic shielding. The ﬁnal Bt value increased
with increasing Bex. For Bex=3.1∼4.0 T, no ﬂux jump took
place. Figure 7(b) shows similar results using the inhomogeneous model, as shown in ﬁgure 4(b). The time evolutionof Bt(t) for Bex lower than 3.5 T was similar to those using thehomogeneous model. However, the ﬂux jump took place for
Bex=4.0 T, for which Bt(t) increased rapidly at 14 ms andreached its peak value of Bt=2.7 T at t=15 ms. In thedescending stage, Bt(t) decreased gradually with increasingtime and eventually reached a negative value of Bt=−0.3 T,
which is a clear evidence of the disturbance of the circulating5Supercond. Sci. Technol. 33 (2020) 044003

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mainly in the area of ﬂux penetration. At 5 ms<t<20 ms,
the magnetic ﬂux suddenly penetrated into the notch area via
a ﬂux jump and the temperature rise gradually increased
across the whole bulk region. At t=20 ms, the magnetic ﬂux
completely penetrates into the center of the ring bulk and thetemperature at the notch area reached around 90 K. These
results in the ascending stage are reasonably understood by
the following positive cycle of the ﬂux penetration and theresultant temperature rise by the ﬂux jump: (1) the notch areain the bulk periphery weakens the magnetic shielding effectand the magnetic ﬂux is concentrated; (2) the ﬂux concentration promotes a temperature rise due to the ﬂux
movement, compared to other areas; (3) the shielding effect isfurther decreased due to the temperature rise; and (4) the ﬂux
penetration and the resultant temperature rise are promoted
more and more. This cycle leads to the ﬂux jump in theascending process.
Figure 9(d) shows the time evolution of the trapped ﬁeld
(the color of Bt=0 T includes the negative value) andtemperature distributions on the bulk surface (z=8.5 mm)
during the descending stage (20 ms<t300 ms). The
magnetic ﬂux escapes mainly from the notch area andthe amount of magnetic ﬂux decreases with increasing time.
The temperature around the notch area remains as high as
90 K even at t=300 ms because the thermal diffusivity ismuch lower, compared with the ﬂux movement. As a result,the Bt proﬁle exhibited the ‘C-shaped proﬁle’. The local andlarge temperature rise promotes a large ﬂux ﬂow and thenleads to the ‘C-shaped proﬁle’ in the trapped ﬁeld. It should
be noted that no ﬂux jump takes place in the ideal homogeneous bulk in this simulation. We found that there was onlya slight difference in the temperature distributions along the zdirection in the inhomogeneous model. In the actual bulks, alarge number of voids, cracks and secondary phases (RE211
and Ag etc) exist. The present simulation results strongly
suggest that such inhomogeneity in the bulk can trigger theﬂux jump easily.

Figure 8. Final trapped ﬁeld distributions on the bulk surface(z=9.5 mm) for Bex=4.0 T at 65 K; (a) homogeneous model,(b) inhomogeneous model.

supercurrent. The Bt(t) behavior qualitatively reproduced
experimental results, as shown in ﬁgure 1, only in the inhomogeneous model.
Figures 8(a) and (b), respectively, show ﬁnal trapped
ﬁeld distributions on the bulk surface (z=9.5 mm) for
Bex=4.0 T at 65 K using the homogeneous and inhomogeneous models. In the homogeneous model, the magnetic ﬂux
was trapped symmetrically across the whole bulk and the Bt
value at the center of the ring bulk was positive. On the other
hand, in the inhomogeneous model, no magnetic ﬂux wastrapped at the notch area. As a result, the ﬁnal Bt proﬁleexhibited the ‘C-shaped proﬁle’, which is similar to theexperimental one, as shown in the inset of ﬁgure 1. In thisway, the ‘C-shaped proﬁle’ was reproduced only using theinhomogeneous model.
3.3. Flux dynamics and thermal behavior in theinhomogeneous model

To clarify the cause of the ‘C-shaped proﬁle’ in the inhomogeneous model, the ﬂux dynamics and thermal behavior
were investigated. Figure 9(a) shows 1/4 of the top view of
the numerical model, in which the notch is contained in the
periphery. Figure 9(b) shows the time evolution of the trapped
ﬁeld, Bt(t), at the center of the bulk surface (r=0 mm,
z=8.5 mm) and applied pulsed ﬁeld, Bex(t), for Bex=4.0 T
at 65 K. The right vertical axis shows the time evolution of
the temperature, T(t), at the notch surface (r=28 mm,
z=8.5 mm). The Bt value increased rapidly, reached a peakvalue of 2.7 T at t=20 ms, and then gradually decreased
with increasing time. A negative Bt value can be observed,
even at the center of the bulk surface, similar to that at thecenter of the ring bulk (r=z=0 mm), as shown inﬁgure 7(b). The temperature rise at the notch surface
increased signiﬁcantly with increasing Bex, reached around
90 K at t=40 ms, and remained up to t=300 ms due to theextremely low heat diffusion for the ceramic material.
Figure 9(c) shows the time evolution of the trapped ﬁeld
(the color of Bt=2 T includes over 2 T) and temperature
distributions on the bulk surface (z=8.5 mm) during theascending stage of PFM (0 ms<t20 ms). At t5 ms,
the magnetic ﬂux was concentrated mainly around the notch
area, where the Bt value at the center of the bulk was nearlyzero. At the same time, the temperature rise increased locally,

3.4. The influence of notch size and position on the flux jump

To clarify the inﬂuence of the notch size and position on theﬂux jump, the ﬂux dynamics and thermal behavior were
investigated during PFM using the inhomogeneous models.
Figure 10(a) shows the typical time evolution of the trapped
ﬁeld, Bt(t), at the center of the ring bulk (r=z=0 mm) for
Bex=3.1∼4.0 T at Ts=65 K using the inhomogeneousmodel with the ‘inner notch’, as shown in ﬁgure 5. The time
evolution of Bt(t) for Bex lower than 3.5 T was similar to thoseusing the inhomogeneous model with the ‘outer notch’, asshown in ﬁgure 7(b). However, the ﬂux jump took place for
Bex=4.0 T and the trapped ﬁeld reached its peak value of
Bt=2.6 T at t=25 ms. The peak time becomes delayed in
the case of the ‘inner notch’, compared with the ‘outer notch’.
In the descending stage, Bt(t) decreased gradually withincreasing time and eventually reached a negative value of
Bt=−0.1 T.
Figure 10(b) shows the time evolution of the trapped
ﬁeld, Bt(t), at the center of the ring bulk (r=z=0 mm) for6Supercond. Sci. Technol. 33 (2020) 044003

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Figure 9. (a) 1/4 of the top view of the inhomogeneous model, containing the notch area. (b) Time evolution of the trapped ﬁeld, Bt(t), at thecenter of the bulk surface (r=0 mm, z=8.5 mm), applied pulsed ﬁeld, Bex(t), for Bex=4.0 T at 65 K and temperature (right vertical axis),
T(t), on the notch surface (r=28 mm, z=8.5 mm) in the inhomogeneous model. Time evolution of the trapped ﬁeld and temperature
distributions on the bulk surface (z=8.5 mm) in (c) the ascending stage (t=1∼20 ms) and (d) the descending stage (t=40∼300 ms) of
the PFM process.

Bex=4.0∼5.0 T at Ts=65 K using the inhomogeneousmodel with the ‘outer tiny notch’. The Bt value for Bex lower
than 4.5 T increases with increasing Bex. These results are incontrast to the ‘outer notch’ shown in ﬁgure 7(b). These
results suggest that it is more difﬁcult for the magnetic ﬂux topenetrate for the smaller notch. However, for Bex=5.0 T, a
ﬂux jump occurred, where the larger magnetic ﬁeld abruptly

penetrated into the bulk center, and reached its peak value of
Bt=4.0 T at t=17 ms. The peak time becomes delayed forthe ‘outer tiny notch’, compared with the ‘outer notch’. In thedescending stage, Bt(t) decreased drastically with increasingtime and eventually reached a small value of Bt=0.3 T.
Although the Bt value at the bulk center was not a negativevalue, the ﬁnal trapped ﬁeld proﬁle exhibited the ‘C-shaped7Supercond. Sci. Technol. 33 (2020) 044003

T Hirano et al

Figure 10. (a) Time evolution of the trapped ﬁeld, Bt, at the center of the ring bulk (r=z=0 mm) for Bex=3.1∼4.0 T at Ts=65 K andapplied pulsed ﬁeld, Bex(t), for Bex=4.0 T using the inhomogeneous model with the ‘inner notch’. (b) Time evolution of the trapped ﬁeld,
Bt, at the center of the ring bulk (r=z=0 mm) for Bex=4.0∼5.0 T at Ts=65 K and applied pulsed ﬁeld, Bex(t), for Bex=5.0 T using theinhomogeneous model with the ‘outer tiny notch’. (c) Final trapped ﬁeld distributions on the bulk surface (z=0 mm) for Bex=5.0 T at
Ts=65 K using the inhomogeneous model with the ‘outer tiny notch’. The contour of the ring bulk is shown in the inset of ﬁgure.

Figure 11. The mapping of the occurrence of the ﬂux jump for (a) the inner notch, center notch and outer notch shown in ﬁgures 5(a)–(c) at

Ts=65 K. The mapping of the occurrence of the ﬂux jump for (b) the inner tiny notch, center tiny notch and outer tiny notch shown inﬁgures 5(d)–(f) at Ts=65 K.

proﬁle’, as shown in ﬁgure 10(c), because of the incomplete
circulating supercurrent. These results indicate that the ﬂux
jump took place even in the case of the ‘outer tiny notch’.
To understand the role of notch triggering the ﬂux jump,
the critical applied ﬁeld was deﬁned as the value at which theﬂux jump takes place, for the inhomogeneous models with
notches of various sizes and positions, during PFM.
Figure 11(a) shows the mapping of the occurrence of the ﬂux
jump for the inner, center and outer notches shown inﬁgures 5(a)–(c). The critical applied ﬁeld for the ﬂux jump

was 4.0, 4.0 and 3.75 T for the inner notch, center notch andouter notch, respectively. These results suggest that a defectexisting at outer side is easier to trigger a ﬂux jump.
Figure 11(b) shows the mapping of the occurrence of the ﬂux
jump for the inner tiny notch, center tiny notch and outer tiny
notch shown in ﬁgures 5(d)–(f). The critical applied ﬁeld forthe ﬂux jump was 5.0, 5.0 and 4.75 T for the inner, center andouter tiny notch, respectively, which was larger than that for
the larger notches. These results suggest that a large appliedﬁeld is necessary to cause a ﬂux jump in the bulk with smaller8Supercond. Sci. Technol. 33 (2020) 044003

T Hirano et al

Figure 12. Time evolution of (a) the electromagnetic hoop stresses, sq EM (t), and (b) thermal hoop stress, sq therm (t), at the inner surface of thering bulk (r=18 mm, z=0 mm) for Bex=4.0 T at Ts=65 K for the homogeneous and inhomogeneous models.

and inner defects. To prevent a ﬂux jump for the ring bulk,
smaller and/or inner defects in bulk materials are desirablefrom the results of these numerical simulations.
A comment should be made here on the inﬂuence of theother parameters on the ﬂux jump behavior. If the operating
temperature decreases, the critical applied ﬁeld, at which theﬂux jump takes place, should increase because of the increasein Jc. As for the inﬂuence of the rise time of the magnetic
pulse on the ﬂux jump, the critical applied ﬁeld should
increase with increasing rise time of the pulsed ﬁeld becauseof the reduction in the temperature rise. Such inﬂuences on
the ﬂux jump will be investigated in the future.

results suggest that it is highly unlikely that the bulk would
fracture due to the electromagnetic stress during PFM.
Figure 12(b) shows the time evolution of the thermalhoop stress, sq therm (t), at the center of the inner surface of thering bulk (r=18 mm, z=0 mm) for Bex=4.0 T at
Ts=65 K for the homogeneous and inhomogeneous models.
The thermal hoop stress originates from the temperature
distribution in the bulk reinforced by an Al alloy ring during
PFM. The sq therm value in the homogenous model was verysmall due to a low temperature rise. On the other hand, in theinhomogeneous model, a maximum tensile hoop stress of+26 MPa was developed by the ﬂux jump just after the peakof Bex(t) at t=20 ms, which was smaller than the fracture
strength of the bulk by bending tests [41–43]. These numerical simulation results of the electromagnetic and thermalhoop stresses suggest that the ‘C-shaped trapped ﬁeld proﬁle’shown in ﬁgure 1 resulted from the ﬂux jump rather than thebulk fracture. It should be noted that the sudden and large
local temperature rise increasing by several decades may
promote the ﬂux jump. A bulk NMR magnet using stacked
REBaCuO ring bulks has been developed, which was magnetized by FCM [4, 46, 47]. If such a bulk NMR magnet canbe magnetized by PFM in the future, adequate measures toavoid such ﬂux jumps may be necessary to mitigate an
inhomogeneous trapped ﬁeld proﬁle.

3.5. Electromagnetic and thermal hoop stresses during PFM

The cause of the ‘C-shaped proﬁle’ in the experimental results
shown in ﬁgure 1 may result from the bulk fracture during
PFM, in which the circulating supercurrent path was cut off,
in addition to the being the origin of a ﬂux jump. To clarify
the possibility of fracture, the mechanical stress behavior wasinvestigated during PFM using inhomogeneous model withouter notch, as shown in ﬁgure 4(b). Figure 12(a) shows time
evolution of the electromagnetic hoop stress, sq EM (t), at thecenter of the inner surface of the ring bulk (r=18 mm,z=0 mm) for Bex=4.0 T at Ts=65 K for the homogeneous and inhomogeneous models. sq EM (t) starts toincrease negatively (compressive stress) and takes a peakvalue around t∼10 ms, at which the applied pulsed ﬁeld takes
a maximum. After that, sq EM (t) decreases and changes to asmall positive value (tensile stress) for each case, which
results from the change of the direction of the induced
superconducting current. Similar numerical results arereported in [37–40]. The maximum tensile stress in thehomogeneous and inhomogeneous models was +2 and+4 MPa, respectively, which are much smaller than thefracture strength of ∼70 MPa estimated by bending measurements [41–43]. On the other hand, the simulated maximum compressive stress of −30∼−40 MPa is about oneorder of magnitude larger than the maximum tensile stress.
The fracture strength of the REBaCuO bulk for compressive
stress was reported to be at least −200 MPa [44, 45]. These

4. Conclusion
We have investigated the electromagnetic and thermal properties of a REBaCuO ring bulk with an inhomogeneous critical current density, Jc, proﬁle during pulsed ﬁeld
magnetization (PFM) using a numerical simulation andcompared these to a bulk with a homogeneous Jc proﬁle. The
important results and conclusions of this study are summarized as follows.
(1) A notch was introduced in the bulk periphery, which
was assumed as a crack existing in the actual bulk
material. A sudden ﬂux penetration (ﬂux jump) took
place through the notch area, and as a result, a large
temperature rise also took place around this notch.
9Supercond. Sci. Technol. 33 (2020) 044003

T Hirano et alConsequently, the ﬁnal trapped ﬁeld proﬁle wassimulated to be ‘C-shaped proﬁle’, which qualitatively
reproduced our previous experimental results.
(2) The inﬂuence of the size and position of the notch on
the occurrence of the ﬂux jump was investigated. It was
conﬁrmed numerically that the larger and the outernotches, which simulates real imperfections, such as a
crack, void or low Jc region, is easier to cause a
ﬂux jump.
(3) On the other hand, in the homogeneous model, no ﬂux
jump phenomenon was observed for all the cases
studied. These results suggest that an imperfection in
the bulk can be a possible starting point of a ﬂux jump.
(4) The electromagnetic and thermal hoop stresses are
simulated in the ring bulk during PFM. A compressive
electromagnetic stress and a tensile thermal stress wereobserved around the peak of the pulsed magnetic ﬁeld,
both of which were lower than the experimentallymeasured fracture strength of the bulk material reported
in the literature.
(5) We conclude that the experimentally observed ‘Cshaped proﬁle’ results from the ﬂux jump, rather thanthe bulk fracture. The size of the present notch is muchlarger than that of the realistic crack, which must be
investigated in the future.

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Acknowledgments
This research is partially supported by a ‘Development of
Systems and Technologies for Advanced Measurement andAnalysis’ from Japan Agency for Medical Research andDevelopment, AMED and by JSPS KAKENHI Grant Nos.
15K04646 and 19K05240. M D Ainslie would like toacknowledge ﬁnancial support from an Engineering andPhysical Sciences Research Council (EPSRC) Early Career
Fellowship, EP/P020313/1. All data are provided in full in
the results section of this paper.

ORCID iDs
Tatsuya Hirano https://orcid.org/0000-0003-1658-914X
Hiroyuki Fujishiro https://orcid.org/0000-00031483-835X
Tomoyuki Naito https://orcid.org/0000-0001-7594-3466
Mark D Ainslie https://orcid.org/0000-0003-0466-3680

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