Superconductor Science and Technology
Supercond. Sci. Technol. 27 (2014) 065019 (7pp)

doi:10.1088/0953-2048/27/6/065019

Numerical simulation of the trapped ﬁeld in
MgB2 bulk disks magnetized by ﬁeld cooling
Hiroyuki Fujishiro, Tomoyuki Naito and Takafumi Yoshida
Faculty of Engineering, Iwate University, 4-3-5 Ueda, Morioka 020-8551, Japan
E-mail: fujishiro@iwate-u.ac.jp
Received 12 February 2014, revised 21 March 2014
Accepted for publication 1 April 2014
Published 15 May 2014
Abstract

We performed a numerical simulation of the trapped ﬁeld properties of MgB 2 bulk disks withdifferent superconducting characteristics magnetized by ﬁeld cooling. The temperaturedependence of the trapped ﬁeld Bz (T ), and the time dependence of the local ﬁeld Bz (t ) andtemperature T (t) were calculated, and compared with experimental results. Since the measured
critical current density Jc (T , B) was used in the simulation, the results of the simulation
reproduce the experimental results well. These results suggest that the value of Bz and the trapped
ﬁeld proﬁle can be estimated precisely for bulks of any shape and under any conditions, such as
the operating temperature and applied ﬁeld, using the measured Jc (T , B) characteristics. The
diameter and thickness dependence of Bz was also calculated.
Keywords: MgB2, ﬁeld-cooled magnetization, simulation(Some ﬁgures may appear in colour only in the online journal)
magnetization (FCM). At present, the maximum trapped ﬁeld
Bz reported has been 2.25 T at 15 K on a single MgB 2 bulk [5]and 3.14 T at 17.4 K in a bulk pair [6]. Pulsed-ﬁeld magnetization (PFM) has also been investigated to magnetize thesuperconducting bulks because of its inexpensive and mobile
setup without the need for a superconducting coil magnet[7, 8]. We have investigated the trapped ﬁeld Bz by PFM on
MgB 2 bulks fabricated by several techniques [9, 10]. However, the maximum trapped ﬁeld was as low as 0.71 T at 15 K
for a single MgB 2 bulk fabricated by a capsule method [10].
The trapped ﬁeld characteristics of the superconducting
bulks vary widely depending on the temperature and magnetic ﬁeld dependence of the critical current density Jc (T , B),
which results from the strength and density of the vortex
pinning centres. The Jc (T , B) characteristics are usually estimated by M − H hysteresis curves for small pieces cut fromthe bulk. The macroscopic Jc characteristics are evaluated by
the trapped ﬁeld Bz (T ) by FCM, because qualitatively, bulkswith higher Jc are believed to trap a higher Bz . For REBaCuO
bulks, the geometry effect on the trapped ﬁeld by FCM
was studied numerically using a sand-pile model and a
Biot–Savart law, and was compared with the experimental
results [11, 12]. The trapped ﬁeld Bz and the proﬁles were

1. Introduction
A superconducting bulk magnet using REBaCuO (RE: rareearth element or Y) material is one of the typical models forpractical applications, such as sputtering cathodes, magnetic
separation, drug delivery systems and so on [1], which produces quasi-permanent magnets on the order of several Tesla.
However, it is difﬁcult to fabricate single-domain REBaCuO
bulk crystals larger than 100 mm in diameter because of the
difﬁculties with crystal growth. MgB 2 bulk also has potential
as a quasi-permanent magnet, and is attractive in a number ofways as it is low-cost, rare-earth-free and light-weight, andhas a homogeneous trapped ﬁeld distribution, which is inclear contrast to the distribution of an REBaCuO bulk magnet. Since the problem of weak links at the grain boundaries
can be ignored in the MgB 2 polycrystal due to their longcoherence length, ξ [2], better and larger polycrystalline
MgB 2 bulk magnets can be achieved below the transition
temperature Tc = 39 K. MgB 2 bulk magnets are expected to be
applicable in magnetically levitated trains (MAGLEVs) [3]and wind power generators [4] using cryocooler cooling or as
part of a liquid hydrogen economy in the near future. Several
groups have fabricated MgB 2 bulks using different techniques, and reported the trapped ﬁeld using ﬁeld-cooled0953-2048/14/065019+07$33.00

1© 2014 IOP Publishing Ltd Printed in the UK

Supercond. Sci. Technol. 27 (2014) 065019

H Fujishiro et alof − 0.22 T min−1 and the time dependences of the trapped
ﬁeld Bz (t ) and the temperature T (t) were recorded. The
temperature dependence of the trapped ﬁeld Bz (T ) was
measured by a ‘sweep method’, that is, after the FCM procedure at Ts , the bulk was warmed slowly with a heating rateof 0.1 K min−1 and the trapped ﬁeld Bz (T ) was measured.
Figure 2 shows the time dependences of the local ﬁeld Bz ( t )and temperature T (t) of bulk-A during FCM at Ts = 13 K . As
the external ﬁeld Bex decreases, Bz also decreases and reaches aﬁnal value. The temperature T gradually increases about
0.8 K, reaches a maximum at the time at which Bex becomes
zero, and decreases with increasing time. Figure 3 shows theexperimental results for Bz (T ) for both bulks. Bz (T ) for bulkA was larger than that of bulk-B. The maximum Bz for bulk-Aand bulk-B was 2.5 T at 13 K and 1.9 T at 16 K, respectively.
After the magnetizing experiments, the bulks were cut
and small pieces about 1 × 1 × 2 mm 3 in size were preparedfor magnetization curve M(H) measurements at 10 and 20 K
under a magnetic ﬁeld up to μ0 H = 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 the extended Bean model, J c = 20 ΔM/a(1-a/3b),where a and b are the dimensions of the plane of the sample
perpendicular to the applied ﬁeld and ΔM is the width of theM(H) loop. Figure 4 depicts the estimated Jc − B characteristics for both bulks at 10 and 20 K.

Table 1. Speciﬁcation of the measured MgB 2 bulks.

Bulk

Diameter(d)Thickness(tB )Bulk-A

26 mm

6.5 mm

Bulk-B

30 mm

10 mm

Bzmax

Tc

Mass
density

2.5 T
at
13 K
1.9 T
at
16 K

38.5 K95%

38.8 K50%

calculated at a ﬁxed temperature using the ﬁxed Jc (B) value,
which suggested that the results of the simulation also
reproduced the experimental results. However, the relationship between Jc (T , B) and Bz (T ) has not been investigatedquantitatively using numerical techniques. For the PFM process, the numerical simulation of the ﬂux dynamics and heatgeneration for REBaCuO and MgB 2 bulks has been investigated to optimize the PFM procedure and to enhance the Bz
value [9, 13, 14].
In the present paper, we perform the numerical simulation of the trapped ﬁeld properties by FCM for homogeneous
MgB 2 bulk disks with different trapped ﬁeld characteristics.
The temperature dependence of the trapped ﬁeld Bz (T ), thetrapped ﬁeld proﬁle, and the temperature rise of the bulk were
calculated, and compared with experimental results. The
diameter and thickness dependences of Bz for the MgB 2 bulks
are also calculated.

3. Modeling and numerical simulation
Based on our experimental setup for FCM shown in ﬁgure 1,
the framework for the numerical simulation was constructed[13]. In the model analysis, we include a thin spacing plate
between the bulk and the cold stage with thermal conductivity
κCT from 5 × 10 −4 to 5 × 10 −1 W K−1 m −1, which represents
both the cooling power of the refrigerator and the thermal
contact. A static magnetic ﬁeld of Bex = 5 T was applied to thebulk above Tc using a solenoid magnet of inner diameter
120 mm, outer diameter 150 mm and height 100 mm, and thebulk was then ﬁeld-cooled to Ts . A descending ﬁeld Bex ( t ) wasapplied at a rate from − 0.052 to − 5.2 T min−1. The physicalphenomena occurring during FCM are described using thefundamental electromagnetic and thermal equations in axisymmetric coordinates.

2. Experimental procedure and results
Two MgB 2 superconducting bulks (hereafter, bulk-A andbulk-B) with different superconducting characteristics wereused in this study. Bulk-A was fabricated by a hot isostaticpressing method; the precursor pellet was prepared by cold
isostatic pressing at 196 MPa using mixed powders of Mg(99.5% in purity, ⩽180 μm in grain size) and B (99% in
purity, 300 mesh in grain size) with a molar ratio of 1.0:2.0.
The pellet was sealed under vacuum in a stainless steel container and was heated at 900 °C for 3 h at an isostatic pressureof 98 MPa. Bulk-B was fabricated by a capsule method,which is described in detail [15]. The speciﬁcation of thebulks and the maximum trapped ﬁeld by FCM are summarized in table 1.
A schematic view of the experimental setup for FCM
is presented in ﬁgure 1. The MgB 2 bulk disk with a stainlesssteel ring was tightly mounted on the cold stage of a
Gifford–McMahon cycle helium refrigerator. A Hall sensor(F W Bell, BHT 921) and a Cernox thermometer wereadhered to the centre of the bulk surface. A static magnetic
ﬁeld of Bex = 5 T was applied to the bulk above Tc using acryo-cooled superconducting solenoid magnet (JASTEC
JMTD-10T100) and the bulk was then ﬁeld-cooled to Ts .
Thereafter, the applied ﬁeld was swept down to zero at a rate⎤
∂ ⎡ν ∂∂ ⎛ ∂A ⎞⎜ν
⎟ = J + JrA) ⎥ +(
0⎢⎣⎦
∂r r ∂r∂z ⎝ ∂z ⎠(1)∂T∂ ⎛ ∂T ⎞1 ∂ ⎛ ∂T ⎞
⎜ rκ
⎟ +
⎜κ
⎟ + Q.
=∂tr ∂r ⎝ ∂r ⎠ ∂z ⎝ ∂z ⎠

(2)

ρC

Here, A represents the magnetic vector potential, J 0 is theforced current density or coil current density and J denotes thesuperconducting current density or induced current density. In
equation (2), κ, C and ρ, respectively, denote the thermal
conductivity, the speciﬁc heat and the mass density2Supercond. Sci. Technol. 27 (2014) 065019

H Fujishiro et al

Figure 1. Schematic view of the experimental setup and the basis for the numerical model of FCM.

Figure 2. Experimental results of the time dependence of the local

ﬁeld Bz ( t ) and temperature T (t) at the centre of the surface of bulk-A
during FCM at Ts = 13 K, when the external ﬁeld Bex ( t ) was
decreased at a rate of −0.22 T/min from 5 to 0 T.

Figure 4. The estimated Jc − B characteristics for both bulks at 10
and 20 K. The solid lines show the ﬁtting curves used in thesimulation (see text).

Figure 3. Experimental results of the trapped ﬁeld Bz (T ) for bothbulks magnetized by FCM.

Figure 5. Thermal conductivity and speciﬁc heat of the MgB 2 bulks
used in the simulation.

3

Supercond. Sci. Technol. 27 (2014) 065019

H Fujishiro et al( = 2590 kg m −3) of the MgB 2 bulks. Figure 5 shows thetemperature dependence of κ and C for the MgB 2 bulks usedin the simulation, in which κ(T) of both bulks was measured
and C(T) was referred from [16].
The J value is given as
J = σE = −σ

∂A
,
∂t

Table 2. Fitting parameters for the Jc − B characteristics ofeach bulk.

(3)

Ts

Jc0 (A/m 2 )B0 (T)a in
eqn (5)α (A/m 2 )in eqn (6)Bulk-A

10 K
20 K
10 K
20 K

6.01 × 109
4.31 × 109
2.36 × 109
1.71 × 109

1.50
1.05
1.30
0.95

1.50
1.50
1.40
1.50

6.70 × 109

Bulk-B

where E is the electric ﬁeld and σ the electric conductivity.
The power-n model (n = 100) was used to describe thenonlinear E-J characteristic of the superconducting bulk:⎛ J ⎞nE = Ec ⎜ ⎟ ,⎝ Jc ⎠

Bulk

2.70 × 109(4)

where J c is the critical current density and E c (1 × 10 −6 V m −1)is the reference electric ﬁeld. In equation (2), heat generation
Q in the bulk is given as Q = JE.
The magnetic ﬁeld dependence of critical current density
Jc (B) for each bulk at 10 and 20 K was ﬁtted, as shown inﬁgure 4, using the following relation [17],⎡ ⎛ B ⎞a ⎤Jc ( B ) = Jc0 exp ⎢ −⎜ ⎟ ⎥ ,⎢⎣ ⎝ B0 ⎠ ⎥⎦(5)

where B0 and a are the ﬁtting parameters, which are summarized in table 2. The temperature dependence of Jc0 underzero ﬁeld is described by the following equation,32⎧⎛ T ⎞2 ⎫⎪⎪⎬−Jc0 = α ⎨1⎜ ⎟⎪ ,⎪⎝ Tc ⎠ ⎭⎩(6)

where Tc is the critical temperature (=39 K) and α is a constantrepresenting Jc at 0 K. Iterative calculations were performed toobtain convergence of σ in the superconductor at each time
step. Commercial software, Photo-Eddy, combined with
Photo-Thermo (Photon Ltd, Japan), was used for analyzing
the combined problem of electromagnetic ﬁelds and heatdiffusion using the ﬁnite element method.
Figure 6. (a) The results of the simulation of the trapped ﬁeld Bz(z = 0 mm) at 10 and 20 K for each bulk. Solid lines show theexperimental results. (b) Cross-sections of the trapped ﬁeld proﬁles
at 20 K for each bulk at z = 0 (bulk surface) and 3 mm.

4. Results of simulation and discussion
4.1. Simulation under experimental conditionsFirstly, the results of the simulation during FCM arepresented under the experimental conditions with thedescending ﬁeld of − 0.22 T min−1. Figure 6(a) shows thetrapped ﬁeld at the centre of the bulk surface Bz (z = 0 mm) as
a function of temperature at 10 and 20 K for each bulk. In thisﬁgure, the experimental results for bulk-A and bulk-B are also
shown. The results of the simulation reproduce the experimental results well. The slight difference between experiments and simulation may result from the inhomogeneity ofJc (B) in the bulk and/or the temperature distribution in
the bulk.

Figure 6(b) shows the cross-section of the trapped ﬁeld
proﬁle Bz ( r ) at z = 0 mm (bulk surface) and 3 mm. During
FCM, the magnetic ﬁeld in the bulk decreased withdecreasing external ﬁeld Bex ( t ) and a bell-shaped trapped
ﬁeld proﬁle materializes, which may originate from thesteep magnetic ﬁeld dependence of Jc . These results are incontrast with those using Beanʼs critical state model, which
results in a conical proﬁle. At z = 3 mm , the absolute valueof Bz decreases and the proﬁles become ﬂatter. In this way,
the trapped ﬁeld and the proﬁles can be estimated at any
4Supercond. Sci. Technol. 27 (2014) 065019

H Fujishiro et al

Figure 7. (a) Maximum temperature rise ΔT max and ﬁnal trapped ﬁeld

Figure 8. (a) Time dependence of the temperature T (t) forBz (z = 0 mm) as a function of the thermal contact conductivity κTC at
20 K for bulk-A. (b) Time dependence of temperature T (t) and local
ﬁeld Bz ( t ) (z = 0 mm) at the centre of the bulk surface forκTC = 5 × 10−3 W K−1 m −1 at 20 K.

descending Bex rates of −0.52, −0.22 and −0.10 T min−1. (b)Maximum temperature rise ΔT max and ﬁnal trapped ﬁeld as afunction of the descending Bex rate during FCM.

(ﬁnal) slightly decreases with decreasing κTC , which isa reasonable result. Using ﬁgure 7(b), κTC in the present
experimental setup, which represents both the coolingpower of the refrigerator and the thermal contact, can beroughly estimated to be 3 × 10−3 W K−1 m −1.
Figure 8(a) presents the results of the simulationof the time dependence of the temperature T (t) for various
descending rates of Bex (t ) during FCM for bulk-A at 20 K
under the condition of κTC = 5 × 10−3 W K−1 m −1. Figure 8(b)shows the maximum temperature rise ΔT max and ﬁnal
trapped ﬁeld Bz (ﬁnal), as a function of the descending rate of
Bex . ΔT max increases with an increasing rate of descending Bex
and, as a result, the value of Bz (ﬁnal) is decreased slightly. In
this way, using the numerical simulation, the maximum
temperature rise ΔT max and ﬁnal trapped ﬁeld Bz (ﬁnal) can be
estimated for various κTC and descending rates of Bex (t ).
However, ΔT max is relatively small due to the small C andhigh κ values of the MgB 2 bulk compared with those for
REBaCuO bulks [19], and the inﬂuence of the temperaturerise on the Bz (ﬁnal) value is also quite small during FCM.

position ( z, r ) and at any temperature Ts using thesimulation.
Figure 7(a) depicts the results of the simulation of
the time dependence of the temperature T (t) and the local
ﬁeld Bz ( t ) at the centre of the bulk surface during FCM
for bulk-A at 20 K for a spacing plate with thermal conductivity κTC = 5 × 10−3 W K−1 m −1. T (t) gradually increases
with decreasing Bex (t ), takes a maximum of 20.5 K justafter the time at which Bex becomes zero, and recovers towards
the initial temperature as time increases. The local ﬁeld Bz (t )decreases according to the decrease in Bex (t ) for t ⩽ 500 s,
but with increasing time, Bz (t ) approaches a ﬁnal value ofabout 2 T according to pinning effects, which are reﬂected in
the Jc − B characteristics of the bulk. The results of thesimulation qualitatively reproduce the experimental results
shown in ﬁgure 2. Even in the FCM process, a small temperature rise occurs due to the slow movement of magnetic
ﬂux in the superconductor, which is much smaller than thatseen during the PFM process with a time scale of milliseconds [18].
The temperature rise during FCM is supposed to varylargely depending on the κTC value used in the simulation.
Figure 7(b) shows the maximum temperature rise ΔT max andﬁnal trapped ﬁeld Bz (ﬁnal), as a function of κTC . ΔT max
increases adiabatically with decreasing κTC . As a result, Bz

4.2. Thickness and diameter dependence of the trapped field

Using the numerical simulation, the magnitude and proﬁle ofthe trapped ﬁeld can be estimated for bulks with any
5Supercond. Sci. Technol. 27 (2014) 065019

H Fujishiro et al

Figure 9. Results of the simulation of the thickness dependence ofthe trapped ﬁeld Bz (z = 0 mm) at the centre of MgB 2 bulk disks of
26 mm and 60 mm in diameter.

dimension without experiments. In this subsection, thethickness and diameter dependence of the trapped ﬁeld of thebulk disk is estimated, using the Jc − B proﬁle of the bulk-A
in the simulation, with the temperature of the bulk ﬁxed at
Ts = 20 K because the temperature rise during FCM is fairly
small, as shown in the previous section. Figure 9 shows the
thickness (tB ) dependence of the trapped ﬁeld Bz (z = 0 mm) at
the centre of MgB 2 bulk disks of diameter d = 26 mm and
60 mm. Bz (z = 0 mm) increases with increasing thickness ofthe bulk for tB ⩽ 20 mm, but is independent of the value of tB
for thicker samples.
Figure 10(a) presents the diameter (d) dependence of thetrapped ﬁeld Bz (z = 0 mm) at the centre of an MgB 2 bulk diskof tB = 6.5 mm. Figure 10(b) shows the cross-section of thetrapped ﬁeld proﬁle Bz ( r ) at z = 0 mm (bulk surface) of thebulk for various diameters. Bz (z = 0 mm) gradually increases
with increasing diameter of the bulk, but this effect of diameter on the value of Bz is not so prominent; for example, Bz(z = 0 mm) for the d = 100 mm bulk is only 24% larger thanthat calculated for the d = 20 mm bulk, even though thediameter has increased ﬁve-fold. In ﬁgure 10(a), the experimental results for bulks with identical Jc − B characteristics
are also shown. The experimental results coincide with theresults of the simulation. The diameter dependence of Bz
comes from the steep Jc − B characteristic, that is, Jc reduces
signiﬁcantly with increasing applied magnetic ﬁeld. For
Beanʼs model, Bz increases linearly with increasing diameterof the bulk. From these results, in the MgB 2 bulk, increasing
the tB and d values is not a valuable solution to enhance the Bz
value, similar to REBaCuO bulks. Also for MgB 2 bulks, theimprovement of the Jc − B characteristic is only a possible
solution to enhance the Bz value. If the Jc − B at the mid-ﬁeld
region is enhanced and does not decay rapidly under anexternal ﬁeld at lower temperature (e.g. 10 K), higher trapped
ﬁelds in larger diameter samples can be obtained.

Figure 10. (a) Diameter dependence of the trapped ﬁeld Bz(z = 0 mm) at the centre of the MgB 2 bulk disk of thickness 6.5 mm.
The experimental results are also shown. (b) Cross-section of thetrapped ﬁeld proﬁle Bz ( r ) at z = 0 mm (bulk surface) of bulks withvarious diameters.

5. Summary
A numerical simulation of electromagnetic and thermal
properties has been performed for cryo-cooled MgB 2 bulk
discs magnetized by ﬁeld cooling, and the simulation results
are compared with the experimental results. The importantresults and conclusions obtained from this study are summarized as follows.
(1) Since measured critical current density Jc (T , B) proﬁleswere used in the simulation, the results of the simulation
reproduce the experimental results well. These results
suggest that the value of Bz and the trapped ﬁeld proﬁlecan be estimated precisely for the bulk of any shape
under various conditions, such as operating temperature
and applied ﬁeld, using the measured Jc (T , B)
characteristics.
(2) The diameter and thickness dependence of Bz was
calculated for MgB 2 bulks of various diameters andthicknesses. The diameter of the bulk did not have asigniﬁcant effect on the value of Bz ; for example, Bz(z = 0 mm) for d = 100 mm bulk is only 24% larger thanthat for d = 20 mm bulk, even though the diameter has
6Supercond. Sci. Technol. 27 (2014) 065019

H Fujishiro et al

increased ﬁve-fold. Increasing the thickness of the MgB 2
bulk is not very effective in enhancing the value of Bz .
For MgB 2 bulks, improving the Jc − B characteristic is apossible way to enhance the Bz value.

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Acknowledgments
This work was supported in part by JSPS KAKENHI grant
number 23560002, JST A-STEP (AS232Z02579B), andHitachi Metals Materials Science Foundation.

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