Superconductor Science and Technology
Supercond. Sci. Technol. 28 (2015) 095018 (8pp)

doi:10.1088/0953-2048/28/9/095018

Drastic improvement of the trapped ﬁeld
homogeneity in a superconducting hollow
bulk by the insertion of a high-Jc
superconducting cylinder for NMR bulk
magnets
Hiroyuki Fujishiro1, Yoshitaka Itoh2, Yousuke Yanagi2 andTakashi Nakamura3
1

Faculty of Engineering, Iwate University, 4-3-5 Ueda, Morioka 020-8551, JapanIMRA Material R&D Co. Ltd, 2-1 Asahi-machi, Kariya 448-0032, Japan
3
RIKEN, 2-1 Hirosawa, Wako 351-0198, Japan2

E-mail: fujishiro@iwate-u.ac.jp
Received 17 June 2015, revised 16 July 2015
Accepted for publication 24 July 2015
Published 18 August 2015
Abstract

Three-dimensional (3D) numerical simulation has been performed for the ﬁeld-cooledmagnetization (FCM) process of REBaCuO superconducting bulk cylinders applicable to thebulk magnet for a 200 MHz (4.7 T) nuclear magnetic resonance (NMR). The inﬂuence of theposition and size of the defect in the bulk on the inhomogeneity of the trapped ﬁeld in the hollow
space has been investigated. The trapped ﬁeld proﬁle changes depending on the position and thesize of the defect, and the inhomogeneity of the trapped ﬁeld can be drastically improved byinserting a superconducting thin cylinder with a critical current density (Jc2) higher than that of
the bulk cylinder (Jc1). The superconducting current ﬂows mainly on the outermost surface of theinserted thin cylinder to compensate for the inhomogeneous trapped ﬁeld caused by the defect.
The insertion of the superconducting thin cylinder with a higher Jc2 is signiﬁcantly effective forthe practical realization of a compact NMR bulk magnet.
Keywords: numerical simulation, superconducting bulks, NMR magnet, trapped ﬁeld
(Some ﬁgures may appear in colour only in the online journal)
1. Introduction

and cryogen-free NMR spectrometer with a medium resolution of 200 MHz (4.7 T)–270 MHz (6.3 T) has been also
desired by many laboratories in both companies and universities. This is because a large superconducting magnet
system with multi-layered superconducting coils cooled byliquid helium is necessary in order to enhance the homogeneity of the magnetic ﬁeld.
The REBaCuO bulk superconductor (RE: a rear earth
element and Y), which can trap a higher magnetic ﬁeld over
17 T [2, 3], can form a uniform trapped ﬁeld in the space ofthe bulk cylinder, because the superconducting persistentcurrent can ﬂow through the bulk using ﬁeld-cooled

The high performance of a nuclear magnetic resonance(NMR) spectrometer has been investigated recently and an
NMR superconducting magnet with a record high ﬁeld of1020 MHz (24.0 T) has been developed [1]. In superconducting magnets used in NMR spectrometers, a simultaneous ultra-high special homogeneity and stability of themagnetic ﬁeld, in addition to an increased magnetic ﬁeld
strength, is critical. The NMR spectrometer is usually used forthe research and development of synthetic organic chemistry,
as it is capable of determining molecular structure. A compact
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1© 2015 IOP Publishing Ltd Printed in the UK

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H Fujishiro et al

magnetization (FCM). In 2007, a compact and cryogen-free
NMR magnet was developed for the ﬁrst time using annularbulk superconductors [4]. The bulk magnet, consisting of two
SmBaCuO bulks with O.D. of 36 mm and I.D. of 7 mm, wascooled by a cryocooler and energized at 3 T (128 MHz); amagnetic ﬁeld homogeneity of 2400 ppm was achieved in thef 1.2 mm×5.0 mm cylindrical space with a magnetic stability lasting as long as 4 months. The ﬁrst NMR spectrum of1
H was also observed using this bulk magnet. In 2011, a new
bulk magnet using six annular EuBaCuO bulks with O.D. of
60 mm, I.D. of 16 mm and height of 70 mm was developed
for the 4.7 T (200 MHz) NMR spectrometer. The 1H NMR
spectra of toluene with a full width at half of the maximum(FWHM) of 0.4 ppm (80 Hz) were observed with a 10 mmdiameter bore at room temperature [5]. The ﬁrst magnetic
resonance imaging (MRI) system was also developed using
annular EuBaCuO bulks [6]. A magnetic ﬁeld homogeneity of
37 ppm was achieved in a f 6.2 mm×9.1 mm cylindricalspace with ﬁrst order shimming, which was performed byadjusting the offset current of the three axes’ gradient coils.
Using the MRI system, a clear three-dimensional (3D) MR
image of a chemically ﬁxed mouse fetus was observed.
Recently, a visualization technique for a precise magnetic
ﬁeld distribution in the bulk hole magnet, which uses an MRIbased method and would be useful for developing highquality superconducting bulk magnets, has been proposed [7].
Using the top-seeded melt growth (TSMG) method,
single-grain REBaCuO bulk superconductors can be grown,
resulting in the formation of growth sector boundaries (GSBs)
with a higher critical current density Jc and growth sector
regions (GSRs) with a relatively lower Jc [8]. As a result, afour-fold trapped ﬁeld inhomogeneity necessarily exists in
the hole of the bulk ring. A superconducting bulk is a composite material, in which RE2BaCuO5 (RE211) non-superconducting particles and Pt and Ag particles are embedded in
the matrix of c-axis-oriented REBa2Cu3O7 (RE123) superconducting phase. Voids and crystal defects such as twinboundaries are also created during the crystal growth. An
inhomogeneous distribution of the secondary phases results in
an inhomogeneous supercurrent ﬂow in the bulk. As a result,
the trapped ﬁeld inhomogeneity exists in the superconductors
themselves. It is necessary to understand how the size andposition of these secondary phases inﬂuence the inhomogeneity of the trapped ﬁeld. We refer to these non-superconducting secondary phases as ‘defects’.
A recent proposal, in which a cylinder wrapped by hightemperature superconducting (HTS) tape with a higher Jc isinserted in the bulk cylinder, was experimentally conﬁrmed tobe effective in reducing the inhomogeneity of the trapped
ﬁeld in the hollow space [9]. The numerical simulation
technique is a valuable tool to understand these problems.
However, there are no reports numerically demonstrating the
problems concerning the effect of the HTS cylinder. We haveconstructed a numerical model for the magnetizing processes
in the superconducting bulk disk during FCM and pulsed ﬁeld
magnetization (PFM) [10, 11].
In this paper, we performed a 3D simulation during FCM
on the bulk cylinder for the application of a 200 MHz NMR

spectrometer. The inﬂuence of the inhomogeneous Jc distribution caused by the defects on the trapped ﬁeld proﬁle wasinvestigated. The drastic improvement on the homogeneity ofthe trapped ﬁeld due to the insertion of a superconducting thin
cylinder was discussed.

2. Modeling and numerical simulation
A 3D model of the REBaCuO bulk cylinder using the
200 MHz NMR bulk magnet was constructed. Figure 1 shows
the framework of a half cylinder model for the numericalsimulation for FCM. The REBaCuO bulk cylinder has an O.
D. of 60 mm, an I.D. of 28 mm and a height of 120 mm. The
magnetizing coil for FCM (O.D. of 150 mm, I.D. of 100 mm
and an inﬁnite height) was placed and a static magnetic ﬁeldof Bex=4.7 T was applied to the bulk cylinder above thesuperconducting transition temperature Tc=90 K. Then thebulk cylinder was cooled to Ts=50 K and the static magnetic
ﬁeld was decreased to zero at a rate of −0.0235 T min−1. The
physical phenomena that occurred during FCM are describedusing fundamental electromagnetic equations. The details of
the numerical simulation are described elsewhere [10, 11].
Brieﬂy, the following power-n model (n=100) was used todescribe the nonlinear E-J characteristic of the superconducting bulk,⎛ J ⎞nE = Ec ⎜ ⎟ ,⎝ Jc ⎠(1 )where Jc is the critical current density and Ec (=10−4 V m−1)isthereferenceelectric
ﬁeld.
The
constant
Jc=Jc1=4.8×108 Am−2 , which is a typical Jc in the
GdBaCuO bulk at Ts=50 K, was assumed for the bulk
cylinder [12]. For simplicity, the Jc is independent of any
applied magnetic ﬁeld, such as a Bean’s model, in thefollowing [13],
dB
dt

(2 )

rotB = m 0 J .

(3 )rotE = -According to equation (2), the electric ﬁeld E is generated in the bulk in the descending process of the external
magnetic ﬁeld. As a result, the current J ﬂows in the bulk
according to equation (1), and the magnetic ﬁeld gradienttakes place according to equation (3).
Photo-Eddy commercial software (Photon Ltd, Japan)
was used to analyze the electromagnetic ﬁeld using the ﬁnite
element method. Iterative calculations were performed toobtain the electromagnetic ﬁeld in the superconductor at eachtime step. In the numerical analysis, the bulk cylinder was
equally divided into 20 elements along the circumferential
direction and 16 elements along the radial direction. Alongthe z direction, a mesh was created as shown in ﬁgure 1. In thehollow space within x=y=±5 mm, a mesh was createdwith 1 mm spacing.
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H Fujishiro et al

Figure 1. (a) Half model of the framework for the numerical simulation of FCM, in which the dimension of the bulk cylinder and the positionof the defect-A and defect-B are shown. (b) The size and position of the ‘defect-Ain (or defect-Bin)’ and ‘defect-Aout (or defect-Bout)’. (c) The
superconducting (SC) thin cylinder inserted inside the bulk cylinder.

was changed from Jc=Jc2=4.8×104 to 4.8×1010 Am−2.
The improvement to the magnetic ﬁeld proﬁle in the hollow
space was investigated along the x- and z-directions.

Two types of unrealistically large defects (defect-A anddefect-B) with a Jc of zero were introduced in the bulk
cylinder in order to emphasize the inﬂuence of the defects on
the inhomogeneity of a magnetic ﬁeld, in addition to a uniform bulk cylinder without any defects. The positions of both
defects are shown in ﬁgure 1, where defect-A is located at thecentre (z=0 mm), which can be understood as the inﬂuenceof the defect near the centre on the inhomogeneity of thetrapped ﬁeld. Defect-B is located at the top surface(z=60 mm) of the bulk cylinder, which can be clariﬁed as
the inﬂuence of the defect far from the centre. The dimensionsof the defects are 2 elements (18 deg.) along the circumferential direction, 15 elements (15 mm) along the radial direction and 8 elements (8 mm) in thickness. Note that the
innermost element between x=14 and 15 mm remains as
superconducting bulk, through which a superconducting
current barely ﬂows along the circumferential direction. In
order to investigate the inﬂuence of defect size and theposition along the x-direction, two types of defects were also
prepared for each defect; as shown in ﬁgure 1(b), ‘defect-Ain’and ‘defect-Bin’ have 8 elements along the radial direction
from x=15 to 23 mm and ‘defect-Aout’ and ‘defect-Bout’
have 8 elements from x=23 to 30 mm.
In order to clarify the effect of inserting another superconducting cylinder inside the bulk cylinder, a superconducting
thin cylinder with an O.D. of 27 mm, I.D. of 26 mm and heightof 120 mm was inserted in the hollow bulk, as shown inﬁgure 1(c). The Jc value of the superconducting thin cylinder

3. Results of numerical simulation
3.1. Influence of the position and size of a defect on theinhomogeneity of the trapped field

Figures 2(a) and (b) present the cross-sections of the macroscopic trapped ﬁeld proﬁle along the x-direction at y=z=0
and along the z-direction at x=y=0 for the uniform,
defect-A and defect-B bulks, respectively. In ﬁgure 2(a),along the x-direction, a trapezoidal Bz proﬁle can be obtainedfor each bulk, which suggests that the applied ﬁeld of 4.7 T to
the bulk cylinder with Jc1=4.8×108 Am−2 is insufﬁcientfor full magnetization. For the defect-A bulk, the chipping in
the Bz proﬁle along the +x-direction can be seen asymmetrically at x=+15 mm in the vicinity of the defect-A. On theother hand, the Bz-x proﬁle for the defect-B bulk shows asimilar symmetric proﬁle to that of the uniform bulk because
the position is far from the defect-B. In ﬁgure 2(b) along thez-direction, the symmetric Bz-z proﬁle can be obtained for
defect-A as well as for the uniform bulks. However, a steep Bz
decrease can be seen at z>+30 mm in the defect-B bulk, as
the defect-B affects the trapped ﬁeld proﬁle near the defect.
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Figure 2. The cross-sections of the macroscopic trapped ﬁeld proﬁlefor the uniform, defect-A and defect-B bulks (a) along the x-directionat y=z=0 and (b) along the z-direction at x=y=0.

Figure 3. Trapped ﬁeld deviation ΔBz around the central region (a)along the x-direction (−4x4 mm) at y=z=0 and (b) along
the z-direction (−5z5 mm) at x=y=0 in the bulk-hole forthe defect-A bulk and uniform bulk. The results of the defect-Ain anddefect-Aout bulks are also shown.

Figures 3(a) and (b) show the microscopic trapped ﬁeld
deviation ΔBz around the central region along the x-direction(−4x4 mm) at y=z=0 and along the z-direction (−5z5 mm) at x=y=0 in the hollows of the uniform anddefect-A bulks, respectively. ΔBz is deﬁned by (Bz-Bz(0))/Bz(0), where Bz(0) shows the Bz value at the centre of thehollow (x=y=z=0). The ΔBz of the defect-A bulk wasvery large along both the x- and z-directions. Although the ΔBz
for the uniform bulk should be zero in principle, the obtained
ΔBz was within 5 ppm along the x- and z-directions due to theerrors derived from signiﬁcant digits in the EFM calculation.
The trapped ﬁeld deviation ΔBz of the defect-Ain anddefect-Aout bulks is also shown in ﬁgure 3. It was found that theΔBz values of the defect-Ain and defect-Aout became smaller
due to the decrease of the defect size. The defect-Aout, locatedfarther from the center has a smaller ΔBz value than that ofdefect-Ain. The relationship between the size and position of thedefect and the inhomogeneity of the trapped ﬁeld proﬁle cannot
be easily explained, since it is a product of the integration of the
3D current distribution. However, qualitatively speaking, thecurrent distribution becomes inhomogeneous around thedefects, where a current density nearly identical to Jc1 might
ﬂow, even on the inner side of the bulk cylinder.

Figures 4(a) and (b), respectively, depict the microscopic
trapped ﬁeld deviation ΔBz around the central region along thex-direction (−4x4 mm) at y=z=0 and along the zdirection (−5z5 mm) at x=y=0 in the hole of theuniform and defect-B bulks. The results for the uniform bulk
were also reused. The ΔBz for the defect-B bulk was small
along the z-direction, while the gradient of the Bz-x relation was
the opposite. This is in clear contrast with those for the defectA. The deviation ΔBz of the defect-Bin and defect-Bout bulks isalso shown in ﬁgure 4. Similarly to the cases in defect-Ain anddefect-Aout, the ΔBz was inﬂuenced by the size and the position of defect-B. However, the inﬂuence of the position isslightly different in the case of defect-A, as shown in ﬁgure 3.

3.2. Improvement effect of inserting the superconducting
cylinder on trapped field uniformity

In order to apply the superconducting bulks to the NMR
spectrometer, it is necessary to enhance the uniformity of thetrapped ﬁeld in the hollow of the bulk cylinder to as high an
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H Fujishiro et al

Figure 5. The trapped ﬁeld proﬁle along the x-direction and the z-

direction in the defect-A bulk with a superconducting cylinder insert,
as a function of the critical current density Jc2 of the inserted
superconducting cylinder.

Figure 4. The microscopic trapped ﬁeld deviation ΔBz around thecentral region (a) along the x-direction (−4x4 mm) at
y=z=0 and (b) along the z-direction (−5z5 mm) at
x=y=0 in the hole of the defect-B bulk and uniform bulk. The
results of the defect-Bin and defect-Bout bulks are also shown.

becoming slightly wider toward the outside region. For
Jc2=4.8×1010 Am−2, which is two orders of magnitudelarger than the Jc1 of the bulk cylinder, the proﬁle along the xdirection still remained, with the ﬂat proﬁle along the zdirection widening toward the outer region as well. In thisway, the homogeneity of the magnetic ﬁeld in the hollow
cylinder can be drastically recovered by inserting a superconducting cylinder with a higher Jc2 than that of the bulk
cylinder with Jc1.
Figures 6(a) and (b) show the trapped ﬁeld proﬁles along
the x- and z-directions, respectively, in the defect-B bulk withthe superconducting cylinder insert as a function of Jc2 of thesuperconducting cylinder. For a Jc2 from 4.8×104 to
4.8×106 Am−2, the cylinder insertion had no inﬂuence on
the trapped ﬁeld proﬁle. For Jc2=4.8×108 Am−2, the sameas Jc1 of the bulk, the inhomogeneity of the trapped ﬁeld
along the x-direction was suppressed slightly and the ﬂatnessof the trapped ﬁeld also widened along the z-direction. For
Jc2=4.8×1010 Am−2, two orders of magnitude larger thanJc1 of the bulk, the trapped ﬁeld along the x-direction andz-direction became ﬂat. It should be noted that the effect ofinserting the superconducting thin cylinder withextent as possible. Our group clariﬁed that inserting a cylinder
wrapped in high-temperature superconducting (HTS) tape isan experimentally effective method to reduce the inhomogeneity of the trapped ﬁeld [9]. In this subsection, the cylinder
wrapped in HTS tape is replaced by the ideal superconducting
thin cylinder with a higher critical current density Jc2, and therelationship between the Jc2 of the inserted cylinder and theuniformity of the trapped ﬁeld was investigated using
numerical simulation.
Figures 5(a) and (b) present the macroscopic trapped
ﬁeld proﬁle along the x- and z-directions in the defect-A
bulk with the superconducting cylinder insert, respectively,
as a function of Jc2 of the inserted superconducting cylinder.
In this case, the Jc of the bulk cylinder is Jc=Jc1=
4.8×108 Am−2. For Jc, from 4.8×104 to 4.8×106 Am−2,
no effect can be seen from the insertion of the superconducting cylinder. For Jc2=4.8×108 Am−2, which is thesame as that of the bulk cylinder, the magnetic proﬁle along
the x-direction became drastically ﬂat. At the same time, theproﬁle along the z-direction also became drastically ﬂat,
5

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Figure 6. The trapped ﬁeld proﬁle along the x-direction and the z-

Figure 7. The trapped ﬁeld deviation ΔBz(±4 mm) along the x-

direction in the defect-B bulk with a superconducting cylinder insert,
as a function of the Jc2 of the inserted superconducting cylinder.

direction (−4x4 mm) at y=z=0 and along the z-direction(−4z4 mm) at x=y=0 in the hole for (a) defect-A and (b)defect-B bulks, as a function of the Jc2 of the inserted superconducting cylinder.

Jc2=4.8×108 Am−2 on the defect-B bulk was slightly
different from that on the defect-A bulk, shown in ﬁgure 5.
This is due to the differences between the distributions of thesuperconducting current depending on the position of thedefect. In this way, the homogeneity of the magnetic ﬁeld
caused by defect-B in the hollow cylinder can be recoveredby the insertion of a superconducting cylinder with Jc2 higher
than that of the bulk cylinder with Jc1.
Figures 7(a) and (b) summarize the trapped ﬁeld deviation ΔBz(±4 mm) along the x-direction (−4 x4 mm) at
y=z=0 and along the z-direction (−4 z4 mm) at
x=y=0 in the hole of the defect-A and defect-B bulks,
respectively, as a function of the Jc2 of the inserted superconducting cylinder. In the defect-A bulk, the improvement
effect of the trapped ﬁeld proﬁle can clearly be obtained witha Jc2 value of the inserted superconducting cylinder higher
than 4.8×108 Am−2 in both the x- and z-directions. The ΔBz
minimum at Jc2=4.8×108 Am−2 is not an essential character, and may come from an error in the FEM calculation. On
the other hand, for the defect-B bulk, the improvement effectcan clearly be obtained with a Jc2 value higher than
4.8×1010 Am−2. The difference may come from therelationship between the conﬁguration of the defect and theJc2 value of the superconducting thin cylinder.

4. Discussion
In order to understand the reason why the insertion of thesuperconducting cylinder with higher Jc2 improved the trapped ﬁeld proﬁle in the hollow space of the bulk cylinder, weanalyzed the superconducting current distribution in the
superconducting thin cylinder. Figures 8(a) and (b) show thecalculated superconducting current distributions in the outermost
surfaceofthethin
cylinderwith
Jc2=4.8×1010 Am−2 inserted in the defect-A and defect-B
bulks, respectively. Schematic views of the current distributions are also shown in ﬁgures 8(c) and (d) for each bulk. For
both bulks, the circular current J with nearly the same valueas Jc2 ﬂows into both ends of the thin cylinder to compensate
for the decrease of the trapped ﬁeld at the bulk edges. In
addition, for the defect-B bulk, the additional circular current
also ﬂows at the edge of the thin cylinder facing defect-B in
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H Fujishiro et al

Figure 8. The results of the simulation of the superconducting current distribution in the outermost surface of the thin cylinder for (a) the

defect-A bulk and (b) the defect-B bulk, in which the Jc2 is 4.8×1010 Am−2. Schematic views of the superconducting current distribution
are also shown in (c) and (d) for each bulk.

order to compensate for the decrease of the trapped ﬁeld
caused by defect-B. On the other hand, for the defect-A bulk,
the superconducting circular current ﬂows in the thin cylinder
facing defect-A, which is smaller than that of the thin cylinder
facing defect-B, to compensate for the decrease in the trapped
ﬁeld due to defect-A. For the innermost surface as well as theinside of the thin cylinder, no remarkably large supercurrent
was found to ﬂow. A large superconducting current to compensate for the decrease in the magnetic ﬁeld due to thedefects mainly ﬂows in the outermost surface of the thin
cylinder with a higher Jc2. The Jc values in REBaCuO-coatedconductors have been enhanced over 1010 Am−2 under themagnetic ﬁeld of several tesla at 50 K [14, 15]. The Jc2 value
as large as 1010 Am−2 can be practically attainable at present.
In this way, the mechanism, in which the homogeneity of thetrapped ﬁeld can be improved via the insertion of a superconducting thin cylinder, can be explained qualitatively by
the rearrangement of the superconducting current within thethin cylinder.
Finally, we discuss the difference between the realisticbulk superconductor and the bulk model in the simulation. In
the present analysis, unrealistically large defects are intentionally introduced in the hollow bulk superconductor inorder to enhance the effect of inserting the superconducting
cylinder on the trapped ﬁeld distribution. In the realistic bulk
superconductors, the defects are much smaller, and more orless uniformly distributed throughout the bulk. This study is
the ﬁrst step towards understanding the relationship between
the size and position of the defect and the trapped ﬁeld

uniformity. We want to apply this technique to the realistic
model for the bulk superconductors.

5. Summary
A three-dimensional (3D) numerical simulation has been
performed for the ﬁeld-cooled magnetization (FCM) processof REBaCuO superconducting bulk cylinders applicable to a
200 MHz NMR (4.7 T) bulk magnet. The inﬂuence of theposition and size of the defect (Jc=0) in the bulk on thehomogeneity of the trapped ﬁeld in the hollow space has been
investigated. The important results and conclusions obtained
from this study are summarized as follows.
(1) The trapped ﬁeld proﬁle changes depending on theposition and size of the defect, as evidenced by thecases of defect-A, situated near the centre, and defect-B,
situated in the bulk edge. The trapped ﬁeld deviation
ΔBz at the central region, which is deﬁned by (BzBz(0))/Bz(0), along the x- and z-directions decreased
with the decreasing size of the defect in both cases. The
defect situated inside the bulk cylinder had a moderateinﬂuence on the inhomogeneity of the trapped ﬁeld
distribution in the hollow space.
(2) The inhomogeneity of a trapped ﬁeld due to defects can be
improved drastically via the insertion of a superconducting
thin cylinder with a higher critical current density than that7

Supercond. Sci. Technol. 28 (2015) 095018

H Fujishiro et al

the ‘Development of System and Technology for Advanced
Measurement and Analysis’ project of the Japan Science andTechnology Agency (JST).

of the bulk cylinder. The insertion effect creates a slightdifference along the x- and z-directions for both defectbulks, which may be caused by the relationship between
the conﬁguration of the defect and the superconducting
current distribution ﬂowing in the thin cylinder.
(3) In the case of defect-B bulk, when inserting thesuperconducting thin cylinder with a Jc2 of two ordersof magnitude higher than that of the bulk cylinder Jc1, it
was found that a large circular current ﬂows in the thin
cylinder facing defect-B to compensate for the decreasein the trapped ﬁeld caused by defect-B, in addition to
the circular current with nearly the same value as Jc2 at
both ends of the cylinder. For the defect-A bulk, only arelatively smaller superconducting current ﬂows in thethin cylinder facing defect-A. Only the superconducting
current ﬂowing in the outermost surface of the thin
cylinder is effective at compensating for the inhomogeneity of the trapped ﬁeld. The mechanism with which
to improve the inhomogeneity of the trapped ﬁeld canbe explained qualitatively by rearranging the superconducting current in the thin cylinder.

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Acknowledgements
The authors thank Dr T Naito of Iwate University for thevaluable discussion. This study was ﬁnancially supported by8