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A new concept of a hybrid trapped field magnet lensTo cite this article: Keita Takahashi et al 2018 Supercond. Sci. Technol. 31 044005

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Superconductor Science and Technology
Supercond. Sci. Technol. 31 (2018) 044005 (9pp)

https://doi.org/10.1088/1361-6668/aaae94

A new concept of a hybrid trapped ﬁeld
magnet lensKeita Takahashi1,3 , Hiroyuki Fujishiro1

and Mark D Ainslie2

1

Department of Physical Science and Materials Engineering, Faculty of Science and Engineering, Iwate
University, Morioka 020-8551, Japan2
Bulk Superconductivity Group, Department of Engineering, University of Cambridge, Cambridge CB2
1PZ, United Kingdom
E-mail: t2216017@iwate-u.ac.jp and fujishiro@iwate-u.ac.jp
Received 22 December 2017, revised 7 February 2018
Accepted for publication 12 February 2018
Published 7 March 2018
Abstract

In this paper, a new concept of a hybrid trapped ﬁeld magnet lens (HTFML) is proposed. The
HTMFL exploits the ‘vortex pinning effect’ of an outer superconducting bulk cylinder, which ismagnetized as a trapped ﬁeld magnet (TFM) using ﬁeld-cooled magnetization (FCM), and the‘diamagnetic shielding effect’ of an inner bulk magnetic lens to generate a concentrated magnetic
ﬁeld higher than the trapped ﬁeld from the TFM in the bore of the magnetic lens. This requires that,during the zero-ﬁeld-cooled magnetization process, the outer cylinder is in the normal state (T>superconducting transition temperature, Tc) and the inner lens is in the superconducting state(T<Tc) when the external magnetizing ﬁeld is applied, followed by cooling to an appropriate
operating temperature, then removing the external ﬁeld. This is explored for two potential cases: (1)exploiting the difference in Tc of two different bulk materials (‘case-1’), e.g. MgB2 (Tc=39 K) and
GdBaCuO (Tc=92 K) or (2) using the same material for the whole HTFML, e.g., GdBaCuO, bututilizing individually controlled cryostats, the same cryostat with different cooling loops or coolants,
or heaters that keep the outer bulk cylinder at a temperature above Tc to achieve the same desired
effect. The HTFML is veriﬁed using numerical simulations for ‘case-1’ using an MgB2 cylinder and
GdBaCuO lens pair and for ‘case-2’ using a GdBaCuO cylinder and GdBaCuO lens pair. As a result,the HTFML could reliably generate a concentrated magnetic ﬁeld Bc=4.73 T with the external
magnetizing ﬁeld Bapp=3 T in the ‘case-1’, and a higher Bc=13.49 T with higher Bapp=10 T in
the ‘case-2’, respectively. This could, for example, be used to enhance the magnetic ﬁeld in the boreof a bulk superconducting NMR/MRI system to improve its resolution.
Keywords: hybrid trapped ﬁeld magnet lens, bulk superconductors, trapped ﬁeld magnets,
magnetic lens, vortex pinning effect, diamagnetic shielding effect, ﬁnite element method(Some ﬁgures may appear in colour only in the online journal)
1. Introduction

introduction of strong pinning centers and the improvement ofcrystal growth techniques, which have resulted in increased critical current density, Jc [1, 2]. As a result, such bulks exhibit
higher trapped ﬁeld capabilities using ﬁeld-cooled magnetization(FCM) and have signiﬁcant potential for practical applications as
high-strength trapped ﬁeld magnets (TFMs) capable of generating
magnetic ﬁeld of several Tesla. Figure 1(a) shows the time
sequence of conventional FCM of superconducting bulks toutilize them as TFMs (the case shown is for a cylindrical ringbulk superconductor). Although the trapped ﬁeld, BT, of REBaCuO bulks, which can be estimated from the Jc(B, T)

The superconducting characteristics of REBaCuO bulks (RE: rareearth element or Y) continue to be enhanced due to the3Author to whom any correspondence should be addressed.

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Supercond. Sci. Technol. 31 (2018) 044005

K Takahashi et al

Figure 1. Time sequence of (a) conventional magnetizing process of ﬁeld-cooled magnetization (FCM) of superconducting bulks to utilize them as
TFMs and (b) magnetizing process of zero-ﬁeld-cooled magnetization (ZFCM) for a conventional superconducting bulk magnetic lens.

characteristics, could be over 20 T at 20 K in a bulk pair [3], themechanical strength of the brittle ceramic material restricts thepractical maximum trapped ﬁeld of such TFMs experimentally.
To date, REBaCuO disk bulks have been shown to trap magnetic
ﬁeld over 17 T by mechanical reinforcement using glass ﬁber
reinforced epoxy resin or shrink-ﬁt stainless steel to reduce theelectromagnetic hoop stress [3, 4]. Such TFMs, in which the ﬁeld
is trapped using the strong ‘vortex pinning effect’ of the material,
require a high, stationary magnetic ﬁeld to magnetize them andthey can trap ﬁelds nearly the same or slightly lower than theapplied ﬁeld, Bapp, using FCM, but such high ﬁelds are only
available from specialized magnets at a limited number of facilities worldwide. In this sense, such high-strength TFMs are notpractical for applications and much research has been carried out
on the pulsed ﬁeld magnetization (PFM) technique as a fast,compact and relatively inexpensive magnetization technique.
However, due to the large temperature rise associated with therapid dynamic movement of the magnetic ﬂux within the bulk,
the trapped ﬁeld capability is severely limited and only ﬁelds up
to around 5 T have been achieved using PFM [5].
On the other hand, a ‘magnetic lens’ using cone-shaped
superconducting bulks has been investigated, in which themagnetic ﬂux is concentrated in the bore of the magnetic lens
using the ‘diamagnetic shielding effect’ of superconducting
materials and the available magnetic ﬁeld in the lens is larger than

the applied ﬁeld generated by the external magnetizing coil [6–9].
Figure 1(b) shows the time sequence of magnetizing process for aconventional bulk superconducting magnetic lens, for which zeroﬁeld-cooled magnetization (ZFCM) is used. The existence of slits
in the superconducting hollow cone is essential to suppress thecurrent along the circumferential direction and to concentrate themagnetic ﬂux. A concentrated ﬁeld of Bc=12.42 T has beenachieved at 20 K for a background ﬁeld of Bapp=8 T using abulk GdBaCuO magnetic lens [10] and Bc=30.4 T has beenachieved at the center of the lens in higher background ﬁeld of
Bapp=28.3 T elsewhere [11]. Using a bulk MgB2 magnetic lens,a concentrated ﬁeld of Bc=2.18 T at 4.2 K has also beenachieved for a background ﬁeld of Bapp=1 T [12]. Mechanicalreinforcement of the magnetic lens and the avoidance of the ﬂux
jump are necessary to achieve the magnetic lens effect stably [13].
Since the magnetic lens effect vanishes after the applied ﬁeld
decreased to zero, the external magnet must be operated continuously, which consumes a large amount of energy.
In this paper, we propose a new concept of a hybrid trapped
ﬁeld magnet lens (HTFML), consisting of a cylindrical bulk
TFM using the vortex pinning effect, combined with a bulk
magnetic lens using the diamagnetic shielding effect. The
HTFML can reliably generate a magnetic ﬁeld at the center of themagnetic lens higher than the trapped ﬁeld in the single cylindrical bulk TFM and the external magnetizing ﬁeld, even after the2Supercond. Sci. Technol. 31 (2018) 044005

K Takahashi et al

Table 1. Numerical parameters for the Jc(B) characteristics of thebulk GdBaCuO material at 20 and 40 K using equation (2).

T (K)
40
20

Jc1 (A m−2)
9

3.5×10
9.0×109

BL (T)
0.9
1.5

Jc2 (A m−2)9

2.7×10
5.4×109

Bmax (T)

α

6.0
8.0

0.8
0.5

superconducting cylinder is made from bulk MgB2 and thesuperconducting magnetic lens is made from bulk GdBaCuO.
This model is abbreviated as ‘case-1’. The bulk MgB2
cylinder is 60 mm in outer diameter (O.D.), 40 mm in innerdiameter (I.D.) and 80 mm in height (H). The shape of themagnetic lens is referred from [13] so that the present numerical framework explains the concept of the HTFML basedon such a typical lens geometry (as shown in ﬁgure 2), but
importantly two slits exist that are 10° wide. The MgB2
cylinder and GdBaCuO lens are magnetized by a solenoid
coil of 170 mm in O.D., 120 mm in I.D. and 200 mm in H.
Another example is also provided, in which both the superconducting cylinder and magnetic lens are made from bulk
GdBaCuO, which is abbreviated as ‘case-2’. The numericalsimulation results for ‘case-1’ and ‘case-2’ are presented in
sections 4.1 and 4.2, respectively.
Electromagnetic phenomena during the magnetization
process are described by the fundamental equations shownelsewhere in detail [16–18]. The E–J power law is assumed todescribe the nonlinear electrical properties of the superconducting bulk:

Figure 2. Numerical model and dimensions of the MgB2 cylinder,
GdBaCuO magnetic lens and solenoid magnetizing coil for ‘case-1’.
The MgB2 cylinder is replaced by a GdBaCuO cylinder in ‘case-2’.

externally applied ﬁeld decreases to zero. This concept requires
that, during the ZFCM process, the outer cylinder is in thenormal state (T> superconducting transition temperature, Tc) andthe inner lens is in the superconducting state (T<Tc) when theexternal magnetizing ﬁeld is applied, followed by cooling to an
appropriate operating temperature, then removing the externalﬁeld. This is explored for two potential cases: (1) exploiting thedifference in Tc of two different bulk materials (‘case-1’), e.g.
MgB2 (Tc=39 K) and GdBaCuO (Tc=92 K) or (2) using thesame material for the whole HTFML, e.g., GdBaCuO, but utilizing individually controlled cryostats, the same cryostat withdifferent cooling loops or coolants, or heaters that keep the outerbulk cylinder at a temperature above Tc to achieve the samedesired effect. The effectiveness and superiority of the HTFML is
veriﬁed using numerical simulations for two cases: ‘case-1’ using
an MgB2 cylinder and REBaCuO lens pair, and ‘case-2’ using a
REBaCuO cylinder and REBaCuO lens pair. The concentrated
magnetic ﬁeld in the HTFML changes depending on the superconducting characteristics of the bulks [14, 15], their shape andsize [12], as well as the magnetizing conditions.

⎛ J ⎞nE = Ec ⎜ ⎟ ,⎝ Jc ⎠(1 )where Ec (=10−4 V m−1) is the characteristic electric ﬁeld.
n=20 for the GdBaCuO bulk and n=100 for the MgB2
bulk are used [19]. The numerical simulation results depend
strongly on the Jc(B, T) characteristics of the superconductor[20–22]. For this study, the Jc(B) characteristics for the
GdBaCuO bulk are based on the equation presented by Jirsa
et al to represent the so-called ‘ﬁsh-tail’ or ‘peak’ effect foundin such superconducting materials [18, 23],
⎡1 ⎛
⎛ B⎞
⎛ B ⎞a ⎞ ⎤B
Jc (B) = Jc1 exp ⎜ - ⎟ + Jc2
exp ⎢ ⎜1 - ⎜⎟ ⎟ ⎥.
⎢⎣ a ⎝⎝ BL ⎠⎝ Bmax ⎠ ⎠ ⎥⎦
Bmax
(2 )

The values for the parameters Jc1, BL, Jc2, Bmax and α at
20 and 40 K used in the model are shown in table 1,
respectively.
The Jc(B) of the MgB2 bulk is described by the following
equation [24],⎡ ⎛ B ⎞b ⎤Jc (B) = Jc0 exp ⎢ - ⎜ ⎟ ⎥ ,⎢⎣ ⎝ B0 ⎠ ⎥⎦2. Numerical simulation framework(3 )where Jc0, B0 and β are ﬁtting parameters, which are summarized in table 2. Isothermal conditions are assumed while
ramping down the ﬁeld because the magnetization process is
slow; hence, no thermal model is included.

The following numerical simulation framework for the magnetizing process of the HTFML has been developed. A
schematic view of the three-dimensional numerical model andthe relevant dimensions are shown in ﬁgure 2, in which the3Supercond. Sci. Technol. 31 (2018) 044005

K Takahashi et al

Table 2. Numerical parameters for the Jc(B) characteristics of thebulk MgB2 material at 20 K using equation (3).

T (K)
20

Jc0 (A m−2)
9

4.3×10

B0 (T)

α

1.1

1.5

the MgB2 cylinder is magnetized by FCM and magnetic
ﬂux is trapped in the cylinder. The magnetic ﬁeld
concentration effect slightly decreases due to thedecrease of external ﬁeld. However, a magnetic ﬁeld
at the center of the magnetic lens still remains due to theexistence of the trapped ﬁeld in the MgB2 cylinder.
(5) As a result, HTFML can reliably generate a magnetic
ﬁeld higher than BT of the single cylindrical TFM and
Bapp, even after Bex=0.
There are some examples of a practical cooling systemusing the difference in Tc of two superconducting components
for an aircraft motor design [25] and a magnetic levitation
application [26].
In ‘case-2’, in which both the superconducting cylinderand magnetic lens are made from bulk GdBaCuO, (1) and (2)above are changed as follows, labeled at (1’) and (2’):

Figure 3. Time step sequence of (a) the temperature, T and (b) theexternal ﬁeld, Bex, and concentrated magnetic ﬁeld, Bc, at the centerof magnetic lens for ‘case-1’, in which an MgB2 cylinder and
GdBaCuO magnetic lens are used. In ‘case-2’, the temperature of the
GdBaCuO lens follows the blue line and the GdBaCuO cylinder
follows the dotted orange line in the upper panel (see text).

(1’) The bulk GdBaCuO cylinder is maintained at 100 K(dotted orange line in ﬁgure 3) and the bulk GdBaCuO
magnetic lens is cooled to TH=40 K (blue line inﬁgure 3). Hence, the bulk GdBaCuO cylinder is in thenormal state and the bulk GdBaCuO lens is in the
superconducting state (step 0).
(2’) The external magnetic ﬁeld, Bex, is increased linearlyover ﬁve steps (steps 1–5) up to Bapp, and the magnetic
ﬁeld, essentially equal to Bapp, completely penetrates
the GdBaCuO cylinder, but the GdBaCuO lens ismagnetized by ZFCM.

3. Magnetizing procedure
First, the magnetizing process is described for the following
time step sequence, from (1) to (5), for the HTFML for ‘case1’, in which the bulk MgB2 cylinder and the bulk GdBaCuO
magnetic lens are used. Figure 3 shows the time sequence of(a) the temperature, T and (b) the external ﬁeld, Bex, andconcentrated magnetic ﬁeld, Bc, at the center of the magnetic
lens. The magnetizing applied ﬁeld, Bapp, corresponds to the
maximum value of Bex.

In the next section, the results of the numerical simulation for ‘case-1’ and ‘case-2’ are presented to prove theeffectiveness of the HTFML.

(1) The bulk MgB2 cylinder and the bulk GdBaCuO lens
are cooled from 100 K to TH=40 K, which is higher
than the superconducting transition temperature of
MgB2, Tc=39 K, but lower than that of GdBaCuO(Tc=92 K). In this stage, the MgB2 cylinder is in thenormal state and the GdBaCuO lens is in the superconducting state (step 0).
(2) The external magnetic ﬁeld, Bex, is ramped up linearly
at 0.222 T min−1 over ﬁve steps (steps 1–5) up to Bapp,
which corresponds to ZFCM of the GdBaCuO lens. The
magnetic ﬁeld, essentially higher than Bapp because of
the shielding effect by the magnetic lens, completely
penetrates the MgB2 cylinder and the magnetic ﬁeld isconcentrated at the center of the lens.
(3) The temperatures of both MgB2 cylinder and GdBaCuO
lens are then decreased to TL=20 K, which is lower
than the Tc of MgB2.
(4) Bex is decreased linearly at 0.222 T min−1 over ﬁve
steps (steps 6–10) down to zero. During this process,4. Simulation results and discussion
4.1. ‘Case-1’: MgB2 cylinder–GdBaCuO lensFigure 4 shows the time step dependence of the magnetic ﬁeld
proﬁle along the x-direction across the center of the lensduring (a) the ascending stage and (b) the descending stage of
ZFCM of the GdBaCuO lens under an applied ﬁeld,
Bapp=3 T in ‘case-1’, which incorporates FCM of the MgB2
cylinder. In ﬁgure 4(a), during the ascending stage from steps
0–5, the concentrated magnetic ﬁeld, Bc, at the center of the
GdBaCuO lens was enhanced with increasing Bex owing to
the diamagnetic shielding effect of the GdBaCuO lens. It canbe found that there is little or no ﬂux penetration in the
GdBaCuO lens region (r=±5∼18 mm) in this caseapplying a relatively low magnetic ﬁeld of 3 T by a magnetizing coil. The magnetic ﬂux intrudes into the lens from inner4Supercond. Sci. Technol. 31 (2018) 044005

K Takahashi et al

Figure 4. Time step dependence of the magnetic ﬁeld proﬁle along

Figure 5. Time step dependence of the magnetic ﬁeld proﬁle along

the x-direction across the center of the lens during (a) the ascending
stage and (b) the descending stage of ZFCM of the GdBaCuO lensunder an applied ﬁeld, Bapp=3 T in ‘case-1’, which incorporates
FCM of the MgB2 cylinder.

the x-direction across the center of the lens during (a) the ascending
stage and (b) the descending stage of FCM of the MgB2 cylinder
without the GdBaCuO lens for an applied ﬁeld, Bapp=3 T at 20 K.

descending stages of FCM for an applied ﬁeld, Bapp=3 T for
only the MgB2 cylinder, without the GdBaCuO lens, in ‘case1’. In ﬁgure 5(a), when only the MgB2 cylinder is considered,the BT value is nearly the same as Bex in the ascending stage(steps 0–5), and attains a maximum value Bex=3 T at step 5
with a uniform magnetic ﬁeld proﬁle along x-direction, whichcorresponds to the applied ﬁeld proﬁle generated by theexternally magnetizing coil. During this ascending stage,
the MgB2 cylinder is in the normal state. In ﬁgure 5(b), in
the descending stage (steps 6–10), the trapped ﬁeld, BT, of the
MgB2 cylinder by FCM decreases slightly with decreasing
Bex, and settles to a ﬁnal value of BT=2.85 T at the ﬁnal step
(step 10) in the bore of MgB2 cylinder, now acting as a TFM
that can continue to provide the trapped ﬁeld quasi-permanently. One of the particular characteristics of the HTFML
device is to utilize the trapped ﬁeld from this TFM, instead ofrequiring a continuously applied ﬁeld from an external
magnetizing coil. Thus, it is useful to be able to reproduce amagnetic ﬁeld proﬁle similar to that which might be produced

periphery rather than outer edge because of the magnetic ﬂux
concentration, which will be discussed later for higher Bapp in‘case-2’. In this ascending stage, the GdBaCuO lens is under
ZFCM conditions at 40 K; however, the MgB2 cylinder is in
the normal state and is not yet utilized as a TFM. In
ﬁgure 4(b), during the descending stage of Bex from steps
5–10, which incorporates FCM of the MgB2 cylinder during
ZFCM of the GdBaCuO lens, Bc decreased with decreasing
Bex, but becomes stable after the external ﬁeld decreases tozero and a magnetic ﬁeld is trapped in the MgB2 cylinder. As
a result, Bc at the center of the GdBaCuO lens settled to
Bc=4.73 T at the ﬁnal step (step 10), which can be realized
quasi-permanently by the novel combination of the ‘vortex
pinning effect’ and ‘diamagnetic shielding effect’ of superconducting bulk materials.
Figures 5(a) and (b), respectively, show the time step
dependence of the magnetic ﬁeld proﬁles along the x-direction across the center of the lens during the ascending and5Supercond. Sci. Technol. 31 (2018) 044005

K Takahashi et al

Figure 7. Concentrated magnetic ﬁeld, Bc, at the center of themagnetic lens as a function of the external ﬁeld, Bex, for Bapp=3 T.
The trapped ﬁeld, BT, at the center of the MgB2 cylinder for the casewithout the GdBaCuO lens extracted from ﬁgure 5 is also shown.

as ﬁgure 4(a), in which Bc at the center of the GdBaCuO lenswas enhanced gradually with increasing Bex in the ascending
stage from steps 0–5. In ﬁgure 6(b), in the descending stage
from steps 6–10, it can be seen that there is some ﬂux
penetration into the inner edge of GdBaCuO lens at aroundr=±5∼7 mm from step 7 even for a relatively low appliedﬁeld of 3 T during ZFCM. The magnetic ﬁeld reachedeventually becomes −0.46 T at the center of the GdBaCuO
lens at the ﬁnal step (step 10) after ZFCM, resulting in a
reduction from the ideal of the GdBaCuO lens shielding effectin those regions where the magnetic ﬂux penetrates during
ZFCM. Thus, to maximize the lens’s shielding, and hence its
ability to concentrate the magnetic ﬁeld, the ﬂux penetration
should be minimized.
Figure 7 shows the concentrated magnetic ﬁeld, Bc, at thecenter of magnetic lens as a function of external ﬁeld, Bex, for
Bapp=3 T, which was extracted from ﬁgure 4. The trapped
ﬁeld, BT, at the center of the MgB2 cylinder was also
extracted from ﬁgure 5 for the case without the GdBaCuO
lens, which clariﬁes the effectiveness of the HTFML. For
only MgB2 cylinder, the BT value is nearly the same as Bex in
the ascending stage from step 0 and attains a maximum valueBex=3 T at step 5. The BT value is then 2.85 T at the ﬁnal
step (step 10) once Bex=0, showing that 3 T is a reasonablevalue to fully magnetize the MgB2 cylinder by FCM at 20 K.
In the case of the HTFML using both the MgB2 cylinder and
GdBaCuO lens, Bc at the center of the GdBaCuO lens wasenhanced up to 6.10 T at step 5 due to the shielding effect oflens and settled to Bc=4.73 T at the ﬁnal step (step 10),which is higher than both Bapp from the magnetizing coil andBT from the MgB2 TFM. These results indicate the superiorityof the proposed HTFML device, which can reliably generate
the concentrated magnetic ﬁeld higher than the applied ﬁeld
by the external magnetizing coil and the trapped ﬁeld of the
TFM, even after removal of the external ﬁeld.

Figure 6. Time step dependence of the magnetic ﬁeld proﬁle along

the x-direction across the center of the lens during (a) the ascending
stage and (b) the descending stage of ZFCM of the GdBaCuO lens
without the MgB2 cylinder for an applied ﬁeld, Bapp=3 T at 20 K.

by a magnetizing solenoid coil when we use the TFM as asource of magnetic ﬁeld for the HTFML, for further concentration of the magnetic ﬂux in the HTFML. This could, forexample, allow the realization of higher resolution in acompact and cryogen-free NMR/MRI system using annular
REBaCuO superconducting bulks [27, 28].
To achieve the highest concentrated ﬁeld in the lens, the
GdBaCuO magnetic lens must, in the ideal case, completely
shield the magnetic ﬁeld from its interior. However, since it isa type II superconductor, some magnetic ﬂux will penetrate
the material, depending on its Jc(B, T) characteristics (andhence operating temperature), as well as its geometry, during
the HTFML magnetizing procedure. Figures 6(a) and (b),respectively, show the time step dependence of the magnetic
ﬁeld proﬁles along the x-direction across the center of the lensduring the ascending and descending stages of ZFCM for an
applied ﬁeld, Bapp=3 T for the only GdBaCuO lens without
the MgB2 cylinder in ‘case-1’. Figure 6(a) is exactly the same
6

Supercond. Sci. Technol. 31 (2018) 044005

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Figure 9. Time step dependence of the concentrated magnetic ﬁeld,
Bc, at the center of the GdBaCuO lens in the HTFML for Bapp=3, 6
and 10 T for ‘case-2’. The external ﬁeld, Bex, generated by thesolenoid magnetizing coil is also shown for each Bapp.

the GdBaCuO lens from steps 0–5 (noting that the MgB2
cylinder is in the normal state). In ﬁgure 8(a), the Bc value wasenhanced up to Bc=16 T at the center of the lens at step 5
when applying Bapp=10 T. The GdBaCuO lens could retain its
shielding effect even in a higher external magnetic ﬁeld of 10 T,
although the magnetic shielding effect weakens with increasing
applied ﬁeld due to further penetration of magnetic ﬂux [11]. In
ﬁgure 8(b), during the descending stage (steps 5–10), the
GdBaCuO cylinder maintains a trapped ﬁeld similar to Bapp at
the ﬁnal step (step 10) as it is magnetized by FCM, but below its
full capability based on its high Jc–B characteristics at 20 K. The
Bc value gradually decreased during the descending stage andsettled to a ﬁnal value of Bc=13.5 T at the center of the lens.
Figure 9 shows the time step dependence of the concentrated magnetic ﬁeld, Bc, and external ﬁeld, Bex, at the centerof the GdBaCuO lens for Bapp=3, 6 and 10 T for ‘case-2’.
Table 3 summarizes the concentrated magnetic ﬁeld, Bc, trapped ﬁeld from the TFM, BT, when only considering the TFM
cylinder (i.e., the lens is not present), at the central position at
the ﬁnal step (step 10), and the concentration ratio, Bc/Bapp, in‘case-2’ extracted from ﬁgure 9 for each actual Bapp. An
accurate value of Bapp is shown as B*app=3.09 T, 6.13 T and
10.18 T in this table, which was named roughly as Bapp=3, 6,
10 T so far. Similar results for ‘case-1’ from ﬁgure 4 are also
shown for comparison. For the lower Bapp=3 T, a magnetic
ﬁeld concentration ratio of Bc/Bapp=1.70 is achieved at theﬁnal step (step 10) in ‘case-2’, which is higher than 1.53 whenusing the MgB2 cylinder in ‘case-1’. This results from thehigher BT value of 3.09 T in ‘case-2’, where the outside
GdBaCuO cylinder was magnetized below its full capability.
Furthermore, in ‘case-2’, the concentration ratio decreased withincreasing Bapp from 1.70 for Bapp=3 T to 1.33 for
Bapp=10 T. In ﬁgure 9, the higher Bapp resulted in a larger ﬂux
creep during the descending stage of FCM of the TFM cylinderand further penetration of magnetic ﬂux in the ZFCM of the
GdBaCuO lens. There is a possibility to achieve further

Figure 8. Time step dependence of magnetic ﬁeld proﬁle along the x-

direction across the center of the lens during (a) the ascending stageand (b) the descending stage for Bapp=10 T in ‘case-2’.

4.2. ‘Case-2’: GdBaCuO cylinder–GdBaCuO lens

In the previous section, the superiority of the HTFML devicein ‘case-1’ using a MgB2 cylinder and REBaCuO lens pair
was veriﬁed, for which the concentrated magnetic ﬁeld wasenhanced up to Bc=4.73 T at the center of the GdBaCuO
lens, which is higher than the trapped ﬁeld of BT=2.8 T
generated by the MgB2 cylinder and the applied ﬁeld,
Bapp=3.0 T, generated by the external magnetizing coil. In
this section, the other example of ‘case-2’ using a REBaCuO
cylinder and REBaCuO lens pair for further concentration of
the trapped magnetic ﬂux is explored.
Figure 8 shows the time step dependence of the magnetic
ﬁeld proﬁle along the x-direction across the center of the lensduring (a) the ascending stage and (b) the descending stage for
Bapp=10 T in ‘case-2’. In this case, it should be noted that thetemperature of the GdBaCuO cylinder and lens must be controlled individually. Similar results were shown previously for‘case-1’ in ﬁgure 4(a), where the magnetic ﬁeld proﬁle wasshown during the ascending stage of ZFCM for Bapp=3 T for7

Supercond. Sci. Technol. 31 (2018) 044005

K Takahashi et al

Table 3. Concentrated magnetic ﬁeld, Bc, trapped ﬁeld by TFM, BT, at the center of the GdBaCuO lens at the ﬁnal step (step 10), andmagnetic ﬁeld concentration ratio, Bc/Bapp, in ‘case-2’ extracted from ﬁgure 9 for each actual applied ﬁeld, B*app. Similar results for ‘case-1’
from ﬁgure 4 are also shown for comparison.

‘case-1’ (MgB2 cylinder)‘case-2’ (GdBaCuO cylinder)

Bapp (T)B*app (T)BT (T)

Bc (T)

Bc/Bapp

3
3
6
10

3.09
3.09
6.13
10.18

2.85
3.09
6.13
10.18

4.73
5.25
9.19
13.49

1.53
1.70
1.50
1.33

enhancement of Bc and Bc/Bapp by optimization of the magnetic
design, including the geometry of the cylinder and lens, and themagnetization conditions, including temperature and appliedﬁeld. The shielding property of the magnetic lens should be also
exploited for further concentration of the trapped ﬁeld, such as
for hollow bulk cylinders [14]. These results show that theeffectiveness of the HTFML would be enhanced in terms of thecharacteristics of the superconducting material(s) used, including the possibility of utilizing new and improved materials,
such as BaFe2As2 (Tc=38 K) [29, 30].

comparatively inferior Jc(B) characteristics of MgB2 and it
must operate at a temperature lower than the superconducting
transition temperature of MgB2, Tc=39 K.
‘case-2’: The all-GdBaCuO HTFML offers higher concentrated ﬁelds at temperatures much higher than 39 K, butdoes require separate cooling of the cylinder and lens parts toobtain the necessary effect and it would weigh more.
This HTFML device could become a standard method fortrapped ﬁeld enhancement in several practical applications
using a superconducting bulk and there is a scope for optimization of the magnetic design, including geometry around twobulks, and magnetization conditions, including temperature andapplied ﬁeld. The effectiveness of the HTFML would beenhanced with improvements in the characteristics of thesuperconducting material(s) used, including the possibility ofutilizing new and improved materials such as BaFe2As2(Tc=38 K). The device could, for example, be used toenhance the magnetic ﬁeld in the bore of a bulk superconducting NMR/MRI system to improve its resolution.

5. Conclusion
A new concept of an HTFML, consisting of a superconducting
bulk cylinder TFM using the vortex pinning effect, combinedwith a bulk magnetic lens using the diamagnetic shielding effect,is proposed, which can reliably generate a magnetic ﬁeld at thecenter of the magnetic lens higher than the trapped ﬁeld by TFM
and the maximum external magnetizing ﬁeld, even after theexternally applied ﬁeld decreases to zero. The effectiveness andsuperiority of the HTFML was veriﬁed using numerical simulations for two examples: (1) an MgB2 cylinder and GdBaCuO
lens pair (‘case-1’) and (2) a GdBaCuO cylinder and GdBaCuO
lens pair (‘case-2’). In ‘case-1’, using the outer MgB2 cylinderand inner GdBaCuO lens pair, the MgB2 cylinder was magnetized by FCM with an applied ﬁeld, Bapp=3 T, during thedescending stage, also corresponding to ZFCM of the GdBaCuO lens. The trapped ﬁeld, BT=2.85 T, in the MgB2 TFM
cylinder was concentrated by the introduction of GdBaCuO lens,and a concentrated magnetic ﬁeld, Bc=4.73 T, was reliably
achieved at the center of the lens. In ‘case-2’, using the outer
GdBaCuO TFM cylinder and inner GdBaCuO lens pair, inwhich the GdBaCuO cylinder is held above Tc and the GdBaCuO lens is cooled below Tc for the ascending stage of magnetization, followed by both bulks being cooled below Tc for thedescending stage, a higher Bc=13.49 T, was reliably achievedat the center of the magnetic lens for Bapp=10 T.
The advantages and disadvantages of each HTFML,
comparing use of the MgB2 cylinder and the GdBaCuO
cylinder are summarized as follows:‘case-1’: The MgB2 HTFML only needs one coolingprocess for the whole device by exploiting the difference in Tc
of the two superconducting materials. Its weight would also
be lower due to the use of the lighter bulk MgB2 cylinder.
However, the trapped ﬁeld capability is limited in comparison
to the GdBaCuO cylinder (‘case-2’) because of the

Acknowledgments
This research is partially supported from JSPS KAKENHI
Grant No. 15K04646. Mark Ainslie would like to acknowledge ﬁnancial support from an Engineering and PhysicalSciences Research Council (EPSRC) Early Career Fellowship
EP/P020313/1. All data are provided in full in the results
section of this paper.

ORCID iDs
Keita Takahashi https://orcid.org/0000-0002-8278-2688
Hiroyuki Fujishiro https://orcid.org/0000-00031483-835X
Mark D Ainslie https://orcid.org/0000-0003-0466-3680

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