Superconductor Science and Technology

PAPER • OPEN ACCESSA conceptual study of a high gradient trapped field magnet (HG-TFM)
toward providing a quasi-zero gravity space on Earth
To cite this article: Keita Takahashi et al 2021 Supercond. Sci. Technol. 34 035001

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Superconductor Science and Technology
Supercond. Sci. Technol. 34 (2021) 035001 (13pp)

https://doi.org/10.1088/1361-6668/abd386

A conceptual study of a high gradient
trapped field magnet (HG-TFM) toward
providing a quasi-zero gravity space on
Earth
Keita Takahashi1, 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: s3119004@iwate-u.ac.jp and fujishiro@iwate-u.ac.jp
Received 23 September 2020, revised 18 November 2020
Accepted for publication 14 December 2020
Published 25 January 2021
Abstract

In this work, we propose a new concept of a high gradient trapped field magnet (HG-TFM). The
HG-TFM is made from (RE)BaCuO bulk superconductors, in which slit ring bulks (slit-TFMs)
are tightly stacked with TFM cylinders (full-TFMs), and state-of-the-art numerical simulations
were used to investigate the magnetic and mechanical properties in detail during and aftermagnetization. A maximum value of the magnetic field gradient product ofBz · dBz /dz = 6040 T2 m−1 was obtained after conventional field cooled magnetization (FCM)with an applied field, Bapp , of 10 T of the HG-TFM with 60 mm in outer diameter and 10 mm in
inner diameter. This value may be the highest value ever reported compared to any othermagnetic sources. The Bz · dBz /dz value increased with decreasing inner diameter of the
HG-TFM and with increasing Bapp during FCM. The electromagnetic stress in the HG-TFM
during the FCM process mainly results from the hoop stress along the circumferential direction.
The simulations suggested that there is no fracture risk of the bulk components during FCM
from 10 T in a proposed realistic configuration of the HG-TFM where both TFM parts aremounted in Al-alloy rings and the whole HG-TFM is encapsulated in a steel capsule. A
quasi-zero gravity space can be realized in the HG-TFM with a high Bz · dBz /dz value in an open
space outside the vacuum chamber. The HG-TFM device can act as a compact and cryogen-freedesktop-type magnetic source to provide a large magnetic force and could be useful in a numberof life/medical science applications, such as protein crystallization and cell culture.
Keywords: bulk superconductors, trapped field magnets, high gradient magnets, finite element
method, magnetic levitation, quasi-zero gravity(Some figures may appear in colour only in the online journal)

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Supercond. Sci. Technol. 34 (2021) 035001

K Takahashi et al

1. Introduction

zero-resistivity due to electromagnetic induction after applying and removing a large magnetizing field. Such (RE)BaCuO
bulks with a critical temperature, T c , over 90 K are the mostpromising materials to be utilized as a trapped field magnet(TFM) in a compact and desktop-type magnetic source. The
record-high trapped field, 17.6 T, was achieved at the centerof a two-stack GdBaCuO bulk pair magnetized by field-cooledmagnetization (FCM) at 26 K using outer shrink-fit stainlesssteel (SS) rings [7] as well as at 23 K using SS laminations[8]. Regarding the validity of bulk TFMs magnetized by FCM,
Vakaliuk et al indicated that the reproducibility of the trappedfield can be verified experimentally under the applied field of
10 T at 50 K, but might not be proved for higher fields over
Bapp = 14 T [9]. The exploration of practical use of such strong
TFMs can be classified into three phases: fabrication, magnetization, and applications [6, 10]. Such experimental work
has showed the significant potential of such TFMs as a powerful magnetic source. However, its configuration would not beappropriate for practical use in applications, which sometimes
require the large magnetic field and/or gradient product to beprovided in an open space inside or outside vacuum chamber.
Recently, the authors proposed a new concept of the hybrid
TFM lens (HTFML), consisting of two bulk parts: an innerbulk lens and outer bulk cylinder [11]. The diamagnetic lenseffect of the inner bulk lens can concentrate the trapped field
generated from the outer bulk TFM cylinder after magnetization. It was predicted numerically that an HTFML consistingof a GdBaCuO bulk lens and GdBaCuO TFM cylinder can
generate a concentrated magnetic field of Bc = 11.4 T at 20 K
that is higher than the external magnetic field of Bapp = 10 T,
quasi-permanently without any additional power consumption[12]. Experimental validation has proved the feasibility of the
HTFML, exploiting an inner GdBaCuO bulk lens and outer
MgB2 TFM cylinder, in which Bc = 3.5 T was achieved for
Bapp = 2.0 T at 20 K [13]. This HTFML device suggests
a new pathway to enhance the trapped field from the viewpoint of the magnetizing method in contrast to conventional
approaches that depend on the superconducting properties ofthe bulk itself.
Additionally, in the case of the HTFML using a
GdBaCuO bulk lens and GdBaCuO TFM cylinder, we
showed that a magnetic field gradient product as high as
Bz · dBz /dz = −3000 T2 m−1 could be generated inside thebulk annuli under the concentrated trapped field of Bc = 11.4 T
at 20 K, which allows for a quasi-zero gravity environment ina cost-effective and efficient way [12]. However, the HTFML
device has some challenges in its construction and the magnetizing process. The temperature of the GdBaCuO bulk lensand the GdBaCuO TFM cylinder must be controlled individually during zero field-cooled magnetization (ZFCM) for thelens part and FCM for the TFM part. A special technique isrequired in order to control the temperature separately and themagnetization process can take several hours. Moreover, it isquite difficult to realize a room temperature bore for practicalapplications exploiting the advantages of the HTFML becauseof a narrow bore less than 10 mm in diameter in the GdBaCuO
lens, which is also encapsulated in a SS holder. This does limit
the effective size for practical scientific experiments.

Zero gravity in space is known to be a characteristic condition in scientific research fields. Up to now, some experimental
research has been carried out exploiting quasi-zero gravity
conditions in the International Space Station as an environmental parameter, which focuses positively on applications
related to life/medical sciences, such as protein crystallization [1] and cell culture [2] without natural convection causedby the gravitational force of the Earth. The so-called ‘spacebiology’ resulting from this research is an important developing field that could contribute significantly to the sustainable
development goals.
A large magnetic field in combination with a large field
gradient can provide a repulsive force against gravity on the
Earth and achieve a counter-balance situation for any diamagnetic materials such as water, common metals and even cells of
the human body. In 1990, magnetic levitation of diamagnetic
materials such as water and plastics was first demonstrated by
Beaugnon et al using a hybrid-type superconducting magnet(HM), in which a magnetic field gradient product, Bz · dBz /dz,
as high as −1923 T2 m−1 was applied along the vertical direction against gravity on Earth [3].
Superconducting technologies have satisfied several
demands for high magnetic fields including magnetic levitation. Nikolayev et al summarizes the specification of existing superconducting magnets worldwide, where the highest
Bz · dBz /dz = −3000 T2 m−1 with a record-high magnetic
field of 30 T has been ever achieved in a HM and superconducting coil magnet (SM) [4]. Cryo-cooled superconducting
magnets without liquid helium that can generate Bz · dBz /dz
up to −400 T2 m−1 with magnetic fields up to 10–12 T are
now used practically at the laboratory scale.
The magnetic force, F m , acting on a material increases inproportional to the magnetic field, Bz , and the magnetic field
gradient, dBz /dz, along the z-axis (upper direction) as follows,z
Fm ∝ Bz · dB
dz .

(1)

To enhance F m , there are some trends related to the magnet
design, i.e. the improvement of Bz and/or dBz /dz. One example
is the gradient-type superconducting magnet developed forprotein crystallization, in which an inverse coil made from
Nb3 Sn was placed on the hybrid coil, consisting of an inner
Nb3 Sn coil and outer NbTi coils. As a result, the magnet cangenerate a Bz · dBz /dz as large as −1500 T2 m−1 around theboundary of both coils [5]. The most significant issue is thatsuch a comparatively large Bz · dBz /dz value has been realized
only in a specialized facility [4]. For practical use of a magnetic quasi-zero gravity condition on ground, it is desirable
for the magnetic source to be lightweight, mobile, and costeffective as a desktop-type apparatus, and preferably cryogenfree, i.e. operating without the need for any coolant such as
liquid helium.
Large, single-grain (RE)BaCuO bulk superconductors (RE:rare earth element or Y) can be utilized as a compact andstrong, quasi-permanent magnet [6]. Magnetic flux is trappedin bulk superconductors by a supercurrent flowing with2

Supercond. Sci. Technol. 34 (2021) 035001

K Takahashi et al

Figure 1. (a) Cross-sectional schematic of the three-dimensional numerical model of the HG-TFM, in which a conventional ring-shaped

TFM cylinder, labelled ‘full-TFM’ (yellow), is sandwiched by two TFM cylinders with slits 10 deg. wide in the diagonal direction, labelled‘slit-TFM’ (orange). The top views of the full-TFM and slit-TFM are also shown in (b) and (c), respectively.

In this study, we propose a new high gradient TFM (hereafter referred to as HG-TFM) to realize a quasi-zero gravity environment, in which slit ring bulks are combined withbulk TFM cylinders, and the magnetic properties were investigated in detail during and after magnetization using stateof-the-art numerical simulations. A maximum value of Bz ·dBz /dz = −6040 T2 m−1 was obtained after conventional
FCM from Bapp = 10 T at 40 K for an HG-TFM with 10 mm
in inner diameter, which may be the highest value ever reported compared to any other magnetic sources. The mechanicalproperties of the HG-TFM were also investigated to propose arealistic configuration of the HG-TFM device reinforced by ametal support structure that is essential to prevent mechanicalfracture of the bulk superconducting material during the FCM
process due to its brittle, ceramic nature.

Figure 2. Magnetizing sequence for FCM of the HG-TFM, whichis the same as that for a conventional TFM. The external field, Bex ,is decreased linearly at a ramp rate of −0.222 T min−1 from
Bapp = 3, 5 or 10 T at a constant temperature at 40 K.

2. Numerical modeling procedure
inner diameter (I D) is assumed to be either 10, 20 or 36 mm
for both TFM components. The height (H) of the full-TFM
is 48 mm, for which three ring-shaped bulks 16 mm in H are
stacked, and each of the slit-TFMs are 16 mm in H. In themodel, the height of the full-TFM is determined so that thetrapped field capability to be 10 T for the HG-TFM with wider
I D = 36 mm and is fixed temporarily for the comparison withvarious I D cases. The number of layers of the full-TFM can be
reduced within the capability required in the device. Each bulk
is mounted separately in an Al-alloy ring 5 mm in thickness (OD = 70 mm, I D = 60 mm, H = 16 mm) adhered by a thin layerof epoxy resin. A thin indium sheet layer (0.1 or 1 mm in thickness) is inserted between each bulk to reproduce the boundary condition between each bulk in the realistic HG-TFM. A

Focusing on magnetic force applications for magnetic levitation [14–16], a higher magnetic field gradient product, Bz ·dBz /dz, is desirable in the system as indicated by equation (1).
The magnetic field gradient (dBz /dz) term should be highly
considered, as well as the trapped field (Bz ) term, to improvethe value of Bz · dBz /dz overall.
Figure 1(a) presents a cross-sectional schematic of thethree-dimensional numerical model of the HG-TFM, in whicha conventional ring-shaped TFM cylinder, labeled ‘full-TFM’,
is sandwiched by two TFM cylinders with slits 10 deg. wide in
the diagonal direction, labeled ‘slit-TFM’. Figures 1(b) and (c)show the top views of the full-TFM and slit-TFM, respectively.
The outer diameter (O D) of the HG-TFM is 60 mm and the3Supercond. Sci. Technol. 34 (2021) 035001

K Takahashi et al

Table 1. Numerical fitting parameters for the assumed J c (B) characteristics of the GdBaCuO bulk used in equation (3), where the β value isadjusted to reproduce the experimental result for the trapped field [12].
T (K)J c1 (A m−2 )BL (T)

J c2 (A m−2 )Bmax (T)αβ40

3.5 × 109

0.9

2.7 × 109

6.0

0.8

0.50

Figure 3. The time step dependence of the magnetic field, Bz , during FCM from Bapp = 3, 5 and 10 T at 40 K at the center of (a) the single

full-TFM (I D = 10 mm, O D = 60 mm, H = 48 mm) and (b) the single slit-TFM (I D = 10 mm, O D = 60 mm, H = 16 mm, slit 10 deg.
wide). The schematic view of the magnetic flux distribution and induced current direction in (c) the single full-TFM and (d) the single
slit-TFM.

solenoid coil of 170 mm in O D, 120 mm in I D and 200 mm
in H is used to apply the magnetic field for the magnetization
process (not shown in figure 1). As described earlier, appropriate mechanical reinforcement must be considered for the bulks
due to their brittle, ceramic nature for improving the reliability of the magnetization process and the device itself. Several studies have investigated the fracture behavior of the bulk
TFMs, which has a high risk of occurring under FCM fromapplied fields over 10 T at temperatures lower than 50 K, inparticular [17–19]. As shown in figure 1(a), the model represents a realistic experimental setup for the mechanical reinforcement of the bulk magnets, which will be performed in thenear future, employing outer Al-alloy rings and an outer SS
capsule of 5 mm in thickness that can provide a compressive
stress over −100 MPa when cooling from room temperature
to the operating temperature to perform FCM. The mechanicalreinforcement could endure up to Bapp = 10 T in actual FCM
experiments [20, 21].
Figure 2 presents the magnetizing sequence for FCM of the
HG-TFM, which is the same as that for a conventional TFM.
The external field, Bex , is decreased linearly at a ramp rateof −0.222 T min−1 . from Bapp = 3, 5 or 10 T at a constanttemperature at 40 K. Once the external field is ramped down

to 0 T for step 5, the static magnetic field 15 min later (forstep 8) is then referred to as the trapped field value, BT . Only
the z-component is considered for magnetic levitation because
gravity acts along the z direction. The electromagnetic phenomena of the bulk superconductor during the magnetization
process are analyzed using the commercial software, PhotoEDDY (Photon Ltd, Japan).
The nonlinear electrical property of the bulk superconductor is derived from the E–J power law as follows,
E = Ec

( )n
J
Jc

,

(2)

where Ec = 10−4 V m−1 is the characteristic electrical field,
n = 20 is an appropriate power-law exponent for GdBaCuO
bulk, and J c is the critical current density. The realistic J c (B)characteristics, including the fish-tail effect under certain magnetic fields, can be represented by the following equation proposed by Jirsa et al [22],
{
(
)
[ (
)α )]}(B
B
Jc (B) = β · Jc1 exp − BBL + Jc2 Bmax
exp α1 1 − Bmax
,
(3)

4

Supercond. Sci. Technol. 34 (2021) 035001

K Takahashi et al

Figure 4. Numerical results for (a) the magnetic field, Bz , and (b) the magnetic field gradient product, Bz · dBz /dz, after magnetization with

Bapp = 10 T at 40 K along the z-axis inside the bulk annuli, comparing three types of TFMs: the HG-TFM, the HTFML and a conventionalring-shaped TFM (single full-TFM).

where each relevant parameter, J c1 , BL , J c2 , Bmax , and α are
fitted from the experimental results for a GdBaCuO bulk reported by Kii et al [23]. The fitting coefficient, β, is employed so
as to modify the deviation from the actual trapped field value(obtained experimentally) after FCM, particularly at temperatures lower than 50 K [12]. The parameters assumed for the
GdBaCuO bulk material for equation (3) are summarized intable 1.

current state-of-the-art J c (B, T) characteristics [8]. The schematic view of the magnetic flux distribution and induced current direction in the single full-TFM is shown in figure 3(c),in which the conventional magnetic flux lines can be seen.
In the single slit-TFM case shown in figure 3(b), when Bex
decreased linearly, the Bz steeply decreased and became a negative value. Finally, the Bz value took a negative maximum atstep 5 and then slightly recovered to the final BT value becauseof the non-linear electrical property of bulk superconductor(see equation (2)). At step 8, BT = −2.8, −3.3, −3.6 T values
were obtained at the center of the single slit-TFM for Bapp = 3,
5 and 10 T, respectively. Figure 3(d) also shows the schematic view of the magnetic flux distribution and induced current direction in the single slit-TFM. An induced current flows
counterclockwise in each piece separated by the slits, which isthe same direction with the full-TFM. However, the downward(−z-direction) magnetic flux would penetrate through the slit
gap and exist even inside the central bore to build the magnetic loop in each piece of the slit-TFMs, which becomes the
opposite direction to that generated by the single full-TFM(+z-direction). The HG-TFM then provides a large magnetic
field gradient around the interface between the full-TFM withupward magnetic flux lines and the slit-TFM with downward
magnetic flux lines.
Figures 4(a) and (b), respectively, show a comparison of thenumerical results for the magnetic field, Bz , and the magnetic
field gradient product, Bz · dBz /dz, after magnetization with
Bapp = 10 T at 40 K along the z-axis inside the bulk annuli,
comparing three types of TFMs: the HG-TFM, the HTFML

3. Electromagnetic analysis results
Before describing the mechanism of the HG-TFM in detail, theelectromagnetic properties of each key component of the HGTFM—the single full-TFM and the single slit-TFM—were
investigated. Figures 3(a) and (b), respectively, show the time
step dependence of the magnetic field, Bz , during FCM from
Bapp = 3, 5 and 10 T at 40 K at the center of the single fullTFM (I D = 10 mm, O D = 60 mm, H = 48 mm) and the single
slit-TFM (I D = 10 mm, O D = 60 mm, H = 16 mm, slit 10
deg. wide). For the single full-TFM case shown in figure 3(a),the final trapped field, BT , after FCM was almost equivalent toeach Bapp because of partial magnetization within the full capability of the full-TFM, which also enables the generation of
a comparatively homogeneous magnetic field inside the bulk
bore. Such a high field is realized by an induced supercurrent
flowing inside the full-TFM as depicted. Nowadays, it is estimated that the trapped field capability for a disk-shaped bulk pair
(O D = 24 mm, H = 24 mm) could be over 20 T, based on5Supercond. Sci. Technol. 34 (2021) 035001

K Takahashi et al

Figure 5. (a) The magnetic field, Bz , and (b) the magnetic field gradient product, Bz · dBz /dz, profiles of the HG-TFM with inner bores of

I D = 10, 20 and 36 mm along the z-direction inside the bulk bore after FCM from Bapp = 10 T at 40 K, compared with those of a single
full-TFM (−24 mm ⩽ z ⩽ 24 mm) and single slit-TFM pair (±z = 25 ∼ 41 mm).

(±z = 25 ∼ 41 mm). In figure 5(a), the slit-TFM pair witha narrower bore of I D = 10 mm generates a higher negative
trapped field (= −3 T) over a wider area at ±z = 20 ∼ 40 mm,
which is effective for improving the HG-TFM performancewith a larger magnetic field gradient. The negative trappedfield of the single slit-TFM pair decreased with increasing
I D value and there is no remarkable negative value for
I D = 36 mm. On the other hand, for the cases of the full-TFM
and HG-TFM, the Bz profile becomes broad with increasing
I D value and the shape of the trapped field profile no longer
varies between the full-TFM and HG-TFM with I D = 36 mm.
Since Bz decays significantly with distance outside of the
TFM, i.e. at ±z ⩾ 24 mm, there is less, or no magnetic flux
controlled in the HG-TFM with wider I D = 36 mm.
In figure 5(b), for the HG-TFM, the peak value of Bz ·dBz /dz was enhanced with decreasing I D at the boundary
position between each TFM around ±z = 24 mm due to thepresence of the slit-TFM. A maximum Bz · dBz /dz value of
6040 T2 m−1 was achieved for I D = 10 mm, which is larger than that for the full-TFM (−3790 T2 m−1 ). It should
be noted that a Bz · dBz /dz value as large as −1400 T2 m−1along the +z-direction of the vertical magnet is required tolevitate a diamagnetic water drop in air [24]. In this sense,
the effective volume for magnetic levitation can be estimatedto be +z = 19–30, 17–30 and 13–27 mm inside the bore of
the HG-TFM with I D = 10, 20 and 36 mm, respectively.
For protein crystallization in a diamagnetic solution, a larger
Bz · dBz /dz value of −4450 T2 m−1 is required to realize anapparent zero-gravity condition without buoyancy flow, which
is much higher than that estimated numerically [25]. For realizing a higher field gradient product, the HG-TFM with a narrower inner bore is desirable, leading to a trade-off between
field gradient product and effective volume.

and a conventional ring-shaped TFM (single full-TFM). The
dimensions of each TFM are shown in the figure, where thedimensions of the full-TFM part were assumed to be identicalfor each case (I D = 10 mm, O D = 60 mm, H = 48 mm).
The dimensions of the magnetic lens for the HTFML can bereferred to elsewhere [12]. In figure 4(a), the trapped field, Bz ,at the center (z = 0 mm) of the HG-TFM and full-TFM should
be comparable with the applied field of Bapp = 10 T. On theother hand, for the HTFML exploiting the diamagnetic lenseffect, Bz can be concentrated into the bulk annuli, in which aconcentrated trapped field of Bc = 11.0 T was achieved at 40 K
at z = 0 mm. In figure 4(b), the HG-TFM shows the highest
Bz · dBz /dz value of −6040 T2 m−1 at the boundary between
each TFM part, z = 24 mm, which is remarkably superior thanthe other two types of TFM. In this sense, to improve the magnetic force efficiently, it would be more desirable to control
the magnetic field gradient (dBz /dz) profile by focusing on thegradient itself rather than improving the trapped field (Bz ). One
significant advantage of the HG-TFM is that its superior magnetic properties could be realized using a conventional FCM
process at one temperature and without any additional costs,
except for the slit-TFM. Note again that figure 4 is an example
assuming a narrow 10 mm bore in inner diameter for each system. The magnetic properties of the HG-TFM would change
depending on the shape of the bulk material (O D, I D, H, slit
angle and so on) and the conditions of the magnetization process, such as the magnetizing temperature and Bapp .
Figures 5(a) and (b), respectively, show the magnetic field,
Bz , and the magnetic field gradient product, Bz · dBz /dz, profiles of the HG-TFM with inner bores of I D = 10, 20 and
36 mm along the z-direction inside the bulk bore after FCM
from Bapp = 10 T at 40 K, compared with those of a single
full-TFM (−24 mm ⩽ z ⩽ 24 mm) and single slit-TFM pair6Supercond. Sci. Technol. 34 (2021) 035001

K Takahashi et al

Figure 6. (a) The final trapped field, BT (= Bz at step 8), at the center (x = y = z = 0) and (b) the maximum field gradient product,|Bz · dBz /dz|, at the boundary position between the full and slit-TFMs of the HG-TFM, as a function of Bapp for various I D cases. The
results are compared to that of the full-TFM cases and other large-scale magnets: superconducting magnets (SM), and hybrid-type
superconducting magnets (HM).

Figures 6(a) and (b) summarize the final trapped field, BT(= Bz at step 8), at the center (x = y = z = 0) and the maximum
field gradient product, |Bz · dBz /dz|, around the boundary position between the full and slit-TFMs of the HG-TFM, as a function of Bapp for various I D cases. The results for the full-TFM
cases are also shown for reference. In figure 6(a), the trappedfield capability is equivalent for all cases; thus, the relation
BT = Bapp , is established based on the typical FCM process.
However, for the magnetic field gradient product shown infigure 6(b), the HG-TFM would work more effectively for
Bapp higher than 5 T and for narrower bores smaller thanI D = 20 mm compared to the full-TFM. Table 2 summarizes
the maxima of the magnetic properties for each case including
the HG-TFM, HTFML and full-TFM, extracted from figures 4
and 6. The HG-TFM with I D = 10 mm would provide amaximum |Bz · dBz /dz| = 6040 T m−2 when BT = 10 T
at 40 K, which may be the record-high value compared tothat of other large-scale magnets: SM ∼ 1500 T m−2 , andHM ∼ 3000 T m−2 using liquid helium, shown in figure 6(b)[4, 26].
In most related studies, based on apparatus using a permanent magnet with a magnetic field ∼1 T, the magnetoArchimedes method must be exploited to reduce the requiredmagnetic force using a paramagnetic medium solution[27, 28]. The HG-TFM no longer needs such a method,and consequently, can provide simpler, medium-free operation with air or water. These advantages of the HG-TFM
can provide the versatility of a quasi-zero gravity space asa desktop-type magnetic source for magnetic levitation processing, which enables the suppression of gravity-induced
convection in a wide range of potential industrial applications
such as protein crystallization and cell culture [1, 2]. It should
be noted that Bz · dBz /dz values of 1980 and 2760 T2 m−1 canbe achieved after FCM from Bapp = 10 T even for the single
full-TFM with I D = 36 and 20 mm (without the slit-TFM),
respectively, as shown in table 2, which are higher than that
achieved by SMs [4]. These results suggest that the single fullTFM can be also applicable for quasi-zero gravity in the open

space outside the vacuum chamber with a range of 15–30 mm
in diameter. There is a limit for the enlargement of the space inthe bore because it is difficult to fabricate good quality bulks>60 mm in O D, resulting in a narrower bore compared to thatof SMs.
In our previous work [12], where the mechanism of magnetic levitation in the quasi-zero gravity space is explained indetail, it was highlighted that a large magnetic field gradientproduct up to 3000 T2 m−1 of that particular HTFML design
could be highly applicable to magnetic separation. Regarding the effectiveness of the magnetic force for crystal growth,
it is shown elsewhere that the magnetic force affects several
features: the magnetic field orientation, convection control,
reduction of sedimentation and so on [14]. However, there
are no theoretical rules yet regarding how the magnetic force
works during the process and it differs between the target
materials used. Thus, appropriate design of the I D of the HGTFM needs to be explored as related to the particular application, where the magnetic force profile in the HG-TFM could
be correlated with the quality of the resultant crystallization.

4. Mechanical stress analysis results
The mechanical fracture of bulk superconductors has beenreported in several studies, where a crack might occur along
the circumferential direction mainly due to the hoop stressinduced during FCM from applied fields over 10 T at temperatures below 50 K [17–19]. To date, to prevent such mechanical fracture during magnetization, mechanical reinforcement has been usefully applied using glass fiber reinforced
epoxy resin [29] or shrink-fit steel [7]. We also proposed a new
hat structure of an outer metal reinforcement ring based on
the results of mechanical stress simulations, in which a large(RE)BaCuO ring-shaped bulk (I D = 36 mm, O D = 60 mm,
H = 20 mm) was experimentally confirmed to endure FCM
from 10 T without fracture [20]. For the HTFML consisting of a (RE)BaCuO lens and (RE)BaCuO TFM cylinder,7

Supercond. Sci. Technol. 34 (2021) 035001

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Table 2. Numerical results of maxima of the magnetic properties for each case including the HG-TFM, HTFML and full-TFM magnetizedat 40 K, extracted from figures 4 and 6.
HG-TFM
Bulk I D (mm)Bz (Bapp ) (T)|Bz ·dBz /dz| (T2 m−1 )(single full-TFM)
Magnetization
Temperature (K)10
10 (10)6040 (3790)

HTFML
20
10 (10)3460 (2760)36
9.8 (10)1980 (1850)1011 (10)2430 (1850)
ZFCM + FCM

FCM
40

Table 3. Assumed mechanical parameters (EY :Young’s modulus

the mechanical properties have been analyzed during themagnetizing process from Bapp = 10 T [30], in which thenecessity of adequate mechanical reinforcement was strongly
suggested, at least for the GdBaCuO TFM cylinder, but possibly the magnetic lens as well. It is also necessary to investigate the mechanical properties of the present HG-TFM as wellduring FCM, where the full-TFM part is tightly sandwichedby the slit-TFM pair.
After calculating the trapped field profile shown in the previous section, the nodal force was then imported into anothersoftware package, Photo-ELAS (Photon Ltd., Japan), for calculating the electromagnetic stress on each mesh element of
the bulk TFMs during magnetization, which enables us toassess the reliability of the HG-TFM in terms of the mechanical reinforcement method.
Elastic behavior in an isotropic material can be expressedby Hooke’s law, in which the stress tensor, σij , is linearly proportional to the strain tensor, εij , as follows
σij = λ · εkk · δij + 2G · εij ,

(4)

E·ν
λ = (1+ν)(1−2ν) ,

(5)

G = 2(1E+ν) ,(6)and ν: Poisson ratio) for numerical stress analysis of the HG-TFM
shown in figure 1 [20, 31].

(RE)BaCuO bulk
Epoxy resin
Al alloy (A7075-T6)
Indium
SS 304EY (GPa)

ν

100
3.0
78
12.7
193

0.33
0.37
0.34
0.45
0.28

main component of the electromagnetic stress results from thehoop stress along the y-direction in the full-TFM part. The
maximum σYY value was almost independent of the bore size
because the trapped field is equivalent for each HG-TFM withdifferent bore sizes as shown in figure 6(a). These numericalresults for the stress profile are similar to that reported for theannular (RE)BaCuO bulk magnet system (full-TFM) for an
NMR spectrometer [21], in which an electromagnetic hoop
stress about 100 MPa was generated at the inner peripheral
region of the annular bulks reinforced by a similar structure.
This means that the present reinforcement technique would
work even for the present HG-TFM system. Considering thecompressive stress of −100 MPa from the reinforcement by
the Al-alloy rings and the SS capsule during the cooling process before FCM, the maximum total stress would be less than
the fracture strength of typical Ag-doped (RE)BaCuO bulks
of 50–70 MPa [32]. As a result, the full-TFM part should beappropriately reinforced and not break [21].
Figures 8(a)–(c) show the contour map of the Von Mises
stress, σVM , the electromagnetic hoop stress along the ydirection, σYY , and the electromagnetic radial stress along thex-direction, σXX , respectively, for the slit-TFM pair of the HGTFM with inner diameters of I D = 10, 20 and 36 mm, justafter FCM from 10 T. For the case of I D = 10 mm, a maximum Von Mises stress of σVM = 54 MPa exists at the surfaces of the slit-TFM pair facing the full-TFM part. The σXX
profile for each I D, as shown in figure 8(c), is similar tothat for σVM . On the other hand, the σYY contribution to the
Von Mises stress, as shown in figure 8(b), is relatively small.
These results suggest that the main component of the electromagnetic stress results from the electromagnetic radial stressalong the x-direction, σXX , in the case of I D = 10 mm andI D = 20 mm. However, for the wider I D = 36 mm, the electromagnetic stress mainly results from the hoop stress along they-direction, σYY . Since an induced current flows in two regions

where λ and G represent Lame’s constants, δij is the Kronecker
delta function, EY is the Young’s modulus, and ν is the Poisson ratio. The mechanical parameters, EY and ν, for each component of the HG-TFM are summarized in table 3, which areassumed in the simulation to be isotropic and in the elastic
region. Note that all components, including the epoxy resin,
indium and air gap, as shown in figure 1, were included in themodeling of the HG-TFM, which should be considered for theaccurate prediction of the mechanical behavior [31].
Figures 7(a) and (b) show a contour map of the Von Mises
stress, σVM , and the electromagnetic hoop stress along the ydirection, σYY , respectively, calculated for the full-TFM partof the HG-TFM with inner diameters of I D = 10, 20 and
36 mm, just after FCM from 10 T (step 5). In figure 7(a),the σVM value, which is a scalar quantity, represents the total
value of the mechanical stresses including all directions. Compared with the maximum σYY value shown in figure 7(b), which
is close to the maximum σVM value of 90 MPa around theinner periphery of the full-TFM part, it was found that the8Supercond. Sci. Technol. 34 (2021) 035001

K Takahashi et al

Figure 7. Contour map of (a) the Von Mises stress, σVM , and (b) the electromagnetic hoop stress along the y-direction, σYY , calculated forthe full-TFM part of the HG-TFM with inner diameters of I D = 10, 20 and 36 mm, at step 5 just after FCM from 10 T.

Figure 8. Contour map of (a) the Von Mises stress, σVM , (b) the electromagnetic hoop stress along the y-direction, σYY , and (c) the

electromagnetic radial stress along the x-direction, σXX , for the slit-TFM pair of the HG-TFM with inner diameters of I D = 10, 20 and
36 mm, just after FCM from 10 T.

in the slit-TFM independently due to the existence of slits, theresultant trapped field and the mechanical stress become lower,
compared to the full-TFM, even if the slit-TFM is magnetizedby FCM from 10 T.

During the magnetization of the HG-TFM exploiting slitTFMs stacked on the full-TFM, an extra stress component
may be generated due to the repulsive force between each
TFM part. Figure 9(a) shows a cross-sectional contour map
9Supercond. Sci. Technol. 34 (2021) 035001

K Takahashi et al

Figure 9. (a) Cross-sectional contour map of the electromagnetic stress along the vertical direction, σZZ , for the HG-TFM with various

inner diameters, just after the FCM process from 10 T. (b) The σZZ distribution along the z-direction at the inner periphery (x = 18 mm) andat the outer periphery (x = 30 mm) for the HG-TFM with I D = 36 mm, extracted from (a).

of the electromagnetic stress along the vertical direction, σZZ ,for the HG-TFM with various inner diameters, just after the
FCM process from 10 T. The positive σZZ value is concentrated mainly in the inner periphery of the slit-TFM facing thefull-TFM, and the magnitude of σZZ increases with increasing
I D These results suggest that a repulsive force exists in thisregion. However, the positive σZZ region vanishes at the outerinterface of the slit-TFM facing the full-TFM (z = ±24 mm,−15 ⩽ x ⩽ 30 mm for I D = 36 mm case), which indicatesthe possibility of an attractive force or very small repulsive
force there. Figure 9(b) shows the σZZ distribution along thez-direction at the inner periphery (x = 18 mm) and at the outer
periphery (x = 30 mm) for the HG-TFM with I D = 36 mm,
which were extracted from figure 9(a). At the inner periphery,the σZZ value changes from positive to negative around theinterface (z = ±24 mm). These results indicate that a repulsive force exists at the inner interface. On the other hand, atthe outer periphery, the σZZ value is negative and continuous

for both the slit-TFM and full-TFM, suggesting that an attractive magnetic force might exist at the outer interface. Similar
trends were also observed for the HG-TFM for I Ds = 10 and
20 mm.
Figure 10(b) shows a contour map of the final trapped field,
BT , of the HG-TFM with I D = 20 mm after FCM from 10 T.
It should be noted that a trapped field along the +z-direction
exists at the periphery of the slit-TFM (region A), which has asimilar BT strength of 10 T as for the full-TFM. These results
suggest that ‘region A’ is magnetically attracted to the magnetized full-TFM. On the other hand, there exists little fieldtrapped downward along the z-direction in the inner periphery
region of the slit-TFM (‘region B’). These results indicate thata repulsive magnetic force exists in the counter side in the fullTFM, which can be understood well from the schematic viewof the magnetic flux lines in figure 10(a).
Figure 10(a) shows a schematic view of the magnetic
flux profile for the HG-TFM after FCM. To understand the10Supercond. Sci. Technol. 34 (2021) 035001

K Takahashi et al

Figure 10. (a) Schematic view of the magnetic flux profile for the HG-TFM after FCM. (b) Contour map of the final trapped field, BT , ofthe HG-TFM with I D = 20 mm after FCM from 10 T. (c) The effective magnetic poles of the HG-TFM after FCM. The outer periphery
region (region A) and inner periphery region (region B) of the slit-TFM can be considered as opposing magnetic poles.

distribution of the magnetic flux lines at the interface visually,
the full-TFM and the slit-TFM pair are placed at a distance.
It can be confirmed visually that the magnetic flux lines coexist at the interface and form the superposition of the magnetic fields that is destructive around the central bore or constructive around the bulk periphery.
Let us describe these results using a magnetic pole model.
Figure 10(c) shows the effective magnetic poles of the HGTFM after FCM, which presents the mechanism of the larger
magnetic gradient of the HG-TFM. The outer periphery region(region A) and inner periphery region (region B) of the slitTFM can be considered as opposing magnetic poles. As a result, region A is attractive (an N–S or S–N pair) and region B
is repulsive (N–N or S–S pair) with respect to the magnetizedfull-TFM at the interface. Since both TFM parts are mounted
in an Al-alloy ring and the whole HG-TFM is encapsulated
in an SS capsule, σZZ acts as a compressive stress, and thatmaxima is around 50 MPa for the full-TFM and 35 MPa forslit-TFM in the HG-TFM with I D = 36 mm. However, in the
narrower bore cases, I D = 10 and 20 mm, σZZ decreases despite the superior Bz · dBz /dz performance. In any case, there
is little fracture risk due to the compressive stress for the HGTFM during FCM from 10 T, since the compressive fracture
strength of (RE)BaCuO bulk is about one order of magnitude
higher than the tensile mechanical strength [33].

quasi-zero gravity space using magnetic force, in which a slit
ring bulk (slit-TFM) was tightly stacked with a TFM cylinder(full-TFM). State-of-the-art numerical simulations were used
to investigate the magnetic and mechanical properties detailduring and after magnetization. The important results and conclusions in this study are summarized as follows.
(a) A maximum value of the magnetic field gradient productof Bz · dBz /dz = −6040 T2 m−1 was obtained after conventional FCM from Bapp = 10 T of the HG-TFM witha 60 mm outer diameter and 10 mm inner diameter. Thisvalue may be the highest ever reported compared to any
other magnetic sources. The superiority of the HG-TFM
is the easier magnetization process including a single temperature and a simpler construction without any additionalcost, but also the higher Bz · dBz /dz can be provided in an
open space outside the vacuum chamber.
(b) The Bz · dBz /dz value increased with decreasing inner diameter of the HG-TFM and with increasing applied field,
Bapp , during FCM. Even if the inner diameter of the HGTFM is 36 mm, a Bz · dBz /dz value of 1980 T2 m−1 can beachieved, which is slightly higher than 1850 T2 m−1 forthe single full-TFM case.
(c) The electromagnetic stress during the FCM process was
also investigated to analyze the mechanical behavior
of HG-TFM device. The electromagnetic stress in the
HG-TFM device during the FCM process mainly resultsfrom the hoop stress along the y-direction, σYY , for the fullTFM.
(d) The outer and inner periphery regions of the slit-TFM
can be considered as opposing magnetic poles. The outer

5. Conclusion
We have proposed a new concept of a HG-TFM. The HG-TFM
is made from (RE)BaCuO bulk superconductors to realize a11Supercond. Sci. Technol. 34 (2021) 035001

K Takahashi et al

periphery region is attractive (N–S or S–N pair) and theinner periphery region is repulsive (N–N or S–S pair) withrespect to the magnetized full-TFM at the interface. However, it was confirmed that there is little fracture risk forthe bulks during FCM from 10 T in the proposed realistic configuration of the HG-TFM, where both TFM parts
are mounted in Al-alloy rings and the whole HG-TFM is
encapsulated in an SS capsule.
(e) A quasi-zero gravity space can be realized using the HGTFM with its large Bz · dBz /dz value in an open spaceoutside the vacuum chamber even without the magnetoArchimedes method. The HG-TFM device is a compact
and cryogen-free desktop-type magnetic source to provide
a large magnetic force and can be applicable to scientificresearch, such as in the life/medical sciences for protein
crystallization and cell culture.

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Acknowledgments
This research is supported by JSPS KAKENHI Grant No.
19K05240 and by Adaptable and Seamless Technology transfer Program through Target-driven R&D (A-STEP) fromJapan Science and Technology Agency (JST), Grant Nos.
VP30218088419 and JPMJTM20AK. M D Ainslie would liketo acknowledge 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
Keita Takahashi  https://orcid.org/0000-0002-8278-2688
Hiroyuki Fujishiro  https://orcid.org/0000-0003-1483835X
Mark D Ainslie  https://orcid.org/0000-0003-0466-3680
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