Superconductor Science and Technology
Supercond. Sci. Technol. 35 (2022) 054003 (11pp)

https://doi.org/10.1088/1361-6668/ac5fe3

Validation of a desktop-type magnet
providing a quasi-microgravity spacein a room-temperature bore of ahigh-gradient trapped field magnet(HG-TFM)
Keita Takahashi1,∗, Hiroyuki Fujishiro2,∗ and Mark D Ainslie3
1

Department of Physics, Faculty of Science, Gakushuin University, 1-5-1 Mejiro, Toshima,
Tokyo 171-8588, Japan2
Department of Physical Science and Materials Engineering, Faculty of Science and Engineering,
Iwate University, 4-3-5 Ueda, Morioka, Iwate 020-8551, Japan3
Bulk Superconductivity Group, Department of Engineering, University of Cambridge,
Cambridge CB2 1PZ, United Kingdom
E-mail: keita.takahashi@gakushuin.ac.jp and fujishiro@iwate-u.ac.jp
Received 4 February 2022, revised 11 March 2022
Accepted for publication 22 March 2022
Published 4 April 2022
Abstract

The concept of a high-gradient trapped field magnet (HG-TFM), which incorporates a hybrid
system of two (RE)BaCuO superconducting bulk components with different functions, wasproposed in 2021 by the authors based on the results of numerical simulations. The HG-TFM asa desktop-type magnet can be a more effective way to generate a higher magnetic field gradientproduct of Bz · dBz /dz (>−1400 T2 m−1 , as calculated for a pure water), which can realize aquasi-microgravity space applicable for Space Environment Utilization on a laboratory scale. In
this study, to validate the quasi-microgravity space in the HG-TFM, a prototype HG-TFM
apparatus has been built using a slit-bulk TFM and stacked full-TFM (without slits) with innerdiameters of 36 mm. After field-cooled magnetization from 8.60 T at 21 K, a trapped field of
BT = 8.57 T was achieved at the center (i.e. at the bottom of a room temperature bore of 25 mm
diameter outside the vacuum chamber), and consequently, a maximum
Bz · dBz /dz = −1930 T2 m−1 was obtained at the intermediate position between the slit-bulk
TFM and the stacked full-TFM. Magnetic levitation was demonstrated successfully for bismuth
particles and a pure water drop, which validates the quasi-microgravity environment in the
HG-TFM. Based on numerical simulation results of the trapped field profile, it is concluded thatthe reason for the instability of the levitated targets is because of the repulsive magnetic force
applied along the horizontal plane. The levitating state can be controllable, for example, bychanging the operating temperature, which would allow objects to levitate statically along thecentral axis.

∗

Authors to whom any correspondence should be addressed.

Original content from this work may be used under the termsof the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and thetitle of the work, journal citation and DOI.
1361-6668/22/054003+11$33.00 Printed in the UK

1© 2022 The Author(s). Published by IOP Publishing Ltd

Supercond. Sci. Technol. 35 (2022) 054003

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Keywords: bulk superconductors, trapped field magnets, high gradient magnets,
magnetic levitation, quasi-zero gravity, magnetic force, quasi-microgravity(Some figures may appear in colour only in the online journal)

1. Introduction

for pure water by equation (3) and this is difficult to achieveusing a conventional permanent magnet or electromagnets.
Up to now, a large magnetic field sufficient for magnetic levitation can be generated by large-scale magnets: superconducting magnets (SMs), which can generate Bz · dBz /dz up
to −500 T2 m−1 with 10 T [7], and hybrid magnets with
Bz · dBz /dz up to −3000 T2 m−1 with 20 T [8]. A cryocooled 10 T SM is more practical on the laboratory scalefrom the viewpoint of its reasonable footprint, and it does
not require any cryogenic coolants such as liquid nitrogen or
liquid helium for operation. Hence, in most studies relating
to magnetic levitation [7, 9] or magnetic separation [10, 11],the magneto-Archimedes method has been employed as toreduce the required Bz · dBz /dz for magnetic levitation of a
specific target by exploiting an additional paramagnetic gas or
liquid (e.g. oxygen or aqueous MnCl2 ). In this sense, there is amajor difficulty in developing a practical magnet that improves
magnetic performance and achieves this with a reasonablefootprint; hence, providing a quasi-microgravity space in an
efficient way.Large, single-grain (RE)BaCuO bulk superconductors (RE: rare earth element or Y) can be used as a so-called
trapped field magnet (TFM) that can provide the trapped field
after it is magnetized. In the field-cooled magnetization (FCM)
process, for example, where an external magnetizing field isramped down at a constant temperature below the superconductor’s critical temperature, a magnetic flux density as large
as several Tesla becomes trapped inside the material due tothe ‘vortex pinning effect’, and the induced supercurrent flows
with zero resistance quasi-permanently. Based on the criticalstate model reported by Bean [12, 13], the magnitude of thetrapped field at the center of the top surface, BT , of a discshaped bulk TFM can be expressed by,High magnetic fields and magnetic field gradients are key parameters for developing a magnet able to levitate diamagnetic
materials such as water, human cells and common metals. Thisso-called magnetic levitation can be achieved in an equilibrium condition, where the magnetic levitation force is appliedsuch that it counteracts Earth’s gravitational force. In scientificresearch fields—for example, in the life/medical sciences forprotein crystallization [1] and cell culture [2, 3]—such magnetic levitation can be utilized as a quasi-microgravity space
without natural convection due to gravity. Various case studies exploiting a quasi-, or real, micro-gravity space have beenreported in the International Space Station (ISS) for the purposes of academic research, science education and space business [4]. This research field is an important developing baserelating to the so-called ‘Space Environment Utilization,’ ledby the Japan Aerospace Exploration Agency and the ISS program that aims to develop innovative technologies for humanliving on Earth as well as on space.
The potential energy per unit volume, U, of a diamagnetic
particle in a static external B-field can be expressed in termsof the total amount of the magnetic energy, UM , and the gravitational energy, Ug [5, 6],1(χs − χm ) B2 + (ρs − ρm ) gz, (1)2µ0√where µ0 is the vacuum permeability, B = B2x + B2y + B2z isthe magnitude of the magnetic field, g is the acceleration due togravity, and z is the vertical position. χs and ρs are the volume
susceptibility and the density of the target substance, respectively. In the same sense, χm and ρm are the parameters for the
medium surrounding the target, such as a liquid or gas. The
total force density balance, Fz , along the z-direction (central
vertical axis) at x = y = 0 mm, is given as,
U = UM + Ug = −∂U(x=y=0)(χs − χm ) dBz
Fz = −=Bz− (ρs − ρm ) g.
∂z
µ0
dz

BT = kµ0 Jc a,(t
k = ln2a(2)ρs − ρm
dBz
=
µ0 g.
dz
χs − χm√1+( a )2t),(5)where J c is the critical current density of the superconducting material, a is the sample radius. k is a geometric factor
to account for the finite thickness of the disc bulk, t [14].
Hence, based on equation (4), a simple way to achieve a higher
Bz · dBz /dz for magnetic levitation is to achieve a higher BT
by improving the superconducting properties (i.e. J c ) or enlarging the diameter of the material. The record-high trapped fieldof 17.6 T was achieved at the center of a stacked disc-shaped
GdBaCuO bulk pair magnetized by FCM with an external fieldof 17.8 T at 26 K using stainless steel (SS) rings for mechanical reinforcement [15]. In subsequent experiments, it was confirmed that laminated structures exploiting SS spacers within
the stacked bulks are more reliable for realizing similar BT
values over 17 T [16].

Hence, the magnitude of Fz is determined by the Bz · dBz /dzvalue, the magnetic field gradient product, which changes
depending on the position in the vertical bore of the levitation
magnet. In the levitating condition, where the vertical component of the magnetic repulsive force is equivalent to the forceof gravity, the equilibrium condition for magneto-Archimedes
levitation can then be solved as,
Bz

a
+
t

(4)

(3)

This relationship requires a large Bz · dBz /dz value to levitate
any substances with a small χs (>−1400 T2 m−1 , as calculated2Supercond. Sci. Technol. 35 (2022) 054003

K Takahashi et al

To develop a practical magnet using bulk superconductors from the viewpoints of both the Bz · dBz /dz performanceand the magnet versatility (e.g. easy to operate on a laboratory scale), the authors proposed a new concept of hybrid-type
TFMs which incorporates two bulk components with different functions: a hybrid trapped field magnet lens (HTFML)
and a high gradient trapped field magnet (HG-TFM). In the
numerical simulations of the HTFML in 2018 [17], a concentrated magnetic field, Bc = 11.0 T, was obtained owing to thediamagnetic lens effect of the inner bulk lens magnetized by
the zero field-cooled magnetization (ZFCM) process, which ishigher than the trapped field from the outer bulk TFM cylinder magnetized by FCM in the descending stage of the ZFCM
process with 10 T at 40 K. A maximum Bc = 9.8 T was validated in experiments with a 7 T external field exploiting aloose contact method [18], where a huge temperature difference between each bulk component, as large as ∼100 K, wasrealized to incorporate two magnetizing processes: FCM of theouter bulk cylinder and ZFCM of the inner magnetic bulk lens,systematically. In contrast, the HG-TFM, which incorporates
slitted ring bulks stacked on a conventional bulk cylinder, has
the advantage of improving the field gradient, dBz /dz, by controlling the trapped field profile with an inverted field from theslit-bulk. Numerical results showed that a maximum value ofBz · dBz /dz = −6040 T2 m−1 was estimated in the case of the
HG-TFM with an inner diameter (I.D.) of 10 mm magnetizedby FCM with 10 T at 40 K [19]. This suggests a more effective
way for magnetic levitation since the HG-TFM requires just a
simple FCM process at a constant temperature and can provide
a higher Bz · dBz /dz value in an open space outside the vacuum
chamber.
In this study, to validate the quasi-microgravity space inthe HG-TFM, we have built a prototype device using a slitbulk TFM and stacked full-TFM (without slits) with I.Ds. of
36 mm. This results in a room temperature bore of 25 mm
diameter. All of bulk components are encapsulated in a SS
capsule for reinforcement and tightly connected to the cold
stage of the cryocooler for cooling (before magnetization) to
the lowest temperature of 21 K. In the present HG-TFM, a
trapped field at the center (i.e. the bottom of the room temperature bore outside the vacuum chamber) of BT = 8.57 T
was achieved by FCM from 8.60 T at 21 K, and consequently,
a maximum Bz · dBz /dz = −1930 T2 m−1 was obtained at
the intermediate position between the slit-bulk TFM and thestacked full-TFM. Magnetic levitation was demonstrated for
bismuth particles and a pure water drop in the room temperature bore of the HG-TFM. Numerical simulation of the trappedfield profile was also carried out to explain the instability of
the levitated targets observed by an optical microscope in theexperiment.

‘slit-TFM’. Figures 1(b) and (c), respectively, show the top
views of the full-TFM and slit-TFM. The outer diameter ofthe HG-TFM is 60 mm and the I.D. is 36 mm. All of thesebulk superconductors were fabricated by using the QMGTM
method (Nippon Steel Corporation, Japan) and each bulk wasmachined mechanically to prepare each of the full-TFM andthe slit-TFM part of the HG-TFM. The total height (H) of thethree stacked full-TFM components is 60 mm (each 20 mmin H), and the single, slit-TFM is 17 mm in H. Note that a
SS spacer of 1 mm in H is necessary to isolate the magnetic
loop of each bulk component. All of the bulk materials areencapsulated in an Al alloy ring 5 mm in thickness and a SS
holder 5 mm in thickness for mechanical reinforcement, asshown in figure 1(a). The numerical assumptions for the mechanical reinforcement were discussed previously in the conceptual paper [19], in which a compressive stress of −100 MPa
provided by both the Al-alloy rings and the SS capsule witha cooling process from 300 K to 40 K would be enough tocounteract the electromagnetic hoop stress of about 100 MPa
during FCM from an applied field of 10 T. In the present experiment, in which a temperature gradient would occur across
such large, four-stacked TFMs of ≈100 mm in total height
H, a copper heat sink is also incorporated for achieving better cooling from the inner periphery of each TFM. Overviewsof the whole apparatus are also presented in the photographs
in figure 2. The outer size of the apparatus including the cryocooler is (35 cm × 35 cm) × 103 cm in height, which includes
a room-temperature bore of 25 mm in diameter and 80 mm indepth.
The temperature was monitored on the top surface of theouter SS capsule using a CernoxTM thermometer and under
the cold stage of the 10 K GM-cycle helium cryocooler usinga Pt–Co thermometer (for details, see section 4). The HG-TFM
was enclosed in a vacuum chamber, in which an ambientpressure was kept below 10−3 Pa using a turbo-molecularpump. A cryocooled 10 T SM (JASTEC, JMTD-10T100) wasused to magnetize the HG-TFM by FCM from an appliedfield of 8.60 T, which was ramped down at a constant rateof −0.05 T min−1 . It was predicted for the HG-TFM thatan applied field above 8 T would be enough to achieve therequired Bz · dBz /dz value as large as −1400 T2 m−1 for magnetic levitation of pure water [19]. During the magnetizing
process, the trapped field was measured using an axial-type
Hall sensor at z = 0 mm, the bottom of the room-temperaturebore corresponding to 2 mm above the center of the full-TFM
component. Hence, the interface of the two bulk components
between the slit-TFM and the full-TFM is at z = 28 mm. The
external magnetic field was calculated from the electric current measured by a shunt resistor. The fundamental diamagnetic targets—bismuth particles and a pure water drop—arelevitated in the room-temperature bore outside the vacuum
chamber to demonstrate the quasi-microgravity space in the
HG-TFM. The in-situ observation of magnetic levitation was
carried out using an optical microscope system (OLYMPUS
Ltd, MK027) connected to a CCD camera. Material properties, including the density and the volume susceptibility, were
measured using a magnetic balance (Sherwood Scientific Ltd,
MSB-MKI) for theoretical calculations using equation (3).

2. Experimental setup
Figure 1(a) presents the cross-sectional schematic view of theexperimental setup of the HG-TFM, in which a stack of ringshaped TFM cylinders, labeled ‘full-TFM,’ is attached to a
separate TFM cylinder with two slits 500 µm in width, labeled3

Supercond. Sci. Technol. 35 (2022) 054003

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Figure 1. (a) Cross-sectional schematic view of the experimental setup of the HG-TFM, in which a stack of three conventional ring-shaped

TFM cylinders, labelled ‘full-TFM’ (yellow), is attached to a separated TFM cylinder with two slits 500 µm in width, labelled ‘slit-TFM’(orange). The top views of the full-TFM and slit-TFM are also shown in (b), (c), respectively.

parameters used in the present modeling were also described
previously in the original numerical study of the HG-TFM[19].
The nonlinear E–J characteristic of the bulk superconductor is represented by a power-n law as follows,( )nJ
E = Ec,(6)Jcwhere Ec (=10−4 V m−1 ) is the characteristic electric field,
and n = 20 is a typical value for (RE)BaCuO bulk superconductors [23, 24]. The critical current density, J c (B), canbe described by the following equation reported by Jirsa et al[25–27],{()
B
B
Jc (B) = β · Jc1 exp −
+ Jc2
BL
Bmax
[ (
(
)α )]}
1
B
×exp
1−
,
(7)
α
Bmax

Figure 2. Side and top views of the whole apparatus. The outer sizeof the apparatus is (35 cm × 35 cm) × 103 cm in height, which
includes a room-temperature bore of 25 mm in diameter and 80 mm
in depth.

where each fitting parameter, J c1 , BL , J c2 , Bmax , and α canbe determined from a realistic experimental J c value. The fitting coefficient, β, was also introduced to reproduce the actual
trapped field value in experiments, as reported in [28], but for
20 K in the present simulation. The fitting parameters for bulk
superconductors used in equation (7) are presented in table 1.
There is a minor discrepancy between the numerical modeland the experimental setup shown in figure 1, for which theslit-TFM has a wider slit, 6 degrees wide, due to a limitationof the number of mesh elements required. Note that the othernon-superconducting components in the HG-TFM (Al alloy,

3. Numerical modelling framework
Based on the experimental setup shown in figure 1, a threedimensional (3D) numerical model of the HG-TFM was constructed using the commercial software Photo-EDDY (Photon
Ltd, Japan), in which the magnetization of the HG-TFM by
the external SM was resolved using the finite element method[20–22]. The simulation procedure and the superconducting

4Supercond. Sci. Technol. 35 (2022) 054003

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Table 1. Numerical fitting parameters for the assumed J c (B) characteristics of the GdBaCuO bulk for equation (7), where the β value isadjusted to reproduce the experimental result for the trapped field by FCM at 20 K.
T (K)J c1 (A m−2 )BL (T)

J c2 (A m−2 )Bmax (T)αβ20

9.0 × 109

1.5

5.4 × 109

8.0

0.5

0.29

Figure 5 shows the time dependence of (a) the temperatures
measured at each position, and (b) the external field, Bex , andthe magnetic field, Bz , during magnetization with an appliedfield, Bapp = 8.60 T. During magnetization, the temperatures
gradually increased with decreasing external field and T top
took a maximum value of 54.9 K. Such thermal instability may
cause flux jumps and mechanical fracture of the bulk cylinder;hence, the temperature should be kept as constant as possible.
In the present setup, a trapped field as large as BT = 8.57 T
can be maintained without flux creep even after completing themagnetization process. This implies that the stacked TFM cylinders could have sufficient capability in terms of their criticalcurrent density for partial magnetization with Bapp = 8.60 T ata relatively high temperature above 50 K for FCM, where theinduced current might flow around the outer peripheral region
and not saturate inside of each of the TFMs.
Figure 6(a) shows the magnetic field, Bz , and the magnetic field gradient product, Bz · dBz /dz, profiles measured by scanning a Hall probe inside the room-temperaturebore of the HG-TFM along the z-direction after FCM from
Bapp = 8.60 T at 21 K. A schematic view of the apparatus that includes the optical observation system for magnetic levitation is also presented. It is shown that the maximum peak value of Bz · dBz /dz = −1930 T2 m−1 occurs at
z = 25 mm, just below the interface of the two bulk components, between the slit-TFM and the full-TFM, and a trappedfield BT = 8.57 T is obtained at z = 0 mm, the bottom of
the room temperature bore. Figure 6(b) shows photographsof the levitated targets: bismuth particles enclosed in a test
tube, and a pure water drop, which is dripped from the tipof a dropper in air. The results of the in-situ observation
can be download online (URL1: https://imgur.com/QSBNlF5,
URL2: https://imgur.com/KX07NtE). Firstly, the levitatedbismuth particles appear stable and collide with the counterside of the test tube against the camera, which floatedat z ∼= 35 mm (±1 mm): here Bz · dBz /dz = −695 to−950 T2 m−1 , obtained from the Bz · dBz /dz profile infigure 6(a). On the other hand, the water drop moves up anddown unstably because of gravity acceleration when it drippedout from the tip of the dropper. Meanwhile, the water drop
was pushed out and collided with the side wall of the innerbore at z ∼= 30 mm (±1 mm): here Bz · dBz /dz = −1308 to−1595 T2 m−1 . Comparing these materials, it is important tonote that the magnetic buoyancy force along the z-direction
should be applied slowly or step by step to levitate stably ata constant z-position as shown for the bismuth particles. Thisis difficult for a pure water drop in liquid form as it is. As
for the stability along the horizontal plane, there is a common
issue that both targets could not remain along the central axis,
x = y = 0 mm, due to a repulsive magnetic force along the

Figure 3. Time sequence of the external field, Bex , at the center ofthe magnetizing solenoid coil, and the operating temperature, T,
during FCM of the HG-TFM in the simulation. The applied field,
Bapp , corresponding to the initial Bex value, was ramped down from8.60 T to 0 T over six steps with a ramp rate of −0.05 T min−1 . The
temperature, T = 20 K, was assumed to be constant and uniform,
assuming isothermal conditions.

SS and copper) were considered as a non-magnetic material(assumed as air) for the present FCM process that is essentiallyquasi-static with a slowly-ramped magnetic field.
Figure 3 presents the conceptional view of time sequence ofthe external field, Bex , at the center of the magnetizing solenoid
coil and the operating temperature, T, during FCM of the HGTFM in the simulation. The applied field, Bapp , corresponding
to the initial Bex value, was ramped down from 8.60 T to 0 T
over six steps with a ramp rate of −0.05 T min−1 . The temperature, T = 20 K, was assumed to be constant and uniform,
assuming isothermal conditions.
4. Experimental results
Figure 4 shows the typical experimental results of the cooling test, for which the temperatures were measured on the
top surface of the HG-TFM, T top , and under the cold stage,
T stage . Note that the magnetizing process is not yet includedin this initial stage. The temperature was cooled down from300 K and reached the lowest values 10 h later: T stage = 17.5 K
under the cold stage and T top = 21.0 K at the top surface ofthe HG-TFM. Hence, the temperature difference between eachmeasurement point was minimized to within 4 K owing to theutilization of the copper heat sink. The use of this heat sink
is recommended to achieve better cooling efficiency for suchstacked TFM cylinders.

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Figure 4. Typical experimental results for the cooling test, for which the temperatures were measured on the top surface of the HG-TFM,

T top , and under the cold stage, T stage . Note that the magnetizing process is not yet included in this initial stage.

Figure 5. (a) Time dependence of the temperatures measured at each position, and (b) the external field, Bex , and the magnetic field, Bz ,during magnetization with an applied field, Bapp = 8.60 T. A Hall probe is placed on the bottom of the room-temperature bore, whichcorresponds to the center of the external SM.

6Supercond. Sci. Technol. 35 (2022) 054003

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Figure 6. (a) The magnetic field, Bz , and the magnetic field gradient product, Bz · dBz /dz, profiles measured by scanning a Hall probe inside

the room-temperature bore of the HG-TFM along the z-direction after FCM from Bapp = 8.60 T at 21 K. A schematic view of the apparatus
that includes the optical observation system for magnetic levitation is also presented. (b) Photographs of the levitated targets: bismuth
particles enclosed in a test tube, and a pure water drop, which is dripped from the tip of a dropper in air. The results of the in-situ
observation can be downloaded online (URL1: https://imgur.com/QSBNlF5, URL2: https://imgur.com/KX07NtE).
Table 2. Material properties of density, ρs , and the volume susceptibility, χs , as well as the calculated Bz · dBz /dz values compared with theexperimental Bz · dBz /dz values obtained from the corresponding levitation positions.
Targets
BismuthPure water

Density ρs (g cm−3 ) Volume susceptibility χs (-, SI) Calculated Bz · dBz /dz (T2 m−1 ) Experimental Bz · dBz /dz (T2 m−1 )
3.54
0.995

−4.93 × 10−5
−8.54 × 10−6

−879
−1376

−695 to −950
−1308 to −1595

tube. Theoretical Bz · dBz /dz values were calculated based onequation (3), using the material properties of air as a solvent
referred from [29], which gives −879 T2 m−1 for the bismuth
particles, and −1376 T2 m−1 for the pure water drop. It was
confirmed that the experimental results for the levitation position for each substance with finite size are reasonable fromthese theoretical assumptions.

horizontal plane, which will be discussed later based on the
numerical results shown in figure 8. These experimental results validate that the present HG-TFM can provide a quasimicrogravity space, where a buoyancy magnetic force along
the z-direction is sufficient to levitate the target materials.
Table 2 summarizes the material properties of density, ρs ,and the volume susceptibility, χs , as well as the calculated
Bz · dBz /dz values, which are compared with the experimental
Bz · dBz /dz values obtained from the corresponding levitation positions. Note that for bismuth particles in solid form,
the effective values of ρs and χs measured by the magneticbalance should become smaller in consideration of the measurement setup, consistent with the actual magnetic levitation
environment, which includes air between each particle in a test

5. Numerical simulation results
Figure 7 shows the numerical results of (a) the magnetic field,
Bz , and (b) the magnetic field gradient product, Bz · dBz /dz,
profiles inside the bore of the HG-TFM (Sim._20 K) and7

Supercond. Sci. Technol. 35 (2022) 054003

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Figure 7. Numerical results of (a) the magnetic field, Bz , and (b) the magnetic field gradient product, Bz · dBz /dz, profiles inside the bore of

the HG-TFM and the stacked-TFM along the vertical, z, direction after FCM from Bapp = 8.60 T. The experimental result of the HG-TFM
presented in figure 6(a) is labelled as ‘Exp._21 K’ for comparison. The other experimental result, in which the HG-TFM was magnetized at
a higher temperature, T top = 40 K, is also given as ‘Exp._40 K’.
Table 3. Comparison of magnetic performance, in terms of the trapped field, BT , and magnetic field gradient product, Bz · dBz /dz, for eachcase of the HG-TFM and only the TFM, extracted from figure 7.
Case
HG-TFM

Only TFM

T (K)

BT (T)

Bz · dBz /dz (T2 m−1 )

21
40
20
20

8.57
8.39
8.49
8.50

−1930
−1840
−1508
−1397

Exp._21 K
Exp._40 K
Sim._20 K
Sim._20 K_TFM

Bz · dBz /dz, profiles inside the HG-TFM and the stacked-TFM
only, after FCM from Bapp = 8.60 T at 20 K. The top schematic views indicate the repulsive, or restoring, magnetic force
along the horizontal, xy-plane, presenting the stability along
the horizontal plane for a target material levitated in different
z-positions. At z > 30 mm in the HG-TFM, the Bz profile curves
outward and forms a convex-shaped profile along the horizontal plane in the bore, which means a repulsive force would
be applied such that the levitated targets collide with a test tube
for bismuth particles or the side wall of the vacuum chamber
for the water drop, as observed in the experiment. On the other
hand, at z < 30 mm, the Bz profile forms a concave-shaped
profile, where a restoring force would be applied such that thelevitated targets remain on the central axis, x = y = 0 mm,
as depicted in the inset. In a related study of magnetic separation using a conventional SM, reported by Ueda et al [30],the magnetic force along the horizontal plane, which differs
depending on the height position, is utilized to separate kindsof plastics in an efficient way. In the case of only the stackedTFM without the negative Bz of the slit-bulk, there is a slightdifference 2 mm above, i.e. z = 32 mm, with an inflection
point that determines the shape of the Bz profiles as convex-,
or concave-shaped. Numerical results imply that the levitatingstate of the targets would be determined by the vector potentialof the magnetic force, which differs along the vertical positionof the HG-TFM. The reason that both targets could not remainalong the central axis as shown in the video comes from a
repulsive magnetic force along the horizontal plane in the bore

the stacked-TFM (Sim._20 K_TFM) along the z-direction
after FCM from Bapp = 8.60 T at 20 K. For the case of‘Sim._20 K_TFM,’ the slit-TFM is considered as air in themodel. The experimental result of the HG-TFM presented infigure 6(a) is labelled as ‘Exp._21 K’ for comparison. The
other experimental result, in which the HG-TFM was magnetized at a higher temperature, T top = 40 K, is also given
as ‘Exp._40 K’. As summarized in table 3, a similar magnetic performance could be obtained for the Exp._40 K casemagnetized at T top = 40 K, but the Bz · dBz /dz peak as largeas −1840 T2 m−1 was equivalent to or a little lower thanthat at 21 K. In the numerical results of the Bz · dBz /dz profiles, Bz · dBz /dz for the Sim._20 K case has a peak valueof −1508 T2 m−1 at the same z-position, z = 25 mm, as
the Exp._21 K case; however, it appears to be much lower
than the experimental value. This discrepancy might comefrom the present 3D modelling, in which the wider slit angleof 6 degrees was employed because of the limitation on thenumber of mesh elements. Furthermore, the negative magnetic field trapped in the slit-TFM should be explored morein additional experimental studies, to clarify the magnetic performance of each bulk component—the slit-TFM and the fullTFM—separately. Only the case of ‘Sim._20 K_TFM’ hada peak in a different z-position, z = 29 mm, owing to theabsence of the slit-TFM, which supports the validity of thehybrid structure of the HG-TFM in the present modelling.
Figure 8 shows the numerical results of (a) the magnetic field, Bz , and (b) the magnetic field gradient product,8Supercond. Sci. Technol. 35 (2022) 054003

K Takahashi et al

Figure 8. Numerical results of (a) the magnetic field, Bz , and (b) the magnetic field gradient product, Bz · dBz /dz, profiles inside the

HG-TFM and the stacked-TFM only, after FCM from Bapp = 8.60 T at 20 K. The top schematic views indicate the repulsive, or restoring,magnetic force along the horizontal, xy-plane, describing the stability along the horizontal plane for a target material levitated in different
z-positions.

6. Conclusionof the HG-TFM. One possible recommendation is to avoid
such a transition zone between convex-, or concave-shaped
profiles at z = 30 mm, and levitate target objects statically on
the central axis at z < 30 mm for the present HG-TFM. The
Bz · dBz /dz profile could be controlled properly, for example,
by increasing the operating temperature after magnetization sothat the intensity of the Bz · dBz /dz peak at z < 30 mm decreases
to the required value for magnetic levitation as calculated byequation (3). The HG-TFM with its superior magnetic capability, Bz · dBz /dz = −1508 T2 m−1 , has an advantage oflevitating a wide range of diamagnetic materials, compared toBz · dBz /dz = −1397 T2 m−1 for the conventional stackedTFM, as shown in figure 8(b).
The concept of the HG-TFM—including the proposed
structure for the cooling and magnetizing process—is validated experimentally, showing that the quasi-microgravity
space can be utilized with superior magnetic performance ina room-temperature bore outside the vacuum chamber. These
results show that a magnetic field source using bulk superconductors would become an important platform for further
applicational studies towards Space Environment Utilization,such as magnetic levitation, magnetic separation, and convection control in a quasi-microgravity.

A prototype HG-TFM was built, in which a slit-ring bulk
superconductor (slit-TFM) was placed on stacked ring bulks(full-TFM), with the aim of validating the device as a desktoptype magnet, providing a quasi-microgravity space in the room
temperature bore of the HG-TFM. Numerical simulation of
the trapped field profile in the HG-TFM was also carried out
to confirm the validity of the magnetic performance of the
HG-TFM measured in the experiment, as well as to explain
the mechanism of the levitating state observed in the demonstration of magnetic levitation. The important results and conclusions in this study are summarized as follows.
(a) The present HG-TFM incorporates the slit-bulk TFM andthe three stacked full-TFM, all with I.Ds. of 36 mm, which
results in a room-temperature bore of 25 mm in diameterand 80 mm in depth. The outer size of the apparatus including the cryocooler is (35 cm × 35 cm) × 103 cm in height.
(b) A trapped field of BT = 8.57 T was achieved at z = 0 mm,
the bottom of the room temperature bore, after the FCM
process from Bapp = 8.60 T at 21.0 K. This resulted ina maximum peak value of the magnetic field gradient9Supercond. Sci. Technol. 35 (2022) 054003

K Takahashi et al

product of Bz · dBz /dz = −1930 T2 m−1 at z = 25 mm,
the interface of the two bulk components, between the slitTFM and the full-TFM. During magnetization, the temperatures gradually increased with decreasing external field
and the temperature of the top surface of the HG-TFM,
T top , took a maximum value of 54.9 K. It was confirmedthat a heat sink had a good effect on both the cooling process and the magnetizing process to keep the temperature
as low as possible in such stacked TFM cylinders.
(c) The in-situ observation of magnetic levitation has validated the quasi-microgravity environment of the HG-TFM
for bismuth particles, which floated at z ∼= 35 mm(±1 mm): here Bz · dBz /dz = −695 to −950 T2 m−1 anda pure water drop, which floated at z ∼= 30 mm (±1 mm):here Bz · dBz /dz = −1308 to −1595 T2 m−1 . It was confirmed that the experimental results for the levitation positions for each substance with finite size are reasonable withtheoretical Bz · dBz /dz values calculated from the material
properties of density and magnetic susceptibility.
(d) Numerical simulation results gave a peak valueof Bz · dBz /dz = −1508 T2 m−1 at the same
z-position, z = 25 mm, for the HG-TFM. Only
the case of ‘Sim._20K_TFM’ had a lower peak ofBz · dBz /dz = −1397 T2 m−1 in a different z-position,z = 29 mm, owing to the absence of the slit-TFM,
which supports the validity of the hybrid structure of the
HG-TFM.
(e) It should be noted that the levitating state of the targets
would be determined by the vector potential of the magnetic force, which differs along the vertical position of the
HG-TFM. As the first step, the magnetic buoyancy force
should be applied slowly or step by step to levitate stably
at a constant vertical position, as observed for the bismuth
particles. This is difficult for a pure water drop in liquid
form as it is. Furthermore, regarding to the horizontal direction, a repulsive magnetic force would be applied witha convex-shaped magnetic profile at z > 30 mm such thatthe levitated targets collide with a test tube or the side wallof the vacuum chamber as observed, while the restoring
magnetic force would be applied with a concave-shaped
profile at z < 30 mm in contrast.

M D Ainslie would like to acknowledge financial support
from an Engineering and Physical Sciences Research Council(EPSRC) Early Career Fellowship, EP/P020313/1.
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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Data availability statement
The data that support the findings of this study are openly
available at the following URL/DOI: https://doi.org/10.17863/
CAM.80920.
Acknowledgments
A portion of this research was supported from JSPS Fellows
Grant No. 21J14069, as well as from JSPS KAKENHI Grant
No. 19K05240, and from Adaptable and Seamless Technology transfer Program through Target-driven R&D (A-STEP)
from Japan Science and Technology Agency (JST), Grant No.
JPMJTM20AK.

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