Simulation study for magnetic levitationin pure water exploiting the ultra-highmagnetic field gradient product of a hybrid
trapped field magnet lens (HTFML)
Cite as: J. Appl. Phys. 127, 185106 (2020); https://doi.org/10.1063/5.0005356
Submitted: 19 February 2020 . Accepted: 19 April 2020 . Published Online: 11 May 2020
K. Takahashi, H. Fujishiro, and M. D. Ainslie

J. Appl. Phys. 127, 185106 (2020); https://doi.org/10.1063/5.0005356© 2020 Author(s).

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Simulation study for magnetic levitation in purewater exploiting the ultra-high magnetic field
gradient product of a hybrid trapped field magnet
lens (HTFML)
Cite as: J. Appl. Phys. 127, 185106 (2020); doi: 10.1063/5.0005356
Submitted: 19 February 2020 · Accepted: 19 April 2020 ·Published Online: 11 May 2020
K. Takahashi,1,a)

H. Fujishiro,1,b)

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CrossMarkand M. D. Ainslie2,c)

AFFILIATIONS
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 Kingdoma)Author to whom correspondence should be addressed: s3119004@iwate-u.ac.jpb)c)E-mail: fujishiro@iwate-u.ac.jp

E-mail: mark.ainslie@eng.cam.ac.uk

ABSTRACT
A hybrid trapped field magnet lens (HTFML) is a promising device that is able to concentrate a magnetic field higher than the applied fieldcontinuously, even after removing an external field, which was conceptually proposed by the authors in 2018. In this study, we propose a
new additional advantage of the HTFML, which could be applicable for magnetic levitation and separation. The HTFML device consistingof a GdBaCuO bulk cylinder and a GdBaCuO magnetic lens, after the magnetization process from an applied field, Bapp = 10 T, can generatea maximum trapped field, Bc = 11.4 T, as well as an ultra-high magnetic field gradient product, Bz⋅dBz/dz, over ±3000 T2/m at Ts = 20 K,
which is higher than that of existing superconducting magnets and large-scale hybrid magnets. Through detailed numerical simulations, the
HTFML device is considered for the magnetic separation of a mixture of precious metal particles (Pt, Au, Ag, and Cu) dispersed in purewater, by exploiting the magneto-Archimedes effect. The HTFML can be realized as a compact and mobile desktop-type superconducting
bulk magnet system, and there are a wide range of potential industrial applications, such as in the food and medical industries.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0005356
I. INTRODUCTION
The realization of superconducting magnets that can generatestationary magnetic fields up to 40 T has been encouraged by many
types of fundamental studies related to magnetism: magnetic levitation1,2 and separation,3,4 crystal growth,5,6 and nuclear magnetic
resonance (NMR).7,8 The magnetic levitation technique, in particular, needs a strong magnetic field from an external magnet torealize the counterbalanced levitated situation against gravity acting
on a diamagnetic material. The technique was first demonstrated
for a diamagnetic water drop, in which the magnetic field gradientproduct, Bz⋅dBz/dz, was required to be as large as −1400 T2/m
along the upper direction of the vertical magnet.9 Such a large
Bz⋅dBz/dz value is required because of the low magnetic

J. Appl. Phys. 127, 185106 (2020); doi: 10.1063/5.0005356
Published under license by AIP Publishing.

susceptibility of water and is difficult to achieve with a conventional
superconducting magnet. Thus, up to now, such magnetic levitation has been realized only by using a comparatively large-scale
hybrid magnet in a specialized facility.10 In the past, such largescale hybrid magnets have been demonstrated with Bz⋅dBz/dz
values around −1500 T2/m or better.11,12 The Bz⋅dBz/dz value for astandard 10 T cryo-cooled superconducting magnet is much lower,
up to −500 T2/m, which has been enhanced with a ferromagnetic
iron yoke up to −1060 T2/m.2 The magneto-Archimedes technique,proposed in 1998, also enables the reduction of the required
Bz⋅dBz/dz for the magnetic levitation of any target objects in a paramagnetic medium gas or liquid (e.g., oxygen or aqueous MnCl2).13
This method is utilized for the magnetic separation system for

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ARTICLEstructural isomers of phthalic acid using oxygen and fluorocarbon
instead of MnCl2 solution, as introduced in Ref. 3. In such magnetic applications, the whole system, including the magnetic source,
separation unit, and medium solution, must be considered, basedon the physical properties of the target objects. It would be also
desirable to avoid using any harmful materials, as well as to construct the separation system in a simple way for ease-of-processing,reliability, and stability. The intensity of the magnetic force, Fm, ispotentially restricted by the performance of the magnet used;however, it is not particularly cost-effective and realistic to build aconventional superconducting magnet or a large-scale magnet inany industrial facilities or laboratories, except for experimental purposes.6 There is still a demand for a new magnetic field source thatcan provide both a stronger magnetic field, Bz, and magnetic field
gradient product, Bz⋅dBz/dz, if it can be done in a cost-effective andefficient way.
Large, single-grain (RE)BaCuO bulk superconductors (RE:rare earth element or Y) can trap or expel magnetic flux, owing to
the induced supercurrent flowing inside the material with zero
resistance, and that functionality depends on the magnetizing
method. Toward practical engineering applications, researchadvances in bulk superconductivity, including the relevant technologies such as material processing, cryogenic systems, and magnetization techniques, are summarized in Ref. 14.
Such superconducting bulks, magnetized by field-cooled magnetization (FCM) as a so-called trapped field magnet (TFM), cantrap magnetic flux inside the bulk due to the “vortex pinningeffect” and generate quasi-permanent magnetic fields over 17 T atmagnetizing temperatures below 30 K. A record-high trapped fieldof 17.6 T has been reliably achieved by FCM at 26 K in the gap
between two GdBaCuO disk bulks of 25 mm in diameter.15,16
Pulsed-field magnetization (PFM) is also known as a key magnetizing technique to develop a desktop and mobile magnetizing
system, for which the dynamic magnetic flux behavior and theresultant heat generation inside the bulk have been carefully investigated to improve trapped fields using PFM.17–19 The record-hightrapped field by PFM is 5.2 T at 28 K on the surface of a 45 mm
diameter GdBaCuO disk bulk,20 and 5.3 T at 30 K in the gap
between two 25 mm diameter GdBaCuO disk bulks.21 In this way,
bulk superconductors have been considered as a strong andcompact magnetic field source, which could realize a desktop-type
mobile magnet for applications in a cryogen-free NMR system22,23
and Lorentz force velocimetry (LFV).24
Based on the critical state model by Bean,25–26 the trappedfield at the center of the top surface of a disk-shaped bulk, BT, canbe expressed asBT  kμ0 Jc d,(1)where μ0 is the permeability of free space, Jc is the critical current
density of the superconducting material, d is the sample diameter,
and k is the geometric factor to account for the finite thickness ofthe disk bulk. Hence, based on Eq. (1), a simple way to realize
higher BT is to improve Jc by improving its superconducting material properties27 or to make its diameter larger.28
Superconducting bulks, magnetized by zero-field-cooled magnetization (ZFCM), can expel the external magnetic flux from its

J. Appl. Phys. 127, 185106 (2020); doi: 10.1063/5.0005356
Published under license by AIP Publishing.

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inside due to its “diamagnetic effect” and, as shown for the magnetic lens in the HTFML, can enhance a magnetic field in thecentral bore of the lens. Several studies have investigated the electromagnetic properties and thermal stability of such magnetic
lenses from the viewpoints of superconducting material selection,geometry, and mechanical reinforcement.29–31 A concentrated fieldof 13 T at 20 K was achieved under a background field of 7 T, inwhich a GdBaCuO lens was impregnated with an epoxy resin toprevent mechanical fracture during its magnetization.31
The authors proposed a hybrid trapped field magnet lens
(HTFML) consisting of two superconducting bulk components—
an outer TFM cylinder and an inner bulk magnetic lens—in2018.32 The HTFML can quasi-permanently generate a concentrated magnetic field, Bc, higher than the applied field, Bapp, froman external magnetizing coil, even after this external field isremoved, exploiting the combination of the “vortex pinning effect”
of the outer TFM and the “diamagnetic lens effect” of the innermagnetic lens. In this conceptual paper, Bc = 13.49 T for
Bapp = 10 T was predicted numerically for a GdBaCuO TFM cylinder and GdBaCuO lens pair.32 The device was verified experimentally, for which a concentrated magnetic field Bc = 3.55 T wasachieved in the central bore of the HTFML using an MgB2 TFM
cylinder and GdBaCuO magnetic lens, magnetized with an appliedfield of Bapp = 2.0 T.33 These studies showed that the trapped fieldof the bulk could be enhanced by appropriate selection of the bulk
materials for the TFM cylinder and lens, as well as the bulk Jc anddiameter, following Eq. (1). In addition to this function, it is considered that the HTFML could generate an ultra-high magnetic
field gradient product, Bz⋅dBz/dz, which could, for example, beapplicable to magnetic separation.
In this study, three-dimensional numerical models of the
HTFML are constructed to investigate the magnetic field performance, in which realistic sizes and shapes of the bulk components
and realistic Jc characteristics are assumed, based on an experimental FCM setup. It is predicted that the HTFML could be highly
applicable to magnetic levitation as a compact and strong magnetic
field source and can generate a magnetic field gradient product,
Bz⋅dBz/dz, over −3000 T2/m, which is six times higher than that ofa conventional 10 T superconducting magnet (−500 T2/m). These
results provide further new understanding of the HTFML device
and its potential wider use in industrial applications as a strong
magnetic field source.

II. NUMERICAL SIMULATION FRAMEWORK
Commercial software, Photo-EDDY (Photon Ltd, Japan), wasused to analyze the magnetic field profile during the magnetization
process, combined with Photo-THERMO to maintain a constanttemperature for each bulk component. Electromagnetic phenomenaduring the magnetization process are described by the fundamental
equations introduced in detail in Refs. 34–36. The E–J power law isassumed to represent the highly nonlinear electrical properties ofthe superconducting material as follows:
E ¼ Ec

 n
J
,
Jc

(2)

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where Ec (=10−4 V/m) is the characteristic electric field, n = 20 is
the power-law exponent for GdBaCuO, and Jc is the critical current
density.
Figure 1 shows a schematic view of the three-dimensional
numerical model of the HTFML device, consisting of the outer
GdBaCuO TFM cylinder and inner GdBaCuO lens pair, and theassumed dimensions are exactly the same as those used in theexperimental verification of the HTFML presented in Ref. 33.
When this HTFML consists of the same superconducting material, e.g., GdBaCuO, for each bulk component, the temperature ofeach part must be controlled independently, for which a so-called“thermal switch” would be required to control the superconducting state or the normal state during the magnetization process,where both the ZFCM process for the inner magnetic lens andthe FCM process for the outer TFM cylinder are used simultaneously. This magnetizing process could be realized by utilizing an
individually controlled cryostat or the same cryostat with different cooling loops for each component. The thermal switch used
could be possibly realized mechanically,37 thermally,38 or
magnetically.
The Jc(B) characteristics of the GdBaCuO bulk are describedby the following equation, proposed by Jirsa et al.:39

FIG. 1. Schematic view of the three-dimensional numerical model of the
HTFML, consisting of the outer GdBaCuO TFM cylinder and inner GdBaCuO
lens pair. The assumed dimensions are exactly the same as those used in theexperimental verification of the HTFML presented in Ref. 33.

J. Appl. Phys. 127, 185106 (2020); doi: 10.1063/5.0005356
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


 

 
B
B
1
B α
þ Jc2
,
1
Jc (B) ¼ β  Jc1 exp 
exp
BL
Bmax
α
Bmax
(3)
where each relevant parameter, Jc1, BL, Jc2, Bmax, and α are fitted
from the experimental Jc(B) profile. In this work, the experimentally measured Jc(B) data for the bulk GdBaCuO reported by Kii
et al. in Ref. 40 are assumed, which come from a small sample. The
fitting coefficient, β, was also employed to reproduce the actual
trapped field from experimental results.
To obtain the β value at each temperature, we refer to ourexperimental FCM result for the GdBaCuO bulk ring. Figure 2 presents the temperature dependence of the trapped field, BT, in the
ring-shaped GdBaCuO bulk (40 mm in inner diameter, 64 mm inouter diameter, and 20 mm in height), which was magnetized by
FCM from an applied field of 10 T at 50 K and then measured in
the heating run as the temperature is increased to 93 K, i.e., above
its Tc.41 The experimental BT(T) curve would have a tendency tosaturate under 50 K due to the full magnetization of the bulk
within its Jc capability. When the Jc(B) characteristics obtained by a
SQUID magnetometer for a small piece of bulk sample (β ¼ 1) areused to estimate the BT value numerically at low temperatures, e.g.,
at 40 K and 20 K, the trapped field was overestimated and this saturation tendency cannot be reproduced. To reproduce the saturation
tendency of BT(T) numerically, β should be adjusted to be 0.50 for
40 K and 0.29 for 20 K, based on the extrapolation of the experimental BT(T) curve at lower temperatures. The parameters related
to the data fitting are summarized in Table I.
Figure 3 shows the time step sequence of the temperature of
the TFM cylinder, TTFM, and the magnetic lens, Tlens, and theexternal magnetic field, Bex, at the center of the magnetic lensduring the magnetizing process. At step 0, before the external field
is applied, the bulk cylinder is cooled down to TH, e.g., 100 K,

FIG. 2. The temperature dependence of the trapped field, BT, in the central
bore of the ring-shaped GdBaCuO bulk (40 mm in inner diameter, 64 mm inouter diameter, and 20 mm in height; referred to as TFM cylinder in Fig. 1),
which was magnetized by FCM from an applied field of 10 T at 50 K and thenmeasured in the heating run as the temperature is increased to 93 K. Numericalresults of BT by FCM from 10 T for β ¼ 1 shown in Eq. (3) and the most suitable BT estimations at 20 K and 40 K are also shown (see text).

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ARTICLETABLE I. Numerical fitting parameters for the Jc(B) characteristics of the bulk
GdBaCuO used in Eq. (3), for 20 K and 40 K, where the β value is adjusted toreproduce the experimental result for the trapped field in Fig. 2 (see text).

T (K)
20
40

Jc1 (A/m2)
99.0 × 10
3.5 × 109

BL (T)
1.5
0.9

Jc2 (A/m2)

Bmax (T)

α

β

8.0
6.0

0.5
0.8

0.29
0.50

9

5.4 × 10
2.7 × 109

above its Tc ( = 92 K) from room temperature, and the bulk lens iscooled to TL (<TH). In this study, two cases—TL = 40 K and
TL = 20 K—are investigated for comparison. The bulk temperature
is assumed to be constant, under isothermal conditions; thus, anyheat generation is ignored during the whole magnetizing process,assuming a near-static magnetic field. The magnetizing field, Bapp,
which corresponds to the maximum values of Bex = 3, 5, and 10 T,
is applied using a solenoid coil. The magnetizing procedure isdivided into two stages: (1) the ascending stage from step 0 to step5, and (2) the descending stage from step 5 to step 10, where theexternally applied field is ramped up and down linearly at±0.222 T/min.
III. NUMERICAL SIMULATION RESULTS
The concentrated magnetic field, Bc, in the HTFML changes
depending on the maximum applied field, Bapp, and magnetizing
temperature, TL, which can also be deduced from the Jc(B, T) characteristics indicated by Eq. (1). In this section, Bapp = 3, 5, and 10 T
was applied at TL = 20 K and 40 K in an identical HTFML device
using all GdBaCuO bulk materials, and those results are used toassess the performance of the HTFML in terms of magnetic field
and magnetic field gradient product, Bz⋅dBz/dz.
First, the magnetic flux dynamics and magnetization mechanisms of the HTFML are explained, by comparing the magnetic
field profiles during the magnetization process for each externallyapplied field. Figure 4 shows the magnetic field profiles along the x(radial)-direction at the middle of the magnetic lens, z = 0 mm,

FIG. 3. Time step sequence of the temperature of the TFM cylinder, TTFM, andmagnetic lens, Tlens, and the external magnetic field, Bex, at the center of themagnetic lens during the magnetizing process, including the ascending stage
from steps 0 to 5, and the descending stage until step 10.

J. Appl. Phys. 127, 185106 (2020); doi: 10.1063/5.0005356
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during the magnetization process for the applied fields of Bapp = 3,
5, and 10 T for TL = 20 K, which is divided into (a) the ascending
stage from steps 0 to 5, and (b) the descending stage from steps 5
to 10. In the ascending stage, when the TFM cylinder is kept in thenormal state at 100 K, but the magnetic lens is already in the superconducting state at 20 K, the diamagnetic effect of the magnetic
lens expels the external field from its inside within its Jc(B) capability. As a result, a concentrated magnetic field of Bc = 4.9 T was
obtained in the bore of the lens for Bapp = 3 T at step 5, whichbecomes higher with increasing Bapp; i.e., Bc = 7.4 T for Bapp = 5 T
and Bc = 12.5 T for Bapp = 10 T. It is worth mentioning here thatsome magnetic flux penetrates into the magnetic lens, positioned
from x = 5 to 15 mm, which becomes more obvious for higher Bapp,and this will degrade the resultant concentrated field of the
HTFML. In the descending stage, which includes the FCM process
for the TFM cylinder at 20 K, the magnetic flux is graduallytrapped in the TFM cylinder positioned at x = 18 to 30 mm withincreasing time step after step 5. The trapped field in the TFM cylinder at step 10 is roughly equal to Bapp (or slightly higher because
the applied field profile is not completely homogeneous), whichcorresponds to the so-called partial magnetization of the TFM cylinder below its full capability [see Fig. 5(b), for example]. The concentrated magnetic field in the magnetic lens appears to degrade
little during the descending stage, compared to the ascending stage,
indicating that the ascending stage of the magnetization would
determine the remnant concentrated field in the HTFML after its
magnetization. These numerical results during the magnetizing
process are helpful toward achieving a higher magnetic field and ahigher Bz⋅dBz/dz value in the HTFML to improve its performance.
Figures 5(a) and 5(b) show the cross-sectional profiles of themagnetic field, Bz, along the z (height)-direction and the induced
current density, Jy, along the y (circumferential)-direction in the
HTFML, after the magnetization process at step 10 for each Bapp at
TL = 20 K. As already shown in Fig. 4(a), the flux penetration into
the magnetic lens is more remarkable for higher applied fields, as
shown in Fig. 5(a). In the case of Bapp = 10 T, there is no region of
0 T inside the magnetic lens, meaning that the magnetic flux
completely penetrates the inside of the lens during the magnetization process. In Fig. 5(b), inside each part of the magnetic lens, theinduced current flows in the opposite direction to that in the TFM
cylinder due to the existence of the slits in the hollow cone (seeFig. 1). The Jy profile also indicates the flux penetration into themagnetic lens for higher Bapp. The magnetic lens can only exhibit
the diamagnetic effect with an induced current flowing predominantly on its surface, whereas the TFM cylinder generates its strongest trapped field by an induced current flowing through the entire
cross-section of the TFM cylinder.
Figures 6(a) and 6(b) show the time step dependence of theconcentrated magnetic field, Bc, in the central bore of the magnetic
lens (x = y = z = 0 mm) and the magnetic field concentration ratio,
Bc/Bapp, respectively, extracted from Fig. 4 for each Bapp at
TL = 20 K. The concentration ratio can be referred to characterize
the diamagnetic lens effect, which is determined by the shape oflens, including the outer and inner diameter, the outer and innerheight, and the angle of the slits.42 From these results, a higher
applied field is needed to achieve a higher concentrated field at any
time step, as shown in Fig. 6(a), although the concentration ratio

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FIG. 4. Magnetic field profiles along the x (radial)-direction at the middle of the magnetic lens, z = 0 mm, during the magnetization process from an applied field Bapp = 3,
5, and 10 T at TL = 20 K, which are divided into (a) the ascending stage from steps 0 to 5 and (b) the descending stage from steps 5 to 10.

gradually degrades as the magnetization process proceeds from step
0 to step 10, as shown in Fig. 6(b). The so-called decay behavior in
the bulk magnet due to flux creep usually exhibits an approximately logarithmic time dependence, as reported in the case only
using the magnetic lens. If the outer TFM cylinder can provide ahomogeneous magnetic field and replace outer magnetizing coilused for magnetizing process, the similar trend would be obtained
also in this HTFML as well, which means a concentrated field of
5 T could be maintained by only about 0.1% drop per hour aftermagnetization; but that decay rate depends on operating

temperature and applied magnetic field.31 It may be needed todevelop a new magnetizing sequence to improve a temporal stabilization of magnetic flux after the magnetizing process, specificallyfor magnetic separation.
Figures 7(a) and 7(b) show the resultant magnetic field, Bz,
profiles (x = y = 0 mm) along the z- (height) direction, after magnetization from Bapp = 10 T at TL = 20 K, and the magnetic field
gradient product, Bz⋅dBz/dz, profile (x = y = 0 mm), respectively.
A schematic image of the HTFML is also shown for reference, indicating the corresponding position in the HTFML. A characteristic

FIG. 5. Cross section of (a) the magnetic field, Bz, profile along the z(height)-direction and (b) the induced
current density, Jy, along the y(circumferential)-direction in the HTFML,
after the magnetization process at step
10 for each applied field, Bapp, at
TL = 20 K.

J. Appl. Phys. 127, 185106 (2020); doi: 10.1063/5.0005356
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FIG. 6. Time step dependence of (a)the concentrated magnetic field, Bc, inthe central bore of GdBaCuO lens (x =y = z = 0 mm), and (b) the magnetic field
concentration ratio, Bc/Bapp, extracted
from Fig. 4 for each Bapp at TL = 20 K.

feature of the HTMFL is based on the existence of magnetic lens
located inside the homogeneous trapped field generated from the
TFM cylinder, which enables the generation of a higher inhomogeneous magnetic field by exploiting the diamagnetic effect of magnetic lens. As a result, the HTFML also has a significant ability to
produce an intentional ultra-high magnetic field gradient productof Bz⋅dBz/dz ±3000 T2/m, just below and above the lens center at
z = ±5 mm, for a concentrated field of Bc = 11. 4 T for Bapp = 10 T.
This value is comparable with that produced by a > 20 T-class,large-scale hybrid magnet.43
Another parameter that determines the HTFML performanceis the magnetizing temperature TL in Fig. 3. Hence, the HTFML
performance was explored for the higher magnetizing temperature
TL = 40 K to compare with TL = 20 K. Figure 8 summarizes theapplied field dependence of the HTFML properties, including (a)the concentrated magnetic field, Bc, at the center of the GdBaCuO
lens, (b) maximum jBz  dBz /dzj and (c) concentration ratio,

Bc/Bapp, for each TL. The difference in Bc is reasonably small foreach TL in Fig. 8(a), but jBz  dBz /dzj in Fig. 8(b) at TL = 20 K is
500 T2/m larger than that at TL = 40 K for the applied field
Bapp = 10 T. A lower TL is desirable for a higher applied field, resulting in a higher Bc and higher jBz  dBz /dzj, although the diamagnetic effect degrades with increasing Bapp for each TL, as shown by
Bc/Bapp in Fig. 8(c). The obtained Bc and the maximum Bz⋅dBz/dz
are summarized for each applied field, Bapp, in Table II. There is a
scope for further improvement of the HTFML performance as astrong magnet in terms of both the magnetic design of each bulk
component and their magnetizing conditions.
IV. POSSIBILITY OF MAGNETIC SEPARATION USINGTHE ULTRA-HIGH MAGNETIC FIELD GRADIENTPRODUCTIn this section, the superiority of the HTFML as a strong
magnet applied to magnetic levitation or separation is discussed.

FIG. 7. Resultant magnetic field, Bz, profiles (x = y = 0 mm) along the z (height)-direction after the magnetization process (step 10) from Bapp = 10 T at TL = 20 K, and themagnetic field gradient product, Bz⋅dBz/dz, profiles (x = y = 0 mm). A schematic image of the HTFML is also shown for reference, indicating the corresponding position in
the HTFML.

J. Appl. Phys. 127, 185106 (2020); doi: 10.1063/5.0005356
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FIG. 8. Applied field dependence of the
HTFML properties for (a) the concentrated magnetic field, Bc, at the centerof the GdBaCuO lens, (b) the maximumof the magnetic field gradient product,jBz  dBz /dzj, and (c) the concentration
ratio, Bc/Bapp, for each TL.

The HTFML could, for example, be highly applicable to magnetic
separation, which needs a large magnetic field gradient product,
Bz⋅dBz/dz, over −2000 T2/m to realize the so-called “apparent zerogravity” condition, even in water or air for any diamagnetic objects
with the magnetic susceptibility ≈−10−5, such as metals, foods, or
plastics.
The total potential energy per unit mass of a substance dispersed in a medium, e.g., liquid or gas, is expressed by44U¼ 

1
(χ  χm )B2 þ (ρs  ρm )gz,
2μ0 s

(4)

liquid or gas. The total magnetic force, Fz, along z (central axis)direction at x = y = 0 mm, is@U(x¼y¼0)¼ Fm þ Fg@z(χ  χ m ) dBz¼ s
Bz
 (ρs  ρm )g:
μ0dz

Fz ¼ 

(5)

Fz is determined by the Bz⋅dBz/dz value, which changes in the vertical bore of the magnet as shown in Fig. 7(b). The equilibrium condition for magneto-Archimedes levitation, which corresponds towhere μ0 is the vacuum permeability, B is the field intensity, g is
the acceleration due to gravity, and z is the vertical position. χ s andρs are the susceptibility and density of the target substance, respectively. χ m and ρm are the same parameters for the medium, such as
TABLE II. Maximum applied field, Bapp, concentrated magnetic field, Bc, at thecenter of the GdBaCuO lens at the final step of the magnetizing process (step 10),magnetic field concentration ratio, Bc/Bapp, and maximum of the magnetic field gradient product, Bz⋅dBz/dz, at TL = 20 K and 40 K.

TL (K)
20

40

Bapp (T)Bc (T)

Bc/Bapp

Bz⋅dBz/dz (T2/m)

10
5
3
10
5
3

11.4
6.45
4.34
11.0
6.07
3.97

1.14
1.29
1.45
1.10
1.21
1.32

−3080
−1650
−943
−2430
−1320
−724

J. Appl. Phys. 127, 185106 (2020); doi: 10.1063/5.0005356
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FIG. 9. Schematic image of magnetic separation using the magnetized HTFML
device, in which the force balance between gravity, Fg , and the magnetic repulsive force, Fm , is realized for each substance in the medium, e.g., water, at particular positions related to the magnetic field gradient product.

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ARTICLEFz ¼ 0, can then be written as
Bz

dBz ρs  ρm
μ g:
¼
dz
χs  χm 0(6)

Figure 9 shows a schematic image of magnetic separation using the
HTFML device. Several kinds of substances of different susceptibility and density are dispersed in the medium, e.g., water in the tube.
When the tube is inserted into the central bore of the magnetized
HTFML device, each substance is re-arranged vertically according
to the differences in χ s and ρs of each substance. Most simply, inthis case, a collecting capillary tube could be used for the recovery
process and should be placed above the optional height of each
substances floated in the glass tube in the HTFML device.
Figures 10(a) and 10(b), respectively, show cross-sectionalviews of Bz and Bz⋅dBz/dz in the xz (vertical)-plane at y = 0 mm
and the xy (horizontal)-plane at z = 0 mm of the HTFML, aftermagnetization from Bapp = 10 T at 20 K. The effective hollow
area for levitation is indicated by the dotted line. As shown in
Fig. 10(a), the Bz⋅dBz/dz value changes from −3000 to +3000 T2/m
along the vertical direction in this area of w10 × ±30 mm2 in heightin the bore of the magnetic lens for the present HTFML.

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For magnetic separation, a large Bz⋅dBz/dz would be needed
for a variety of cases, especially if the density of the target object isheavy, such as platinum (ρs = 21.5 g/cm3), and/or its magnetic susceptibility is exceedingly low, such as water (χm = −9.03 × 10−6).
The use of harmful paramagnetic medium liquids, such as MnCl2
solution, should be avoided for food and medical use.3
Hereafter, the levitation position of the target object in the
HTFML is estimated for kinds of precious metals (Pt, Au, Ag
and Cu) based on the magneto-Archimedes effect described by
Eq. (4). The relevant physical parameters of the objects, referredfrom Ref. 2, are presented in Table III. The parameters foraqueous MnCl2 solution with concentrations of MnCl2, 50 wt. %and 12 wt. %, and 0 wt. % (i.e., pure water), are also referredfrom Refs. 2, 3, and 13. The required magnetic field gradientproduct, Bz⋅dBz/dz, for levitation increases with decreasing concentration of MnCl2 in the solution. All of the precious metals
can levitate in the HTFML for MnCl2 concentrations of
50 wt. % and 12 wt. %. However, for the 0 wt. % MnCl2 solution(pure water), only copper needs a higher Bz⋅dBz/dz value ofover −7000 T2/m in water due to its high density and low susceptibility, which exceeds the capability of the present HTFML
device. As a result, copper sinks in water, but the other precious

FIG. 10. Cross-sectional views of the resultant Bz, and Bz⋅dBz/dz along (a) the xz (vertical)-plane at y = 0 mm and (b) the xy (horizontal)-plane at z = 0 mm of the HTFML,
after magnetization from Bapp = 10 T at 20 K.

J. Appl. Phys. 127, 185106 (2020); doi: 10.1063/5.0005356
Published under license by AIP Publishing.

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TABLE III. Density and susceptibility of the example metals and calculated Bz⋅dBz/dz values to levitate them in aqueous MnCl2 solutions with concentrations 50 wt. % and
12 wt. % MnCl2, and 0 wt. % (i.e., pure water). The parameters for aqueous MnCl2 solutions are also shown for reference from Refs. 2, 3, and 13.

Calculated Bz⋅dBz/dz (T2/m)
Metal
PlatinumGold
SilverCopperDensity (ρs) (g/m3)
21.5
19.3
5.56
8.96

Susceptibility (χs)

MnCl2 (50 wt. %)MnCl2 (12 wt. %)MnCl2 (0 wt. %; pure water)

1.05 × 10−1.70 × 10−4
−2.91 × 10−5
−2.25 × 10−5

−805
−380
−118
−216

−2388
−590
−230
−417

+2210
−1401
−2800
−7282

MediumDensity (ρm) (g/m3)

Susceptibility (χm)

MnCl2 (50 wt. %)MnCl2 (12 wt. %)MnCl2 (0 wt. %; pure water)

1.33
1.10
1.00

4.13 × 10−4
2.10 × 10−4
−9.03 × 10−6

−4metals can stably levitate at different heights in the bore of the
HTFML.
Figure 11 shows the estimated levitation position of each precious metal in the pure water solution in the central bore of the
HTFML magnetized by Bapp = 10 T at 20 K. Magnetic levitation
would be realized in the case when the object is stably lifted frombeneath by the repulsive magnetic force; that is, the effective area
exists above a negative peak of Bz⋅dBz/dz at z = +5 mm for a diamagnetic object and above a positive peak at z = −5 mm for a paramagnetic object. For the precious metals in Table III, magnetic
levitation could be realized at z = +11 mm for the diamagnetic gold(Bz⋅dBz/dz = −1401 T2/m) and at z = +6 mm for the diamagnetic
silver (Bz⋅dBz/dz = −2800 T2/m) above the negative Bz⋅dBz/dz peak.
On the other hand, magnetic levitation could be realized atz = −8 mm for the paramagnetic platinum (Bz⋅dBz/dz = +2210 T2/m)
above the positive Bz⋅dBz/dz peak. However, copper would settle at

FIG. 11. Estimated levitation positions for each precious metal in the pure water
solution in the central bore of the HTFML magnetized by Bapp = 10 T at 20 K
(see text).

J. Appl. Phys. 127, 185106 (2020); doi: 10.1063/5.0005356
Published under license by AIP Publishing.

the bottom of the tube because it is outside of the present HTFML’s
magnetic separation capability.
With respect to the recovery method, a suitable one has notbeen proposed yet, and it should be determined on the basis of theseparation method and target material(s). A separation and recovery experiment was reported by Ando et al.,44 in which glass particles floated by the magneto-Archimedes method were collectedusing movable partitions into each compartment in their device.
Thus, the superiority of the HTFML to achieve magnetic separation using its ultra-high magnetic field gradient product has been
proposed from the following viewpoints:(1) Realization of an ultra-high magnetic field gradient product,
Bz⋅dBz/dz: there are a limited number of facilities worldwide
that can generate a Bz⋅dBz/dz ∼ 2000 T2/m for magnetic levitation. Figure 12 summarizes the relationship between theapplied magnetic field, Bapp, and the magnetic field gradientproduct, Bz⋅dBz/dz, for superconducting magnets (SMs) andlarge-scale hybrid magnets (HMs) around the world,2,11,45–50
which are compared to the numerical results for the present
HTFML device. The HTFML would generate quasipermanently, an ultra-high magnetic field gradient producthigher than that of conventional SMs and large-scale HMs,
even for a lower applied field of 10 T, which is enhanced dueto the presence of the magnetic lens.
(2) Compact size and mobility: this kind of a high magnetic fieldsource utilizing the HTFML would be a compact and mobile
desktop-type superconducting bulk magnet system. Thus, it
may be possible to improve the accessibility to strong magnetic
fields in a variety of food, medical, and industrial applications.
(3) Mass productivity: Superconducting bulk magnets such as
REBaCuO, fabricated by the melt-growth technique, and
MgB2, fabricated by the reactive Mg liquid infiltration, areavailable commercially and are being used in commercial products. In addition, a cryogenic refrigerator enables thermal
control of the bulks in a compact and light-weight cryostat.
The HTFML magnet, which consists of these technologyintensive components, is superior in terms of mass productivity

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FIG. 12. Relationship between theapplied magnetic field, Bapp, and themagnetic field gradient product,jBz  dBz /dzj, for superconducting
magnets (SMs) and large-scale hybrid
magnet(HM)
aroundtheworld,2,11,45–50 which are compared to
the numerical results for the HTFML.

to carry out high-throughput continuous separation in devices
with different magnetic field gradient products.
(4) Cost efficiency: Unlike a conventional separation techniquebased on chemical processes, this HTFML would not needother energy consumptions for magnetic separation due tosuperconducting current that keeps flowing inside the bulk
quasi-permanently, if the operating temperature is preserved,
for which the bulk is kept as below Tc in the HTFML. It is possible to establish and operate the HTFML system using acryogen-free refrigerator, which only needs electric power consumption for operation as a more cost-effective way without
any coolants such as liquid nitrogen or liquid helium.

V. CONCLUSION
In this study, we have proposed and analyzed a new advantageof the HTFML by numerical simulations, other than the ability toprovide a continuous, concentrated magnetic field higher than theapplied field reported previously: exploiting the HTFML’s ultrahigh magnetic field gradient product, Bz⋅dBz/dz, which could behighly applicable to magnetic levitation and separation as acompact and strong magnetic field source. The HTFML device
consisting of a GdBaCuO TFM cylinder and GdBaCuO magnetic
lens after magnetization from 10 T can generate a concentrated
magnetic field, Bc, of 11.4 T and a Bz⋅dBz/dz over ±3000 T2/m,
which is higher than that of other superconducting magnets (SMs)and large-scale hybrid magnets (HMs) around the world. The usefulness of the HTFML device in this respect was discussed basedon numerical results for the magnetic separation of a mixture of
precious metal particles (Pt, Au, Ag, and Cu) dispersed in purewater, exploiting the magneto-Archimedes effect. The HTFML
device enables the design of a compact and mobile desktop-type
superconducting bulk magnet system and can improve the accessibility to strong magnetic fields in a variety of food, medical, andindustrial applications.
ACKNOWLEDGMENTS
This research is supported by the Adaptable and Seamless
Technology transfer Program through Target-driven R&D

J. Appl. Phys. 127, 185106 (2020); doi: 10.1063/5.0005356
Published under license by AIP Publishing.

(A-STEP) from the Japan Science and Technology Agency (JST),
Grant No. VP30218088419 and by the JSPS KAKENHI Grant No.
19K05240. M. D. Ainslie would like to acknowledge financial
support from an Engineering and Physical Sciences ResearchCouncil (EPSRC) Early Career Fellowship (No. EP/P020313/1). Alldata are provided in full in the results section of this paper.
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