Superconductor Science and Technology
Supercond. Sci. Technol. 28 (2015) 075009 (9pp)

doi:10.1088/0953-2048/28/7/075009

Numerical modelling and comparison of
MgB2 bulks fabricated by HIP and inﬁltration
growthJ Zou1, M D Ainslie1, H Fujishiro2, A G Bhagurkar3, T Naito2, N Hari Babu3,
J-F Fagnard4, P Vanderbemden4 and A Yamamoto5
1

Bulk Superconductivity Group, Department of Engineering, University of Cambridge, Trumpington
Street, Cambridge CB2 1PZ, UK
2
Department of Materials Science and Engineering, Faculty of Engineering, Iwate University, 4-3-5 Ueda,
Morioka 020-8551, Japan
3
Brunel Centre for Advanced Solidiﬁcation Technology, Brunel University, Uxbridge UB8 3PH, UK
4
SUPRATECS and Department of Electrical Engineering and Computer Science B28, Sart-Tilman,
B-4000 Liege, Belgium
5Department of Applied Chemistry, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656,
Japan
E-mail: mda36@cam.ac.uk
Received 2 March 2015, revised 28 April 2015
Accepted for publication 30 April 2015
Published 26 May 2015
Abstract

MgB2 in bulk form shows great promise as trapped ﬁeld magnets (TFMs) as an alternative tobulk (RE)BCO materials to replace permanent magnets in applications such as rotatingmachines, magnetic bearings and magnetic separation, and the relative ease of fabrication of
MgB2 materials has enabled a number of different processing techniques to be developed. In thispaper, a comparison is made between bulk MgB2 samples fabricated by the hot isostatic pressing(HIP), with and without Ti-doping, and inﬁltration growth (IG) methods and the highest trapped
ﬁeld in an IG-processed bulk MgB2 sample, Bz = 2.12 at 5 K and 1.66 T at 15 K, is reported.
Since bulk MgB2 has a more homogeneous Jc distribution than (RE)BCO bulks, studies on such
systems are made somewhat easier because simpliﬁed assumptions regarding the geometry and
Jc distribution can be made, and a numerical simulation technique based on the 2D axisymmetricH-formulation is introduced to model the complete process of ﬁeld cooling (FC) magnetization.
As input data for the model, the measured Jc(B,T) characteristics of a single, small specimen
taken from each bulk sample are used, in addition to measured speciﬁc heat and thermal
conductivity data for the materials. The results of the simulation reproduce the experimental
results extremely well: (1) indicating the samples have excellent homogeneity, and (2) validating
the numerical model as a fast, accurate and powerful tool to investigate the trapped ﬁeld proﬁleof bulk MgB2 discs of any size accurately, under any speciﬁc operating conditions. Finally, thepaper is concluded with a numerical analysis of the inﬂuence of the dimensions of the bulk
sample on the trapped ﬁeld.
Keywords: bulk MgB2, numerical modelling, hot isostatic pressing, inﬁltration growth, trapped
ﬁeld magnets, ﬁeld cooling, superconductivity(Some ﬁgures may appear in colour only in the online journal)0953-2048/15/075009+09$33.00

1© 2015 IOP Publishing Ltd Printed in the UK

Supercond. Sci. Technol. 28 (2015) 075009

J Zou et al

1. Introduction

magnetization. As input data for the model, the measured
Jc(B,T) characteristics of a single, small specimen taken fromeach bulk sample are used, in addition to measured speciﬁcheat and thermal conductivity data for the materials, and theresults of the simulation reproduce the experimental results
extremely well. The numerical model provides a fast, accurate
and powerful tool to investigate the trapped ﬁeld proﬁle ofbulk MgB2 discs of any size accurately, under any speciﬁc
operating conditions. Finally, the diameter and thickness
dependence of the trapped ﬁeld proﬁle is assessed with a viewof optimizing the geometry of the bulks for such materials
acting as TFMs in practical applications.

Large, single-grain (RE)BCO (where RE = rare earth element
or yttrium) bulk superconductors have signiﬁcant potential totrap large magnetic ﬁelds over 17 T at temperatures below
30 K [1, 2] and up to 3 T at the technologically important
temperature of 77 K [3]. Such materials fabricated into discshapes are typical candidates to replace permanent magnets inapplications such as rotating machines [4], magnetic bearings[5–7] and magnetic separation [8]. The ﬁeld trapping abilityof these materials depends critically on the capability tomaintain a high critical current density (Jc) in the presence oflarge magnetic ﬁelds [9], a large shielding current loop, and awell-aligned grain microstructure [10]. However, (RE)BCO
bulk superconductors can unfortunately suffer from an inhomogeneous Jc distribution during the growth process and it is
difﬁcult to fabricate large, single-domain (RE)BCO bulk
superconductors with homogeneous properties over 100 mm
in diameter [11].
Superconductivity in MgB2 was ﬁrst reported in 2001 in[12]. Although the critical temperature, Tc, for this material
is low (39 K), requiring a lower operating temperature(15−20 K), which can lead to thermal instability/ﬂux jumps,
and a more complex cryogenic system than that required for(RE)BCO bulks, the material is cheaper, lighter weight andhas a more homogeneous Jc distribution. Hence, MgB2 inbulk form shows great promise as trapped ﬁeld magnets(TFMs) as an alternative to (RE)BCO materials.
The relative ease of fabrication of MgB2 materials, aswell as their long coherence length [13], lower anisotropy andstrongly linked supercurrent ﬂow in untextured polycrystalline samples [14, 15], has enabled a number of different
processing techniques to be developed. Signiﬁcant improvements continue to be made in terms of in-ﬁeld Jc and trapped
ﬁeld capability:

2. Sample fabrication and properties
2.1. HIP method

Three MgB2 bulk superconductors were prepared using a HIP
method [11], which are annotated as HIP#22, HIP#38 and
HIP-Ti20% in this paper. The HIP#22 and HIP#38 samples
were fabricated as described in [11], where a precursor pellet
was prepared by cold isostatic pressing at 196 MPa using
mixed powders of Mg (99.5% purity, ⩽180 μm grain size),and B (99% purity, ⩽50 μm grain size) with a molar ratio of
1.0:2.0. For the HIP-Ti20% sample, Ti (99% purity, ⩽45 μm
grain size) was added to the Mg and B powders with a molar
ratio of Ti:Mg:B = 0.2:0.8:2.0, and then the precursor pellet
was prepared. In the HIP-Ti20% sample, a small amount of Ti
is reacted and substituted for the Mg-site and, as a result, mostof the Ti is precipitated as a metal Ti phase [18]. All of thesepellets were sealed under vacuum in a stainless steel container
and were heated at 900 °C for 3 h under an isostatic pressureof 98 MPa. The samples were then embedded in a stainlesssteel ring, prepared especially for FC magnetization with aslightly larger inner diameter than the sample, using epoxy
resin (Stycast 2850TM). The outer diameter of the ring was
ﬁxed as 65 mm for all the three samples to match the coldhead and each sample was tightly anchored onto a cold stageof a Gifford–McMahon cycle helium refrigerator for FC
magnetization.

• 5.4 T at 12 K was achieved in a single 20 mm diameter
MgB2 bulk fabricated by hot-pressing ball-milled Mg
and B powders [16];• 4 T at 11 K and 3 T at 20 K with a pair of 30 mm
diameter MgB2 bulks fabricated by conventional in situ
reaction [17];• 3.6 T at 13.2 K and 2.8 T at 20 K with a single 38 mm
diameter Ti-doped MgB2 bulk fabricated by the hot
isostatic pressing (HIP) method [18];• 3.14 T at 17.5 K with a pair of 25 mm diameter MgB2
bulks fabricated by uniaxial hot pressing [19];• 1.5 T at 16.4 K with a 30 mm diameter MgB2 bulk
fabricated by a capsule method [20]; and• 1.3 T at 15 K with a 55 mm diameter sample (with a
6 mm central hole) fabricated by the reactive Mg liquid
inﬁltration (Mg-RLI) technique [21].

2.2. IG method

For the IG processed samples, crystalline boron powder (98%purity, <38 μm particle size, phase: rhombohedral) and subμm MgB2 powder (Pavezyum) were thoroughly mixed with aratio (by weight) of 70:30. The resultant mix of powders wasused to make a cylindrical precursor (32 mm diameter, 6 mm
thickness) under a uniaxial load of 10 MPa and the inclusionof the MgB2 powder in the precursor allows for improvementof the wettability of the system. Magnesium was separately
melted and held at 700 °C in a graphite crucible. The precursor was kept in a porous steel enclosure, which was thenslowly transferred to the crucible containing liquid Mg. The
use of the porous steel enclosure ensures that no part of the
precursor touches the walls of the crucible and that it does not
ﬂoat on top of the Mg melt, which can result in non-uniform

In this paper, a comparison is made between bulk MgB2
samples fabricated by the HIP and inﬁltration growth (IG)methods and the highest trapped ﬁeld in an IG-processed bulk
MgB2 sample is reported. A numerical simulation techniquebased on the 2D axisymmetric H-formulation is then introduced to model the complete process of ﬁeld cooling (FC)2

Supercond. Sci. Technol. 28 (2015) 075009

J Zou et al

inﬁltration. Boron has poor wetting with liquid Mg. This,
together with a 25% volume shrinkage in the MgB2 phaseformation, can also result in non-uniform inﬁltration, which
can lead to further defects, such as continuous Mg channels(as the MgB2 phase begins to form in the precursor pellet, theresulting shrinkage makes way for the surrounding liquid Mg[22]). Such samples have very low trapped ﬁelds as theconcentric current loops that form in a type II superconductor
are impeded by these non-superconducting Mg channels.
Thus, wettability of the precursor is important to obtain
homogeneous inﬁltration. In this sample, a fraction of presynthesized MgB2 powder is used as a wetting enhancer(mixed thoroughly with Boron), so continuous Mg channels
are eliminated.
The assembly is then subject to the thermal proﬁle shownin [22] and reacted at 850 °C for four hours. A cover gas
mixture of N2 + SF6 with a volume ratio of 95:5 was maintained to minimize oxidation of the Mg. The SF6 gas is signiﬁcantly denser than air/oxygen and is also inert towards
liquid Mg; thus, a continuous ﬂow of this gas ensures formation of a layer of SF6-rich atmosphere on the surface of the
Mg melt and protects the melt from oxidation. After thereaction is complete, the surrounding Mg was removed bymachining and the sample was recovered.

Figure 1. Temperature dependence of the normalized, average in-ﬁeld critical current density Jc calculated for each bulk MgB2 sample
under analysis.

each sample are shown in ﬁgure 1. For the reader’s reference,
the original, raw trapped ﬁeld data is presented in section 3.2(ﬁgure 7), where a comparison is made with the simulated
results from the numerical model. From ﬁgure 1, it can beobserved that among the three HIP samples, HIP-Ti20% has,
as well as a higher trapped ﬁeld, the highest normalized Jc,
indicating the positive effect the Ti-doping has on improving
the superconducting properties of HIP samples [18].
Although the IG sample has a comparatively lower trapped
ﬁeld/Jc in comparison with the HIP samples, its trapped ﬁeld
is the highest reported to date, and the IG process offers
signiﬁcant advantages due to its lower reaction temperatureand capability to produce large samples in polycrystalline
form of different shapes that can carry large current, andhence, provide large trapped ﬁelds [25].

2.3. Sample information

After the initial FC magnetization trapped ﬁeld measurements
were carried out, a small, rectangular prism-shaped specimen
was taken from each bulk sample (approximately 1 × 1 ×
2 mm3 in size from the HIP samples and 2 × 2 × 3 mm3 fromthe IG sample) to measure the Jc(B,T) characteristics. Magnetization curve M(H) measurements at 10 K, 20 K and 30 K
(5 K, not 10 K, for the IG sample) were carried out using acommercial SQUID magnetometer (MPMS-5T). The Jc(B,T)
characteristics were estimated from the M(H) hysteresis loopby employing the extended Bean model [23]. Detailed
information on all of the samples under analysis is provided
in table 1.
In order to make a fair and adequate comparison between
all of the samples made by the HIP and IG processes, acorrection factor is used to normalize the measured trapped
ﬁelds Bz(T) for the samples due to their different dimensions.
Similar to the method used in [16, 24], the measured trapped
ﬁelds were normalized to consider the bulk (or average), inﬁeld critical current density for each sample, averaged overthe sample volume. The original, measured trapped ﬁeld isdivided by the factor, −kμ0a, as given by equation (1), where
k is the geometric constant given by equation (2) to accountfor different sample radii and thicknesses:
Btrap = kμ 0 Jc R,
⎛
R+
tB
k=
⋅ ln ⎜⎜
2R⎝⎞
R 2 + tB 2 ⎟
,
⎟
tB
⎠

3. Numerical simulation of FC magnetization
The development of appropriate, practical magnetization
techniques is crucial to the success of bulk superconductors
acting as TFMs in practical applications, and there are threemain methods of magnetization in common use: zero ﬁeld
cooling (ZFC), FC and pulsed ﬁeld magnetization (PFM)[24]. In ZFC, the temperature of bulk samples is lowered
down below their critical temperature, Tc, before the application and removal of the external magnetizing ﬁeld. The
ﬁeld required to trap the maximum possible ﬁeld with the FC
method, where the superconductor is cooled below Tc in thepresence of the external ﬁeld, is half of the ﬁeld required forZFC. The PFM technique can be considered a type of ZFC,
but the duration of the applied magnetic pulse is very short,on the order of milliseconds [26]. Compared with ZFC and
FC techniques, the PFM technique provides a compact,
mobile and relative inexpensive way to magnetize the bulk
samples. However, the trapped ﬁeld produced by PFM is
generally much smaller than the other two techniques because

(1)
(2)

where R and tB are the sample radius and thickness,
respectively. The normalized results for the temperaturedependence of the average, in-ﬁeld critical current density for3Supercond. Sci. Technol. 28 (2015) 075009

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Table 1. Bulk MgB2 sample information.

SampleHIP#22

HIP#38

HIP-Ti20%

IG1

Tc (K)
Diameter, d (mm)Thickness, tB (mm)Aspect ratio (d/tB)
Relative mass density
Maximum trapped ﬁeld, Bz (FC at 20 K) (T)Jc (FC at 20 K) (A m–2)aReference

38.5
22
18
1.2
93%
1.92
2.94 × 108
[11]

38.5
38
7
5.4
93%
2.09
2.76 × 108
[11]

39
36
7
5.1
94%
2.77
4.23 × 108
[11]

37.5
32
6
5.33
90%
1.34
2.08 × 108[22]aCalculated from the measured trapped ﬁelds presented in ﬁgure 7 using equations (1) and (2).

of the signiﬁcant temperature rise generated by fast magnetic
ﬂux propagation during this magnetizing process [27]. In thissection, a numerical modelling technique is introduced tomodel the complete process of FC magnetization.
In practical applications of superconducting materials inwhich changes in temperature are non-trivial, such as thePFM of bulks, temperature-dependent modelling is inevitable
to simulate accurate temperature, current and magnetic ﬁeld
distributions. Numerical simulation is also a powerful tool to
optimize the design of the magnetization ﬁxture and determine the optimal activation ﬁeld when using various magnetization techniques [28]. The more homogeneous Jc
distribution of bulk MgB2 makes studies on such systems
somewhat easier because simpliﬁed assumptions can be maderegarding the geometry and the Jc distribution in comparison
to (RE)BCO bulks, i.e., a 2D axisymmetric model can be used
and the assumptions regarding Jc are less dependent on theposition of sub-specimens taken from the sample [28, 29].
3.1. Modelling framework and assumptions

The numerical model developed here combines the electromagnetic and thermal equations governing the behaviour ofthe superconducting material, based on the 2D axisymmetricH-formulation [30–32] implemented using the commercial
FEM software package COMSOL Multiphysics 4.3a. The ac/dc module of COMSOL is employed for the electromagnetic
analysis and the Heat Transfer module is used for the thermal
analysis, which are coupled together as described below andin [24]. A schematic view of the numerical model, including
the cold head and vacuum chamber, is shown in ﬁgure 2.
In the 2D axisymmetric H-formulation, the governing
equations are derived from Maxwell’s equations—namely,
Faraday’s (3) and Ampere’s (4) laws:

×E+

(

)

d μ 0 μr H
dB
=×E+
= 0,
dt
dt

 × H = J,

Figure 2. Schematic view of the numerical model of the MgB2 bulk
superconductor, including the cold head and vacuum chamber, forthe simulation of FC magnetization.

The temperature-dependence of Jc below Tc is described
as equation (5)1.5⎡⎛ T ⎞2 ⎤⎢⎥Jc0 (T ) = α 1 − ⎜ ⎟,⎢⎣⎝ Tc ⎠ ⎥⎦(5)

where α is the critical current density extrapolated to T = 0 K
and Tc ≈ 39 K in the case of MgB2.
Suitable parameters for the thermal properties must also
be assumed for the materials and input into the model. The
copper cold head is assumed to have a density of
8940 kg m−3, and a temperature-dependent speciﬁc heat andthermal conductivity over the temperature range 0–100 K as
given in [33]. For the MgB2 material, the density is assumed
to be 2590 kg m−3, and the speciﬁc heat and thermal conductivity for each sample is shown in ﬁgures 3 and 5,
respectively. All data presented in ﬁgures 3 and 5 are measured values from a small sample, except for the speciﬁc heat

(3)
(4)

where H = [Hr, Hz] represents the magnetic ﬁeld components,
J = [Jφ] represents the current density and E = [Eφ] represents
the electric ﬁeld. μ0 is the permeability of free space and forthe superconducting, cold head and vacuum chamber subdomains, the relative permeability is simply μr = 1.
4

Supercond. Sci. Technol. 28 (2015) 075009

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Figure 5. Experimentally measured κ (thermal conductivity) data for

Figure 3. Experimentally measured C (speciﬁc heat) data for the
HIP#22, HIP#38 and IG1 samples, and the estimated C for the HIPTi20% sample.

the MgB2 samples.

that of MgB2, which is also shown for reference in ﬁgure 4
from [37].
A suitable approximation must also be made for the Jc(B,T) characteristics of the bulk as shown by equation (6), which
was presented in [38, 39]:
⎛ B ⎞aJc (B , T ) = Jc0 (T ) exp ⎜ − ⎟ ,⎝ B0 ⎠(6)

where B0 and a are the ﬁtting parameters for eachtemperature. The measured Jc(B,T) characteristics from asingle, small specimen taken from each bulk sample and theestimated curves from the data ﬁtting that are used as input
data for the model are shown in ﬁgure 6. The data ﬁtting
parameters are summarized in table 2.
It is assumed that the electric ﬁeld E (Eφ) is parallel to thecurrent density J (Jφ) [40, 41] such that J = σE or E = ρJ,
where σ and ρ are the conductivity and resistivity, respectively, and both highly nonlinear for the superconductor. The
electrical behaviour of the superconducting material is modelled by the E–J power law [42, 43], where E is proportional
to J n, as shown in equation (7).

Figure 4. Reference speciﬁc heat values for Ti [35], Mg [36] andMgO [37].

for the HIP-Ti20% sample, which is estimated using thespeciﬁc heat of Ti and the standard HIP samples with a ratioof MgB2:Ti of 0.8:0.2. This results in a slightly higher speciﬁc heat than the undoped samples (HIP#22 and #38, whichshow good agreement with the classical reference for MgB2[34]), due to the presence of Ti, which has a higher speciﬁcheat than MgB2. Reference data for the speciﬁc heat of Ti is
provided in ﬁgure 4 from [35]. These data, as well as the coldhead thermal properties, are input into the thermal modelusing a direct interpolation, similar to a look-up table, in
COMSOL.
Interestingly, the speciﬁc heat of the IG sample is signiﬁcantly higher than that of the HIP samples. After processing via IG, there exists a portion of unreacted Mg and MgO
in the sample (reference values are 5% MgO and 13% Mg inthe sample investigated in [22]) and it is the unreacted Mg
that causes an increase in speciﬁc heat, due to the signiﬁcantly
higher speciﬁc heat of Mg, which is shown in ﬁgure 4 from[36]. The speciﬁc heat of MgO is slightly lower, but similar toE=

E0
Jc(B , T )J
Jc(B , T )

n −1

J,

(7)

where E0 = 1 μV cm–1 is the characteristic electric ﬁeld and n
is an appropriate value for the superconductor (in thismodelling work, n = 21).
Since the real FC magnetization process is being modelled, and this process involves cooling the bulk in-ﬁeld froma temperature above its transition temperature, this requires
valid information on the superconductor’s electromagnetic
properties above Tc. In order to avoid non-convergencearound the transition temperature, we assume a smoothtransition from the superconducting state to normal state,
shown as equation (8) [44]
ρsc ⋅ ρnormal,(8)
ρ (T ) =ρsc + ρnormal
5Supercond. Sci. Technol. 28 (2015) 075009

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Figure 6. The measured Jc(B,T) characteristics from a single, small specimen taken from each bulk sample and the estimated curves from thedata ﬁtting that are used as input data for the models: (a) HIP#22, (b) HIP#38, (c) HIP-Ti20%, and (d) IG1. The data ﬁtting parameters are
summarized in table 2.
Table 2. Data ﬁtting parameters for the Jc(B,T) characteristics of each bulk sample based on a single, small specimen.

SampleTs

Jc0 (A m–2)

B0 (T)

a, From equation (6)α, From equation (5)HIP#22

10 K
20 K

4.42 × 109
3.1 × 109

1.53
1.05

1.35
1.35

4.9 × 109

HIP#38

10 K
20 K
30 K

4.97 × 109
3.48 × 109
1.43 × 109

1.41
1.09
0.57

1.35
1.51
1.82

5.5 × 109

HIP-Ti20%

10 K
20 K
30 K

5.24 × 109
3.67 × 109
1.51 × 109

2.08
1.57
0.75

1.63
1.75
1.98

5.8 × 109

IG1

5K
20 K
30 K

3.8 × 1092.47 × 109
1.02 × 109

1.27
0.94
0.39

1.09
1.38
1.59

3.9 × 109in the normal state (ρnormal = 3 × 10−8 Ωm) can be foundin [11].
The simulation of FC magnetization is hence split into
three separate time domains:

where ρ is the resistivity of the superconductor at any
particular temperature, ρnormal is the temperature-dependent
resistivity of MgB2 in the normal state above its transition
temperature Tc ≈ 39 K, and ρsc is the resistivity below thistemperature based on equation (7) and E = ρJ. Therefore,
when T < Tc, the electrical resistivity of the bulk sample in themodel tends towards ρsc and when T > Tc, ρ is approximatelyρnormal. The details regarding the assumption of the resistivity(1) (0 ⩽ t ⩽ x1) Apply a ramped external ﬁeld to its
maximum magnitude (Bex) while the temperature is6Supercond. Sci. Technol. 28 (2015) 075009

J Zou et al

Figure 7. Comparison of the experimental measurements of the

Figure 8. Simulated results for the thickness dependence of thetrapped ﬁeld Bz with temperature and the simulation results at 10, 20
and 30 K for the HIP samples and at 5, 20 and 30 K for the IG
sample.

trapped ﬁeld Bz at the centre of the MgB2 bulks (HIP-Ti20% andIG1) of diameter 30 mm at z = 0 mm (top surface) and z = 3 mm at an
operating temperature of 20 K.

maintained at Tex, which is larger than Tc (Tex = 100 K
in this paper)(2) (x1 ⩽ t ⩽ x2) Slow cooling of the bulk to an appropriate
operating cooling temperature Top (i.e., 30, 20 and 10 K
for the HIP samples, and 30, 20 and 5 K for the IG
samples). Meanwhile, the external ﬁeld is held at Bex.
(3) (x2 ⩽ t ⩽ x3) Once the operating cooling temperature hasstabilized, slowly ramp the applied ﬁeld down from Bex
down to 0 T.

resulting in a higher trapped ﬁeld than pristine, undoped
samples processed with the same technique. The criticaltemperature of such samples is also unaffected by the Tidoping [18]. Secondly, the trapped ﬁeld of Bz = 2.12 T at 5 K
and 1.66 T at 15 K for the IG1 sample is the highest reported
trapped ﬁeld in a bulk MgB2 sample processed by IG.
In the following section, the numerical model is used toinvestigate the inﬂuence of the sample geometry on thetrapped ﬁeld for samples of different diameters and thickness
to optimize the samples geometrically.

This 2D asymmetric model in this case is essentially areverse-engineered superconducting fault current limitermodel with appropriate modiﬁcations, and the modelling
framework allows for FC magnetization, but can be easily
adapted for ZFC and PFM, which are described separately in[30] and [24], respectively.

3.3. Influence of bulk dimensions on trapped field

Using the numerical model in this paper, the trapped ﬁeld
proﬁle can be estimated for both HIP and IG samples of any
dimension, based on valid assumptions, without the need forfurther experiments. Since the HIP-Ti20% sample has thehighest trapped ﬁeld among the HIP-processed bulks, ananalysis of the thickness and diameter dependence of thetrapped ﬁeld for the HIP-Ti20% and IG1 samples is carriedout for different sample sizes.
Figure 8 shows the thickness dependence of trapped ﬁeld
at the centre at z = 0 mm (top surface of the bulk) and atz = 3 mm for the HIP-Ti20% and IG1 samples. The diameterof both samples is ﬁxed at 30 mm and the operating temperature is 20 K. For both samples for a ﬁxed diameter, it canbe seen that increasing the thickness of the sample results in aslight increase in trapped ﬁeld, but this saturates as the sample
thickness approaches or is greater than the sample diameter,
which agrees with experimental results for the (RE)BCO
samples analysed in [45]. A similar trend is observed for thetrapped ﬁeld at a distance z = 3 mm above the top surface,indicating that improving the Jc of the sample is the mosteffective method to increase the trapped ﬁeld as found in [39].
Figure 9 shows the diameter dependence of the trapped
ﬁeld at the centre at z = 0 mm and z = 3 mm for the same
samples with a ﬁxed thickness of 10 mm. Similarly, the3.2. Comparison of experimental and simulation results

Figure 7 shows a comparison of the simulation results withthe temperature variation of the experimentally measured
trapped ﬁeld, Bz, at a height of z = 1 mm above the centre of
the bulk surface. In the simulation, an applied ﬁeld Bex = 5 T
was applied to each sample to replicate the same experimental
conditions under which the samples were magnetized (see, forexample, [18] for the details of the FC magnetization for the
HIP samples). The time domains (see section 3.1) were split
using x1 = 2 s, x2 = 60 s, x3 = 180 s; this corresponds to a ramp
down rate of the applied ﬁeld of 41.67 mT s–1. The simulation
reproduces the experimental results extremely well, andadditionally, the fact that the experimental results can be
reproduced with only a single, small specimen taken fromeach bulk sample indicates that all of the bulk samples have ahighly homogeneous Jc distribution.
Two important results can also be observed from theexperimental results. Firstly, doping an MgB2 bulk with Ti
signiﬁcantly increases the Jc of the HIP-processed sample,
7Supercond. Sci. Technol. 28 (2015) 075009

J Zou et al

investigate the superconducting properties and estimate thetrapped ﬁeld proﬁle of bulk MgB2 discs of any size accurately, under any speciﬁc operating conditions.
Finally, the inﬂuence of the geometric dimensions of abulk sample on trapped ﬁeld is analysed numerically, basedon the properties of the HIP-Ti20% and IG1 samples. For aﬁxed diameter, increasing the thickness of the sample results
in a slight increase in trapped ﬁeld, but this saturates as the
sample thickness approaches or is greater than the sample
diameter. For a ﬁxed thickness, the trapped ﬁeld increases
initially with increasing sample diameter, but also saturates
when the diameter is several times the thickness. Therefore,
enhancing the ﬂux pinning in order to improve the Jc(B,T)
characteristics of such bulk superconductors is the mosteffective method to increase the trapped ﬁeld.
Figure 9. Simulated results for the diameter dependence of thetrapped ﬁeld Bz at the centre of the MgB2 bulks (HIP-Ti20% andIG1) of thickness 10 mm at z = 0 mm (top surface) and z = 3 mm at
an operating temperature of 20 K.

Acknowledgments
JZ would like to acknowledge the support of Churchill College, Cambridge, the China Scholarship Council and theCambridge Commonwealth, European and International
Trust. MA would like to acknowledge the support of a Royal
Academy of Engineering Research Fellowship. HF would
like to acknowledge support in part by a Grant-in-Aid for
Scientiﬁc Research from the Ministry of Education, Culture,
Sports, Science and Technology, Japan. This research was
also supported in part by a Royal Society International
Exchanges Scheme grant, IE131084. J-FF would like to thankthe Ministry of Higher Education through the ResearchCouncil of the University of Liege (Action de Recherches
Concertées grant, ARC 11/16-03).

trapped ﬁeld increases initially with increasing sample diameter, but saturates when the diameter is several times the
thickness. However, increasing the diameter does not result inthe same proportional increase in trapped ﬁeld, which was
also concluded in [39]. Based on Bean’s model (equation (1)),
the trapped ﬁeld increases linearly by increasing the diameterof the bulk, but this analysis is limited, even when taking into
account the geometric factor k given by equation (2), becauseof the assumption of constant Jc, whereas realistically for the
MgB2 material, and as taken into account in this modelling
framework, there is a strong suppression of Jc for increasing
magnetic ﬁelds. This also leads to the conclusion thatenhancing the ﬂux pinning in order to improve the Jc(B,T)
characteristics of such bulk superconductors is the mosteffective method to increase the trapped ﬁeld.

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4. Conclusion
MgB2 in bulk form shows great promise as TFMs as analternative to (RE)BCO materials to replace permanent
magnets in applications such as rotating machines, magnetic
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is made between bulk MgB2 samples fabricated by the HIP,
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the experimental results extremely well: (1) indicating thesamples have excellent homogeneity, and (2) validating thenumerical model as a fast, accurate and powerful tool to8Supercond. Sci. Technol. 28 (2015) 075009

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