Optimisation of stacked, bulk high temperature superconductors for trapped-field magnet applications

It is necessary to fabricate (RE)BCO bulk high temperature superconductors in the form of individual single grains in order to maximise the length scale over which current flows, and hence the trapped magnetic field. However, inherent difficulties in the grain growth process place limitations on the diameter and height of the single grain that may be achieved by existing melt processes. A practical approach to increase the height of the sample and the trapped field at its surface is by assembling individual single grains in a stack formation with their ab planes aligned parallel, primarily to avoid the expensive process of fabricating large, individual monoliths. The trapped fields observed at the top and bottom surfaces of a single grain sample are frequently different since both the superconducting and physical properties of single grain (RE)BCO samples are generally non-uniform. This leads to challenges in determining how to spatially arrange stacks of single grain samples to generate the largest and most uniform trapped field overall. In this study, we report the optimisation of two-stack configurations involving a total of 8 individual GdBCO/Ag single grains. The samples were arranged in four pairs and configured with different surfaces in contact in the assembly of the stack. The primary superconducting properties for trapped field and total flux distributions were measured at 77 K and compared for each stack arrangement. The initial results indicate that surfaces with inferior flux trapping properties (measured in terms of the overall trapped field value) of a two-sample stack should be positioned at the middle of the assembly to achieve the best overall trapped field and higher total flux at the external, and therefore, usable surface of the stack sample. A numerical modelling method that incorporates different Jc-B characteristics for the top and bottom layers of a single grain to take account of the variability in physical properties and spatial non-uniformity confirmed the optimised experimental arrangement of the stacked bulk samples. Furthermore, the optimisation of single grains of ring geometry to achieve a longer and wider uniform magnetic field zone inside the bore was also performed.

It is necessary to fabricate (RE)BCO bulk high temperature superconductors in the form of individual single grains in order to maximise the length scale over which current flows, and hence the trapped magnetic field. However, inherent difficulties in the grain growth process place limitations on the diameter and height of the single grain that may be achieved by existing melt processes. A practical approach to increase the height of the sample and the trapped field at its surface is by assembling individual single grains in a stack formation with their ab planes aligned parallel, primarily to avoid the expensive process of fabricating large, individual monoliths. The trapped fields observed at the top and bottom surfaces of a single grain sample are frequently different since both the superconducting and physical properties of single grain (RE)BCO samples are generally non-uniform. This leads to challenges in determining how to spatially arrange stacks of single grain samples to generate the largest and most uniform trapped field overall. In this study, we report the optimisation of two-stack configurations involving a total of 8 individual GdBCO/Ag single grains. The samples were arranged in four pairs and configured with different surfaces in contact in the assembly of the stack. The primary superconducting properties for trapped field and total flux distributions were measured at 77 K and compared for each stack arrangement. The initial results indicate that surfaces with inferior flux trapping properties (measured in terms of the overall trapped field value) of a two-sample stack should be positioned at the middle of the assembly to achieve the best overall trapped field and higher total flux at the external, and therefore, usable surface of the stack sample. A numerical modelling method that incorporates different J c -B characteristics for the top and bottom layers of a single grain to take account of the variability in physical properties and spatial non-uniformity confirmed the optimised experimental arrangement of the stacked bulk samples. Furthermore, the optimisation of single grains of ring geometry to achieve a longer and wider uniform magnetic field zone inside the bore was also performed. * Author to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Introduction
Single grain, (RE)BCO bulk high temperature superconductors have significant potential for use in a variety of technologically sustainable applications due to their ability to trap magnetic fields as high as 17.6 T at 26 K, which is an order of magnitude higher than the field generated by conventional, iron-based permanent magnets [1,2]. However, inherent difficulties in the grain growth process place a significant limitation on the size of both the diameter and height of a single grain that may be achieved by existing melt processes. A practical approach for increasing the height of the single grains is to stack single grain samples with their ab planes aligned parallel to each other, given their highly anisotropic superconducting properties [3][4][5] and that supercurrents flow almost entirely within the ab planes. As a result, it is not necessary to fabricate large, individual monoliths, which are both expensive and particularly susceptible to high fabrication failure rates.
Both the superconducting and physical properties of single grain (RE)BCO samples are non-uniform. In particular, J c values at different positions in a single grain vary significantly throughout the sample [6,7]. This is understandable because the current density is determined by local flux pinning forces, which are, in turn, related largely to the microstructure of the single grain. As a result, the measured trapped fields at the top and bottom surfaces of single grain samples are frequently different (from the Biot-Savart Law, the measured magnetic field at any point in space outside the sample arises as the volume integral of the local currents at every point in the sample). This leads to issues of how to arrange stacks of single grain samples spatially to generate the largest and most uniform trapped field overall. In this study, we report the optimisation of two-stack configurations involving eight individual GdBCO/Ag single grains.
The samples were arranged in four pairs and configured with different (top and bottom) surfaces in direct contact. The primary superconducting properties for applications of trapped field and the distribution of field at the top and bottom surfaces of the samples were measured and compared at 77 K. The initial results show that surfaces with inferior flux trapping properties (i.e. measured in terms of the trapped field value) of a two-sample set should be positioned at the middle of the assembled stack to achieve the best overall trapped field and higher total flux at the external, and therefore, accessible and usable surface. The optimised configuration also achieved higher and more uniform trapped fields.
Numerical modelling is widely used as a fast and efficient way of better understanding the physical properties of bulk superconductors and can provide insights into sample behaviour that are often difficult to obtain experimentally. A single J c -B characteristic is typically used when modelling the trapped field of an individual (RE)BCO single grain to simplify the analysis. This generally yields qualitative results with occasional quantitative agreement for larger scale changes in variables [8] such as heat [9], pulsed field magnetisation [10] or the composite make-up of the sample [11]. In other words, bulk, single grain materials have been treated generally as uniform in most of the modelling studies to date. The nonuniformity of single grains has been studied more recently and the J c -B distribution at 77 K of the whole cross-section of a GdBCO single grain of diameter 20 mm has been reported [6,7]. For example, the J c -B characteristics of all 24 subspecimens in a single grain cross-section, illustrated schematically in figures 1(a) and (b), have been estimated from the M-H loops measured at 77 K using the extended Bean model [12,13]. Figure 1(a) shows a photograph of a typical single grain with four-fold facet lines; the so-called ab planes are parallel to each other and are roughly parallel to the top surface of the sample. A slice was cut from the top to the bottom of the single grain along the red dashed lines shown for the M-H loop measurements. Figure 1(b) is a schematic illustration of the half of the slice that was further cut into 24 small subspecimens labelled from 1ta, to 4ta and then from 1tf to 4tf, as described in [6]. The results [6] showed that all J c -B characteristics of the 24 specimens are different; J c -B decreases generally along the c-growth direction from top to the bottom of the single grain but varies much less along the a/b direction, if edge effects of the sample are ignored (which may be machined away).
The present modelling assumes that J c -B in each ab layer is the same and that two experimental J c -B characteristics are sufficient to describe the current-carrying ability of the single grain, with higher values represented by The results of the numerical modelling based on the use of these two J c -B curves fit the experimental data well. Furthermore, the modelling of two stacks of single grain discs and two 20 mm inner diameter (ID), 37 mm outer diameter (OD) single grain rings (i.e. the same arrangement as the experiment) suggests that that surfaces with inferior flux trapping properties (measured in terms of the trapped field value) of a two-sample arrangement should be positioned at the centre of the assembled stack to achieve the best overall trapped field and to achieves a longer and wider uniform trapped field zone at the top or bottom surfaces of complete single grains or within the bore of a stack of rings. A schematic illustration of the half of the slice that was further cut into 24 small sub-specimens labelled from 1ta to 4ta and then from 1tf to 4tf, as described in [6].

Fabrication of GdBCO/Ag single grains, 25 mm in diameter
The top-seeded-melt-growth process [14][15][16] was used to grow large single grains of GdBCO/Ag. Gd-123 (Toshima, 99.9% purity), Gd-211 (Toshima, 99.9% purity), CeO 2 (Aldrich, 99.9% purity), BaO 2 (Aldrich, 97% purity) and AgO 2 (Alfa Aesar, 99.9% purity) powders were mixed for two hours with a composition (75 wt% Gd-123 + 25 wt% Gd-211) + 0.5 wt% CeO 2 + 1.0 wt% BaO 2 + 10 wt% Ag 2 O by an auto-electronic pestle and mortar. The mixed precursor powder of mass 45 g was weighed and poured into a die of 30 mm diameter and compacted using an uniaxial press. Either a generic [17,18] or thin-film seed, both of which have a higher melting temperature and the same lattice structure as the material to be seeded, was placed on the top surface of the as-pressed, green sample. The pressed pellet and seed were then put into a box-furnace for melt-processing. This involved heating the sample to between 1055 • C and 1060 • C, holding for 1.0 h, cooling to 1015 • C at a rate of 100 • C h −1 , cooling slowly to 1008 • C at between 0.8 • C h −1 and 1.0 • C h −1 , cooling to 980 • C at between 0.4 • C h −1 and 0.2 • C h −1 and, finally, furnace cooling to room temperature. The samples were oxygenated subsequently at temperatures of between 420 • C and 450 • C for ten days to allow the lattice structure of the samples to transition from tetragonal (non-superconducting) to orthorhombic (superconducting). Figure 2 shows photographs of the top surfaces of the eight samples used in this study.
The top and bottom surfaces of the as-grown, fully oxygenated samples were polished (or cut) flat before measurement of their trapped fields. The vector component of the trapped field parallel to the z direction of each single grain was measured after field-cooling (FC) the sample to 77 K in an applied field of 1.4 T and then ramping down the field linearly over a period of 100 s. A hand-held Hall probe was placed above the top surface of each sample in order to locate the position of the maximum trapped magnetic field. The complete trapped field profile was then measured using a rotating array of 18 Hall probes located 1.5 mm above the sample surface. A sample was chosen for this research where the maximum trapped field at the top surface was 0.9 ± 0.1 T, which is consistent with typical well-performing (RE)BCO samples at 77 K [14,19]. The maximum trapped field of each sample is indicated in figure 2, along with its height, H. Note that these eight samples were not all processed in the same batch, and the trapped fields at their surfaces are all different, reflecting the non-uniformity and sample-to-sample variation experienced in real situations in practical applications. It can be seen that the trapped fields vary from 0.81 T to 1.050 T at one of the surfaces and the height  of the samples varies from 7.35 mm to 10.60 mm. The sample used in Stack II-1 contains a visible sub-grain, which limits the maximum trapped field at its bottom surface to 0.479 T.

Trapped field measurements of stacks of four pairs of samples in four combinations
Trapped field measurements of the two-sample stacks were performed in the same manner as the measurements on a single grain sample described above. There are four possible top and bottom surface combinations of any two samples in a single stack arrangement. For example, Stack I which is shown schematically in figure 3, contains two samples with two top surfaces T1, T2 and two bottom surfaces B1, B2. The combination shown in the diagram has the bottom surface of sample 1 (B1) in contact with the top surface of sample 2 (T2) and is therefore labelled as I-T1B1T2B2. As a result, Stack I has four possible combinations: I-T1B1B2T2, I-B1T1T2B2 and I-B1T1B2T2. There are, therefore, 16 combinations of top and bottom surfaces for four stacks of sample pairs. The trapped field measurements of these stacks form the focus of this study in an attempt to understand the issues in optimising their arrangement for practical applications.

Modelling a stack of single grains assuming two different Jc-B characteristics for the top and bottom layers
It is well-known that the trapped field of (RE)BCO single grains of the same dimensions varies significantly, even within the same batch of bulk samples. State of the art, batchprocessed single grains of 25 mm in diameter can typically trap fields of 0.9 ± 0.05 T at 77 K [14]. The variation in trapped field between different batches of samples is even greater, as can be seen from the measured trapped fields of the samples shown in figure 2. This behaviour is to be expected, given that trapped field values are determined by the Biot-Savart law, which is effectively a 3D integral of J c -B and its relevant distance to the position where the trapped field is measured, and that J c -B varies with position within an individual single grain sample. J c -B is sensitive to the generally unknown presence and distribution of nano-size, non-superconducting defects or particles within the single grain microstructure, which form effective, but variable strength, flux pinning centres. A recent study has illustrated starkly the potential variation in J c -B at 77 K in a sample of 20 mm diameter in which values of J c of between 4.5 × 10 8 A m −2 and 0.5 × 10 8 A m −2 were observed at different positions across the radius of the single grain at an applied field of 0.1 T [6]. It is reasonable to infer, therefore, that the fluctuation of the trapped field from sample to sample is linked strongly to the observed non-uniform J c -B in single grain samples.
In order to better understand how to stack optimally nonuniform single grains with different J c -B characteristics, a 2D axisymmetric finite-element model, implemented using the commercial finite element software package COMSOL Multiphysics, was used to study the trapped field of different bulk superconductor stacks following FC magnetisation. This differs from the approach taken by most models that assume a single J c -B characteristic, which, by definition, yields the same trapped field for individual single grains of the same geometry. Ideally, a position-dependent J c -B would be used to model the trapped field of non-uniform superconducting single grains. However, it is very difficult to determine a suitable single function that represents the full 3D variation in J c -B for all types of single grain, so this is not a practical solution. Single grains are anisotropic in nature with a layered structure and, when magnetised, contain supercurrents that flow mainly within the ab planes. As a result, stacks of slices cut parallel to the ab plane of the parent single grain with a 45 • offset were investigated to address the in-plane non-uniformity of J c -B in the ab planes, and within which J c can vary by ±10% [7] between the facet lines and the a/b direction. More uniform and higher trapped fields were observed both experimentally and by modelling for this off-set arrangement of ring-shaped single grains [20] following pulse-field magnetisation and for disc-shaped single grains [7] following field cooling magnetisation. In order to account for non-uniformity in the c direction, it is reasonable to assume that J c -B is unchanged in each layer based on the experimental results that show J c -B varies much less along the a/b direction [6]. In this study, therefore, we employ two average J c -B characteristics in different layers in each single grain sample as a trade-off between an overly simple assumption and an assumption that is too complex to implement practically. J c -B(1) is assumed for the top part of the sample, corresponding to the average J c -B measured at positions (1a) and (1b), illustrated in figure 1(b). J c -B(2) is assumed for the bottom part of the sample, corresponding to the average J c -B of layers B, C and D layers in figure 1(b) in [6], as shown here in figures 1(c) and (d). Furthermore, it is assumed that the thickness of the layers with different average J c -B characteristics determines the trapped field (see figure 3(b)). In the present study, the thickness of each single grain sample in Stack III was determined initially by fitting the model to experimental results and then stacking samples with the resultant assumed spatial extent of the J c -B characteristics to determine the optimised stacking combination. Finally, the same optimised configuration was confirmed for a stack of two single grain rings of 20 mm ID and 37 mm OD (20ID37OD) to obtaining uniform trapped fields at the middle of the bore of the stack of rings. Figure 4 shows the maximum trapped fields (B t max ) and their spatial distribution at the top and bottom surfaces of the samples of Stack III (Stack III-1 and Stack III-2) for different contact combinations of top and bottom surfaces. It can be seen that the original samples, Stack III-1 and Stack III-2, perform well in terms of the maximum trapped field (0.907 T and 1.034 T at 77 K) and that the heights of the samples are similar. These values are typical of well-grown GdBCO/Ag single grains of 25 mm in diameter fabricated at both Cambridge [14] and Nipon Steel [19].

Experimental results and discussion
The maximum trapped fields, B t max , observed for the twosample stacks are either unchanged or improved marginally compared to the original single grain samples for the same surface. These results are not surprising and are similar to those reported previously [21] because the trapped field saturates for aspect ratios of 0.67 < H/R < 1 (where H is the height of the cylindrical single grain and R is the radius of the sample). The effect of the thickness t on the trapped field at the top surface of the single grain is more apparent from Chen's formula (1) below (which is the direct integral of the Biot-Savart law for a finite long cylindrical sample for the situation where J c is constant) when z is zero and for constant r, which reduces equations (1) and (2). In other words, B trap increases monotonically when t increases for a sample of fixed diameter r, but increases more slowly as t increases, levelling-off at a sufficiently high value of t/r, at which point B trap saturates. The total flux along the c direction of the sample continues to increase as t increases further.
where r is the sample radius (m), t is the sample thickness (m), z is the height of the position above the top surface (m), µ 0 = 4π × 10 −7 (A m 1 ) and B trap is the maximum trapped field (T).
Although tall, individual single grains are difficult to fabricate, and particularly when the diameter of the sample is large, the stacking together of shorter sample to generate a structure of optimum aspect ratio can achieve a higher trapped field and higher total flux for practical applications.
The highest field trapped at the surfaces of the four combinations of the sample stacks was observed at the bottom surface of Stack III-T1B1B2T2: 1.030 T (labelled in bold red in figure 4). This surface corresponds to the top of Stack III-2, for which the measured trapped field was also the greatest. The largest trapped field at both the top and bottom surfaces of Stack III was observed for the Stack III-T1B1B2T2 configuration, which contains two bottom surfaces positioned at the middle of the assembled stack. This arrangement achieved the best overall trapped field and highest total flux at the external usable surface. However, the flux contour maps of the two constituent single grains are not perfect in that their contour lines do not constitute perfectly concentric rings, although they (Stack III-1 and Stack III-2) are clearly amongst the best of the eight samples studied here. The trapped field for a simplified 2D uniform high temperature superconducting material can be expressed as, ⃗ B t =´⃗ J c × d⃗ r, where J c is the critical current density at some position within the sample and r is the radius at that position. The flux contour lines will form perfectly concentric rings if J c is uniform along the same radius. The highest fields trapped at the surfaces of the four combinations of Stack II in figure 6 were observed at the top surface of Stack II-T1B1B2T2, 1.044 T (labelled in bold red in figure 6), and the bottom surface of Stack II-T1B1B2T2, 1.031 T. These surfaces correspond to the top surfaces of Stack II-1 and the bottom surface of Stack II-2, respectively, which are the surfaces with the higher trapped fields for the single grain samples. This suggests that the best configuration of stacks is determined by arranging the surfaces based on their trapped field properties and that surfaces with higher trapped field in each single grain in the Stack II configuration should be arranged as external surfaces in order to achieve enhanced performance for the whole stack. Figure 7 shows that the highest fields trapped at the surfaces of the four combinations of Stack IV were observed at the bottom surface of Stack IV-T1B1B2T2, 1.040 T (labelled in bold red in figure 7), and the bottom surface of Stack IV-T1B1B2T2, 1.060 T. These surfaces correspond to the top surfaces of Stack IV-1 and Stack IV-2, respectively, which are the surfaces with the highest trapped fields for the single grain samples. Sample Stack II-1 is not a single grain due to the presence of a sub-grain at the bottom surface. The contour map in figure 7-1(c) shows the non-uniformity of the surface field of the single sample (Stack IV-1). However, in comparison, the resulting contour maps shown in figure 7-5 and   value) in a two-sample set in the middle of the assembled stack to achieve the best overall trapped field and highest total flux at the external, and therefore, usable surface. The optimised experimental configuration also results in a more uniform trapped field.

Modelling results and discussion
The numerical modelling in this work is based on a magnetostatic, 2D axisymmetric finite-element model [8,22], implemented in COMSOL Multiphysics, using the Magnetic Fields interface in the AC/DC module. The 'External Current Density' node was used to assume that a current density of J c -B flows through the 2D axisymmetric cross-section of the bulk single grain with no flux creep. Therefore, the model is Bean-like [12] that takes into account J c -B and obtains a selfconsistent solution by solving Ampere's law using the magnetic vector potential A [23].
Two  figure 1(c). A recent study has established that porosity significantly affects the average J c -B of a single grain and, specifically, that porosity increases along the c direction from the top to the bottom surface of the sample (except at the very bottom of the sample). As a result, single grains with higher trapped field have a lower porosity [24]. Hence, it is reasonable to assume that it is the spatial extent, or thickness, of J c -B(1) that determines the magnitude of the trapped field at the centre of the top surface of the single grain. The spatial extent of J c -B(1) in Stack III-1 along the c direction was determined by running the model for various assumptions to be 1.5 mm in thickness, whereas that for Stack III-2 was determined to be 3.5 mm, primarily because the modelled trapped fields for these thicknesses reproduce accurately those at the top and bottom surfaces of each single grain sample in Stack III. It can be seen from figures 8(a) and (b), which show the modelled maximum trapped fields at the top (green curves at 0 mm-the centre of the sample) and bottom (red curves at 0 mm) surfaces, that the results of the model are similar to those measured experimentally for the samples III-1 and III-2, presented in figure 4-1 and 2(a). The agreement between the results of the model using two J c -B curves and the experimental data probably reflects the layered structure of the single grains, where the difference in J c -B is greater along the c-axis at the centre of the single grain than that in the a/b direction [6]. These results suggest that it is both reasonable and feasible to use two J c -B characteristics along the c direction of a single grain to model the trapped field performance of the sample geometries investigated here, even though the J c of the samples is not intrinsically uniform [6].
The results of the model for a stack of two single grains under the same experimental configurations and with the same thicknesses assumed for J c -B (1) and J c -B (2), are shown in figure 9. Parts (a)-(d) of this figure show the distributions of the trapped fields across the radius at the top (green), middle  (blue) and bottom (red) of stack arrangements III-T1I1T2B2, III-T1B1B2T2, III-B1T1T2B2 and III-B1T1B2T2. It can be seen that the arrangement T1B1B2T2 for Stack III achieved the highest trapped fields in both the measured and modelled data (see figure 4-4(a)).
The results of the model obtained by arranging non-uniform single grains in various configurations based on their field trapping ability suggest clearly that it is possible to optimise the trapped field at the external surface of the stack. The optimum arrangement to obtain a higher trapped field at the top or bottom surface of a stack of two bulk, single grains is to place the surfaces with lower trapped field at the middle of the stack. The results of the model were confirmed by experiment.
The same approach was then applied to single grain rings (20 mm ID, 37 mm OD and 10 mm in height) for which the requirements for field uniformity are typically much stricter (e.g., in NMR/MRI applications). Interestingly, the trapped field within the bore of the rings of the T1B1B2T2 arrangement shown in figure 10(b) (blue curve in the range of 0 < distance from the centre < 10 mm) exhibits the lowest variation in trapped field in the z direction, from 0.8 T to 0.9 T.  It is well known that the uniformity of the magnetic field increases with the length of a wire-wound solenoidal magnet along the z, or axial, direction. The modelling of the optimum arrangement of ring-shaped single grains of height 10 mm was extended to include taller rings, and specifically to a stack of two 20 mm thick, 20ID37OD samples. The results show that a near-flat trapped field is achieved at the centre of the bore of the stack (blue curve at 0 < distance from centre < 10 mm), as shown in figure 11. This trapped field profile is both higher in magnitude and more uniform (i.e. flatter) than that shown in figure 10(b). The green and red curves in figure 11 show the trapped fields at a position 10 mm above and below the centre of the bore. It can also be seen that these are both higher in value and flatter than those in figure 10(b). These results suggest that the modelled trapped field can be very uniform within the bore of a stack of taller rings, where −10 mm < distance from centre <10 mm and height within ±10 mm from the centre. Further research is needed to achieve uniform fields in FC, zero-FC and pulse magnetisation processes and the model described here represents an efficient tool for evaluating these magnetising conditions. However, it is clear that uniform, practical trapped fields can be achieved by stacking or arranging (RE)BCO single grains with nonuniform J c -B characteristics in an optimised configuration.

Conclusions
Stacking single grain bulk superconductors is a practical approach for achieving better combined superconducting properties for trapped-field magnet applications. In this study, non-uniform samples with different trapped fields at their surfaces were arranged in four pairs and configured with different surfaces in contact with each other in assemblies of two single grain sample stacks. The trapped field, which is fundamental to most applications of bulk, single grain superconductors, was measured at 77 K for different samples and stack arrangements and compared. The results show that surfaces with inferior flux trapping properties (in terms of the magnitude of trapped field) of a two-sample set should be positioned at the middle of the assembled stack to achieve the best overall trapped field and highest total flux at the external, and therefore, usable surface. The optimised stack configuration resulted in a more uniform trapped field. Simple finite-element modelling, assuming two J c -B characteristics to reflect the non-uniformity of the single grains at the top layers and the bottom layers of the stack, was used to verify the stack arrangement optimised experimentally for two single grain samples. The model was extended successfully to predict the optimum arrangements for stacking two non-uniform single grain rings to achieve a uniform trapped field, both horizontally and vertically, within the bore of the stack. In summary, uniform vertical and horizontal trapped fields can be achieved experimentally by stacking optimally non-uniform single grains, and these results are confirmed by a simple finite element model by using two different J c -B characteristics. The applicability of this relatively simple modelling technique is anticipated to be applicable to more complicated and practical situations, such as pulse magnetisation, zero-field cooling and the modelling of composite materials for engineering applications.

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.96495.