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In this paper, an improved interpolating element-free Galerkin method (IIEFGM) with nonsingular weight function is introduced to calculate the magnetic field distribution of electrical machine. Since the shape function satisfied the property of Kronecker-delta function, the IIEFGM can impose the boundary conditions directly. Meanwhile, the IIEFGM adopts nonsingular weight function, which means the weight function of the conventional element-free Galerkin method (EFGM) can be used. To demonstrate the effectiveness of IIEFGM, the magnetic field distribution of a permanent magnet coreless disc machine is analyzed. The selection of weight function and the nodes distribution are discussed, the comparison with Finite element method (FEM) shows the IIEFGM has high computational accuracy.&lt;br/&gt;I. Introduction&lt;br/&gt;Meshless methods are the important supplement to the finite element method in solving boundary problems, and element-free Galerkin method is the most widely used method. The mainly difference between the EFGM and the FEM is the construction method of the shape function, however, since the shape function of EFGM does not satisfy the property of Kronecker-delta function, the imposition of boundary conditions required additional methods, such as penalty method. To solve this problem, reference [1] proposed an interpolating element-free Galerkin method (IEFGM), however, the method adopt singular weight, which makes the computation difficult. Reference [2] proposed IIEFGM with nonsingular weight function, which has the advantages of both the FEM and the IEFGM. In this paper,a permanent magnet coreless disc machine (PMCDM) is taken as an example [3] the IIEFGM is introduced to calculate the magnetic field distribution of the PMCDM, and the comparison with the FEM validated the effectiveness of the IIEFGM.&lt;br/&gt;II The foundation of the IIEFGM&lt;br/&gt;In conventional EFGM, the basis function is defined as: p&lt;sub&gt;1&lt;/sub&gt;=1, p&lt;sub&gt;2&lt;/sub&gt;=x, p&lt;sub&gt;3&lt;/sub&gt;=y (m=3), and the trial function is u&lt;sup&gt;h&lt;/sup&gt;(x)=p&lt;sup&gt;T&lt;/sup&gt;(x)a(x). The coefficients a(x) can be obtained by minimizing the weighted least-square function, then the shape function N(x) can be derived. To make the shape function satisfied the property of the Kronecker-delta function, the IIEFGM modified the basis function. Then the boundary conditions can be imposed directly, and the negative exponential function is selected as the weight function.&lt;br/&gt;III Magnetic field calculation of the PMCDM&lt;br/&gt;Fig.1 shows the magnetic field distribution of the PMCDM by using IIEFGM, the problem domain is divided into the radial part and the circular part. Fig.2 depicts the main flux density on the airgap center line. It can be seen that the calculated results obtained by IIEFGM are in excellent agreement with the results with the FEM.&lt;br/&gt;IV Conclusion&lt;br/&gt;This paper introduced an improved interpolating element-free Galerkin method. The Galerkin weak form and the discretized equation are derived, the flow of the algorithm is optimized, and a weight function with higher accuracy is given. Based on IIEFGM, the magnetic field distribution of a PMCDM is analyzed, numerical results verified the effectiveness of the IIEFGM. The study results provide references for the application of IIEFGM.

**References:**

[1] Ren H , Cheng Y ,&quot;The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems,&quot; Engineering Analysis with Boundary Elements, Vol. 36(5), p.873-880, 2012.&lt;br/&gt;[2]Wang J F , Sun F X , Cheng Y M, &quot;An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems,&quot; Chinese Physics B, Vol. 21, No.9, 2012.&lt;br/&gt;[3]Gu C L , Wu W , Shao K R, &quot;Magnetic field analysis and optimal design of DC permanent magnet coreless disc machine,&quot; IEEE Transactions on Magnetics, Vol. 30(5), p. 3668-3671, 1994.

**Images:**

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/16d3efc5f003de67c4a8d3fbbb91d924.jpg)
Fig.1 Magnetic field distribution. (a) circular direction. (b) radial direction

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/4fb8054f5ba97bdbc11fc680df6f9a1b.jpg)
Fig.2 Main flux density on airgap center line. (a) circular direction. (b) radial direction

INTERMAG 2021

 Application of an improved interpolating element-free Galerkin method in magnetic field calculation

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In this paper, an improved interpolating element-free Galerkin method (IIEFGM) with nonsingular weight function is introduced to calculate the magnetic field distribution of electrical machine. Since the shape function satisfied the property of Kronecker-delta function, the IIEFGM can impose the boundary conditions directly. Meanwhile, the IIEFGM adopts nonsingular weight function, which means the weight function of the conventional element-free Galerkin method (EFGM) can be used. To demonstrate the effectiveness of IIEFGM, the magnetic field distribution of a permanent magnet coreless disc machine is analyzed. The selection of weight function and the nodes distribution are discussed, the comparison with Finite element method (FEM) shows the IIEFGM has high computational accuracy. I. Introduction Meshless methods are the important supplement to the finite element method in solving boundary problems, and element-free Galerkin method is the most widely used method. The mainly difference between the EFGM and the FEM is the construction method of the shape function, however, since the shape function of EFGM does not satisfy the property of Kronecker-delta function, the imposition of boundary conditions required additional methods, such as penalty method. To solve this problem, reference [1] proposed an interpolating element-free Galerkin method (IEFGM), however, the method adopt singular weight, which makes the computation difficult. Reference [2] proposed IIEFGM with nonsingular weight function, which has the advantages of both the FEM and the IEFGM. In this paper,a permanent magnet coreless disc machine (PMCDM) is taken as an example [3] the IIEFGM is introduced to calculate the magnetic field distribution of the PMCDM, and the comparison with the FEM validated the effectiveness of the IIEFGM. II The foundation of the IIEFGM In conventional EFGM, the basis function is defined as: p1=1, p2=x, p3=y (m=3), and the trial function is uh(x)=pT(x)a(x). The coefficients a(x) can be obtained by minimizing the weighted least-square function, then the shape function N(x) can be derived. To make the shape function satisfied the property of the Kronecker-delta function, the IIEFGM modified the basis function. Then the boundary conditions can be imposed directly, and the negative exponential function is selected as the weight function. III Magnetic field calculation of the PMCDM Fig.1 shows the magnetic field distribution of the PMCDM by using IIEFGM, the problem domain is divided into the radial part and the circular part. Fig.2 depicts the main flux density on the airgap center line. It can be seen that the calculated results obtained by IIEFGM are in excellent agreement with the results with the FEM. IV Conclusion This paper introduced an improved interpolating element-free Galerkin method. The Galerkin weak form and the discretized equation are derived, the flow of the algorithm is optimized, and a weight function with higher accuracy is given. Based on IIEFGM, the magnetic field distribution of a PMCDM is analyzed, numerical results verified the effectiveness of the IIEFGM. The study results provide references for the application of IIEFGM.

**References:**

[1] Ren H , Cheng Y ,"The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems," Engineering Analysis with Boundary Elements, Vol. 36(5), p.873-880, 2012. [2]Wang J F , Sun F X , Cheng Y M, "An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems," Chinese Physics B, Vol. 21, No.9, 2012. [3]Gu C L , Wu W , Shao K R, "Magnetic field analysis and optimal design of DC permanent magnet coreless disc machine," IEEE Transactions on Magnetics, Vol. 30(5), p. 3668-3671, 1994.

**Images:**

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/16d3efc5f003de67c4a8d3fbbb91d924.jpg)
Fig.1 Magnetic field distribution. (a) circular direction. (b) radial direction

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/4fb8054f5ba97bdbc11fc680df6f9a1b.jpg)
Fig.2 Main flux density on airgap center line. (a) circular direction. (b) radial direction

poster

Introduction: Analyzing and processing the non-linear and non-stationary vibration signals is a critical issue in the motor bearing fault diagnosis. In order to solve this problem, Empirical Mode Decomposition (EMD) was proposed, which decomposes the signal in Intrinsic Mode Functions (IMFs) [1]. However, the EMD has the drawback of the so called “mode mixing” phenomenon, which means that the signal components with different amplitudes in one IMF, or the signal components with a similar amplitude in different IMFs. To address this issue, an effective solution was proposed, which suppresses the mode mixing by adding the Gaussian white noise in the original signal, named as “Ensemble Empirical Mode Decomposition (EEMD)” [2-3]. But it brings in some new harmonics in IMFs. To overcome these problems, the Complete Ensemble Empirical Mode Decomposition (CEEMD) was put forward, which adds the opposite Gaussian white noise in the original signal for the EMD respectively [4]. However, the amplitude of the Gaussian white noise is usually estimated by the practical experience rather than calculating by equations. Besides, the inappropriate Gaussian white noise could not mitigate the mode mixing effectively. The essence of mode mixing is the coupling between IMFs, and IMFs are not orthogonal to each other. This paper focuses on the mode mixing, and optimizes the amplitude of the white noise by the genetic algorithm (GA) to minimize the orthogonality among IMFs. Optimization Method: Mutual information is an indictor which symbolizes the correlation between the data. The mutual information between IMFs obtained by the CEEMD will be affected and changed, when the amplitude of the white noise varies. Therefore, the white noise amplitude (All of these values are multiples of the standard deviation of the signal) is used as the input function, and the minimum mean of the mutual information between IMFs is used as the objective function in the GA. As a consequence, it can suppress the mode mixing phenomenon theoretically, when the amplitude of white noise is the minimum mean of the mutual information. Results: The bearing vibration fault signal used in this paper comes from the Case Western Reserve University [5]. In Fig.1, the values in the bar denote the mean of mutual information of IMFs corresponding to different white noise amplitudes, where 0.1103 is the white noise amplitude selected by GA and 0.1 to 0.4 are the values selected by the experience. The mean of mutual information is 0.5728 when the amplitude of the white noise is 0.1103, which is smaller than the amplitudes of 0.1, 0.2, 0.3 and 0.4 that the mean of mutual information is 0.705, 0.6230, 0.6998 and 0.7093, respectively. Fig.2 shows the amplitude of IMFs in figure Fig.2 (a) and the spectrum diagram of IMFs in figure Fig.2 (b) decomposed by the CEEMD with the optimized white noise. It can be seen from the figure Fig.2 (b) that the main components of each IMF are distributed at different frequencies. Meanwhile, it shows in the spectrum of IMF3 that the five times frequency (539.1 Hz) component of the bearing outer race fault. It proves the effectiveness of the proposed method in motor bearing fault diagnosis. Detailed results and discussion will be given in the full paper.

**References:**

[1] N. E. Huang, Z. Shen, S. R. Long, et al., Proc. R. Soc. Lond. A Math. Phys. Sci., vol. 454, p. 903-995, (1998). [2] Z. WU and N. E. Huang, Adv. Adapt. Data Anal., vol. 01, p. 1-41, (2008). [3] P. Flandrin, G. Rilling and P. Gonçalvès, IEEE Signal Process. Lett., vol. 11, p. 112-114, Feb. (2004). [4] F. Liu, J. Gao and H. Liu, IEEE Access, vol. 8, p. 19810-19819, (2020). [5] C. W. R. University,''https://csegroups.case.edu/bearingdatacenter/pages/12k-drive-end-bearing-fault-data," (2014).

**Images:**

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/c71a8b8dc1e20c7f4b86ef846db3f898.jpg)
Fig. 1 The mean mutual information of different amplitude of the white noise.

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/348217bb9979a412c80e3a633833c3b6.jpg)
Fig. 2 IMFs decomposed by the CEEMD with the optimized white noise. (a) Amplitude of IMFs. (b) Spectrum diagram of IMFs.

 Adaptive Suppression of Mode Mixing in CEEMD Based on Genetic Algorithm for Motor Bearing Fault Diagnosis

In a standard measurement of diamagnetic susceptibility χDIA (per unit mass), it is necessary to prepare a bulk sample that weigh more than ~0.5 mg. In the present report, we show that χDIA of a small particle is determined from its parabolic movement that is induced by a field gradient generated by a Niobium magnetic circuit, without the necessity of knowing mass m of particle. This principle of measuring χDIA is based on a recent finding that acceleration of a translating particle induced in an area of magnetic field gradient is independent to m; it is uniquely determined by intrinsic χDIA value of material [1][2]. This characteristic derives from a well-known fact that a magnetic body force induced in a particle is proportional to m. The observed results overturn a common recognition that ordinary diamagnetic particles are magnetically inert unless a strong field in applied in the experiment [3]; the result indicate that most solid particles composed of diamagnetic or weak paramagnetic materials may potentially show translation at a low field produced by a permanent magnet, which is usable in practical applications. It was proposed that material identification of a particle is possible[4], by collating the obtained χDIA value with a compiled list of published data [5]; unlike many conventional methods of chemical analysis, the identification is conducted without consuming the small sample. In the present experiment, field-induced parabolic translations were observed in three diamagnetic materials, namely graphite, bismuth and diamond. The particles were released at an initial position as shown in Fig.1 with negligibly small initial velocity, and the particles translated in an area with magnetic field gradient in normal gravity. Fig 2 shows the example of the trajectory observed for a graphite particle and a bismuth particle which proceeded in the xz-plane of Fig.1; they were observed by a high-speed camera (CASIO EX-F1, Japan). The χDIA value of each particles were determined from a relationship between velocity of particle v (x,z) and field intensity B (x, z); the theoretical equation used in the calculation was deduced from an assumption that the variance of potential energy (i.e., magnetic & gravitational) of a particle between two arbitrary positions in the trajectory was completely converted to kinetic energy. The χDIA values obtained in this manner was consistent with the published values [5]. The accuracy of the χDIA values obtained by the present apparatus was significantly improved compared to that obtained in the previous research [3]; the improvement was achieved by optimizing the field distribution of the magnetic circuit, thus decreasing the fluctuation of v(x) values. The accuracy was also increased by precisely identifying the initial position of sample improving the observation using the hi-vision camera. Accordingly, accuracy of χDIA obtained for a submillimetre-particle was improved to a level of ~ 4×10-7emu/g, and variance of χDIA between different materials can be distinguished by the proposed method; in previous studies, the accuracy remained at a level of ~10×10-6emu/g [1][2][4]. The experimental techniques achieved in this study will be a milestone in reducing the measurable size of the sample to micrometre scale. According to recent measurements, the velocities of ferro- (ferri-)magnetic particles [6] and paramagnetic particles [2] during their field-induced translation were also independent to m. Hence, similar motions as reported in the present study is generally expected to occur in all categories of magnetic material by a relatively low magnitude of field-gradient. It is expected from the results shown in Fig.2 that an ensemble of heterogeneous particles can be separated into fractions of different materials [2][4] by releasing the ensemble from the initial position in Fig.1. By improving the compact system, a non-destructive chemical analysis can be conducted shortly after collecting the heterogeneous particles at the site of material explorations in various research fields; such method of pre-treatment will improve the efficiency of reaching a target sample in the research. &lt;div id="UMS_TOOLTIP" style="position: absolute; cursor: pointer; z-index: 2147483647; background-color: transparent; top: -100000px; left: -100000px; background-position: initial initial; background-repeat: initial initial;"&gt; &lt;/div&gt;

**References:**

[1] C. Uyeda, K. Hisayoshi and S. Kano, J. Phys. Soc. Jpn. Vol.79, p.064709 (2010). [2] K. Hisayoshi, C. Uyeda and K. Terada, Sci. Rep. vol.6, p.38431 (2016) [3] G. Maret and K. Dransfield, Adv. Appl. Phys. Vol.46, p.143 (1985) [4] C. Uyeda, K. Hisayoshi and K. Terada, Sci. Rep. Vol.9, p.3971 (2019). [5] R. Gupta, Landolt Bornstein. New Series II, Vol.46 “Diamagnetism”, p.445 (19830 [6] C. Uyeda, K. Kuwada and K. Hisayoshi, J. Jpn. Soc. Powder Metall. Vol.61, p.S78 (2014) &lt;div id="UMS_TOOLTIP" style="position: absolute; cursor: pointer; z-index: 2147483647; background-color: transparent; top: -100000px; left: -100000px; background-position: initial initial; background-repeat: initial initial;"&gt; &lt;/div&gt;

**Images:**

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/e2339049a76984bb17bb885247799039.jpg)
Fig.1 A top view (above) and a side view (below) of a NdFeB magnetic circuit developed in the present study to observe parabolic translation of a diamagnetic particle in terrestrial gravity.

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/22d65a0cb18c05ff3ed752e42a3fb932.jpg)
Fig.2 Trajectories of a graphite particle and a bismuth particle that translated from the initial position described inFig.1. The particles translated in the xz plane in the figure. The χ values obtained from the trajectories was 49×10-7emu/g for graphite and was 11×10-7emu/g for bismuth; the values were both consistent with the published values [4].

 Diamagnetic Susceptibility of a Single Particle Detected from its Parabolic Movement Caused by Magnetic Field gradient and Terrestrial Gravity

With the increasing demand for new high-tech and energy-efficient devices, multi-stimuli functional materials with strong interplay between their structural, magnetic, and electronic subsystem appear at the forefront of research in material science [1,2]. They form a new family of promising materials for applications in different technological areas, like heat-assisted magnetic recording (HAMR), thermal energy storage, novel spintronic devices, and magnetic refrigeration. In a first-order phase transition (FOPT) material [3], the changes caused by external stimuli not only impact one subsystem of the solid (e.g., change of magnetization induced by a magnetic field), but also give rise to significant transformations occurring in other subsystems (e.g., expansion of the crystal lattice, change in electrical resistivity, heating of the sample, etc.). However, the contributions of the different subsystems are usually characterized independently and in different samples, environment, or even laboratories. This makes difficult to assemble scattered pieces of a puzzle into one coherent picture, and this is probably a reason why the mechanisms of magneto-structural coupling remain till now poorly understood. Therefore, it is important to change the existing paradigm and to develop a novel experimental approach allowing the characterization of the magnetic materials exhibiting a FOPT by combining various experimental techniques, including: ● field/temperature dependencies of magnetization both macroscopically (magnetometry) and in the element selective manner (XMCD) ● changes occurring in the crystal lattice (X-ray diffraction) and in the sample as whole (dilatometry) ● thermal response of the material (calorimetry). The key point is that all these measurements should be performed simultaneously on the same sample and under strictly identical experimental conditions. Such an experimental setup is to be implemented at the beamline ID12 at the European Synchrotron Radiation Facility (ESRF) in Grenoble (France) in the framework of the project ULMAG – Ultimate MAGnetic characterization. It will serve as a versatile tool for combined macroscopic and microscopic characterization of magnetic materials open to all users. In our presentation, we describe the ULMAG setup, which is inspired from a previously purpose-built device “MaRS” at TU Darmstadt [4]. The MaRS is based on a commercial VSM option of QD-PPMS [4]. However, in ULMAG, the sample must remain static for the experiments with x-rays at the beamline, impeding the VSM possibility. Thus, the magnetic measurements are performed by means of a GaAs Hall probe situated as close as possible to the sample. This allows to estimate the stray-field produced by the surface of the sample, mirroring the surface magnetization of the sample. The dilatometry and calorimetric properties are measured by means of strain gauges and thermometer, respectively, as described in our previous study [4]. A first prototype was built based on the commercial resistivity option of the QD-PPMS. The sensors are connected to external instruments controlled by the MultiVu software implemented in the QD-PPMS. This prototype serves as a proof of concept and the general scheme of the sample with the different sensors are presented in Fig. 1. At the ESRF, we are building a similar prototype which will have in addition the X-Ray beam (see Fig. 1). With the current ULMAG setup, we can probe under the same conditions and simultaneously the (i) magnetic, (ii) magneto-volumetric and (iii) caloric changes as function of the magnetic field (up to 8 T) and temperature (4-350 K). Preliminary experiments are performed on a quasi-equiatomic FeRh sample with the antiferromagnetic (AFM) to ferromagnetic (FM) transition occurring around 300 K (see Fig. 2). Results show that contrary to what is usually assumed, magnetic and magneto-volumetric changes (parallel and perpendicular) do not evolve hand in hand but rather occur at different fields (or temperatures for temperature induced transition, not shown here). Such peculiarities were also observed in other samples (La-Fe-Si and FeRh) measured with the MaRS [4-6]. The combination of XMCD with the macroscopic properties reported above is planned to be performed in early 2021 at the beamline ID12 of the ESRF. This will be a first achievement to complete a new way of characterizing magnetocaloric materials. In the future, the use of this set-up will be extended to study other types of magnetic compounds like magnetostrictive materials or permanents magnets and beyond. Aknowledgements: This work was funded by the Federal Ministry of Education and Research (BMBF) under the grant number BMBF-Projekt05K19RD2.

**References:**

[1] O. Gutfleisch, et al, Magnetic materials and devices for the 21st century: stronger, lighter, and more energy efficient. Adv Mater, vol. 23, p. 821 (2011). [2] O. Gutfleisch, et al., Mastering hysteresis in magnetocaloric materials, Phil. Trans. R. Soc. A, vol. 374, p. 20150308 (2016). [3] J. Y. Law, et al., A quantitative criterion for determining the order of magnetic phase transitions using the magnetocaloric effect. Nat Commun, vol. 9, p. 2680 (2018). [4] D. Yu. Karpenkov et al. Pressure Dependence of Magnetic Properties in La(Fe,Si)13: Multistimulus Responsiveness of Caloric Effects by Modeling and Experiment Phys. Rev. Applied, vol. 13, p. 034014 (2020) [5] K. Skokov et al, to be published [6] K. Skokov et al, to be published

**Images:**

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/5f2a294dbf80c7e20867ad734e257ab5.png)
Fig.1: Schematic of the ULMAG setup for simultaneous measurements of magnetic stray field, magnetovolume change, caloric change and XMCD/XRD, expected to be implemented at ID12 at ESRF (Grenoble, France).

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/daf9ffa9613d66fd4cf451c0943050ce.jpg)
Fig. 2: Simultaneous measurements of the stray field (hall probe), the magnetostriction (strain gage // and ⊥), and the calorimetry (thermometer); showing a magnetic field induced AFM—FM transition at 245 K for a quasi-equiatomic FeRh.

 “ULMAG – Ultimate Magnetic Characterization,” A Novel Experimental Setup for the Beamline ID12 at the European Synchrotron Radiation Facility (ESRF)

An apparatus to obtain magnetic anisotropy Δχ (per unit mass) of a diamagnetic or a paramagnetic crystal is newly desinged and opperated in microgravity (μg) condition. In this experiment, rotational harmonic oscillation of a milimeter size crystal was observed, which was caused by a magnetic ansiotropy energy -½ｍΔχB2 of the crystal with mass m; here oscilation was induced by a homogeneous field B produced in area of μg [1][2]. The period of this oscillation τ is described as τ= 2π (I/m)½Δχ-½B-1, in condition that angle of amplitude is small. Note that τ is independent to m in the above equation, and if oscillation is observable, Δχ can be obtained by measuring τ, I/m and B no matter how small crystal particle may be [1]-[3]. Using a standard comertial apparatus, detection of Δχ is prevented in a small sample due to the background signal from the sample holder, and also due to the difficulty in measuring m. Hence in using the coventional methods, it is nccesary to prepare a single crystal with a milimeter-size to obtain an accurate Δχ value. It is noted that the obtainable size of a single crystal is below sub-milimeter level in many materials, and it is presently difficult to obtain reliable Δχ values in these materials. As described in Fig.1, a pair of ferrite magnetic plate (10cm x 5cm x 1cm) was introduced in the present study to generate B (B = 0.051T). By introducing these plates, area of homogeneous field was enhanced to a spherical space of φ~3 cm , and the mm-size sample released in μg area could remain inside the homogeneous area of during the τ measurement. Note that a solid particle released in μg space tends to freely trasnlate, and in previous researches [3], the samplefrequently moved away from the small homogeneous space of φ~1.0cm during the τ measurement. Due to this distervance, success rate of of Δχ measurement was less than 20%. In the present experiment, Δχ values were obtained in milimetre size crystals of paramagnetic chrorite and a diamagnetic graphite following the aforementioend propcedure. The milimetre-size sample used in the experiment was separated from the bulk single crystal using a wire saw and a titanum knife to avoid ferromagnetic contaminations. The sample was placed on a sample stage that was located at the the field center as shown in Fig,1. A short microgravity condition (duration &lt; 0.5s) was supplied by conducting a free fall of the apparatus which was installed in a wodden rack [3]. Shortly after the biginning of the free fall, the sample stage was slightly lifted from its initial position, which was effective in inducing a smooth release of the small sample in a diffuse area. The rotational-oscillation of the samples were observed by a high-speed camera (ZWO ASI290MC), which could observe and presere the sample motions with a spatial resolution of 0.004 cm and a time resolution of 0.033 fps. Fig.2 shows a time dependent image of a graphite crystal showing that field-induced rotational oscillation occurs at a low field intensity produced by the ferrite plates; it was generally considerd that dinamic motion of diamagneitc and paramagnetic material were relaized by a existance of ultra high magnetic field [4]. The experimental values of τ and I are obtained from the images, and are used in the calculation of Δχ according to the aforementioned procedure. The values obtained in this manner are +(7.7±0.8)x10-6emu/g and -(2.2±0.6) x10-5emu/g for chrorite and graphite, respectively; these values are consistent with the publised data measured by the conventional torque methods [5]. According to the aforementioned principle of the proposed Δχ measurement, further improvement of sensitivity is possible, specifically by increasing the magnitude of B and/or τ; improvement is also possible by decreasing the value of I/m. The experimental techniques achieved in the present study will be a milestone in reducing the measurable size of the sample down to micrometre scale. Diamagnetic anisotropy originate from the 3-dimensional distribution of localized electrons that compoase a material. In case of organic materials, origin of Δχ were explained in terms of a molecular-orbital method; consistency of this method was examined by comparing theoretical and experimental values of Δχ [5]. Similar investigation was not performed in other types of solid materials because the number of experimental Δχ values were limited due to the difficultty of detecting the small Δχ values. By improving the method proposed in the present report, Δχ having a magnitude below 1x10-9emu/g canbe obtained for oxides with high crystal symmetry. The improvement of Δχ sensitivity will increase the number of reliable data to clatify the relationship between diamagnetic anisotropy and spatial electron distribution assinged to individual diamagnetic crystals [2][5].

**References:**

[1] C.Uyeda, Phys. Chem. Mineral 20, 77–81(1993) [2] C.Uyeda, K.Hisatoshi, T. Tanaka, Appl. Phys. Lett. 28, 094103 (2005). [3] C. Uyeda et al., J. Phys. Soc. Jpn. 79, 064709 (2010). [3] (2011) [4] G.Maret & K. Dransfield, Adv. Appl. Phys. 46, 143 (1985). [5] R. Gupta, Landolt Bornstein. New Series II, 46 “Diamagnetism”, p 445 (1983).

**Images:**

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/6362df752c85ff6b2c33c946a8e00ee1.jpg)
An apparatus developed to detect magnetic anisotropy Δχ in a small crystal by observing its field-induced trasnslation in a short μg condition, which is supplied by a comopact drop shaft shown in the right portion of figure.

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/aa8f5055640f26be8ec99f1782214b75.jpg)
Time dependent image of a graphite crystal showing field-induced rotational oscillation by a field produced by a ferrite magnetic circuit. The c-axis of graphite is normal to the sample plane showing oscillation with respect to B. Period of oscillation τ is obtained from these images, and is used to calculate Δχ of the crystal.

 Magnetic anisotropy detected in a small crystal by observing its rotational oscillation caused by a ferrite magnetic circuit

Transcranial magnetic stimulation (TMS) is a non-invasive and painless treatment for brain disorders. Rodent models of TMS provide a possibility to reveal the mechanism of TMS at cellular and molecular level. However, their application has been hampered by lack of stimulating coils with appropriate focus and intensity. In this study, we analyzed various types of coils and proposed conductive shield and iron core materials to improve the coil performance. The results showed that combination of Halo coil and figure of eight coil with shield and iron core could generate better stimulus focus (29.2 mm2) and intensity (87.6 V/m). Keywords: Transcranial magnetic stimulation; Coils; Focus; Mouse model I. Introduction At present, most commercial animal coils generally cannot meet the requirements of stimulus depth, stimulus intensity and stimulus focus simultaneously. Therefore, it is necessary to further develop a special TMS coil for small animals [1, 2]. II. Method In this study, a mouse model was constructed with three-compartment body-skull-brain model of conductivities 0.3344, 0.0203 and 0.106 S/m. The mouse and coil models were shown in Fig.1. The shield conductor (Cu) with a thickness of 1 mm had a window in the middle. The iron core was made of sheets of silicon steel. All the coils were kept at a distance of 5 mm from the highest point of the mouse head. The stimulus intensity is defined as the max of the electric field (Emax), and the focus was defined as the half power region (HPR). III. Results The results showed that Halo coil, shield and iron core could complement each other well. The combination of Halo coil and figure of eight coil with shield, icon core could get a better stimulus focus (29.2 mm2) and a higher stimulus intensity (87.6 V/m), which had increased about 79% and 54% respectively compared to the general commercial coil, as shown in Fig.2. IV. Conclusion Our results suggested that the combination of Halo coil and figure of eight coil with shield and icon core could improve TMS coil performance. We provided an alternative method to deliver TMS to small animals by constructing a coils with better stimulus intensity and stimulus focus.

**References:**

[1] Meng Q, Jing L, Badjo J P, et al. A novel transcranial magnetic stimulator for focal stimulation of rodent brain [J]. Brain Stimulation, 2018, 11(3): 663-665. [2] Alekseichuk I, Mantell K, Shirinpour S, et al. Comparative modeling of transcranial magnetic and electric stimulation in mouse, monkey, and human [J]. Neuroimage, 2019, 194: 136-148.

**Images:**

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/c980cf69f9b7ab8e29272aa563f9908a.jpg)
Fig.1 Mouse and coil models of this study

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/ff8235f037f228bd4581f42851cc12cf.jpg)
Fig.2 Electric field induced in the brain of the mouse

 Design of Transcranial Magnetic Stimulation Coils for Mouse with Improved Stimulus Focus and Intensity

Magnetic hysteresis loop measurement [1]-[4] is crucial for the development of advanced magnetic materials. From the magnetization versus magnetizing field (M-H) curve, one can extract the useful magnetic properties, which are important for applications in sensors, transformers, actuators, and miscellaneous electrical devices. The M-H measurements were usually done by using the vibrating sample magnetometer [1] or the SQUID magnetometer [2], for which the magnetizing field are virtually static during the measurement. The induction-coil-based hysteresis loop and susceptibility measurement systems [3], [4] are rapid and low-cost devices capable of determining the alternating-current (AC) magnetic properties, which is important for inferring the frequency-dependent features such as the energy loss due to eddy currents in conducting materials. The convention hysteresis loop measurement device [3] is designed for the magnetic induction versus field (B-H) measurement of toroidal samples. The alternating susceptibility measurement system (AC susceptometer) is capable of determining the frequency dependent initial magnetization and susceptibility. But the induction coil can’t detect the environmental direct-current (DC) field. For the soft magnetic materials, the non-zero environmental magnetic field can significantly alter the measured result, which must be compensated by using a magnetic sensor to detect the unwanted magnetic interference. To achieve higher accuracy in the M-H measurement, we proposed a M-H measurement system combining the signal processing method of B-H measurement and the coil design of the AC susceptometer. The interference from the environmental magnetic field is eliminated by using a giant magnetoresistance (GMR) sensor as the zero field detector, as shown in Fig. 1. The system consists of solenoidal magnetization and pickup coils. The pickup coil consists of reference and sensing coils symmetrically positioned with respect to the excitation coil. The sensing coil detects the magnetization of the sample, whereas the reference coil is wound in the opposite direction to null out the magnetizing signal. The signal output of the integrator is proportional to the magnetization of the sample. The current in the magnetization coil is directly proportional to the magnetizing field strength H. However, the DC level of the magnetizing field at the sample is usually shifted by the environmental field, which can’t be observed by monitoring the current in the coil. To solve the problem, the GMR sensor, which is the GF708 spin-valve sensor [5] from Sensitec GmbH, is used for the zero field detector. The sensor has a very high sensitivity but the dynamic range less than 2 Oe. The output of GMR is used as the trigger for taking the M-H curve to eliminate the disturbance from the environmental DC field, as shown in Fig. 2. The 1-Hz sinusoidal current signal measured by a resistor shows a peak-to-peak amplitude of 0.96 A, corresponding the magnetizing field amplitude of 6.7 mT. When the environmental DC field was completely eliminated in the pickup coil, the observed output signal of the GMR has a zero-field-crossing point defined by the trigger level, shown as in Fig. 2. The trigger point to activate the M-H measurement is at the intersection of the trigger level line and the GMR signal curve for “No external field”. When the DC field (of about 0.77 mT) was applied, the trigger point shifted by 40 ms in comparison with the zero-field curve, while the current waveform in the coil remained unchanged. The new trigger point to activate the M-H measurement is at the intersection between the trigger level line and the GMR signal curve with “External field”. The coercivity and the asymmetry pinning field of the GMR sensor induces a constant time shift in the trigger point, which can be corrected by employing a calibration procedure provided that the sweeping magnetizing field always saturates the output of GMR on both field polarities. With the GMR zero-field detector to set the trigger point, the high-quality M-H curve can be obtained by taking averaging for multiple cycles of the sweeping magnetizing field. In this way, the accuracy of the M-H curve measurement for the soft magnetic materials is successfully improved by eliminating the interference from the environmental field without using a costly magnetic shield chamber. This work is supported by the Ministry of Science and Technology of Taiwan under Grant No. MOST108-2221-E992-083MY2.

**References:**

[1] S. Foner, “Vibrating sample magnetometer”, Review of Scientific Instruments, 27(7), 548-548 (1956). [2] R. Nagendran, N. Thirumurugan, N. Chinnasamy, M. P. Janawadkar, & C. S. Sundar, “Development of high field SQUID magnetometer for magnetization studies up to 7 T and temperatures in the range from 4.2 to 300 K”, Review of Scientific Instruments, 82(1), 015109 (2011). [3] V. Munoz, M. C. Martinez, & V. Such, “Improved systems for the measurement of hysteresis loops: DC and AC characterization”, Journal of Physics E: Scientific Instruments, 20(7), 861 (1987). [4] R. B. Goldfarb & J. V. Minervini, “Calibration of ac susceptometer for cylindrical specimens”, Review of scientific instruments, 55(5), 761-764 (1984). [5] C. Reig, S. Cardoso, & S. C. Mukhopadhyay, “Giant magnetoresistance (GMR) sensors”, In Smart Sensors, Measurement and Instrumentation, 6(1), pp.1-301 (2013).

**Images:**

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/e67c168e41c249ae72cd97327beac3a7.jpg)
Fig. 1. Schematics of the magnetization measurement system with GMR zero-field detector
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![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/53502c98b622d9f578af31fde8a199d5.jpg)
Fig. 2. The trigger points of the GMR zero-field detector for the M-H measurement system
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 Magnetization measurement system with giant magnetoresistance zero-field detector

Many scholars have found that magnetic anomaly signals can reflect the characteristics of magnetic target, such as distance, speed, magnetic moment and other related information [1-3]. This provides a new idea for target recognition by matching magnetic anomaly signals. However, the magnetic anomaly signals in the detection environment are weak and easily disturbed by the strong background noise of the geomagnetic filed, resulting in a low signal-to-noise ratio (SNR) of the signals output by the magnetic sensor [4], which brings great challenges to the traditional signal matching method. In this paper, a method combining OBF (Orthogonal Basis Function) decomposition and EDR (Edit Distance on Real sequence) is proposed to match magnetic anomaly signals under low SNR. The calculation process of OBF-EDR method is divided into three steps (Fig. 1). Step 1: The OBF method is used to decompose the magnetic anomaly signals. According to the characteristics that the background noise is not correlated with the basis function, three sets of discrete basis function coefficient sequences with higher SNR can be obtained [5]. Step 2: The sequences of discrete basis function coefficient are taken as the calculation object, and the EDR method with strong anti-noise ability is used to measure and get their similarity results. Step 3: The matching result of the magnetic anomaly signals is calculated through the similarity results of the three sets of discrete basis function coefficient sequences by the weight method. The simulation shows that the curves of matching accuracy of OBF-EDR and EDR method have the same changing trend with the change of signals’ SNR, but the curve of matching accuracy of OBF-EDR method moves to the direction of lower SNR compared with that of EDR (Fig. 2). It indicates that the OBF-EDR method can match the magnetic anomaly signals correctly at a lower SNR than the EDR method. What’s more, when the matching accuracy of OBF-EDR reaches 0.9, the required signal’s SNR is -2dB, which is 7dB lower than that of EDR method. The above conclusions prove the advantage of the OBF-EDR method to matching magnetic signals under low SNR.

**References:**

 [1] Y. Shen, J. Wang, and J. Shi, J. Magn. Magn. Mater., 484, 164-171, (2019). [2] Y. Shen, D. Hasanyan, and J. Gao, Smart Mater. Struct., 22, 095007, (2013). [3] J. Wang, Y. Shen, and R. Zhao, J. Magn. Magn. Mater., 505, 166761, (2020). [4] O. Faggioni, M. Soldani, and A. Gabellone, J. Appl. Geophys. 72, 46-56, (2010). [5] B. Ginzburg, L. Frumkis, and B. Kaplan, Sens. Actuator A-phys., 102, 67-75, (2002).

**Images:**

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/342a1c3ef4df12f64377b274bb525671.jpg)
Fig.1. OBF-EDR method calculation process.

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/5e76e72a07c2b094b80ba3d24bfe86fb.jpg)
Fig.2. Matching accuracy of OBF-EDR and EDR method to match the magnetic anomaly signals under different SNR.

 A Matching Method for Magnetic Anomaly Signals Under Low Signal-to-Noise

Biphasic or monophasic pulsed magnetic fields are commonly used in non-invasive stimulation of biological systems [1]. Biphasic systems bring the advantage of using both the positive and negative pulsed fields, thereby leading to twice the amount of signal obtained by using monophasic pulsed field systems [2]. With a focus on medical applications, biphasic pulses have been used in transcranial magnetic stimulation applications [2, 3] for neuronal excitation and in magnetic resonance applications for inducing proton transitions [4, 5]. Typically, commercial field generator systems with sophisticated data acquisition and processing mechanisms are used for such medical applications. However, with the recent trends towards portable diagnostics, a redesign of the pulsed field generator systems for improved portability is required. In this work, design and challenges of creating biphasic low-pulsed magnetic fields, which may be utilized for portable NMR applications, is discussed. Since portable solutions rely on permanent magnet geometries, the strength of static field is low and correspondingly, the strength of the pulsed field is typically low (less than 0.1T). Prior portable pulsed field generators have been designed to obtain repetitive pulses at both high (up to 0.5T) [2,3] and low-field strengths (0.05T) [4]. However, obtaining high-frequency (1-10MHz) pulsed sinusoidal fields (both monophasic or biphasic) is a challenge for most portable systems. The monophasic pulse generators are useful in obtaining higher pulsed fields and amplitudes with limits on the number of nuclear spins that may be excited during an excitation cycle. Meanwhile, biphasic pulses can provide uniform excitation but have lower field amplitudes. Both the techniques are equally valuable and appropriate pulse sequencing techniques can be designed to make such pulsed field generators valuable additions in pulsed NMR applications. This work highlights the design parameters and challenges of both techniques and details the design of a biphasic portable pulsed field generator operational between 0.5 – 2 MHz. The primary challenge of biphasic pulsed generators is the rapid switching between the different phases. The designed biphasic pulse generator (Figure 1) includes two switching devices and intertwined inductors connected to a common terminal and pulsed with a 180 degree offset. The two switches operate such that only one inductor is active at a particular time. The biphasic generator is designed to obtain a maximum of +/- 15A through the inductor (Figure 2) at an operational frequency of 1MHz. Synchronizing the phase offset between the two switching devices ensures simultaneous operations. In summary, different aspects of the switching device, load, operational frequency and phase synchronization needed to obtain stable biphasic pulsed magnetic fields will be highlighted in this paper.

**References:**

 [1]. N. M. Shupak, F. S. Prato and A. W. Thomas, "Therapeutic uses of pulsed magnetic-field exposure: A review," in URSI Radio Science Bulletin, vol. 2003, no. 307, pp. 9-32, Dec. 2003. [2]. J. Selvaraj, P. Rastogi, N. Prabhu Gaunkar, R. L. Hadimani and M. Mina, "Transcranial Magnetic Stimulation: Design of a Stimulator and a Focused Coil for the Application of Small Animals," in IEEE Transactions on Magnetics, vol. 54, no. 11, pp. 1-5, Nov. 2018, Art no. 5200405. [3]. N. R. Bouda, M. Mina and R. J. Weber, "High-Current Magnetic Field Generator for Transcranial Magnetic Stimulation Applications," in IEEE Transactions on Magnetics, vol. 50, no. 11, pp. 1-4, Nov. 2014, Art no. 5400204. [4]. N. Prabhu Gaunkar, J. Selvaraj, L. Bauer, M. Mina, R. Weber and D. Jiles, "Design and Experimental Implementation of a Low Frequency Pulsed Magnetic Field Generator," in IEEE Transactions on Magnetics, vol. 53, no. 11, pp. 1-4, Nov. 2017, Art no. 6501004. [5]. N. Prabhu Gaunkar, "Developments in static and pulsed magnetic field systems for detection of magnetic resonance in non-uniform magnetic fields" (2018). Graduate Theses and Dissertations, Iowa State University, 17291.

**Images:**

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/01008017854c7244c2b8306e8ac882b5.jpg)
Schematic of biphasic pulsed field generator

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/d82b09b7d55f78fed164814ed756efac.jpg)
Current through inductor at a frequency of 1MHz (estimated from current sense resistor)

 Design of stable biphasic pulsed magnetic fields for portable diagnostic applications

Progress in the development of nanomagnetism has forced the scientific community to pay close attention to 3D magnetic nanostructures [1]. Currently, the use of X-ray nanotomography makes it possible to determine temporal evolution of the magnetization [2, 3]. Since both spatial and temporal resolution of experimental methods is limited, it is natural to compare the experimental results with the computer simulation predictions. This approach allows resolve both the fine spatial structure of vortex aggregates and the fast processes of topological rearrangements. In our simulations, we take advantage of instability of a domain wall motion in a magnetic field above a certain threshold value. In continuous films this instability generates various dynamic vortex structures and spin waves [4, 5]. In the case of antidot films, new types of vortex-induced topological rearrangements of magnetization can be observed in the region of the array of holes. We simulate the process of magnetization reversal in a permalloy cuboid with the dimensions X = 800 nm, Y = 200 nm (the film thickness), Z = 400 nm, with periodic boundary conditions along the z axis. This axis is the easy magnetization axis in the film plane and the y axis is perpendicular to the film plane. We have considered a single rectangular hole with size d as well as arrays with three or nine such holes, where d = 25, 50, or 100 nm. Micromagnetic simulations of magnetic dynamics were performed by solving the Landau-Lifshitz-Gilbert (LLG) equation using mumax3 solver [6]. We used a 3D orthogonal uniform mesh of size a = 3.125 nm. To visualize the magnetization topological structure we employed an approach based on the calculation of topological charges of two types [4, 5]. They satisfy the laws of conservation and we have classified permissible transformations of the magnetization distributions at the film boundaries. When a domain wall moves in a sample with holes, various scenarios of transformations of vortex structures are realized. We have specified the initial distribution of magnetization as a specially formed configuration containing a C-shaped (vortex) domain wall (DW) near the boundary of the sample. The axis of the DW (the z axis) is in the plane of the sample. To the right (left) of the DW, the magnetization is directed against (along) the direction of the z axis. After turning on the DC magnetic field H = 200 Oe along the z axis, the DW begins to move, becoming the S–shaped one. Further movement of the DW is accompanied by the generation of vortex structures leading to the complex dynamics of magnetization reversal in the antidot array. (a) In the case of large holes (d = 100 nm), after the DW reaches the antidot array boundary, the vortex structures disappear and the process of magnetization reversal stops. The final steady-state configuration of the magnetization in the three-hole array is as follows: when traversing the contours of the boundaries of the holes clockwise, the magnetization makes a full turn clockwise (counterclockwise) for the middle (lower) hole. This means that the holes are carriers of topological charges and are topologically equivalent to local vortex and antivortex. (b) In the case of medium-sized holes (d = 50 nm), the displacement of the magnetization reversal region is accompanied by the appearance of vortices near the holes. Some vortices connect the lateral boundary surfaces of adjacent holes. (c) In the case of small holes, in the regions between the holes, emerging magnetization distributions look like fragments of DWs. The appearance of vortex filaments similar to the vortex core of C-shaped DWs is typical. Evolution of the 3D magnetization distribution leads to a number of peculiar instant topological configurations. (a) Two vortex filaments connect the boundary surfaces of the film and two other filaments connect the lateral boundaries of the hole with the boundaries of the film (Fig 1a). At the upper boundary of the film, there are two local vortices and one local antivortex (ends of vortex filaments on the surface, Fig. 1b). Additionally, there is an antivortex-like configuration around the hole. At the lower boundary, there is one local vortex, two local antivortices, and a vortex-like configuration around the hole (Fig. 1c). (b) One vortex filament connects the boundary surfaces, the other filament connects the intra-film boundary of the hole and the boundary of the film. There is a local vortex and an antivortex-like hole at the upper boundary, and a vortex-antivortex pair at the lower boundary. (c) A fully antivortex-like hole is combined with a local vortex at the upper boundary and vortex-like hole at the lower boundary.

**References:**

[1] A. Fernández-Pacheco, R. Streubel, O. Fruchart, et al. Nature Comm. Vol. 8, 15756 (2017). [2] C. Donnelly, S. Finizio, S. Gliga, et al. Nat. Nanotechnol. Vol. 15, p. 356 (2020). [3] C. Donnelly, K. L. Metlov, V. Scagnoli, et al. Nat. Physics Lett. https://doi.org/10.1038/s41567-020-01057-3 (2020). [4] V. V. Zverev, E. Zh. Baykenov, I. M. Izmozherov. Phys. Met. Metallogr. Vol. 120, p. 1299 (2019). [5] V. V. Zverev, E. Zh. Baykenov, I. M. Izmozherov. Phys. Solid State, Vol. 61, p. 2041 (2019). [6] A. Vansteenkiste, J. Leliaert, M. Dvornik et al. AIP Advances, Vol. 4, 107133 (2014).

**Images:**

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/f1b905260d9cf801ccb2bec91c5c2329.png)
Fig. 1. A snapshot of the vortex structure (a) inside the film, (b) at the upper boundary (top), (c) at the lower boundary (bottom).

 Dynamical Rearrangements of 3D Vortex Structures in Moving Domain Walls in Continuous and Antidot Patterned Permalloy Films

Nickel thin films (100 nm) deposited on 128o Y-cut: LiNbO3 exhibit a well-defined magnetic anisotropy upon annealing at 325o C [1]. Simulating the magnetization of these films using a temperature dependent Stoner-Wohlfarth model indicates that the films have a very narrow angular distribution of anisotropy axes [2]. Here we report a direct measurement of the angular distribution of anisotropy axes by the angular remanence technique [3]. The angular range over which the easy axis remanence drop significantly is a measure of the angular distribution of the anisotropy axes. For annealed nickel thin films on 128o Y-cut: LiNbO3 the FWHM of the remanence as a function of angle (α) is approximately 0.44o .

**References:**

[1] S.A. Mathews, N.S. Bingham, R.J. Suess, K.M. Charipar, R.C.Y. Auyeung, H. Kim, and N.A. Charipar, IEEE Trans. Mag., Vol. 55, Issue 2, (2019) [2] S.A. Mathews, A. Ehrlich, and N. A. Charipar, “Hysteresis branch crossing and the Stoner-Wohlfarth model”, Scientific Reports, 10:15141 (2020). [3] H.J. Richter and H. Hibst, Jour. Mag. Mag. Mat., 95, 118-122 (1991).

**Images:**

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/a8bbb46dcce86d5d88cbc7706792ebd7.jpg)
Figure 1: Absolute value of easy axis remanence as a function of α (angle between the hard axis and the saturating field). Open squares represent measured data. The solid line represents the model, assuming an ensemble of Stoner-Wohlfarth particles with two normal (Gaussian) distributions of anisotropy orientation.

Application of an improved interpolating element-free Galerkin method in magnetic field calculation

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