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Rare heart substituted iron garnets (REIG) are characterized by a high magnetic susceptibility that results in a large magnetization in the presence of small magnetic fields [1]. The magnetic microstructures are influenced by natural or artificial defects; in particular, domain walls (DWs) undergo to nucleation, segmentation and pinning in correspondence to them. In REIG with in-plane spontaneous magnetization, the Néel spike domain structure is one of the most common magnetic microstructure nucleated at microdefects and its charged DWs can produce not negligible stray fields above the REIG. These DWs are interesting both for fundamental studies and for innovative applications such as micro positioning by contactless actuators [2].&lt;br/&gt;To make the most out of this, performing a quantitative analysis is crucial. The experimental characterization by means of quantitative magneto-optical imaging with an indicator film [3] showed that stray fields are detectable above the surface of (Lu, Bi) substituted iron garnets, characterized by a spontaneous in-plane magnetization around 150 kA/m, in correspondence of DWs delimiting a Néel spike that is generated by a micro-crack [4].&lt;br/&gt;In order to understand how the externally detected stray fields are correlated to the pattern of the internal magnetization, here we propose 2D and 3D models based on FEM developed through the COMSOL Multiphysics&lt;sup&gt;©&lt;/sup&gt; software. These models take into account the magnetic microstructure of the Néel spike, i.e., either single DWs that run parallel into the bulk of the domain and a couple of DWs at the Néel spike tip (2D) or DWs that merge in a single point (3D). This approach allows us to describe the magnetic microstructure by means of CAD defined patterns, whose complex structure can be implemented easily in the FEM simulation. It turns out that the results are in agreement both with experiments and with previous numerical calculations performed with a simplified model in MATLAB&lt;sup&gt;©&lt;/sup&gt; [4]. The comparison with experiments allowed us also to determine the distance of measurement between the indicator film of MOI experiments and the REIG under investigation.&lt;br/&gt;Figure 1 shows linear profiles across the DWs, along with experimental data. The distance of measurement is about 7 μm and the magnetic induction at the tip of the Néel spike is twice the value found for the single charged DW at the domain boundary. The simplicity of the 2D model limits the computational cost required; on the other hand, the 3D simulation can be useful for reproducing in detail the stray field distribution generated by the complex structure of the Néel spike domain, as shown in Figure 2.&lt;br/&gt;In summary, we validated these models for reproducing the stray field distribution above charged DWs, opening the way for the design of artificial structures in order to obtain controlled magnetic patterns, thanks to the powerful 3D engine of the COMSOL Multiphysics&lt;sup&gt;©&lt;/sup&gt; software.

**References:**

[1] L. E. Helseth, M. Kurniawan, R. W. Hansen, and T. H. Johansen, PHYSICAL REVIEW B77, 064433 (2008)&lt;br/&gt;[2] L. E. Helseth, R. W. Hansen, T. H. Johansen, and T. M. Fischer, Appl. Phys. Lett.85, 2556 (2004)&lt;br/&gt;[3] F. Laviano, D. Botta, A. Chiodoni, R. Gerbaldo, G. Ghigo, L. Gozzelino, S. Zannella and E. Mezzetti, 16, (2003) 71; Physica B 403 (2008) 293&lt;br/&gt;[4] A. Napolitano, C. Ragusa, S. Guastella, S. Musumeci, P. Rivolo, F. Laviano, JMMM, Vol 504, 2020, 166556&lt;br/&gt;[5] O. Khan, C. Ragusa, and F. Khan, IEEE transactions on magnetics, vol. 50, no. 11, November 2014

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7f0da98d3c563c66b54b53692a817a42.jpg)
Figure 1. Cross sectional profiles of the stray field, simulated and experimental values, for a single DW (a) and the tip of the Nèel spike (b).

![](https://assets.underline.io/uploads/markdown_image/1/image/2928dd60fc86135c15716d29f81cc2da.jpg)
&lt;i&gt;Figure 2. Magnetic field distribution of the stray field above a Néel spike: 3D plot of the magnetic induction intensity that clearly shows the stray field peak in correspondence to the tip of the Nèel spike. The distance of measurement was chosen on the basis of the previous comparison between experimental data and 2D simulations.&lt;/i&gt;

INTERMAG 2021

 Modeling stray-field distribution generated by domain-walls in rare-earth substituted iron garnets

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Rare heart substituted iron garnets (REIG) are characterized by a high magnetic susceptibility that results in a large magnetization in the presence of small magnetic fields [1]. The magnetic microstructures are influenced by natural or artificial defects; in particular, domain walls (DWs) undergo to nucleation, segmentation and pinning in correspondence to them. In REIG with in-plane spontaneous magnetization, the Néel spike domain structure is one of the most common magnetic microstructure nucleated at microdefects and its charged DWs can produce not negligible stray fields above the REIG. These DWs are interesting both for fundamental studies and for innovative applications such as micro positioning by contactless actuators [2]. To make the most out of this, performing a quantitative analysis is crucial. The experimental characterization by means of quantitative magneto-optical imaging with an indicator film [3] showed that stray fields are detectable above the surface of (Lu, Bi) substituted iron garnets, characterized by a spontaneous in-plane magnetization around 150 kA/m, in correspondence of DWs delimiting a Néel spike that is generated by a micro-crack [4]. In order to understand how the externally detected stray fields are correlated to the pattern of the internal magnetization, here we propose 2D and 3D models based on FEM developed through the COMSOL Multiphysics© software. These models take into account the magnetic microstructure of the Néel spike, i.e., either single DWs that run parallel into the bulk of the domain and a couple of DWs at the Néel spike tip (2D) or DWs that merge in a single point (3D). This approach allows us to describe the magnetic microstructure by means of CAD defined patterns, whose complex structure can be implemented easily in the FEM simulation. It turns out that the results are in agreement both with experiments and with previous numerical calculations performed with a simplified model in MATLAB© [4]. The comparison with experiments allowed us also to determine the distance of measurement between the indicator film of MOI experiments and the REIG under investigation. Figure 1 shows linear profiles across the DWs, along with experimental data. The distance of measurement is about 7 μm and the magnetic induction at the tip of the Néel spike is twice the value found for the single charged DW at the domain boundary. The simplicity of the 2D model limits the computational cost required; on the other hand, the 3D simulation can be useful for reproducing in detail the stray field distribution generated by the complex structure of the Néel spike domain, as shown in Figure 2. In summary, we validated these models for reproducing the stray field distribution above charged DWs, opening the way for the design of artificial structures in order to obtain controlled magnetic patterns, thanks to the powerful 3D engine of the COMSOL Multiphysics© software.

**References:**

[1] L. E. Helseth, M. Kurniawan, R. W. Hansen, and T. H. Johansen, PHYSICAL REVIEW B77, 064433 (2008) [2] L. E. Helseth, R. W. Hansen, T. H. Johansen, and T. M. Fischer, Appl. Phys. Lett.85, 2556 (2004) [3] F. Laviano, D. Botta, A. Chiodoni, R. Gerbaldo, G. Ghigo, L. Gozzelino, S. Zannella and E. Mezzetti, 16, (2003) 71; Physica B 403 (2008) 293 [4] A. Napolitano, C. Ragusa, S. Guastella, S. Musumeci, P. Rivolo, F. Laviano, JMMM, Vol 504, 2020, 166556 [5] O. Khan, C. Ragusa, and F. Khan, IEEE transactions on magnetics, vol. 50, no. 11, November 2014

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7f0da98d3c563c66b54b53692a817a42.jpg)
Figure 1. Cross sectional profiles of the stray field, simulated and experimental values, for a single DW (a) and the tip of the Nèel spike (b).

![](https://assets.underline.io/uploads/markdown_image/1/image/2928dd60fc86135c15716d29f81cc2da.jpg)
Figure 2. Magnetic field distribution of the stray field above a Néel spike: 3D plot of the magnetic induction intensity that clearly shows the stray field peak in correspondence to the tip of the Nèel spike. The distance of measurement was chosen on the basis of the previous comparison between experimental data and 2D simulations.

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HDDR process can not only refine the coarse R2Fe14B grains (~ 100 μ m) into fine grains (~300 nm), but also induce these fine grains to be oriented along the c-axis. The grain refinement can improve the coercivity of the magnetic materials, and the grain orientation is helpful to obtain high remanence, so the anisotropic R2Fe14B materials with high coercivity, remanence and maximum energy products can be obtained after HDDR process. Recent researches show that the coercivity of HDDR R2Fe14B magnetic materials can be enhanced to ~20 kOe by refining the grain size of the materials or by diffusion of the eutectic alloys with the low melting point, but there is still a big gap between the coercivity and its anisotropic field (70 kOe). This indicates that the coercivity of HDDR magnetic materials still has a lot of room to improve, especially the grain size of HDDR powder is close to that of R2Fe14B single domain particles. In order to further improve the coercivity, it is necessary to study the coercivity mechanism of HDDR magnetic materials. In this study, the initial magnetization curve of thermally demagnetized HDDR anisotropic Pr-Fe-B sample was firstly investigated. The results show that the magnetization increase moderately with external magnetic field, but it is not only different from the rapid rise of magnetization of sintered R-Fe-B magnets but also different from the slow increase of melt-spun R-Fe-B powders at the beginning and then the sudden rapid increase at a certain critical field. Therefore, it is not easy to determine the type of the coercivity mechanism for the powders. In order to study the coercivity mechanism of the above HDDR anisotropic powders in detail, the temperature dependence of coercivity of the sample was measured. The coercivity decrease monotonically with the increase of temperature. According to the micromagnetic model, the coercivity Hc can be generally expressed Hc=αφαk2K1/MS-NeffMs. Due to the preferential nucleation of inhomogeneous regions in the magnetic powder, a correction factor αk should be introduced. In addition, because the grain alignment in the magnetic powder is not completely consistent, the correction factor αφ is also introduced. The dependence of the anisotropy constants K1, and saturation magnetization (Ms) of the main phase Pr2Fe14B on the temperature are all from the literature. The measured data of HDDR Pr-Fe-B and the theoretical formula were compared to analyze that the coercivity mechanism is controlled by the nucleation of the reversed domain or by the uniform rotation of the magnetic moment. （1）Uniform rotation model: Firstly, in order to analyze whether the coercivity mechanism belongs to uniform rotation of the magnetic moments, the dependence of HC/MS on 2K1/Ms2 was investigated. It is found that the experimental data only fall on a straight line in a part of the temperature region. Even in the section of the line, the slope of the line is not 1, which does not meet the theoretical requirements of uniform rotation. So we can exclude the possibility that the magnetization process is uniform rotation of the magnetic moments. （2）Strongly correlated reverse domain nucleation model When the grains in the magnetic powder are strongly magnetically coupled, a reversion of a misaligned grain would drive the reversion of the neighboring grains. Therefore, the coercivity of the material depends on the grains with the smallest nucleation field. In this case, αφ= αφmin, The coefficient can be calculated for a given temperature. The value of this angle is about 40 ~ 60 degrees on the basis of the calculation results of the external field angle corresponding to the minimum nucleation field αφmin at 50K ~ 350K. Fig.1 shows the HC/MS vs 2αφminK1/Ms2 curve of the powder. The purpose of this curve is to analyze whether the coercivity mechanism of the above powder is the nucleation mechanism of strongly magnetically coupled grains. It can be seen from the figure that the experimental data points basically fall on a straight line, which means that under the framework of micromagnetism, the nucleation mechanism with the magnetically coupled grains can describe the magnetization reversal process of the HDDR powder. Fig.2 shows magnetic domain patterns of the HDDR treated Pr-Fe-B alloys. In the case of the HDDR treated alloys, very fine maze-like magnetic domains were observed and these domains extend over more than 1 μm in the original large single particles with the mean size of 10 μm (Fig.2 (b) and (c)). This domain pattern shows a characteristic of interaction domains, and each domain consists of several fine grains, which are magnetically coupled to form large interaction domains. The evolution of the magnetic domain of HDDR anisotropic Pr-Fe-B magnetic powder under the external applied magnetic field (0 T → 2 T → 0 T → -2 T → 0 T → 2 T) was also investigated. Only the disappearance of the labyrinth domain composed of multiple grains and the appearance of the magnetic domain under the condition of positive magnetization field and negative reverse magnetization field were observed, while the domain wall movement was not observed, which further implies that the nucleation of demagnetization is the main mechanism controlling the coercivity of anisotropic HDDR Pr-Fe-B powder.

**References:**

[1] T. Takeshita, R. Nakayama, in 11th Workshop on Rare Earth Magnets and their Applications, Pittsburg, USA, 49 (1990). [2] K. D. Durst and H. Kronmüller, J. Magn. Magn. Mater. 68, 63 (1987). [3] H. Kronmüller, K. D. Durst, and M. Sagawa, J. Magn. Magn.Mater. 74, 291（1988）. [3] W. L. Chen, W. Liu, and Z. D. Zhang, Phy. Rev. B, 72，224419 (2005)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/573cdeaf9631d4c74e5e89dbcdd1548c.jpg)
Fig.1 The dependence of Hc/Ms on 2𝛼𝜑minK1/MS2 of HDDR powders

![](https://assets.underline.io/uploads/markdown_image/1/image/6c2d8f8fab2a230db89ba7a5b00ecf97.jpg)
Fig. 2 Magnetic domain Kerr images of Pr-Fe-B alloys (a) before and (b) after the HDDR treatment. (c) zoom in of red rectangle region in the figure (b)

 Study on magnetization reversal processes of anisotropic HDDR Pr2Fe14B-type magnetic materials

Cobalt as a substitute for Iron, and combinations of different rare earths in alloy can increase the Curie temperature and coercivity of Nd2Fe14B type ferromagnets[1]. Complex ferromagnets, with atomic scale ordering, require high resolution simulations which are computationally expensive due to individual treatment of atoms. The compromise of the atomistic model[2], allows nanometer scale simulations with reasonable efficiency, but requires input parameters often derived from ab initio calculations, or Density functional theory, creating a multi-scale model. Complex materials often have several input parameters, such as magnetic exchange and magnetocrystalline anisotropy constants, which cannot be easily evaluated using these methods. As a result, often the requirement is to fit the exchange parameter to macroscopic experimental data. For simple materials this can be evaluated with few test simulations, as the relationship between exchange and macroscopic properties behaves linearly for single element materials, as in the mean field model. Complex materials can have complex relationships between exchange and macroscopic properties, so many test simulations are often required to perform a full sweep of the parameter space. This body of work details the evaluation of exchange and anisotropy constants for 2:14:1, or Nd2Fe14B type rare earth ferromagnets, using the VAMPIRE atomistic software modelling package. Treating this as an optimisation problem reduces the number of atomistic simulations required for a given numerical precision of constant. Evaluation of exchange constants was performed using the Curie temperature as a figure of merit with several different methods of evaluating the Curie temperature compared for robustness and consistency, with the optimisation of these exchange constants performed with a combination of linear searching and root finding algorithms. These methods have often been compared for various rock magnets[3]. The search uses a locally linear assumption, to prime the low temperature quantum rescaling in VAMPIRE, and to better parallelise the initial stages of the search. Similar approaches are taken to calculate the anisotropy constants of Nd2Fe14B to reproduce the First Order Magnetic Phase Transition[4] at high external fields and low temperatures. The exchange constant of various rare-earth ferromagnets is reproduced, yielding the correct macroscopic Curie temperatures without reliance on density functional theory/ab initio calculations. Lanthanum was used as the prototype for the Fe-Fe exchange, and this was fixed throughout the other ferromagnets. The Curie temperatures calculated are in good agreement with J.F.Herbst, 1991 [5]. The First Order Magnetic Phase Transition was effectively reproduced, but requires additional tuning for a complete analysis of the relationship between second and fourth order anisotropy constants and the subsequent macroscopic magnetic behaviour.

**References:**

[1] Wei, L, et al. J. Less Common Metals 126 (1986); [2] R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler, M. O. A.Ellis and R. W. Chantrell, Journal of Physics: Condensed Matter,vol. 26, 103202, (2014); [3] K. Fabian,V. P. Shcherbakov, S.A. McEnroe, Geochemistry, Geophysics, Geosystems, 14, 947, (2013); [4] J. M. Cadogan, J. P. Gavigantt, D. Givordt and H. S. Li, J. Phys. F: Met. Phys. 18 779 (1988); [5] J. F. Herbst Rev. Mod. Phys. 63, 819 (1991)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/4b9dd3a6e04c6e1367a35ec04d9cfb05.jpg)
The exchange paramaters calculated using the optimsiation method result in appropriate Curie temperatures when evaluated using the VAMPIRE atomsitic modelling software package. The Curie temperature calculated from this simulation compares well to the values from literature[5] (Herbst, 1991). 
 
Ce2Fe14B has not been calculated, as the rare earth behaves sufficiently differently to Lanthanum as to prevent using Lanthanum as the base of Cerium's Fe-Fe exchange in this manner. 
 
The magnitude of the exchange is presented, the heavier rare earths are ferrimagnetic in the 2:14:1 phase.

 Optimisation of atomistic modelling parameters for Nd2Fe14B-type rare earth ferromagnets, using the Curie temperature and first order magnetic phase transitions as figures of merit

High-performance, high-efficiency motors are indispensable for the development of electric vehicles and the realization of electric aircraft. The key is the development of permanent magnets that can withstand use even at high temperatures. The reported Sm-Fe-based magnet compounds [1-3] have a high Curie temperature, and the values of the magnetic anisotropy field and saturation magnetization are higher or comparable to those of Nd2Fe14B. In this work, we focused on the Induction thermal plasma(ITP) process as a new process for obtaining anisotropic nano-sized powder of these permanent magnet compounds. The ITP process is a physical bottom-up process that can synthesize nano-sized single-crystal powders due to the high quenching effect during the process [4, 5]. When this process can be applied to prepare a fine permanent magnet anisotropic powder of around 100 nm, which is difficult to obtain by the conventional method, can be synthesized, leading that an anisotropic magnet having a large coercive force can be obtained. Therefore, as the first step, we tried to produce nano-sized anisotropic Sm-Fe-N alloy powder and to evaluate the feasibility of preparing the nano-sized powder by the ITP process. The mixed powder of Sm powder (D50 = 18 μm) and Fe powder (particle size about 3 μm, High Purity Chemical Laboratory) was used as a starting powder for the ITP process. The ITP process was performed under the condition of process pressure of 100 kPa. The input power was 6 kW. Since Sm metal powder as a starting material is not commercially available, we prepared micron-sized Sm powder by using a crucible-less skull gas atomizing device (manufactured by Sinfonia Technology Co., Ltd.). Though the ITP process, the obtained nano-sized powder was sealed in a quartz capillary in a glove box, and the crystal structure was evaluated by a powder X-ray diffraction experiment with Aichi Synchrotron BL5S2 without oxidizing the powder. In addition, SEM and STEM were used for microstructure analysis. From the XRD profile, SmFe2 and Sm-Fe with TbCu7-type structure were identified as Sm-Fe alloy phases. From the SEM image, the average particle size of the SmFe alloy was about 100 nm. The Sm-Fe alloy particles were found to be single crystals by high-resolution STEM observation. In addition, many particles consist of Fe-core/Sm-shell and Fe-core/SmFe-shell were also observed. Combine the results obtained from the XRD measurement and the STEM observation, it was found that the Fe detected by XRD is not the Fe particle but the Fe core in the core-shell structure. This is because the vapor pressure of Sm is more than 10000 times higher than that of Fe, so in the thermal plasma process, Fe nucleates first and then Sm and Fe or Sm agglomerates around the Fe particles. After the nitriding process by the heat treatment at an elevated temperature of 400C in N2 flow for 15min, nitrogen was surely introduced into the Sm-Fe lattice, which is confirmed from the lattice expansion. For the first time in this study, we succeeded in synthesizing Sm-Fe-N anisotropic particles with a particle size of 100 nm or less using the ITP process. Especially, the preparation of the anisotropic particle with TbCu7-type structure will lead to the next generation high-performance permanent magnet.

**References:**

[1] T Iriyama, K Kobayashi, N Imaoka et al., IEEE Trans. Magn., Vol.28, p.2326 (1992) [2] S. Sakurada, A. Tsutai, T. Hirai et al., J. Appl. Phys., Vol.79, p4611 (1996) [3] Y. Hirayama, Y.K. Takahashi, S. Hirosawa et al., Scr. Mater., Vol.138, p.62 (2017) [4] Y. Hirayama and K. Takagi, J. Alloys. Compd., Vol.792, p.594 (2019) [5] Y. Hirayama, M. Shigeta, Z. Liu et al., Available at http://dx.doi.org/10.2139/ssrn.3692994

**Images:**

 Nano-sized anisotropic Sm-Fe-N particle preparation by induction thermal process

Permanent magnets are a key component of a wide range of everyday devices, including motors, actuators, transducers and sensors. Depending on the application needs, magnets are based on high performance rare-earth transition metal phases or on more moderate performance hard ferrite phases, and they may be sintered or polymer bonded. While the remanence of bonded magnets is necessarily limited by dilution of the magnetic material in a non-magnetic matrix, they are cheaper in both material and fabrication costs. What is more, the mechanical properties of the magnets can be tuned according to the choice of matrix. Traditional fabrication routes for polymer bonded magnets include calendering, injection moulding, extrusion, and compression bonding [1] and there has been much recent activity concerning 3D printing of polymer bonded magnets [2,3,4]. Down-sizing magnets opens new opportunities for their use in micro-scaled devices with applications in fields as diverse as telecommunications, energy management and bio-medicine. Sputtering combined with micro-patterning has been used to make fully dense magnets which are a micro-scale equivalent to sintered magnets while various techniques including casting and screen printing combined with photo-lithography have been used to make polymer bonded micro-magnets [5]. Sputtered films may be isotropic or textured, depending on the deposition temperature, whereas micro-scaled polymer bonded magnets reported to date are isotropic. Here we present a process developed to fabricate arrays of anisotropic, flexible bonded micro-magnets. The study focuses on micro-magnets prepared using commercial single crystalline hard ferrite (Br: 195 mT, Hc: 123 kA/m) or Sm-Fe-N (Br: 1.31 T, Hc: 889 kA/m) powders mixed with polydimethylsiloxane (PDMS). This binder was chosen so as to have flexible micro-magnets that can be used to study the influence of mechanical forces on cell development [6]. The innovative aspect of this process, schematized in Fig. 1, consists in the possibility to transfer a specific pattern from a silicon mould, made by deep reactive ion etching (DRIE), to a magnetically aligned powder-polymer composite. The composite is maintained within the mould during the alignment step by coverage with a glass slide. A first soft curing step is performed with the cover-slide in place, then a final curing is performed to attach the micro-magnets to a PDMS base. The procedure has been used to fabricate arrays of micro-magnets of thickness ranging from 50 µm to 300 µm and width in the range from 20 µm to 4000 µm. As a magnetic field source, bulk magnets, a superconductive coil and pulsed magnetic field systems were tested. The degree of alignment was assessed by measuring the hysteresis loops of fully cured micro-magnet arrays with the measuring field applied parallel or perpendicular to the direction of the alignment field. Note that these measurements were made prior to removal of the arrays from the Si mould, to prevent distortion of the flexible micro-magnets under the torque induced by the measuring field. As an example, easy and hard axes hysteresis loops made on an array of SmFeN-bonded square micro-pillars with in-plane dimensions of 200x200 µm2 and a height of 300 µm, aligned in a magnetic field of 800 mT, are compared in Fig. 2 a. As can be expected, higher intensity fields were needed to align the SmFeN particles compared to the ferrite particles. The stray fields produced by aligned and non-aligned micro-magnet arrays were measured using an in-house developed Scanning Hall Probe Microscope [7]. In Fig. 2 b we present the z-component of the stray field produced at a height of 50 µm above the SmFeN micro-magnet array featured in Fig. 2 a. In this presentation, we will give details of structural and magnetic characterisation as well as simulations of arrays of ferrite and SmFeN bonded micro-magnets. We will also report on the actuation of these micro-magnets under the influence of a 3D magnetic field. Finally, we will present prospects for the further development and use of such micro-magnet arrays.

**References:**

[1] J. Ormerod and S. Constantinides, Journal of Applied Physics, Vol. 81, p.4816 (1997) [2] C. Huber, C. Abert and F. Bruckner, Appl. Phys. Lett. , Vol. 109, p.162401 (2016) [3] L. Li, A. Tirado and I. C. Nlebedim, Scientific Reports, Vol. 6, p.36212 (2016) [4] K. Sonnleitner, C. Huber and I. Teliban, Appl. Phys. Lett., Vol. 116, p.092403 (2020) [5] N.M. Dempsey, "Nanoscale Magnetic Materials and Applications", J.P. Liu et al.(eds), p.661, (Springer 2009) [6] N.M. Dempsey, " Handbook of Magnetic Materials", J.M.D. Coey et al. (eds), Springer, (in-press) [7] G. Shaw, R.B.G. Kramer and N. M. Dempsey, Rev Sci Inst, Vol.87, p.113702 (2016)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3a78eeb2c7e66ef3542f55dd6887a55f.jpg)
Fig. 1 Schematic of the process developed for the fabrication of arrays of flexible, anisotropic bonded micro-magnets.

![](https://assets.underline.io/uploads/markdown_image/1/image/a01987af8c35d1a195ab777fd1bb4d16.jpg)
Fig. 2 (a) Hysteresis loops of an array of SmFeN (30 vol %) - PDMS (70 vol %) bonded micro- magnet pillars with in-plane dimensions of 200x200 µm2 and a height of 300 µm. (b) Z-component of the stray magnetic field produced by this array, measured using a micro-Hall probe scanned at a height of 50 µm above the magnet array.

 Fabrication and characterisation of polymer-bonded flexible anisotropic micro-magnet arrays

The CoPt and FePt alloys, when chemically ordered in the L10 phase [1], are among the magnetic materials displaying the highest magnetic anisotropy constant. Therefore, they are perfect candidates for ultra-high density magnetic storage applications, provided nanoparticles can be prepared in such a high anisotropy phase. In the bulk L10 phase both alloys exhibit close magnetic and structural properties. Indeed, the tetragonalization of the unit cell, measured by the c/a ratio is almost the same and equal to 0.97 for both alloys while their anisotropy constant is equal to 5 and 7 MJ.m-3 respectively for CoPt and FePt [1]. The first part of this work is then to determine and compare the intrinsic magnetic anisotropy constant of CoPt and FePt L10 nanoparticles. For this purpose, we dilute, as far as possible, mass-selected nanoparticles (around 3 nm diameter) in an inert carbon matrix to avoid coalescence and to make magnetic interactions negligible. To promote the chemically L10 ordered phase the samples have to be annealed at 600°C during two hours. The magnetic properties are then determined using SQUID magnetometer and a global fitting procedure of the entire set of magnetic measurements [2,3] (see Fig. 1). In a second part we try to explain their significant differences using extended X-ray absorption fine structure measurements and atomic relaxation calculations (see Fig. 2). We conclude that the very high magnetic anisotropy measured in FePt nanoparticles [4] is related to the weak atomic relaxation in this alloy, while for chemically ordered CoPt nanoparticles the strong atomic relaxation in the Co plans [5] seems to reduce drastically the magnetic anisotropy.

**References:**

[1] P. Andreazza et al., Surf. Sci. Rep. 70, 188 (2015) [2] A. Tamion et al., Phys. Rev. B 85, 134430 (2012) [3] A. Hillion et al., Phys. Rev. B 88, 094419 (2013) [4] P. Capiod et al., Phys. Rev. Lett. 122, 106802 (2019) [5] N. Blanc et al., Phys. Rev. B 87, 155412 (2013)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/bd365171dd94e366a992d84cc0f1e80e.jpg)
Fig. 1: Low temperature hysteresis loop measured for a diluted assembly of CoPt nanoparticles (left) and FePt nanoparticles (right), after annealing. The dots correspond to measurements while the line is obtained with a global fit taking into account the particle size and magnetic anisotropy distributions. Note that despite the particles are similar (around 3.5 nm diameter), the coercivity is much larger for FePt nanoparticles.

![](https://assets.underline.io/uploads/markdown_image/1/image/3e51fbabcc3e62496eafd8c27717286a.jpg)
Fig. 2: Comparison between the mean c/a ratio (computed from interatomic distances: homoatomic and heteroatomic bonds) around transition metal atoms and Pt atoms, from density functional calculations of relaxed CoPt or FePt clusters. N represents the number of atoms used to build perfect truncated octahedrons. The lines are guide to the eye, while the dashed line indicates the c/a ratio in the bulk L10 phase.

 Magnetic anisotropy of chemically ordered CoPt and FePt nanoparticles, why is it so different?

In the 1980s ferrous materials were the first subject of extensive investigation on magnetic field annealing (MFA). Hereby, the potential of using high external magnetic fields as an additional degree of freedom for materials synthesis and tailoring of microstructures in solid state was demonstrated. Since then, due to the progression in magnetic materials and superconducting magnets, processing in magnetic field has become both less costly and more viable. Hence, MFA is achieving renewed attention nowadays. The processing technique might even evolve into a valuable tool on broad industrial level, since it provides the opportunity to save energy by lowering heat treatment temperatures and shortening processing times. Furthermore, the application of an external high magnetic field as an additional independent processing parameter allows for stabilization of metastable phases and creation of texturized microstructures that are not accessible on zero-field conditions. Nevertheless, a lot of experimental work and modeling is still required to entirely unveil the mechanisms of MFA for gaining control on such material manipulation. Mn3Ga is a very interesting and highly suitable candidate for exploring MFA. The intermetallic compound occurs in three different crystal structures, all of them possessing rich magnetic properties attractive for different kinds of application. Regarded as a rare-earth-free permanent magnet in potential application, especially the tetragonal phase meets the particular requirements of being exceptionally prone to manipulation by MFA. Due to the ferrimagnetic ordering it exhibits a much higher magnetic moment compared to the other Mn3Ga polymorphs. Therefore, it is supposed to be thermodynamically favored in its formation under influence of an external magnetic field. Additionally, the tetragonal structure shows high anisotropy in susceptibility, which is conditional to establish a texture upon MFA. As already reported by Ener et al., the phase transition from hexagonal to tetragonal Mn70Ga30 can be accelerated remarkably by combining severe plastic deformation and MFA [1]. In contrast, the study at hand adds the investigation of the influence of different degrees of deformation plus the impact of MFA on the phase transformation starting from a predominantly cubic structure of the parent material Mn3Ga. For this purpose annealing was performed at 400°C, lasting for at least 0.5 hours and up to 14 days. In the case of MFA samples, a magnetic flux density of 7 T was applied. By X-ray diffraction the different stages of phase transformation were observed. Further characterization regarding microstructure and magnetic properties was achieved using light and electron microscopy as well as SQUID magnetometry. The results indicate that the properties after long-term annealing are very similar, independent of previous state of deformation and use of magnetic field. More specifically, after 14 days of annealing samples attained a phase-pure tetragonal structure and show equal remanence. Nevertheless, within the first 24 hours of annealing the differences are enormous with respect to phase formation, microstructure and magnetic properties. In case of zero-field annealing of highly deformed samples or MFA of undeformed samples, reaching the tetragonal structure is postponed by a pronounced intermediate formation of the hexagonal phase. Apart from that, combining previous deformation and MFA accelerates the phase transformation effectively. Furthermore, with increasing degree of deformation the coercivity can be increased stepwise. However, the development of coercivity over annealing time is complex for both zero-field annealing and MFA. An attempt to solve this complex behavior by relating microstructure and coercivity is presented. Finally, the phase transformation behavior of Mn3Ga under influence of an external magnetic field is correlated to changes in defect mobility and development of local deviations in composition (see Fig. 1).

**References:**

[1] S. Ener, K.P Skokov, D.Y. Karpenkov, M.D. Kuz’min and O. Gutfleisch, Journal of Magnetism and Magnetic Materials, 382, 265-270 (2015)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/b5271fcdca9ea6f7afc41d135535331f.jpg)
Fig. 1: EDS line scan of highly deformed sample after 1 h of MFA 
A) SEM picture showing local course of scan (bright areas in hexagonal, grey areas in tetragonal structure) 
B) compositional mismatch along line scan

 Manipulation of Phase Transformation and Microstructure in Mn3Ga Using Magnetic Field Annealing

Recently, the topic of rare-earth-free permanent magnets has received considerable attraction for the alternative to rare-earth magnets due to the resource crisis and the high cost of rare-earth metals. Among rare-earth-free permanent magnetic materials, the low-temperature phase (LTP) of MnBi has been considered as a potential alternative because it exhibits an unusual positive temperature coefficient of coercivity in the temperature range 153-550 K. Because of the above magnetic property, the MnBi-LTP has a higher coercivity compared to Nd2Fe14B permanent magnetic material at elevated temperature, and it was recently reported that the coercivity of MnBi-LTP reached up to 28.3 kOe at 530K. According to previous researches, this excellent coercivity characteristic of the MnBi-LTP is attributed to its high magneto-crystalline anisotropy (K1≈1.6 x 106 J/m-3 at 300 K)[1]. In spite of such excellent intrinsic magnetic properties, the MnBi-LTP bulk magnets practically fabricated have a significantly lower energy product compared to its theoretical value (BHmax ≈ 16 MGOe at 300 K) due to engineering issues that occur during the mechanical milling process such as decomposition of the magnetic phase, manganese oxidation, and agglomeration. At present, several approaches including high or low energy ball-milling (HEBM or LEBM), surfactant-assisted ball-milling (SABM) have been reported to avoid those issues. However, those approaches based on a ball milling process at room temperature have inevitably a restriction in reducing the powder size as well as agglomerating the powder by cold welding phenomenon. Consequently, this leads to the degradation of the energy product by lowering the total anisotropic characteristics of the MnBi-LTP bulk magnets. Therefore, it is essential to synthesize the particle size of the MnBi-LTP to be close to that of its single magnetic domain (≈ 500 nm) in order to achieve the MnBi-LTP bulk magnets with improved anisotropic characteristics. In this regard, it has been proposed that the cryo-milling method is able to be one of the approaches to synthesize the MnBi-LTP particle satisfying the above-required size. It is well known that the cryo-milling method is able effectively to reduce the particle size as well as to avoid cold welding and agglomeration of the power because it carried out at a temperature lower than 123 K, where materials become brittle and friable[2]. It has been reported that the cryo-milling method is successfully synthesized nanocrystalline structures in ductile materials such as Fe, Ni, Al, Zn, etc. However, researches into the cry-milling of MnBi alloys have been rarely conducted. In this study, the Mn54Bi46-LTP powders were synthesized by the cryo-milled method combined with the addition of a surfactant (stearic acid) as well as calcium, and a schematic illustration of the process is shown in figure 1. As shown in the figure, a stainless-steel vial containing a mixture of powder, balls, surfactant, and calcium was vibrated with a certain frequency, and the temperature was maintained at 77 K during the milling process by the flow of liquid nitrogen. In addition, surfactant (stearic acid) and calcium were added in order to suppress the particle agglomeration and manganese oxidation, respectively. Figure 2 shows the Hysteresis loops for the cryo-milled Mn54Bi46 powder according to the addition of surfactant and calcium. All of the milling conditions have an identical ball-to-powder ratio of 18:1, and stearic acid and calcium were added 0.225g and 0.01g to Mn54B46 powder of 0.9 g, respectively. As shown in the figure, the cry-milled Mn54B46 powder without any addition has a Hc of 14.1 kOe with a remanence (Mr) of 40.7 emu/g and a magnetization (Ms,2.5T) of 40.7 emu/g, while the cry-milled Mn54B46 powders with the addition of stearic acid have a higher both Mr of 51.0 emu/g and Ms,2.5T of 53.2 emu/g. These results, especially the increase of Mr/Ms,2.5T ratio, indicate that the powders were effectively separated by the addition of stearic acid during the milling process. In addition, the result of improved Ms,2.5T means that the decomposition rate of the magnetic phase decreases as the impact energy received by the powder is alleviated by the stearic acid on the powder surface. Furthermore, from the result of magnetic properties under the calcium addition conditions, it is found that calcium addition plays an important role in the purification of Mn54Bi46-LTP powder by suppressing manganese oxidation. Based on these results, further investigation on the optimization of the cryo-milling process and the magnetic properties of bulk magnets manufactured with cryo-milled powders will be discussed.

**References:**

[1] V. N. Antonov and V. P. Antropov, Low Temperature Physics, vol. 46, no. 1, p.1 (2020) [2] N. K. Katiyar, K. Biswas, and C. S. Tiwary, International Materials Reviews, p.1 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/f2a8abc3721902719bd228a4fc2e173e.jpg)
Fig. 1. A schematic illustration of the cryo-milling process used in this study.

![](https://assets.underline.io/uploads/markdown_image/1/image/ad03f42558598a49f705daf6a85b3f5b.jpg)
Fig. 2. Hysteresis loops for the cryo-milled Mn54Bi46 powder measured parallel to the alignment direction.

 Magnetic properties of bulk magnets manufactured by the cryo-milled Mn54Bi46 powder

Introduction The metastable α”-Fe16N2 is considered as a candidate for rare earth free semi-hard magnetic materials [1]. It is necessary to improve magnetocrystalline anisotropy of α”-Fe16N2 to obtain higher coercivity Hc [2]. Many theoretical calculations and thin film experiments substituting Fe with some elements such as Co, Ni, Mn, Al, etc. have been done. For example, Co is suggested to increase in magnetocrystalline anisotropy [3]. As one of the synthesis methods of α”-Fe16N2 nanoparticles, it is well known that the nitridation of α-Fe nanoparticles produced by reduction of iron hydroxide such as α-FeOOH. Previously, we have reported that the influence of substituting Co, Al for Fe by synthesizing α”-(Fe, Co)16N2 or α”-(Fe, Al)16N2 nanoparticles [5] and revealed that Co substitution results in the temperature regimes for reduction and nitridation were confirmed to very narrow limits, on the other hand Al substitution brings the wider temperature regimes. In this paper, in order to investigate the complex substitution effect of Fe by Co and Al, we challenged to the synthesis of α”-(Fe0.95Co0.02Al0.03)16N2 nanoparticles by hydrogen reduction of α-(Fe0.95Co0.02Al0.03)OOH as a starting material and subsequent nitridation. Experimental We used commercially available α-(Fe1-xMx)OOH (M= Co, Al, x= 0, 0.02, 0.05) and α-(Fe0.95Co0.02Al0.03)OOH nanoparticles as a starting material. The conditions for reduction and nitridation were systematically examined to obtain higher yield of α”-(Fe1-xMx)16N2 or α”-(Fe0.95Co0.02Al0.03)16N2 according to the following procedure. The α-(Fe1-xMx)OOH nanoparticles were first reduced at 300 - 500 °C for 4 hours in a H2 gas flow. Nitridation was subsequently performed at 150 - 180 °C for 5 hours in an NH3 gas flow. Crystallite diameter Dc of reduced α-Fe1-xMx nanoparticles was measured as average diameter of nanoparticles using Scherrer’s formula based on X-ray diffraction (XRD) with Co Kα radiation. Morphological changes were observed using transmission electron microscopy (TEM). For magnetic properties measurements, these nanoparticles were packed into a disk shape (6 mm diameter, 1 mm thick) with resin. Magnetization M15k and coercivity Hc were measured using vibrating sample magnetometer (VSM) with a maximum applied magnetic field of 15k Oe. Results Fig. 1 (a) shows the regions in the nitridation temperature versus reduction temperature diagrams corresponding to where the α”-phase was produced. M15k and Hc values of the assembled nanoparticles are also indicated. α”-phase was produced under the limited conditions of reduction temperature of 340 °C and nitridation temperature of 170 °C for starting from α-(Fe0.98Co0.02)OOH. On the contrary, α”-phase was produced under broad conditions of reduction temperatures of 400 - 500 °C and nitridation temperatures of 155 - 170 °C for starting from α-(Fe0.98Al0.02)OOH. In the case of starting from α-(Fe0.95Co0.02Al0.03)OOH, α”-phase was produced under the conditions of reduction temperatures of 400 - 500 °C and nitridation temperature of 160 °C. The conditions of α”-phase production were broadened by Al addition to Co. Fig. 1 (b) shows TEM images for the nanoparticles after nitridation starting from α-(Fe0.98Co0.02)OOH, α-(Fe0.95Al0.05)OOH and α-(Fe0.95Co0.02Al0.03)OOH. These nanoparticles are synthesized with hydrogen reduction at 340 °C for 4 hours and subsequent nitridation at 170 °C for 5 hours for α-(Fe0.98Co0.02)OOH, and with hydrogen reduction at 400 °C for 4 hours and subsequent nitridation at 160 °C for 5 hours for α-(Fe0.98Co0.02)OOH and α-(Fe0.95Co0.02Al0.03)OOH, respectively. The particle size of nanoparticles after nitridation starting from α-(Fe0.98Co0.02Al0.03)OOH are between the ones of starting from (Fe0.98Co0.02)OOH and α-(Fe0.95Al0.05)OOH. This result suggested that Al was effective to avoid sintering during reduction. Fig. 2 (a) shows the relationship between M15k and Hc for the nanoparticles after nitridation starting from α-(Fe0.98Co0.02)OOH, α-(Fe0.95Al0.05)OOH and α-(Fe0.95Co0.02Al0.03)OOH. In the case of starting from α-FeOOH was also shown. Higher Hc were obtained by the complex of Co and Al compared to Co or Al alone. Fig. 2 (b) shows Hc change as a function of Dc of those nanoparticles. This result indicates that there is a good relationship between Hc and Dc regardless additive composition. Hc increases with decrease in Dc. In the case of nanoparticles after nitridation starting from α-(Fe0.95Co0.02Al0.03)OOH gives the smallest Dc of 22 nm and highest Hc of 1800 Oe. It is assumed that Al addition to Co is effective to prevent crystalline growth which leads to high Hc of α”-phase. To clarify the reason of increase in Hc, further precise investigation of the distribution of Al and the relationship between anisotropy field of nanoparticles after nitridation will be presented in this conference.

**References:**

[1] R. Skomski et al., Scr. Mater., 112 (2016) 3 [2] M. Tobise et al., J. Magn. Soc. Jpn, 41 (2017) 58 [3] Y. Jiang et al., AIP Adv., 6 (2016) 056007 [4] L. Ke et al., Phys. Rev. B, 88 (2013) 024404 [5] M. Tobise et al., AIP Adv., 9 (2019) 035233

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/af71dcf495e025c51610d79bc24cb5e1.jpg)
Fig. 1. (a) The regions in the nitridation temperature versus reduction temperature diagrams corresponding to where the α”-phase was produced and M15k and Hc values of the nanoparticles are also indicated. (b) TEM images for the nanoparticles after nitridation starting from the material α-(Fe0.98Co0.02)OOH, α-(Fe0.95Al0.05)OOH and α-(Fe0.98Co0.02Al0.03)OOH.

![](https://assets.underline.io/uploads/markdown_image/1/image/b01b30df78a588a568b32b8156ed275d.jpg)
Fig. 2. (a) The relationship between M15k and Hc for the nanoparticles after nitridation starting from the material α-(Fe0.98Co0.02)OOH, α-(Fe0.95Al0.05)OOH and α-(Fe0.98Co0.02Al0.03)OOH. (b) Hc change as a function of crystalline diameter Dc of those nanoparticles.

 Synthesis of α”-(Fe, M)16N2 nanoparticles obtained by hydrogen reduction and subsequent nitridation starting from α-(Fe, M)OOH (M= Co, Al)

— In this paper, a toolbox baptized “PM-Demag” taking into account the permanent magnet (PM) demagnetization phenomenon has been developed. It is based on a nonlinear demagnetization model which considers the thermal state of the PM and reproduces the knee rounded shape of the B(H) curve. Validations against experimental measurements performed on commercial NdFeB PM are presented. Index Terms— Demagnetization, electrical machines, finite element method, permanent magnet. Nowadays, permanent magnets (PM) are widely used in industry, especially in electrical machines. Their use is expected to grow especially in the context of the energy transition of the 21st century. Like soft magnetic materials, they exhibit nonlinear behavior and hysteresis. The part of the curve situated in the second quadrant of the hysteresis cycle, commonly called the demagnetization curve is of interest because it defines the PM properties and operation. It mainly depends on the temperature and on the applied external field. These two parameters strongly influence the magnetic properties of the PM and can cause reversible or irreversible demagnetization [1]. In the special case of a PM electrical machine, an overheating or a short circuit could lead to partial or total demagnetization of the magnets [2]. In this framework, considering the demagnetization phenomena is necessary to precisely predict the change in the PM characteristics and then the machine performances. The knee in the PM demagnetization curve is crucial for a good knowledge of the magnet state. This state depends on the position of the system operating point (intersection between the load line and the demagnetization curve) with respect to the position of the knee. When the operating point is located in the linear zone, above the knee point (reversible part), the PM keeps its initial performances. However, if the operating point is situated below the knee point (irreversible part), the PM must be considered to be partially demagnetized, the remanent flux density decreases and the PM will operate along a minor hysteresis loop which is often approximated by a recoil line [3]. This phenomenon often occurs in electrical machines (especially on the PM surfaces and edges) and could degrade their performances. The degradation degree depends on the temperatures that the machine could reach and on the PM quality. In this work, a toolbox baptized “PM-Demag” based on a non-linear exponential demagnetization model is developed. The model considers the thermal state and it is expressed as a function of the remanent flux density Br, the coercive field Hc, the relative permeability µr as well as an adjustable coefficient K. All these parameters are themselves expressed as functions of the temperature T. The model parameters are identified from the measurements carried out on a dedicated PM hysteresis meter (HyMpulse of Metis®) for different NdFeB magnets. For example, Fig. 1.a shows the measured B(H) and J(H) demagnetization curves for the N35 NdFeB PM and for different temperatures ranging from room temperature (RT) to 140°C. Figure 1.b shows the simulated B(H) demagnetization curves. The developed “PM-Demag” toolbox will be implemented in code_Carmel, which is a home-made finite element method (FEM) software. Application to an electrical machine is envisaged for the extended paper.

**References:**

[1] &lt;!--[endif]--&gt;Furlani, E.P. (2001) Permanent Magnet and Electromechanical Devices: Materials, Analysis, and Applications. Academic Press, Waltham. &lt;!--[if !supportLists]--&gt;[2] &lt;!--[endif]--&gt;P. Zhou, D. Lin, Y. Xiao, N. Lambert and M. A. Rahman, "Temperature-Dependent Demagnetization Model of Permanent Magnets for Finite Element Analysis," in IEEE Transactions on Magnetics, vol. 48, no. 2, pp. 1031-1034, Feb. 2012, doi: 10.1109/TMAG.2011.2172395. [3] S. Hamidizadeh, N. Alatawneh, R. R. Chromik and D. A. Lowther, "Comparison of Different Demagnetization Models of Permanent Magnet in Machines for Electric Vehicle Application," in IEEE Transactions on Magnetics, vol. 52, no. 5, pp. 1-4, May 2016, Art no. 2100304, doi: 10.1109/TMAG.2015.2513067.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/dcae2862acc82b67848b462f70ff229a.jpg)
Experimental results [B(H) in solid line].

![](https://assets.underline.io/uploads/markdown_image/1/image/b146435ffa1d737ffdbeeac7ad9eb82f.jpg)
Simulated B(H) demagnetization curves.

 Permanent Magnet Non-linear Demagnetization Model for FEM Simulation Environment

The influence of Ce3+ ions substitution for Fe3+ ions in Ni0.42Zn0.58CexFe2-xO4 (x = 0, 0.02, 0.04, 0.06, 0.08, 0.10) compounds on the selected magnetic properties and structure were investigated. The samples were synthesized by means of wet glycine-nitrate technology annealed at the temperature 1200°C for 6 hours. The wide spectrum of characterization methods were used for the study of relevant properties of synthetized materials. X-ray diffraction patterns were recorded by a diffractometer using conventional X-ray tube (CuKα radiation, λ = 1.5418 Å). EDAX were performed by the SEM JEOL 7600-F equipped with both EDX and WDX system. Temperature dependences of magnetic susceptibility was measured using bridge method. The hysteresis loops were recorded using computer controlled digital hysteresisgraph at the same amplitude of applied field Hmax = 1500 A/m, measured at the frequency f = 50 Hz and at room temperature. The determination of real µ' and imaginary µ'' parts of complex (relative) permeability have been performed by means of an impedance analysis procedure using a vector network analyser Keysight E5063A in the frequency range from 1 MHz to 3 GHz. Results and Discussion XRD results indicate a presence of multi-phase structure. The major is found to be spinel phase and secondary, increasing with the addition of Ce3+ ions, corresponds to CeO2. The final number of possible substitution of Ce3+ ions for Fe3+ ions in the B sites, and the difference in their ionic radii, is the cause of the formation of aggregates on the grain boundary as secondary phase [1]. The distribution of elements and their amounts in weight percentages for NiZnCe samples were studied by means of EDAX analysis. The real content of Ni2+, Zn2+, Ce3+, Fe3+ and O2- shown in Tab. I confirmed the substitution of Fe3+ ions by Ce3+ ions, since the increase of Ce3+ content decreases the amount of Fe3+. Temperature dependences of magnetic susceptibility were recorded within the range from room temperature up to 450°C at rise of 4°C/min. Ferrite powder specimens of amount approximately 50 mg were measured; susceptibility values were related to the same weight (1 mg) for all cases. These dependences are decreasing monotonously and they exhibit one strong drop to paramagnetic state. From that, we can conclude that no unwilled phase is present in the ferrite to a significant extent. Curie temperature TC was determined from the point of inflexion at strong decrease of temperature dependence. Even the paramagnetic substituent (Ce3+) ions occupy B-sites dominantly at expense of Fe3+ ions, it does not change the A-B exchange interactions, because TC remains almost constant [2]. The hysteresis loops exhibit quite complicated behavior. By adding larger amount of Ce3+ ions, the maximum flux density Bmax at given maximum applied field sometimes raises and sometimes decreases comparing to the non-substituted ferrite. Note that the value of Bmax at given maximum applied field Hmax = 1500 A/m approaches the largest value for x = 0.08 (only slightly larger than for x = 0.04). The differences can be explained by the fact that the saturation magnetization is an intrinsic parameter (thus, structurally independent) meanwhile the Bmax corresponds to structurally sensitive behavior of material when magnetizing and approaching the saturation. In addition, coercive field is nearly linearly increasing with increment of x. This behavior is similar to that observed in case of Cu-substituted ferrites with the similar chemical formula (except for Cu instead of Ce), [3]. This may be justified by the complexity of the spinel structure, presence of other crystalline phases, etc. In Fig. 1 the frequency dependences of real µ' and imaginary µ'' parts of complex (relative) permeability µ = µ' - jµ'' are presented for prepared spinel ferrites. The obtained frequency dispersion of complex permeability is caused by the domain wall and ferrimagnetic (or spin precession) resonance [4], [5]. The achieved results may be explained as follows. Magnetic properties of nickel-zinc ferrites as representatives of spinel-like ferrites are affected not only by the method of preparation and chemical structure (and composition), but also by properties such as porosity, crystallite size (or grain size), lattice impurities (or defects), concentration of cations, etc. Also the super exchange interactions may strongly change the resultant properties: in spinel ferrite system AA and BB interactions are poorer than AB interaction. In the prepared ferrites, the shrinkage of lattice is present because of the larger ionic radius of cations such as Ce3+ (1.143Å), Zn2+ (0.74Å) and Ni2+ (0.69Å) in comparison with Fe3+ (0.65Å). The magnetic moments of these cations are as follows (in Bohr magnetons): 2.5µB, 0µB, 2.3µB and 5µB. Values of magnetic moments relates the cation distribution of Ni0.42Zn0.58CexFe2-xO4 ferrite system (Ce3+ ions have preference on B sites): (Zn0.582+Fe1-x3+)A[Ni0.422+Cex3+Fe1+x3+]BO42-. Due to the values of ionic radius and magnetic moment, Ce3+ ions replace more Fe3+ ions in the crystal lattice and this leads to the raise of coercive field which in turns changes the value of permeability [5], [6]. Conclusion From the above experimental results, it can be seen that adding even a small amount of cerium element has evident influence on selected properties. Consequently, by adding defined amount of Ce3+ ions, the desired parameters can be easily controlled by a straightforward way and thus tailored for any particular application required in industrial practice.

**References:**

[1] R. S. Yadav, J. Havlica, J. Masilko, J. Magn. Magn. Mater., Vol. 399, pp. 109-117 (2016) [2] M. A. Ahmed, S. F. Mansour, S. I. El-Dek, Solid State Ionics, Vol. 181, pp. 1149-1155 (2010) [3] J. Sláma, A. Grusková, M. Ušáková, J. Magn. Magn. Mater., Vol. 321, pp. 3346-3351 (2009) [4] M. Šoka, R. Dosoudil, M. Ušáková, Acta Phys. Pol. A, Vol. 131, pp. 690-692 (2017) [5] R. Dosoudil, M. Ušáková, A. Grusková, IEEE Trans. Magn., Vol. 50, no. 4 (2014) [6] M. N. Akhtar, M. A. Khan, J. Magn. Magn. Mater., Vol. 460, pp. 268-277 (2018)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/eccf1c3546b0486f5775ad28c48f37bb.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/2358bea70a2a67441e24abb7319de58f.jpg)

Modeling stray-field distribution generated by domain-walls in rare-earth substituted iron garnets

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