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Magnetite is one of the oldest known magnetic materials. At room temperature, magnetite exhibits an inverse spinel structure with Fe&lt;sup&gt;3+&lt;/sup&gt; ions split between tetrahedral and octahedral sites, and Fe&lt;sup&gt;2+&lt;/sup&gt; ions occupying only octahedral sites. As it is well known, with decreasing temperature, magnetite undergoes a first order magnetic transition accompanied by a metal-insulator transition: the so-called Verwey transition (~120 K) [1]. However, there is another magnetic transition taking place at even lower temperatures (&amp;lt; 50 K), whose underlaying nature is still not well understood. In bulk magnetite, a few studies have attributed this transition to different causes, such as the freezing of conduction electrons or rearrangement of electrons within domain walls [2,3]. Of particular interest is the case of magnetite nanoparticles, whose magnetic properties have been extensively studied in the last decades due to their multiple applications, especially in biomedicine [4]. However, a clear understanding on the presence of this transition in these nanoparticles has remained elusive [5].&lt;br/&gt;Therefore, this work has been focused on the study of the low T transition in several types of magnetite nanoparticles, with different sizes, shapes, and purities. These nanoparticles have been structurally and morphologically characterized by using a combination of X-ray Diffraction and Transmission Electron Microscopy. Different oxidation states of the nanoparticles (i.e., into maghemite) have been revealed. Their magnetic response has been analyzed both as a function of the temperature (M vs. T, 5-300 K) and the magnetic field (M vs. H, up to 5 T). Some of these nanoparticles have exhibited a magnetic response close to the bulk, with saturation magnetization ~90 Am&lt;sup&gt;2&lt;/sup&gt;/kg (300 K) and a clear Verwey transition. The low T transitions have been studied by AC susceptibility. In all cases, we have observed 1 or 2 magnetic transitions below 50 K, even for single domain nanoparticles. The nature of these transitions has been further investigated by using different spin relaxation models. Arrhenius law fittings suggest electron hopping freezing (τ&lt;sub&gt;0&lt;/sub&gt; = 10&lt;sup&gt;-8&lt;/sup&gt; – 10&lt;sup&gt;-9&lt;/sup&gt; s, and E&lt;sub&gt;B&lt;/sub&gt;/k&lt;sub&gt;B&lt;/sub&gt; = 200-300 K) for the nanoparticles with magnetic behavior closer to the bulk.

2022 Joint MMM-INTERMAG

Magnetite Nanoparticles: A Detailed Analysis of the Low Temperature Magnetic Transitions

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INTERMAG 2021

Magnetite is one of the oldest known magnetic materials. At room temperature, magnetite exhibits an inverse spinel structure with Fe3+ ions split between tetrahedral and octahedral sites, and Fe2+ ions occupying only octahedral sites. As it is well known, with decreasing temperature, magnetite undergoes a first order magnetic transition accompanied by a metal-insulator transition: the so-called Verwey transition (~120 K) [1]. However, there is another magnetic transition taking place at even lower temperatures (&lt; 50 K), whose underlaying nature is still not well understood. In bulk magnetite, a few studies have attributed this transition to different causes, such as the freezing of conduction electrons or rearrangement of electrons within domain walls [2,3]. Of particular interest is the case of magnetite nanoparticles, whose magnetic properties have been extensively studied in the last decades due to their multiple applications, especially in biomedicine [4]. However, a clear understanding on the presence of this transition in these nanoparticles has remained elusive [5]. Therefore, this work has been focused on the study of the low T transition in several types of magnetite nanoparticles, with different sizes, shapes, and purities. These nanoparticles have been structurally and morphologically characterized by using a combination of X-ray Diffraction and Transmission Electron Microscopy. Different oxidation states of the nanoparticles (i.e., into maghemite) have been revealed. Their magnetic response has been analyzed both as a function of the temperature (M vs. T, 5-300 K) and the magnetic field (M vs. H, up to 5 T). Some of these nanoparticles have exhibited a magnetic response close to the bulk, with saturation magnetization ~90 Am2/kg (300 K) and a clear Verwey transition. The low T transitions have been studied by AC susceptibility. In all cases, we have observed 1 or 2 magnetic transitions below 50 K, even for single domain nanoparticles. The nature of these transitions has been further investigated by using different spin relaxation models. Arrhenius law fittings suggest electron hopping freezing (τ0 = 10-8 – 10-9 s, and EB/kB = 200-300 K) for the nanoparticles with magnetic behavior closer to the bulk.

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The 15th Joint MMM-INTERMAG Conference (2022 Joint) is sponsored jointly by AIP Publishing and the IEEE Magnetics Society. Members of the international scientific and engineering communities interested in recent developments in fundamental and applied magnetism are invited to attend and contribute to the technical sessions. The technical program will include invited and contributed papers in oral and poster sessions, invited symposia, a plenary session, and an evening session, with about 1500 presentations overall. This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

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NdFeB sintered magnet shows the highest energy product ((BH)max) at room temperature of all developed magnetic materials, but the drastic decrease in magnetic anisotropy field (HA) of Nd2Fe14B (2:14:1) phase with rising temperature limits its application at high temperature [1]. In recent years, the grain boundary diffusion (GBD) has been reported to be an effective method to significantly enhance coercivity (iHc) with minor sacrificing remanence (Br), and also to reduce the usage of heavy rare earth (HRE) elements simultaneously [2]. The formation of core-shell structure, the distribution of HRE only at the surface of 2:14:1 phase, through GBD of HRE-containing alloys, could inhibit the nucleation of reverse domain, and thus enhance iHc [3]. In this work, the dual-alloy (DA) sintering method together with GBD treatment are adopted to further enhance coercivity and modify the magnetic properties of NdFeB sintered magnets. (BH)max of 51.4 MGOe and iHc of 14.9 kOe are obtained for the original NdFeB magnet made by sintering Ce85Al15 doped NdFeB powders. Magnetic isolation due to the distribution of Ce and Al at grain boundary increases of the coercivity effectively, yet few Ce entering into grain interior decreases Br, (BH)max, and Curie temperature slightly. In addition, coercivity of the above magnets can be further enhanced to 22.7-25.1 kOe by GBD with different amount of Tb70Cu30 powders. The clear core-shell structure, which consists of (Nd, Tb)2Fe14B phase with higher HA as the shell and Nd2Fe14B phase as the core, is observed. CeAl helps to form continuous and smooth grain boundary. The diffusion source of TbCu could infiltrate into the magnet along such smooth grain boundary and form homogeneous core-shell structure. As a result, coercivity are remarkably enhanced from 14.9 kOe for DA sintered magnet with 2 wt.% CeAl to 25.1 kOe for the magnet GBD with TbCu. Most importantly, various levels of the NdFeB magnets, such as 52M, 35-50SH, and 45UH, could be attained by simply modifying the amounts of CeAl for DA sintered magnets and/or GBD with TbCu powders.

Magnetic properties of Ce85Al15 doped NdFeB sintered magnet by grain boundary diffusion of Tb70Cu30 powders

In order to reveal the demagnetization mechanism of NdFeB magnets and establish the relationship between microstructure and magnetic property, this paper focuses on the structural change of magnetic domain and the demagnetization behavior of NdFeB magnets under various demagnetization factors. Temperature, constant demagnetizing field and alternating demagnetizing field are applied to NdFeB magnets by high and low temperature alternating test chamber, demagnetization test device, programmable DC power supply and programmable AC power supply. Demagnetization test device is made up of 2000 turns copper winding winded on a laminated hollow rectangular steel core. Magnetic properties of the studied material are determined by magnetic domain size, amplitude and open-circuit remanence. The magnetic domain structure and open-circuit remanence are measured by atomic force microscope and Tesla meter. When the sample is demagnetized, the open-circuit remanence of the measuring point in the magnetization direction decreases, and the open-circuit remanence of the measuring point perpendicular to the magnetization direction increases. In the domain imaging, the domain size in the magnetization direction becomes fine and the amplitude decreases, while the domain size perpendicular to the magnetization direction expands and the amplitude increases. The change of magnetic domain structure is consistent with the macroscopic demagnetization law. As the domain size decreases, the number of domains increases and the area of domain wall increases. As a transition region with weak magnetism, the increase of domain wall area will lead to the decrease of macro magnetic properties. When the alternating demagnetizing field is 1000Hz at 140 °C, the remanence losses in the magnetization direction reaches 47.9%, and the domain size increases from 2.35 μm reduced to 1.15 μm. This work is helpful to understand the demagnetization mechanism of NdFeB magnets.

![](https://assets.underline.io/uploads/markdown_image/1/image/6e029230f24bf115f2a0be497a2d004f.jpg)

Magnetic domain of NdFeB magnet demagnetized by an alternating magnetic field with frequency of 1000Hz at 140 °C.

![](https://assets.underline.io/uploads/markdown_image/1/image/a78fa86255ef22db4c513685daf75636.jpg)

Open-circuit remanence losses and domain size of NdFeB magnets demagnetized by multiple levels of alternating fields and temperatures.




Magnetic Domain Structure and Remanence Losses of NdFeB Magnets Under the Influence of Demagnetization Factors

Because the degree of vibration in the permanent magnet synchronous motor (PMSM) is dependent on the magnetization distribution in the permanent magnet. The accurate estimation of magnetization distribution at the previous stage of the practical design eliminates the manufacturing of PMSM with poor performance. To nondestructively estimate the magnetization distribution in the permanent magnet, some numerical methods using the magnetic flux density measured on the outside region of the permanent magnet have been proposed [1], [2]. Because there are countless combinations of magnetization distribution in the permanent magnet, these methods can effectively remove the non-physical distributions. However, since those performances depend on the parameter tunings, the non-physical distributions cannot be completely removed. Then, the deep learning-based estimation method is proposed. In this paper, the deep feedforward neural network is applied to the estimation method. The network is configured by the framework of Keras [3], which is the python-based library of deep learning. The number of network layers is set to 9 layers. The number of neurons in the input layer is set to 344, which is identical to the summation of x- and y- direction of magnetic flux density on 172 measuring points on the outside of the 2-D permanent magnet model. The number of neurons in the output layer is set to 816, which is identical to the number of cells for the magnetization. The target distribution of magnetization is focused on the parallel orientation. The training data are generated by changing the angle of parallel direction every 0.01 degree within [0, 360] with the uniform intensity of magnetization. The weighted coefficients are determined by AdaGrad [4]. The input data (magnetic flux density) is generated by Biot-Savart law, and the 5 % error of the magnetization intensity is superposed on all data, as shown in Fig. 1. While the non-physical distribution tends to be estimated using the conventional methods [1], [2], the practical distribution is estimated using the proposed deep learning system, as shown in Fig. 2. The complicated magnetization distributions in 3D field will be estimated in the full paper.

![](https://assets.underline.io/uploads/markdown_image/1/image/7c8243f47a10aa08684a096da45db3f1.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/37d9c1618dae2eb45dea4d383d086e70.jpg)

Deep Learning-based Estimation Method of Magnetization Distribution in Permanent Magnet

The 2:17-type Sm-Co sintered magnet has excellent magnetic properties, which is widely used in aviation modern high-tech fields [1-2]. It was reported that the cell size increased with increasing isothermal aging time [3]. The effect of Fe-rich content on the phase structure and element distribution was rarely discussed. In this study, the phase structure, elementals distribution and magnetic properties of Fe-rich magnets were studied. The magnets with nominal composition Sm(Co0.91-xFexCu0.07Zr0.02)7.6 (x=0.22, 0.30, 0.35) were prepared, and the corresponding final magnets were named Fe0.22, Fe0.30 and Fe0.35. At the solid solution, the 2:17H phase, 2:17R phase and Zr-rich(1:3R) phase diffraction peaks are detected in the Fe0.30 and Fe0.35 magnet, while the Fe0.22 magnet bears typic 1:7H phase. Furthermore, the characteristic peaks [Fig. 1(a)] of 2:17H, 2:17R and 1:3R phases enhanced with the increase of Fe content, indicating that the high Fe content is not conducive to obtain homogeneous solid solution resulting in poor magnetic properties. The powder XRD patterns [Fig. 1 (b)] of three final magnets are almost the same, except the 1:3R phase can be detected in the Fe0.30 and Fe0.35 magnets. Fig. 2 shows the elemental distribution of the Fe0.22 and Fe0.35 magnets. Compared with the uniform element distribution of Fe0.22 magnet [Fig. 2(a-e)], there are large-sized Sm and Cu enrichment in Fe0.35 magnet, as marked in Fig. 2 (g) and (f). The enrichment regions in Fe0.35 magnet consumes a lot of Sm and Cu, leading to the destruction of the continuity of the cell wall and incomplete cellular structure which is revealed by TEM results. The poor cellular structure should be the main reason for the low coercivity of the Fe0.35 magnet. Our results confirmed that the 1:7H main phase in the solid solution was destroyed with too large Fe content. In addition, enrichment of Cu and Sm elements and incomplete cell structure formed for the large Fe content magnet, resulting in poor magnetic properties.

![](https://assets.underline.io/uploads/markdown_image/1/image/c0ed8d5a60db8ed5927cb50958e1edd0.jpg)

Fig. 1 (a) and (b) the powder XRD patterns of final magnets and solid solution magnets with different Fe contents

![](https://assets.underline.io/uploads/markdown_image/1/image/ae847a024e473dd590c03713f82c09e3.jpg)

Fig. 2 Element distribution of (a)-(e) Fe0.22 and (f)-(j) Fe0.35 magnet

The Phase Structure and Properties of Fe-riched 2:17-type Sm-Co Sintered Magnets

RT12 compounds with the ThMn12 crystal structure have been recently studied as potential candidates for high-performance magnets, due to their low rare-earth content which are considered as critical elements. These compounds with high Fe content, are expected to be strong magnets, because of their large magnetization, high Curie temperature as well as strong uniaxial magnetocrystalline anisotropy when R=Sm. Arc-melted alloys with composition Sm1Fe10.5−xCoxMo1.5(x =0,1,2) were prepared with the tetragonal ThMn12 structure. The cast samples were first high-energy-ball-milled followed by hot-compaction at 650 degree celsius under 275 MPa. Annealing of compacted samples was done from 700 to 1000 degree celsius to optimize the magnetic properties. The effects of Co on the magnetic properties, including the Curie temperature, saturation magnetization, anisotropy field, and energy product, have been investigated. It was found that the Curie temperature shows a linear relationship with the Co substitution; an increasing Co content increases Ms but decreases Hc, The Sm1Fe10.5−xCoxMo1.5 alloy with x =1 has an Hc=2782 Oe and shows the highest energy product(2.63 MGOe) after annealing at 750 degree celsius for 48 h. This work was supported by the U.S. Department of Energy under Grant DE-FG02-90ER45413.

![](https://assets.underline.io/uploads/markdown_image/1/image/ac144c27ad737697673ba2a9f28002d2.jpg)

Table 2: Lattice parameters and magnetic properties of arc-melted alloys Sm1Fe10.5−xCoxMo1.5 at 300K, with x =0, 1, 2

![](https://assets.underline.io/uploads/markdown_image/1/image/d8abe03bb48bec5c706bc98217bb2394.jpg)

Table 2: Magnetic properties of hot compacted nanocrystalline Sm1Fe10.5−xCoxMo1.5 magnets, with x =0, 1, 2, annealed at 750 degree celsius

Magnetic properties of nanocrystalline Sm1Fe10.5−xCoxMo1.5 alloys with x =0,1,2

FePt nanoparticles (NPs) with fct (L10) structures have a wide promising application in high-density magnetic storage media, catalysis, and biomedicine [1, 2], because of their high magnetocrystalline anisotropy, catalytic activity, and good biocompatibility. Ag addition can promote ordering transition and directly obtain L10-FePt NPs with high structural ordering [3-5]. However, the ordering mechanism of the NPs by Ag addition and the ordering transition process are rarely reported. Herein, we report the direct synthesis of L10-FePt NPs by Ag addition. The ordering transition process was investigated, and the ordering mechanism was discussed. The metal precursors Pt(acac)2, Fe(acac)3, and AgAc were dissolved in HDA in the presence of HDD as reductant. And then the mixed solution was heated to 360 °C for 180 min. Finally, the product was collected by centrifugation with ethanol. The phase structure and magnetic properties of a series of products sampled at 360 °C for different reaction times (from 0 min to 180 min) are shown in Fig. 1 and Fig. 2, respectively. The precursors of Ag and Pt were firstly reduced, and form AgPt phase at beginning of the reaction. With reacting for 10 min, the precursor of Fe was reduced, and the FePt NPs were obtained. The products were mainly combined of elemental Ag and L10-FePt phases. The magnetic properties of the products changed from diamagnetic to ferromagnetic. With extending the reaction time, the both content of Ag and L10-FePt phases continuously increased. The coercivity of products enhanced from 0.9 kOe to 6.3 kOe with increasing the reaction time from 10 min to 180 min. These results indicate that Ag atoms are easy to migrate and enrich because of their low surface energy and high mobility. The Ag migration helps the formation of a large number of defects and vacancies in the crystal lattice, which can promote the rearrangement of atoms, and reduce the ordering transition barrier. Therefore, Ag addition can result into improvement of structural ordering of the FePt NPs.


![](https://assets.underline.io/uploads/markdown_image/1/image/81f76544f273f94ac19ed3a8359a5ab6.jpg)


Fig.1 The XRD patterns of products obtained at 360 °C for different reaction times

![](https://assets.underline.io/uploads/markdown_image/1/image/166d467e2867a3c4344affaf327827f5.jpg)


Fig. 2 The hysteresis loop of products obtained at 360 °C for different reaction times

Ordering Mechanism of L10-FePt Nanoparticles Synthesized by Ag Addition

Nanocomposite magnets composed of Nd2Fe14B and α-Fe nanoparticles are not only expected to increase the (BH)max compared to Nd-Fe-B magnets [1], but also effective in reducing the use of HREs [2]. Recently, we have reported on NdFeB/Mo/FeCo multilayered magnet films formed by a two-step method [3]. However, long-time multi-step annealing (MSA) using a halogen lamp was necessary to realize this magnet film. It is well known that method using a laser is an excellent technique that can significantly reduce the process time and is expected to improve the magnetic properties. In this study, we report the formation of nanocomposite NdFeB/Mo/FeCo magnet films using infrared laser annealing (iLA) technique. The Nanocomposite [NdFeB(30 nm)/Mo(0.3 nm)/FeCo(5 nm)/Mo(0.3 nm)]5 multilayered magnet films were prepared by UHV sputtering [3] followed by iLA processing. It is equipped with a CW semiconductor laser with a wave length of 939.8 nm and a spot size of about 5 mm in diameter on surface of films. Fig. 1 shows in-plane magnetization curves of the multilayered films as-deposited and iLA processed at 760°C. Both films with and w/o Mo interlayer before iLA processing showed soft magnetism with small coercivity and weak perpendicular magnetic anisotropy (PMA). The PMA of the film with the Mo interlayer is larger than that of the film without the Mo interlayer. Magnetic properties of both films were drastic changed from soft magnetism with PMA to hard magnetism with in-plane magnetic anisotropy after iLA processed at 760°C. Coercivity of in-plane of iLA processed film with Mo interlayer was 10 kOe which was double of the film w/o Mo interlayer. After iLA process, we can see that the squareness of film was improved by the introduction of Mo interlayer. These results show that the iLA process is effective in forming in-plane anisotropic nanocomposite NdFeB/Mo/FeCo magnet films even though the processing time of the iLA process is 1/100 of the processing time of the MSA process [3].


![](https://assets.underline.io/uploads/markdown_image/1/image/4abf4bd1b86901df35a7bb50f4fa6b41.jpg)


An in-plane magnetization curves of [Nd-Fe-B/Mo(tMo = 0 nm, 0.3 nm)/FeCo]5 multilayered films as-deposited state and iLA processed at 760°C.

Infrared Laser Annealing of Nanocomposite Nd-Fe-B/Mo/FeCo Multilayered Magnet Films

Effects of La3+ and Cu1+ co-doping on structural and magnetic properties of M-type strontium hexaferrite have been studied by using crystallographic and magnetic measurements. Stoichiometric Sr1-xLaxFe12-x/2Cux/2O19 (x=0, 0.25, 0.5, 0.65) samples have been prepared by the conventional ceramic method using La2O3, SrCO3, Fe2O3, and CuO powders with repeated sintering at 1300 °C for 24 hours. EDS analysis showed that the ratios of metal elements were in good agreement with compositions of the samples. XRD patterns showed that when x ≥ 0.7, LaFeO3, SrFeO3-δ, and CuFe2O4 coexisted along with the M-type hexaferrite. Lattice parameters deduced from the GSAS refinement shows that a slightly increases while c decreases with increasing x (Table 1). Variations of the saturation magnetization MS and the coercivity HC with x was obtained from room temperature VSM measurement as in Table 1. Whereas the HC decreases monotonically with x, the MS increases with the La3+ - Cu1+ co-doping throughout the doping range examined. Effective magnetic anisotropy constants K are calculated as well from the hysteresis curves using the law of approach to saturation method. K shows gradual increase with increasing x as listed in Table 1. XPS Cu2p scan shows that dominant valence state of Cu is +1. There were no trace observed for Cu2+ or metallic Cu (Fig. 1(a)). Apparently, monovalent Cu1+ has no spin magnetic moment. In view of the fact that diamagnetic substitution of either 4f1 or 4f2 site give rise to positive contribution to MS, this behavior strongly suggests that Cu1+ ions preferentially replace Fe3+ ions at those sites. Finally, Mössbauer spectra show gradual decrease in 4f2 intensity, in support of the octahedral 4f2 site preference of Cu1+. In addition, one extra sextet with Hhf ≈ 341 kOe was detected from the Mössbauer spectra, which were explained by the 12k site Fe3+ that have more than one 4f2 Cu1+ neighbors (Fig. 1(b)).

![](https://assets.underline.io/uploads/markdown_image/1/image/cd4821a42565474b6f0714cd6daacbb7.jpg)

Various parameters of Sr1-xLaxFe12-x/2Cux/2O19 (x=0, 0.25, 0.5, 0.65)


![](https://assets.underline.io/uploads/markdown_image/1/image/a24b2b3f1b09efbbee07fd50283fefb2.jpg)

(a) XPS Cu 2p scan for Sr0.5La0.5Fe11.75Cu0.25O19 and (b) room temperature Mössbauer spectra of Sr0.35La0.65Fe11.675Cu0.325O19

Structural, Magnetic, and Mössbauer Spectroscopic Study of La3+ and Cu1+ Co-doped M-type Strontium Hexaferrite

Exchange-coupled hard/soft ferrite nanoparticles are prospective to squeeze out a part of expensive magnets based on rare-earth elements. However, known exchange-coupled composite ferrite nanoparticles often suffer from a lack of a powerful enough hard-magnetic core, a high defectivity of magnetic phases, and a poor interface between them. Herein we demonstrate the first efficient synthesis of sandwiched nanomagnets, which exhibit the pronounced exchange-coupling effect. This work is featured by using individual high-coercive strontium hexaferrite nanoplates prepared by a borate glass crystallization method as cores for the composite particles. High crystal quality of the hexaferrite cores as a substrate provides an epitaxial growth of CoFe2O4 layers on the 001 facets from the organic high-boiling solvent and results in enhancing the remanence magnetization and the maximum energy product of the composite material. The obtained composite nanoparticles are characterized by well-defined epitaxial interfaces between the constituent phases. The exchange coupling results in strongly different magnetic properties compared to ones expected for a mechanical mixture of hexaferrite and cobalt ferrite. At room temperature, the magnetic properties are determined by the uniaxial magnetocrystalline anisotropy of the hexaferrite, while upon cooling the situation changes to the triaxial magnetocrystalline anisotropy typical for the spinel cobalt oxide. In the temperature range 5 – 300 K the composite shows significant improvement of the (BH)max values of both initial phases. The results of this work open new prospects for fabrication of multilayer oxide heterostructures with a synergetic performance which expands applications of exchange-coupled composites. The research was supported by Russian Science Foundation grant no 20-73-10129

Sandwiched CoFe2O4/SrFe11.5Al0.5O19/CoFe2O4 nanoparticles with exchange-coupling effect

Manipulations of micro-sized magnetic particles by an external field have been widely studied for the applications in MEMS and biofluidics. A popular way of manipulation is forming a microchain containing several particles in a liquid environment. The chain can be well controlled by a designated external field to perform rotation [1] or oscillation [2] for desired purposes. Nevertheless, the induced hydrodynamic drag would affect the formation and motion of the chain, such as the well studied chain rupture in a dynamic field. In the present paper, study of motion synchronicity of a micro-magnetic-particle chain in a rotating field is focused. To achieve synchronous rotation with the external field, force balance acting on the chain is required. Experiments are performed firstly to identify actual rotating frequency of an un-ruptured chain, denoted as fc, subjected to various field frequency fH. As the results shown in figure, for a 3-particle chain, motion synchronicity can only preserved up to a critical field frequency. Beyond this critical field frequency, the average rotating frequency of the chain starts to decay. The critical field frequency increases if magnetic field strength H is raised. The mechanism of this asynchronous motion can be understood by the competition of magnetic torque and hydrodynamic torque. The predicted critical frequencies, based on two approximations reported in the literatures [3,4], i.e., fc1 and fc2 shown in the figure, agree well with the experiments. In addition, temporal dynamical motion is also presented to illustrate the asynchronous motion.

![](https://assets.underline.io/uploads/markdown_image/1/image/fa77da4a4f02a4fcfe4bf92f2f888ada.jpg)

Actual rotating frequency of a 3-particle chains fc subjected to various field frequency fH. Marks of circle (O) and cross (X) are the predicted critical frequency for correspondent field strength H. The linear lines inductae synchronous motion of the chain and external field. Beyond a critical frequency, the actual rotating frqeuency of chain fc starts to drop. The predicted critical frquencies for different field strength H agree well with the experiments.

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Magnetic properties of Ce85Al15 doped NdFeB sintered magnet by grain boundary diffusion of Tb70Cu30 powders

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