France



As a single solid element, the magnetoelectric (ME) gyrator has verified Tellegen&#39;s conjecture about the gyrator. In recent years, a number of attempts have been made to reduce losses in the energy conversion process. It is well known that the properties of spinel ferrites are dependent on such as composition, cation substitutions. A small contents of rare earth doping can modify the magnetic and magnetostrictive properties by replacing the tetrahedral (A) or octahedral (B) sites, which is beneficent to ME coupling&lt;sup&gt;1&lt;/sup&gt;. It has been reported that Dy elements possess high magnetocrystalline anisotropy due to the strong anisotropy of the &lt;i&gt;4f&lt;/i&gt; electron clound, thus inducing strong magnetostriction&lt;sup&gt;2&lt;/sup&gt;. However, due to the high loss in the energy conversion process of the pure spinel ferrite ME gyrator, it is necessary to improve the ability of energy capture by means of the change of magnetic properties caused by Dy&lt;sup&gt;3+&lt;/sup&gt; doping. Therefore, the research on the improvement of gyration properties caused by high magnetostriction induced by Dy&lt;sup&gt;3+&lt;/sup&gt; doped ferrite has been initiated. Here, a variety of ME gyrators composed of nickel zinc ferrite and piezoelectric ceramic under various Dy&lt;sup&gt;3+&lt;/sup&gt; concentration were fabricated with coil around it and systematically characterized of magnetic properties,dynamic piezomagnetic coefficient, resonance ME response, &lt;i&gt;V/I-I/V&lt;/i&gt; conversion ratio and power conversion efficiency, and each stage involved in the magneto-mechanical-electric energy conversion process under influences of Dy&lt;sup&gt;3+&lt;/sup&gt; concentration was clarified. Fig.1 shows the &lt;i&gt;V/I&lt;/i&gt; conversion ratio spectra of the ME gyrators with frequency, applied flied and Dy&lt;sup&gt;3+&lt;/sup&gt; concentration, respectively. It can be clearly seen that under the resonant frequency and optimal bias, the maximum &lt;i&gt;V/I&lt;/i&gt; of 1610 V/A occurring at sample with x=0.04 while the minimum of 998 V/A occurring at the one with x=0. Compared with pure spinel ferrite, the conversion effect of the ME gyrators is improved by 161% by Dy&lt;sup&gt;3+&lt;/sup&gt; doping. The experimental results attributed to the strong ME coupling is caused by the high magnetocrystalline anisotropy of Dy&lt;sup&gt;3+&lt;/sup&gt;. Therefore, it can be concluded that Dy&lt;sup&gt;3+&lt;/sup&gt; doping reduces the loss in the energy conversion process of ME gyrator and promotes the improvement of gyration properties. Consequently, the high gyration properties of ME gyrator has important potential application for miniaturized &lt;i&gt;V/I&lt;/i&gt; converters for power electronics.

**References:**

&lt;sup&gt;1&lt;/sup&gt;&lt;sup&gt; &lt;/sup&gt;A. Anwar, S. Zulfiqar, and M. A. Yousuf, Journal of Materials Research and Technology-Jmr&amp;T. Vol. 9, p.5313-5325 (2020)&lt;br/&gt;&lt;sup&gt;2&lt;/sup&gt;&lt;sup&gt; &lt;/sup&gt;T. Y. Jin, W. Wu, and C. B. Jiang, Scripta Materialia. Vol. 74, p.100-103 (2014)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/606efb870463e67f36a0cefd9acfefa5.jpg)
Fig.1 &lt;i&gt;V/I&lt;/i&gt; conversion of the ME gyrators with different Dy&lt;sup&gt;3+&lt;/sup&gt; concentrations under optimal bias in the frequency range of 54kHz-66kHz. The insets are the &lt;i&gt;V/I&lt;/i&gt; as a function of the Dy&lt;sup&gt;3+&lt;/sup&gt; concentration and Applied Flied respectively.



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 Improvement of gyration effects by dysprosium doping in spinel ferrite/piezoelectric magnetoeletric gyrators

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As a single solid element, the magnetoelectric (ME) gyrator has verified Tellegen's conjecture about the gyrator. In recent years, a number of attempts have been made to reduce losses in the energy conversion process. It is well known that the properties of spinel ferrites are dependent on such as composition, cation substitutions. A small contents of rare earth doping can modify the magnetic and magnetostrictive properties by replacing the tetrahedral (A) or octahedral (B) sites, which is beneficent to ME coupling1. It has been reported that Dy elements possess high magnetocrystalline anisotropy due to the strong anisotropy of the 4f electron clound, thus inducing strong magnetostriction2. However, due to the high loss in the energy conversion process of the pure spinel ferrite ME gyrator, it is necessary to improve the ability of energy capture by means of the change of magnetic properties caused by Dy3+ doping. Therefore, the research on the improvement of gyration properties caused by high magnetostriction induced by Dy3+ doped ferrite has been initiated. Here, a variety of ME gyrators composed of nickel zinc ferrite and piezoelectric ceramic under various Dy3+ concentration were fabricated with coil around it and systematically characterized of magnetic properties,dynamic piezomagnetic coefficient, resonance ME response, V/I-I/V conversion ratio and power conversion efficiency, and each stage involved in the magneto-mechanical-electric energy conversion process under influences of Dy3+ concentration was clarified. Fig.1 shows the V/I conversion ratio spectra of the ME gyrators with frequency, applied flied and Dy3+ concentration, respectively. It can be clearly seen that under the resonant frequency and optimal bias, the maximum V/I of 1610 V/A occurring at sample with x=0.04 while the minimum of 998 V/A occurring at the one with x=0. Compared with pure spinel ferrite, the conversion effect of the ME gyrators is improved by 161% by Dy3+ doping. The experimental results attributed to the strong ME coupling is caused by the high magnetocrystalline anisotropy of Dy3+. Therefore, it can be concluded that Dy3+ doping reduces the loss in the energy conversion process of ME gyrator and promotes the improvement of gyration properties. Consequently, the high gyration properties of ME gyrator has important potential application for miniaturized V/I converters for power electronics.

**References:**

1 A. Anwar, S. Zulfiqar, and M. A. Yousuf, Journal of Materials Research and Technology-Jmr&T. Vol. 9, p.5313-5325 (2020) 2 T. Y. Jin, W. Wu, and C. B. Jiang, Scripta Materialia. Vol. 74, p.100-103 (2014)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/606efb870463e67f36a0cefd9acfefa5.jpg)
Fig.1 V/I conversion of the ME gyrators with different Dy3+ concentrations under optimal bias in the frequency range of 54kHz-66kHz. The insets are the V/I as a function of the Dy3+ concentration and Applied Flied respectively.

poster

Materials with a giant magnetocaloric effect near ambient temperature have been a subject of intense investi-gation in the last 20 years due to their high technological potential as magnetic refrigerators or heat pumps, and as thermomagnetic regenerators for thermal energy harvesting1. HVAC (heating, ventilation and air conditioning) technologies based on the magnetocaloric effect could lead to more environmentally friendly systems, due to higher energy efficiency and the lack of greenhouse gases1. These materials are characterized by their strong spin-lattice coupling that manifests itself in first-order phase transitions (FOPT) during which the materials undergo simultaneous or closely coupled magnetic and structural transitions. These transitions may be controlled by extrinsic (temperature, magnetic field, pressure, time) and intrinsic (composition, morphology) parameters. In the referred applications, the devices require relatively high operating frequencies (of the order of 10 Hz) in order to maximize their efficiency. Such a technological requirement implies the utilized materials should complete their FOPT rapidly (for 10Hz operating frequencies, the FOPT timescale should be in the order of few milliseconds), however, unfortunately, there are relatively few studies dedicated to these materials transition kinetics2–8. From the few reports dedicated to this topic, the ones focused on the La-Fe-Si family stand out. In these, it has been shown that the shape and the interactions between particles influence the magnetic hysteresis and the time-scale of the transitions significantly4. In the Gd-Si-Ge family of materials, the Gd5Ge4 composition has been subject of relevant kinetic studies which unveiled its magnetic glass state and a kinetically arrested AFM-FM transition5. To our knowledge there is only one preliminary study for the Gd5Si2Ge2 composition, where it has been shown that the FOPT transition can last up to 300 seconds to be completed - an impressively long time considering the required operating frequencies mentioned above6. Heusler alloys such as Ni-Mn-In exhibit a giant inverse-magnetocaloric effect along their AFM-PM transition, and display typically faster transition kinetics7. Our work will thus fit in an effort to shed light on the kinetics of the FOPTs which occur in these most prominent giant magnetocaloric material families. To accomplish this, we will thoroughly study the kinetics of FOPT, encompassing the magnetic transitions (through time-dependent SQUID measurements) and the (de) coupled structural transitions time-dependencies, their electron transport kinetics as a function of temperature, magnetic field, composition and micro-structure and then attempt to theoretically model the observed phenomena. The growing surge of interest in these unusual transitions is not only due to the pursuit of application-ready materials, but also to attempt to provide answers to purely scientific questions such as how the transition kinetics of the different material families compare and how they change with different magnetovolume couplings and different morphologies (macro, micro and nanostructured). Bulk samples of the above mentioned materials families were produced and their single-phase nature, magnetic and structural properties were assessed via Scanning Electron Microscopy, SQUID magnetometry and X-ray Diffraction. As preliminary work, we have followed the time dependent magnetic field (SQUID) measurement protocol done in8 for a bulk (6x2.5x2 mm) Gd5Si2Ge2 sample. We analyze the evolution of the magnetization of the sample through time, while subjected to initially increasing and then paused fields: the fields are increased with a constant sweep rate of 70 mT/s until a given Hpause value, which is maintained for 300 seconds. Not only is magnetic hysteresis detected, but also a significant difference is observed between magnetization vs time curves obtained by two different measurement protocols: 1) a thermal “reset”, where in between measurements the sample is heated up to 45K above Tc=270K and cooled down again before the following measurement, and 2) no thermal “reset”, where the sample temperature is unchanged before subsequent measurements (see figure 1). In both, one can see once the field is stopped (at t=tpause), the rate of increase of the magnetization slows down, but keeps evolving in time. We made measurements for different temperatures a few degrees above Tc (T=283K in the presented results) so that the sample begins in its paramagnetic state and the transition to the high magnetization state must be induced by the applied field. At the given temperature, the field is increased at 70 mT/s rate until a given field value is reached (HPause) and the M(H) curves show a relaxation behavior - vertical lines in figure 2. As can be seen, the relaxation has non monotonous behaviour, increasing until a maximum at the critical field value around 2 T, and then decreasing again for higher Hpauses. The measurements followed protocol 1, where the relaxation is more significant.

**References:**

1) V. Franco, J. S. Blázquez and J. J. Ipus, Prog. Mater. Sci. Vol. 93 p. 112–232 (2018) 2) V. Basso, M. Piazzi and C. Bennati, Physica Status Solidi B Vol. 255, Issue 2 (2017) 3) A. Biswas, Y. Mudryk and A. K. Pathak, J. Appl. Phys. 126, 243902 (2019) 4) M. G. Zavareh, Y. Skourski and K. P. Skokov, Phys. Rev. Applied 8, 014037 (2017) 5) J. D. Moore, G. K. Perkins and K. Morrison, J. Phys.: Condens. Matter Vol. 20, 465212 (2008) 6) J. Leib, J. E. Snyder and T. A. Lograsso, J. Appl. Phys. 95, 6915 (2004) 7) L. Pfeuffer, T. Gottschall and T. Faske, Phys. Rev. Materials 4, 111401(R) (2020) 8). E. Lovell , A. M. Pereira and A. D. Caplin, Adv. Energy Mater. Vol. 5, Issue 6 (2015)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/33ca740aa8e7cb74167544f6139bf403.jpg)
The evolution of the magnetization for a measurement with protocol 1) (M(t) - thermal "reset") and another with protocol 2) (M(t) - no "reset") and the applied field H(t) increasing at 70mT/s and pausing at Hpause=2.2 T.

![](https://assets.underline.io/uploads/markdown_image/1/image/f853dc79d741d8ca99e0daab54dcdaf2.jpg)
The M vs H curves for different maximum paused fields, Hpause. In the inset, the M vs T curves (for constant field of 0.1 T) are shown. The vertical bar at 283K highlights the temperature at which the M vs H curves were obtained.

 Kinetics of First-Order Phase Transitions in Magnetocaloric Materials

The use of a significant share of consumed energy is dedicated to refrigeration and heating systems. Hence, innovating the current state of refrigeration and heating systems to improve their efficiencies is a critical challenge to be tackled, especially under the quest of global warming mitigation. Magnetocaloric (MC) systems are one of the most promising innovations, as their theoretical coefficient of performance (COP) is significantly larger than that of vapor-compression systems. MC systems rely on the reversible magnetocaloric effect where applying (or removing) magnetic fields (H) changes the thermodynamic state of the material in such a way that, in adiabatic processes, the temperature increases (or decreases) [1]. Although the development of magnetocaloric systems is now in a relatively mature phase, the geometrical and operating optimization is still a time consuming task. One way of optimizing a system is by predicting what are the performance values if using a set of geometrical and operating parameters. In that respect, the recent interest of artificial intelligence lead to the development of several statistical learning techniques and respective algorithm implementations [2]. These techniques have recently been used in a wide number of reserarch fields. One specific class of statistical techniques concerns supervised learning, where regressors (or classifiers) are trained based on datasets with continuous (or bolean) output values [2]. In this work, different statistical learning regressors are applied, and critically compared, in predicting the three main performance values for a generic magnetocaloric heat pump: no load temperature span, heating power and COP [4]. Four different regressors were used: ordinary least squares (ols), ridge, lasso, and K-nearest neighbors (KNN). The used dataset was obtained with the recently developed heatrapy framework to compute caloric systems [5]. The minimum average relative error of the temperature span, heating power and COP were 23%, 29% and 31%, respectively, as can be seen in Fig. 1. The Lasso regressor shows the best performance with the order of the polynomials. The best order of polynomials range between 3, for the heating power, to 5, for the COP. The worse performance in predicting the three performance values occurs when using the KNN regressor. Furthermore, the application of regressors to the dataset is more adequate to evaluate the temperature span rather than energetic performance values.

**References:**

[1] A. Kitanovski et al., Magnetocaloric Energy Conversion. Springer (2015). [2] C. M. Bishop, Pattern recognition and machine learning. Springer (2006). [3] C. Aprea et al., Int. J. Refrig. 82 (2017) 238. [4] D. J. Silva et al., Energy and AI, 2 (2020) 100030. [5] D. J. Silva et al., SoftwareX 7 (2018) 373.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/4552fbdd65b2086aba0ca2c6a67e6fc9.jpg)
Figure 1: Minimum average relative error of the temperature span, heating power and COP for the different used regressors.

 Predicting the performance of magnetocaloric heat pumps using statistical learning algorithms

Magnetic refrigeration, which is a way to cool a matter with a magnetic field, was discovered more than 100 years ago. For the last several decades, the magnetic refrigeration has been intensively studied for possible application of refrigeration near room temperature. More recently, the magnetic refrigeration research field was also opened for low target temperature for the purpose of hydrogen liquefaction with condensation temperature of 20 Kelvin. However, in most of the studies, high magnetic fields, which is not lower than several Tesla, were necessarily used to obtain a large entropy change, leading to large energy costs to operate a superconducting magnet. Alternatively, we here propose high efficient cooling technique where small magnetic field changes superimposed on a static bias magnetic field can obtain one order magnitude higher cooling efficiency than the typical magnetocaloric materials. In the case of Ho, the cooling efficiency reaches ΔSM/Δμ0H = −31 J kg−1K−1T−1. The temperature range with high efficiency can be adjusted by varying the bias magnetic field. We hope that the presently provided small field controlled magnetic refrigeration technique will be a standard method, equipped with permanent magnets, for hydrogen liquefaction as an alternative to conventional gas compression-based cooling.

**References:**



**Images:**

 High Efficient Magnetic Refrigeration using Static Bias Magnetic Field

The RTX compounds, where R is a 4f-metal, T and X are d- and p-elements, exhibit various physicochemical properties and have potential for practical application [1]. The GdFeSi, GdTiSi and GdMnSi compounds crystallize into the same tetragonal CeFeSi (P4/nmm)-type structure [1,2]. This structure is composed of alternating (001) layers with the sequence: Gd-Si-T2-Si-Gd-Gd-Si-T2-Si-Gd (T = Fe, Co, Ti, Mn). This is why the exchange interactions between atoms Gd–Gd, Gd–T, T–T in GdTSi depend on the lattice parameter c, whereas interaction T–T depends on the lattice parameter a as well. The hybridization of Si p and T 3d states causes filling of the 3d band and results in the absence of magnetic moment of T = Fe or Co in the GdTSi compound [1-5]. Mn is a unique example of the 3d-transition metal T with a magnetic moment in the GdTSi compounds. Significant changes in the magnetic and structural parameters of these compounds were observed upon a mutual partial substitution of d-metals. In this work, the intermetallic compounds GdTi0.05Fe0.95-xMnxSi, x=0-0.95 with a tetragonal crystal structure of the CeFeSi (P4/nmm)-type have been studied. It is shown in Fig.1 that the lattice parameter c and the Curie temperature TC increase, whereas the lattice parameter a is almost unchanged in the GdTi0.05Fe0.95-xMnxSi system when the concentration of Mn increases. The similar variations of the lattice parameters and TC were earlier obtained for the GdFe1-xMnxSi system [3], although c and TC are noticeably smaller as compared with GdTi0.05Fe0.95-xMnxSi, whereas a is about the same. Hence, an increase of TC in the systems is caused by short-range Mn-Mn exchange interactions in the basal plane, whereas higher TC in GdTi0.05Fe0.95-xMnxSi is caused by stronger indirect exchange interactions Gd-Gd. The increase in the interatomic distances along the c axis due to large Ti substitution atoms should weaken the Gd-Gd, Gd-T and T-T exchange interactions and should decrease TC in GdTi0.05Fe0.95-xMnxSi as compared with GdFe1-xMnxSi. At the same time, the partial substitution of Ti for Fe or Mn decreases a number of the 3d “spin-down” electrons in the system, because Ti, Mn and Fe atoms have 2, 5 and 6 of the 3d electrons, respectively. As a result, a number of the 3d electrons with “spin up” non-compensated by the 3d electrons with “spin down” increases. Owing to this, indirect exchange interactions between Gd ions become stronger, because they are realized through polarization of the conduction electrons. The difference between the numbers of the 3d electrons for Ti (2) and Mn (5) or Fe (6) is less for the compositions x &gt; 0.5, where the concentration of Mn prevails, than for the compositions x &lt; 0.5, where the concentration of Fe prevails. As a result, TC for GdTi0.05Fe0.95-xMnxSi exceeds TC for GdFe1-xMnxSi by 10-15 K for the compounds with x&gt; 0.5 and by 50-70 K for the compounds with x &lt;0.5. Fig.2 shows that the saturation magnetization Msat decreases non-monotonically in the GdTi0.05Fe0.95-xMnxSi system with a minimum at intermediate compositions. Usually, the R–T intermetallic compounds with a heavy rare earth element R are characterized by strong negative R–T exchange interaction [1]. From this point of view, one should expect the antiparallel arrangement of Gd and Mn moments and a linear decrease in magnetization in the GdTi0.05Fe0.95-xMnxSi system with an increase in Mn concentration. It is obvious that this model cannot explain the existence of the minimum on the dependence Msat.(x). A similar change of Msat.(x) with a minimum was published for the GdFe1-xMnxSi system [4]. It was shown that the GdMnSi compound is characterized by strong exchange interactions inside the Gd and Mn subsystems and weaker Gd–Mn interaction, since the atoms of different types are stacked in isolated (001) layers [2,6]. As a result, a non-collinear ferrimagnetic structure is formed in GdMnSi in which the moments of Gd ions are oriented along the tetragonal axis of the compound and the moments of Mn ions are oriented at an angle to it. Apparently, a similar structure exists in the GdFe1-xMnxSi and GdTi0.05Fe0.95-xMnxSi systems. Variations of Mn concentration or strength of external magnetic field can produce an essential change in the relative orientation of magnetic moments of the Gd and Mn sublattices and non-monotonic variation of Msat.(x). The isothermal magnetic entropy change -ΔSM (i.e. the magnetocaloric effect MCE) was calculated from the magnetization isotherms M(H) using the well-known Maxwell relation [1]. The peak value of -ΔSM decreases with the appearance Mn in the GdTi0.05Fe0.95-xMnxSi system and varies within a narrow range of 1.2-1.6 J/kgK at 17 kOe field for all Mn contents. At the same time, TC varies in a wide range around room temperature: from 220 to 360 K. Thus, the MCE of the GdTi0.05Fe0.95-xMnxSi compounds can be of practical interest. Acknowledgements Support by RSF (Project No. 18-72-10098) is acknowledged.

**References:**

[1] S. Gupta and K.G. Suresh, J. Alloys Comp., Vol. 618, p.562 (2015) [2] R. Welter, G. Venturini and B. Malaman, J. Alloys Comp., Vol. 189, p.49 (1992) [3] S.A. Nikitin, T.I. Ivanova and I.A. Tskhadadze, J. Alloys Comp., Vol. 280, p.16 (1998) [4] S.A. Nikitin, T.I. Ivanova and I.A. Tskhadadsze, Acta Phys. Pol., Vol. A 91, p.463 (1997) [5] M. Napoletano, F. Capera and P. Manfrinetti, J. Mater. Chem., Vol. 10, p.1663 (2000) [6] I.A. Ovtchenkova, S.A. Nikitin and T.I. Ivanova, J. Alloys Comp., Vol. 451, p.450 (2008)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/9de671cd9d1da85ef442536dd7b098b7.jpg)
Fig.1. Dependences of the lattice parameters a (■) and c (●) and the Curie temperature TC (△) of the GdTi0.05Fe0.95-xMnxSi compounds versus Mn content.

![](https://assets.underline.io/uploads/markdown_image/1/image/182009ac8016f05e56d7616efee6545a.jpg)
Fig.2. Concentration dependence of the saturation magnetization Msat. measured on free powder samples of the GdTi0.05Fe0.95-xMnxSi compounds at T = 4 K. Points (●) – experiment, line – interpolation.

 Substitution of Fe by Ti and Mn in GdFeSi

HoFeO3 is one of the most interesting representatives of the rare-earth orthoferrites RFeO3 with orthorhombic structure possessing different useful for potential applications physical properties. Dzyaloshinskii-Moriya interaction (DMI) which leads to a weak ferromagnetic (WF) moment in the Fe sublattice, might play an essential role in their occurrence. A spontaneous electric polarization occurs in HoFeO3 at ~ 210 K [1]. At lower temperatures, strong magneto-caloric effect makes the compound to a promising candidate for the magnetic cooling in the cryogenic gases liquefying technology. Three peaks in the entropy-change occur for field variation 0-7 T: ΔSM=9 J/Kg K at 53 K, ΔSM=15 J/Kg K at 10 K and ΔSM=18 J/Kg K at 3 K [2]. The first peak is associated with a spin reorientation in the Fe subsystems. Below 53 K the Fe sublattice starts to polarize the Ho. Thus, the second and third peaks should be associated with some processes related also to the Ho subsystem. The crystal structure of HoFeO3 is described by space group Pbnm. Below 637 K, the Fe sublattice orders antiferromagnetic (AFM) along the a-axis, and WF moment along c-axis (magnetic phase Gm4). At zero field, the compound has two spin reorientation transitions: at 53 K where the Fe moments rotate from a to b-axis (magnetic phase Gm1) and at 35 K where the magnetic moments align almost along c-axis (magnetic phase Gm2). Ho orders at a temperature of 4 K [3]. Polarized Neutron diffraction (PND) experiments were performed at the Heinz Maier-Leibnitz Zentrum (Garching, Germany) on the diffractometer POLI [4]. The used instrumental setup with a 2.2 T cryogen-free, compact high-Tc superconducting magnet is presented in Ref. [5]. The evolution of the magnetic structure in HoFeO3 with temperature and magnetic field in the range of 300-47 K and 0.15-2.2 T was investigated. In particular, the effect of the magnetic field on the reorientation transition from Gm4 to Gm1 symmetry, which takes place in zero field at 53 K, has been studied in detail. Originating from the mmm Laue class of HoFeO3, the Bragg reflections to which the Fe sublattice contributes can be separated in four sets: F: (h + k even, l even), A: (h + k even, l odd), C: (h + k odd, l even) and G: (h + k odd, l odd). The asymmetry values A=(I+-I-)/( I++I-) for 275 Bragg reflections were measured at 65 K. Here, I± is the measured intensity for the two antiparallel spin orientations of the incoming neutron beam. Using these values, we could refine the precise orientation of the AFM moments in HoFeO3 in the Gm4 phase (Fig. 1). Our results are in good agreement with the previously published structure at 70 K [6] but superior in details. To follow the magnetic phase transition explicitly, the asymmetries of 70 selected peaks (mostly of G-type) were measured in 2 K steps between 67-47 K. Below 53 K in the lowest field (0.15 T), the magnitude of the x-component of the Fe moment decreases abruptly whilst that of the y-component increases. This behavior is consistent with the reorientation transition to the Gm1 structure observed in zero field. Further increasing the applied field lowers the temperature of the transition, until it does not occur within the scanned temperature range for 2.2 T. However, an attempt to fit the intermediate phase ( Fig. 1-2) as a linear combination of different volume fractions of Gm1 and Gm4 phases did not lead to a converging solution. In particular, the observed presence of significant asymmetry values for G type reflections below 53 K suggests, that the applied field may introduce some distortion of the purely AFM Gm1 structure. This is confirmed by the occurrence of a G-type z-component in the intermediate phase, which is forbidden in both Gm1 and Gm4. Taking also into account the occurrence of magnetic moments on the Ho positions, we can finally conclude that the observed intermediate state is a correlated phase with monoclinic symmetry. Additionally, in the Gm4 phase the magnetic field aligns the WF moment along its direction. This lifts the degeneracy between 180 degrees AFM domains, since the orientation of the AFM structure to the WF moment is fixed by the DMI. Conversely, the sign of the DMI can be determined by the refined precise orientations of the AFM moments in the Gm4 phase from PND data (Fig. 2). By performing a proper averaging over the four Fe sublattices of the Gm4 structure, we could reduce the DMI interaction to two independent parameters D12 and D14, pointing in y-direction. Dnm denotes the interaction of the Fe sublattice n to m, with sublattice notations from Ref. [6]. With this clear convention, positive signs for D12 and D14 could be deduced. Corresponding DMI and exchange interactions magnitudes were determined in our previous work by inelastic neutron scattering [7]. One may conclude that the competition between the external magnetic field, the DMI and the exchange interactions between different magnetic sublattices leads to a complex picture of phase transitions in rare-earth orthoferrites causes also their useful functionality.

**References:**

1. K. Dey, A. Indra, S. Mukherjee, S. Majumdar, J. Strempfer, O. Fabelo, E. Mossou, T. Chatterji, and S. Giri. Phys. Rev. B 100, 214432 (2019) 2. M. Shao et al. / Solid State Communications 152 (2012) 3. T. Chatterji, M. Meven, P. J. Brown, AIP ADVANCES 7, 045106 (2017) 4. V. Hutanu, M. Meven, S. Masalovich, G. Heger, and G. Roth, J. Phys.: Conf. Ser. 294 12012 (2011) 5. H. Thoma,W. Luberstetter, J. Peters, and V. Hutanu, J. Appl. Cryst. 51 17-26 (2018) 6. T Chatterji et al, J. Phys.: Condens. Matter 29 385802 (2017) 7. A.K.Ovsianikov, I.A.Zobkalo, W.Schmidt, S.N.Barilo, S.A.Guretskii, V.Hutanu, JMMM, Volume 507, 166855 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/8a7a1f6a643a1d20fc4e008a7653ed53.jpg)
Fig.1: Magnetic structure evolution by the reorientation phase transition in HoFeO3 near 50 K with magnetic field along [001]. An intermediate phase braking orthorombic simmetry occurs between the magnetic phases Gm4 and Gm1. Here: blue balls - Fe ions, green - Ho, red - O, colored arrows denote ordered magnetic moments.

![](https://assets.underline.io/uploads/markdown_image/1/image/6f114888b1245ad04fa87bafc996b68a.jpg)
Fig. 2: Temperature/field map for the measured asymmetry of G-type peak (3 0 -1). Redish area denotes magnetic phase Gm4, blue - Gm1, celestial-greenish - intermediate monoclinic phase.

 Breaking the magnetic symmetry by reorientation transition near 50 K in multiferroic, magnetocaloric HoFeO3.

Magnetic refrigeration based on magnetocaloric effect (MCE) is attractive cooling technology because of its high energy efficiency and environmental friendliness. MCE is defined as the changes in temperature and entropy by changing a magnetic field. Magnetic refrigeration for near room temperature needs the magnetocaloric materials with large magnetic entropy change (ΔSm) near room temperature. Hf1-xTaxFe2 crystallizes the hexagonal structure. The magnetism of Hf1-xTaxFe2 changes by Ta substitution. For example, Hf1-xTaxFe2 for 0.1&lt;x&lt;0.3 shows the successive transitions from ferromagnetic order to antiferromagnetic order on cooling [1]. For x=0.17, large ΔSm at 225 K is reported [2]. The magnetic transition temperature (TM) decreases with applying pressure [3]. On the contrary, the volume expands by inserting the interstitial atom, TM would be decrease. In this work, we study the B doping effect on the magnetic properties of Hf1-xTaxFe2. The Hf0.83Ta0.17Fe2Bz (z=0, 0.1, and 0.5) alloys were synthesized by arc-melting under Ar atmosphere. The crystal structure was confirmed by the powder X-ray diffraction (XRD) technique. Magnetic properties were measured in the range of 150≤T≤270 K up to 5 T. The XRD pattern of Hf0.83Ta0.17Fe2 indicates a single phase as the hexagonal structure. For z=0.1, the minor phases as HfO2 and HfB2 were observed. For z=0.5, the intensity of the peak of HfB2 phase increased and is comparable to those of Hf0.83Ta0.17Fe2 phase. The scattering angles of Hf0.83Ta0.17Fe2 shifted higher, indicating the volume shrinking. These results suggest that the unit cell volume decreased due to deficiency of Hf by forming HfB2 in B-doped Hf0.83Ta0.17Fe2. Figure 1 shows the temperature dependence of magnetization (M) of Hf0.83Ta0.17Fe2Bz (z=0 and 0.1) in a magnetic field of 0.1 T. For z=0, M increased suddenly at 215 K with decreasing temperature. The values of M for z=0.1 increased at 200 K. The obtained data indicate that the TM decreased by B-doping. The result is consistent with the volume reduction. The maximum values of ΔSm for z=0 and 0.1 are both 5 J/K kg, which is approximately one-third of the previously reported data [2]. In summary, introducing B into Hf1-xTaxFe2, B elements reacted Hf atoms to form HfB2. The volume and TM decreased due to the deficiency of Hf.

**References:**

[1] Y. Nishihara and Y. Yamaguchi, J. Phys. Soc. Jpn. Vol. 51, p.1333 (1982). [2] J. F. Herbst, C. D. Fuerst and R. D. McMichael, J. Appl. Phys. Vol. 79, p.5998 (1996). [3] K. Hoshi, J. Phys. Soc. Jpn. Vol. 57-9, p.3112 (1988).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/20e9a98a4f97b628bf140ad7928bd159.jpg)
Fig. 1. Magnetization versus temperature of Hf0.83Ta0.17Fe2Bz (z=0 and 0.1) in a magnetic field of 0.1 T.

 Magnetocaloric effect in Hf1-xTaxFe2By

One of the possible practical applications of magnetocaloric effect (MCE) is eco-friendly magnetic refrigeration [1]. A condition for the successful operation of a magnetic refrigerator is a high efficiency of heat transfer between the working body and the surroundings. This is facilitated by an increase in the surface to the volume ratio for the working body. The material in the form of powder meets this requirement [2]. Therefore, the features of MCE in powder samples are the subject of active research [3-5]. Ball milling is a well-established technique for producing different materials in powder shape. This process contributes to the reduction of the crystal size down to nanometric scales and the accumulation of defects in grain boundaries [6]. For magnetic materials, this causes a decrease in magnetization and slow change in magnetization near the phase transition temperature. This lowers the MCE [3]. On the other hand, nanostructuring causes the effect of a Curie temperature distribution. In turn it leads to a broadening of the peak of the temperature change of the magnetic entropy (ΔSM) and an increase in a refrigeration capacity (RC) value, which can be defined as the product of the maximum value of the peak of magnetic entropy change and the width of ΔSM peak at its half-height [1]. In addition, nanostructuring can enhance the field dependence of the magnetocaloric effect [5-7]. Appropriate annealing of polycrystalline materials can to some extent restore the perfection of their crystal structure and magnetization behavior, thereby establishing an advantageous balance between the height and width of ΔSM peak, which determine the RC value [8]. In this work, we present the result of detailed study of influence of thermal treatment on structure, magnetic and magnetocaloric properties of Gd powder obtained by ball milling. Ball milling was performed with a mixer/mill using hardened-steel vials in acetone. Gd powder samples were annealed in vacuum at different temperatures up to 600 K. The structure of the samples was studied by X-ray diffraction technique using automatic PHILIPS X’PERT PRO diffractometer operating with Cu-Ka radiation. The magnetic entropy change was calculated using the Maxwell relation follow the standard procedure [1]. The temperature dependence of the magnetic entropy change measured for Gd powder samples annealed at different temperature shows that ΔSM peak is constricted in each case (Fig. 1). The value of the ΔSM peak height and the RC values are decreased with the increase of the temperature (Fig. 2). Observed MCE behavior was understood taking into account the changes in the crystal structure including strain values and contribution of the partial surface oxidation as a result of the heat treatment. At the same time, heat treatment at the temperature of 500 K results in the shift of MCE peak exactly for the room temperature point that might be very useful for applications. This study was supported by the Ministry of Science and Higher Education of the Russian Federation (project No. FEUZ -2020-0051) and Elkartek AVANSITE grant of the Basque Government.

**References:**

[1] V. Franco, J.S. Blázquez, J.J. Ipus, at al. Prog. Mater Sci. 93 (2018) 112. [2] Q. Gao, B.F. Yu, C.F. Wang, at al. Int. J. Refrig. 29 (2006) 1274. [3] J.S. Blázquez, J.J. Ipus, L.M. Moreno-Ramírez, at al. Metall. Mater. Trans. E 2 (2015) 131. [4] S. Wolf, T.M. Riedemann, J. Barclay, at al. Powder Technol. 359 (2020) 331. [5] A.V. Svalov, A.V. Arkhipov, S.V. Andreev, at al. Mater. Lett. 284 (2021) 128921. [6] J.S. Blázquez, J.J. Ipus, L.M. Moreno-Ramírez, at al. J. Mater. Sci. 52 (2017) 11834. [7] D. Doblas, L.M. Moreno-Ramírez, V. Franco, at al. Mater. Design 114 (2017) 214. [8] S.V. Taskaev, V.D. Buchelnikov, A.P. Pellenen, at al. J. Appl. Phys. 113, (2013) 17A933.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a2f6aa8431127d60a2c6250253cd8552.jpg)
Fig 1. The magnetic entropy changes versus the temperature for Gd powder samples in initial state (curve I) and after annealing at different temperatures (curves II, III, IV, V and VI for 350 K, 450 K, 500 K, 550 K and 600 K, respectively). Magnetic field changes from 0 to 7 T.

![](https://assets.underline.io/uploads/markdown_image/1/image/d536f7c3aa1aa8bf5e20e19fa17c0027.jpg)
Fig 2. RC versus the magnetic field for Gd powder samples in initial state (curve I) and after annealing at different temperatures (curves II, III, IV and V for 350 K, 450 K, 500 K and 550 K, respectively).

 Magnetocaloric properties of ball milled Gd powder subjected to heat treatments

Recent studies on multicomponent systems and high entropy alloys offer enormous motivation and scope to study the magnetic and magnetocaloric properties of a multicomponent Laves phase intermetallic compound, RAl2, where R site is occupied by five equiatomic heavy rare earth elements [1]. Polycrystalline intermetallic compound Gd0.2Tb0.2Dy0.2Ho0.2Er0.2Al2 has been prepared by arc-melting under argon atmosphere and the single phase nature was confirmed by powder X-ray diffraction (XRD) study carried out at room temperature. DC magnetization has been measured in the temperature range of 2 K to 300 K in applied magnetic fields up to 70 kOe. Electrical resistivity, heat capacity and thermoelectric power have been measured as a function of temperature from 300 K to about 4 K. Magnetization data reveal that Gd0.2Tb0.2Dy0.2Ho0.2Er0.2Al2 compound orders ferromagnetically at ~80 K (TC) [Fig. 1]. This ordering temperature is approximately equal to the mean value of the ferromagnetic ordering temperatures of the single rare earth component compounds GdAl2, TbAl2, DyAl2, HoAl2 and ErAl2 respectively. The Arrot plots and the nature of the temperature variation of magnetization suggest second order nature for this transition at 80 K. In the present work, electrical resistivity, thermopower and specific heat of the title compound are studied. Electrical resistivity shows metallic behaviour and one does see an anomaly around TC. Electrical resistance vs field is measured at a few selected temperatures in fields up to 50 kOe. A weak, negative magnetoresistance is observed below TC. The heat capacity vs temperature plot shows a weak second order transition around 80 K. However, thermoelectric power did not register the magnetic anomaly strongly. The isothermal magnetic entropy change calculated using the magnetization-field data near TC reveal a broad magnetocaloric effect around TC, as one would expect for ferromagnetic material showing a second order transition. These results are compared with that of the individual heavy rare earth Laves phase dialuminides in order to understand the collective behavior of magnetic rare earth moments in such a multicomponent system. This along with the studies on high entropy alloys should pave a new way in the materials design for magnetics and magnetocalorics.

**References:**

P. K. Jesla, J. Arout Chelvane, A. K. Nigam and R. Nirmala, AIP conference proceedings 2265 (2020) 030565

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a9d5598e21fb3da4e0e8f55d874357e8.jpg)
Fig. 1 Zero-field-cooled (ZFC) and field-cooled cooling and warming (FCC and FCW) Magnetization vs temperature data of Gd0.2Tb0.2Dy0.2Ho0.2Er0.2Al2 measured in 100 Oe field [Inset dM/dT vs T plot]

 Magnetic and Transport properties of Multicomponent Laves Phase Intermetallic Compound Gd0.2Tb0.2Dy0.2Ho0.2Er0.2Al2

A new type of rare-earth giant magnetostrictive transducer, working in ultrasonic frequencies, is designed by the structural dynamic model and the dynamic coupling model. Based on the vibration, wave and frequency equations, the structural dynamic model of the transducer is established. By using COMSOL Multiphysics software parametric design method, considering the influence of materials, shapes and dimensions on the output characteristics of the transducer, the optimum mechanical structure of the ultrasonic transducer is determined. The magnetic circuit structure is analyzed by the dynamic coupling model based on the Jiles-Atherton model, Maxwell’s equations and magneto-mechanical coupling effect. The prototype of the ultrasonic transducer is fabricated. The relationship between the output characteristics and the frequency is studied. The results demonstrate that the ultrasonic transducer can work in the range of 18-22 kHz, and exhibits higher magneto-mechanical conversion efficiency at the frequency of 19.5 kHz. I. Introduction The rare-earth giant magnetostrictive transducer has the advantages of high power, large amplitude and high response speed and it can be used in the fields of accurate actuation and ultrasonic machining [1]. At present, the transducers used in ultrasonic applications are mainly piezoelectric ceramic transducer [2]. Compared with piezoelectric materials, rare-earth giant magnetostrictive material, Terfenol-D (Tb0.7Dy0.3Fe2), has greater strain, smaller hysteresis, and lower saturation magnetic field. Wang designed a giant magnetostrictive transducer (GMT) with a resonant frequency of 2 kHz [3]. Karunanidhi designed a GMT. The resonant frequency was 3 kHz. The transducer with mechanical amplifier and permanent magnet biasing was designed and realized [4]. Sheykholeslami proposed an approach to design a GMT. The resonant frequency was 8.25 kHz. This approach can make the transducer have high mechanical quality factor [5]. Fang introduced a design and finite element analysis method of a GMT. The working frequency of the transducer is 15.2 kHz [6]. Karafi designed a new hybrid longitudinal–torsional GMT with a resonant frequency of 20.6 kHz. The output displacement of the transducer was 0.6 µm [7]. References [3]-[5] studied the design method and output characteristics at relatively lower frequency magnetostrictive transducers. Reference [7] designed the giant magnetostrictive ultrasonic transducers, but the output displacement is small. So, how to design and manufacture a rare-earth giant magnetostrictive transducer with large output displacement, working in ultrasonic frequencies, is still a crucial problem for the development of transducers. II. The magnetic circuit and mechanical structure design of the ultrasonic transducer The magnetic circuit design needs to improve the magnetic flux density of magnetostrictive rod and reduce losses of the magnetic circuit. Based on the Jiles-Atherton model, Maxwell’s equations and magneto-mechanical coupling effect, the magnetostrictive rod, magnetic yoke and magnetic circuit structure of the ultrasonic transducer are designed. It is found that the magnetic circuit structure of two Terfenol-D rods with coils in series connection is more suitable for the ultrasonic transducer. The radius of the Terfenol-D rod is 7.5 mm, and the length is 102 mm. To reduce the effect of dynamic losses, the Terfenol-D rod is machined into 2 mm slices and glued together by epoxy resin. The width of the magnetic yoke is 16 mm and the length is 60 mm. Based on the vibration, wave and frequency equations, the structural dynamic model of the ultrasonic transducer is established. By using COMSOL Multiphysics software parametric design method, considering the influence of materials, shapes and dimensions on the output characteristics of the ultrasonic transducer, the optimum mechanical structure of the ultrasonic transducer is determined. The whole mechanical structure of the transducer is analyzed, and the resonant frequency of the transducer is obtained. A detailed and complete design scheme is given and the main body of the ultrasonic transducer consists of two Terfenol-D rods, driving coils, disc spring, adjusting nut, vibrating horn, magnetic yoke and shell, as shown in Fig.1. III. Experimental results The prototype of the ultrasonic transducer was fabricated. The relationship between the output characteristics and the frequency is experimentally studied by using laser displacement sensor (LK-H008, KEYENCE). Output displacement amplitude versus frequency is shown in Fig.2. The amplitudes of exciting current are 1A and 2A. The frequencies are 16.5 kHz-22.5 kHz. The experimental results have obvious resonance peaks, and their changing tendencies are consistent. When f&lt;19.5 kHz, the amplitudes of vibration increase with increasing frequencies; When f=19.5 kHz, the output amplitude reaches the maximum value, the amplitude reaches 6.07 μm at 1A and 9.46 μm at 2A exciting current respectively; When f&gt;19.5 kHz, the amplitudes of vibration decrease with frequency increasing. It can be seen that the measured resonant frequency is 19.5 kHz, and the resonant frequency almost does not change at the different applied magnetic field.

**References:**

[1] J. J. Scheidler and V. M. Asnani, Smart Materials and Structures, vol. 26, p. 035057 (2017) [2] X. Wei, Y. Yang and W. Yao, Sensors, vol. 17, p. 2253, (2017) [3] W. Wang and P. J. Thomas, Sensors and Actuators A: Physical, vol. 263. p. 113, (2017) [4] S. Karunanidhi and M. Singaperumal, Sensors and Actuators A: Physical, vol. 157, p. 185, (2010) [5] M. R. Sheykholeslami, Y. Hojjat and S. Cinquemani, Smart Structures & Systems, vol. 17, p. 313, (2016) [6] S. Fang, Q. Zhang and H. Zhao, Micromachines, vol. 9. p. 98, (2018) [7] M. R. Karafi, Y. Hojjat and F. Sassani, Smart Materials & Structures, vol. 22, p. 065013, (2013)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/f023af9851d401207c90b3e5c84b22a2.jpg)
Fig.1 (a) Structure of the ultrasonic transducer (b) Prototype of the ultrasonic transducer

![](https://assets.underline.io/uploads/markdown_image/1/image/01445a34938883b4dcd818b6e049f4f1.jpg)
Fig.2 Output displacement amplitude versus frequency at different exciting current

 The Design and Output Characteristics of Ultrasonic Transducer Based on Rare-earth Giant Magnetostrictive Material

Magneto-elastic beams in a cantilever configuration represents a quite common layout for a vibration energy harvester and its design is aimed to increase performances in terms of geometrical characteristic and power conversion [1-4]. Cantilever-like harvesters are employed when a quite constant vibration source is available, dealing to periodic steady state conditions. But the analysis of such a system is still a multiphysics problem, involving the mechanical deformation of the elastic body coupled with the electromagnetic problem through the magneto-elastic characteristics of the active material. In [1] and [4] the importance of cantilever’s geometry has been outlined, showing the importance of the neutral axis location for the output power optimization, also providing [4] a qualitative analysis describing the phenomenon. In the same contribution, the focus was on the design and development of the harvester through an experimental analysis of several layouts. The effort was so addressed on the identification of the converted electrical source supplying the output electrical circuit by the standard tools of circuit’s theory (Thevenin’s theorem), which was designed to optimize load resistance and efficiency for the whole device. The modeling of the harvester as a magneto-mechanical converter requires more effort, e.g. coupling the beam’s dynamics through the Euler-Bernoulli equation and the magneto-elastic characteristics of the beam. Despite interesting contributions exploiting 3-D FEM algorithms to describe the beam dynamics [3],[5],[6], in the present contribution attention is focused on the definition of a semi-coupled model where the mechanical behavior (in periodic steady state conditions) is taken into account through a model, described synthetically in Fig. 1b, and referred to the cantilever beam, described in Fig. 1a, supplied by a vertical sinusoidal acceleration on the left tip, in ACsinusoidal steady state, with amplitude a. The well-known solution (not provided for brevity) describes the behavior of the beam deformation w(z,t) which provides the moment bending and therefore the longitudinal stress and the strain along the cantilever, [7] and, through the assumed linear magneto-elastic coupling of the magnetostrictive alloy, the flux linkage and the emf at the coil terminals, the as shown in Fig. 2. The detailed derivation of eqns. requires some algebra and will be detailed in the paper. The emf measured at coil’s ends is related to the beam’s deformation dynamics and to the cantilever’s geometry. In particular, when the neutral axis is exactly in the middle of the magnetostrictive sheet (x1 = - x2) the total emf is zero. This result seems to fit with the experimental evidence of. In Fig. 2b where the output power on a 5.6Ω resistive load is plotted vs frequency, with a = 1g, 2g, 3g, 4g, for three different cantilever layouts. In the first one a magnetostrictive (Fe-Ga) lamination is “sandwiched” by two Al sheets (C2) and realizes a configuration with neutral axis at almost the middle of the active alloy. The second is conversely realized by an Al sheet “sandwiched” by two Fe-Ga laminations (C1), while in the last a single Al sheet is sticked to a Fe-Ga lamina (C3). In both C1 and C3 cases, the neutral axis lays outside the magnetostrictive laminas. All configurations share the same two bias magnets on the upper side of the cantilever. In linear assumptions, the RMS power in the plot is proportional to ε2 and according to the model shows a much lower converted power of C2 case with respect to other configurations. An accurate analysis on the influence of geometric and physical parameters of the cantilever by the proposed modeling will be discussed in the paper. Such plot confirms that a simplified magnetostrictive cantilever’s modeling could provide information on the parameters that impact the harvester design and represents a simple but effective generalization of [4]. Further, without significative effort, a natural introduction of a nonlinear magneto-elastic behavior could be plugged in.

**References:**

[1] T. Ueno, S. Yamada, Performance of energy harvester using iron–gallium alloy in free vibration, IEEE Trans. Magn. 47 (10) (2011) 2407–2409. [2] J.-H. Yoo, A.B. Flatau, A bending-mode Galfenol electric power harvester, J. Intell. Mater. Syst. Struct. 23 (6) (2012) 647–654. [3] Z. Deng, M.J. Dapino, Modeling and design of Galfenol unimorph energy harvesters, Smart Mater. Struct. 24 (12) (2015) 125019. [4] V. Apicella, C. S. Clemente, D. Davino, D. Leone, C. Visone, Magneto-mechanical optimization and analysis of a magnetostrictive cantilever beam for energy harvesting, J. of Magnetism and Magnetic Materials, 475, (2019), 401-407. [5] Mojtaba Ghodsi, Hamidreza Ziaiefar, Morteza Mohammadzaheri, Amur Al-Yahmedi, Modeling and characterization of permendur cantilever beam for energy harvesting, Energy, 176 (2019), 561-569. [6] M. Borowiec, A. Syta, G. Litak, Energy Harvesting Optimizing with a Magnetostrictive Cantilever Beam System, International Journal of Structural Stability and Dynamics Vol. 19, No. 05, 1941002 (2019). [7] S. H. Crandall, Mechanics of Solids, McGraw-Hill, Third Edition.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7cfc0d5eff81b7d787608d9a833e4c02.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/6787b6aca4d66b15409c7c97beb82ee6.png)

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