United States

In the present work, we combine magnetometry and magnetic small-angle neutron scattering to study the influence of the microstructure on the macroscopic magnetic properties of a nanocrystalline Ni bulk sample, which was prepared by straining via high-pressure torsion. As seen by magnetometry, the mechanical deformation leads to a significant increase of the coercivity compared to nondeformed polycrystalline Ni. The neutron data reveal a significant spin-misalignment scattering caused by the high density of crystal defects inside the sample, which were created by the severe plastic deformation during the sample preparation. The corresponding magnetic correlation length, which characterizes the spatial magnetization fluctuations in real space, indicates an average defect size of 11 nm, which is smaller than the average crystallite size of 60 nm. In the remanent state, the strain fields around the defects cause spin disorder in the surrounding ferromagnetic bulk, with a penetration depth of around 22 nm. The range and amplitude of the disorder is systematically suppressed by an increasing external magnetic field. Under certain assumptions, the analysis of the penetration depth allows one to determine the magnitude of the effective magnetic anisotropy. For HPT Ni, we find an increased anisotropy (factor of 4), presumably due to magnetoelastic interactions, relative to the single-crystalline ground state. Our findings are supported and illustrated by micromagnetic simulations, which, for the particular case of nonmagnetic defects (holes) embedded in a ferromagnetic Ni phase, further highlight the role of localized spin perturbations for the magnetic microstructure of defect-rich magnets such as high-pressure torsion materials.&lt;br/&gt;&lt;br/&gt;This work was supported by the financial support from the National Research Fund of Luxembourg (Pride MASSENA Grant No. 10935404 and CORE SANS4NCC Grant), from KAKENHI (Grant No. 19K05102) and from the Japan Science and Technology Agency (Grant No. JPMJSK1511).

2022 Joint MMM-INTERMAG

Revealing defect-induced spin disorder in nanocrystalline

#### Welcome to the 67th Annual Conference on Magnetism and Magnetic Materials, MMM 2022!

The Conference was held from October 31 to November 4 and offers both in-person and on-demand content.

MMM 2022 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 tutorial, and several special sessions. 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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MMM 2022

This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

In the present work, we combine magnetometry and magnetic small-angle neutron scattering to study the influence of the microstructure on the macroscopic magnetic properties of a nanocrystalline Ni bulk sample, which was prepared by straining via high-pressure torsion. As seen by magnetometry, the mechanical deformation leads to a significant increase of the coercivity compared to nondeformed polycrystalline Ni. The neutron data reveal a significant spin-misalignment scattering caused by the high density of crystal defects inside the sample, which were created by the severe plastic deformation during the sample preparation. The corresponding magnetic correlation length, which characterizes the spatial magnetization fluctuations in real space, indicates an average defect size of 11 nm, which is smaller than the average crystallite size of 60 nm. In the remanent state, the strain fields around the defects cause spin disorder in the surrounding ferromagnetic bulk, with a penetration depth of around 22 nm. The range and amplitude of the disorder is systematically suppressed by an increasing external magnetic field. Under certain assumptions, the analysis of the penetration depth allows one to determine the magnitude of the effective magnetic anisotropy. For HPT Ni, we find an increased anisotropy (factor of 4), presumably due to magnetoelastic interactions, relative to the single-crystalline ground state. Our findings are supported and illustrated by micromagnetic simulations, which, for the particular case of nonmagnetic defects (holes) embedded in a ferromagnetic Ni phase, further highlight the role of localized spin perturbations for the magnetic microstructure of defect-rich magnets such as high-pressure torsion materials. This work was supported by the financial support from the National Research Fund of Luxembourg (Pride MASSENA Grant No. 10935404 and CORE SANS4NCC Grant), from KAKENHI (Grant No. 19K05102) and from the Japan Science and Technology Agency (Grant No. JPMJSK1511).

technical paper

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.

The 2022 Joint Conference will offer both in-person and prerecorded on-demand content.

For tickets to the event, please register at the following link: https://magnetism.org/register

Please register for the Joint MMM-Intermag conference to join the event.

Single-phase multiferroic materials that allow the coexistence of ferroelectric and magnetic ordering above room temperature are highly desirable, motivating an ongoing search for mechanisms for unconventional ferroelectricity in magnetic oxides. Here, we report an antisite defect mechanism for room temperature ferroelectricity in epitaxial thin films of yttrium orthoferrite, YFeO3, a perovskite-structured canted antiferromagnet [1]. A combination of piezoresponse force microscopy, atomically resolved elemental mapping with aberration corrected scanning transmission electron microscopy and density functional theory calculations reveals that the presence of YFe antisite defects facilitates a non-centrosymmetric distortion promoting ferroelectricity. This mechanism is predicted to work analogously for other rare earth orthoferrites, with a dependence of the polarization on the radius of the rare earth cation. Furthermore, a vertically aligned nanocomposite consisting of pillars of a magnetoelastic oxide CoFe2O4 embedded epitaxially in the YFeO3matrix exhibits both robust ferroelectricity and ferrimagnetism at room temperature, as well as a noticeable strain-mediated magnetoelectric coupling effect [2]. Our work uncovers the distinctive role of antisite defects in providing a novel mechanism for ferroelectricity in a range of magnetic orthoferrites and further augments the functionality of this family of complex oxides for multiferroic applications.

Defect mechanism for ferroelectricity in orthoferrites

One of the most promising ways of improvement of thermoelectric energy conversion leads via band gap engineering towards the enlargement of the density of states at the Fermi level, e.g. via a co-doping to induce a resonant level in the electric transport relevant band [1]. Cr in PbTe is a particularly interesting example since Cr atoms form mixed valence Cr2+/3+ donor centers resonant with the PbTe conduction band [2]. We report on detailed temperature and magnetic field dependence of magnetization of PbTe doped with transition metal Cr2+/3+. The material is studied by a SQUID magnetometer in order to quantitatively determine the contribution of single substitutional Cr2+/3+ as well as of various Cr-Te magnetic nanocrystals, including their identification. The applied experimental procedure reveals the presence of about 1019 cm-3 paramagnetic Cr ions, of which 2/3 are the Cr3+ ions formed via self-ionization of Cr2+ resonant donors. The magnetic finding agrees with previous Hall effect studies thus providing a new experimental support for the proposed electronic structure model of PbTe:Cr system with resonant Cr2+/3+ state located (at low temperatures) about 100 meV above the bottom of the conduction band. Below room temperature a ferromagnetic-like signal points to the presence of Cr-rich nanocrystalline precipitates. Two most likely candidates, namely: Cr2Te3 and Cr5Te8 are identified upon dedicated temperature cycling of the sample at the remnant state. As an ensemble, the nanocrystals exhibit (blocked) superparamagnetic properties. Also HR-TEM confirmed the presence of numerous single nanometer size precipitates of Cr-Te compounds. The magnetic susceptibility of both n- and p-type PbTe in the temperature range 100 &lt; T &lt; 400 K has been established. These magnitudes are essential to properly account for the high temperature magnetic susceptibility of PbTe:Cr [3]. This study has been supported by the National Science Centre (Poland) through project OPUS (UMO - 2017/27/B/ST3/02470) and by the National Science Centre for Development (Poland) through grant TERMOD No TECHMATSTRATEG2/408569/5/NCBR/2019 and by the Foundation of Polish Science through the IRA Programme co-financed by EU within SG OP.

Magnetic Constitution of PbTe:Cr - Experimental Aspects

We find the magnetic susceptibility of NH4Al(SO4)2-12H2O below its ferroelectric Curie temperature of TC = 58 K can be tuned to become positive or negative simply by controlling the cooling speed. We explain this by taking into consideration that reorienting NH4+ ions possess weak magnetic moments that are linearly coupled to dipole moments which in turn are aligned by the crystal field with cooling but at the same time experiencing competition from geometric frustration. With fast cooling such as a rate of 1.0 K/min, the magnetic moments are not able to become ordered in time so the system remains mostly diamagnetic. In contrast, with a slow cooling rate of 0.1 K/min, the magnetic moments are allowed to access a more ordered state so the magnetic susceptibility becomes positive. From such, four -ferroic states can be accessed below TC by applying either a positive or negative bias field and cooling the sample slow or fast to obtain polarization and magnetization of different polarities. Specific heat and dielectric constant measurements near TC indicate the origin of multiferroic behavior as originating from proton orbital degeneracies. Our findings identify an entire new class of multiferroic materials in the ammonium alums and possibly other hydrogen-bonding compounds.

Multiferroic Behavior in NH4Al(SO4)2-12H2O: Magnetism and Ferroelectricity Based on Proton Orbital Degeneracies

Magnetic materials in which it is possible to control the topology of their magnetic order in real space or the topology of their magnetic excitations in reciprocal space are highly sought-after as platforms for alternative data storage and computing architectures. Here we show that multiferroic insulators, owing to their magneto-electric coupling, offer a natural and advantageous way to address these two different topologies using laser fields. We demonstrate that via a delicate balance between the energy injection from a high-frequency laser and dissipation, single skyrmions—archetypical topological magnetic textures—can be set into motion with a velocity and propagation direction that can be tuned by the laser field amplitude and polarization, respectively. Moreover, we uncover an ultrafast Floquet magnonic topological phase transition in a laser-driven skyrmion crystal and we propose a new diagnostic tool to reveal it using the magnonic thermal Hall conductivity.

![](https://assets.underline.io/uploads/markdown_image/1/image/7db0278173bbc5cba45a88101bb6611a.jpg)

Real- and reciprocal-space topology can be controlled by lasers. (a) The topological spin structure of a skyrmion carrying in-plane electric polarization Ρ undergoes translational motion under a circularly polarized laser. (b) An ultrafast topological phase transition occurs in the Floquet magnon band structure of skyrmion crystals due to the effective magnetic field induced by circularly polarized laser irradiation.



Laser-Controlled Real and Reciprocal Space Topology in Multiferroic Insulators

Ferrites and magnetic nanoparticles (NPs) are widely used in sectors such as energy, electronics and biomedicine [1,2]. This work shows an optimum combination of co-precipitation, milling and reduction process of ferrite precursors (CoFe2O4 and NiFe2O4) to obtain CoFe and NiFe NPs, respectively, with tailored magnetic properties and improved heat generation for magneto-thermal applications. Ferrite precursor powders have been synthesized by co-precipitation [3] followed by nanostructuring through self-developed ultrafast milling (24 min). Reduction at different temperatures under Ar+H2 has led to NPs with a tuneable composition going from pure ferrite to CoFe and NiFe NPs, with core-shell nanostructures (CoFe2O4-CoFe and NiFe2O4-NiFe) at intermediate reduction stages. An optimized temperature has resulted in CoFe and NiFe NPs with a high saturation magnetization of 235 and 149 emu/g, respectively. Milling has allowed for diminishing the reduction temperature in 100 °C compared to that required for non-milled particles, which is of relevance in view of upscaling (Fig. 1) [4]. The magneto-thermal response of the NPs has been studied by exposure to alternating magnetic fields at 103 kHz for 5 min. Remarkably, an increase of 28% in the heat generated (under an applied field of 20 mT) has been obtained for the reduced milled CoFe NPs in comparison to the non-milled NPs (Fig. 2), due to particle refinement and high saturation magnetization. The milled NiFe NPs has shown a lower heat generation due to agglomeration effects. This cost-efficient and scalable approach eases the synthesis of NiFe and CoFe NPs for magneto-thermal applications. Acknowledgments. Authors acknowledge financial support from EU M-ERA.NET through COSMAG project (PCI2020-112143); from MICINN through NEXUS project (PID2020-115215RB-C21); and from CM through NanoMagCOST project (P2018/NMT-4321). A.J.C.-H. acknowledges support from "La Caixa" by the Doctoral INPhINIT Incoming program (LCF/BQ/DI20/1178002).

![](https://assets.underline.io/uploads/markdown_image/1/image/3d3878bca20e35efefe648bf0daa5801.jpg)

Fig.1. (a) Room temperature hysteresis loops and (b) XRD patterns of milled cobalt ferrite NPs reduced at different temperatures.

![](https://assets.underline.io/uploads/markdown_image/1/image/1e335e0631e7215287a7642ecf21df92.jpg)

Fig.2. Magneto-thermal measurements of milled and non-milled CoFe NPs.

Scalable Synthesis of CoFe and NiFe Nanoparticles from Ferrite Precursors with Tailored Magneto-Thermal Properties by a Combined Co-precipitation, Milling and Reduction Process

Metal Amorphous Nanocomposites (MANCs) are an evolving class of soft magnetic materials consisting of an Fe- or Co-based crystalline phase embedded in an amorphous matrix. In thin-film form, they possess highly tunable magnetic properties, low eddy current and hysteresis losses, and are capable of performing under high-frequency magnetic field reversal. Therefore they are good candidates for a range of applications including use in power electronics, transformers, and electric motors[1, 2]. We investigated the structural, static and dynamic properties of magnetron sputtered and vacuum annealed (5 minutes and 20 minutes at 520oC) films of CoFeMnNbSiB using Transmission Electron Microscopy (TEM), Atom Probe Tomography (APT), Vibrating Sample Magnetometry (VSM) and Ferro-Magnetic Resonance (FMR). The sputtered films did not show crystallinity, however, an Mn-rich columnar structure that spanned the thickness of the film parallel to the growth direction was observed. After annealing for 5 minutes and 20 minutes, the columnar structure transformed to a partially crystallized microstructure with Co and Fe enriched crystallized phases with ~7nm grain size enclosed by an amorphous intergranular region as shown in Fig. 1. The magnetization dynamics of the films were measured using a custom-built instrument with a frequency range of 1-65 GHz and fields up to 1.6 T. The FMR spectrum of annealed samples shows two overlapping resonances (Fig. 2) due to the two phases present in the samples whereas the as-sputtered sample shows only one resonance. The effective magnetization remained constant whereas the linewidth increased from 13 to 55 Oe after annealing. The magnetic coercivity was observed to be (~26 Oe) independent of annealing while the saturation magnetization increased from 828 to 953 emu/cm3. The observed changes in magnetic properties with annealing indicate that short-time annealing can be useful in optimizing the performance of these MANCs and tailoring properties for various applications.

![](https://assets.underline.io/uploads/markdown_image/1/image/8e0cc2fd197e33d1445aaa05db782123.jpg)

Fig. 1.a) TEM image of 20 minutes annealed sample b) Magnified section of a crystalline region

![](https://assets.underline.io/uploads/markdown_image/1/image/5a0af14f69f8ff951824fdad5f91dd49.jpg)

Fig. 2. FMR spectrum of 20 minutes annealed sample at 40 GHz reflecting two overlapping peaks due to dual-phase in the sample

Structural and magnetic characterization of Co-based soft magnetic amorphous nanocomposite films

Magnetic wires can present outstanding magnetic properties such as giant magnetoimpedance (GMI) or magnetic bistability and related single domain wall (DW) propagation. Magnetically soft glass-coated magnetic microwires prepared by the Taylor-Ulitovsky technique with metallic nucleus covered by flexible, insulating and biocompatible glass are quite demanded for a great number of applications [1]. Recenly, we showed that stress-annealing allows magnetic softening, GMI effect improvement and even grading the magnetic anisotropy in Fe-rich microwires [2]. Although generally thinner wires present certain advantages related to the industrial application, for some applications (like non-contact stress monitoring or magnetic tags) thicker wires are highly requested. Recently, preparation of thicker (above 50 μm in diameter, d) glass-coated Fe-rich microwires is reported [3]. In this paper, we present our recent experimental results on preparation and processing of “thick” Co-rich magnetic microwires (d = 94 μm) with graded magnetic anisotropy. From previous publications related to the effect of stress-annealing on magnetic anisotropy of Fe-rich microwires, is known that the stress-annealing induced anisotropy depends on the annealing temperature, Tann [2]. Such graded magnetic anisotropy has been obtained by stress-annealing of Co-rich microwire at variable annealing temperature. The hysteresis loops for different sections of the microwire have been evaluated using a short (2 mm long) movable pick-up coil. A gradual change in the hysteresis loop (measured by the short pick-up coil) of the sample stress-annealed at variable Tann is observed (see Fig.1). Such microwire, presents also variable squareness ratio, Mr/M0, along the microwire length, L (see Fig.1). Obtained graded anisotropy has been attributed to a gradual modification of the domain structure along the microwire. Previously, for preparation of thin films with a graded magnetic anisotropy, rather sophisticated methods were used, including a change in the chemical composition during the deposition of thin films [4]. Accordingly, we propose a rather simple route to design graded magnetic anisotropy in a magnetic microwire.

![](https://assets.underline.io/uploads/markdown_image/1/image/57c8ead2c65eb757399987363b53f20c.jpg)

Graded magnetic anisotropy in Co-rich microwires

Fe-Nb-Cr-B rapidly quenched melt-spun ribbons (MSRs) show glassy structures in the as-quenched state and Curie temperatures (TC) of 290-330 K depending on the Cr content [1]. The medical applications are usually requiring submicron powders, which can be easily targeted to the sick body organs by means of the external magnetic field [2]. The present work focusses on the preparation of Fe67.2Nb0.3Cr12.5B20 submicron powders by high-energy ball milling from MSRs precursors, using different milling agents: oleic acid (OA), sodium oleate (NAOL) and stearic acid (SA). The main challenge is to preserve the glassy structure from MSRs through the milling process, because only the glassy structure guarantees the low TC. MSRs were annealed at 4000C for 1 h to facilitate the milling. The ratio stainless steel balls:powder was of 50:1. The powders milled in OA and NAOL in the ratio of 1:1 have been washed in n-heptane and dried at 700C in vacuum, while for the ones milled in SA, in the ratio of 5:1, trichloroethylene was used for removing the surfactant. After 150 h of milling in OA the powders have ~100 nm, XRD and UHR-TEM investigations indicating the presence of only the glassy structure and the absence of precipitating crystalline phases. The milling up to 250 h reduces the size of the powders to &lt; 50 nm, while the glassy structure is preserved, as confirmed also by the same TC value (~450C) as for the precursor MSRs. The milling in NAOL keeps the powders in the range of 800÷900 nm, while TC remains ~450C. The milling over 100 h in NAOL results in an increase of TC to 650C and of Ms to 45 Am2/kg, compared with MSRs powders milled in OA (Ms = 35 Am2/kg). The increase of TC and Ms is even more obvious for the powders milled in SA (1100C and 112 Am2/kg, respectively). Moreover, the powders milled in SA are not changing their size even after 50 h of milling, i.e. ~5 µm, comparable with the ones of the precursor MSRs. However, the HR-SEM images indicate round shaped powders after milling in SA, compared with the starting lamellar pieces of MSRs, while the XRD and SAED patterns emphasize similar glassy structures for powders and MSRs. Financial support by the NUCLEU Program (PN 19 28 01 01) is gratefully acknowledged.

Influence of wet and dry milling conditions on the evolution of microstructure and TC of Fe-Cr-Nb-B glassy powders

The possibility of the powderization of the precursor with a single amorphous phase was investigated in the FeBNbP and the FeBNbPSi nanocrystalline alloy powders. The Fe90-XBxNb7P3 (x=9, 10), the Fe79.5B9.5Nb7P3Si1 and the Fe82-xB9Nb6P3Six (x=2, 3) powders were prepared by the gas atomization method using high pressure water for rapid quenching. The crystallization by abnormal grain growth was shown by cross sectional images and SAED pattern in the as-atomized Fe81B9Nb7P3 particle. On the other hand, the as-atomized particle as a precursor exhibiting low Hc of 141 A/m with single amorphous phase was observed from a cross sectional image and SAED pattern in the Fe80B10Nb7P3 nanocrystalline alloy powder. In addition, the stability of amorphous phase in the FeBNbP nanocrystalline alloy powder was also significantly improved by the addition of Si. The as-atomized Fe79.5B9.5Nb7P3Si1 nanocrystalline alloy powder with Si of 1 at% was decreased the crystallization peak compared with the Fe80B10Nb7P3 nanocrystalline alloy powder without Si. In particular, the as-atomized Fe79B9Nb6P3Si3 alloy powder with Si of 3% as a precursor powder of nanocrystalline alloy exhibited the excellent magnetic softness of low Hc of 53 A/m compared to ordinary Fe73Si11B11Cr3C2 amorphous and Fe73.5Si13.5B9Nb3Cu1 nanocrystalline alloy powders. Furthermore, the nanocrystallized Fe79B9Nb6P3Si3 powder at 873K achieved both high Bs of 1.41T and low Hc of 37A/m compared to ordinary amorphous, and nanocrystalline alloys. Therefore, we concluded that the novel FeBNbPSi nanocrystalline alloy powder with both high Bs and low Hc was developed for magnetic core of inductor products.

![](https://assets.underline.io/uploads/markdown_image/1/image/550cdf016aa2f60c6e842e8e68bc0cee.jpg)

Fig.1 (a) SEM image of the particle cross section with the Fe81Nb7B9P3, (b) TEM bright field image of amorphous and crystallization region with the Fe81B9Nb7P3, (c), (d), (e) SAED pattern of a region with A, B and C, (f) SEM image of the particle cross section with the Fe80B10Nb7P3, (g) TEM bright field image of amorphous region with the Fe80B10Nb7P3, (h) SAED pattern of D region.

![](https://assets.underline.io/uploads/markdown_image/1/image/a6df95996d718f997f1d5f7c18ef6299.jpg)

Table. 1 Average particle size and magnetic characteristics of Fe90-XBxNb7P3 (x=9, 10), Fe79.5 B9.5Nb7P3Si1 and Fe82-xB9Nb6P3Six (x=2, 3) powders with ordinary FeSiBCCr and FeSiBNbCu powders.

Development of Novel Fe Based Nanocrystalline FeBNbPSi Alloy Powder with High Bs of 1.41T

Amorphous ferromagnetic glass-coated microwires present various properties suitable for technological applications, such as great magnetic softness, ultra-fast magnetization switching and domain wall propagation, inductive heating capabilities or Giant Magnetoimpedance, GMI, effect [1, 2]. Thus, they have been widely studied in the past few years [1]. As-prepared Fe and Co based microwires differ in some of these properties. Fe microwires are observed to present magnetic bistability and rather low GMI values, while Co ones possess nearly non-hysteretic magnetization curves and higher GMI [1], making the latter a more attractive candidate for GMI related technologies. However, Co is a critical raw material, making Fe-based microwires preferable for large scale applications. Thus, search of appropriate postprocessing for magnetic softness and GMI improvement is essentially relevant. Among the prospective methods allowing these goals are stress-annealing and Joule heating. As the latter is more effective for tuning of GMI in Co-rich microwires [1], systematic studies of Joule heating on magnetic properties and GMI of Fe-rich ones are scarce. Accordingly, influence of Joule heating on magnetic properties and GMI effect of Fe75B9Si12C4 glass-coated microwires was studied. The GMI ratio, ΔZ/Z, is defined as: ΔZ/Z = [Z(H) - Z(Hmax)] / Z(Hmax) A decrease in coercivity of Joule heated samples is observed, while hysteresis loop maintain rectangular shape (Fig 1). However, a remarkable increase in ΔZ/Z is observed (Fig 2). Observed GMI ratio improvement has been discussed considering magnetic anisotropy by induced magnetic field during Joule heating and internal stress relaxation. Remarkable GMI effect improvement observed in Joule heated Fe-rich microwires, along the fact that Fe is a more common, cheap metal than Co, makes them suitable for magnetic sensor applications.

![](https://assets.underline.io/uploads/markdown_image/1/image/0c6d500e18157a87372c9dde525ab3a3.jpg)

Fig. 1. Hysteresis loops of Fe75B9Si12C4 microwires Joule heated at 20mA for different times. Inset: coercivity dependence on annealing time.

![](https://assets.underline.io/uploads/markdown_image/1/image/4f8abb050b87d288134fc3e6171b6a1e.jpg)

Fig. 2. GMI ratio of as-prepared (a) and Joule heated for 3min (b) Fe75B9Si12C4 microwires measured at different AC frequencies.

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