Unraveling the magnetic softness in Fe-Ni-B-based nanocrystalline material by magnetic small-angle neutron scattering.

Magnetic small-angle neutron scattering is employed to investigate the magnetic interactions in (Fe0.7Ni0.3)86B14 alloy, a HiB-NANOPERM-type soft magnetic nanocrystalline material, which exhibits an ultrafine microstructure with an average grain size below 10 nm. The neutron data reveal a significant spin-misalignment scattering which is mainly related to the jump of the longitudinal magnetization at internal particle-matrix interfaces. The field dependence of the neutron data can be well described by micromagnetic small-angle neutron scattering theory. In particular, the theory explains the 'clover-leaf-type' angular anisotropy observed in the purely magnetic neutron scattering cross section. The presented neutron data analysis also provides access to the magnetic interaction parameters, such as the exchange-stiffness constant, which plays a crucial role towards the optimization of the magnetic softness of Fe-based nanocrystalline materials. © Mathias Bersweiler et al. 2022.

Introduction

Since the pioneering work of Yoshizawa et al. (1988 ▸), the development of novel Fe-based nanocrystalline soft magnetic materials raised considerable interest owing to their great potential for technological applications (Petzold, 2002 ▸; Makino et al., 1997 ▸). The most well known examples are FINEMET- (Yoshizawa et al., 1988 ▸), VITROPERM-(Vacuumschmelze GmbH, 1993 ▸) and NANOPERM-type (Suzuki et al., 1991 ▸) soft magnetic alloys, which find widespread application as magnetic cores in high-frequency power transformers or in interface transformers in the ISDN-telecommunication network. For a brief review of the advances in Fe-based nanocrystalline soft magnetic alloys, we refer the reader to the article by Suzuki et al. (2019 ▸).

More recently, an ultra-fine-grained microstructure combined with excellent soft magnetic properties was obtained in HiB-NANOPERM-type alloys (Li et al., 2020 ▸). The magnetic softness in such materials can be attributed to the exchange-averaging effect of the local magnetocrystalline anisotropy K 1. This phenomenon has been successfully modeled within the framework of the random anisotropy model (RAM) (Herzer, 1989 ▸, 1990 ▸, 2007 ▸; Suzuki et al., 1998 ▸), and becomes effective when the average grain size D is smaller than the ferromagnetic exchange length , where A ex is the exchange-stiffness constant and φ0 is a proportionality factor of the order of unity which reflects the symmetry of K 1. In this regime, the RAM predicts that the coercivity H C scales as , where n = 3 or n = 6 depending on the nature of the magnetic anisotropy [see, for example, the work by Suzuki et al. (1998 ▸, 2019 ▸) for details]. Therefore, an improvement of the magnetic softness comes about by either reducing D and/or increasing L 0.

In the context of increasing L 0, the quantitative knowledge of A ex could help to further develop novel Fe-based soft magnetic nanocrystalline materials. However, up to now, most of the research activities in this field are focused on the overall characterization, e.g. via hysteresis-loop measurements (coercivity, saturation magnetization and permeability) and magnetic anisotropy determination (crystalline, shape or stress related) (McHenry et al., 1999 ▸; Herzer, 2013 ▸; Suzuki et al., 2019 ▸). One reason for this might be related to the fact that many of the conventional methods for measuring A ex (e.g. magneto-optical, Brillouin light scattering, spin-wave resonance or inelastic neutron scattering) require thin-film or single-crystal samples.

In the present work, we employ magnetic field-dependent small-angle neutron scattering (SANS) to determine the magnetic interaction parameters in (Fe0.7Ni0.3)86B14 alloy, specifically, the exchange-stiffness constant and the strength and spatial structure of the magnetic anisotropy and magnetostatic fields. The particular alloy under study is a promising HiB-NANOPERM-type soft magnetic material, which exhibits an ultra-fine microstructure with an average grain size below 10 nm (Li et al., 2020 ▸). Magnetic SANS is a unique and powerful technique to investigate the magnetism of materials on the mesoscopic length scale of ∼1–300 nm [e.g. nanorod arrays (Grigoryeva et al., 2007 ▸; Günther et al., 2014 ▸; Maurer et al., 2014 ▸), nanoparticles (Bender et al., 2019 ▸, 2020 ▸; Bersweiler et al., 2019 ▸; Zákutná et al., 2020 ▸; Kons et al., 2020 ▸; Köhler et al., 2021 ▸), INVAR alloy (Stewart et al., 2019 ▸) or nanocrystalline materials (Ito et al., 2007 ▸; Mettus & Michels, 2015 ▸; Titov et al., 2019 ▸; Oba et al., 2020 ▸; Bersweiler et al., 2021 ▸)]. For a summary of the fundamentals and the most recent applications of the magnetic SANS technique, we refer the reader to the literature (Mühlbauer et al., 2019 ▸; Michels, 2021 ▸).

This paper is organized as follows: Section 2 provides some details of the sample characterization and the neutron experiment. Section 3 summarizes the main expressions for the magnetic SANS cross section and describes the data-analysis procedure to obtain the exchange constant and the average magnetic anisotropy field and magnetostatic field. Section 4 presents and discusses the experimental results, while Section 5 summarizes the main findings of this study.

Experimental

The ultra-rapidly annealed (Fe0.7Ni0.3)86B14 alloy (HiB-NANOPERM-type) was prepared according to the synthesis process detailed by Li et al. (2020 ▸). The sample for the neutron experiment was prepared by employing the low-capturing isotope 11B as the starting material. The average crystallite size was estimated by wide-angle X-ray diffraction (XRD) using a Bruker D8 diffractometer in Bragg–Brentano geometry (Cu Kα radiation source). The magnetic measurements were performed at room temperature using a Cryogenic Ltd vibrating sample magnetometer equipped with a 14 T superconducting magnet and a Riken Denshi BHS-40 DC hysteresis loop tracer. The crystallization and Curie temperatures were determined by means of differential thermal analysis (DTA) and thermo-magneto-gravimetric analysis (TMGA) on Perkin Elmer DTA/TGA 7 analyzers under a constant heating rate of 0.67 K s−1. For the neutron experiments, six (Fe0.7Ni0.3)86B14 ribbons with a surface area of 12 × 20 mm and a thickness of ∼15 µm were stacked together, resulting in a total sample thickness of ∼90 µm. The neutron measurements were conducted at the instrument SANS-1 at the Swiss Spallation Neutron Source at the Paul Scherrer Institute, Switzerland. We used an unpolarized incident neutron beam with a mean wavelength of λ = 6.0 Å and a wavelength broadening of Δλ/λ = 10% (full width at half-maximum). All neutron measurements were conducted at room temperature and within a q-range of about 0.036 nm−1 ≤ q ≤ 1.16 nm−1. A magnetic field H 0 was applied perpendicular to the incident neutron beam (H 0 ⊥ k 0). Neutron data were recorded by decreasing the field from the maximum field available of 8.0 to 0.02 T following the magnetization curve (see Fig. 2). The internal magnetic field H i was estimated as , where M S is the saturation magnetization and N d is the demagnetizing factor, which was determined based on the analytical expression given for a rectangular prism (Aharoni, 1998 ▸). Neutron data reduction (corrections for background scattering and sample transmission) was conducted using the GRASP software package (Dewhurst, 2018 ▸).

Micromagnetic SANS theory

Unpolarized SANS

Based on the micromagnetic SANS theory for two-phase particle–matrix-type ferromagnets developed by Honecker & Michels (2013 ▸), the elastic total (nuclear + magnetic) unpolarized SANS cross section dΣ/dΩ at momentum-transfer vector q can be formally written as (H 0 ⊥ k 0): where corresponds to the (nuclear + magnetic) residual SANS cross section, which is measured at complete magnetic saturation, and denotes the purely magnetic SANS cross section. In Equations (1)–(3), V is the scattering volume; b H = 2.91 × 108 Å−1 m−1 relates the atomic magnetic moment to the atomic magnetic scattering length; and represent the Fourier transforms of the nuclear scattering length density N(r) and of the magnetization vector field M(r), respectively; θ specifies the angle between H 0 and q ≃ q{0, sin(θ), cos(θ)} in the small-angle approximation; and the asterisks (*) denote the complex conjugated quantities. is the Fourier transform of the saturation magnetization profile M S(r), i.e. at complete magnetic saturation [compare Equation (2)]. For small-angle scattering, the component of the scattering vector along the incident neutron beam, here q, is smaller than the other two components q and q, so that only correlations in the plane perpendicular to the incoming neutron beam are probed.

In our neutron-data analysis, to experimentally access dΣmag/dΩ, we subtracted the SANS cross section dΣ/dΩ measured at the largest available field (approach-to-saturation regime; compare Fig. 2 ▸) from measured at lower fields. This specific subtraction procedure eliminates the nuclear SANS contribution , which is field independent, and therefore where Δ represents the differences of the Fourier components at the two selected fields (low field minus highest field).

Figure 2

(a) Normalized positive magnetization branch measured at room temperature (semi-logarithmic scale). Color-filled circles: M/M S values for which the SANS measurements have been performed. The approach-to-saturation regime, defined as M/M S ≥ 90%, is indicated by the red-shaded area. Inset: plot of the magnetization as a function of 1/H i (black circles). Red dashed line: linear regression for (linear–linear scale). (b) Normalized magnetization curve measured using a Riken Denshi BHS-40 DC hysteresis loop tracer, revealing a coercivity of μ0 H C ≃ 0.0049 mT (linear–linear scale).

Approach-to-saturation regime

In the particular case of the approach-to-saturation regime, where , and which implies therefore in Equation (4), dΣ/dΩ can be re-written as: where and correspond to the magnetic scattering contributions due to perturbing magnetic anisotropy fields and magnetostatic fields, respectively. More specifically, the anisotropy-field scattering function depends of the Fourier coefficient of the magnetic anisotropy field, whereas the scattering function of the longitudinal magnetization is related to the Fourier coefficient . For an inhomogeneous material of the NANOPERM-type, the latter quantity is related to the magnetization jump ΔM at internal (e.g. particle–matrix) interfaces. We would like to emphasize that the q dependence of S H and S M can often be described by a particle form factor (e.g. sphere) or a Lorentzian-squared function. The corresponding (dimensionless) micromagnetic response functions R H and R M are given by and The dimensionless function depends on the internal magnetic field H i and on the exchange length .

Estimation of the magnetic interaction parameters

Most of the time it is more convenient to analyze the (over 2π) azimuthally averaged SANS cross sections instead of the 2D ones. By performing an azimuthal average of the response functions [Equations (8) and (9)] with respect to the angle θ, i.e. , and by assuming S H and S M to be isotropic (θ-independent), the SANS cross section dΣ/dΩ can be written as: where and For a given set of parameters A ex and M S, the numerical values of R H and R M are known at each value of q and H i. Because of the linearity of Equation (10) in R H and R M, one can obtain the values of (as the intercept) and S H and S M (as the slopes) at each q-value by performing a (weighted) non-negative least-squares fit of the azimuthally averaged SANS cross sections dΣ/dΩ measured at several H i. Treating A ex in the expression for as an adjustable parameter during the fitting procedure allows us to estimate this quantity. The best-fit value for A ex is obtained from the minimization of the (weighted) mean-squared deviation between experiment and fit: where the indices μ and ν refer to the particular q and H i-values, denotes the uncertainties in the experimental data, N = N μ N ν corresponds to the number of data points, and dΣexp/dΩ and dΣsim/dΩ are the azimuthally averaged SANS cross section determined from the neutron experiments and numerically computed using Equation (10), respectively. We would like to point out that the best-fit value for A ex represents an average over the sample volume.

Finally, the numerical integration of the determined S H(q) and S M(q) over the whole-q space according to the work by Honecker & Michels (2013 ▸) yields the mean-square anisotropy field 〈∣H p∣2〉 and the mean-square longitudinal magnetization fluctuation , respectively. Since the neutron experiments are performed within a finite q-range and since both integrands do not exhibit any sign of convergence, one can only obtain a lower bound for both quantities by numerical integration. Moreover, it is important to realize that the specific neutron data analysis described above does not represent a ‘continuous’ fit of dΣ/dΩ in the conventional sense, but rather the point-by-point reconstruction of the theoretical cross sections based on the experimental data.

Results and discussion

Fig. 1 ▸ displays the wide-angle XRD results of the (Fe0.7Ni0.3)86B14 ribbons. The XRD pattern exhibits only the reflections from the f.c.c.-Fe(Ni) phase, as expected for this particular composition (Li et al., 2020 ▸), and therefore confirms the high-quality synthesis of the sample. The values of the lattice parameter a and the average crystallite size D were estimated from the XRD data refinement using the LeBail fit method (LBF) implemented in the FullProf suite (Rodríguez-Carvajal, 1993 ▸). The best-fit values are summarized in Table 1 ▸. Both values are consistent with the data in the literature [compare the work by Anand et al. (2019 ▸) and Li et al. (2020 ▸) for a and D, respectively]. As previously discussed, the origin of the exceptionally fine microstructure observed in (Fe0.7Ni0.3)86B14 alloys may be qualitatively attributed to the ultrafast nucleation kinetics of the f.c.c.-Fe(Ni) phase (Li et al., 2020 ▸).

Figure 1

XRD pattern for (Fe0.7Ni0.3)86B14 ribbons, a HiB-NANOPERM-type soft magnetic nanocrystalline material (black crosses; Cu Kα radiation). Red solid line: XRD data refinement using the LBF method implemented in the FullProf software. The bottom orange solid line represents the difference between the calculated and experimental intensities.

Table 1

Summary of the structural and magnetic parameters for (Fe0.7Ni0.3)86B14 alloy (HiB-NANOPERM-type soft magnetic nanocrystalline material) determined by wide-angle XRD, magnetometry, DTA, TMGA and SANS

Parameter	(Fe0.7Ni0.3)86B14 alloy
a (nm)	∼0.359
D (nm)	7 ± 1
μ0 M S (T)	1.34 ± 0.20
μ0 H C (mT)	∼0.0049
T_{\rm C}^{\rm am} (K)	720
A ex pJ m−1	10 ± 1
ξM (nm)	2.4 ± 0.2
L 0 (nm)	∼50
μ0〈∣H p∣2〉1/2 (mT)	∼0.3
μ0〈∣M z ∣2〉1/2 (mT)	∼24

Fig. 2 ▸(a) presents the positive magnetization branch on a semi-logarithmic scale (measured at room temperature), while the hysteresis loop on a linear–linear scale, and between ±0.03 mT, is displayed in Fig. 2 ▸(b). The data have been normalized by the saturation magnetization M S, which was estimated from the linear regression for [see inset in Fig. 2 ▸(a)]. The values of M S and H C (see Table 1 ▸) are in agreement with those reported in the literature (Li et al., 2020 ▸). Defining the approach-to-saturation regime by M/M S ≥ 90%, we can see that this regime is reached for μ0 H i ≳ 65 mT. Moreover, the extremely small value for H C combined with the high M S confirms the huge potential of (Fe0.7Ni0.3)86B14 alloy as a soft magnetic material, and suggests that in the framework of the RAM (Herzer, 2007 ▸), H C should fall into the regime where (Suzuki et al., 2019 ▸).

Fig. 3 ▸ shows the DTA and TMGA curves for the amorphous (Fe0.7Ni0.3)86B14 alloy. Two exothermic peaks are evident on the DTA curve reflecting the well known two-stage reactions, where f.c.c.-Fe(Ni) forms at the first peak followed by decomposition of the residual amorphous phase at the second peak. The sharp drop of the TMGA signal just before the second stage crystallization corresponds to the Curie temperature of the residual amorphous phase ( ≃ 720 K). This value, which reflects the exchange integral in our sample (see below), is consistent with those determined for amorphous Fe86B14 samples prepared under similar conditions (Zang et al., 2020 ▸).

Figure 3

Results of DTA (red solid line) and TMGA (blue solid line) for amorphous (Fe0.7Ni0.3)86B14 alloy. The arrows mark the crystallization and Curie temperatures.

Fig. 4 ▸ (upper row) shows the experimental 2D total (nuclear + magnetic) SANS cross sections dΣ/dΩ of the (Fe0.7Ni0.3)86B14 ribbons at different selected fields. As can be seen, at μ0 H i = 7.99 T (near saturation), the pattern is predominantly elongated perpendicular to the magnetic field direction. This particular feature in dΣ/dΩ is the signature of the so-called ‘ -type’ angular anisotropy [compare Equation (2)]. Near saturation, the magnetic scattering resulting from the spin misalignment is small compared with that resulting from the longitudinal magnetization jump at the internal (e.g. particle–matrix) interfaces. By reducing the field, the patterns remain predominantly elongated perpendicular to the magnetic field, but at the smaller momentum transfers q an additional field-dependent signal is observed ‘roughly’ along the diagonals of the detector, suggesting a more complex magnetization structure. Fig. 4 ▸ (middle row) presents the corresponding 2D purely magnetic SANS cross sections dΣmag/dΩ determined by subtracting dΣ/dΩ at μ0 H i = 7.99 T from the data at lower fields. In this way, the maxima along the diagonals of the detector become more clearly visible, thereby revealing the so-called ‘clover-leaf-type’ angular anisotropy pattern. This particular feature was also previously observed in NANOPERM-type soft magnetic merials (Honecker et al., 2013 ▸), and is related to the dominant magnetostatic term S M × R M in the expression for dΣmag/dΩ [compare Equations (8) and (9)]. More specifically, the jump in the magnitude of the saturation magnetization at the particle–matrix interfaces, which can be of the order of 1 T in these type of alloys (Honecker et al., 2013 ▸), results in dipolar stray fields which produce spin disorder in the surroundings. Fig. 4 ▸ (lower row) displays dΣmag/dΩ computed using the micromagnetic SANS theory [Equations (5)–(9)] and the experimental parameters summarized in Table 1 ▸. As is seen, the clover-leaf-type angular anisotropy experimentally observed in Fig. 4 ▸ (middle row) can be well reproduced using micromagnetic theory.

Figure 4

Experimental 2D total (nuclear + magnetic) SANS cross section dΣ/dΩ of (Fe0.7Ni0.3)86B14 alloy at the selected fields 7.99, 2.99, 0.59, 0.29 T (upper row), and the corresponding purely magnetic SANS cross section dΣmag/dΩ (middle row). Experimental dΣmag/dΩ were obtained by subtracting dΣ/dΩ at the (near-) saturation field of 7.99 T from the data at the lower fields. The applied (internal) magnetic field H i is horizontal in the plane of the detector . Lower row: computed dΣmag/dΩ based on the micromagnetic SANS theory [Equations (5)–(9)] at the same selected field values as above, and using the experimental parameters given in Table 1 ▸. Note that dΣ/dΩ and dΣmag/dΩ are plotted in polar coordinates with q (nm−1), θ (°) and intensity (cm−1).

Fig. 5 ▸(a) displays the (over 2π) azimuthally averaged dΣ/dΩ, while the corresponding dΣmag/dΩ are shown in Fig. 5 ▸(b). By decreasing μ0 H i from 7.99 T to 10 mT, the intensity of dΣ/dΩ increases by almost two orders of magnitude at the smallest momentum transfers q. By comparison to Equations (1)–(4), it appears obvious that the magnetic field dependence of dΣ/dΩ can only result from the mesoscale spin disorder (i.e. from the failure of the spins to be fully aligned along H 0). As is seen in Fig. 5 ▸(b), the magnitude of dΣmag/dΩ is of the same order as dΣ/dΩ, supporting the notion of dominant spin-misalignment scattering in (Fe0.7Ni0.3)86B14 alloy.

Figure 5

(a) Magnetic field dependence of the (over 2π) azimuthally averaged total (nuclear + magnetic) SANS cross section dΣ/dΩ of (Fe0.7Ni0.3)86B14 alloy. (b) The corresponding purely magnetic SANS cross section dΣmag/dΩ (log–log scale).

Fig. 6 ▸ shows the magnetic SANS results determined from the field-dependent approach described in Section 3.3. In the present case, to warrant the validity of the micromagnetic SANS theory, only dΣ/dΩ measured for μ0 H i ≳ 65 mT (i.e. within the approach-to-saturation regime, compare Fig. 2 ▸) were considered. We have also restricted our neutron data analysis to , since the magnetic SANS cross section is expected to be field-independent for q ≥ q max (Michels, 2021 ▸). In Fig. 6 ▸(a), we plot the (over 2π) azimuthally averaged dΣ/dΩ along with the corresponding fits based on the micromagnetic SANS theory [Equation (10), black solid lines]. It is seen that the field dependence of dΣ/dΩ over the restricted q-range can be well reproduced by the theory. Fig. 6 ▸(b) displays the (weighted) mean-squared deviation between experiment and fit, χ2, determined according to Equation (13), as a function of the exchange-stiffness constant A ex. In this way, we find A ex = (10 ± 1) pJ m−1 (see Table 1 ▸). The comparison with previous studies is discussed in the next paragraph for more clarity. Fig. 6 ▸(c) displays the best-fit results for dΣres/dΩ, S H and S M. Not surprisingly, the magnitude of dΣres/dΩ (limit of dΣ/dΩ at infinite field) is smaller than the dΣ/dΩ at the largest fields [compare Fig. 6 ▸(a)], supporting the validity of the micromagnetic SANS theory. Furthermore, the magnitude of S H is about two orders of magnitude smaller than S M, suggesting that the magnetization jump ΔM at internal particle–matrix interfaces represents the main source of spin disorder in this material. The estimated values for the mean-square anisotropy field and the mean-square magnetostatic field in terms of Equation (14) are 0.3 and 24 mT, respectively. These values qualitatively support the notion of dominant spin-misalignment scattering due to magnetostatic fluctuations. The q-dependence of S M can be described using a Lorentzian-squared function [blue solid line in Fig. 6 ▸(c)] from which an estimate for the magnetostatic correlation length ξM = 2.4 ± 0.2 nm is obtained. This value compares favorably with the value of l M = (2A ex/μ0 M S 2)1/2 = 3.7 nm [using A ex = 10 pJ m−1 and μ0 M S = 1.34 T (taken from Table 1 ▸)], which reflects the competition between the exchange and magnetostatic energies.

Figure 6

Results of the SANS data analysis of (Fe0.7Ni0.3)86B14 alloy. (a) Magnetic field dependence of the (over 2π) azimuthally averaged total (nuclear + magnetic) SANS cross section dΣ/dΩ plotted in Fig. 5 ▸(a) along with the corresponding fits (black solid lines) based on the micromagnetic SANS theory [Equation (10)]. (b) Weighted mean-squared deviation between experiment and fit, χ2, determined using Equation (13) as a function of the exchange-stiffness constant A ex. Inset: Fe-composition dependence of the magnetocrystalline anisotropy K 1 in Fe1−Ni alloys [data taken from the literature (Tarasov, 1939 ▸; Hall, 1960 ▸)]. Black dashed line: linear regression of K 1(x). (c) Best-fit results for the residual scattering cross section dΣres/dΩ (red diamonds), the scattering function S H (orange open circles) and S M (blue open circles). Blue solid line: fit of S M assuming a Lorentzian-squared function for the q-dependence.

We would like to emphasize that our experimental value for A ex = 10 pJ m−1 is about 2–3 times larger than those reported in NANOPERM-type soft magnetic materials (Honecker et al., 2013 ▸). Since the Curie temperature of the residual amorphous phase in our nanocrystalline (Fe0.7Ni0.3)86B14 sample is well above 700 K (see Fig. 3 ▸ and Table 1 ▸), while that of the Fe89Zr7B3Cu1 sample used in the previous study (Honecker et al., 2013 ▸) was as low as 350 K, the local exchange stiffness in the grain boundary amorphous phase in HiB-NANOPERM-type alloys is expected to be higher than that in NANOPERM-type alloys. This finding could explain the origin of the larger A ex value reported in the present study. Therefore, one can expect an improvement of the magnetic softness in HiB-NANOPERM thanks to the ensuing increase of the ferromagnetic exchange length L 0. It is well established that nonmagnetic and/or ferromagnetic additives and the annealing conditions strongly affect the microstructural and magnetic properties of Fe-based nanocrystalline materials (McHenry et al., 1999 ▸; Herzer, 2007 ▸, 2013 ▸; Suzuki et al., 2019 ▸) and therefore have a strong impact on their magnetic softness. Using A ex = 10 pJ m−1 (this study), K 1 ≃ 9.0 kJ m−3,1 and φ0 ≃ 1.5 (Herzer, 2007 ▸), we obtain L 0 ≃ 50 nm. This value for L 0 is in very good agreement with the typical length scale of ∼30–50 nm previously reported in soft magnetic Fe-based alloys. Moreover, the comparison of the average grain size D = 7 nm with the L 0 value, here D ≪ L 0, also confirms that in the framework of the random anisotropy model (Herzer, 1989 ▸, 1990 ▸, 2007 ▸; Suzuki et al., 1998 ▸), the exchange-averaged magnetic anisotropy 〈K〉 falls into the regime where 〈K〉 ∝ D 3. This finding is also consistent with the (experimental) D 3-dependence of H C reported in Fe–B-based HiB-NANOPERM alloys (Suzuki et al., 2019 ▸; Li et al., 2020 ▸).

Conclusions

We employed magnetic SANS to determine the magnetic interaction parameters in (Fe0.7Ni0.3)86B14 alloy, which is a HiB-NANOPERM-type soft magnetic material. The analysis of the magnetic SANS data suggests the presence of strong spin misalignment on a mesoscopic length scale. In fact, the micromagnetic SANS theory provides an excellent description of the field dependence of the total (nuclear + magnetic) and purely magnetic SANS cross sections. The clover-leaf-type angular anisotropy patterns observed in the magnetic SANS signal can be well reproduced by the theory. The magnitudes of the scattering functions S H and S M allow us to conclude that the magnetization jumps at internal particle–matrix interfaces and the ensuing dipolar stray fields are the main source of the spin-disorder in this material. Our study highlights the strength of the magnetic SANS technique to characterize magnetic materials on the mesoscopic length scale. The structural and magnetic results (summarized in Table 1 ▸) provide valuable information on the (Fe0.7Ni0.3)86B14 ribbons, and further confirm the strong potential of Fe–Ni–B-based HiB-NANOPERM-type alloys as soft magnetic nanocrystalline materials. In the context of the random anisotropy model, we demonstrated that the magnetic softness in this system can be attributed to the combined action of the small particle size (D = 7 nm) and an increased exchange constant (A ex = 10  pJ m−1) resulting in an enhanced exchange correlation length L 0.

The data that support the findings of this study are available from the corresponding author upon reasonable request.