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 (Fe_0.7Ni_0.3)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.

1. 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 K₁. 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 K₁. 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 L₀.

In the context of increasing L₀, 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 (Fe_0.7Ni_0.3)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.

2. Experimental

The ultra-rapidly annealed (Fe_0.7Ni_0.3)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 (Fe_0.7Ni_0.3)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 H₀ was applied perpendicular to the incident neutron beam (H₀ ⊥ k₀). 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).

3. Micromagnetic SANS theory

3.1. 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 (H₀ ⊥ k₀):

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 H₀ 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_x, is smaller than the other two components q_y and q_z, so that only correlations in the plane perpendicular to the incoming neutron beam are probed.

In our neutron-data analysis, to experimentally access dΣ_mag/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/MS values for which the SANS measurements have been performed. The approach-to-saturation regime, defined as M/MS ≥ 90%, is indicated by the red-shaded area. Inset: plot of the magnetization as a function of 1/Hi (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 μ0HC ≃ 0.0049 mT (linear–linear scale).

3.2. 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 .

3.3. 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_μN_ν corresponds to the number of data points, and dΣ^exp/dΩ and dΣ^sim/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.

4. Results and discussion

Fig. 1 displays the wide-angle XRD results of the (Fe_0.7Ni_0.3)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 (Fe_0.7Ni_0.3)₈₆B₁₄ alloys may be qualitatively attributed to the ultrafast nucleation kinetics of the f.c.c.-Fe(Ni) phase (Li et al., 2020).

Table 1
Summary of the structural and magnetic parameters for (Fe_0.7Ni_0.3)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
μ0MS (T)	1.34 ± 0.20
μ0HC (mT)	∼0.0049
(K)	720
Aex pJ m^−1	10 ± 1
ξM (nm)	2.4 ± 0.2
L0 (nm)	∼50
μ0〈∣Hp∣^2〉^1/2 (mT)	∼0.3
μ0〈∣Mz∣^2〉^1/2 (mT)	∼24

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.

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 μ₀H_i ≳ 65 mT. Moreover, the extremely small value for H_C combined with the high M_S confirms the huge potential of (Fe_0.7Ni_0.3)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 (Fe_0.7Ni_0.3)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 (Fe_0.7Ni_0.3)₈₆B₁₄ ribbons at different selected fields. As can be seen, at μ₀H_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 dΣ_mag/dΩ determined by subtracting dΣ/dΩ at μ₀H_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 dΣ_mag/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 dΣ_mag/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 Hi 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 dΣ_mag/dΩ are shown in Fig. 5(b). By decreasing μ₀H_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 H₀). As is seen in Fig. 5(b), the magnitude of dΣ_mag/dΩ is of the same order as dΣ/dΩ, supporting the notion of dominant spin-misalignment scattering in (Fe_0.7Ni_0.3)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 μ₀H_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 dΣ_res/dΩ, S_H and S_M. Not surprisingly, the magnitude of dΣ_res/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 ξ_M = 2.4 ± 0.2 nm is obtained. This value compares favorably with the value of l_M = (2A_ex/μ₀M_S²)^1/2 = 3.7 nm [using A_ex = 10 pJ m⁻¹ and μ₀M_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, χ^2, determined using Equation (13) as a function of the exchange-stiffness constant Aex. Inset: Fe-composition dependence of the magnetocrystalline anisotropy K1 in Fe1−xNix alloys [data taken from the literature (Tarasov, 1939; Hall, 1960)]. Black dashed line: linear regression of K1(x). (c) Best-fit results for the residual scattering cross section dΣres/dΩ (red diamonds), the scattering function SH (orange open circles) and SM (blue open circles). Blue solid line: fit of SM 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 (Fe_0.7Ni_0.3)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 L₀. 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), K₁ ≃ 9.0 kJ m⁻³,¹ and φ₀ ≃ 1.5 (Herzer, 2007), we obtain L₀ ≃ 50 nm. This value for L₀ 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 L₀ value, here D ≪ L₀, 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〉 ∝ D³. This finding is also consistent with the (experimental) D³-dependence of H_C reported in Fe–B-based HiB-NANOPERM alloys (Suzuki et al., 2019; Li et al., 2020).

5. Conclusions

We employed magnetic SANS to determine the magnetic interaction parameters in (Fe_0.7Ni_0.3)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 (Fe_0.7Ni_0.3)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 L₀.

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

Footnotes

¹Estimated by assuming a linear regression of K₁ in Fe_1−xNi_x alloys for an Fe composition x between 0 and 0.4 at% [see inset in Fig. 6(b), data taken from the literature (Tarasov, 1939; Hall, 1960)].