Multiple magnetic scattering in small-angle neutron scattering of Nd–Fe–B nanocrystalline magnet

We have investigated the influence of multiple scattering on the magnetic small-angle neutron scattering (SANS) from a Nd–Fe–B nanocrystalline magnet. We performed sample-thickness- and neutron-wavelength-dependent SANS measurements, and observed the scattering vector dependence of the multiple magnetic scattering. It is revealed that significant multiple scattering exists in the magnetic scattering rather than the nuclear scattering of Nd–Fe–B nanocrystalline magnet. It is considered that the mean free path of the neutrons for magnetic scattering is rather short in Nd–Fe–B magnets. We analysed the SANS data by the phenomenological magnetic correlation model considering the magnetic microstructures and obtained the microstructural parameters.

then the magnetic scattering intensity I mag (q) ∝ dΣ mag /dΩ(q) is obtained. Hereafter, the scattering vector q is represented as a projection q onto the detector plane, i.e. the y-z plane.

Results and Discussion
Experimental observation of magnetic multiple scattering in Nd-Fe-B nanocrystalline magnet.
Nuclear and magnetic scattering. We performed sample-thickness-and neutron wavelength-dependent experiment, to reveal the extent to which multiple scattering in a Nd-Fe-B nanocrystalline magnet. Figure 1 shows scattering intensities I(q) for the Nd-Fe-B nanocrystalline magnet in the thermally demagnetized state with different sample thickness t (t = 0.24, 0.48 and 0.90 mm) and neutron wavelength λ (λ = 0.5 and 0.81 nm) measured at room temperature. For the thermally demagnetized state, I(q) includes both the nuclear scattering I nuc (q) and the magnetic scattering I mag (q) that arises from neutrons scattered at the magnetic microstructures. Typical SANS pattern for the thermally demagnetized state is shown in the inset of Fig. 1. SANS patterns for the Nd-Fe-B nanocrystalline magnet exhibit anisotropic intensity as a result of the anisotropic shape of Nd 2 Fe 14 B grains and the anisotropic contribution of the magnetic scattering along the c-perpendicular and c-parallel directions, respectively, as we reported previously [7][8][9][10] . The scattering intensity along the c-perpendicular direction (θ = 90°) includes the magnetic scattering from the magnetic microstructures along the nominal c-axis 9 . On the other hand, the scattering intensity along the c-parallel direction (θ = 0°) includes the magnetic scattering from the spin misalignment which comes from the orientational fluctuation of the Nd 2 Fe 14 B grains 10 . Hereafter, we focus on the scattering intensity along the c-perpendicular direction, which includes a large magnetic scattering contribution from the magnetic microstructures. Absolute intensities of I(q) for λ = 0.81 nm is comparable for different thicknesses because the intensities are normalized for t. By comparing I(q) for λ = 0.81 nm for different t, the intensity is suppressed for thicker samples in the lower q region. Inflection points of I(q) for λ = 0.81 nm and for λ = 0.5 nm shift toward higher q with increasing sample thickness. These are characteristics of multiple scattering 16,26,27 . For any thickness, I(q) for λ = 0.81 nm and λ = 0.5 nm overlap in the higher q (Porod) region.
Nuclear scattering. The effect of multiple scattering on nuclear scattering was measured at elevated temperature. One can observe only nuclear scattering signals when measured above T C because Nd 2 Fe 14 B becomes paramagnetic and magnetic interaction disappears. Figure 2(a) shows the nuclear scattering intensities I nuc (q) along the c-perpendicular direction for different t (t = 0.1 and 0.5 mm) and λ (λ = 0.5, 0.81 and 1.15 nm) observed at T > T C . Nuclear scattering intensities seem to be identical for all sample thicknesses and neutron wavelengths investigated. It indicates that the multiple-scattering effect on the nuclear scattering is negligible at least in the observed q region in this experiment (0.02-0.4 nm −1 ). It is suggested that the multiple-scattering behavior observed in the ferromagnetic state shown in Fig. 1 originates from the magnetic scattering.
Magnetic scattering. The magnetic scattering intensity I mag (q) was obtained by subtracting I nuc (q) from the scattering intensity obtained in the thermally demagnetized state I(q): I mag (q) = I(q) − I nuc (q). Figure 2(b) shows I mag (q) along the c-perpendicular direction for different t (t = 0.1 and 0.5 mm) and λ (λ = 0.5, 0.81 and 1.15 nm). Magnetic scattering intensities show different q-dependences for different sample thickness and neutron wavelength. Arrows in Fig. 2(b) indicate the critical q points below which I mag (q) for t = 0.5 mm and t = 0.1 mm behave differently. Suppression of the intensity in the low q region is marked for the thicker sample measured with longer wavelength neutrons which are one characteristic of the multiple-scattering effect. These characteristics are similar to multiple nuclear scattering in the literature 26,27 , however, the shape of I mag (q) for the thin sample (t = 0.1 mm) is largely independent of the λ investigated.
Multiple magnetic scattering. Figure 3 shows the magnetic to nuclear scattering intensity ratio I mag /I nuc (q) along the c-perpendicular direction for the Nd-Fe-B nanocrystalline magnet with different sample thickness t (t = 0.1 and 0.5 mm) and neutron wavelength λ (λ = 0.5, 0.81 and 1.15 nm). It is shown that the magnetic scattering intensities are 1-5 times larger than the nuclear scattering intensities especially in the low q region (below q ~ 0.1 nm −1 ). In particular, I mag /I nuc for the thick sample (t = 0.5 mm) show maxima at specific q indicated by arrows in Fig. 3. These maxima coincide with the critical q points at which I mag (q) for t = 0.5 mm and t = 0.1 mm differ as shown in Fig. 2(b). The maximum I mag /I nuc value of I mag /I nuc ~ 5 does not depend on the neutron wavelength λ. These results suggest the most significant contribution to the multiple scattering effects in these ferromagnetic materials arises from the magnetic scattering associated with magnetic microstructures.   6). The measurement was carried out on the V4 instrument.
I nuc (q) and I mag (q) are proportional to the square of the nuclear scattering-length density (SLD) contrast (Δ ρ nuc ) 2 and that of the magnetic SLD contrast (Δ ρ mag ) 2 , respectively. We estimate (Δ ρ nuc ) 2 and (Δ ρ mag ) 2 to discuss I mag /I nuc (q). It is known that the Nd-Fe-B nanocrystalline magnet is composed of Nd 2 Fe 14 B grains and grain boundary phases containing metallic Nd-rich phase 32 . We assume metallic hcp Nd for grain boundary phase. Nuclear SLD ρ nuc for Nd 2 Fe 14 B (7.76 g/cm 3 ) and hcp Nd (7.01 g/cm 3 ) are evaluated to be 6.613 × 10 14 m −2 and 2.251 × 10 14 m −2 , respectively. The square of the nuclear SLD contrast between Nd 2 Fe 14 B and hcp Nd is estimated to be (Δ ρ nuc ) 2 ≃ 1.90 × 10 29 m −4 . The square of the magnetic SLD is given as follows 2 : The square of the magnetic SLD is about 3-times higher than that the nuclear SLD in Nd 2 Fe 14 B, however it does not explain why I mag (q) becomes up to 5-times higher than I nuc (q) as shown in Fig. 3. Therefore, it is suggested that the number of scattering events at the magnetic domain boundaries is much larger than that of the nuclear scattering at the grain boundaries.
By comparing I nuc (q) and I mag (q) for the same q range for the nuclear and magnetic scattering, it is evident that the multiple scattering more significantly affects the magnetic scattering in the Nd-Fe-B nanocrystalline magnet. The neutron MFP, i.e. the length on which the neutron beam intensity reduces to 1/e, in Nd 2 Fe 14 B (7.76 g/cm 3 ) for the wavelength of 0.5, 0.81 and 1.15 nm, are 0.85, 0.54, and 0.39 mm, respectively, when only the nuclear scattering is considered. Thus, the sample thickness of t = 0.5 mm is thinner than or comparable to the MFP at λ = 0.5 nm and λ = 0.81 nm. However, significant magnetic multiple scattering were observed in these samples. Thus, the "magnetic" MFP, which appears to be shorter than "nuclear" MFP, should also be considered. To prevent the multiple magnetic scattering as well as multiple nuclear scattering, it is necessary to prepare sufficiently thin samples to preserve the single-scattering approximation regime demonstrated in this study. However, it is noted that preparing sufficiently thin samples results in its challenges regarding maintaining sample integrity, particularly if the material is brittle. Also, as the sample thickness is reduced, the ratio of surface-to-bulk increases and surface-dependent influences on the overall structure and the influence of both magnetic and nuclear scattering should be regarded.
Multiple magnetic scattering in SANS for the Nd-Fe-B nanocrystalline magnet is clearly experimentally observed. While we have discussed its origin within the context of the scattering length density and the neutron mean free path, more detailed theoretical studies for multiple magnetic scattering are desired.

Analysis of SANS data with a phenomenological magnetic correlation model. It is essential to
prepare thin enough samples that consider not only the nuclear MFP but also the magnetic MFP to suppress the multiple scatterings. However, this may be challenging to prepare sufficiently thin samples in the actual experiment for a variety of reasons. Also, there is a finite probability of multiple scattering for any sample with finite thickness because the scattering arises from a stochastic process. Therefore, it is important to explore an applicability of an analysis method of magnetic SANS in which this is considered. Silas and Kaler reported sample-thickness-and scattering-contrast-dependent, i.e. the multiple-scattering-dependent effects in SANS in microemulsions 34 . They plotted phenomenological parameters against the relative scattering probability and they obtained certain values on the extrapolating their data to zero thickness.
We propose a simple phenomenological magnetic correlation model for Nd-Fe-B nanocrystalline magnet. The phenomenological model is derived by considering features of the maze-like magnetic domains, i.e. alternating domains of opposite magnetization direction with periodicity and the shorter-range magnetic correlation within grains. The magnetic correlation function γ(r) is given as follows: First term exp(− r/ξ) describes the magnetic correlation function from intra-grain magnetic interactions and the correlation length ξ correlates to the average radius of the Nd 2 Fe 14 B grains. The second term, the Bessel function J 0 (2πr/d), shows the magnetic domain structure originated from long-range magnetic interactions. Magnetic domains in the thermally demagnetized state of the permanent magnet materials exhibit maze-like or labyrinthine structures [35][36][37][38] and the correlation function of the labyrinthine structure are known to be J 0 (2πr/d) 35,36 where d is the periodicity. Equation (5) yields the scattering intensity as follows: where a 2 , c 1 , c 2 , and bkg (incoherent background) are parameters. The parameters d and ξ are represented by It should be noted that our magnetic correlation model for Nd-Fe-B nanocrystalline magnet is mathematically identical to the Teubner-Strey model which is usually adopted for microemulsion systems 39 . Application of the model to SANS of Nd-Fe-B nanocrystalline magnet has been reported previously 7 . We performed model fitting to the SANS data and the results of the model to I mag (q) are shown in Fig. 2(b) as solid and dotted curves.
The periodicity d and the correlation length ξ for different λ are plotted as a function of the sample thickness t in Fig. 4. For any neutron wavelength, d and ξ are smaller for the thicker sample than those for the thinner sample. Thus d and ξ are underestimated if one uses a SANS data with significant magnetic multiple scattering. Extrapolation of the linear regression of d and ξ to t → 0 yields almost the same values for different λ. The extrapolation to t = 0 should, therefore, serve as reasonable approximations for the true values of d and ξ at the single-scattering regime 34 . The correlation length at t = 0, ξ ~ 110 nm, in the present case is interpreted as the radius of the Nd 2 Fe 14 B grains, and the diameter 2ξ ~ 220 nm is consistent with the diameter of Nd 2 Fe 14 B grains of 160-300 nm which is determined by transmission electron microscopy 32 . On the other hand, the magnetic periodicity at t = 0, d ~ 420 nm, is explained by magnetic domains formed by magnetically coupled grains, i.e. so-called interaction domains 38,40 . These results indicate the applicability of the phenomenological model and the extrapolation to the zero thickness to retrieve parameters for magnetic correlation function.
In conclusion, significant multiple-scattering effects have been observed in the magnetic scattering, rather than the nuclear scattering, in a Nd-Fe-B nanocrystalline magnet. A phenomenological model fitting approach was applied to the magnetic scattering and the magnetic periodicity, d, and the correlation length, ξ, were obtained. It is revealed that the analysis yields the anticipated values for the bulk magnetic domains in the thermally demagnetized state of Nd-Fe-B nanocrystalline magnet.

Methods
Sample preparations. Nd-Fe-B nanocrystalline magnet samples were made from rapidly quenched meltspun ribbons. The melt-spun ribbons were crushed into powders of a few hundred micrometers and then sintered at 873 K under a pressure of 100 MPa. This sintered bulk was hot-deformed with a height reduction of ~80% to develop the (001) texture of the Nd 2 Fe 14 B phase. Nd 2 Fe 14 B grains are stacked along the c-directions with some degree of orientational fluctuation. Typical grain sizes are 160-300 nm and 50-110 nm in the c-perpendicular and c-parallel directions, respectively 32 . All samples were thinned to specific sample thickness, t, of between 0.1 and 0.9 mm. Samples were thermally demagnetized by heating up to 673 K (above T C = 586 K of Nd 2 Fe 14 B 41 ) in a vacuum furnace.  SANS experiments. Small-angle neutron scattering experiments were performed on the QUOKKA instrument at the OPAL research reactor at the Australian Nuclear Science and Technology Organisation (ANSTO) 42 and on the V4 instrument at the BER-II research reactor at Helmholtz-Zentrum Berlin (HZB) 43 . Figure 5 shows the schematic of the SANS experimental setup. Unpolarized neutron beams with wavelength, λ, of 0.5, 0.81 and 1.15 nm were used. The sample temperature T was set to RT and above T C of Nd 2 Fe 14 B. Data reduction was performed using the NCNR SANS reduction procedure for IGOR Pro 44 and BerSANS 45 , respectively.