We experimentally report a hitherto unseen angular anisotropy in the polarized small-angle neutron scattering (SANS) cross section of a magnetically strongly inhomogeneous material. Based on an analytical prediction using micromagnetic theory, the difference between the spin-up and spin-down SANS cross sections is expected to show a spin-disorder-induced anisotropy. The effect is particularly pronounced in inhomogeneous magnetic materials, such as nanoporous ferromagnets, magnetic nanocomposites, or steels, which exhibit large nanoscale jumps in the saturation magnetization at internal pore-matrix or particle-matrix interfaces. Analysis of the experimental neutron data constitutes a method for determining the exchange-stiffness constant. Our results for the nuclear-magnetic interference terms contained in the polarized magnetic neutron scattering cross section might also be of relevance to other neutron techniques.

Taking advantage of recent progress in neutron instrumentation and in the understanding of magnetic-field-dependent small-angle neutron scattering, here, we study the three-dimensional magnetization distribution within an isotropic Nd-Fe-B bulk magnet. The magnetic neutron scattering cross section of this system features the so-called spike anisotropy, which points towards the presence of a strong magnetodipolar interaction. This experimental result combined with a damped oscillatory behavior of the corresponding correlation function and recent micromagnetic simulation results on spherical nanoparticles suggest an interpretation of the neutron data in terms of vortex-like flux-closure patterns. The field-dependent correlation length Lc is well reproduced by a phenomenological power-law model. While the experimental neutron data for Lc are described by an exponent close to unity (p = 0.86), the simulation results yield p = 1.70, posing a challenge to theory to include vortex-vortex interaction effects.

We employ micromagnetic simulations to model the effect of pore-type microstructural defects on the magnetic small-angle neutron scattering cross section and the related pair-distance distribution function of spherical magnetic nanoparticles. Our expression for the magnetic energy takes into account the isotropic exchange interaction, the magnetocrystalline anisotropy, the dipolar interaction, and an externally applied magnetic field. The signatures of the defects and the role of the dipolar energy are highlighted and the effect of a particle-size distribution is studied. The results serve as a guideline to the experimentalist.

The development of higher-order micromagnetic small-angle neutron scattering theory in nanocrystalline materials is still in its infancy. One key challenge remaining in this field is understanding the role played by the microstructure on the magnitude and sign of the higher-order scattering contribution recently observed in nanocrystalline materials prepared by high-pressure torsion. By combining structural and magnetic characterization techniques, namely X-ray diffraction, electron backscattered diffraction and magnetometry with magnetic small-angle neutron scattering, this work discusses the relevance of higher-order terms in the magnetic small-angle neutron scattering cross section of pure iron prepared by high-pressure torsion associated with a post-annealing process. The structural analysis confirms: (i) the preparation of ultra-fine-grained pure iron with a crystallite size below 100 nm and (ii) rapid grain growth with increasing annealing temperature. The analysis of neutron data based on the micromagnetic small-angle neutron scattering theory extended to textured ferromagnets yields uniaxial magnetic anisotropy values that are larger than the magnetocrystalline value reported for bulk iron, supporting the existence of induced magnetoelastic anisotropy in the mechanically deformed samples. Furthermore, the neutron data analysis revealed unambiguously the presence of non-negligible higher-order scattering contributions in high-pressure torsion iron. Though the sign of the higher-order contribution might be related to the amplitude of the anisotropy inhomogeneities, its magnitude appears to be clearly correlated to the changes in the microstructure (density and/or shape of the defects) induced by combining high-pressure torsion and a post-annealing treatment.

The MATLAB-based software tool MuMag2022 is presented for the analysis of magnetic-field-dependent unpolarized small-angle neutron scattering (SANS) data of bulk ferromagnets such as elemental nanocrystalline ferromagnets, magnetic nanocomposites or magnetic steels. On the basis of the micromagnetic theory for the magnetic SANS cross section, the program analyzes unpolarized total (nuclear and magnetic) SANS data within the approach-to-saturation regime. The main features of MuMag2022 are the estimation of the exchange-stiffness constant, and of the strength and spatial structure of the magnetic anisotropy field and the magnetostatic field due to longitudinal magnetization fluctuations. MuMag2022 is open source and available as a standalone executable for Windows at https://mumag.uni.lu.

Magnetic nanoparticles offer unique potential for various technological, biomedical, or environmental applications thanks to the size-, shape- and material-dependent tunability of their magnetic properties. To optimize particles for a specific application, it is crucial to interrelate their performance with their structural and magnetic properties. This review presents the advantages of small-angle X-ray and neutron scattering techniques for achieving a detailed multiscale characterization of magnetic nanoparticles and their ensembles in a mesoscopic size range from 1 to a few hundred nanometers with nanometer resolution. Both X-rays and neutrons allow the ensemble-averaged determination of structural properties, such as particle morphology or particle arrangement in multilayers and 3D assemblies. Additionally, the magnetic scattering contributions enable retrieving the internal magnetization profile of the nanoparticles as well as the inter-particle moment correlations caused by interactions within dense assemblies. Most measurements are used to determine the time-averaged ensemble properties, in addition advanced small-angle scattering techniques exist that allow accessing particle and spin dynamics on various timescales. In this review, we focus on conventional small-angle X-ray and neutron scattering (SAXS and SANS), X-ray and neutron reflectometry, gracing-incidence SAXS and SANS, X-ray resonant magnetic scattering, and neutron spin-echo spectroscopy techniques. For each technique, we provide a general overview, present the latest scientific results, and discuss its strengths as well as sample requirements. Finally, we give our perspectives on how future small-angle scattering experiments, especially in combination with micromagnetic simulations, could help to optimize the performance of magnetic nanoparticles for specific applications.

The small-angle neutron scattering data of nanostructured magnetic samples contain information regarding their chemical and magnetic properties. Often, the first step to access characteristic magnetic and structural length scales is a model-free investigation. However, due to measurement uncertainties and a restricted q range, a direct Fourier transform usually fails and results in ambiguous distributions. To circumvent these problems, different methods have been introduced to derive regularized, more stable correlation functions, with the indirect Fourier transform being the most prominent approach. Here, the indirect Fourier transform is compared with the singular value decomposition and an iterative algorithm. These approaches are used to determine the correlation function from magnetic small-angle neutron scattering data of a powder sample of iron oxide nanoparticles; it is shown that with all three methods, in principle, the same correlation function can be derived. Each method has certain advantages and disadvantages, and thus the recommendation is to combine these three approaches to obtain robust results.

Volume 33 • Number 2 • 2022 15 Science Snapshot Resolving the Complex Spin Structure in FeBased Soft Magnetic Nanocrystalline Material by Magnetic Small-Angle Neutron Scattering Mathias Bersweiler1, Michael P. Adams1, Inma Peral1, Joachim Kohlbrecher2, Kiyonori Suzuki3, and Andreas Michels1 1 Department of Physics and Materials Science, University of Luxembourg, Luxembourg, Grand Duchy of Luxembourg 2 Laboratory for Neutron Scattering, ETH Zurich & Paul Scherrer Institut, Villigen, Switzerland 3 Department of Materials Science and Engineering, Monash University, Clayton, VIC, Australia

Shell ferromagnetism is a new functional property of certain off-stoichiometric Ni-Mn-In Heusler alloys, with a potential application in non-volatile magnetic memories and recording media. One key challenge in this field remains the determination of the structural and magnetic properties of the nanoprecipitates that are the result of an annealing-induced segregation process. Thanks to its unique mesoscopic length scale sensitivity, magnetic small-angle neutron scattering appears to be a powerful technique to disclose the microstructure of such annealing-induced nanoprecipitates. In this study, the microstructure of a zero-field-annealed off-stoichiometric Ni50Mn45In5 Heusler alloy is investigated by unpolarized magnetic small-angle neutron scattering. The neutron data analysis reveals a significant spin-misalignment scattering, which is mainly related to the formation of annealing-induced ferromagnetic nanoprecipitates in an antiferromagnetic matrix. These particles represent a source of perturbation which, due to dipolar stray fields, gives rise to canted spin moments in the surroundings of the particle-matrix interface. The presence of anticorrelations in the computed magnetic correlation function reflects the spatial perturbation of the magnetization vector around the nanoprecipitates. The magnetic field dependence of the zero crossing and the minima of the magnetic correlation function are qualitatively explained using the law of approach to ferromagnetic saturation for inhomogeneous spin states. More specifically, at remanence, the nanoprecipitates act magnetically as one superdefect with a correlation length that lies outside the experimental q range, whereas near saturation the magnetization distribution follows each individual nanoprecipitate. Analysis of the neutron data yields an estimated size of 30 nm for the spin-canted region and a value of about 75 nm for the magnetic core of the individual nanoprecipitates.

We report the results of an unpolarized small-angle neutron scattering (SANS) study on Mn-Bi-based rare-earth-free permanent magnets. The magnetic SANS cross section is dominated by long-wavelength transversal magnetization fluctuations and has been analyzed in terms of the Guinier-Porod model and the distance distribution function. This provides the radius of gyration which, in the remanent state, ranges between about $220-240 \, \mathrm{nm}$ for the three different alloy compositions investigated. Moreover, computation of the distance distribution function in conjunction with results for the so-called $s$-parameter obtained from the Guinier-Porod model indicate that the magnetic scattering of a Mn$_{45}$Bi$_{55}$ sample has its origin in slightly shape-anisotropic structures.

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. 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.

Magnetic field dependent unpolarized small-angle neutron scattering (SANS) experiments demonstrate that high-pressure torsion (HPT) straining induces spin misalignments in pure Ni, which persist in magnetic fields up to 4 T. The spin-misalignment scattering patterns are elongated perpendicular to the applied magnetic field due to an unusual predominant longitudinal sin(2)theta-type angular anisotropy. Such a contribution cannot be explained by the conventional second-order (in spin-misalignment amplitude) micromagnetic SANS theory in the approach-to-saturation regime, nor can its magnitude relative to the other features of the cross sections be explained by the third-order micromagnetic SANS theory. This indicates that the high density of crystal defects induced via HPT straining in Ni makes such higher-order effects in the micromagnetic SANS cross sections observable.

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.

In the quest to image the three-dimensional magnetization structure we show that the technique of magnetic small-angle neutron scattering (SANS) is highly sensitive to the details of the internal spin structure of nanoparticles. By combining SANS with numerical micromagnetic computations we study the transition from single-domain to multidomain behavior in nanoparticles and its implications for the ensuing magnetic SANS cross section. Above the critical single-domain size we find that the cross section and the related correlation function cannot be described anymore with the uniform particle model, resulting, e.g., in deviations from the well-known Guinier law. In the simulations we identify a clear signature for the occurrence of a vortexlike spin structure at remanence. The micromagnetic approach to magnetic SANS bears great potential for future investigations, since it provides fundamental insights into the mesoscale magnetization profile of nanoparticles.

In the quest to image the three-dimensional magnetization structure we show that the technique of magnetic small-angle neutron scattering (SANS) is highly sensitive to the details of the internal spin structure of nanoparticles. By combining SANS with numerical micromagnetic computations we study the transition from single-domain to multi-domain behavior in nanoparticles and its implications for the ensuing magnetic SANS cross section. Above the critical single-domain size we find that the cross section and the related correlation function cannot be described anymore with the uniform particle model, resulting e.g. in deviations from the well-known Guinier law. We identify a clear signature for the occurrence of a vortex-like spin structure at remanence. The micromagnetic approach to magnetic SANS bears great potential for future investigations, since it provides fundamental insights into the mesoscale magnetization profile of nanoparticles.

We investigated a polycrystalline sample of the ferrimagnetic compound Tb_0.15Co_0.85 by magnetometry and small-angle neutron scattering (SANS). The magnetization curve at 300 K is characteristic for soft ferrimagnets but at 5 K the hysteresis indicates the existence of magnetic domains. The magnetic SANS signal suggests that at 300 K the Tb and Co moments are correlated over large volumes within the micrometer-sized grains with correlation lengths > 100 nm. At 5 K, however, the magnetic SANS analysis reveals a reduced correlation length of around 4.5 nm, which indicates the formation of narrow magnetic domains within the ferrimagnet with one dimension being in the nm range. We attribute the observed changes of the domain structure to the temperature-dependence of the magnetic properties of the Tb sublattice.

One of the key challenges in magnetism remains the determination of the nanoscopic magnetization profile within the volume of thick samples, such as permanent ferromagnets. Thanks to the large penetration depth of neutrons, magnetic small‐angle neutron scattering (SANS) is a powerful technique to characterize bulk samples. The major challenge regarding magnetic SANS is accessing the real‐space magnetization vector field from the reciprocal scattering data. In this study, a fast iterative algorithm is introduced that allows one to extract the underlying 2D magnetic correlation functions from the scattering patterns. This approach is used here to analyze the magnetic microstructure of Nanoperm, a nanocrystalline alloy which is widely used in power electronics due to its extraordinary soft magnetic properties. It can be shown that the computed correlation functions clearly reflect the projection of the 3D magnetization vector field onto the detector plane, which demonstrates that the used methodology can be applied to probe directly spin textures within bulk samples with nanometer resolution.

Magnetic nanoparticles offer a unique potential for various biomedical applications, but prior to commercial usage a standardized characterization of their structural and magnetic properties is required. For a thorough characterization, the combination of conventional magnetometry and advanced scattering techniques has shown great potential. In the present work, we characterize a powder sample of high-quality iron oxide nanoparticles that are surrounded with a homogeneous thick silica shell by DC magnetometry and magnetic small-angle neutron scattering (SANS). To retrieve the particle parameters such as their size distribution and saturation magnetization from the data, we apply standard model fits of individual data sets as well as global fits of multiple curves, including a combination of the magnetometry and SANS measurements. We show that by combining a standard least-squares fit with a subsequent Bayesian approach for the data refinement, the probability distributions of the model parameters and their cross correlations can be readily extracted, which enables a direct visual feedback regarding the quality of the fit. This prevents an overfitting of data in case of highly correlated parameters and renders the Bayesian method as an ideal component for a standardized data analysis of magnetic nanoparticle samples.

Mathias Bersweiler ,1 Philipp Bender ,1 Laura G. Vivas,1 Martin Albino,2 Michele Petrecca ,2,3 Sebastian Mühlbauer,4 Sergey Erokhin ,5 Dmitry Berkov ,5 Claudio Sangregorio ,2,3 and Andreas Michels 1 1Physics and Materials Science Research Unit, University of Luxembourg, 162A Avenue de la Faïencerie, L-1511 Luxembourg, Grand Duchy of Luxembourg 2Università degli Studi di Firenze, Dipartimento di Chimica “U. Schiff,” Via della Lastruccia 3, 50019 Sesto Fiorentino, Italy 3ICCOM-CNR via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy 4Heinz Maier-Leibnitz Zentrum (MLZ), Technische Universität München, D-85748 Garching, Germany 5General Numerics Research Lab, An der Leite 3B, D-07749 Jena, Germany

We report the results of an unpolarized small-angle neutron-scattering (SANS) study on Mn-Zn ferrite (MZFO) magnetic nanoparticles with the aim to elucidate the interplay between their particle size and the magnetization configuration. We study different samples of single-crystalline MZFO nanoparticles with average diameters ranging between 8 to 80 nm, and demonstrate that the smallest particles are homogeneously magnetized. However, with increasing nanoparticle size, we observe the transition from a uniform to a nonuniform magnetization state. Field-dependent results for the correlation function confirm that the internal spin disorder is suppressed with increasing field strength. The experimental SANS data are supported by the results of micromagnetic simulations, which confirm an increasing inhomogeneity of the magnetization profile of the nanoparticle with increasing size. The results presented demonstrate the unique ability of SANS to detect even very small deviations of the magnetization state from the homogeneous one.