Using small-angle scattering to guide functional magnetic nanoparticle design

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.

Dr Dirk Honecker is instrument scientist at the ISIS Neutron and Muon Source (UK) since 2020. Before joining ISIS, he conducted a research stay on analysing magnetic nanostructures with small-angle scattering and micromagnetic methods in the nanomagnetism group, University of Luxembourg, and he was employed to facilitate and expand studies on magnetism at the massive dynamic q-range small-angle diffractometer of the Institut Laue-Langevin, France. A few examples include magnetic nanowire arrays, the magnetisation distribution in nanostructured alloys, the magnetic disorder within nanoparticles, and the interparticle coupling in clusters.
Dr Philipp Bender received his PhD in physics in 2013 from the Saarland University (Germany). Aer working as a postdoctoral researcher at the Saarland University, he went in 2015 to the Universidad de Cantabria to participate within the EU project Nanomag, and in 2018 he joined the University of Luxembourg as a research scientist. In 2020, he started working as an instrument scientist at the neutron spin-echo spectrometer RESEDA at the Heinz Maier-Leibnitz Zentrum (MLZ), Germany. His research focused on the application of magnetic nanoparticles for biomedicine and their characterization by magnetometry and magnetic small-angle neutron scattering.

Introduction
Magnetic nanoparticles (MNPs) possess unique tunable properties and can be manipulated (e.g. moved or rotated) by external magnetic elds, making them ideal candidates for various technical, biomedical, and energy applications. For each application, the employed particles must fulll specic requirements, especially regarding their magnetic properties. The magnetism of MNPs can be controlled by various parameters including their morphology, chemical composition, and arrangement. Some prominent technological applications are their usage in data-storage devices 1 or ferrouids, 2 which are stable and dense colloidal suspensions of MNPs used for seals and dampers. Ferrouids rely on superparamagnetic particles stabilized by surfactants that create (reversible) anisotropic aggregates in a magnetic eld, whereas the particles envisioned for data-storage devices need strong magnetic anisotropy leading to high coercivities to guarantee thermal stability. Recently, Liu et al. 3 demonstrated the reversible paramagneticto-ferromagnetic transition by jamming MNPs at oil-water interfaces producing congurable and permanently magnetized emulsion droplets. This concept can be applied, e.g., to synthesize magneto-responsive nematic liquid crystals. 4 Typical life science applications, on the other hand, include biosensors, 5 magnetic resonance 6 and particle imaging, 7 remote cell control, 8 magnetic drug targeting and delivery, 9 as well as magnetic separation. 10 For each purpose the particle properties need to be optimized: whereas some applications may need superparamagnetic particles (e.g., magnetic resonance imaging), others prefer large particle moments (e.g., biosensors and magnetic separation). Another prominent biomedical application, which is highly dependent on the used particles is magnetic hyperthermia. 11 The working principle of magnetic hyperthermia is to inject MNPs into tumors and heat them by applying alternating magnetic elds to kill the surrounding tumor cells. Commercial heating coil setup provide a xed frequency (of around 100-1000 kHz) and amplitude (of around 5-20 mT). For clinical application, regulatory requirements for biocompatibility restrict the material choice mainly to iron oxides, and thus, only particle morphology and structural arrangement are tuneable parameters to achieve appreciable heating under physiological conditions. The unique ability of MNPs to transduce heat in alternating elds on the nanoscale 12 is also utilized for catalysis and energy applications, such as CO 2 hydrogenation 13 and electrolysis (i.e. water splitting), 14 in which case the material is not necessarily restricted to iron oxides.
Different advanced synthesis routes exist to design the ideal particles for each application, as we will illustrate exemplarily in the following for magnetic hyperthermia and particle separation. Generating sufficient heat with alternating elds requires iron oxide MNPs with a signicant dynamic susceptibility at high frequencies/low amplitudes. Recent works indicate that for magnetic hyperthermia large, defect-rich MNPs are great candidates, 15 including so-called nanoowers 16 and nanocubes with interfacial defects. 17 In general, defect-engineering is a promising approach for particle design since it allows manipulating intrinsic MNP properties either by introducing structural defects such as point defects, 18 by doping, 19 by combining crystalline and amorphous parent material, 20 or by controlling twin structures in the nanocrystals. 21 Core-shell particle systems are another particle type explored for magnetic hyperthermia applications 22,23 as the heating power can be adjusted by tuning the interface coupling. 24 The control over the magnetic properties via exchange coupling makes core-shell particle systems compelling for various applications in life science 25 and technology. 26 Particles that are suited for magnetic hyperthermia are commonly also suitable for magnetic particle imaging as the signal is created by alternating magnetic elds. 27 In contrast, for magnetic separation, ideal particles have a vanishing intrinsic coercivity but a high static magnetic susceptibility together with a high load capacity and selective and reversible binding affinity. Particle clusters, 28 magnetic microspheres 29 or large multi-core systems of selfassembled nanoparticles, so-called supraparticles, 30 are desired as they have large moments and can be thus easily extracted by magnetic gradient elds. Magnetic separationwhich can be used, e.g. for water purication 31 or bioseparation -oen necessitates surface modication to achieve the specic binding of the target compound to the MNPs. 32 Particle surfactants can affect the magnetic properties of the functionalized MNP ensembles, which need to be controlled during particle synthesis. 33 It is worth mentioning that supraparticles are well-suited for magnetic separation and also envisioned for a wide array of applications related to sustainability. 34 Nowadays, the synthesis of supraparticles or mesocrystals can be well controlled 35 allowing the preparation of particle systems with a wide range of magnetic properties and additional functionalities. 36 In addition to the synthesis of complex, multifunctional particle systems, the shape and size of the individual nanocrystals can be easily controlled for many materials, 37 including iron oxides magnetite/maghemite, 38 and hematite. 39 This allows the preparation of shape anisotropic nanoparticles, which are great candidates for various applications as the magnetic properties can be controlled by particle morphology. 40 Elongated ferromagnetic nanoparticles can be applied as nanoprobes for nanorheological approaches when their magnetic moment is preferentially aligned along the long rotation axis due to shape anisotropy. 41 More exotic magnetization states can be found, e.g. in hollow particles, 42 nanorings, 43 nanotubes, 44 and other shape-anisotropic hollow particles, 45 or nanodots 46 and nano-octopods. 47 In many cases, the MNPs consist of the typical 3d ferromagnetic elements Fe, Co, Ni, alloys (FePt, FePd) or their oxides, with iron oxides being the most prominent example. Recently, MNPs of 4f-intermetallic alloys (e.g. TbCu 2 (ref. 48) or GdCu 2 (ref. 49)) gained interest because their magnetic order can be easily tuned with particle size and microstrain. Rare-Earth intermetallics are promising candidates for magnetocaloric applications by tuning the strength of magnetic coupling and modifying the contributions of frustrated and disordered magnetic moments. 50 Furthermore, they excel by a high saturation magnetization and hysteresis-less magnetic response which is desirable for various technological applications. 51 To sum up, a large catalog of different MNP systems exists with unique functionalities tailored with the structural and magnetic properties. The latter can be achieved by either varying the material, the morphology of the individual nanoparticles, or interparticle interactions. As illustrated in Fig. 1, to optimize a given MNP sample for a specic application, one needs to interrelate their macroscopic properties with their chemical and magnetic nano-/microstructure.
To describe the magnetic response of MNP on the macroscopic level, it is oen assumed that the individual particles or crystals are single-domains with a homogeneous internal magnetization prole. Single-domain means that all the atomic magnetic moments m a within the particle are aligned parallel to each other. 104 Hence, the total particle moment can be represented by a macrospin m ¼ P m a , and thus MNPs may be regarded macroscopically as simple dipoles. The moment m uctuates with a characteristic relaxation time s between energy minima due to thermal activation and it can be distinguished between superparamagnetic (measurement time t m [ s) and thermally blocked (t m ( s) particles. Only spherical and defectfree particles with diameters below a material-specic singledomain size can be considered model-like, homogeneously magnetized particles. Real MNP samples always deviate to a certain extent from this oversimplied picture. The intra-and inter-particle magnetization proles depend on various parameters, including the particle size and shape, 105,106 as well as dipolar interactions with neighboring particles. 107 Furthermore, structural defects within MNPs are very common, 108 especially at the particle surface, 109 which can critically affect their magnetic properties. Magnetization or Mössbauer spectroscopy measurements indicate the existence of non-negligible spin disorder in MNPs. 110 The precise determination of the internal 3D magnetization prole remains a key challenge in MNP research and is necessary to fully understand the complex interrelations between the structural and magnetic properties.
Possible approaches to determine the morphology and 3D magnetization prole of nanoscopic systems are electron microscopy-based techniques. Transmission or scanning electron microscopy (TEM, SEM) are applied to identify the particle shape and size of MNPs. With scanning transmission electron microscopy (STEM) a more detailed picture of the atomic structure can be obtained, e.g. resolving anti-phase boundaries within individual MNPs with atomic precision. 108 On the other hand, Lorentz transmission electron microscopy (LTEM) allows detecting the stray eld magnetization of MNPs with nanometer-resolution. 111 Higher spatial resolution than LTEM is provided by electron holography, which is sensitive to the entire nanoparticle spin conguration. 112 This technique helped to image the dipolar coupling in planar arrangements of MNPs. 113 Electrons can only probe locally and thin individual structures or arrangement of few particles due to the restricted penetration lengths.
To investigate large MNP assemblies and samples with embedded and buried MNPs, we advocate small-angle scattering techniques using either X-rays or neutrons. These techniques cover the technologically relevant mesoscale mesoscale ($1-1000 nm) with nanometer-resolution and enable a structural characterization and the determination of the 3D Fig. 1 The macroscopic properties and consequently the applicability of MNP systems (e.g. for technological, biomedical, or energy applications) are defined by their structural arrangement and magnetic interactions on a mesoscopic length scale ranging from a few hundred nanometers down to the individual MNPs. The magnetic properties of the MNPs depend on the particle morphology, chemical composition, and atomic structure. Small-angle scattering allows to determine the chemical and magnetic structure over the mesoscopic length scale and to connect it with the macroscopic ensemble properties. magnetisation prole of particles and large particle assemblies. For so matter systems and structural biology, small-angle scattering is well established as an advanced characterization technique sampling a statistically relevant number of nanoparticles. 114 A general, comprehensive introduction on the nonmagnetic theoretical foundations and specics of both X-ray and neutron small-angle scattering is given in the textbook by Hamley 115 . The last decade has seen a continuous drive for new sample environments and continuous development of smallangle scattering based instruments with polarized beam options dedicated for magnetism. Concerning neutron scattering, the review article by Mühlbauer et al. 116 provides an overview on dominantly bulk magnetic systems, like magnetic alloys and oxides, noncollinear magnetic structures, skyrmions and ux-line lattices. This review will close a gap and shows the benets and recent advances of magnetic small-angle scattering using both neutrons and X-rays to resolve the structural features and essential magnetic information on magnetic nanoparticles and assemblies. The presented examples demonstrate both the exibility of the techniques and the breadth of the covered topics e.g. to study the formation of nanoparticles and their assemblies under in situ conditions, linking the structure with the magnetic response, and probing the magnetization dynamics by covering the relevant timescales.
In this review, we focus on the following techniques: conventional small-angle X-ray and neutron scattering (SAXS and SANS), X-ray and neutron reectometry (XRR and NR), grazing-incidence SAXS and SANS (GISAXS and GISANS), X-ray resonant magnetic scattering (XRMS), and neutron spin-echo (NSE) spectroscopy. Table 1 lists suitable instruments for these techniques, which can be found at large-scale facilities worldwide, together with their main characteristics, explaining the kind of samples typically characterized with each given technique, and which information is retrievable. In the following sections, we present typical examples for each technique regarding MNP characterization and highlight recent outstanding works to show the reader the possibilities to use small-angle scattering to guide functional particle design. Additionally, new developments regarding the application of micromagnetic simulations for the analysis of magnetic smallangle scattering data are presented. Finally, we summarize and discuss the topic, and present our perspectives and visions for future research avenues and developments in this eld.
2 Conventional small-angle X-ray and neutron scattering Small-angle scattering allows gathering detailed information on the mesoscopic length scale from about 1 to several hundred nanometers about the chemical composition and magnetization distribution in magnetic nanoparticles. This allows for example investigating interparticle correlations in aggregates and the formation of superstructures depending on parameters such as particle concentration and applied magnetic, electric, or ow elds. The chemical composition and density are associated with the scattering length density (SLD) r(r), which for X-rays measures the electron charge density and the isotopic composition for the so-called nuclear scattering of neutrons, respectively.
The measured SAXS and nuclear SANS intensity can be simply written as I(q) fjN(q)j 2 , where N(q) is the Fourier transform of r(r) and q is the scattering vector which is given as the difference between the incoming and scattered wave vector with the magnitude q ¼ 4p l sin q for small scattering angles 2q, where l is the wavelength of the incoming radiation. Note that small-angle scattering experiments are conducted either in transmission geometry as illustrated in Fig. 2a, or in reection geometry (Fig. 2b). Oen, instead of the 2D scattering pattern, 1D data I(q) are analyzed, e.g. the radial average, which can be written in the case of spherical symmetry as I(q) f Ð p(r)sin(qr)/ (qr)dr. Here, p(r) ¼ r 2 C(r) is the so-called pair distance distribution function, which is connected to the two-point densitydensity (Debye) correlation function C(r) of the scattering length density prole. The correlation function is derived from experimental data by inverse Fourier transforms and provides essential information regarding particle shape, size, density prole, and particle interference, and-in particular for neutrons-the magnetization distribution and magnetic inhomogeneities caused by spatial variations in magnetic parameters. 118 Alternatively, in the case of an ensemble of identical, spherical symmetric scatterers, the intensity can be written as the product of the particle form factor P(q) and the structure factor S(q), which can arise from particle interference. 119 For several other particle geometries, analytical functions for P(q) exist. For sufficiently narrow size distributions, the analysis of higher-order form factor oscillation allows resolving ner details of the surface conguration and irregular particle shape, e.g. the degree of truncation and roundness of cuboids. 120 The polydispersity of the nanoparticle ensembles is considered with a corresponding density distribution function. The relevant structural parameters of the particles and the corresponding distribution function can be determined by model ts of the reciprocal scattering data. To t experimental data or to calculate the expected scattering signal of a variety of sample systems, open-source soware exists. Fig. 3 shows exemplarily the computed data for a polydisperse ensemble of spherical NPs. For more details regarding the structural characterization of nanoparticle systems by small-angle scattering, we refer to the review article by Li et al. 121 Magnetic small-angle neutron scattering originates from nanoscale variations in magnitude and orientation of the magnetization in a material. The dipole-dipole nature implies that magnetic neutron scattering only observes the magnetization component perpendicular to the scattering vector. The detected 2D SANS pattern of magnetic samples thus contains additionally to the nuclear scattering contribution a superposition of the Fourier transforms of the three Cartesian coordinates of the 3D magnetization prole M(r) ¼ [M x (r), M y (r), M z (r)]. The weighting of the three contributions of M(q) depends on the measurement mode (i.e. unpolarized, half-polarized, or fully polarized) as discussed in detail in the review of Mühlbauer et al. 122 Complementary, resonant X-ray magnetic scattering relies on measuring the scattered X-ray beam to examine magnetic proles of specic magnetic elements in a material. 123 Analyzing the spin asymmetry in the scattering over an absorption edge for circularly polarized X-ray photons allows obtaining a magnetic scattering contrast that scales with the net magnetic moment. This particular technique will be introduced in Section 5. In this section, we will present an overview of MNP studies that involved conventional SAXS and SANS.

Structural morphology
SAXS delivers representative statistical data on the morphology of MNPs, particle sizes and size distributions, and the number of individual particles/crystals within aggregates. Studying clusters or aggregates of MNPs is especially relevant with life science applications in mind, as MNPs tend to agglomerate in biological environments such as cells. This motivated Guibert et al. 124 to study the inuence of aggregation of around 12 nm iron oxide MNPs on their magnetic hyperthermia performance via SAXS. They could correlate an increasing aggregation of the MNPs with a decrease in magnetic heating. In contrast, the aggregation of smaller iron oxide MNPs enhances magnetic hyperthermia performance. 125 This shows, that depending on the size of the individual MNPs a clustering can either increase or decrease their magnetic heating, which is relevant information regarding a rational particle design for life science applications. To analyze the structure of MNP aggregates, either model ts or inverse Fourier transforms can be applied.
Laboratory-based, commercial X-ray facilities provide routine access for small-angle scattering as a primary characterization tool for structural information on nanostructures like the correlations between the positions of nanoparticles 126 and the packing density. 127 Szczerba et al. 128 for example determined the size and composition of aggregates of iron oxide MNPs, socalled multi-core particles. The form factor of spheres for the MNPs, a mass fractal structure factor for the aggregates, and log-normal size distributions provided nanoscopic insights into the structure of the multi-core particles (see Fig. 4a). Also, for less dened systems, the aggregate size can be directly inferred via inverse Fourier transform. 125 Alternatively, the form factor of the particles can be a priori assumed, and the size distribution can be extracted by a numerical inversion approach analogous to the inverse Fourier transform. This allowed for example the determination of the size distribution of partially aggregated MNPs. 129 Another type of highly investigated MNPs for biomedical applications are core-shell nanoparticles. An analysis of inorganic-core/organic-shell systems is oen difficult due to the low electron density contrast, e.g. of a hydrated polymer shell against water. Exceptionally, SAXS has been utilized to investigate the density prole of highly graed poly(ethylene glycol)coated iron oxide MNPs and the temperature-induced contraction of the shell. 133 For SANS, contrast variation by hydrogendeuterium isotope substitution enables to determine the particle morphology and in particular highlight the structure of the surfactant. 134,135 For example, Koll et al. 130 studied the structure of superparamagnetic iron oxide MNPs encapsulated with a highly stable diblock copolymer shell by SANS and SAXS Fig. 2 Small-angle scattering setup in (a) transmission geometry with a position-sensitive detector placed downstream, which is protected by a beamstop (white area) at the center against the direct incoming beam (orange arrows). (b) Reflection geometry with the direct beam (red arrows) grazing the sample under an angle a i . Specular reflection (orange arrow) is seen at a i ¼ a f above the direct beam on the 2D detector. Interface inhomogeneities give rise to scattering on the vertical incidence line at a different angle than the incident angle. The grazing-incidence small-angle scattering (indicated by the blue arrows) probes the morphology and alignment of nanostructures in the thin film. As an example, the scattering of a square lattice of spheres is shown. The comparison with the form factor P(q) of a single sphere with radius 10 nm shows how the polydispersity smears out the form factor oscillations at high-q, i.e. the Porod range. (c) The corresponding pair distance distribution function p(r) is determined by an inverse Fourier transform of I(q). The data were computed with SasView, 117 which provides tools and models to fit data to various particle geometries.
using contrast variation (see Fig. 4b). They could show that their encapsulation process results in a high polymer shell stability, which makes it a great approach for the preparation of MNPs for drug delivery systems. Contrast variation is also useful to study the inuence of surfactant concentration on the stability and aggregation of ferrouids. [136][137][138] For charge stabilized maghemite MNPs without surfactant, contrast variation allows separating nuclear and magnetic characteristic radii in a waterbased medium. The surface is covered with around 1/3 of absorbed citrate molecules that mediate the electrostatic repulsion as shown in Avdeev et al. 139 For the above works, the data were in most cases analyzed by model ts. This is not possible for more complex particles, in this case inverse Fourier transform can be applied to obtain useful information regarding the particle structure of core-shell-type systems. 140 For X-rays, contrast variation is achieved by varying the wavelength close to the absorption edge of a particular atom. This anomalous scattering provides element specic sensitivity, e.g. to reveal the chemical composition of internal material boundaries Fe 3 O 4 core Mn 2 O 3 shell nanoparticles. 141 Small-angle scattering is a non-destructive method that enables studying a system in its pristine state in comparison to microscopy approaches needing specialized grid preparation, e.g. magnetotactic bacteria in aqueous dispersion. 131 These bacteria biomineralize magnetite nanoparticles in specialized organelles (magnetosomes) in a linear arrangement, which enable them to orient along and migrate with the geomagnetic eld (see Fig. 4c). When dispersed in water, the bacteria align along an externally applied magnetic eld resulting in anisotropic scattering patterns. In Bender et al. 132 an inverse Fourier transform was applied (here using the singular value decomposition) to extract the underlying 2D correlation function, which nicely reects the linear chain of aligned magnetosomes (see Fig. 4d). The particle size can be controlled by the cultivation conditions of the magnetotactic bacteria with a typical core radius of 20 nm and a water-impenetrable organic membrane shell of 3 nm as seen with neutron contrast variation study. 142 The MNPs are arranged in a slightly bent chain and the observed misalignment of the particle moment at low magnetic elds is a consequence of magnetic dipolar interactions, magnetic particle anisotropy, and the acting elastic force of the cytoskeleton. 143 The bacteria prove high stability against distortion by magnetic forces as demonstrated with polarized SANS on a xed freeze-dried powder of the bacteria, 144 which make them great candidates for various biomedical, 145 environmental 146 applications. Recently, for example, their use was suggested for magnonic devices 147 and for magnetic actuation. 148

Magnetic structure of particles
Half-polarized SANS (SANSPOL) with an incoming polarized beam, but no analysis of the scattered neutrons, provides an extra contrast given by the interference between nuclear and magnetic scattering and allows ltering backgrounds such as non-magnetic contributions and spin-misalignment scattering arising from moments deviating randomly from the eld axis. SANSPOL gives access to very weak magnetic contributions, e.g. to infer the existence of a magnetic dead layer in magnetic Fe 3 O 4 glass ceramics, 149 to reveal the diffusion mediated Nb enrichment around nanoprecipitates in a metallic matrix, 150 and to investigate the magnetization distribution of ferrimagnetic iron oxide compounds with their nuclear SANS contribution oen more than an order of magnitude larger than the magnetic signal. A comprehensive contrast-variation study with varying degree of deuteration of the toluene solvent allows highlighting the penetrability of the surfactant shell surrounding the Co nanoparticles. 150 The magnetic scattering length density determined by polarized SANS is a quantitative measure of the magnetization prole. For particles below the material-dependent singledomain size, the particle is commonly described as a coherently magnetized ferromagnet (Stoner-Wohlfarth particle). In this case, small-angle scattering is described in terms of a particle-matrix approach consisting of a form factor describing the shape, size, and orientation of nanoparticles and a structure factor depending on the particle interactions.
In the interior of iron and iron oxide particles, commonly a reduced magnetization compared to bulk material is observed 136,139 that is mainly associated with the presence of antiphase boundaries. 154 Disch et al. 151 used SANSPOL to determine the eld dependence in iron oxide nanocubes and nanospheres coated with oleic acid (Fig. 5a). The particle core exhibits a reduced magnetization of 76% and a gradual demagnetization towards the surface. The reduced core magnetization has been associated with microstructural defects within the particle interior resulting in deviations from the perfect ferrimagnetic order, 155 resulting in spin canting and random disorder of half the atomic moments as indicated by nuclear resonant scattering. 156 Taking advantage of the spatial sensitivity of SANSPOL, Zákutná et al. 152 determined that the coherently magnetized core of Co-doped ferrite MNPs grows with the magnetic eld (Fig. 5b), i.e. inducing magnetic order in the structurally disordered particle surface. The magnetic nature of the outer layer was further investigated with SANS with polarization analysis (POLARIS). The neutron-spinresolved measurements indicate uncorrelated, disordered surface spins. The surface conguration and chemical environment play an important role in the disorder of the surface spins. For instance, a coating of iron oxide MNPs with a silica shell enhances the magnetic properties of the surface regions. 157 POLARIS also allows observing deviations from single domain behavior to complex spin structures with the presence of ordered misaligned moments from the magnetic eld axis. The technique measures the neutron spin-state aer scattering at the samples allowing to separate magnetic from nuclear scattering using a polarized 3 He gas cell as analyzing neutron lter. For a dense self-assembled face-centered cubic superlattice of 9 nm Fe 3 O 4 nanoparticles, Krycka et al. revealed the existence of considerable, temperature-dependent spin canting of 20-30 in a 1 nm surface region. 158,159 A combination of polarized SANS and X-ray magnetic circular dichroism (XMCD) spectroscopy allowed to reveal the evolution of magnetic order in multiphase core-shell nanoparticles. The electronic state and stoichiometry is relatively unaffected with temperature as observed with XMCD, however polarised SANS proposes magnetic disorder of the oxide shell near the blocking temperature and indicates alignment of the metallic Fe core in the eld and reversed magnetization of the surrounding Fe oxide at lower temperatures. 160 Strong spin canting ( Fig. 5c) is observed in densely packed core-shell Fe 3 O 4 @Mn x Fe 3Àx O 4 . 153 Atomistic simulations indicate that the effect originates from reduced exchange interaction or Dzyaloshinskii-Moriya interaction between the core and shell phase. With increasing particle size, inhomogeneous magnetization states occur, not only at the surface but also within the core of MNPs. In Bersweiler et al. 161 the purely magnetic SANS signal for Mn-Zn-ferrite samples shows a transition of a nearly homogenous magnetization prole for 28 nm particles to a vortex-like conguration for 38 nm particles. FePtcore/iron-oxide-shell particles can exhibit a vortex-like intraparticle magnetization conguration reducing dipolar interactions between the particles when no magnetic elds are applied. 162 To identify such complex inhomogeneous conformations, magnetic simulations are a key component. The combination of SANS with micromagnetic numerical approaches is discussed in Section 7.
Multifunctional core-shell nanoparticles enrich the design capabilities for advanced applications. For instance, Wang et al. 163 reported the synthesis of colloidal, superparamagnetic particles from iron oxide nanocube. The nanoparticle assemblies exhibit a spherical or cubic shape in a controllable manner by varying the surface tension and the interaction energy between the nanocubes and are highly crystalline as demonstrated with TEM and SAXS (Fig. 6a). In magneto-uorescent supraparticles, a larger colloidal vesicle encapsulates selfassembled CdSe-CdS core-shell quantum dots with superparamagnetic magnetite MNPs. 164 These supraparticles have a size of 100 nm with an ordered, closed pack core of MNPs surrounded by a shell of randomly distributed quantum dots aer thermal annealing. In Bender et al. 140 a combination of SAXS and SANS revealed that 9 nm iron oxide nanoparticle cores accumulate in the surface layer of a 160 nm polystyrene sphere resulting in a characteristic variation in the scattering length density contrast from core, shell, and solvent. A weak dipolar interaction between the particle moments is indicated by the magnetic moment distribution extracted from isothermal magnetization. The data analysis used a model-independent indirect transformation of small-angle scattering and isothermal magnetization data to extract the structural and magnetic distribution. Such multicore particles are especially interesting for magnetic separation as they have a vanishing coercivity but large effective moments in presence of magnetic elds.
Field-sensitive ferrogels on the other hand are envisioned for so actuators. Helminger et al. 165 produced biocompatible ferrogels using gelatin gels with embedded magnetite nanoparticles and applied SANS contrast variation experiments to explore the nanoparticle packing in the gelatin gel network. To optimize the performance of ferrogel-based so actuators and other functional materials, a good binding between the MNPs and the matrix is necessary as well as a homogeneous MNP distribution. In Bonini et al. 166 cobalt-ferrite MNPs were dispersed in polyacrylamide gels and a combination of SAXS and polarized SANS revealed the high quality of their samples.

Magnetic interparticle correlations
Small-angle scattering is widely applied to study colloidal stability and ordering in dispersions of nanoparticles. The Brownian motion of magnetic nanoparticles becomes directional with the competition between repulsive stabilizing isotropic forces and the anisotropic dipolar magnetic interactions. Small-angle scattering with neutron polarisation analysis allowed to measure the alignment in a concentrated cobalt ferrouid. 167 Even in the absence of particle agglomeration and chaining, long-range concentration uctuations along the eld indicating a strong anisotropy of the Brownian motion are observed under magnetic eld. 168 Using a combination of scattering methods and reverse Monte Carlo simulations, Nandakumaran et al. 169 explored the chain formation mechanism under magnetic eld in solution. Eventually, dipolar magnetic interactions in ferrouids induce the formation of superstructure ranging from short-range ordered aggregates via chain-like structures 170,171 to eld-induced pseudocrystalline ordering in concentrated ferrouids. 172,173 Interparticle forces sensitively affect the rheology of colloidal dispersions. The magnetoviscous effect, i.e. the strong increase in viscosity with magnetic eld, has been investigated with in situ magnetorheology and small-angle scattering investigating the orientational order. 174,175 Shear-thinning behavior is explained by the disruption of the so aggregates for high Fig. 6 (a) The collective properties in superparticles strongly depend on the packing order ranging from amorphous to supercrystalline with well-defined interparticle spacing. Colloidal superparticles of iron oxide nanocubes adopt a simple-cubic superlattice structure. The SAXS pattern consists of the corresponding diffraction peaks. Reprinted with permission from Wang et al. 163 Copyright 2012 by the American Chemical Society. (b) SANS patterns of the magnetic-field induced transition from an isotropic, non-ordered colloid to the selfassembly of 3D fcc supercrystals in a 0.1 vol% dispersion of 17 nm iron oxide nanoparticles. Taken from Fu et al. 173 -Published by The Royal Society of Chemistry. (c) SAXS and nuclear SANS scattering reflect the spatial distribution of 10 nm coated iron oxide nanoparticles in a powder. The feature at q ¼ 0.58 nm À1 is associated with the distance of neighboring particles. The extended high q-range for SAXS shows higher-order oscillations of the particle form factor. The scattering at low q reflects interparticle correlations within the clusters. The fielddependent, purely magnetic neutron scattering cross-section I SF resolves the directional correlations between the particle moments. At small magnetic fields, a maximum at 0.12 nm À1 evolves that indicates dipolar interactions in particle clusters up to 70 nm with a competition between positive and negative moment correlations. (Right) Monte Carlo simulations support the preferential alignment between neighboring moments and dominant anticorrelations for next-nearest moments despite thermal fluctuations. Reprinted figure with permission from Bender et al. 182 Copyright 2018 by the American Physical Society.
enough shear ows. 176 Similarly, driven by particle surface charges, the viscosity can be modied with electric elds in transformer-oil-based ferrouids. 177 Magnetic-eld oriented aggregates of nanoparticles offer a potential way to obtain strongly anisotropic magnetic properties due to the particle alignment and to cast reinforced nanocomposite materials with anisotropic mechanical properties. 178 Magnetic eld-induced self-organization is facilitated by large structural and magnetic anisotropies, as shown for elongated hematite nanospindles producing nematically ordered assemblies under a directing static or dynamic eld. 179 Anisotropic metallic MNPs as constituents in ferrouids may result in a strongly enhanced magnetoviscous effect in comparison to conventional ferrouids. 180,181 Field-dependent SANS is further used to detect collective magnetic correlations among particles in disordered assemblies and ordered particle nanocrystals. [183][184][185] Small-angle scattering accesses the characteristic length scales connected with interparticle correlations and magnetic interactions between nanoparticles (Fig. 6). In Dennis et al. 186 the internal magnetic structure of clusters of MNPs was determined by polarized SANS and connected to their performance for magnetic particle imaging and hyperthermia applications. This emphasizes the signicant inuence of internal coupling either by dipolar or by exchange interactions on the magnetism of MNP clusters or multi-core particles. Dextran coated iron oxide multi-core particles, e.g., show a domain structure extending over a stack of parallelepiped structural grains as observed with polarized SANS. 187 Magnetic nanoowers consisting of sintered iron oxide crystallites are another example of hierarchical nanostructures and are great candidates for magnetic hyperthermia applications thanks to exceptionally high heating rates. Polarized SANS conrms a preferentially superferromagnetic coupling of the crystallites in a nanoower resulting essentially in singledomain particles but with a slight spin disorder due to the grain boundaries and other structural defects. 188 Furthermore, in the case of a dense powder of such nanoowers positive correlations between neighboring particle moments were observed creating locally a supraferromagnetic structure. 189 This is in contrast to conventional, spherical MNPs where the moments of interacting but superparamagnetic particles tend to align more in an antiferromagnetic-like manner. 182 The magnetic correlations between interacting particles' moments are reected in the magnetic structure factor, which will deviate from the scattering of the structural arrangement. 190 Interestingly, the interparticle coupling can enhance the magnetic heating of nanoower samples as shown by Sakellari et al. 191 which illustrates the potential of dipolar interactions to (i) drive particle arrangement and to (ii) modify the static and dynamic magnetization behavior of MNP assemblies. In general, interparticle interactions are an additional control parameter to produce collective magnetism, and which can be monitored by neutron scattering. This makes magnetic SANS an invaluable tool to study nanoscopic magnetic correlations in a large variety of MNP samples and other magnetically nanostructured systems. 192

Time-resolved in situ measurements
Time-resolved studies with time resolution less than 100 ms are routinely possible on X-ray and neutron beamlines. This ranges from the observation of spontaneous nucleation and growth of particles, changes due to oxidation over time scales of several days, to the reorientation and switching behavior of the particle moment and the dynamic assembly of superstructures with a magnetic eld.
A common route to synthesize iron oxide nanoparticles involves precipitating an iron precursor in an alkaline, aqueous solution. 196,197 In situ SAXS helps to identify different reaction pathways that may change with synthesis temperature. 198 The formation pathway involves intermediate metastable precursors before nucleation and growth of nanoparticles. [199][200][201] Ex situ analysis typically requires sample preparation steps like centrifugation and drying, which potentially can lead to artifacts, e.g. a change in the particle size. 202 Continuous ow reactors, in which the reagents are pumped and mixed under well-controlled reaction conditions, realize large-scale and reproducible co-precipitation syntheses. 203,204 Further, the local position along the reaction tube coincides with the reaction time. This allows observing in situ the transient reaction states aer mixing and to study the growth mechanism 205,206 and changes in the magnetic behavior. 207 In situ synchrotron measurements are suitable to follow closely the reaction kinetics and precursor state during the synthesis of iron oxide nanoparticles. 208 For instance, Kabelitz et al. 209 followed the formation of maghemite nanoparticles from ferric and ferrous chloride with triethanolamine as a stabilizing agent in an aqueous solution. At various time steps during the synthesis, samples of a few ml were extracted from the reaction solution and placed in an acoustic levitator to perform X-ray absorption near-edge spectroscopy (XANES) and SAXS. XANES allows determining the oxidation state during the reaction, while the SAXS data detect the growth of particles. The magnetic iron oxide forms rapidly within seconds aer mixing the chloride precursor with the NaOH base solution under the abrupt pH change. The co-precipitation is sensitive to local uctuations of the reaction conditions and affects reproducibility. In situ timeresolved, simultaneous SAXS/WAXS studies under supercritical uid conditions shed light on the synthesis process at 300 bar and above 300 C allowing to choose suitable residence time to obtain narrow size distributions. 210 Thermal decomposition with high boiling point organic solvents allows synthesizing very monodisperse iron oxide nanoparticles economically in large scale quantities. 211,212 By continuously sampling the reaction mixture through a X-ray transparent sample chamber (Fig. 7a), combined SAXS/WAXS experiments resolve the formation of iron oleate complexes, their thermal decomposition to intermediate clusters, and nanoparticle nucleation and growth. 193 For a reproducible process, an in-depth understanding of the reaction mechanism during each step of nanoparticle formation is needed. The development of in situ scattering set-ups gives fundamental insights into the nucleation and particle growth kinetics, e.g. identifying transient amorphous phases and particle aggregation processes in the iron oleate heat-up synthesis, not accounted for in the classical description. 213 Post-processing steps may be required for purication and potentially phase transfer to polar solvents via ligand exchange. 214 Stable, aqueous dispersion of nanoparticles based on amphiphilic polymers are further functionalizable with selected macromolecules, 215 e.g. for targeted drug delivery. The choice of surfactant can alter structural and magnetic properties. 216 The stability of the aqueous particle dispersion and absence of interparticle correlations, expected aer successful phase transfer, is easily conrmed using small-angle X-ray scattering. 217 Controlled evaporation of the particle dispersion results in the formation of nanoparticle superlattices 218 as discussed further in Section 4. To increase the colloidal and chemical stability, magnetic particles can be coated with a protective silica layer, which physically separates the magnetic cores and helps to avoid agglomeration. Time-resolved SAXS has the potential to investigate in situ the growth kinetics of silica coating on magnetite nanoparticles under various reaction conditions, e.g. the dependence of precursor concentration on the coating process and in particular controlling the shell thickness in relation to the magnetic volume fraction and superparamagnetic relaxation. 219 Apart from monitoring the growth kinetics of MNPs, timeresolved experiments allow determining the dynamic ordering and relaxation processes of magnetic nanoparticles in magnetic elds, e.g. the formation of chain-like aggregates in a dilute dispersion, which align with an external magnetic eld. The analysis of the scattering cross-section in Huang et al. 220 indicates that 20% of the particles form two-bead chains under an external magnetic eld. The arrangement is completely reversible when the magnetic eld is absent. For Co nanoparticles concentrated up to 6 vol% dispersed in oil, polarized SANS shows the emergence of sixfold symmetric scattering peaks with a magnetic eld indicating reversible pseudocristalline hexagonal order over domains of 100-150 nm, estimated by the width of the correlation peak. 221 The order disappears at zero eld and the particles arrange in uncorrelated dipolar chains composed of a few particles. The correlation disappears on the timescale of seconds when the eld is switched off. 222 The decay times increase signicantly with a eld, indicating the stabilizing inuence of dipolar interaction on the particle moments relaxation. 223 Time-resolved unpolarized and polarized small-angle neutron scattering with AC eld and for temperatures down to the freezing temperature of the solvent demonstrates that the magnetic reorientation process of Co and Fe 3 O 4 ferrouids is composed of moment relaxation characteristic for Brownian rotation of the magnetic cores with nite viscosity or by Néel type relaxation in the frozen state and a variable volume fraction of arrested, static moments, which can be aligned along with a preferred orientation. 224,225 For anisotropic magnetic particles and aggregates, the distorted scattering pattern of a rotating sample in a static magnetic eld or a rotating magnetic eld allows estimating the rotational diffusion coefficient in the characteristic range up to 1000/s. 226 Continuous beam measurements are restricted to AC frequencies below a few hundred Hz. Neutrons passing the sample at a given time arrive at the detector as time-shied events due to the spread in velocity resulting in a smeared oscillation amplitude of the signal. Time resolution is hence limited to 1 ms.
Regarding SANS, faster relaxation times are accessible with a phase-lock technique called TISANE, which synchronizes microsecond short neutron pulses from high-speed choppers with a periodic stimulus like oscillatory shear, electric or magnetic elds extending the probed frequency range to the kHz regime (Fig. 7c). Similar to stroboscopic data acquisition, the scattering signal is observed over many periods to obtain sufficient counting statistics. A TISANE chopper system has been installed at a few neutron instruments, like SANS-I at FRM-II, D22 at ILL, and NG-7 SANS at NIST. In a concentrated Co ferrouid at ambient temperature, eld-induced ordering occurs on timescales of 100 ms determined by Brownian rotation, locally ordered domains of 100 nm size driven by a dipolar-eld governed ordering process are created at a later stage within a few seconds of applying an external magnetic eld. 195 The pulsed beam technique has been further used to study the reorientation dynamics of colloidal dispersions of Ni nanorods in oscillating elds. 80 A recent experiment at NIST investigated hematite nanospindles dispersed in D 2 O. These anisometric nanoparticles show a signicantly different and more complex reorientation behavior. Sufficiently large nanospindles have a tendency for uniaxial anisotropy overcoming thermal uctuation of the magnetic moment within the basal plane. 227 With the magnetic easy axis xed in the basal plane, the hematite spindles orient and rotate with their long axis perpendicular to an applied eld. 228 From an analysis of the frequency-dependent phase delay of the scattering amplitude to the oscillating magnetic eld, one can extract the rotational diffusion coefficient of opaque, dense magnetic particle dispersions. 229 The structure-directing inuence of static and dynamic magnetic elds 179,230,231 can induce self-assembly of nanocrystals with translational and orientation order that exhibit strongly anisotropic properties that can be used for optical lters, and nanometer-scale viscoelasticity sensors. 232 Furthermore, doping ferronematic liquid crystals with elongated MNPs such as the hematite spindles is a promising approach to improve their performance, and thus understanding the relaxation dynamics of these composites is crucial for potential applications, e.g., in at panel displays. 233 4 Reflectometry and grazingincidence scattering Nanoparticle assemblies revealing interparticle correlations are typically obtained either by bottom-up self-organization or topdown lithographic techniques. Commonly, a solid substrate supports the sample and provides connement with a structuredirecting inuence. The scattering under grazing incidence geometry, or reection geometry (Fig. 2b), has several advantages for such supported nanostructures. First, if the incident angle a i of the incoming X-ray or neutron beam is small, its large footprint illuminates a relatively large sample area. This results in a much larger scattering volume than in transmission geometry, so that even very thin layers with nanoscale thickness can be studied. Moreover, close to the critical angles for total reection of the surface and the substrate dynamical scattering effects can be exploited to enhance the scattering intensity and exclusively illuminate the interlayer such that the scattering pattern will be highly sensitive to its structure.
Reectometry, off-specular scattering, and grazing-incidence small-angle scattering (GISAS) are closely related experimental techniques exploiting the reection geometry for the characterization of thin lms and interfaces. Excellent literature is available that gives fundamental knowledge about these techniques and the associated scattering theory. [236][237][238] Here, we will focus on an overview of the scattering geometries and the different information gained from specular/off-specular reectometry and grazing-incidence scattering using X-ray and neutron scattering probes. Scattering in the specular condition (with scattering angle equal to the incidence angle, a f ¼ a i , Fig. 8a) probes the structural and magnetic depth prole of thin lms, multilayers, and laterally nanostructured materials. Lateral sample inhomogeneities, such as interface roughness or magnetic domains in the mm regime, give rise to off-specular scattering (with scattering angle unequal to the incidence angle). 239 In contrast, lateral structures in the nm length scale result in intensity registered outside the scattering plane, denoted as GISAS (Fig. 2b).
Polarized neutron scattering measures the change in the polarization state of the scattered neutron: the sample magnetization parallel to an external magnetic eld causes non-spin-ip reectivity, whereas neutron polarization reversal indicates perpendicular magnetization components, which allows resolving variations of the magnetization vector. 240 Polarized neutron reectometry (PNR) has revealed the dipolar magnetic particle coupling in nanoparticle assemblies observed as domain-like congurations at remanence 241 and through varying layer density. 242 Polarized GISANS, on the other hand, is an emerging technique that will enhance our understanding of lateral interparticle coupling. 243 The strong potential of reectometry and grazing incidence scattering techniques towards depth-and laterally resolved nanostructures nds wide application in the structural and magnetic characterization of MNP assemblies in different dimensions: ranging from nanoparticle monolayers 244 through multilayers 245 to highly ordered 3D superstructures such as mesocrystals. 246,247 The following subsections will give an overview of recent achievements using reectometry (4.1) and grazing incidence scattering (4.2), emphasizing also timeresolved studies of in situ self-organization phenomena. 235,248

Structural and magnetic depth prole
Neutron reectometry has the unique strength to assess depthresolved structural information, e.g. from buried interfaces, multilayer systems, and nanostructured polymer templates. In combination with off-specular scattering, Lauter et al. 249 revealed the structure transformation from cylindrical to lamellar structure in nanocomposite lms of diblock copolymer and magnetite nanoparticles. For assemblies of nanoparticle arrays, the mean distance between the mm sized supercrystals is accessible using off-specular scattering. 250 In a combined XRR, GISAXS, and PNR study, Mishra et al. 245 analyzed the structural and magnetic ordering of spin-coated nanoparticle lms and monolayers. Next to a hexagonal lateral order revealed by GISAXS, a clear modulation of the depth prole was found using XRR, indicating a layered nanoparticle stacking with a linear density gradient between the substrate and top layer. PNR revealed dipolar interparticle coupling and formation of local domains, resembling a so ferromagnetic state in remanence. Neutron reectometry allows studying the self-organization of nanoparticles from dense ferrouids at the solid-liquid interface. Vorobiev et al. 234 revealed the eld dependence of the layered structure of ferrouids near a solid substrate with a dense wetting layer (Fig. 8a). The different depth proles resulting from either static conditions, under shear, and with a magnetic eld, are accessible using neutron reectometry. 251 Moreover, the ferro-uid properties such as particle surface functionalization strongly affect the rst adsorption layers on the solid substrate. 252,253 Theis-Bröhl et al. 254 revealed the inuence of different substrates on the wetting layer and subsequent layer formation from ferrouids and the magnetization depth prole in the obtained assemblies. Saini et al. 242 elucidated the impact of a magnetic substrate on the iron oxide nanoparticle layer formation using PNR, highlighting the importance of particle size and the resulting magnetic moment. Saini and Wolff 255 further used a magnetic eld to induce a microshearing effect on small quantities of magnetic polymer nanocomposites that improved the crystallization behavior of nonmagnetic surfactant micelles in water.

Lateral and 3D nanostructure
Long-range ordered arrays of nanoparticles, such as mesocrystals or supercrystals, typically assemble as a two- dimensional powder on the substrate with only the substrate normal as the preferred direction. As a result, all (hkl) reections of the sample can be detected using GISAS at the same time. 246 In contrast, for an individual, single-crystalline array of nanoparticles, the sample needs to be rotated around the substrate normal to successively fulll the Bragg condition for different lattice planes. Lateral structural information can be unambiguously derived from the so-called Yoneda line that emerges at a scattering angle equal to the critical angle of total reection of the sample (a f ¼ a c ). For shallow incidence angles a i , both diffraction and refraction processes contribute to the scattering pattern. This leads to two distinct reections for each lattice plane (hkl), which can be indexed by a combination of Bragg's and Snell's laws. 256,257 The full GISAXS pattern, including the diffuse scattering resulting from mosaicity, originates from a series of different combinations of reection and scattering events. For a correct description, multiple scattering effects are considered in the frame of the distorted wave Born approximation (DWBA). 258,259 In the case of a rough surface, e.g. for islands of nanoparticle assemblies on the substrate, the condition of an ideally at sample is no longer valid, and the conventional born approximation (BA) applies. 260 GISAXS is widely employed to follow the self-organization of nanoparticles into two-and three-dimensional arrangements. For arrangements of semiconductor nanocrystals, GISAXS has been applied to gain insight into oriented attachment, 261 the inuence of the swelling behavior of the surrounding ligand shell, 262 and two-dimensional nanoparticle organization at liquid/air 263 and uid/uid 264 interfaces. GISAXS allows comparing quantitatively the quality (substrate coverage, grain size, packing density, and lattice disorder) of FePt nanoparticle monolayers for different Langmuir deposition and spin coating techniques. 244 The structure-directing inuence of the particle shape has been investigated for cuboidal maghemite nanoparticles, revealing the signicant impact of the degree of corner truncation 246,256 and overall nanoparticle size 247 on the formed mesocrystal structures, ranging from body-centered tetragonal to face-centered cubic and simple cubic structure types. A rich structural diversity of binary nanocrystal superlattices composed of iron oxide and gold nanoparticles and the inuence of lattice contraction upon solvent evaporation was reported by Smith et al. 265 For binary assemblies of CoFe 2 O 4 -Fe 3 O 4 , SAXS discerned the long-range ordering and high phase purity that results in coherent magnetic switching mediated by the enhanced dipolar coupling. 266 Lu et al. 267 report binary superlattices composed of two different nanoparticle sizes and how excess nanoparticles of one size regime may be expelled and grow separately into locally monodisperse nanoparticle superlattices.
Using the high ux of synchrotron beamlines, GISAXS is a suitable tool for the in situ investigation of the dynamic crystallization processes during the self-organization of nanoparticles. For Au and PbS quantum dots, time-resolved GISAXS enabled a distinction between lateral and three-dimensional growth during the initial stages of superlattice formation as well as overall lattice contraction effects during long-term aging. 268 Evaporation of a colloidal nanoparticle dispersion can stimulate spontaneous MNP self-assembly yielding highly ordered and extended nanoparticle superlattices. The process is determined by a complex interplay between various interactions between particles and substrate, added surfactant molecules, drying kinetics of the solvent, and interfacial energy between surfaces. Siffalovic et al. 269 revealed the three-phase (liquidsolid-air) drop contact line as the origin of iron oxide nanoparticle self-organization during drop-casting. The early stages of the self-assembly process of iron oxide nanoparticles in a fast-drying colloidal drop were studied with GISAXS with a temporal resolution down to milliseconds, leading to assemblies with a perfect hexagonal close-packed array within domains with less than 100 nm extension. 270 Hu et al. 271 applied vertical scans with transmission SAXS near the liquid interface to identify the region of concentrated NPs and the degree of order above the interface. Mishra et al. 272 reported the inuence of wetting vs. dewetting properties of the solvent on the lm morphologies of iron oxide nanoparticles on solid substrates. Josten et al. 235 studied the formation of iron oxide mesocrystals by a drop-casting approach using an evaporation cell designed for in situ GISAXS with controlled evaporation rates (Fig. 8b), identifying four different stages: from a concentrating dispersion to formation, growth, and nally rearrangement of the superlattice. The onset of superlattice formation happens suddenly within seconds indicated by the appearance of sharp structure peaks between two sequential measurements. A detailed analysis quanties the ratio of ordered and disordered particle fractions and yields information on the growth kinetics and structural evolution of the superlattice. Nanoparticle selforganization into 1-, 2-, and 3-dimensional assemblies within the solution is accessible using transmission SAXS. 273 Eliminating the structure-directing inuence of the substrate, levitation of drops by ultrasonic waves allows monitoring the twostep assembly of nanoparticles into mesocrystals using timeresolved transmission SAXS (Fig. 7b). The particles cluster in intermediate, dense, but disordered precursors that rapidly transform into large mesocrystals. 194,274 Given the well-progressed application of GISAXS to the structural characterization of MNP assemblies, the next important step will be applying grazing-incidence scattering techniques to the understanding of the magnetic morphology in nanoparticle arrangements. Schlage et al. 275 combined the structural analysis of in situ grown magnetic antidot arrays using GISAXS with magnetic characterization using nuclear resonance X-ray scattering, revealing the magnetization of the growing iron nanostructure and the impact of an iron oxide capping layer. GISANS in combination with polarized neutrons will be a suitable tool to investigate interparticle interactions, such as short-range coupling between nanoparticles in layered assemblies as demonstrated for Co nanoparticles by Theis-Bröhl et al. 276,277 Wolff et al. 243 applied a magnetic eld to induce a microshearing effect on small quantities of magnetic polymer nanocomposites that improved the crystallization behavior of nonmagnetic surfactant micelles in water. The authors suggest a time-resolved and polarized GISANS experiment to elucidate the shear-induced magnetic structure formation and lateral interparticle coupling. Shear alignment of polymer micelles can serve as a template to impose crystalline order and to fabricate ordered so magnets.

X-ray magnetic scattering and spectroscopy
Synchrotron X-ray radiation offers an advanced tool to probe magnetic correlations in nanostructured materials, such as MNP monolayers and is a unique technique to investigate nanoscale magnetism in the presence of high neutron absorbers such as Sm or Gd. Due to their high brilliance, synchrotron X-rays enable the detection of scattering signals produced by very thin magnetic layers and low amounts of magnetic materials in a relatively short time, which is oen challenging to probe with neutrons. In addition, the ability to tune the X-ray energy to specic magnetic resonances provides element selectivity and the magneto-optical contrast necessary to obtain magnetic information. Also, switching the X-ray polarization allows separating the magnetic scattering signal from the charge distribution signal, yielding information on the magnetic and structural correlations, separately. In this section, we will describe two X-ray techniques to study nanostructured magnetic systems, with examples of MNPs. X-ray magnetic circular dichroism (XMCD) allows identifying magnetic resonances and extracting information about the spin and orbital moments of a system, but without information on spatial correlations. X-ray resonant magnetic scattering (XRMS) provides additionally spatio-temporal information about nanoscale magnetic correlations. 280 Also, we will show how coherent-XRMS can provide unique information about the local magnetic disorder and the dynamics of uctuations in MNP assemblies.

Magnetic resonances via XMCD
First predicted and demonstrated in the mid-1980's, 281,282 X-ray magnetic circular dichroism (XMCD) exploits the polarization degree of X-rays to probe the magnetic state of matter via absorption spectroscopy. [283][284][285] XMCD is the X-ray equivalent to the magnetic Faraday and Kerr effects observed with visible light in transmission and reection geometries, respectively. XMCD is a spectroscopy technique, where the X-ray energy E is scanned across specic absorption edges of the material. X-ray absorption (XAS) spectra are recorded at opposite helicities of the circular polarization in order to measure the direction of the atomic magnetic moment relative to the polarization vector of the X-rays. The two resulting XAS spectra m AE (E) are then subtracted to estimate the standard dichroic ratio R D ¼ m þ ðEÞ À m À ðEÞ m þ ðEÞ þ m À ðEÞ .
For best magneto-optical contrast, XMCD is typically measured near absorption edges of electronic energy bands for which spin-orbit coupling is present. For transition metals, such as Mn, Fe, Co, Ni, XMCD is routinely performed at the L 2 and L 3 edges where electrons transition from the 2p 1/2 and 2p 3/2 bands, respectively, up to the 3d valence band. For rare-earth elements, such as Sm, Gd, Tb, Dy, Yb, XMCD is usually carried out at the M 2 , M 3 , M 4 or M 5 edges where electrons transition from the 3p 1/2 , 3p 3/2 , 3d 3/2 or 3d 5/2 bands, respectively, up to the 4f valence band. In this band notation, the subscript corresponds to the quantum number j, resulting from the combination of the spin s and orbital l angular momenta. The strength and shape of the absorption signal m(E) (which reects the electronic density of state in a specic orientation of the magnetic moment associated with j) change when the helicity of the polarization is switched. The resulting dichroic ratio R D informs on the distribution of magnetic resonance energies, for each magnetic ion and each crystallographic site present in the material. Also, information about the spin s and orbital l moments can be extracted by applying the sum rules. 286,287 Combining XAS spectra collected at opposite polarization helicities yields the imaginary part f 2 of the charge (f c ) and magnetic (f m ) atomic scattering factors: f c/m ¼ f 1,c/m + if 2,c/m as illustrated in Fig. 9. The real part f 1 can subsequently be obtained via Kramers-Kronig transformation. 278 A few XMCD studies of MNPs have been carried out so far, including on Co, 278 296 and nanoscale ferrite produced by bacteria and tailoring their magnetic properties by controlled chemical doping. [297][298][299] For bimagnetic core-shell systems, element-specic XMCD contributed to determining the interaction between the iron core and spin-canted ferrimagnetic iron oxide shell, 300 measuring the magnetothermal behavior, 301 explaining the enhanced anisotropy originating from mixed-oxide interfacial layer 302 and the distribution and coupling of cations. 303 The high sensitivity of XMCD revealed also the induced ferromagnetic order of Ag atoms at the interface of crystalline Fe nanograins and clusters 304 and the polarization of Au nanoparticles by dilute Fe nanoparticles 305 in granular thin lms. Fig. 9 shows examples of XAS/XMCD spectra collected at the Co-L 2,3 edges on 9 nm Co MNPs 278 (Fig. 9b) and at the Fe-L 3 edges on 8 nm Fe 3 O 4 MNPs 279 (Fig. 9c). The various elements can be separated by identifying the characteristic absorption edges. This allows in particular to investigate the magnetic contributions of mixed ferrites. 306 Regarding the data collected at the Fe-L 3 edge on Fe 3 O 4 MNPs (Fig. 9c), the XMCD spectrum exhibits a characteristic W shape with three narrow peaks, which correspond to the various tetrahedral and octahedral sites occupied by the Fe 2+ and Fe 3+ ions in the spinel crystallographic structure. A quantitative analysis of the XMCD signal using the sum rules indicates that the orbital moment M L is quenched and that the magnetization is supported by the spin M S moment. The value of M S is found to be around 2.5 (2.7) m B / Fe 3 O 4 at 300 K (20 K), which is smaller than the value measured in bulk Fe 3 O 4 , suggesting nano-sizing effects and spin canting at the surface of the Fe 3 O 4 MNPs.
These examples demonstrate how XMCD can yield useful quantitative information on the magnetic resonances, magnetic atomic scattering factors f c/m , orbital moment M L , and spin moment M S , all spatially averaged over the MNP material. However, XMCD does not provide any information on spatial correlations. Such information can be obtained locally on discrete nanostructures via imaging, e.g. photoemission electron microscopy with XMCD (XMCD-PEEM) to resolve the magnetization state of single particles, 307 or by X-ray scattering techniques over particle ensembles, as discussed in the following.
It is worth mentioning that XMCD is only present when the material exhibits a non-zero net magnetization (for example, ferromagnetic materials). For anti-ferromagnetic materials, where the net magnetization sums up to zero, the technique of X-ray magnetic linear dichroism (XLMD) is applied instead of XMCD.

Spatial and temporal correlations via XRMS
X-ray resonant magnetic scattering (XRMS) exploits the X-ray polarization to probe spatial magnetic correlations in the matter via scattering. 308,309 To optimize the magneto-optical contrast, the energy of the X-rays needs to be nely tuned to resonance edges of the magnetic element(s), which are oen shied relative to the theoretical tabulated edges, due to the electronic state of the excited atom in the local chemical and magnetic environments. Additionally, magnetic resonances may be sharper than electronic resonances. For these reasons, and given potential energy shis in beamline calibrations, it is generally recommended to perform XMCD measurements prior to XRMS identing the exact magnetic resonance edges to optimize the magneto-optical contrast.
Unlike for XMCD, where integrated absorption spectra are measured either as drain current from the sample, total uorescence yield or as transmitted photon intensity (see Fig. 9a), the XRMS intensity is recorded as a function of the scattering momentum transfer q (see Fig. 10a) and necessitates scanning the photo-detector spatially or using two dimensional detection downstream.
Historically, XRMS was rst used in the hard X-ray regime, using L-absorption edges of rare-earth elements, for example to characterise magnetic nanostructures in granular amorphous GdFe alloys. 310 It is also worth mentioning that with sufficient magnetic contrast, non-resonant magnetic scattering of hard Xrays may be used to obtain information about existing atomic magnetic ordering. 311 One advantage is the deep penetration of hard X-rays compared to so X-rays, allowing to study thicker, even mm thick materials with XRMS in transmission geometry.
However, XRMS is nowadays mostly used in the so X-ray range to study nanoscale magnetic structures, such as magnetic multilayers, 312-317 magnetic domains in ferromagnetic thin lms, 318-326 noncollinear spin textures in magnetic thin lms, 327-332 magnetic ordering in antiferromagnetic system 333 and exchange-coupled magnetic layers. 334,335 When tuned to the Co and Fe-L 3 edges, the X-ray wavelength is around l ¼ 1.6 nm and 1.75 nm, respectively (so X-rays). These wavelengths are perfectly suited for the study of MNP assemblies, as it gives access to spatial ranges from a few nanometers up to about 100 nm depending on the resolution and the angular extent of the detector. So far, only a few studies have utilized XRMS to probe magnetic nano-objects, such as Co/Pt nanowire lattices, 336 FePd single nanowires, 337 Co MNPs, 278 patterned nanomagnet arrays, 338 and Fe 3 O 4 MNPs, 339 as illustrated in Fig. 10 and 11a-c. For all these systems, the X-ray energy is tuned to either Co-L 2,3 or Fe-L 2,3 edges in the so X-ray range. Because these materials are nano-structured, so X-rays are well suited to access their inherent nanoscale magnetic correlations.
For isotropic materials that do not show any preferential direction, one can use a point detector, typically a photodiode mounted on a rotating arm that allows scanning the scattering angle 2q through a wide range from 0 up to 90 similar to powder X-ray diffraction (XRD). However, most of the XRMS signal is oen concentrated in the small angle region. An example of a scan in the small-angle regime is illustrated in Fig. 10b, where the XRMS signal is recorded at the Co-L 3 edge on 9 nm Co MNPs. 278 To extract magnetic correlations, I(q) data sets are recorded at le and right circular (I AE ) as well as linear (I lin ) X-ray polarization, and at various magnetic eld values B. Using a point detector allows to study the dependence of the scattered intensity I(B) with the magnetic eld B at a xed q value, as illustrated in Fig. 10c. The two-dimensional detection mode, similar to single crystal detection, provides 2D lateral spatial resolution, particularly informative for non-isotropic nanostructured materials. In this case, a two-dimensional detector such as a CCD camera is placed either directly downstream if the material is probed in transmission (Fig. 9a), or positioned at some angle if the material is probed in reection (Fig. 2b). When done in transmission, the probed scattering angle 2q is typically limited to small values, making XRMS fall in the small-angle X-ray scattering (SAXS) category, analog to small-angle neutron scattering (SANS). Fig. 11 shows examples of 2D XRMS patterns collected on self-assemblies of Fe 3 O 4 MNPs with the energy E tuned to the rst magnetic peak within the Fe-L 3 edge ( Fig. 11a and b), and on assemblies of Co MNPs with energy tuned to the Co-L 3 edge (Fig. 11c). Because the MNP materials are nanostructured, the scattering signal is actually a combination of conventional (Thomson) charge scattering (induced by structural correlations) and magnetic scattering (induced by magnetic correlations). The shape of the scattering signal is an isotropic ring ( Fig. 11a and b), due to the fact that the scattering signal is collected over a large portion of the sample, averaging many various orientations of short-range MNP assemblies. The radius of the ring informs on the average inter-particle distance, whereas its width is related to the correlation lengths. On the other hand, the scattering pattern in Fig. 11c shows a set of local peaks, revealing in this case long-range ordered lattices of NPs. To extract information about magnetic periodicities and magnetic correlation lengths, some further steps are necessary, exploiting the dependence of the signal with varying magnetic eld as well as polarization dependence, as explained below.
The scattering intensity I(q) collected in opposite helicities of circular polarization can be decomposed as a mix of charge and magnetic scattering amplitudes, A c and A m , and be written as 319,339 In this expression, the respective charge/magnetic scattering  amplitudes A c/m ¼ f c/m s c/m (q) are made of the atomic scattering factor f c/m and the structure factor s c/m (q) that contains information on the charge/magnetic structure of the material. The factor s c/m (q) is essentially the Fourier transform of the charge/ magnetic density functions.
To access the information on magnetic correlations specically, one method consists in looking at the reconstructed linear intensity I lin (q) ¼ I + + I À ¼ jA c j 2 + jA m j 2 , and following its eld dependence by collecting data at different eld values, typically at B ¼ 0 and at a saturating eld B max . If the charge component A c is not eld-dependent, one can access the magnetic component A m by measuring the difference DI lin (q) 278 This method, illustrated in Fig. 10b and c, only works if the magnetic scattering is comparable to charge scattering so the variation can be effectively measured.
If jA m j 2 ( jA c j 2 , a more suited approach consists in exploiting the dichroic difference and calculating the magnetic ratio, . In the small magnetic scattering approximation, one can show that R M z 2ReðA c A m Þ jA c j 2 fjA m jfjs m ðqÞj, thus giving access to the magnitude of the magnetic scattering amplitude jA m j directly rather than its square jA m j 2 . 339 The q-dependence of the magnetic scattering is a measure for spatial magnetic correlations, such as inter-particle ferromagnetic and antiferromagnetic couplings, which correspond to a preferential parallel and antiparallel alignment of magnetic particle moments, respectively. 340 Finally, coherent X-ray resonant magnetic scattering (C-XRMS) makes use of the high brilliance and coherence of synchrotron radiation, which produces wavefronts highly correlated in space and in time. Coherence may be achieved in two ways: longitudinally (temporal), by selecting a monochromatic wavelength band, or laterally (spatial) via pinhole ltering and reducing the source size. Under coherent illumination, the interference between scattered beams from different parts of the material produces a speckle pattern (Fig. 11d). The particular position and shape of the speckle spots reect the local charge distribution and magnetic structure in the material. The speckle pattern is a unique ngerprint of the nanoscale correlations, e.g. the size and distribution of the magnetic domain conguration. 341 C-XRMS is sensitive to short-range inter-particle ordering as well as on slow dynamical behavior in MNP assemblies. 342 In the cross-correlation process, two speckle patterns A and B are compared by superimposing them with some gradual lateral shi and multiplying them pixel-by-pixel. This operation (denoted by the symbol Â) produces a correlation pattern, such as the one shown in Fig. 11e. By integrating the signal under the peak ( P operation) and normalizing by the auto-correlations of the two speckle patterns, 343,344 one obtains a normalized correlation coefficient r ¼ P ðA Â BÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P ðA Â AÞ P ðB Â BÞ p which varies between 0 (no correlation) and 100% (same exact pattern). The coefficient r can be estimated over the entire speckle pattern, or on specic regions and q values, e.g. characteristic for short-range magnetic correlations in the plane of a monolayer of MNPs. 345 The dependence of r on an applied magnetic eld and temperature shows the evolution of these magnetic correlations throughout a magnetic hysteresis and crossing phase transitions. Similar to comparing the correlation coefficient r for two images A and B, another, more common approach, known as X-ray photon correlation spectroscopy (XPCS), consists in estimating the correlation coefficient 346,347 DtÞi hI A ðtÞi 2 and following its evolution in time.
XPCS allows accessing collective diffusion coefficients 348 and investigating the slow dynamics of uctuating magnetic congurations at the mesoscopic scale with eld and temperature. 349 Fig. 11f shows g 2 (t) measured at the Fe-L 3 edge on Fe 3 O 4 MNPs at different temperatures crossing the superparamagnetic blocking transition.

Neutron spin-echo techniques
Neutron spin-echo (NSE) spectroscopy is an ultra-high energy resolution technique, offering energy resolutions better than 1 neV 102,350 by encoding the generalized velocities of a neutron with the precession angle (Larmor labeling) in a magnetic eld.
Since the development of the rst NSE instrument in the 1970s, several more techniques have been established that utilize the neutron spin precession to address the scattering process with unique energy and momentum transfer resolution. Amongst these are neutron resonant spin-echo (NRSE) and modulation of intensity with zero effort (MIEZE) that utilize spin ippers in resonance with the precessional frequency of a neutron instead of large solenoids used in classical NSE. 351 Fig. 12 shows schematically the three different NSE spectrometers. While classical NSE still holds the records for reaching the highest temporal correlations (and therefore highest energy resolution), NRSE offers the possibility to reach shorter Fourier times with the use of a eld subtraction coil. 352 The MIEZE technique on the other hand is best suited for measurements of magnetic samples as well as measurements requiring an applied magnetic eld or other depolarizing sample conditions as all spin manipulation takes place exclusively before the sample region. NSE techniques measure the intermediate scattering function S(q, t) directly (Fig. 13a), while other inelastic neutron scattering techniques such as backscattering and time-of-ight spectroscopy measure its Fourier transform, the dynamic structure factor S(q, u). Typical timescales accessible with spin-echo techniques are 2 ps to 50 ns (NSE) and 0.1 ps to 20 ns (MIEZE). First and foremost, NSE techniques have been used to study diffusion and relaxation processes of MNPs in dispersion as ferrouids, colloids, or ferrogels, and embedded in a metallic matrix. Ferrouids with different concentrations of magnetite MNPs (size range 3-5 nm) measured with classical NSE, showed that the energy transfer of the nuclear scattering scales with q 4 , rather than with q 2 , which is typical for translational diffusion. 353 The authors explained this so mode through dipolebond fractals, which break by applying a magnetic eld. The magnetic subsystem is superparamagnetic at low elds and the spin dynamics seem to follow a Néel relaxation process. Larger elds induce stronger uctuations at small q. 353 The same group employed a mixture of isotopic 54 Fe and natural magnetite MNPs (size z30 nm) to eliminate the contribution of the particle pair correlation function due to the varying scattering amplitudes. 354 The self-scattering of individual particles reveals a stretched relaxation at low q. The pair-correlation, however, shows an oscillation-like behavior at large q and a mixed motion of oscillation and diffusion at small q. The period of oscillation (for both q) was found to be 40 ns with an amplitude of 10Å, which is of the order of magnitude of the length of the surfactant molecule. 354 Similar behavior was found for a magnetite ferrouid (MNP size z10 nm) and a magnetite ferrogel containing the same amount of MNPs. 355 The dynamics can be described as a motion of chain-like structures composed of random dipole bonds between particles. For an aqueous ferro-uid constituted of maghemite MNPs (size ¼ 9.8 nm) with a globally repulsive interparticle potential it was found that for low q the collective diffusion is accelerated due to the repulsive interactions between the particles. When a eld is applied, a small anisotropy in the diffusion coefficient is predicted by the simulation. This is however not observed in the NSE measurements either due to polydispersity, or the relatively small dipolar interaction. 356 Lebedev et al. 357 have studied the diffusion of single-domain particles of MnZn-ferrite (size $10 nm) in dodecane. As presented in Fig. 13b, the temperature-dependent intermediate scattering function nicely shows the existence of two distinct relaxation processes, which can be attributed to a fast and slow diffusion process. This example demonstrates the strength of NSE techniques for the characterization of a colloidal suspension of MNPs (such as ferrouids) as relaxation processes can be determined at different q-values over a large timescale (several orders of magnitude). For ferromagnetic Fe(Cu) nanoparticles embedded in a silver matrix of a heterogeneous alloy, NSE provides direct evidence of the interparticle correlations of correlated interparticle magnetic relaxation. 358 For magnetic measurements, classical NSE is complicated as the sample region needs to be decoupled from the precession eld areas to conserve the Larmor labeling between the two spectrometer arms. Since all spin manipulation occurs before the sample position in MIEZE it is ideally suited for measurements of samples under magnetic elds. The MIEZE method has recently been developed at various institutes around the world, including the Reactor Institute Del, the ISIS neutron and muon source, 83  MIEZE can further play a crucial role in the investigation of magnons in MNP systems. Such collective magnetic excitations were predicted by Krawczyk and Puszkarski 366 and have rst been experimentally detected by Krycka et al. 367 using the tripleaxis spectrometer BT7 at NIST. 368 Tartakovskaya et al. 369 describe theoretically the dispersion of spin waves in two-and threedimensional MNP systems by calculations based on a linear Fig. 12 Schematic depiction of the spin manipulation components for a classical neutron spin-echo spectrometer (NSE), longitudinal field neutron resonance spin-echo (L-NRSE), and longitudinal modulation of intensity with zero effort (L-MIEZE) from left to right. The red arrow indicates the orientation of the guide field that maintains the neutron polarization along the neutron beam path from left to right. The upstream p/2 flipper starts the neutron spin precession in the plane perpendicular to the guide field. For NSE, the p flipper reverses the spin states such that for an empty beam the initial polarization is recovered at the second p/2 flipper, which projects the neutron spin back along the field direction to measure the resulting beam polarization with a polarization analyzer and a neutron detector. For NRSE and MIEZE, the radio-frequency (RF) spin flippers produce magnetic fields rotating with neutron Larmor frequency that (de)accelerates neutrons yielding a spin-echo on the detector. combination of atomic orbitals (LCAO) model taking into account dipolar interactions. They predict a phase transition that is caused by the competition between dipolar interaction and uniaxial anisotropy and formulate an expression for the relaxation times and their relation to the dimensionality of a MNP system. MIEZE is an excellent tool for studying magnetic excitations as demonstrated on RESEDA (MLZ) on magnons in bulk Fe 370,371 and on uctuating skyrmion lattices. 372 One advantage of NSE techniques is that the energy resolution is decoupled from the width of the used wavelength band. Therefore, a wavelength selector with a broad wavelength band dl l ¼ 10% can be used instead of a monochromator (as is used for example at triple-axis spectrometers) providing a much higher incoming ux onto the sample without sacricing energy resolution. Furthermore, the large dynamic range $ 1 ps to 10 ns and the availability of a small angle scattering option make MIEZE an interesting candidate to study intra-as well as interparticle dynamics, like the relaxation processes at the interfaces between particle core and shell. 153 In dispersion, these effects are obscured by the presence of Brownian diffusion. Hence, to investigate the moment dynamics one should aim to embed the MNPs in a hard matrix but that is rather transparent to neutrons, such as aluminium or zirconium. On the other hand, studying the diffusion and relaxation of MNPs in viscoelastic matrices can be an interesting research eld in itself in particular concerning biomedical applications (e.g., regarding the interaction of MNPs with physiological/cellular environments) or so actuators and robotics (e.g., regarding the binding of MNPs in ferrogels and elastomers).

Micromagnetic simulations
A model-based analysis of magnetic scattering data (either in magnetic SANS, neutron reectometry, GISANS, or XRMS) is feasible only for a few selected sample microstructures (e.g. with a low polydispersity and a well-dened particle morphology). To circumvent this complexity, the application of micromagnetic simulations was proposed recently to translate the reciprocalspace data into real space. This indirect approach for data analysis was so far used for magnetic SANS and bears great potential for the other small-angle scattering techniques. Additionally, dynamic micromagnetic simulations can be performed to evaluate time-resolved measurements and NSE data. In the following, we give a brief overview of the state-of-the-art characterization of MNPs using micromagnetic simulations. The 'classical' (mesoscopic) approach to micromagnetics 374 predicts the static magnetization conguration or the magnetization dynamics with a spatial resolution of a few nanometers. Commonly, ve contributions to the total magnetic Gibbs free energy are taken into account: the Zeeman energy in the external magnetic eld, magnetocrystalline anisotropy, symmetric exchange, the antisymmetric Dzyaloshinskii-Moriya interaction (DMI), and the magnetodipolar energy. On the nanometer length scale, one can ignore the discrete atomic structure and a continuum approximation suffices to resolve the details of typical magnetization structures both in thin lm and bulk systems. Finally, the minimization of the total magnetic energy is performed, e.g. by an optimized version of a gradient method employing the dissipation part of the Landau-Lifshitz equation of motion for magnetic moments. 375 As for all physical theories, the violation of applicability limits (e.g. attempts to achieve a better resolution than allowed by mesoscopic theories) leads to physically irrelevant results. This remark is in particular important for small-sized magnetic particles, where atomistic modeling is oen the more appropriate choice, e.g. to observe the effect of antiferromagnetic bonds across antiphase boundaries within MNPs, 108 or DMIinduced canted spins across core/shell Fe 3 O 4 /Mn x Fe 3Àx O 4 MNPs. 153 Numerical micromagnetics utilizes two different approaches for spatial discretization, nite-difference (FDM) and niteelement methods (FEM). FDM uses commonly a simple rectangular grid that opens up the possibility to apply fast Fourier transformation (FFT) to compute the long-range magnetodipolar interaction energy. A drawback of FDM consists in the difficulty to adequately represent non-at surfaces (e.g. the surface of a spherical nanoparticle) so that an increased computational effort is required to describe curved surfaces with a very ne discretization mesh. A novel micromagnetic approach based on a polyhedral nite-element mesh has been developed by Erokhin et al. 376 This enables the study of the magnetization distribution of a wide range of mono-and polycrystalline materials as well as MNPs embedded in a nonmagnetic matrix. Moreover, the approach allows for a highly exible mesh generation combined with an efficient FFT-based magnetodipolar energy calculation. The methodology was successfully applied to explain the details of the magnetization reversal in Fe-based so magnetic alloys, 377 Nd-Fe-B nanocomposites possessing a core-shell grain structure, 378 and to optimize the structural properties of permanent-magnet materials based on ferrite alloys. 379 Fig. 14 presents examples of MNP systems that can be efficiently discretized by a polyhedron mesh keeping the total number of elements at a computationally reasonable level. These examples include spherical particle systems with different volume fractions, a gradual transition from oblate to prolate particle shapes, clustered particles of polyhedron shapes, and core-shell microstructures.
In the following, we discuss the magnetic SANS response of an ensemble of spherical, defect-free Fe MNPs. For such a system, the magnetization reversal process occurs exclusively via the coherent magnetization rotation for particle sizes below the critical single-domain size D cr ¼ 15 nm for Fe. 373 Only for a dilute ensemble of particles with diameters smaller than D cr and having a cubic magnetic anisotropy, one can expect that the magnetization behavior resembles the analytical result for randomly-oriented Stoner-Wohlfarth particles; 380 for all other cases, micromagnetic modeling is required. As an example, Fig. 15 depicts the magnetization reversal of Fe MNPs with a diameter of D ¼ 40 nm and a magnetic volume fraction of x p ¼ 15%. The hysteresis curve (Fig. 15, blue line) shows the typical behavior of a so magnetic material, which starts to demagnetize in positive external elds due to the inuence of the magnetodipolar interaction. This inuence has a twofold impact: a strong self-generated demagnetization of the particle and a weaker interparticle coupling. To quantify the average magnetization state of the particles as a function of an applied eld, the absolute magnetization dependence jM i j (red line) can be used, where is the total magnetization of the ith-particle, and N p denotes the number of particles in the simulation volume. Therefore, the quantity jMj À M s describes the average deviation of the particle's magnetization distribution from the single-domain state.
In particular, the minimum of the jMj(B)-dependence corresponds to a highly nonuniform magnetization state. As shown in Fig. 15b, this state corresponds to a vortex-type spin conguration, which is predominant at remanence. With the real-space magnetization congurations obtained from micromagnetic simulations, one can directly compare 2D SANS patterns from modeling and experiment. This is possible because both techniques address the same resolution range $1-100 nm. In magnetic neutron scattering, one measures a weighted sum of magnetization components, and the numerical simulations allow one to decrypt the response of individual magnetic contributions and to study the effect of variations in microstructure and magnetic parameters. 381 This fact renders numerical micromagnetics the go-to tool to attain a comprehensive SANS data interpretation of 3D magnetic structures. Fig. 16a represents the results of such a calculation for an ensemble of spherical 40 nm-sized Fe particles. Shown is the total magnetic SANS cross-section at remanence and at saturation (x p ¼ 15%). While at large elds the main contribution to the total scattering originates from the longitudinal magnetization component e M z , at remanence all Fourier components play an important role in the formation of the resulting SANS pattern. Nonzero transversal magnetization components e M x;y at small momentum-transfer vectors indicate interparticle correlations, while for q-ranges associated to the particle interior it indicates an inhomogeneous spin structure and eventually the formation of a multidomain state. As was shown in Vivas et al., 373 the radially-averaged total SANS cross-section can be used for the further analysis of the magnetization conguration. For example, its Fourier transformation to the so-called pair-distance distribution function demonstrates a quantitative different behavior at saturation and in the remanent state. At saturation, the p(r) for almost homogeneously magnetized spheres in the dilute limit (x p ¼ 5%) coincides with the analytical solution, whereas the vortex-like congurations at remanence produce an oscillating pair-distance distribution function (Fig. 16b, upper row). In this way, it is seen that the p(r)-representation has an advantage over the azimuthallyaveraged total magnetic SANS cross-section due to the much larger sensitivity of p(r) to the details of the magnetization conguration. Applying the same approach to the relatively dense ensemble (x p ¼ 15%) of spherical Fe particles of the same size, we observe another distinctive feature (Fig. 16b, lower row). Both (saturated and remanent) curves exhibit a second maximum due to the increased inuence of the interparticle magnetodipolar interaction. The position of the rst local maximum in the saturated regime coincides with the x p ¼ 5% case, which is an indication for the fully magnetized state of spheres of any volume fraction. The peak shi from saturation to remanence, as for the dilute case, is again attributed to an inhomogeneous magnetization conguration of the particles.
The above example suggests that magnetic SANS is highly sensitive to the internal magnetization prole of MNPs. Thus, micromagnetic simulations are a valuable tool to analyze experimental data, as shown in Bersweiler et al., 161 where micromagnetic simulations were employed to interpret measured magnetic SANS proles. The combination of simulations and experiments conrmed that MnZn-ferrite MNPs with diameters of around 10 nm are in a homogeneously magnetized single-domain state, with increasing diameter the magnetization conguration deviates from the collinear alignment. For the analysis of nuclear small-angle scattering data of biomacromolecules, the usage of ab initio 382 and molecular dynamics simulations 383 are already well established for several years. Considering that the capabilities of micromagnetic modeling have dramatically increased in recent years thanks to the deployment of graphical processing units (GPU), which speed up calculations considerably, 384 we believe that also for  The total magnetic SANS cross-section in the perpendicular scattering geometry (x p ¼ 15%). (b) Azimutally-averaged total magnetic SANS cross-sections and corresponding distance distribution functions p(r) for x p ¼ 5% and x p ¼ 15%. Reprinted figure with permission from Vivas et al. 373 Copyright 2020 by the American Physical Society. the analysis of magnetic small-angle scattering data micromagnetic simulations will become the standard approach.

Summary and perspectives
Small-angle scattering of X-rays and neutrons is sensitive to the chemical composition and magnetization prole of nanostructured samples on a mesoscopic length scale from about 1 to a few hundred nanometers. The strength of small-angle scattering is that it can be employed to directly interrelate the macroscopic behavior and magnetic properties with their nuclear-magnetic nano-/microstructure. As shown and discussed for various examples, such a multiscale characterization can help to optimize the MNP systems for specic applications. Additionally, advanced small-angle scattering techniques exist that allow time-resolved experiments. Such dynamic measurements are especially useful to monitor time-dependent synthesis processes, and they allow to characterize MNPs intended for applications where the relaxation dynamics of the system play a decisive role.
Despite small-angle scattering being such a powerful tool, it is signicantly underused by the MNP community, which may be attributed to its reputation for being difficult and inaccessible as the scattering signal is presented in reciprocal space. In contrast to other, localized characterization techniques, however, scattering assesses the genuine compositional and structural arrangement averaged over large sample volumes. The data analysis is highly sensitive to, e.g. size distributions that smear out characteristic features of the signal expected from individual, perfect shapes. This is an advantage as it allows assessing distribution functions and dispersity of MNP ensembles from a single measurement. As a result, SAXSperformed routinely nowadays with laboratory instrumentsis predestined for determining particle size distributions of MNPs and has become the standard approach for a precharacterization of commercial MNP suspensions. To extract the size distributions, the particles should have a well-dened shape so that the intensity can be tted with the appropriate model function. In the case of less well-dened particles, model-independent approaches can be used to reveal the autocorrelation functions of the scattering length density prole from the SAXS data by inverse Fourier transforms, which allows gaining valuable information about the average particle morphology and density prole.
Similar to SAXS and nuclear SANS, magnetic SANS data are either analyzed in reciprocal space via model ts (at least for well-dened and nearly monodisperse samples) or in real space by applying inverse Fourier transforms. Magnetic neutron scattering is very sensitive to differences in the magnetization distribution and allows observing weak disorder superposed on the average magnetization. Magnetic scattering is sensitive to the magnetization density perpendicular to the scattering vector. Thus, the extracted correlation functions do not result from a scalar composition prole (as for nuclear scattering). The magnetic correlation function measures the variation in strength as well as spatial orientation of the magnetization vector eld over the characteristic distances, 385 which makes their interpretation in terms of model functions challenging.
To directly relate data to physically motivated magnetization congurations, micromagnetic simulations have emerged as a tool to interpret magnetic small-angle scattering experiments. This is a new research avenue that will gain steam in upcoming years thanks to the ever-increasing computational speed 384 and accessibility of open-source simulation tools. 386 It is easy to imagine that also neutron reectometry, GISANS, and also XRMS will benet from a combined data analysis with micromagnetic simulations in upcoming years. In general, it can be said that improved tools for data analysis (such as soware and simulations) are key to making small-angle scattering techniques more accessible for non-expert users. In particular, micromagnetic simulations have the huge advantage of basically translating the oen non-intuitive reciprocal data into comprehensible real-space images. However, micromagnetics assumes a continuous magnetization, and thus to simulate the internal magnetization prole of ne particles, e.g., to elucidate the inuence of structural defects, spin coupling across an interface, and magnetic frustration, atomistic simulations may be better suited. 153 Regarding the calculation of interparticle correlations in MNP assemblies to resolve short-and long-range order Monte-Carlo simulations can be employed as demonstrated recently for articial spin ice. 387 For data tting, tools based on Bayesian statistics are developed to improve the predicitive power of model ts of neutron reectometry 388 and SANS 389 data and maximize the information density. Similar to modelts also inverse Fourier transforms are usually ill-posed problems, and thus similiar approaches have been introduced in recent years to nd the most probable solution 390 or by using image processing inspired approaches such as the singular value decomposition 132 or fast iterative method to reveal the real-space 2D correlations. 391 In this context, it is safe to assume that also machine learning will massively contribute in upcoming years. In fact, for neutron reectometry rst approaches are published which aim to achieve an automatized data analysis to extract parameters from scattering data without expert knowledge. 392,393 This concept can be easily transferred to other scattering techniques such as SANS to classify the most appropriate model. 394 Regarding future experiments, we think that combining small-angle scattering experiments with micromagnetic simulations could be particularly useful to investigate exotic nanoparticles such as hollow particles 42 including nanorings, 43 nanotubes, 44 and other shape-anisotropic hollow particles, 45 or nanodots 46 and nano-octopods. 47 Recently, it was shown that the coercivity of MNPs is enhanced by the exchange coupling at the interface of ferrimagnetic and antiferromagnetic selfassembled monolayers. 395 For this and similar systems, reectometry, GISANS, and XRMS can reveal the correlation between structural and magnetic ordering between the layers over a large size range with nanometer resolution. Micromagnetic simulations allow linking the microscopic structure to the corresponding macroscopic magnetic properties. Thus, these studies will help to optimize the particle properties for specic applications.
In addition to static simulations, dynamic micromagnetic simulations can be performed, which will be invaluable for the analysis of time-resolved data. So far, most stroboscopic smallangle scattering measurements of MNPs were performed to monitor structural changes and the diffusional motion of nanoparticles. Such experiments can be easily extended to study clustering dynamics and in combination with dynamic micromagnetic simulations would reveal the structural and magnetic properties of supraparticles 30 or particle clusters. 28 The big strength of stroboscopic small-angle scattering techniques (e.g. SANS or coherent-XRMS) is that they can also resolve the internal magnetization dynamics of the MNPs. Stroboscopic small-angle scattering techniques have the potential to access unique information regarding the relaxation dynamics of MNP samples when combined with micromagnetic simulations. Such experiments enable to study structure formation of MNPs in alternating magnetic elds, similar to what is already done in static elds, or resolve the time-modulated internal magnetization prole of MNPs to compare it with their static prole. By combining simultaneous magnetic hyperthermia experiments with small-angle scattering, 396 a fundamentally new understanding of the microscopic dynamics governing magnetic heating could be obtained and systematic studies (varying the particles sizes, morphology, etc.) will help to optimize the particle properties for hyperthermia applications. These simultaneous, in situ measurements would further give insight into the local modications while heating MNPs 12 and catalytic applications such as CO 2 hydrogenation 13 and electrolysis. 14 In this context, we believe that NSE and especially MIEZE-SANS will gain signicant popularity for MNP characterization once the methodology is better established. With MIEZE, spin dynamics in the picosecond to nanosecond regime are accessible. This time range is especially interesting to study transversal, intra-well moment relaxation in MNPs. 397 Recent experiments revealed that the magnetic heating is signicantly increased for some samples compared to classical magnetic hyperthermia with GHz-frequencies elds and low eld amplitudes of around 0.2 mT. 398 The extremely high heating rates of 150 K s À1 were achieved when the excitation frequency matched the characteristic transversal relaxation frequency of the particle moments. Additionally, within densely packed MNP assemblies, spin-waves may form due to a dipolar coupling between the precessing spins. 367 The expected time and length scales are accessible with MIEZE-SANS and should be investigated similar to studies performed on bulk ferromagnets. 370 Finally, we want to mention another small-angle scattering technique that may be useful for future MNP studies. Similar to NSE, which encodes changes in neutron velocity due to energy transfer, Spin-echo SANS (SESANS) utilizes the Larmor precession of a neutron to resolve minute scattering angles. SESANS allows observing microstructures in a single measurement over three orders of magnitude from nm to several mm, well beyond the conventional SANS regime. It has been successfully used to study the structural properties of non-magnetic NP systems, such as porous silica particles. 399 SESANS can also be applied to magnetic samples with sufficient magnetic scattering contrast. 400 SEMSANS, a comparable approach to MIEZE with neutron spin manipulation before the sample, enables the measurement of spatial magnetic correlations with exible and varying magnetic eld congurations. 401 The technique could be benecial to characterize large MNPs and dense assemblies thereof, including magnetic superstructures, and to study their eld-dependent response.

Author contributions
DH and PB conceived the idea, devised the project, and draed the manuscript. PB wrote the introduction, and summary, and perspectives. MB contributed the SasView computations and designed the artwork. SE and DB wrote the section on micromagnetic simulation. KC wrote the overview on X-ray magnetic scattering and spectroscopy. DAV contributed with DH to the sections on small-angle X-ray and neutron scattering and timeresolved in situ measurements. AQ and SD authored the part on Reectometry and grazing-incidence scattering. JKJ took the lead on neutron spin-echo techniques. AM provided critical feedback on all techniques and the micromagnetic simulations. All authors critically read and discussed the manuscript.

Conflicts of interest
There are no conicts to declare.