research papers
On the angular anisotropy of the randomly averaged magnetic neutron scattering
of nanoparticles^{a}Department of Physics and Materials Science, University of Luxembourg, 162A avenue de la Faiencerie, L1511 Luxembourg, Grand Duchy of Luxembourg
^{*}Correspondence email: michael.adams@uni.lu
The magnetic smallangle neutron scattering (SANS) e.g. uniaxial, cubic), an anisotropic magnetic SANS pattern may result, even in the remanent state or at the coercive field. The case of inhomogeneously magnetized particles and the effects of a and interparticle correlations are also discussed.
of dilute ensembles of uniformly magnetized and randomly oriented Stoner–Wohlfarth particles is calculated using the Landau–Lifshitz equation. The focus of this study is on the angular anisotropy of the magnetic SANS signal as it can be seen on a twodimensional positionsensitive detector. Depending on the symmetry of the magnetic anisotropy of the particles (Keywords: nanoscience; magnetic scattering; computational modeling; nanostructure; neutron scattering; angular anisotropy; Stoner–Wohlfarth particles.
1. Introduction
In a magnetic smallangle neutron scattering (SANS) experiment, the angular intensity distribution of the scattered neutrons on the twodimensional positionsensitive detector usually provides the first information on the magnetic microstructure of the sample under study. When an external magnetic field B is applied and varied during the experiment, such images can yield useful information on the degree of magnetic saturation (at large fields), on the presence of cloverleafshaped angular anisotropies (at intermediate fields), or whether or not the magnetic moments are randomly distributed (at remanence or at the coercive field). The angular anisotropy of the magnetic SANS can have many origins (Michels, 2021), e.g. (a) it can be due to the dipolar nature of the interaction between the of the neutron and the that is formed by the unpaired electrons of the sample, (b) it might be related to the magnetic interactions within the sample such as the magnetodipolar energy between magnetic moments, anisotropic exchange interaction or magnetic anisotropy, or (c) it can be due to the presence of a texture in the microstructure of the material.
One of the simplest examples is an ideal Langevin superparamagnet, which, by definition, consists of randomly oriented noninteracting singledomain nanoparticles (macrospins) that are embedded in a rigid nonmagnetic matrix. The magnetic behavior of this system is determined by the B/T ratio, where T is the absolute temperature. At remanence and not too low T, due to the randomizing effect of the thermal energy, the macrospins are randomly oriented in the matrix and the average system magnetization vanishes. The ensuing magnetic SANS is then isotropically distributed in the detector plane. Applying a magnetic field induces an average magnetization, which may result in the appearance of an angular anisotropy of the scattering pattern [see the discussion in Section 5.1.2 of Michels (2021)].
Here, we consider the case of a statistically isotropic dilute ensemble of identical magnetic nanoparticles, so that case (c) is excluded as a source of the scattering anisotropy. Temperature effects are not taken into account. The particles are assumed to be in a singledomain state during the magnetizationreversal process, which implies that the only samplerelated angular anisotropy that eventually becomes visible on the detector is due to magnetic anisotropy. As we will see below, cases (a) and (b) can be disentangled from one another. The magnetic nanoparticle ensemble is treated within the well known Stoner–Wohlfarth model (Stoner & Wohlfarth, 1948), which is a workhorse in magnetism, since it is the simplest approach for producing hysteresis effects. The Stoner–Wohlfarth model considers a system of noninteracting singledomain particles in the presence of an applied magnetic field. The particles exhibit magnetic anisotropy, which may have its origin in dipolar shape anisotropy and/or in spinorbitinteractionrelated magnetocrystalline anisotropy. We analyze the role played by the magnetic anisotropy for the angular anisotropy of the twodimensional magnetic SANS of Stoner–Wohlfarth particles. The Landau–Lifshitz (LL) equation of motion for the magnetization is employed to determine the magnetic equilibrium state and to calculate the corresponding magnetic SANS signal and the pairdistance distribution function.
This article is organized as follows. Section 2 displays the well known equations for the magnetic SANS of a dilute ensemble of uniformly magnetized singledomain particles that are rigidly embedded in a nonmagnetic matrix. We consider the two most often used scattering geometries, which have the externally applied magnetic field either perpendicular or parallel to the incoming neutron beam. Section 3 briefly recapitulates the basic expressions of the Stoner–Wohlfarth model, while Section 4 presents and discusses the results for the SANS observables. We comment on the case of inhomogeneously magnetized particles and on the effect of interparticle correlations (dense packing). Finally, Section 5 summarizes the main findings of this work. In the supporting information of this article, we provide several movies that feature the average magnetization, the two and onedimensional magnetic SANS cross sections, as well as the pairdistance distribution function, correlation function, and anisotropy parameter during the magnetizationreversal process assuming different magnetic anisotropy symmetries (compare the six cases in Table 1).
2. Magnetic SANS of a dilute ensemble of singledomain particles
Magnetic SANS experiments are usually conducted with the external magnetic field B either applied perpendicular (⊥) or parallel (∥) to the wavevector k_{0} of the incoming neutron beam [compare Figs. 1(a) and 1(b)]. For these two scattering geometries, the macroscopic elastic magnetic SANS dΣ_{M}/dΩ at momentumtransfer or scattering vector q can be expressed as (Michels, 2021)
and
In these equations, V is the scattering volume, b_{H} = 2.91 × 10^{8} A^{−1}m^{−1} denotes the magnetic scattering length in the smallangle regime, are the Cartesian Fourier components of the magnetization vector field M(r) = [M_{x}(r), M_{y}(r), M_{z}(r)], the index i refers to the orientation of particle i, the index k keeps track of the appliedfield value, the asterisk denotes the complexconjugated quantity and the 〈...〉_{i} notation is explained below in equation (7). In the perpendicular and parallel scattering geometries, the scattering vector is given by and , where the angle θ is measured between q_{⊥} and B ∥ e_{z} and q_{∥} and e_{x}, respectively. Note that B ∥ e_{z} in both geometries. Equations (1) and (2) neglect interparticle interference effects and are valid for a dilute scattering system.
In the main part of this article we exclusively focus on the perpendicular scattering geometry, and we refer to the supporting information for movies that show results for the parallel geometry.
The Fourier transform of the magnetization vector field of a nanoparticle is defined by
which, for a uniformly magnetized particle, can be simplified to
In going from equation (3) to equation (4), we have expressed the (constant) magnetization vector as M = M_{0}m, where M_{0} is the saturation magnetization and m is a unit vector along M. The remaining integral in equation (4) is the well known formfactor integral (over the volume V_{p} of the particle), which is analytically known for many particle shapes. For spherical particles (with radius R), equation (4) can be further simplified to
where j_{1}(z) denotes the firstorder spherical Bessel function. For more complicated particle shapes (e.g. cylinders or flat discs), an additional average over the particle orientation might be required to obtain the only qdependent form factor. As a reminder, we consider a dilute system of N identical spherical singledomain particles, where, at a given value of the applied field B^{k}, each particle i has its own random orientation of magnetic easy axes with respect to B (to be further specified in Section 3). For this situation, we may express the Fourier components as
The magnetic SANS cross sections at the kth magnetic field value, averaged over all the random easyaxis orientations i, are then given by equations (1) and (2), where the bracket notation of the mean operator is defined as follows:
and similar for the other Fourier components. Since the spherical Bessel function is a scalar prefactor to the Fourier transform of the magnetization vector, we can simplify the magnetic SANS cross sections as follows:
and
such that the magnetic SANS kth value of the magnetic field as
directly follows from the (real valued) realspace correlation functions of the components of the magnetization vector. We therefore define the crosscorrelation matrix corresponding to thewhere more explicitly written the components are defined as
such that the magnetic SANS cross sections are written as
and
In the general case of an inhomogeneous magnetization distribution, i.e. m = m(r), these formulations of the magnetic SANS cross sections only correspond to the first term in a Taylor series expansion. From equations (12) and (13) we see that the magnetic SANS gives quite different insights into the magnetization structure than the hysteresis loop: the latter contains information about the firstorder moments, while the former yields information about the secondorder moments of the magnetization vector field. The magnetic SANS of uniformly magnetized particles is anisotropic (θ dependent) when the terms in the brackets on the second lines of equations (12) and (13) add up to yield a resulting net θ dependence. This statement can be further specified by noting the symmetry of the equations in the parallel scattering geometry [equation (13)], which is absent in the perpendicular case [equation (12)]. For B ∥ k_{0} the two transversal magnetization components lie in the detector plane, whereas for B ⊥ k_{0} only one transversal component lies in the detector plane and the other one is pointing along the incidentbeam direction [compare Figs. 1(a) and 1(b)]. Since for the hereconsidered Stoner–Wohlfarth system with B ∥ e_{z} in both scattering geometries we have Γ_{xx} = Γ_{yy} and Γ_{xy} = Γ_{xz} = Γ_{yz} = 0 (see Appendix A), it becomes immediately clear that the twodimensional dΣ_{M, ∥}/dΩ is isotropic at all fields, while dΣ_{M, ⊥}/dΩ is generally anisotropic. Moreover, for a constant magnetization vector field, the orientationally averaged formfactor integral in equation (4) can be analytically or numerically computed for many particle shapes, with the result that the prefactor in equations (12) and (13) is only a function of the magnitude of q. Therefore, the above result – isotropy of dΣ_{M, ∥}/dΩ and general anisotropy of dΣ_{M, ⊥}/dΩ – is true for arbitrary particle shapes and also in the presence of a distribution of particle sizes, as long as all the randomly oriented particles are in a singledomain state.
The azimuthally averaged SANS cross sections are straightforwardly obtained as
and
Here, for the 2π azimuthal average, we see that the yz and xy crosscorrelation terms vanish, and only the autocorrelation terms remain. The pairdistance distribution functions are obtained as
and
The related correlation functions c^{k}(r) = p^{k}(r)/r^{2} are
and
The are real numbers, which depend on the applied field and on the symmetry of the magnetic anisotropy of the particles. We then see that – within the present Stoner–Wohlfarth approach – the I(q), p(r) and c(r) are identical for the perpendicular and parallel scattering geometries, except for a numerical prefactor.
To quantify the angular anisotropy of the twodimensional magnetic SANS B ⊥ k_{0}) the following number [compare Fig. 1 with equation (12)]:
we introduce (forFor the parallel scattering geometry, where B is perpendicular to the detector plane, we calculate A^{k} similar to equation (20) as the ratio of integrated intensities along the horizontal and vertical directions on the detector [compare with equation (13)]:
In the following, the quantities A_{⊥} and A_{∥} are denoted as the anisotropy parameters. As discussed before, since Γ_{xx} = Γ_{yy}, we find that A_{∥} = 1 at all fields, while generally A_{⊥} ≠ 1 (see Appendix A). When the Γ_{zz} dominates, we have A_{⊥} < 1 and the twodimensional magnetic SANS signal exhibits a dominant anisotropy, whereas for dominant Γ_{xx} and Γ_{yy} (A_{⊥} > 1) we observe a type angular anisotropy.
3. Recap: Stoner–Wohlfarth model
In this chapter, we recapitulate the basic ideas of the Stoner–Wohlfarth model (Stoner & Wohlfarth, 1948), which considers a magnetically anisotropic singledomain particle in the presence of an applied magnetic field B (assumed here to be parallel to the z direction of a Cartesian laboratory coordinate system). The origin of the magnetic anisotropy can be due to shape anisotropy and/or magnetocrystalline anisotropy. Here, we consider identical particles possessing magnetocrystalline anisotropy only. Note also that spherical particles do not exhibit shape anisotropy and thermal effects are ignored. By denoting with the magnetic anisotropy energy density in the (global) laboratory coordinate system, the total energy density ω^{ik} of a particle i at field k is commonly expressed as
where M_{0} is the saturation magnetization of the material. The two most common forms of magnetocrystalline anisotropy either exhibit uniaxial (u) or cubic (c) symmetry. The corresponding mathematical expressions for the magnetic anisotropy energy densities, in the local coordinate frame of the particle, are the following:
and
where K_{u} and K_{c} are the temperaturedependent anisotropy constants (in J m^{−3}). Depending on their relative magnitude and the signs of the anisotropy constants, different easy axes are obtained (Kronmüller & Fähnle, 2003). The corresponding effective magnetic fields (in Tesla) are then readily obtained as
and
The different (random) particle orientations i are obtained by rotations in threedimensional space (change of basis). This is accomplished by using a rotation matrix R_{i} that is parametrized by (random) Euler angles γ_{i}, δ_{i} and ε_{i} (with 0 ≤ γ_{i} ≤ 2π, 0 ≤ δ_{i} ≤ π and 0 ≤ ε_{i} ≤ 2π) (Goldstein, 1991), so that the total effective field is calculated as follows:
where
and the superscript T refers to the transpose of the matrix. Here we adopt a z–y–z rotation sequence. The procedure of obtaining the Euler angles starts with uniformly distributed random numbers a_{i}, b_{i} and c_{i} in the threedimensional unit cube, such that 0 ≤ a_{i}, b_{i}, c_{i} ≤ 1. As a randomnumber generator we use the lowdiscrepancy Sobol sequences (https://www.mathworks.com/help/stats/sobolset.html). In order to achieve a uniform distribution of random angles on the unit sphere, we use the following transformations:
and
To obtain the static equilibrium magnetization, we insert the expression for into the LL equation (Bertotti, 1998), which describes the magnetization dynamics:
where γ_{G} = 1.76 × 10^{11} T^{−1}s^{−1} is the gyromagnetic ratio and η is the damping constant. Following the temporal evolution of the LL equation, the static spin structure of nanomagnet i at field k can be obtained. Repeating (at fixed k) these simulations N times for different easyaxis orientations allows us to compute the averages that determine the magnetic SANS More specifically, the hysteresis loop of the ensemble of spherical nanomagnets then follows from the averaged magnetization projected along the z direction,
In addition to , we also calculate the field dependence of the transversal magnetization components, and , as well as the field loops of the components of the crosscorrelation matrix Γ_{αβ} (with α, β ∈ {x, y, z}) [equation (11)]. As we have seen in Section 2, these are of particular relevance for the magnetic SANS cross section.
In the numerical computations, we used the following parameters: η = 3 × 10^{11} T^{−1}s^{−1} and an integration time step of 5 × 10^{−15} s. Typically, K = 2000 discretization points for the applied magnetic field and N = 10 000 samples of different orientations for the easy axes of the particles (angles γ_{i}, δ_{i} and ε_{i}) were used. For further details on the SANS simulation methodology using the LL equation, we refer to Adams et al. (2022a).
4. Results and discussion
In our analysis, we consider the following six cases for Stoner–Wohlfarth particles with uniaxial and cubic anisotropy: (i) k_{u1} = +1 and k_{u2} = 0, (ii) k_{u1} = −1 and k_{u2} = 0, (iii) k_{u1} = −0.5 and k_{u2} = +0.5, (iv) k_{c1} = +1 and k_{c2} = 0, (v) k_{c1} = −1 and k_{c2} = 0, and (vi) k_{c1} = −1 and k_{c2} = +9. The k_{u, c} (in Tesla) are related to the K_{u, c} (in J m^{−3}) via k_{u, c} = 2K_{u, c}/M_{0} [compare equations (25) and (26)]. Minimization of the anisotropy energy densities [equations (23) and (24)] shows that these combinations of anisotropy constants correspond to the following well known easyaxis orientations in hexagonal and cubic single crystals (Kronmüller & Fähnle, 2003): (i) easy c axis, (ii) easy basal plane, (iii) easy cone with opening angle , (iv) 〈100〉 directions, (v) 〈111〉 directions and (vi) 〈110〉 directions.
Table 1 contains the values of the second moments Γ_{αα} of the components of the magnetization vectors at selected points on the hysteresis loop (remanence and coercivity), and for the different anisotropy symmetries. The hysteresis loops and the full field dependencies of the autocorrelations are shown in Appendix A. As an example, for uniaxial Stoner–Wohlfarth particles of case (i), Fig. 2 depicts the results for the SANS observables. Inspection of the table entries for the Γ_{αα} and comparison with the magnetic SANS cross sections [equations (12) and (13)] reveals that only case (i) yields, in the perpendicular scattering geometry, an isotropic twodimensional SANS image at remanence. In all other cases, we find (for B ⊥ k_{0}) an anisotropic magnetic SANS pattern at remanence and at the coercive field. By contrast, in the parallel scattering geometry (B ∥ k_{0}), we observe (since Γ_{xx} = Γ_{yy}) an isotropic magnetic dΣ_{M, ∥}/dΩ at all fields during the magnetizationreversal process.

Considering case (i), we see that the angular anisotropy of the twodimensional dΣ_{M, ⊥}/dΩ changes strongly between saturation ( type), remanence (isotropic) and the coercive field ( type) [Fig. 2(a)], while the azimuthally averaged dΣ_{M, ⊥}/dΩ changes relatively little between these situations [Fig. 2(c)]. Decreasing the field from saturation (where Γ_{xx} = Γ_{yy} = 0 and Γ_{zz} = 1) to zero field and to B_{c}, we observe an increase of the onedimensional dΣ_{M, ⊥}/dΩ and of the pairdistance distribution function p_{⊥}(r) [Fig. 2(d)] and of the correlation function c_{⊥}(r) [Fig. 2(e)] [compare equations (16) to (19)]. From these results it may be concluded that, depending on the anisotropy symmetry of Stoner–Wohlfarth particles, an anisotropic magnetic SANS pattern is (generally) obtained; see Fig. 2(f) for the anisotropy parameter A_{⊥}(B). In experimental studies, where often the twodimensional total (nuclear and magnetic) dΣ_{⊥}/dΩ is analyzed, one should therefore be cautious in assuming that an isotropic pattern is to be expected at characteristic field values such as at remanence or at the coercive field. To access the purely magnetic SANS in unpolarized experiments, the subtraction of the total dΣ_{⊥}/dΩ at a field close to magnetic saturation from the data at lower fields might help. As shown in the work by Bersweiler et al. (2019) on Mn—Zn ferrite nanoparticles, an isotropic total dΣ_{⊥}/dΩ at zero field can then turn into an anisotropic purely magnetic signal, in this way providing access to the magnetic correlations. Of course, polarization analysis also yields the purely magnetic SANS albeit with much more effort regarding the experimental setup and the datareduction procedure.
We refer to the supporting information of this article, where several movies that feature the average magnetization and the SANS observables during the magnetizationreversal process are provided. More specifically, for the six cases specified in Table 1, we display, for both scattering geometries, the two and onedimensional magnetic SANS cross sections dΣ_{M, ⊥}/dΩ and dΣ_{M, ∥}/dΩ, the pairdistance distribution functions p_{⊥}(r) and p_{∥}(r), the correlation functions c_{⊥}(r) and c_{∥}(r), and the anisotropy parameters A_{⊥} and A_{∥}.
As mentioned already in Section 2 when discussing equations (12) and (13), within the present Stoner–Wohlfarth approach, the presence of a results in the smearing of the formfactor oscillations (i.e. affects the q dependence) but leaves the angular anisotropy of dΣ_{M, ⊥}/dΩ unchanged.
So far the discussion has been based on uniformly magnetized Stoner–Wohlfarth particles. For nonuniformly magnetized nanoparticles, where the magnetization vector field m = m(r) is a function of the position r within the particle, the dΣ_{M, ⊥}/dΩ at remanence or at the coercive field is generally also expected to depend on the angle θ in the detector plane. Nonuniformities in the magnetization distribution of nanoparticles are e.g. caused by surface anisotropy, vacancies or antiphase boundaries (Nedelkoski et al., 2017; Ijiri et al., 2019; Zákutná et al., 2020; Lak et al., 2021; Köhler et al., 2021; Honecker et al., 2022; Adams et al., 2022b,a; Sinaga et al., 2023). Micromagnetic simulations that take into account the relevant interactions such as isotropic exchange, antisymmetric exchange, magnetic anisotropy, Zeeman energy and the magnetodipolar interaction are an important tool for advancing the understanding of magnetic SANS of nanomagnets (Michels, 2021). Unfortunately, due to the nonlinearity of the underlying integrodifferential equations of micromagnetics, numerical simulations have to be carried out.
As an example, we show in Fig. 3 selected results that feature an anisotropic (randomly averaged) dΣ_{M, ⊥}/dΩ at remanence; for details on the micromagnetic SANS simulation methodology see the works of Adams et al. (2022a) and Sinaga et al. (2023). Fig. 3(a) showcases the results of atomistic SANS simulations, where the focus is set on the effect of the Néel surface anisotropy on the spin structure and ensuing magnetic SANS signal of randomly oriented nanoparticles. This particular form of surface anisotropy arises because in a nanomagnet a significant fraction of atoms belong to the surface (with no neighbors on one side), and their magnetic properties such as exchange and anisotropy can be strongly modified relative to the bulk atoms. The snapshot of the realspace spin structure clearly reveals a significant spin disorder in the nearsurface region of the nanoparticle, with a corresponding characteristic type anisotropic magnetic SANS pattern. Fig. 3(b) displays the results for a random ensemble of spherical nanoparticles. In this system (with no surface anisotropy), the magnetization distribution is determined by the dipolar interaction energy, which gives rise to a vortextype spin texture. The randomly averaged dΣ_{M, ⊥}/dΩ also exhibits a pronounced θ dependence.
The simulation results in Fig. 3 were obtained for a dilute set of nanoparticles, i.e. interparticle correlations are not taken into account. When the in a sample increases, positional correlations become important, which is taken into account by the S(q) = , where r_{ij} = r_{j} − r_{i} denotes the vector connecting the position vectors of particles i and j, and the bracket 〈...〉 refers to an orientational average. For magnetic particles, whether uniformly or nonuniformly magnetized, additional correlations become relevant, resulting in the appearance of a magnetic This was realized by Honecker et al. (2020), who showed that the magnetic can deviate significantly from the nuclear (positional) for magnetically interacting nanoparticle ensembles; see also Hayter & Pynn (1982) and Pynn et al. (1983), where the of a magnetically saturated ferrofluid was derived. Since correlations between the particle magnetizations are magnetic field dependent and also anisotropic (Gazeau et al., 2002; Honecker et al., 2020), extending the present Stoner–Wohlfarth approach to higher concentrations (increased magnetodipolar interaction) does presumably not change the main statement of the present work, namely that the magnetic SANS of a randomly oriented nanoparticle ensemble is, in the B ⊥ k_{0} geometry, generally anisotropic. This assertion is supported by the results of largescale micromagnetic simulations on magnetic nanocomposites (Erokhin et al., 2012; Michels et al., 2014), which clearly show that the dipolar interaction results in an anisotropic magnetic SANS in the perpendicular geometry.
5. Conclusions
We have analyzed the angular anisotropy of the magnetic SANS
of spherical Stoner–Wohlfarth particles using the Landau–Lifshitz equation. Depending on the symmetry of the magnetic anisotropy of the particles (uniaxial, cubic), an anisotropic randomly averaged magnetic SANS pattern may result in the perpendicular scattering geometry, even in the remanent or fully demagnetized state. The magnetic scattering in the parallel geometry is, as expected, isotropic. Inhomogeneously magnetized nanoparticles also generally exhibit an anisotropic randomly averaged magnetic SANS response. From the experimental point of view, the subtraction of the total unpolarized (nuclear and magnetic) scattering at saturation from data at lower fields might help to access the intrinsic anisotropy of the particles. Likewise, this can also be achieved by onedimensional polarization analysis via the measurement of the spinflip SANS Since the present Stoner–Wohlfarth simulations are relatively easy to implement, we recommend carrying them out in parallel to experimental investigations on magnetic nanoparticles.APPENDIX A
Magnetization curves and crosscorrelation values of spherical Stoner–Wohlfarth particles with uniaxial and cubic anisotropy
Figs. 4 and 5 show the hysteresis loops and the field dependencies of the second moments of the components of the magnetization vectors, Γ_{αα}(B), from the Stoner–Wohlfarth model for uniaxial and cubic anisotropy. The six cases specified in Table 1 are considered. All crosscorrelations Γ_{αβ} with α ≠ β vanish, and .
Supporting information
Movie for case (i). DOI: https://doi.org/10.1107/S205225252300180X/fs5216sup1.mp4
Movie for case (ii). DOI: https://doi.org/10.1107/S205225252300180X/fs5216sup2.mp4
Movie for case (iii). DOI: https://doi.org/10.1107/S205225252300180X/fs5216sup3.mp4
Movie for case (iv). DOI: https://doi.org/10.1107/S205225252300180X/fs5216sup4.mp4
Movie for case (v). DOI: https://doi.org/10.1107/S205225252300180X/fs5216sup5.mp4
Movie for case (vi). DOI: https://doi.org/10.1107/S205225252300180X/fs5216sup6.mp4
Acknowledgements
Mathias Bersweiler and Ivan Titov are acknowledged for critically reading the manuscript.
Funding information
We thank the National Research Fund of Luxembourg for financial support (AFR Grant No. 15639149 and PRIDE MASSENA Grant).
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