🔬 mesoscale physics

Micromagnetic simulation of neutron scattering from spherical nanoparticles: Effect of pore-type defects

This paper utilizes micromagnetic simulations to investigate how pore-type defects and dipolar interactions influence the magnetic small-angle neutron scattering signatures and pair-distance distribution functions of spherical nanoparticles, providing experimentalists with valuable guidelines for interpreting their data.

Original authors: Evelyn Pratami Sinaga, Michael P. Adams, Mathias Bersweiler, Laura G. Vivas, Eddwi H. Hasdeo, Jonathan Leliaert, Philipp Bender, Dirk Honecker, Andreas Michels

Published 2026-08-19

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CC BY 4.0

Original authors: Evelyn Pratami Sinaga, Michael P. Adams, Mathias Bersweiler, Laura G. Vivas, Eddwi H. Hasdeo, Jonathan Leliaert, Philipp Bender, Dirk Honecker, Andreas Michels

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

To understand the magnetic heart of modern technology, from the hard drives in our computers to the sensors in our smartphones, scientists must look inside the tiny building blocks of these materials. These blocks are often nanoparticles, spheres of magnetic metal so small that a single one is invisible to the naked eye. Inside these spheres, the atoms act like billions of tiny compass needles, all trying to align with a magnetic field. For decades, scientists have used a technique called small-angle neutron scattering to peek inside these particles. They fire a beam of neutrons at the sample and watch how the particles bounce off the magnetic fields created by the atoms inside. The pattern of this bounce reveals the internal structure of the magnetism. However, a major problem has long plagued this method: the standard way of analyzing the data assumes that every compass needle inside the nanoparticle points in the exact same direction, like a perfectly synchronized army. In reality, especially in larger particles, these needles often twist and turn into complex, swirling patterns to save energy. This mismatch between the simple assumption and the complex reality has made it difficult to accurately interpret what is happening inside these tiny magnets.

A team of researchers has now taken a fresh approach to solve this puzzle by building a detailed digital model of these nanoparticles. Instead of relying on the old, simplified assumption that the magnetism is uniform, they used powerful computer simulations to map out exactly how the magnetic needles behave when the particle contains imperfections. Real-world nanoparticles are rarely perfect; they often contain tiny holes or pores where the magnetic material is missing. The researchers wanted to know how these empty spaces, which act like voids in the magnetic structure, change the way the neutrons scatter. They created a virtual sphere of iron, about 40 nanometers in diameter, and filled it with a grid of tiny cubes. To mimic the real world, they randomly removed some of these cubes to create pores, representing defects that might exist in a physical sample. They then ran simulations to see how the magnetic needles arranged themselves in the presence of these holes and an external magnetic field, calculating the resulting neutron scattering pattern for each scenario.

The simulations revealed that the internal structure of the magnetism is far more intricate than previously thought. In a perfect, defect-free sphere of this size, the magnetic needles do not simply line up in a straight row. Instead, driven by the long-range forces between them, they curl into a vortex, a swirling pattern that minimizes the energy of the system. When the researchers introduced the pore-type defects, this swirling pattern did not disappear; rather, it became slightly disordered and random, but the fundamental vortex structure remained surprisingly robust. The presence of these holes did not completely scramble the magnetic order. This finding is crucial because it suggests that the signature of these complex internal swirls is strong enough to be detected even when the material is not perfect. The researchers found that the way the neutrons scatter changes in a specific, predictable way when these vortex structures are present, creating a distinct pattern that differs significantly from the smooth, uniform patterns predicted by older models.

The study also examined how the size of the particles affects these results. In the real world, a sample of nanoparticles is never made of spheres that are all exactly the same size; there is always a mix of small and large ones. The team simulated a collection of particles with a range of sizes to see if this variety would wash out the clear signals of the internal magnetic structures. They found that while a mix of sizes does blur the details slightly, the unique oscillating patterns caused by the vortex structures and the defects remain visible. This means that experimentalists do not need perfectly uniform samples to see these effects. The simulations showed that the scattering patterns contain specific "fingerprints" of the internal spin disorder and the pore defects, which can be identified even in messy, real-world samples.

One of the most important outcomes of this work is a clear demonstration that the old, simple models are no longer sufficient. The researchers showed that assuming a uniform magnetization leads to a misunderstanding of the data. The dipolar interaction, a long-range force between the magnetic moments, is the primary driver behind these complex, non-uniform structures. Without accounting for this force, the simulations would have predicted a simple, uniform state that does not match reality. By including this force and the presence of defects, the new model provides a much more accurate guide for interpreting experimental data. The researchers emphasize that their results serve as a roadmap for experimentalists, helping them distinguish between the effects of particle size, the presence of defects, and the complex internal swirling of the magnetic spins.

The study concludes that while the pore defects do introduce disorder, they do not destroy the underlying magnetic architecture of the nanoparticle. The vortex-like structures persist, and their influence on the neutron scattering data is significant. This insight allows scientists to move beyond the limitations of the "superspin" model, which treats the entire particle as a single, uniform magnet. Instead, they can now look for the specific signatures of these internal textures. The work suggests that to truly understand magnetic nanoparticles, one must look at the three-dimensional vector field of the magnetization, rather than just a simple average. While the current simulations treat defects as completely empty holes, the researchers note that a more realistic treatment might involve reducing the magnetic strength in those areas rather than removing it entirely. Nevertheless, the findings provide a solid foundation for future research, offering a way to decode the complex magnetic landscapes hidden inside the tiny spheres that power our technology.

Technical Summary: Micromagnetic Simulation of Neutron Scattering from Spherical Nanoparticles with Pore-Type Defects

Problem Statement
Magnetic small-angle neutron scattering (SANS) is the primary method for investigating spin structures on the mesoscopic length scale (1–100 nm) within magnetic materials. Experimental data increasingly reveal complex, nonuniform, canted, or core-shell spin textures in magnetic nanoparticles. However, standard data analysis relies heavily on structural form-factor models borrowed from nuclear SANS, which assume uniform magnetization (macrospin models). These models fail to account for the spatial nanometer-scale variations in the orientation and magnitude of the magnetization vector field. While analytical and numerical studies suggest that non-uniformity must be considered, closed-form analytical solutions are restricted to limiting cases (e.g., approach-to-saturation). Furthermore, the specific signatures of microstructural defects—such as surface anisotropy, vacancies, and antiphase boundaries—on magnetic SANS observables remain to be fully characterized.

Methodology
The authors employ numerical micromagnetic simulations using the GPU-based open-source software MuMax3 to model spherical iron (Fe) nanoparticles. The study extends previous work on defect-free particles to include:

Pore-type defects: Modeled as randomly assigned computational cells with zero saturation magnetization ( $M_s = 0$ ), representing nonmagnetic pores.
Particle-size distribution: A lognormal distribution function is applied to simulate ensembles of particles with varying diameters.

The simulations minimize the total magnetic Gibbs free energy, accounting for four contributions: Zeeman energy (external field), dipolar (magnetostatic) interaction energy, magnetocrystalline anisotropy energy (cubic), and isotropic exchange energy. Material parameters for Fe are used ( $M_s = 1700$ kA/m, $A = 1.0 \times 10^{-11}$ J/m, $K_1 = 47$ kJ/m $^3$ ).

The magnetic SANS cross section ( $d\Sigma_M/d\Omega$ ) is calculated by computing the Fourier transform of the equilibrium magnetization vector field $\mathbf{M}(\mathbf{r})$ and applying the standard magnetic SANS cross-section equation for unpolarized neutrons in a perpendicular scattering geometry. The pair-distance distribution function $p(r)$ and correlation function $c(r)$ are derived from the azimuthally averaged cross section. The study considers an ensemble of randomly oriented particles to match experimental conditions, ignoring interparticle interference and temperature effects.

Key Results

Spin Structures and Dipolar Interactions: For 40-nm particles (larger than the critical single-domain size of ~13.6 nm), the dipolar interaction drives the formation of vortex-type spin configurations in defect-free particles. Introducing pore-type defects (concentrations $x_d$ up to 20%) progressively disorders these vortex structures, leading to more random spin configurations.
Hysteresis and Magnetization: Increasing defect concentration reduces the reduced remanence (from ~0.11 to ~0.04) and coercivity (from 35 mT to 18 mT) while increasing the saturation field. The inclusion of dipolar energy is shown to be critical; without it, the system behaves as a coherent-rotation Stoner-Wohlfarth particle, whereas with dipolar energy, distinct hysteresis regions corresponding to vortex nucleation and multi-domain transitions appear.
SANS Observables ( $d\Sigma_M/d\Omega$ ):
- At the remanent state, the scattering patterns are horizontally elongated due to the vortex structure. The Fourier components exhibit angular anisotropy driven by dipolar interactions.
- Defects induce a shift of form-factor oscillations to larger momentum transfers ( $q$ ) and smear these features, mimicking the effect of a particle-size distribution or instrumental resolution.
- At high fields (near saturation), the scattering becomes dominated by the longitudinal Fourier component ( $|f_{Mz}|^2$ ) and exhibits the characteristic $\sin^2\theta$ anisotropy of a saturated microstructure.
Pair-Distance Distribution ( $p(r)$ ): The vortex-type spin structure manifests as an oscillatory $p(r)$ function. This oscillatory signature is found to be remarkably stable against the introduction of pore-type defects. The presence of defects and the resulting internal spin disorder cause a shift in the global minimum of $p(r)$ to larger distances and a reduction in the maximum amplitude, but the overall oscillatory shape persists.
Particle-Size Distribution: Introducing a lognormal size distribution ( $\sigma$ ) smears the form-factor oscillations in $d\Sigma_M/d\Omega$ . The $p(r)$ function is more sensitive to size distribution variations than the correlation function $c(r)$ due to the $r^2$ weighting factor. However, the oscillatory low-field features in $p(r)$ remain robust even with size distributions.

Significance and Claims
The paper claims that the magnetic SANS signal and correlation functions of nonuniformly magnetized nanoparticles cannot be adequately described by superspin models assuming homogeneous spin microstructures. The study highlights that:

Dipolar interactions are the origin of complex magnetization structures (like vortices) and the resulting anisotropic scattering patterns.
Defect Signatures: While pore-type defects significantly alter the real-space spin structure, their direct signature in the SANS observables ( $d\Sigma_M/d\Omega$ , $p(r)$ ) is relatively subtle compared to the dominant effect of the particle's outer surface and the dipolar-induced nonuniformity. The oscillatory $p(r)$ feature associated with vortex structures appears stable against these specific defects.
Methodological Shift: The work advocates for a paradigm shift in analyzing magnetic SANS data, moving from finding a scalar structural profile to determining the three-dimensional vector field of magnetization $\mathbf{M}(\mathbf{r})$ via energy minimization.

The authors conclude that their results serve as a guideline for experimentalists to identify nonuniform vortex-type spin structures and to interpret deviations from standard form-factor models, particularly in the presence of microstructural defects and size distributions. They note that future work could involve more realistic defect modeling (e.g., reduced but non-zero $M_s$ in defects) and the inclusion of magnetoelastic interactions and surface anisotropy boundary conditions.

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