Angular anisotropy landscape of vortex ensembles in polarized small-angle neutron scattering

Abstract

We present a symmetry-resolved classification of two-dimensional spin-flip small-angle neutron scattering (SANS) patterns arising from dilute ensembles of spherical nanoparticles hosting magnetic vortex states. Based on a linear vortex ansatz with an axially symmetric distribution of vortex axes and the corresponding analytical expression for the orientationally averaged spin-flip SANS cross section, we show that the angular scattering patterns organize into four distinct symmetry regimes: a four-fold anisotropy corresponding to coherent field-aligned magnetization, vertical and horizontal two-fold anisotropies associated with aligned and isotropically distributed vortex ensembles, and an isotropic ring-like condition separating the two two-fold regimes. The corresponding symmetry boundaries are obtained analytically and define a compact symmetry landscape in the parameter space of vortex amplitude and vortex-axis distribution width. Comparison with a nonlinear vortex profile shows that these symmetry regions are robust with respect to the detailed radial structure of the vortex core. The angular anisotropies are therefore governed primarily by rotational symmetry and by the statistical distribution of vortex axes, providing a compact and model-transparent classification framework of experimental polarized SANS data.

Figures (3)

• Figure 1: (a) Vortex configuration inside a spherical particle. The color scale encodes the longitudinal magnetization component $m'_z$. (b) Representation of the vortex-axis ensemble on the unit sphere. The red double cone defines the angular cutoff angle $\alpha_{\mathrm{c}}$ with respect to the $\mathbf{e}_z$ axis of the global laboratory frame. The blue spherical caps correspond to the allowed statistical distribution of vortex axes. CW and CCW vortices populate the upper and lower caps with equal probability, reflecting the absence of chirality selection.
• Figure 2: Symmetry-resolved map of the (normalized) two-dimensional spin-flip SANS cross section for a linear vortex ensemble. Each panel shows the analytically calculated spin-flip scattering pattern [up to $(qR)_{\mathrm{max}} = 5.0$] obtained from Eq. (\ref{['eq:SpinFlip2DSANScrossSection']}) for a given combination of the vortex amplitude parameter $m_1 R / m_0$ (vertical axis) and the cone angle $\alpha_{\mathrm{c}}$ (horizontal axis), which describes the statistical distribution of vortex axes within a rotational cone. The model represents an ensemble of linearly varying vortex structures whose axes are distributed within a cone of opening angle $\alpha_{\mathrm{c}}$, rather than a single vortex configuration. The background colors indicate the dominant angular anisotropy of the scattering pattern: four-fold symmetry $\propto 1 - \cos(4\theta)$ (light blue), vertical two-fold symmetry $\propto 1 - \cos(2\theta)$ (dark blue), horizontal two-fold symmetry $\propto 3 + \cos(2\theta)$ (orange), isotropic ring-like scattering $\propto 1$ (green), and intermediate transition regions. The map provides a controlled visual classification tool for experimental two-dimensional spin-flip SANS data, linking observed angular anisotropies to specific regions in the parameter space of vortex amplitude and axis distribution.
• Figure 3: Same as Fig. \ref{['fig2']}, but based on the nonlinear hyperbolic vortex magnetization ansatz Eq. \ref{['eq:HyperbolicVortex']} and numerically computed using the NuMagSANS software package Adams2026NuMagSANS. In this calculation, we used spherical nanoparticles with a radius of $R = 20 \, \mathrm{nm}$ (discretized into a $2\times2\times2 \, \mathrm{nm}^3$ grid) and $1024$ samples for the rotation angles $\alpha, \beta$. The vortex parameter $\nu$ in Eq. \ref{['eq:HyperbolicVortex']} is selected as $\nu = \mu = m_1 R/m_0$, as the hyperbolic vortex \ref{['eq:HyperbolicVortex']} reduces (in a first-order Taylor expansion at $\rho'=0$) to the linear vortex ansatz \ref{['eq:LinearVortexAnsatz']}. The resulting two-dimensional scattering cross sections are shown up to a maximum scattering vector of $q_{\mathrm{max}} = 0.25 \, \mathrm{nm}^{-1}$. The code used to generate the data is available on Zenodo Adams2026_Landscape.