Magnetic small-angle neutron scattering by a nanocrystalline ferromagnet with anisotropic exchange interaction

Abstract

A micromagnetic framework for magnetic small-angle neutron scattering (SANS) is presented that accounts for weak symmetric anisotropic exchange in centrosymmetric nanocrystalline ferromagnets. The exchange interaction is expressed via a general fourth-rank tensor decomposed into isotropic and deviatoric parts. We start with the exchange energy and effective field, assuming weakly fluctuating in space saturation magnetization, solve micromagnetic problem to find spatial distribution of local magnetization vector and compute the averaged (over random orientations of nanocrystals) SANS cross sections. The isotropic part reproduces the classical Heisenberg-type SANS response, while non-zero deviatoric part of the exchange tensor gives rise to new angular harmonics in the magnetic SANS cross section. As a specific example, analytical response functions for an exchange tensor with hexagonal symmetry in perpendicular and parallel scattering geometries are derived. The results provide a basis for identifying and quantifying symmetric exchange anisotropy in magnetic SANS experiments.

Figures (4)

• Figure 1: Typical geometry of a magnetic SANS experiment. Neutrons with incident wave vector $\mathbf{k}$ are scattered into the direction $\mathbf{k}'$, giving rise to the scattering vector $\mathbf{q}=\mathbf{k}'-\mathbf{k}$. The applied magnetic field $\mathbf{H}$ can be oriented either perpendicular or parallel to the beam direction. In the perpendicular geometry, $\mathbf{H}$ is directed along the $Z$ axis, and the detector plane coincides with the $Y$--$Z$ plane.
• Figure 2: Two-dimensional maps in coordinates of $t=h_q/(h_q+1)$ and $\alpha$ of the standard (top row: $h_q^2R_H$ and $h_qR_M$) and hexagonal-exchange-induced (bottom row: $h_qR_{\xi}$, $h_q^2 R_{\xi \kappa}$ and $h_q^2 R_{\xi^2}$) response functions multiplied by a corresponding power of $h_q$ to compensate for the decay at high fields.
• Figure 3: Two-dimensional maps of the standard (isotropic Heisenberg-type) response functions in $(y_q,z_q)$ coordinates at the fixed reduced field parameter $t=h_q/(h_q+1)=0.5$. These plots serve as reference patterns for comparison with the anisotropic response functions shown in Fig. \ref{['fig:anisotropic_maps']}.
• Figure 4: Two-dimensional maps of the anisotropic response functions in $(y_q,z_q)$ coordinates at different values of the reduced field parameter $t=h_q/(h_q+1)$. Top row: linear contribution $R_{\xi}$. Middle row: mixed contribution $R_{\xi\kappa}$. Bottom row: quadratic contribution $R_{\xi^2}$. Columns correspond to $t=0.1$, $t=0.5$, and $t=0.9$, respectively. The evolution of the angular structure with increasing field illustrates the distinct symmetry properties and scaling behavior of the individual anisotropic terms.