Third-order effect in magnetic small-angle neutron scattering by a spatially inhomogeneous medium

Abstract

Magnetic small-angle neutron scattering (SANS) is a powerful tool for investigating nonuniform magnetization structures inside magnetic materials. Here, we consider a ferromagnetic medium with weakly inhomogeneous uniaxial magnetic anisotropy, saturation magnetization, and exchange stiffness, and derive, to second order in the amplitudes of the inhomogeneities, the micromagnetic solutions for the equilibrium magnetization textures. Further, we compute the corresponding magnetic SANS cross section up to the third order. For the special case of scattering geometry where the incident neutron beam is perpendicular to the applied magnetic field, twice the cross section along the direction orthogonal to both the field and the neutron beam cancels the cross section along the field direction in the second order. This cancellation does not depend on the defect shape and amplitudes of the exchange inhomogeneities. Hence, such a cross-section difference has only a third-order contribution in the amplitudes of the inhomogeneities. It provides a separate gateway for a deeper analysis of the sample's magnetic structure. We derive and analyze analytical expressions for the dependence of this difference on the scattering-vector magnitude for the case of spherical Gaussian inhomogeneities.

Magnetic Small-Angle Neutron Scattering

“…The main task is to derive expressions for these functions based on a particular microstructural model. In this section we briefly summarize the recent developments regarding the analytical computation of the cross sections using the theory of micromagnetics Metlov and Michels, 2015;Mettus and Michels, 2015;Metlov and Michels, 2016;Michels et al, 2016). Micromagnetics is a phenomenological continuum theory which has been developed in order to compute the magnetization vector field M of an arbitrarily shaped ferromagnetic body, provided that the applied magnetic field, the geometry of the ferromagnet, and the magnetic material's parameters are known (Brown, 1963;Aharoni, 1996;Kronmüller and Fähnle, 2003).…”

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

“…These are borrowed from nuclear SANS and do not properly account for the existing spin inhomogeneity inside magnetic nanoparticles. On the other hand, analytical as well numerical computations of the magnetic SANS cross section [17][18][19][20][21][22][23][24][25][26][27] strongly suggest that for the analysis of experimental magnetic SANS data the spatial nanometer scale variation of the orientation and magnitude of the magnetization vector field must be taken into account, and that macrospin-based models-assuming a uniform magnetization-are not adequate.…”

“…Theoretical descriptions of magnetic SANS are based on Brown's static equations of micromagnetics [28], which are a set of nonlinear partial differential equations for the magnetization along with complex boundary conditions on the sample's surface [29]. Therefore, closed-form analytical results for the SANS cross section are restricted to special limiting cases such as the approach-to-saturation regime, where the governing equations can be linearized [17,[21][22][23]26,27]. Recently, we have carried out numerical micromagnetic computations to study the magnetic SANS cross section of microstructural-defect-free spherical nanoparticles during their transition from the single-domain to the multidomain state [30,31].…”

Magnetic neutron scattering by magnetic vortices in thin submicron-sized soft ferromagnetic cylinders

“…Nevertheless, despite the “success” of the magnetic SANS technique, the underlying theoretical framework is still at an early stage and a more fundamental understanding needs to be developed in order to solve the new challenges that magnetism-based nanotechnologies are dealing with. Whereas for bulk ferromagnets the theory of magnetic SANS has recently been developed 20 21 , there exists the open problem of calculating the magnetic SANS cross section of isolated magnetic nanoparticles embedded in a nonmagnetic matrix . This is the prototypical sample microstructure in most magnetic SANS experiments.…”