  Magnetic small-angle neutron scattering (SANS) is a powerful tool for
investigating nonuniform magnetization structures inside magnetic materials.
Here, considering a ferromagnetic medium with weakly inhomogeneous uniaxial
magnetic anisotropy, saturation magnetization, and exchange stiffness, we
derive the second-order (in the amplitude of the inhomogeneities) micromagnetic
solutions for the equilibrium magnetization textures and compute the
corresponding magnetic SANS cross sections up to the next, third order. We find
that in the case of perpendicular scattering (the incident neutron beam is
perpendicular to the applied magnetic field) if twice the cross section along
the direction orthogonal to both the field and the neutron beam is subtracted
from the cross section along the field direction, the result has only a
third-order contribution (the lower-order terms are canceled). This difference
does not depend on the amplitude of the exchange inhomogeneities and 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
combination on the scattering-vector magnitude for the case of spherical
Gaussian inhomogeneities.
