  We investigate the signature of magnetic surface anisotropy in nanoparticles
in their spin-flip neutron scattering cross section. Taking into account the
isotropic exchange interaction, an external magnetic field, a uniaxial or cubic
magnetic anisotropy for the particle's core, and several models for the surface
anisotropy (N\'eel, conventional, random), we compute the spin-flip small-angle
neutron scattering (SANS) cross section from the equilibrium spin structures
obtained using the Landau-Lifshitz equation. The sign of the surface anisotropy
constant, which is related to the appearance of tangential- or radial-like spin
textures, can be distinguished from the momentum-transfer dependence of the
spin-flip signal. The data cannot be described by the well-known and often-used
analytical expressions for uniformly magnetized spherical or core-shell
particles, in particular at remanence or at the coercive field. Based on a
second-order polynomial expansion for the magnetization vector field, we
develop a novel minimal model for the azimuthally-averaged magnetic SANS cross
section. The theoretical expression considers a general magnetization
inhomogeneity and is not restricted to the presence of surface anisotropy. It
is shown that the model describes very well our simulation data as well as more
complex spin patterns such as vortex-like structures. Only seven expansion
coefficients and some basis functions are sufficient to describe the scattering
behavior of a very large number of atomic spins.
