  We address magnetic neutron scattering in the presence of local
non-centrosymmetric asymmetries of the magnetization density. Such
inversion-symmetry breaking, combined with the absence of time-reversal
symmetry, can be described in terms of magnetoelectric multipoles which form
the second term after the magnetic dipole in the multipole expansion of the
magnetization density. We provide a pedagogical review of the theoretical
formalism of magnetic neutron diffraction in terms of the multipole expansion
of the scattering cross-section. In particular, we show how to compute the
contribution of magnetoelectric multipoles to the scattering amplitude starting
from ab initio calculations. We also provide general guidelines on how to
experimentally detect long-ranged order of magnetoelectric multipoles using
either unpolarized or polarized neutron scattering. As a case study, we search
for the presence of magnetoelectric multipoles in CuO by comparing theoretical
first-principle predictions with experimental spherical neutron polarimetry
measurements.
