  Analytic and numerical studies on curved magnetic nano-objects predict
numerous exciting effects that can be referred to as magneto-chiral effects,
which do not originate from intrinsic Dzyaloshinskii-Moriya interaction or
interface-induced anisotropies. In constrast, these chiral effects stem from
isotropic exchange or dipole-dipole interaction, present in all magnetic
materials, which acquire asymmetric contributions in case of curved geometry of
the specimen. As a result, for example, the spin-wave dispersion in round
magnetic nanotubes becomes asymmetric, namely spin waves of the same frequency
propagating in opposite directions along the nanotube exhibit different
wavelenghts. Here, using time-resolved scanning transmission X-ray microscopy
experiments, standard micromagntic simulations and a dynamic-matrix approach,
we show that the spin-wave spectrum undergoes additional drastic changes when
transitioning from a continuous to a discrete rotational symmetry, i.e. from
round to hexagonal nanotubes, which are much easier to fabricate. The polygonal
shape introduces localization of the modes both to the sharp, highly curved
corners and flat edges. Moreover, due to the discrete rotational symmetry, the
degenerate nature of the modes with azimuthal wave vectors known from round
tubes is partly lifted, resulting in singlet and duplet modes. For comparison
with our experiments, we calculate the microwave absorption from the
numerically obtained mode profiles which shows that a dedicated antenna design
is paramount for magnonic applications in 3D nano-structures. To our knowledge
these are the first experiments directly showing real space spin-wave
propagation in 3D nano objects.
