  We theoretically investigate spin dynamics and $L_3$-edge resonant inelastic
X-ray scattering (RIXS) of Chromium with commensurate spin-density wave (SDW)
order, based on a multi-band Hubbard model composed of 3$d$ and 4$s$ orbitals.
Obtaining the ground state with the SDW mean-field approximation, we calculate
the dynamical transverse and longitudinal spin susceptibility by using
random-phase approximation. We find that a collective spin-wave excitation seen
in inelastic neutron scattering hardly damps up to $\sim$0.6 eV. Above the
energy, the excitation overlaps individual particle-hole excitations as
expected, leading to broad spectral weight. On the other hand, the collective
spin-wave excitation in RIXS spectra has a tendency to be masked by large
spectral weight coming from particle-hole excitations with various orbital
channels. This is in contrast with inelastic neutron scattering, where only
selected diagonal orbital channels contribute to the spectral weight. However,
it may be possible to detect the spin-wave excitation in RIXS experiments in
the future if resolution is high enough.
