  We calculate the K-edge indirect bimagnon resonant inelastic X-ray scattering
(RIXS) intensity spectra of the bicollinear antiferromagnetic order known to
occur in the $\alpha$-FeTe chalcogenide system. Utilizing linear spin wave
theory for this large-S spin system we find that the bimagnon spectrum contains
four scattering channels (two intraband and two interband). We find from our
calculations that for suitable energy-momentum combination the RIXS spectra can
exhibit a one-, two- or three- peak structure. The number of peaks provides a
clue on the various bimagnon excitation processes that can be supported both in
and within the acoustic and optical magnon branches of the bicollinear
antiferromagnet. Unlike the RIXS response of the antiferromagnetic or the
collinear antiferromagnetic spin ordering, the RIXS intensity spectrum of the
bicollinear antiferromagnet does not vanish at the magnetic ordering wave
vector $(\pi/2,-\pi/2)$. It is also sensitive to next-next nearest neighbor and
biquadratic coupling interactions. Our predicted RIXS spectrum can be utilized
to understand the role of multi-channel bimagnon spin excitations present in
the $\alpha$-FeTe chalcogenide.
