  We have employed resonant x-ray magnetic scattering to specifically probe the
magnetic order of the rare-earth ions in multiferroic $\mathrm{TbMn_2O_5}$. Two
energy resonances were observed, one originated from the E1-E1 dipolar
transition and the other from the E2-E2 quadrupolar transition. These
resonances directly probe the valence 5d band and the partially occupied 4f
band, respectively. First, full polarization analysis, which is a measurement
of the scattered polarization as a function of incident polarization, confirmed
a spin polarization of the terbium valence states (probed by the E1-E1
transition) by the $\mathrm{Mn^{4+}}$ spin density in the commensurate phase.
Second, full polarization analysis data were collected in the low-temperature
incommensurate and commensurate phases when tuned to the E2-E2 resonance. By
employing a least-squares fitting procedure, the spin orientations of the
terbium ion sublattice were refined.
