  Incomplete cancellation of collinear antiparallel spins gives rise to
ferrimagnetism. Even if the oppositely polarized spins are owing to the equal
number of a single magnetic element having the same valence state, in
principle, a ferrimagnetic state can still arise from the crystallographic
inequivalence of the host ions. However, experimental identification of such a
state as ferrimagnetic is not straightforward because of the tiny magnitude
expected for M and the requirement for a sophisticated technique to
differentiate similar magnetic sites. We report a synchrotron-based resonant
x-ray investigation at the Fe L2,3 edges on an epitaxial film of CaFe2O4, which
exhibits two magnetic phases with similar energies. We find that while one
phase of CaFe2O4 is antiferromagnetic, the other one is ferrimagnetic with an
antiparallel arrangement of an equal number of spins between two distinct
crystallographic sites with very similar local coordination environments. Our
results further indicate two distinct origins of an overall minute M; one is
intrinsic, from distinct Fe3+ sites, and the other one is extrinsic, arising
from defective Fe2+ likely forming weakly-coupled ferrimagnetic clusters. These
two origins are uncorrelated and have very different coercive fields. Hence,
this work provides a direct experimental demonstration of ferrimagnetism solely
due to crystallographic inequivalence of the Fe3+ as the origin of the weak M
of CaFe2O4.
