InMnO$_3$ is a peculiar member of the hexagonal manganites h-RMnO$_3$ (where R is a rare earth metal element), showing crystalline, electronic and magnetic properties at variance with the other compounds of the family. We have studied high quality sa
mples synthesized at high pressure and temperature by powder neutron diffraction. The position of the Mn ions is found to be close to the threshold $it{x}=1/3$ where superexchange Mn-Mn interactions along the $it{c}$ axis compensate. Magnetic long range order occurs below $T_{rm N}$= 120(2) K with a magnetic unit cell doubled along $it{c}$, whereas short range two dimensional dynamical spin correlations are observed above $T_{rm N}$. We propose that pseudo-dipolar interactions are responsible for the long period magnetic structure.
We have performed high resolution neutron diffraction and inelastic neutron scattering experiments in the frustrated multiferroic hexagonal compounds RMnO3 (R=Ho, Yb, Sc, Y), which provide evidence of a strong magneto-elastic coupling in the the whol
e family. We can correlate the atomic positions, the type of magnetic structure and the nature of the spin waves whatever the R ion and temperature. The key parameter is the position of the Mn ions in the unit cell with respect to a critical threshold of 1/3, which determines the sign of the coupling between Mn triangular planes.
The magnetic ordering of the hexagonal multiferroic compound YbMnO$_3$ has been studied between 100 K and 1.5 K by combining neutron powder diffraction, $^{170}$Yb Mossbauer spectroscopy and magnetization measurements. The Yb moments of the two cryst
allographic sites order at two different temperatures, the $4b$ site together with the Mn moments (at $T_N simeq$85 K) and the $2a$ site well below (at 3.5 K). The temperature dependences of the Mn and Yb moments are explained within a molecular field model, showing that the $4b$ and $2a$ sites order via Yb-Mn and Yb-Yb interactions respectively. A simple picture taking into account the local Mn environment of the Rare earth R ($4b$) ion is proposed to couple R and Mn orders in hexagonal RMnO$_3$ manganites. The nature and symmetry of the R-Mn interactions yielding the R order are discussed.
