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In present study, the magnetic structure and spin reorientation of mixeddoped orthoferrite Nd$_{0.5}$Dy$_{0.5}$FeO$_3$ have been investigated. Similarto both parent compounds (NdFeO$_3$ and DyFeO$_3$), the magnetic structure ofFe$^{3+}$ belongs to ${\Gamma}_{4}$ irreducible representation (G$_{x}$,F$_{z}$) at room temperature. The experimental measurements confirmed the spinreorientation where magnetic structure of Fe$^{3+}$ changes from ${\Gamma}_{4}$to ${\Gamma}_{2}$(F$_{x}$, G$_{z}$) between 75 and 20 \,K while maintainingG-type configuration. Such a gradual spin reorientation is unusual since thelarge single ion anisotropy of Dy$^{3+}$ ions causes an abrupt${\Gamma}_{4}$${\rightarrow}$ ${\Gamma}_{1}$(G$_{y}$) spin reorientation inDyFeO$_3$. Between 20 and 10 \,K, the Fe$^{3+}$ magnetic structure isrepresented by ${\Gamma}_{2}$ (F$_{x}$, G$_{z}$). Unexpectedly, magneticstructure of Fe$^{3+}$ with ${\Gamma}_{4}$ representation re-emerges below10\,K which also coincides with the development of rare-earth(Nd$^{3+}$/Dy$^{3+}$) magnetic ordering having C$_{y}$ configuration withmagnetic moment of 1.8 ${\mu}_{B}$. The absence of any signature of secondorder phase transition in the specific heat confirms the role of$R$(Nd$^{3+}$/Dy$^{3+}$)-Fe$^{3+}$ exchange interaction in the observed"rare-earth ordering" unlike DyFeO$_3$ where Dy$^{3+}$ ordering takes placeindependently to the magnetic ordering of Fe$^{3+}$ magnetic structure. Our(DFT+U+SO) calculations show that the C-type arrangement of rare-earth ions(Nd$^{3+}$/Dy$^{3+}$) with ${\Gamma}_{2}$ configuration for Fe$^{3+}$ momentsis the ground state whereas ${\Gamma}_{4}$ phase is energetically very close.Nd-Fe and Nd-Dy exchange interactions, estimated from DFT, are observed to havesignificant roles in the rare earth ordering and Fe spin reorientationcorroborating our experimental results.

Successive spin reorientation and rare earth ordering in Nd$_{0.5}$Dy$_{0.5}$FeO$_{3}$: Experimental and $Ab$-$initio$ investigations