In present study, the magnetic structure and spin reorientation of mixed doped orthoferrite Nd$_{0.5}$Dy$_{0.5}$FeO$_3$ have been investigated. Similar to both parent compounds (NdFeO$_3$ and DyFeO$_3$), the magnetic structure of Fe$^{3+}$ belongs to ${Gamma}_{4}$ irreducible representation (G$_{x}$, F$_{z}$) at room temperature. The experimental measurements confirmed the spin reorientation where magnetic structure of Fe$^{3+}$ changes from ${Gamma}_{4}$ to ${Gamma}_{2}$(F$_{x}$, G$_{z}$) between 75 and 20 ,K while maintaining G-type configuration. Such a gradual spin reorientation is unusual since the large single ion anisotropy of Dy$^{3+}$ ions causes an abrupt ${Gamma}_{4}$${rightarrow}$ ${Gamma}_{1}$(G$_{y}$) spin reorientation in DyFeO$_3$. Between 20 and 10 ,K, the Fe$^{3+}$ magnetic structure is represented by ${Gamma}_{2}$ (F$_{x}$, G$_{z}$). Unexpectedly, magnetic structure of Fe$^{3+}$ with ${Gamma}_{4}$ representation re-emerges below 10,K which also coincides with the development of rare-earth (Nd$^{3+}$/Dy$^{3+}$) magnetic ordering having C$_{y}$ configuration with magnetic moment of 1.8 ${mu}_{B}$. The absence of any signature of second order 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 place independently 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$^{3+}$ moments is the ground state whereas ${Gamma}_{4}$ phase is energetically very close. Nd-Fe and Nd-Dy exchange interactions, estimated from DFT, are observed to have significant roles in the rare earth ordering and Fe spin reorientation corroborating our experimental results.
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