Critical Behavior in Double-Exchange Ferromagnets of Pr0.6Sr0.4MnO3 Nanoparticles

In this paper, we present a detailed study on the critical property around the ferromagnetic (FM)-paramagnetic phase transition of Pr_0.6Sr_0.4MnO₃ nanoparticles (NPs) with different crystallite sizes (d = 39-88 nm), which were synthesized by the combination of the solid-state reaction and mechanical milling methods. The magnetic results show that the samples undergo the second-order phase transition. The values of the Curie temperature (T_C) slightly decrease from 291 to 286 K while the saturation magnetization (MS) decreases from 89.7 to 76.3 emu/g and the coercivity (H_C) increases from 322 to 585 Oe with decreasing d from 88 to 39 nm, respectively. Based on the modified Arrott plot method and the isothermal magnetization data around T_C, we have determined the values of critical exponents (β, y, and S) and T_C for the samples. The critical exponents β = 0.486 - 0.495, y = 1.009 - 1.042, and S = 3.038 - 3.118 obtained are close to those expected for the mean field theory (β = 0.5, y = 1, and S = 3). This proves that long-range FM interactions are present in the NPs. Furthermore, β values are smaller than 0.5, indicating the existence of magnetic inhomogeneity in the NPs. Besides the investigation into the critical behavior, the magnetocaloric effect of the NPs is also investigated.