Assessment of the critical behavior near the FM to PM phase transition in cubic Ni_0.3−xCu_xZn_0.7Fe₂O₄ spinel ferrite

In this study, we report a thorough examination of the critical behavior of the cubic Ni_0.3−xCu_x Zn_0.7Fe₂O₄ (x = 0.0, 0.1 and 0.2) spinel ferrite which are crystallized with different irregular shapes of particle size (spherical,polygonal and cubic). The magnetic transition is ferromagnetic to paramagnetic state for all Cu-doped compounds. Different theoretical analyses, as the modified Arrott plot, Kouvel–Fisher method, and critical isotherm, have been used to identify the values of the ferromagnetic transition temperature T_C, and the critical exponents of β, γ and δ. All the obtained values of critical parameters were found to be close to the 3D-Heisenberg theoretical model, which implies short-range magnetic interactions in Cu-doped compounds. The reliability of critical parameters was further checked through different methods, such as the Widom scaling relationship and the magnetic equation.