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Structural, electronic, and magnetic properties of modified cubic spinelcompound LiNi$_{0.5}$Mn$_{1.5}$O$_{4}$ are studied via x-ray diffraction,resistivity, DC and AC magnetization, heat capacity, neutron diffraction,$^7$Li nuclear magnetic resonance, magnetocaloric effect, magnetic relaxation,and magnetic memory effect experiments. We stabilized this compound in a cubicstructure with space group $P4_{3}32$. It exhibits semiconducting characterwith an electronic band gap of $\Delta/k_{\rm B} \simeq 0.4$ eV. Theinteraction within each Mn$^{4+}$ and Ni$^{2+}$ sub-lattice and betweenMn$^{4+}$ and Ni$^{2+}$ sublattices is found to be ferromagnetic (FM) andantiferromagnetic (AFM), respectively which leads to the onset of aferrimagnetic transition at $T_{\rm C} \simeq 125$~K. The reduced values offrustration parameter ($f$) and ordered moments reflect magnetic frustrationdue to competing FM and AFM interactions. From the $^7$Li NMR shift vssusceptibility plot, the average hyperfine coupling between $^7$Li nuclei andNi$^{2+}$ and Mn$^{4+}$ spins is calculated to be $\sim 672.4$~Oe/$\mu_{\rmB}$. A detailed critical behaviour study is done in the vicinity of $T_{\rm C}$using modified-Arrott plot, Kouvel-Fisher plot, and universal scaling ofmagnetization isotherms. The estimated critical exponents correspond to the 3DXY universality class. A large magneto-caloric effect is observed with amaximum isothermal change in entropy $\Delta S_m \simeq - 11.3$~J/Kg-K and amaximum relative cooling power $RCP \simeq 604$~J/Kg for 9~T magnetic fieldchange. The imaginary part of the AC susceptibility depicts a strong frequencydependent hump at $T=T_{\rm f2}$ well below the blocking temperature $T_{\rmb}\simeq120$~K. The Arrhenius behaviour of frequency dependent $T_{\rm f2}$ andthe absence of ZFC memory confirm the existence of superparamagnetism in theferrimagnetically ordered state.

Unconventional superparamagnetic behavior in the modified cubic spinel compound LiNi$_{0.5}$Mn$_{1.5}$O$_{4}$