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Blume-Emery-Griffiths model is used to study the effects of the bilinear and biquadratic exchange interactions as well as crystal field interaction on the magnetic properties of core-surface smart nanoparticles. From the minimization of the free energy in pair approximation in Kikuchi version, a set of equations for the bond variables and magnetization are derived. Based on the numerical solutions of these equations, magnetization and hysteresis curves are obtained. Besides the first- and second-order phase transitions, martensitic and austenitic phase regions are observed in the phase diagrams of homogeneous and composite nanoparticles and the origin of martensitic transitions (MT)-austenitic transitions (AT) is investigated. It is found that MT-AT occurred for a nonzero biquadratic exchange parameter. On the other hand, nonzero single-ion anisotropy caused the hysteretic splitting in core-surface type nanoparticles. Martensitic and austenitic results of nanoparticles agree with the experimental results in literature.

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One important issue raised in magnetism studies is the thermal response of various magnetic properties. This topic is known as the magnetic thermal hysteresis MTH which is principally associated with magnetic phase transitions. The MTH is of particular interest for both quantum and applied physics researches on magnetization of nanomaterials. Hysteresis of the temperature-induced structural phase transitions in some materials and nanostructures with first-order phase transitions reduces useful magnetocaloric effect to transform cycling between martensite M and austenite A phases under application. In additional, the size, surface and boundary effects on thermal hysteresis loops have been under consideration for the development of research on nanostructured materials. Experimental data indicate that nanostructured materials offer many interesting prospects for the magnetization data and for understanding of temperature-induced M-A phase transitions. In this chapter, we have presented a review of the the latest theoretical developments in the field of MTH related to the structural phase transitions for the core-surface nanoparticles based on the fundamental formulation of pair approximation in Kikuchi version.