Bond energies for the
Abstract
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 temperatureinduced structural phase transitions in some materials and nanostructures with firstorder 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 temperatureinduced MA 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 coresurface nanoparticles based on the fundamental formulation of pair approximation in Kikuchi version.
Keywords
 Thermal hysteresis
 Nanoparticles
 Martensite
 Austenite
 Pair approximation
1. Introduction
The phenomenon of hysteresis is encountered in many areas of physics. It is associated with the delay of the dynamic response of cooperative systems to external perturbation. During a heating–cooling process in a system, thermal hysteresis (TH) commonly appears accompanying phase transitions. In particular, it is regarded as a signature of the firstorder phase transition. But, the TH is less known than the magnetic hysteresis (MH), which is another type of hysteresis in magnetism [1]. Rao and Pandit [2] studied both TH and MH, and they found that TH seems to belong to a different universality class than MH in the same relaxational model.
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) or TH in magnetization, which is principally associated with magnetic phase transitions (MPTs). Many bulk materials with MPTs have been discovered to exhibit significant MTH behaviour. Among those materials, well known samples are spin crossover compounds [3–8], manganites [9], superconductors [10, 11], magnetic multilayers [12, 13], ferrimagnetic and metamagnetic alloys [14, 15], polycrystalline samples [16], manganite thin films [17] and manganite perovskites [18]. Thermal hysteresis occurs in these samples because the transitions between various magnetic phases occur at different temperatures for the heating and cooling processes. Furthermore, the temperature span of the TH can be substantially reduced by applying an external magnetic field. The thermoelastic austenite–martensite transformations in Heusler and shape memory alloys were also characterized by the TH loops [19–21].
Apart from the above investigations of bulk systems, the MTH is of particular interest for both quantum and applied physics researches on magnetization of nanomaterials. Several groups studied some important physical properties of various types of nanostructures using the properties of TH loops. For example, from the thermal response of conductivity for the gold nanoparticles (NPs), a phase transition phenomenon was revealed by a temperature criticality by Sarkar et al. [22]. Based on Isinglike treatment of the MTH and using Monte Carlo algorithm, Kawamoto and Abe [23] obtained smaller hysteresis width when the volume of the spin crossover NPs was decreased. Also, using the same treatment, the simulated hysteretic loops become closer to the experimental ones [24, 25].
On the other hand, hysteresis of the temperatureinduced structural phase transitions in nanostructures with firstorder phase transitions reduce useful magnetocaloric effect to transform cycling between martensite (M) and austenite (A) phases under application. In addition, 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 temperatureinduced martensite/austenite phase transitions.
In this chapter, we shall review the latest theoretical developments in the field of MTH related to the structural phase transitions for the core/surface (
2. Basics of the theoretical model
2.1. Definition of a nanoparticle with core/surface morphology
For a noninteracting spherical nanoparticle, arrays of spins are generally considered on a hexagonal lattice in 2D, as shown in Fig. 1 [26]. This structure may also be extended to hexagonal closed packed (hcp) lattice for any threedimensional (3D) case which is not covered in Fig. 1 but is illustrated in a recent publication by Yalçın et al. [27]. Similarly, a square lattice in 2D (shown in Fig. 2) is enlarged to simple cubic lattice (sc) in 3D for a cubic nanoparticle. For the number of shells in both structures, each lattice is related to the radius (
2.2. Blume–Emery–Griffiths model
The Blume–Emery–Griffiths (BEG) model is one of the wellknown spin lattice models in equilibrium statistical mechanics. It was originally introduced with the aim to account for phase separation in helium mixtures [30]. Besides various thermodynamic properties, the model has been extended to study the structural phase transitions in many bulk systems. By means of meanfield theory (MFT) and Monte Carlo (MC) simulations, magnetostructural phase transitions in some alloys were described via degenerate BEG models in terms of magnetoelastic interactions [31]. In the light of above applications, we have recently used the ordinary BEG model for the investigation of MT/AT transitions in NP systems and observed the behaviours of the MTH loops [28, 29]. In the following, we mention briefly the definition of the BEG Hamiltonian and show clearly how it is modified for the
For a spin configuration
where
The model Hamiltonian (Eq. 1) is now divided into three parts for the
where
2.3. Fundamental formulation of pair approximation in Kikuchi version
In the pair approximation proposed by Kikuchi [34], each spin case is indicated by
Using the pair correlations between spins, another types of internal (or bond/pair) variables denoted by
and connected with state variables through the relations
In order to determine an expression for the bond variables, we define the interaction energy (
where
The minimization of Eq. (8) with respect to
where
Here,
where bond energies
Using the numerical solutions of Eq. (9) by the iteration technique, the MTH curves are drawn easily from Eq. (12). For some selected exchange energies, the MTH behaviours and the temperatureinduced M–A phase transitions within the








































3. Calculations and discussion
3.1. Thermal hysteresis for hexagonal nanoparticles
We firstly analyze the MTH loops for the homogeneous hexagonal nanoparticles (HMHNPs) using
For understanding the features associated with the MTH behaviours in composite hexagonal nanoparticles (CMHNPs), we have proceeded to calculate the magnetization as a function of temperature for the same system, but using
3.2. Thermal hysteresis for cubic nanoparticles
Another class of small particles known in the literature are called cubic nanoparticles (CNPs). The CNPs attach facetoface to the surrounding particles. They can form a 2D square array instead of hexagonal ones (Fig. 2). The square array is one of the closedpacked arrangements and the CNPs with a square array have large surface to volume ratios. Owing to their different electronic properties, in recent years, the CNPs received much attention for applications mostly in material science, sensor technology and semiconductor devices. But, the magnetic properties have been very sensitive to the particle shape due to dominating role of surface anisotropy in its magnetization. Compared to the spherical nanoparticles (SNPs), the flat surface of CNPs enabled the surface metal captions to possess a more symmetric coordination. So, the surface anisotropy in the CNPs should be much smaller than the one in HNPs and SNPs. If the magnetic anisotropy of the CNPs is cubic, all such six directions are magnetically identical, and hence the magnetically ordered assembly is greatly simplified. The anisotropy of the growth rate can be ascribed to a different adhesion of the stabilizer on the growing surface. The stabilizer species is the only parameter which was changed to obtain cubical shapes instead of spheres. Because of the above properties, the singledomain CNPs with surface anisotropy were widely investigated using various numerical methods [38–42].
For the sake of comparison with that of HNPs, we here reestablish the MTH curves of the homogeneous cubic nanoparticles (HMCNPs) and composite cubic nanoparticles (CMCNPs) by using the same procedure as in the preceding section (Figs. 3 and 4) and focus on the important difference between the martensitic and austenitic transition temperatures, derived from these hysteresis loops, for the CNPs and HNPs. The general aspects of the MTH loops for the CNPs displayed in Figs. 5 and 6 are similar to one for the HNPs given in Figs. 3 and 4. But, the loops are calculated using bigger particles (
In general, the thermal hysteresis becomes weaker but, nevertheless, does not disappear completely with increasing NP sizes for the CMNPs. All loops widen as
4. Conclusion
In this chapter, we have presented a review of thermal reversal properties of the
Our review draws a number of important physical properties for the MTH curves regarding the sign of the quadrupolar interactions (
Acknowledgments
One of the authors (RE) acknowledges the financial support from the Scientific Research Projects Coordination Unit of Akdeniz University.
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