Abstract We use exact recursion relations to study the magnetic properties of the half-integer mixed spin-5/2 and spin-3/2 Blume-Capel Ising ferromagnetic system on the two-fold Cayley tree that consists of two sublattices A and B. Two positive crystal-field interactions Δ1 and Δ2 are considered for the sublattice with spin-5/2 and spin-3/2 respectively. For different coordination numbers q of the Cayley tree sites, the phase diagrams of the model are presented with a special emphasis on the case q = 3, since other values of q reproduce similar results. First, the T = 0 phase diagram is illustrated in the (D A = Δ1/J,D B = Δ2/J) plane of reduced crystal-field interactions. This diagram shows triple points and coexistence lines between thermodynamically stable phases. Secondly, the thermal variation of the magnetization belonging to each sublattice for some coordination numbers q are investigated as well as the Helmoltz free energy of the system. First-order and second-order phase transitions are found. The second-order phase transitions become sharper and sharper when D A or D B increases. The first-order transitions only exist for some appropriate non-zero values of D A and/or D B. The corresponding transition lines never connect to the second-order transition lines. Thus, the non-existence of tricritical points remains one of the key features of the present model. The magnetic exponent β 0 of the model is estimated and found to be ¼ at small values of D A = D B = D and β 0 = ½ at large values of D. At intermediate values of D, there is a crossover region where the magnetic exponent displays interesting behaviours.
Abstract We use exact recursion relations to study the magnetic properties of the half-integer mixed spin-5/2 and spin-3/2 Blume-Capel Ising ferromagnetic system on the two-fold Cayley tree that consists of two sublattices A and B. Two positive crystal-field interactions Δ1 and Δ2 are considered for the sublattice with spin-5/2 and spin-3/2 respectively. For different coordination numbers q of the Cayley tree sites, the phase diagrams of the model are presented with a special emphasis on the case q = 3, since other values of q reproduce similar results. First, the T = 0 phase diagram is illustrated in the (D A = Δ1/J,D B = Δ2/J) plane of reduced crystal-field interactions. This diagram shows triple points and coexistence lines between thermodynamically stable phases. Secondly, the thermal variation of the magnetization belonging to each sublattice for some coordination numbers q are investigated as well as the Helmoltz free energy of the system. First-order and second-order phase transitions are found. The second-order phase transitions become sharper and sharper when D A or D B increases. The first-order transitions only exist for some appropriate non-zero values of D A and/or D B. The corresponding transition lines never connect to the second-order transition lines. Thus, the non-existence of tricritical points remains one of the key features of the present model. The magnetic exponent β 0 of the model is estimated and found to be ¼ at small values of D A = D B = D and β 0 = ½ at large values of D. At intermediate values of D, there is a crossover region where the magnetic exponent displays interesting behaviours.
Purpose The authors investigate the magnetic properties of a mixed spin-3/2 and spin-2 Blume-Capel model on square and cubic lattices with two different single-ion anisotropies. Design/methodology/approach To study the critical behavior of a mixed spin-3/2 and spin-2 system, the authors have used a real space renormalization group approximation and specifically the Migdal-Kadanoff technique. The authors give the phase diagrams for two different cases: (1) on the (Δ/|J|, 1/|J|) plane with ΔA = ΔB = Δ, and (2) on the (ΔA/|J|, 1/|J|) and (ΔB/|J|, 1/|J|) planes for selected values of ΔB/|J| and ΔA/|J|, respectively. Findings The phase diagrams obtained show that the system exhibits both second- and first-order phase transitions as well as tricritical points for some values of the anisotropies. Moreover, using the variation of the free energy and its derivative at low temperatures, the authors have seen the appearance of first-order transitions at very low temperatures. Originality/value Few investigations of mixed spin-3/2 and spin-2 systems with crystal field have been realized. For this reason, the authors use the renormalization group approach to complete the work done on these systems. In absence of an exact solution, this contributes to the synthesis of the approximation results on mixed spins models.
The exact recursion relations are employed to study the magnetization and phase transition properties by assuming that each site of a Bethe lattice is occupied by two spin-3/2 atoms forming diatomic molecules in the Blume–Capel model. The atoms interact ferromagnetically via three distinct bilinear coupling parameters: intra-site [Formula: see text], nearest-neighbor [Formula: see text] and diagonal inter-site [Formula: see text]. The crystal field (D) is also active at all sites of the Bethe lattice of coordination number three. The thermal variations of the magnetizations are examined at the core molecule to obtain its phase transition properties. The phase diagrams mapped on the [Formula: see text] and [Formula: see text] planes show that the model presents second- and first-order phase transitions. The obtained first-order phase transition line terminating at the isolated endpoints separating the phases ([Formula: see text]) from ([Formula: see text]) is the result of the usual BC phase diagram on the [Formula: see text] plane. The phase diagrams obtained on the [Formula: see text] planes give either only second-order lines, second- and first-order lines, or second- and two first-order lines for given parameters that are not seen in the usual spin-3/2 models.
Phase diagrams and the thermal variations of the order‐parameters in the mixed spin‐1 and spin‐23 ising model on the Bethe lattice
Abstract The mixed spin‐1 and spin‐23 Blume‐Capel Ising ferromagnetic system (J > 0) with two crystal‐field interactions, DA for spin‐1 and DB for spin‐23, is studied on the Bethe lattice using the exact recursion equations for the central‐spin with spin‐23. The exact expressions for the order‐parameters, the Curie temperature and the free energy are obtained in terms of the recursion relations. The phase diagrams on the (JkTc,JDA) and (JkTc,JDB) planes are studied for selected values of DB/J and DA/J, respectively, for the coordination numbers q = 3, 4, and 6. Besides the second‐order phase transition lines, the lines of the first‐order phase transitions terminating either at a tricritical point or at an isolated critical point are found. The thermal variations of the order‐parameters are studied with two formulations, both obtained on the Bethe lattice for the mixed spin‐1 and spin‐23 Ising system using the exact recursion relations, but one with the central‐spin with spin‐1 and the other is for the central‐spin with spin‐23 for q = 3, thus, it is shown that the choice of the central‐spin on the Bethe lattice has no effect on the phase diagrams.
Using the recursion method,the phase transition and the thermodynamic behavior of the mixed spin-1/2 and spin-2\3 Ising system on the two-fold Cayley tree are studied for the ferromagnetic and ferrimagnetic cases,respectively.The relations of sublattice magnetizations MA,MB and the total magnetization M with the temperature are obtained by the numerical calculation.After comparing the results of the two cases,it is found that the behavior of M is similar to that of MA and that the sublattice magnetizations on the temperature differ between the ferromagnetic and ferrimagnetic cases.There is no tricritical behavior for the two cases.
The dynamic magnetic properties of the mixed-spin (3/2,7/2) Blume–Capel Ising system on an interpenetrating square lattice are investigated using the path probability method in the presence of a time-dependent oscillating magnetic field. The thermal variations of the dynamic magnetizations, stationary magnetization oscillations, compensation behavior and dynamic phase diagrams are examined for both ferrimagnetic and ferromagnetic exchange interactions. The system exhibits dynamically ordered and disordered phases separated by first- and second-order dynamic phase transition lines, together with dynamic tricritical points. For the ferrimagnetic case, coexistence regions and P-, Q-, R-, [Formula: see text] and N-type compensation behaviors are observed, whereas W-, or M-type characteristics are absent within the investigated parameter ranges. Increasing the crystal-field interaction stabilizes the ordered phase, while increasing the magnetic-field amplitude promotes dynamic disordering and bistability. In contrast, the ferromagnetic case exhibits a simpler phase structure, with broader ordered regions, weaker coexistence effects and no compensation behavior. The results demonstrate that the interplay among exchange interactions, crystal-field interactions and oscillating magnetic field governs the nonequilibrium phase topology and dynamic critical behavior of the mixed-spin (3/2,7/2) system.
The thermodynamic properties of the spin S=3/2 ferromagnetic Ising model in the presence of transverse and longitudinal crystal fields (equivalent to the Blume-Capel model with a transverse crystal field) have been studied by using two different approaches: (i) a zero-temperature mapping of the system onto a spin-1/2 quantum Ising model in longitudinal and transverse fields, together with time-independent quantum perturbation theory; and (ii) a standard mean-field approximation within the framework of the Bogoliubov inequality for the free energy. A very rich phase diagram, with different kinds of multicritical behavior, has been obtained. The results show first- and second-order transition lines, tricritical and tetracritical points, critical end points with a two-phase coexistence, double critical end points, and also double noncritical end points. Additionally, the behavior of the magnetization as a function of temperature, over a wide range of values of both longitudinal and transverse crystal fields, has also been analyzed in detail. While large magnitudes of the longitudinal crystal field select the z-spin components either in their states ±3/2 or ±1/2, it is surprising that a large transverse crystal field induces the spin component in the z direction to values ±1, which are completely different from any expected natural component. This comes indeed as a result of the zero-temperature mapping of the ground state with the superposition of the states 3/2 and -1/2, in one sector of the Hilbert space, and the states -3/2 and 1/2 on the other disjoint sector of the Hilbert space. This superposition for a large transverse crystal field prevails even for finite temperatures, implying that the exact critical points are obtained for the model on the one-dimensional lattice and the two-dimensional square lattice, and quite accurate estimates can be achieved for the three-dimensional simple cubic lattice.