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physica status solidi (b)

Phase diagrams and the thermal variations of the order‐parameters in the mixed spin‐1 and spin‐$3 \over 2$ ising model on the Bethe lattice

Erhan Albayrak

Published Aug 22, 2003• 15 citations
Publisher

Abstract

Abstract The mixed spin‐1 and spin‐$3 \over 2$ Blume‐Capel Ising ferromagnetic system (J > 0) with two crystal‐field interactions, DA for spin‐1 and DB for spin‐$3 \over 2$, is studied on the Bethe lattice using the exact recursion equations for the central‐spin with spin‐$3 \over 2$. 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 $\bigg ({{kT_c}\over{J}}, \,{{D_A}\over{J}}\bigg )$ and $\bigg ({{kT_c}\over{J}}, \,{{D_B}\over{J}}\bigg )$ 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‐$3 \over 2$ 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‐$3 \over 2$ 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.

DOI
10.1002/pssb.200301812

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MIXED SPIN-1 AND SPIN-32\frac{3}{2}23​ BLUME–CAPEL ISING FERRIMAGNETIC SYSTEM ON THE BETHE LATTICE
The mixed spin-1 and spin-[Formula: see text] Blume–Capel Ising ferrimagnetic system for the central spin with spin-1 is studied on the Bethe lattice using the exact recursion equations. The exact expressions for the magnetization, the quadrupolar moment, the Curie temperature and the free energy are found and the phase diagrams are constructed on the Bethe lattice with the coordination numbers q = 3, 4 and 6 for the various values of the single-ion anisotropy constants dA = DA/J for spin-1 and dB = DB/J for spin-[Formula: see text]. The existence of a tricritical point is investigated for different values of q and the single-ion anisotropy constants. The phase diagrams in the (kTc/J, dA) plane for the central spin are obtained for two different cases; (1) dA = dB and (2) dA is varied for selected values of dB. The results are compared with those of other approximate methods.
The mixed spin-1/2 and spin-3/2 Blume–Capel (BC) model is considered on the Bethe lattice (BL) with randomly changing coordination numbers (CN) and examined in terms of exact recursion relations. A couple of two different CNs are changed randomly on the shells of the BL in terms of a standard–random approach to obtain the phase diagrams on possible planes of the system parameters. It is found from the thermal analysis of the order-parameters that the model only gives the second-order phase transitions as in the regular mixed case. As the probability of having larger CN increases, the temperatures of the critical lines also increase as expected.
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.
The mixed spin-1/2 and spin-1 Blume–Capel model is studied with randomly alternated coordination numbers (CN) on the Bethe lattice (BL) by utilizing the exact recursion relations. Two different CNs are randomly distributed on the BL by using the standard–random (SR) approach. It is observed that this model presents first-order phase transitions and tricritical points for variations of CNs 3 and 4, even if these behaviors are not displayed for the regular mixed-spin on the BL. The phase diagrams are mapped by obtaining the phase transition temperatures of the first- and second-order on several planes.
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.
A bilayer Ising model consisting of two Bethe lattices, each of which is coupled with crystal fields of different strengths and each with a branching ratio of q Ising spins with one of the layers having only spin-1 and the other having only spin-3/2, is laid over the top of the other and the two layers are tied together via an interaction between the vertically aligned spins. After obtaining the ground-state (GS) phase diagrams on different possible planes depending on the given system parameters, the changes in the order-parameters and the free energy are investigated by use of the exact recursion relations in a pairwise approach to calculate the phase diagrams of the model. The ferromagnetic ordering in each of the layers and ferromagnetic or antiferromagnetic ordering of the adjacent nearest-neighbor (NN) spins of the layers are considered. The system presents both second- and first-order phase transitions. The lines of the first-order phase transitions end on either the stable or unstable tricritical points or at the isolated critical points. The model also displays one or two compensation temperatures when the bilinear interaction of the upper layer with spin-1 can compete with that of the lower layers with spin-3/2.