The Bethe lattice treatment of sound attenuation for a spin- 3/2 Ising model

The sound attenuation phenomena is investigated for a spin- 3/2 Ising model on the Bethe lattice in terms of the recursion relations by using the Onsager theory of irreversible thermodynamics. The dependencies of sound attenuation on the temperature (T$T$), frequency (w$w$), Onsager coefficient (γ$\gamma$) and external magnetic field (H$H$) near the second-order (_Tc)$\left(T_c\right)$ and first-order (_Tt)$\left(T_t\right)$ phase transition temperatures are examined for given coordination numbers q$q$ on the Bethe lattice. It is assumed that the sound wave couples to the order-parameter fluctuations which decay mainly via the order-parameter relaxation process, thus two relaxation times are obtained and which are used to obtain an expression for the sound attenuation coefficient (α)$\left(\alpha \right)$. Our investigations revealed that only one peak is obtained near _Tt$T_t$ and three peaks are found near _Tc$T_c$ when the Onsager coefficient is varied at a given constant frequency for q=3$q=3$. Fixing the Onsager coefficient and varying the frequency always leads to two peaks for q=3,4$q=3,4$ and 6 near _Tc$T_c$. The sound attenuation peaks are observed near _Tt$T_t$ at lower values of external magnetic field, but as it increases the sound attenuation peaks decrease and eventually disappear.     

Spin-1/2 Ising model on a AFM/FM two-layer Bethe lattice in a staggered magnetic field

A bilayer spin-1/2 Ising model consisting of two superposed Bethe lattices with antiferromagnetic/ferromagnetic interactions is studied by the use of exact recursion relations in a pairwise approach in the presence of an external staggered magnetic field. Besides the ground state phase diagrams calculated in different possible planes of the model parameters space, the thermal variations of the order-parameters and the free energy are investigated to obtain the temperature-dependent phase diagrams of the model for different values of the coordination numbers q. Our calculations reveal that depending on the strength of the model parameters, the model exhibits a variety of interesting phase transitions and therefore phase diagrams.     

The temperature-dependent phase diagrams of the spin-3/2 Ising model on a FM/AFM two-layer lattice with a crystal field

The temperature-dependent phase diagrams of the spin-3/2 Ising model on a two-layer Bethe lattice with ferromagnetic (FM)/antiferromagnetic (AFM) intra-layer and either FM or AFM type inter-layer interactions are investigated under a constant magnetic field (H) and in the presence of a crystal field (D) by using exact recursion equations in a pairwise approach for coordination numbers q=3,4 and 6, in detail. In the light of the ground-state (GS) phase diagrams, the temperature-dependent phase diagrams of the model are obtained by studying the thermal variations of the order parameters, response functions and free energy. Then, they are illustrated on the (kT/J₁,J₃/J₁) and (kT/J₁,J₂/J₁) planes for the given system parameters. It is observed that the system exhibits first- and second-order phase transitions for all q values, and hence, in some cases, tricritical points. The existence of critical-end points and that of isolated points are also observed. The re-entrant behavior owes its presence to the two Néel temperatures, T_N, that are present for all q.     

The spin-1 Blume-Capel model with random crystal field on the Bethe lattice

The dependence of the phase diagrams on the random crystal field (RCF) is investigated for the spin-1 Blume-Capel (BC) model on the Bethe lattice. The calculations are carried out in terms of the recursion relations for the coordination number z=4 which corresponds to the square lattice. The model presents tricritical points which are observed at lower negative crystal fields and higher temperatures for higher probabilities p and which vanish at lower p’s. The effect of randomness is illustrated for p=0.5 and shown that it changes the phase diagrams drastically from random to non-random systems. The reentrant behavior is also observed for appropriate p values.     

Spin-3/2 Ising model AFM/AFM two-layer lattice with crystal field

The spin-3/2 Ising model is investigated for the case of antiferromagnetic (AFM/AFM) interactions on the two-layer Bethe lattice by using the exact recursion relations in the pairwise approach for given coordination numbers q=3, 4 and 6 when the layers are under the influences of equal external magnetic and equal crystal fields. The ground state (GS) phase diagrams are obtained on the different planes in detail and then the temperature-dependent phase diagrams of the system are calculated accordingly. It is observed that the system presents both second- and first-order phase transitions for all q, therefore, tricritical points. It is also found that the system exhibits double-critical end points and isolated points. The model also presents two Néel temperatures, T_N, and the existence of which leads to the reentrant behaviour.     

The spin-1 Ising model on a two-layer Bethe lattice with FM/AFM interactions

Two-layer Bethe lattice with the Ising spins of the top layer having only ferromagnetic (FM) interactions and the bottom layer having only antiferromagnetic (AFM) interactions are allowed to interact with the interlayer interaction of either FM or AFM type. The model is studied by using the exact recursion relations in a pairwise approach for given coordination numbers q=3, 4 and 6 with equal external magnetic fields acting on the layers. The phase diagrams of the model are obtained on different planes for given system parameters by studying the ground state (GS) phase diagrams and the thermal variations of the order-parameters and the response functions, i.e. the susceptibility and the specific heat, in detail. The model presents second- and first-order phase transitions, and where their lines are combined is the tricritical point. The critical end points also exist. The reentrant behavior is also seen when the model presents two Néel temperatures.     

The hysteresis loops of FM/AFM two-layer Bethe lattice

The distinct hysteresis loops (HLs) of ferromagnetic/antiferromagnetic (FM/AFM) two-layer Bethe lattice with the Ising spins of the top layer having only FM interactions and the bottom ones having only AFM interactions with the interlayer coupling is either FM or AFM type are studied by using a pairwise approach. The sublattice magnetizations are studied by increasing and decreasing the external magnetic field (H) to obtain the HLs. The shapes of the HLs are strongly dependent on the competitions among the system parameters and on the phase configurations. The HLs are formed only when the AFM-type interactions are involved. The small loops of hysteresis are also formed because of the reentrant behavior in the FM region.     

Sandwiched trilayer of Bethe lattices in the form of spin-(1/2,1,1/2)

The sandwiched trilayer of Bethe lattices in the form of the spins with spin-(1/2,1,1/2) Ising model is studied in terms of the recursion relations with either ferromagnetic or antiferromagnetic type bilinear interactions between the nearest-neighbor (NN) spins. The ground-state (GS) phase diagrams are obtained and it was found that the model presents six different GS phase configurations. In order to obtain the phase diagrams, the thermal variations of the order-parameter, spin-spin correlation functions and free energy are analyzed and different topological phase diagrams are obtained. It was found that the system exhibits different critical behaviors such as, second- and first-order phase transitions, tricritical and bicritical points for the values of the coordination numbers q=3,4 and 6.     

Mixed spin-1/2 and spin-1 Ising model with uniaxial and biaxial single-ion anisotropy on the Bethe lattice

The mixed spin-1/2 and spin-1 Ising model on the Bethe lattice with both uniaxial as well as biaxial single-ion anisotropy terms is solved exactly by combining star-triangle and triangle-star mapping transformations with exact recursion relations. Magnetic properties (magnetization, phase diagrams, and compensation phenomenon) are investigated in detail. Particular attention is focused on the effect of uniaxial and biaxial single-ion anisotropies that basically influence the magnetic behavior of the spin-1 atoms.      

Trilayer Bethe lattices in the form of spin-(1/2,3/2,1/2)

The critical behaviors of the trilayer Bethe lattice in the form of spin-(1/2,3/2,1/2) with the exterior two layers consisting of spin-1/2 and the interior one having only spin-3/2 Ising spins are studied in terms of the recursion relations with either ferromagnetic or antiferromagnetic bilinear exchange interactions between the nearest-neighbor spins. The ground-state phase diagrams are calculated and it is found that the model presents six different ground state phase configurations. The thermal variations of the order-parameter and free energy are investigated, therefore, topologically different phase diagrams of the model are obtained. It is found that the model exhibits second- and first-order phase transitions, tricritical and bicritical points for the values of the coordination numbers q=3, 4 and 6 and, the reentrant behavior for q=4 and 6, respectively.     

Dynamic hysteresis loops of the spin-2 bilayer Ising model

Based on the mean-field theory and Glauber-type stochastic dynamics, the dynamic hysteresis loops (DHLs) of the spin-2 Ising model are studied on the bilayer square lattice. The DHLs are given for different values of temperature, crystal-field, exchange interaction and oscillating field frequency. It is found that the physical parameters have a strong effect on the shape and number of the DHLs. The results are compared with some theoretical and experimental works and found in a qualitatively good agreement.     

The transverse spin-1 Ising model with random interactions

The phase diagrams of the transverse spin-1 Ising model with random interactions are investigated using a new technique in the effective field theory that employs a probability distribution within the framework of the single-site cluster theory based on the use of exact Ising spin identities. A model is adopted in which the nearest-neighbor exchange couplings are independent random variables distributed according to the law P(J_ij)=pδ(J_ij−J)+(1−p)δ(J_ij−αJ). General formulae, applicable to lattices with coordination number N, are given. Numerical results are presented for a simple cubic lattice. The possible reentrant phenomenon displayed by the system due to the competitive effects between exchange interactions occurs for the appropriate range of the parameter α.     

Phase diagrams of the mixed spin-1 and spin-1/2 Ising-Heisenberg model on the diamond-like decorated Bethe lattice

The ground-state and finite-temperature behavior of the mixed spin-1 and spin-1/2 Ising-Heisenberg model on the diamond-like decorated Bethe lattice is investigated within the framework of two rigorous methods: the decoration-iteration transformation and exact recursion relations. The model under consideration describes a hybrid classical-quantum system consisting of the Ising and Heisenberg spins, which interact among themselves either through the Ising or XXZ Heisenberg nearest-neighbor interaction. Both sublattice magnetizations of the Ising and Heisenberg spins are exactly calculated with the aim to examine phase diagrams, thermal variations of the total and sublattice magnetizations. The finite-temperature phase diagrams form continuous (second-order) phase transition lines only, which exhibit a small reentrant region if the diamond-like decorated Bethe lattice with a sufficiently high coordination number is considered.     

Dynamic phase transitions in the kinetic spin-1 Blume-Capel model on the Bethe lattice

The stationary states of the kinetic spin-1 Blume-Capel (BC) model on the Bethe lattice are analyzed in detail in terms of recursion relations. The model is described using a Glauber-type stochastic dynamics in the presence of a time-dependent oscillating external magnetic field (h) and crystal field (D) interactions. The dynamic order parameter, the hysteresis loop area and the dynamic correlation are calculated. It is found that the magnetization oscillates around nonzero values at low temperatures (T) for the ferromagnetic (F) phase while it only oscillates around zero values at high temperatures for the paramagnetic (P) phase. There are regions of the phase space where the two solutions coexist. The dynamic phase diagrams are obtained on the (kT/J,h/J) and (kT/J,D/J) planes for the coordination number q=4. In addition to second-order and first-order phase transitions, dynamical tricritical points and triple points are also observed.     

Random crystal field effects for spin-3/2 Blume-Capel model

The effects of crystal field (Δ) which is randomly turned on with probability p or turned off with probability 1−p on the sites of the Bethe lattice are investigated to study the critical behavior of the spin-3/2 Blume-Capel model. The order-parameters are obtained in terms of the recursion relations and then the possible phase diagrams are calculated at finite temperatures for given system parameters p, Δ and coordination numbers z=3, 4, 5 and 6. It was found that the phase diagrams change drastically for given probability and randomness. The first-order phase transitions are only found for higher p values and the lines of which get shorter and eventually disappear as p decreases. The critical lines do not present any qualitative differences with z, but they are seen at higher temperatures for higher z.     

The crystal-field dependency of sound attenuation in the spin-3/2 ising model

The effects of crystal-field (D) on sound attenuation are considered for the spin-3/2 Ising model by using Onsager theory of irreversible thermodynamics. It is assumed that the sound wave couples to the order-parameter fluctuations, therefore, it decays mainly via order-parameter relaxation process. The order-parameters, magnetization and quadrupole moment, are defined in terms of exact recursion relations (ERR) on Bethe lattice (BL). After our analysis, two relaxation times are obtained and they are used to calculate the sound attenuation coefficient (α). Consequently, the critical behaviors of sound attenuation coefficient are investigated in terms of frequency (w) and Onsager coefficient (γ) for the coordination numbers q=3, 4 ad 6 near the second-order (T_c) and first-order (T_t) phase transition temperatures in the ferromagnetic phase regions for the negative and positive D values and the results are presented on the (k_BT/J, α) planes. It is found that the peaks about T_t’s are observed at the same temperature, but the peaks about T_c’s are observed shifted to lower and higher temperatures in increasing (w)’s and (γ)’s, respectively. In addition, the peaks are also obtained near the tricritical points for all q.     

The antiferromagnetic Ising model for a bilayer Bethe lattice

The totally antiferromagnetic Ising model is analyzed on a bilayer Bethe lattice in detail by studying the order-parameters, response functions, i.e. susceptibility and specific heat, and free energy by using the recursion relations in a pairwise approach. The ground state phase diagrams of the model are also obtained on the (_J2/|_J1|,_J3/q|_J1|)$(J_2/|J_1|,J_3/q|J_1|)$ plane for given values of H/q|_J1|$H/q\left|J_1\right|$ and on the (H/q|_J1|,_J3/q|_J1|)$(H/q|J_1|,J_3/q|J_1|)$ plane for given _J2/|_J1|$J_2/\left|J_1\right|$. As a result, we have obtained the temperature-dependent phase diagrams for various values of the coordination number q   on the (_J3/|_J1|,kT/|_J1|)$(J_3/|J_1|,\mathit{kT}/|J_1|)$ and (H/|_J1|,kT/|_J1|)$(H/|J_1|,\mathit{kT}/|J_1|)$ planes for given values of the rest of the system parameters.     

The mixed spin-1/2 and spin-1 Ising system on a two-layer Bethe lattice

Two layered magnetic Bethe lattice with varying coordination number q is introduced and numerically studied via exact recursion relations within a pairwise approach. The system is influenced by competing interlayer and intralayer nearest-neighbour (NN) coupling interactions and also by the crystal and external magnetic fields. Cases where both layers are ferromagnetic or one is ferro and the other antiferromagnetic are considered. System configurations’ energy calculations are used to devise some ground state phase diagrams that have proven useful for the investigation of the very low temperature behaviour of the model. Analysis of the thermal behaviours of the total magnetization within the model parameters’ space yield interesting phase diagrams which display fascinating properties, in particular the presence of tricritical points. Increasing negative values of the crystal field strength stabilizes the disordered paramagnetic phase and sometimes gives rise to wavy transition lines.