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Journal of Statistical Mechanics Theory and Experiment

The crystal field effects on sound attenuation for a spin-1 Ising model on the Bethe lattice

Tunç Cengiz, Erhan Albayrak

Published Jul 6, 2012• 1 citation
Publisher

Abstract

The sound attenuation phenomenon is investigated by using the Onsager theory of irreversible thermodynamics for a spin-1 Ising model with the inclusion of the crystal field effects on the Bethe lattice. The recursion relations are calculated in a transcendental form to obtain the order-parameters and then the sound attenuation is analyzed. The relationships of sound attenuation with temperature, frequency and Onsager coefficient are examined near the second- and first-order phase transition temperatures, T c and T t respectively, for given negative crystal field values and coordination numbers on the Bethe lattice. The sound wave couples to the order-parameter fluctuations which decay mainly via the order-parameter relaxation process. Thus, two relaxation times are obtained which are used to find an expression for the sound attenuation coefficient. The attenuation maxima are found near the second- and first-order phase transition temperatures in the ferromagnetic and quadrupole phase regions, respectively, for the coordination numbers q = 3,4 and 6. The attenuation peaks are observed at the same temperature before T t and are found to be shifted to lower (higher) temperatures with increasing value of frequency (Onsager coefficient) before T c for any crystal field values. The attenuation peaks are found at lower values and at higher temperatures with negatively increasing crystal field values in the quadrupolar phase regions. In addition, the sound attenuation peaks are also studied at some tricritical points for q = 3,4 and 6 for some critical values of the crystal field.

DOI
10.1088/1742-5468/2012/07/p07004

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The sound attenuation phenomenon is investigated by using the Onsager theory of irreversible thermodynamics for a spin-1 Ising model with the inclusion of the crystal field effects on the Bethe lattice. The recursion relations are calculated in a transcendental form to obtain the order-parameters and then the sound attenuation is analyzed. The relationships of sound attenuation with temperature, frequency and Onsager coefficient are examined near the second- and first-order phase transition temperatures, T c and T t respectively, for given negative crystal field values and coordination numbers on the Bethe lattice. The sound wave couples to the order-parameter fluctuations which decay mainly via the order-parameter relaxation process. Thus, two relaxation times are obtained which are used to find an expression for the sound attenuation coefficient. The attenuation maxima are found near the second- and first-order phase transition temperatures in the ferromagnetic and quadrupole phase regions, respectively, for the coordination numbers q = 3,4 and 6. The attenuation peaks are observed at the same temperature before T t and are found to be shifted to lower (higher) temperatures with increasing value of frequency (Onsager coefficient) before T c for any crystal field values. The attenuation peaks are found at lower values and at higher temperatures with negatively increasing crystal field values in the quadrupolar phase regions. In addition, the sound attenuation peaks are also studied at some tricritical points for q = 3,4 and 6 for some critical values of the crystal field.
The temperature ( T ), frequency ( w ), Onsager coefficient (γ) and external magnetic field ( H ) dependencies of the sound attenuation near the phase transition temperature ( T c ) is studied for the spin-1 Ising model on the Bethe lattice for given coordination numbers q in terms of the recursion relations by using Onsager theory of irreversible thermodynamics. It was assumed that the sound wave couples to the order-parameter fluctuations which decay mainly via the order-parameter relaxation process. Two relaxation times are obtained and which are used to find an expression for the sound attenuation coefficient (α). It was found that α presents only one peak for all frequencies when q =6, and two peaks for q =3 and 4 at low frequencies. In addition, α increases with the increasing values of frequency and with the decreasing values of external magnetic field.
Critical behaviors of the sound attenuation in a spin-1 Ising system with bilinear (J) and biquadratic (K) interactions are investigated within the framework of cluster variation method and Onsager theory of irreversible thermodynamics. The sound wave is assumed to couple mainly to the order parameter fluctuations which decay via order parameter relaxation processes. Two relaxation times are obtained and an expression is found for the sound attenuation coefficient (α) in terms of these relaxation times. The temperature behavior of the sound attenuation near the phase transition temperatures (Tc) is analyzed according to various values of Onsager coefficients (γij) and sound frequency (ω). For T<Tc it is found that the maxima of the attenuation shifted to lower temperatures with increasing ω and γij (i≠j) values. For T>Tc the data give evidence that there is no relaxational contribution to sound attenuation coming from order parameter fluctuations. On the other hand, a convergence is found in attenuation just below the critical and the tricritical points as (Tc−T), while a jump-discontinuity is observed for the first-order behavior. The frequency variation of the sound attenuation is also investigated and in addition to ω2-attenuation dependence observed in the hydrodynamic regime it is observed that in the high frequency region the attenuation is independent of ω and the ratio of two interaction parameters (J/K).
The influence of the crystal field on the phase diagrams of the bilayer spin-1 Ising model on the Bethe lattice is studied in terms of the intralayer coupling constants J1 and J2 of the two layers and interlayer coupling constant J3 between the layers for given values of the coordination number q by using the recursion relation scheme. The six distinct ground-state configurations of the model are obtained on the (J_{2}J_{1},J_{3}qJ_{1}) plane with J1>0 , the ferromagnetic coupling, for given values of the crystal field. Then, the phase diagram of the system is obtained on the (kTJ_{1},J_{3}J_{1}) plane for given values of the crystal field and alpha=J_{2}J_{1} with q=4 corresponding to the square lattice in real lattice systems. It was found that the system presents both first- and second-order phase transitions, therefore, tricritical points. The paramagnetic phase was also divided into two phases, P+ and P_{-} , by studying the thermal behavior of the quadrupolar moments of the two layers.
The field-theoretic description of dynamic critical effects of the disorder on acoustic anomalies near the temperature of second-order phase transition is considered for three-dimensional Ising-like systems.The calculations of sound attenuation in high temperature phase for pure and diluted Ising-like systems near the critical point are presented.The dynamic scaling function for critical attenuation coefficient is calculated.The effect of the quenched disorder on asymptotic behaviour of the critical ultrasonic anomalies is discussed.
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.
Two-layer Bethe lattice whose lattice sites are occupied with spin-3/2 atoms is solved exactly by using the 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 ferromagnetic (FM) and antiferromagnetic (AFM) interactions for the spins of the upper and lower layers, respectively, and either FM or AFM type interactions between the adjacent spins of the layers are assumed.The phase diagrams of the model are studied on different planes for given system parameters by obtaining 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.It was found that the model presents both second-and first-order phase transitions.The reentrant behavior is seen when the model presents two N éel temperatures for higher q values.The existence of the tricritical point and critical end points is also confirmed.