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Journal of Suzhou University

Relaxation phenomena near the second-order phase transition

Cao Hai-xia

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

The spin-1 Ising system with bilinear and biquadratic interactions has been studied by the combination of the quantum statistical mechanics with the two-spin cluster approximation in face-center-cubic lattice. By use of the Qnsager theory, The behaviors of the spin-1 system such as the relaxation times, the sound attenuation were obtained near the second-order phase transition.

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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).

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International Journal of Modern Physics B•Feb 10, 2006•3 citations
The spin-1 Ising system with bilinear and biquadratic interactions has been studied by the combination of the quantum statistical mechanics with the two-spin cluster approximation in face-center-cubic lattice. By use of the Qnsager theory, The behaviors of the spin-1 system such as the relaxation times, the sound attenuation were obtained near the second-order phase transition.
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).
Dynamics of a spin-3/2 Ising system Hamiltonian with bilinear and biquadratic nearest-neighbour exchange interactions is studied by a simple method in which the statistical equilibrium theory is combined with the Onsager's theory of irreversible thermodynamics. First, the equilibrium behaviour of the model in the molecular-field approximation is given briefly in order to obtain the phase transition temperatures, i.e. the first- and second-order and the tricritical points. Then, the Onsager theory is applied to the model and the kinetic or rate equations are obtained. By solving these equations three relaxation times are calculated and their behaviours are examined for temperatures near the phase transition points. Moreover, the z dynamic critical exponent is calculated and compared with the z values obtained for different systems experimentally and theoretically, and they are found to be in good agrement.
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.
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.
A statistical-mechanical study of a solvable spin-one system with bilinear and biquadratic interactions is presented for various values of α, the ratio of the bilinear to biquadratic interaction. First, equations to determine the order-parameters and the free energy in the equilibrium are derived from the partition function to investigate the phase transitions occurring the model. The magnetic susceptibility is then calculated, and an argument about the critical exponents at the second-order phase transition is given for the case of the critical ratio α=2/3, below which the first order phase transition takes place.
Analytical expressions of the order parameters near the transition temperatures in the spin-3/2 Ising system with bilinear (J) and biquadratic (K) interactions are presented for various values of J/K. First, we obtain the free energy expression and the equations to determine order parameters by using the mean-field approximation. Then, the order parameters are expressed in the vicinity of the transition temperatures in which these expressions are very important to study the dynamics of the system by means of Onsager's theory of irreversible thermodynamics. Hence, we investigate the phase transitions occurring in the system and also obtain two tricritical points analytically. Finally, the specific heat and magnetic susceptibility are calculated and an argument about the critical exponents at the second-order phase transitions and tricritical points is given.