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

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 secondand 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.

Acoustic Signatures of the Phases and Phase Transitions in the Blume Capel Model with Random Crystal Field

“…The theoretical treatment of the sound propagation by mode-mode coupling method predicts that the sound attenuation coefficient scales as ∼ for the sound frequencies that are quite larger than the characteristic frequency of the order parameter fluctuations and an extensive study of the frequency dispersion of the critical attenuation of KMnF 3 above 186.2 K measured = 0.13 ± 0.05 [53][54][55]. In addition, similar frequency dependence of the sound attenuation coefficient has been reported for various model systems such as spin-1 and spin-3/2 Ising models on the Bethe lattice [39,40], zero crystal field BEG model [37], and metamagnetic Ising model [38] within mean-field theory.…”

“…Further, the temperature at which attenuation maxima take place moves to lower temperatures with rising frequency. The frequency shift of the attenuation maxima is also observed in the experimental and theoretical studies of the ultrasonic investigation of liquid helium [56,57] and Rb 2 ZnCl 4 and K 2 SeO 4 [58] as well as other Ising models [37][38][39][40] and elastically isotropic Ising system above the critical point on the basis of a complete stochastic model [59].…”

“…The attenuation peak broadens and its amplitude decreases and shifts to lower temperatures with decreasing , while = 0.2. We should note that, for systems that have two or more order parameters, the relation between the generalized forces and currents may be written in terms of a matrix of phenomenological rate coefficients and the effect of the nondiagonal Onsager coefficient has been investigated for BEG and metamagnetic Ising models in the mean-field approximation [37,38] and spin-3/2 and spin-1 Ising models on the Bethe lattice [39,40]. The critical behavior of the sound attenuation coefficient in the neighborhood of the second-order phase transition points is characterized by the dynamical sound attenuation critical exponent .…”

“…Later, the absorption of sound in the spin-3/2 Ising model on the Bethe lattice is obtained and its temperature variance is analyzed near the phase transition points [39]. Recently, Cengiz and Albayrak studied the sound attenuation phenomena for a finite crystal field BEG model on the Bethe lattice in terms of the recursion relations by using the Onsager theory [40]. Due to mathematical complexity, the properties of critical sound propagation have not been studied in any spin system with random bond, random magnetic field, or random crystal field terms in the Hamiltonian expression.…”