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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 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.

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The ground-state (GS) phase diagrams of spin-3/2 Ising model for ferromagnetic/antiferromagnetic (FM/AFM) interactions on a two-layer lattice are studied under a constant magnetic field by varying the crystal field values for general coordination number q in detail. Thus, it is found that the system exhibits twenty-nine topologically different GS configurations according to their GS phase diagrams obtained in the (J2/J1, J3/qJ1) planes. As the crystal field becomes more and more negative, the remaining phase configurations in the phase diagrams include only spin ±1/2 values as expected.