The anomalous behavior of the specific heat for β-glycine was analyzed in terms of the compressible Ising model approximately 5 K below and above the ferrelectric-paraelectric phase transition temperature of TC = 252 K. The obtained value of the critical exponent α = 0.12 in the ferroelectric phase (T < TC) was consistent with that predicted from the 3-d Ising model (α = 0.13), while the obtained value of α = 0.32 in the paraelectric phase (T > TC) was consistent with that predicted from the 2-d potts model (α = 0.30). Some thermodynamic quantities such as the internal energy, the entropy and the free energy of β-glycine were then predicted in terms of these extracted values of the critical exponents close to the phase transition temperature of TC. Our calculated values of the thermodynamic quantities are in good agreement with the observed data.
The chemical shift of the N(2)(CH 3 ) 4 ion, which has been found to exhibit the similar anomalous behavior of the monoclinic angle Aβ, was related to the order parameter to evaluate the temperature dependence of the linewidth (damping constant) for 14 N nuclear magnetic resonance spectrum of this crystal in terms of the dynamic Ising models, namely the pseudospin-phonon-coupled (PS) and the energy fluctuation (EF) models. The results from both PS and EF models were successful to explain the abnormal behavior of the linewidth in the vicinity of the phase-transition temperature of T C = 287.6 K, when compared with the observed linewidth of the transverse acoustic soft mode in this crystal. As an extension of this work, the 14 N relaxation time and the values of the activation free energy were calculated as a function of temperature. The results indicate that the ferroelastic-paraelastic phase transition in this compound is of the order-disorder type.
A power-law formula deduced from the Ising model was used to analyze the temperature dependence of the specific heat Cp and the Gibbs free energy G of [N(CH3)4]2ZnBr4 compound in the vicinity of the phase transition temperature of TC = 287.2 K. Obtained values of the critical exponents α from the Gibbs free energy were consistent with that predicted from 2-d potts model (α = 0.3), while obtained values of α from the specific heat in both ferroelastic and paraelastic phases were consistent with that predicted from the mean field theory (α = 0) in the vicinity of the phase transition temperature. This is an indication of that [N(CH3)4]2ZnBr4 compound undergoes a second order type phase transition.Also, the enthalpy (H) and the entropy (S) of this crystal were calculated in terms of the extracted values of the critical exponent in both ferroelastic and paraelastic phases.
The isothermal mode Grüneisen parameter [Formula: see text] of some Raman modes in [Formula: see text]Ti x O 3 (PZT, [Formula: see text]) were calculated as a function of pressure by means of the observed pressure-dependent volume data of PZT ([Formula: see text]) crystal from the literature at room temperature of 298[Formula: see text]K. Those calculated values of [Formula: see text] were then used to compute the pressure dependence of the Raman modes in PZT ([Formula: see text]) ceramic studied here. The observed and calculated values of the Raman wavenumbers in PZT were in good agreement, which indicates that the isothermal mode Grüneisen parameter can also be used to predict the pressure-dependent wavenumbers of some other perovskite-type crystals. Additionally, the pressure dependence of the thermodynamic quantities such as isothermal compressibility [Formula: see text], thermal expansion [Formula: see text] and the specific heat [Formula: see text] of PZT ([Formula: see text]) ceramic were predicted at constant temperature of 298[Formula: see text]K. Here, the experimentally measurable thermodynamic quantities calculated for PZT ([Formula: see text]) ceramics provide theoretically a significant opportunity for testing.
The temperature dependence of the Brillouin frequency and the linewidth of the LA mode is studied for the concentration of x=0.45 in the ferroelectric phase of PbZr 1−x Ti x O 3 (PZT-x) within the temperature range of 443 to 656 K (T C =657 K). Using the experimental data for the Brillouin frequency of the LA mode as an order parameter below T C the temperature dependence of the linewidth (FWHM) is calculated by the pseudospin-phonon coupled model and the energy fluctuation model for the lead titanate zirconate (x=0.45). Additionally, the activation energies are compared from the damping constant (linewidth) using both models for the temperature range of 443 to 656 K in the ferroelectric phase of PbZr 1−x Ti x O 3 . Our calculated values for the damping constant are in agreement with the observed data for the ferroelectric phase of PbZr 1−x Ti x O 3 single crystals. The activation energies calculated from both models are much higher in the ferroelectric phase than the value of k B T C =0.056 eV at the transition temperature for PbZr 1−x Ti x O 3 (x=0.45).
The temperature dependence of the damping constant is calculated below the transition temperature (T 0=222K) in the ferroelectric phase of hexagonal barium titanate. The damping constant of the coupled soft-optic and acoustic mode which causes an intense central peak in the light scattering spectra, is calculated using the soft mode-hard mode coupling model and the energy fluctuation model for barium titanate. By considering the bilinear coupling between acoustic and soft-optic modes, which is related to the order parameter, the inverse of relaxation time is estimated as a function of temperature below T0 in barium titanate. Our results are compared with the experimental data obtained from the light scattering spectra of this ferroelectric compound.
The damping constant is calculated as a function of pressure at room temperature using the normalized intensity as an order parameter near the cubic-tetragonal phase transition in SrTiO3. The observed X-ray diffraction data are used for the normalized intensities to calculate the damping constant (Γ) from the pseudospin-phonon (PS) coupled model and the energy fluctuation (EF) model, which is fitted to the observed FWHM data from the literature for comparison. Using the calculated Γ values, the pressure dependence of the inverse relaxation time (τ−1) is predicted close to the cubic-tetragonal phase transition in SrTiO3. Our calculated damping constant from both models explains the observed FWHM satisfactorily and our prediction of the inverse relaxation time can also be compared with the experimental measurements when they are available in the literature.
The damping constant (linewidth) of the longitudinal acoustic (LA) mode is calculated as a function of temperature using the observed Brillouin frequencies of this mode from the literature for the ferroelectric-paraelectric transition ( TC = 657 K) in PbZr1-xTixO3 ( x = 0.45 ). For this calculation of the damping constant, the pseudospin-phonon coupled model and the energy fluctuation model are used by fitting to the observed data for the Brillouin frequencies of the LA mode in the ferroelectric ( ) and paraelectric ( T > TC) phases of this compound ( x = 0.45 ). Values of the activation energy are deduced for both ferroelectric and paraelectric phases. The relaxation time is also obtained by means of fitting to the observed data from the literature for the inverse relaxation time at various temperatures in the paraelectric phase of PbZr1-xTixO3. The temperature dependences of the damping constant and of the relaxation time with the values of the activation energy that we have calculated indicate that the pseudospin-phonon coupled model and the energy fluctuation model are capable of describing the ferroelectric-paraelectric transition ( TC = 657 K) in PbZr1-xTixO3 ( x = 0.45 ) adequately.
