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).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 Journal of Chemical PhysicsMar 20, 200320 citations

Critical behavior of energy-energy, strain-strain, higher-harmonics,and similar correlation functions

Roland R. Netz, Amnon Aharony A

The structure factor associated with general biquadratic correlation functions is calculated for an n-component order parameter using \ensuremath{\epsilon}-expansion techniques in d=4-\ensuremath{\epsilon} dimensions. The results apply to energy-energy and strain-strain correlations as well as to correlations of higher harmonics in density wave systems. We find the correlations of these secondary order parameters to be characterized by a correlation length \ensuremath{\xi}-|Ax=\ensuremath{\xi}-| ${\mathrm{Ax}}_{0}$ [(T- ${\mathrm{T}}_{\mathrm{c}}$ )/ ${\mathrm{T}}_{\mathrm{c}}$ ${]}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$, with the same n-dependent exponent \ensuremath{\nu} as for the correlation length characterizing fluctuations of the primary order parameter, which is denoted by \ensuremath{\xi}. The amplitude ratio X-|Ax\ensuremath{\equiv}(\ensuremath{\xi}-| ${\mathrm{Ax}}_{0}$ /${\ensuremath{\xi}}_{0}$ ${)}^{2}$ is universal, and we obtain ${\mathrm{X}}_{\mathrm{T}}$ =${\ensuremath{\gamma}}_{\mathrm{T}}$/6\ensuremath{\gamma}+O(${\mathrm{\ensuremath{\epsilon}}}^{3}$) for quadratic order parameters transforming like a traceless spin tensor in n-component space (with ${\ensuremath{\gamma}}_{\mathrm{T}}$ characterizing the divergence of the corresponding susceptibility) and ${\mathrm{X}}_{\mathrm{E}}$ =\ensuremath{\alpha}/6\ensuremath{\gamma}+O(${\mathrm{\ensuremath{\epsilon}}}^{3}$) for energy-energy correlations, where \ensuremath{\alpha} and \ensuremath{\gamma} denote the usual specific heat and susceptibility critical exponents, respectively. The universal amplitude ratio for the second harmonic in density wave systems is given by ${\mathrm{X}}_{\mathrm{T}}$ with n=2 and takes the value ${\mathrm{X}}_{2}$ =\ensuremath{\epsilon}/20-${\mathrm{\ensuremath{\epsilon}}}^{2}$/100+O(${\mathrm{\ensuremath{\epsilon}}}^{3}$), thus being very small. This naturally explains previously puzzling experimental results for the critical behavior of the second harmonic structure factor at the nematic--smectic- ${\mathrm{A}}_{2}$ transition of a thermotropic liquid crystal. Applications to sound attenuation in liquids or solids close to critical transitions and to colloidal interactions in near-critical binary mixtures are briefly discussed.The structure factor associated with general biquadratic correlation functions is calculated for an n-component order parameter using \ensuremath{\epsilon}-expansion techniques in d=4-\ensuremath{\epsilon} dimensions. The results apply to energy-energy and strain-strain correlations as well as to correlations of higher harmonics in density wave systems. We find the correlations of these secondary order parameters to be characterized by a correlation length \ensuremath{\xi}-|Ax=\ensuremath{\xi}-|${\mathrm{Ax}}_{0}$[(T-${\mathrm{T}}_{\mathrm{c}}$)/${\mathrm{T}}_{\mathrm{c}}$${]}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$, with the same n-dependent exponent \ensuremath{\nu} as for the correlation length characterizing fluctuations of the primary order parameter, which is denoted by \ensuremath{\xi}. The amplitude ratio X-|Ax\ensuremath{\equiv}(\ensuremath{\xi}-|${\mathrm{Ax}}_{0}$/${\ensuremath{\xi}}_{0}$${)}^{2}$ is universal, and we obtain ${\mathrm{X}}_{\mathrm{T}}$=${\ensuremath{\gamma}}_{\mathrm{T}}$/6\ensuremath{\gamma}+O(${\mathrm{\ensuremath{\epsilon}}}^{3}$) for quadratic order parameters transforming like a traceless spin tensor in n-component space (with ${\ensuremath{\gamma}}_{\mathrm{T}}$ characterizing the divergence of the corresponding susceptibility) and ${\mathrm{X}}_{\mathrm{E}}$=\ensuremath{\alpha}/6\ensuremath{\gamma}+O(${\mathrm{\ensuremath{\epsilon}}}^{3}$) for energy-energy correlations, where \ensuremath{\alpha} and \ensuremath{\gamma} denote the usual specific heat and susceptibility critical exponents, respectively. The universal amplitude ratio for the second harmonic in density wave systems is given by ${\mathrm{X}}_{\mathrm{T}}$ with n=2 and takes the value ${\mathrm{X}}_{2}$=\ensuremath{\epsilon}/20-${\mathrm{\ensuremath{\epsilon}}}^{2}$/100+O(${\mathrm{\ensuremath{\epsilon}}}^{3}$), thus being very small. This naturally explains previously puzzling experimental results for the critical behavior of the second harmonic structure factor at the nematic--smectic-${\mathrm{A}}_{2}$ transition of a thermotropic liquid crystal. Applications to sound attenuation in liquids or solids close to critical transitions and to colloidal interactions in near-critical binary mixtures are briefly discussed.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsMar 1, 19977 citations

Critical ultrasound attenuation in isotropic systems

Barbara Drossel, Franz Schwabl

The European Physical Journal BMar 1, 19940 citations

{\mathrm{Ax}}_{0}

[(T-

{\mathrm{T}}_{\mathrm{c}}

{\mathrm{T}}_{\mathrm{c}}

${]}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$, with the same n-dependent exponent \ensuremath{\nu} as for the correlation length characterizing fluctuations of the primary order parameter, which is denoted by \ensuremath{\xi}. The amplitude ratio X-|Ax\ensuremath{\equiv}(\ensuremath{\xi}-|

{\mathrm{Ax}}_{0}

/${\ensuremath{\xi}}_{0}$

{)}^{2}

is universal, and we obtain

{\mathrm{X}}_{\mathrm{T}}

=${\ensuremath{\gamma}}_{\mathrm{T}}$/6\ensuremath{\gamma}+O(${\mathrm{\ensuremath{\epsilon}}}^{3}$) for quadratic order parameters transforming like a traceless spin tensor in n-component space (with ${\ensuremath{\gamma}}_{\mathrm{T}}$ characterizing the divergence of the corresponding susceptibility) and

{\mathrm{X}}_{\mathrm{E}}

=\ensuremath{\alpha}/6\ensuremath{\gamma}+O(${\mathrm{\ensuremath{\epsilon}}}^{3}$) for energy-energy correlations, where \ensuremath{\alpha} and \ensuremath{\gamma} denote the usual specific heat and susceptibility critical exponents, respectively. The universal amplitude ratio for the second harmonic in density wave systems is given by

{\mathrm{X}}_{\mathrm{T}}

with n=2 and takes the value

{\mathrm{X}}_{2}

=\ensuremath{\epsilon}/20-${\mathrm{\ensuremath{\epsilon}}}^{2}$/100+O(${\mathrm{\ensuremath{\epsilon}}}^{3}$), thus being very small. This naturally explains previously puzzling experimental results for the critical behavior of the second harmonic structure factor at the nematic--smectic-

{\mathrm{A}}_{2}

transition of a thermotropic liquid crystal. Applications to sound attenuation in liquids or solids close to critical transitions and to colloidal interactions in near-critical binary mixtures are briefly discussed.