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Physical Review A

High-Frequency Sound Attenuation and Dispersion in the Critical Region

Kyozi Kawasaki

Published Mar 1, 1971• 19 citations
Publisher

Abstract

A general theory is presented for sound attenuation and dispersion near the critical point in the region of frequencies which are much greater than the characteristic frequency of order-parameter dynamics, but are much smaller than the frequency of sound with the wavelength equal to the range of correlation of local order-parameter fluctuations. The method used is the most general version of the mode-mode coupling theory plus the static and dynamic scaling hypotheses and some general properties of time-correlation functions. If we ignore the small critical exponent $\ensuremath{\alpha}$ associated with the heat-capacity singularity, we find that the attenuation behaves as $f{\ensuremath{\epsilon}}^{0}$ and the dispersion (relative sound-velocity change with frequency) as ${f}^{0}{\ensuremath{\epsilon}}^{0}$, where $f$ is the dimensionless frequency and $\ensuremath{\epsilon}$ the dimensionless temperature distance from the critical point.

DOI
10.1103/physreva.3.1097

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Physical Review E•Sep 17, 2009•2 citations
A general theory is presented for sound attenuation and dispersion near the critical point in the region of frequencies which are much greater than the characteristic frequency of order-parameter dynamics, but are much smaller than the frequency of sound with the wavelength equal to the range of correlation of local order-parameter fluctuations. The method used is the most general version of the mode-mode coupling theory plus the static and dynamic scaling hypotheses and some general properties of time-correlation functions. If we ignore the small critical exponent \ensuremathα\ensuremath{\alpha}\ensuremathα associated with the heat-capacity singularity, we find that the attenuation behaves as f\ensuremathϵ0f{\ensuremath{\epsilon}}^{0}f\ensuremathϵ0 and the dispersion (relative sound-velocity change with frequency) as f0\ensuremathϵ0{f}^{0}{\ensuremath{\epsilon}}^{0}f0\ensuremathϵ0, where fff is the dimensionless frequency and \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ the dimensionless temperature distance from the critical point.
The usual static scaling laws are generalized to nonequilibrium phenomena by making assumptions on the behavior of time-dependent correlation functions near the critical point of second-order phase transitions. At any temperature different from Tc{T}_{c}Tc​, the correlation functions are assumed to reflect the hydrodynamic behavior of the system, for sufficiently long wavelengths and low frequencies. As the critical temperature is approached, however, the range of spatial correlations in the system diverges, and the domain of applicability of hydrodynamics is reduced to a vanishingly small region of wavelengths and frequencies. The dynamic-scaling assumptions lead to predictions for the behavior of the hydrodynamic parameters near Tc{T}_{c}Tc​, as well as for the form of the correlation functions for macroscopic distances and times, outside the hydrodynamic range. In particular, singularities are predicted to occur in the temperature dependence of transport coefficients, and anomalies are expected in the frequency spectrum of certain operators, which are observable by inelastic scattering of neutrons or light. A distinction is made between the restricted dynamic-scaling hypothesis, which refers to the order parameter only, and extended dynamic scaling, which applies to other operators and involves stronger assumptions. Applications are discussed to antiferromagnets, ferromagnets, the gas-liquid critical point, and the $\ensuremath{\lambda}$ transition in superfluid helium. Specific experiments are suggested to test the scaling assumptions, and existing experimental evidence is briefly reviewed. Finally, a comparison is made with other theories of dynamical behavior near critical points.
The nonasymptotic critical behavior of sound attenuation coefficient has been studied in an elastically isotropic Ising system above the critical point on the basis of a complete stochastic model including both spin-energy and lattice-energy modes linearly coupled to the longitudinal sound mode. The effect of spin-lattice relaxation on the ultrasonic attenuation is investigated. The crossover between weak-singularity behavior t−2αt^{-2 α}t−2α and strong singularity behavior t−(zν+α)t^{-(z ν+α)}t−(zν+α) is studied. A new high-frequency regime with singularity of the type t−zν+αt^{-z ν+α}t−zν+α is discovered in the magnetic systems. This new regime corresponds to an adiabatic sound propagation and is very similar to the ones in binary mixture and liquid helium. A new frequency-dependent specific-heat being the harmonic average of the bare lattice and critical spin specific-heats is introduced. It was shown that such specific-heat descibes the process of equilibrization between spin and lattice subsystems and includes the most important features of critical sound attenuation. In some regions of coupling constants the acoustic self-energy can be very well approximated solely by this quantity.
Dynamics of critical fluctuations associated with the liquid-gas transition are studied employing the method developed in the previous works of this series. The CDV within which dynamics can be asymptotically closed have been found which correspond to the sound wave modes. However, these CDV are not sufficient to account for all the anomalies (divergences) of the transport coefficients. Rapid random motions also contribute to the divergences through various transport modes, thus leading to partial violation of the dynamical scaling law. The results for the anomalies essentially confirm the work of Kadanoff and Swift. The sound propagation is also studied. In some high frequency regions the sound speed is found to exceed the usual adiabatic value by a constant factor, and the attenuation does not depend upon frequency. The case of the critical mixture is also briefly discussed.
The critical dynamics at elastic phase transitions of second order is studied by the renormalization-group theory. The dynamical theory is based on a stochastic equation of motion for damped phonons. For systems with one-dimensional soft sectors usual dynamical scaling holds with the critical dynamical exponent z=2z=2z=2. For two-dimensional sectors logarithmic corrections appear. Novel features are found for an isotropic-phonon model. The dynamical susceptibility and the characteristic frequency depend singularly on an irrelevant parameter, which, however, diverges at the fixed point. Consequently dynamical scaling breaks down; e.g., the characteristic frequency is no longer a homogeneous function of the wave number kkk and the inverse correlation length ${\ensuremath{\xi}}^{\ensuremath{-}1}$. Instead it is a homogeneous function of these variables and an irrelevant parameter. Consequently, the critical dynamics can be characterized by the relaxational exponent $z=2+c\ensuremath{\eta}$ at Tc{T}_{c}Tc​, while in the hydrodynamic region the sound frequency is characterized by $z=2\ensuremath{-}\frac{1}{2}\ensuremath{\eta}$ and the damping by yet another exponent. This breakdown of scaling is also reflected by the fact that different fixing conditions, i.e., different choices of the frequency, lead to different values of zzz. All of these apparently different transformations lead to the same modified dynamical scaling relations.
Dynamical scaling functions above Tc for the characteristic frequencies and the dynamical correlation functions of the order parameter and the conserved density of model C are calculated in one loop order. By a proper exponentiation procedure these results can be extended in order to consider the changes in these functions using the fixed point values and exponents in two loop order. The dynamical amplitude ratio R of the characteristic frequencies is generalized to the critical region. Surprisingly the decay of the shape functions at large scaled frequency does not behave as expected from applying scaling arguments. The exponent upsilon of the decay does not change when going from the critical to the hydrodynamic region although the shape functions change. The value of upsilon for the order parameter is in agreement with its value in the critical region, whereas for the conserved density it is equal to 2, the value in the hydrodynamic region.