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Acta Physica Polonica A

Spin-Lattice Relaxation Effects in Critical Sound Propagation

A. Pawlak

Published Aug 1, 1997• 1 citation
Publisher

Abstract

The nonasymptotic critical properties of sound propagation are studied in compressible Ising system above Τ.In the present paper we analyse a model where in addition to the coupling to two order-parameter fluctuations tle sound mode couples linearly to the fluctuations of spin-energy and lattice-energy densities.Both subsystems exchange energy with the rate determined by the bare spin-lattice relaxation time: The total energy may be conserved or not.The crossover between insulator-like behaviour t -2 α a n d m e t a l -l i k e b e h a v i o u r t i t -( z v ± α ) i n u l t r a s o n i c a t t e n u a t i o n i s i n v e s t i g a t e d according to the value of ultrasonic freqnency, the reduced temperature t, bare relaxation times and various coupling constants.

DOI
10.12693/aphyspola.92.449

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Physical review. B, Condensed matter•May 1, 1993•5 citations
The nonasymptotic critical properties of sound propagation are studied in compressible Ising system above Τ.In the present paper we analyse a model where in addition to the coupling to two order-parameter fluctuations tle sound mode couples linearly to the fluctuations of spin-energy and lattice-energy densities.Both subsystems exchange energy with the rate determined by the bare spin-lattice relaxation time: The total energy may be conserved or not.The crossover between insulator-like behaviour t -2 α a n d m e t a l -l i k e b e h a v i o u r t i t -( z v ± α ) i n u l t r a s o n i c a t t e n u a t i o n i s i n v e s t i g a t e d according to the value of ultrasonic freqnency, the reduced temperature t, bare relaxation times and various coupling constants.
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.
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 temperature ( T ), frequency ( w ), Onsager coefficient (γ) and external magnetic field ( H ) dependencies of the sound attenuation near the phase transition temperature ( T c ) is studied for the spin-1 Ising model on the Bethe lattice for given coordination numbers q in terms of the recursion relations by using Onsager theory of irreversible thermodynamics. It was assumed that the sound wave couples to the order-parameter fluctuations which decay mainly via the order-parameter relaxation process. Two relaxation times are obtained and which are used to find an expression for the sound attenuation coefficient (α). It was found that α presents only one peak for all frequencies when q =6, and two peaks for q =3 and 4 at low frequencies. In addition, α increases with the increasing values of frequency and with the decreasing values of external magnetic field.
The dynamical response of a spin-glass is studied at the mean-field level for models in which the total magnetization is conserved. The first model considered is totally dissipative, like the conventional relaxational ones, but the Langevin equation is diffusive. In the $k\ensuremath{\rightarrow}0$ limit, the diffusion constant is unaffected by the proximity of the spin-glass transition, but the region of kkk space in which hydrodynamics is valid shrinks to the origin as $T\ensuremath{\rightarrow}{T}_{g}$. In the rest of kkk space, the dynamics are effectively the same as in the relaxational model, with a relaxation rate $\ensuremath{\propto}T\ensuremath{-}{T}_{g}$. At Tg{T}_{g}Tg​, spin correlations have the ${t}^{\ensuremath{-}\frac{1}{2}}$ behavior of the relaxational model. Mode coupling is added in the second model, and a self-consistent calculation gives a transport coefficient $\ensuremath{\propto}{[\mathrm{ln}(T\ensuremath{-}{T}_{g})]}^{\frac{1}{2}}$ [and $\ensuremath{\propto}{(\mathrm{ln}\ensuremath{\omega})}^{\frac{1}{2}} \mathrm{at} {T}_{g}$]. Sound propagation is also examined, and the sound-damping rate is found to have similar logarithmic behavior. The sound speed varies smoothly through Tg{T}_{g}Tg​. Below Tg{T}_{g}Tg​ we find new ${\ensuremath{\omega}}^{\ensuremath{-}\frac{1}{3}}$ singularities in the transport coefficient, spin-correlation function, and sound damping.
The Landau-Ginsburg free-energy functional (effective Hamiltonian) is derived for compressible Ising systems where the Ising spins are coupled with quantum phonon fields and a transverse field is present. In the zero transverse field case, the phonon dynamical modes also disappear from the effective Hamiltonian, which then compares with those studied by Sak and De Moura et al. in the non-quantum compressible Ising systems. In the presence of dynamics in the spin system (compressible transverse Ising system), the phonon modes get effectively coupled in the Hamiltonian and while at non-zero temperatures this does not affect the critical behaviour (which remains the same as in the non-quantum case), the presence of these dynamical modes effectively increases the lattice dimensionality by unity at zero-temperature transitions (T c =0) where one observes the critical behaviour corresponding to a d+1 dimensional non-quantum compressible Ising model. The critical behaviour of a cooperative Jahn-Teller phase transition when T c to 0 is also discussed.
The energy relaxation in the two-dimensional Ising model, with temperature greater than Tc{\mathit{T}}_{\mathit{c}}Tc​ and in a spin system having a 5120\ifmmode\times\else\texttimes\fi{}5120 lattice, is studied using supercomputer-conducted Monte Carlo simulation. The critical exponents of the linear and nonlinear relaxation times of the internal energy are respectively estimated to be ${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathit{l})}$=2.064\ifmmode\pm\else\textpm\fi{}0.026 and ${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathrm{nl})}$=1.132\ifmmode\pm\else\textpm\fi{}0.080, being consistent with the Racz scaling law ${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathit{l})}$-${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathrm{nl})}$=1-\ensuremath{\alpha}, where \ensuremath{\alpha} is the critical exponent of the specific heat. Based on the author's previous result in which the critical exponent of the linear relaxation for magnetization was determined as ${\mathrm{\ensuremath{\Delta}}}_{\mathit{M}}^{(\mathit{l})}$=2.076\ifmmode\pm\else\textpm\fi{}0.005, it is strongly suggested that ${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathit{l})}$=${\mathrm{\ensuremath{\Delta}}}_{\mathit{M}}^{(\mathit{l})}$.