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Physical Review

Diffusion Constants near the Critical Point for Time-Dependent Ising Models. I

Kyozi Kawasaki

Published May 6, 1966• 616 citations
Publisher

Abstract

The diffusion constant of spins in ferromagnets or of molecules in binary mixtures near the critical point is discussed employing a time-dependent Ising model in which spin interactions are replaced by certain temperature-dependent transition probabilities of spin exchange. The spin diffusion constant is calculated with the single approximation of replacing a reduced spin distribution function by its value for local equilibrium with a given inhomogeneous spin density. The behavior of the diffusion constant near the critical point is dominated by a factor ${\ensuremath{\chi}}^{\ensuremath{-}1}$, where $\ensuremath{\chi}$ is the magnetic susceptibility. This problem is also studied with the use of the Bethe lattice. The effects of surrounding spins on the transition probability for spin exchange are found to be essential for obtaining the critical slowing-down near the critical point. In view of this, Koci\ifmmode \acute{n}\else \'{n}\fi{}ski's calculation of the spin diffusion constant is critically discussed.

DOI
10.1103/physrev.145.224

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The diffusion constant of spins in ferromagnets or of molecules in binary mixtures near the critical point is discussed employing a time-dependent Ising model in which spin interactions are replaced by certain temperature-dependent transition probabilities of spin exchange. The spin diffusion constant is calculated with the single approximation of replacing a reduced spin distribution function by its value for local equilibrium with a given inhomogeneous spin density. The behavior of the diffusion constant near the critical point is dominated by a factor \ensuremathχ\ensuremath−1{\ensuremath{\chi}}^{\ensuremath{-}1}\ensuremathχ\ensuremath−1, where \ensuremathχ\ensuremath{\chi}\ensuremathχ is the magnetic susceptibility. This problem is also studied with the use of the Bethe lattice. The effects of surrounding spins on the transition probability for spin exchange are found to be essential for obtaining the critical slowing-down near the critical point. In view of this, Koci\ifmmode \acute{n}\else '{n}\fi{}ski's calculation of the spin diffusion constant is critically discussed.
The nature of the local-equilibrium approximation employed in our previous work is examined with the use of the correlation-function expression for the general frequency-dependent transport coefficient for systems described by master equations. The transport coefficient is found to be a nondecreasing function of frequency which in the high-frequency limit reduces to the transport coefficient calculated in the local-equilibrium approximation. Thus the local-equilibrium approximation provides an upper bound to the zero-frequency transport coefficient. For the case of spin diffusion, combined with the result of our previous work, this limit implies that the diffusion constant must vanish at least as fast as the inverse of the magnetic susceptibility near the Curie point, which establishes the existence of the critical slowing down for our model of spin diffusion.
Employing a time-dependent Ising model of the kind used in the preceding works of this series, we have studied the self-diffusion constant. In the local-equilibrium approximation, the self-diffusion constant is expressed in terms of certain equal-time spin correlation functions. At the critical point the self-diffusion constant is shown to have a finite value, and it is conjectured that it may have an infinite slope as a function of temperature at the critical point. The two relevant experiments are discussed.
A statistical theory of the localized model of ferromagnetism is developed by considering the effective density matrices for a single spin, and two nearest-neighbor spins, respectively. The Ising Model and the Heisenberg model, and even the case of the anisotropic exchange interaction, are treated within a unified mathematical scheme. The result is reduced to that of the Bethe approximation in the case of the Ising model, and the constant coupling approximation in the case of the Heisenberg model. The method is generalized to calculate the static spin-correlation function. It is obtained in Ornstein-Zernike type. In the nonequilibrium state, the expectation value of the spin decays by the homogeneous damping as well as the spin-diffusion process. This relaxation equation is different from that by Van Hove because of the existence of the homogeneous damping. The diffusion constant does not vanish even at the Curie point TC, contrary to Van Hove's result, although the damping constant vanishes at TC. The latter fact causes the critical slowing down of the regression of fluctuation. The solution of this relaxation equation gives the time-dependent spin-correlation function. The method can be applied to the antiferromagnetic case, too.
Applying the master equation to an Ising model, equations of motion are obtained for the average of a single spin and for the average of a pair of spins. These equations relax to equilibrium values that predict a self-consistency relationship among the internal field, temperature, and exchange energy. The critical temperature obtained is that of the Bethe approximation, and is extended to include a static external field in the magnetization direction. With this, the relaxation of an Ising model near the Curie temperature has been investigated in a self-consistent manner. The results are similar to those of Suzuki and Kubo, but in the Bethe approximation instead of the Bragg-Williams approximation, and hence predict the relaxation of the correlation as well. In particular we studied a nearly Ising spin system that is a system with relatively small isotropic transverse coupling. An oscillating, linear, and external magnetic field applied transverse to the strongly coupled direction of the spins leads to lowering of the critical temperature.
Based on the phenomenological approach, loci of relaxation time and magnetic dispersion maxima near the critical regime in a spin-1/2 mean-field Ising model were performed. The shift in temperature (T) of relaxation time (τ) maximum was detected and its behavior near the second-order transition points are presented at different magnetic field values (h) and different lattice coordination numbers (q). An expression for the dynamic (or complex) susceptibility (χ = χ1 - iχ2) is also derived. The temperature dependence of the magnetic dispersion (χ1) and magnetic absorption (χ2) factors have been studied near the critical regime. It is found that the maximum of χ1 as a function of frequency (ω) and kinetic coefficient (L) obeying an approximately exponential increases and decreases in T- ω and T-L planes near the critical region.