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Physical Review Letters

Phase Transition in a Compressible Ising Ferromagnet

Herbert Wagner

Published Jul 6, 1970• 55 citations
Publisher

Abstract

Within the droplet model for the Ising spin system it is shown that the magnetic specific heat ${C}_{p}$ and the susceptibility ${\ensuremath{\chi}}_{p}$ for constant vanishing pressure and magnetic field remain finite in a compressible lattice as the critical point ${T}_{c}$ is approached from below. This regularization of ${C}_{p}$ and ${\ensuremath{\chi}}_{p}$ is caused by the long-range indirect spin-spin interaction, which is a consequence of the dependence of the exchange integral on the lattice spacing.

DOI
10.1103/physrevlett.25.31

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Within the droplet model for the Ising spin system it is shown that the magnetic specific heat Cp{C}_{p}Cp​ and the susceptibility \ensuremathχp{\ensuremath{\chi}}_{p}\ensuremathχp​ for constant vanishing pressure and magnetic field remain finite in a compressible lattice as the critical point Tc{T}_{c}Tc​ is approached from below. This regularization of Cp{C}_{p}Cp​ and \ensuremathχp{\ensuremath{\chi}}_{p}\ensuremathχp​ is caused by the long-range indirect spin-spin interaction, which is a consequence of the dependence of the exchange integral on the lattice spacing.
Using a Ginzburg-Landau model, we study the phase transition behavior of compressible Ising systems at constant volume by varying the temperature TTT and the applied magnetic field hhh. We show that two phases can coexist macroscopically in equilibrium within a closed region in the $T\text{\ensuremath{-}}h$ plane. Its occurrence is favored near tricriticality. We find a field-induced critical point, where the correlation length diverges, the difference of the coexisting two phases and the surface tension vanish, but the isothermal magnetic susceptibility does not diverge in the mean field theory. We also investigate phase ordering numerically.
Using the Ising Model, the phase transition between ferromagnetic and paramagnetic phases of a 2-D square spin lattice are explored. A phase transition into the ferromagnetic phase occurs when the temperature of the system falls below some critical value. The ferromagnetic phase is characterized by a non-zero magnetization on the macroscopic level. Using the Mote Carlo method to obtain system's configurations (i.e., sets of microstates) in thermal equilibrium, we will explore system's macroscopic properties such as magnetization, magnetic susceptibility, and specific heat capacity as functions of temperature. We will evaluate the transition temperature, and discuss how the lattice size affects the results.
Critical exponents for a one-dimensional general continuous Ising model with long-range ferromagnetic interactions decaying as 1/r1+σ are calculated using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between −1 and 1. The critical point behavior is investigated. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents; the singularities in the specific heat and susceptibility depend on σ. For σ=1, there are extremely weak finite size effects: the first-order and the second-order cumulants of the order parameter yield ν=2.42(1). The susceptibility exponent γ=2.259(9). Results for σ=0.7 will be shown and discussed.
A scheme for calculating the thermodynamic characteristics of a 3D one-component spin system in the temperature range below the critical temperature is described on the basis of the sextic measure density (ρ6 model) taking into account the first confluent correction. A microscopic analog of the Landau free energy is calculated. The coefficients of average spin moment, leading critical amplitudes, and the amplitudes of the confluent correction to specific heat and susceptibility of the system are calculated for different values of effective radius of the exponentially decreasing interaction potential. Plots of temperature dependence of entropy and other thermodynamic characteristics in the vicinity of Tc are given for various values of effective radius. The variation of the free energy of the system at the phase transition point, average spin moment, and specific heat with increasing ratio of the potential effective radius to the constant of a simple cubic lattice is traced. The results of calculations and their comparison with the data obtained by other authors show that the ρ6 model provides a better quantitative description of the critical behavior of a 3D Ising ferromagnet than the ρ4 model.
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.