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Low Temperature Physics

Description of critical behavior of Ising ferromagnet in the ρ6 model approximation taking into account the confluent correction. II. Region below the phase transition point

Ihor V. Pylyuk

Published Dec 1, 1999• 3 citations
Publisher

Abstract

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.

DOI
10.1063/1.593847

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A microscopic description of the critical behavior of systems belonging to the universality class of the three-dimensional (3D) Ising model is developed within the collective variables (CV) approach. The higher non-Gaussian approximation (the sextic distribution for the modes of spin-moment density oscillations or the ρ6model) is used. A specific feature of the partition function calculation for an Ising-like system is the inclusion of the correction for the potential averaging. This correction leads to the modified recurrence relations (RR) for the ρ6model and a nonzero critical exponent of the correlation function η. The RR between the coefficients of the effective sextic distributions are written and analyzed. A technique for determining the small critical exponent η is elaborated in the higher non-Gaussian approximation. It is shown that the renormalized critical exponent of the correlation length has a tendency to a reduction in the case when the exponent η is taken into account.

International Journal of Modern Physics B•Jun 30, 2014•1 citation
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 behavior of a 3D Ising system at temperatures above Tc is studied in the approximation of sextic distribution for modes of spin moment density oscillations (ρ6 model). An original method is developed in this higher non-Gaussian approximation for calculating the thermodynamic characteristics of the 3D Ising model near Tc taking into account the first confluent correction. The contributions to the thermodynamic functions of an Ising ferromagnet (free energy, entropy, internal energy, and specific heat) from the short-wave and long-wave modes of spin density oscillations are considered separately. A nonuniversal factor determined by microscopic characteristics of the system is singled out in the expressions for leading critical amplitudes and the amplitudes of the confluent correction. Numerical estimates of the critical region size, phase transition temperature, leading critical amplitudes, and the amplitudes of the correction to scaling of specific heat and susceptibility of the system are given for different values of effective radius of the exponentially decreasing interaction potential.
A microscopic approach to the investigation of the three-dimensional Ising-like system thermodynamics below the phase transition temperature T c is schematically presented within the framework of the higher non-Gaussian approximation (ρ 6 model) taking into account the corrections to scaling. A microscopic analogue of the Landau free energy is calculated. Explicit expressions for the thermodynamic characteristics are obtained as functions of the temperature and microscopic parameters of the system. The thermodynamic characteristics near T c and their amplitudes are given for various values of the effective radius of the exponentially decreasing interaction potential (including the values corresponding to the nearest-neighbour interaction, the interaction between the nearest and next-nearest neighbours, and that between the nearest, next-nearest, and third neighbours). The evolutions 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 simple cubic lattice constant are plotted. The results of calculations and their comparison with other authors' data show that the ρ 6 model provides a better quantitative description of the critical behaviour of a three-dimensional Ising magnetic than the ρ 4 model.
The microscopic approach to calculating and investigating the three-dimensional (3D) Ising-like system thermodynamics above and below the phase transition temperature Tc is developed using the collective variables method.Mathematical description is performed within the framework of the higher non-Gaussian approximation ( 6 model) taking into account the corrections to scaling.A nonuniversal factor depending on microscopic parameters of the system is singled out in the expressions for leading critical amplitudes and correction-to-scaling amplitudes.Numerical estimates of the critical region size, phase transition temperature as well as the graphs of temperature dependences of entropy and other thermodynamic characteristics near Tc are given for various values of eective radius of the exponentially decreasing interaction potential.The variation of the free energy of the system at the phase transition point, average spin moment, and speci c heat with increasing ratio of the potential eective radius to the simple cubic lattice constant is traced.The results of calculations and their comparison with the other authors' data show that the 6 model provides a more adequate quantitative description of the critical behaviour of a 3D Ising-like system than the 4 model.
The behaviour of a three-dimensional Ising-like system at temperatures above the critical value of T c is studied in the approximation of the sextic distribution for modes of spin density oscillations (ρ 6 model). The original collective variables method is developed in this higher non-Gaussian approximation for calculating the thermodynamic characteristics of the system near T c taking into account the first confluent correction. The calculations are illustrated by an example of a simple cubic lattice and an exponentially decreasing interaction potential. The main distinctive feature of the method is the separate inclusion of the contributions to the thermodynamic functions from the short-wave and long-wave spin density oscillation modes. The dependences of the phase transition temperature, leading critical amplitudes, and correction-to-scaling amplitudes for the specific heat and susceptibility on the microscopic parameters of the system are investigated.
The original collective variables method for calculating the thermodynamic characteristics of a three-dimensional Ising-like system is developed in the approximation of the sextic distribution for modes of spin-density oscillations $({\ensuremath{\rho}}^{6}$ model), taking into account the first confluent correction. The calculations within the framework of this higher non-Gaussian approximation are performed in the temperature regions above and below the critical value of Tc{T}_{c}Tc​ and are illustrated by an example of a simple cubic lattice and an exponentially decreasing interaction potential. The collective variables method makes it possible to obtain universal and nonuniversal quantities by using a unified approach. The dependence of the critical region size and phase-transition temperature on the microscopic parameters is investigated. A microscopic analog of the Landau free energy is calculated. Explicit expressions for the thermodynamic characteristics are obtained as functions of temperature and microscopic parameters of the system. It is shown that each of the leading critical amplitudes and correction-to-scaling amplitudes can be represented as a product of a universal factor not depending on the microscopic parameters of the system and a nonuniversal factor depending on these parameters. The thermodynamic characteristics near Tc{T}_{c}Tc​ and their amplitudes are given for various values of the effective radius of the exponentially decreasing interaction potential (including the values corresponding to the nearest-neighbor interaction and the interactions between the nearest and next-nearest neighbors and between the nearest, next-nearest, and third neighbors). The results of calculations and their comparison with other authors' data show that the ${\ensuremath{\rho}}^{6}$ model provides a better quantitative description of the critical behavior of a three-dimensional Ising ferromagnet than the ${\ensuremath{\rho}}^{4}$ model.
A microscopic description of the critical behavior of systems belonging to the universality class of the three-dimensional (3D) Ising model is developed within the collective variables (CV) approach. The higher non-Gaussian approximation (the sextic distribution for the modes of spin-moment density oscillations or the ρ6model) is used. A specific feature of the partition function calculation for an Ising-like system is the inclusion of the correction for the potential averaging. This correction leads to the modified recurrence relations (RR) for the ρ6model and a nonzero critical exponent of the correlation function η. The RR between the coefficients of the effective sextic distributions are written and analyzed. A technique for determining the small critical exponent η is elaborated in the higher non-Gaussian approximation. It is shown that the renormalized critical exponent of the correlation length has a tendency to a reduction in the case when the exponent η is taken into account.