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arXiv:0711.3156 · Phase Transitions

Three-dimensional Ising-like system in an external field: Microscopic calculation of the free energy in the higher non-Gaussian approximation

Ihor V. Pylyuk

Published Jan 1, 2007• 2 citations
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Abstract

An analytic method for deriving the free energy of a three-dimensional Ising-like system near the critical point in a homogeneous external field is developed in the the ρ6 model approximation. The mathematical description proposed for temperatures T>Tc (Tc is the phase-transition temperature in the absence of an external field) is valid for fields near , where the scaling variable is of the order of unity and power series in this variable are not effective. At the limiting field , the temperature and field effects on the system in the vicinity of the critical point are equivalent. The total free energy is obtained as a function of temperature, field and microscopic parameters of the system without using series expansions in the scaling variable. In addition to leading terms, the expression for the free energy includes the terms determining the temperature and field confluent corrections.

DOI
10.1080/01411590701315419

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Journal of Physics Condensed Matter•Nov 1, 2002•7 citations
An analytic method for deriving the free energy of a three-dimensional Ising-like system near the critical point in a homogeneous external field is developed in the the ρ6 model approximation. The mathematical description proposed for temperatures T>Tc (Tc is the phase-transition temperature in the absence of an external field) is valid for fields near , where the scaling variable is of the order of unity and power series in this variable are not effective. At the limiting field , the temperature and field effects on the system in the vicinity of the critical point are equivalent. The total free energy is obtained as a function of temperature, field and microscopic parameters of the system without using series expansions in the scaling variable. In addition to leading terms, the expression for the free energy includes the terms determining the temperature and field confluent corrections.
Method of Calculating the Free Energy of Three-Dimensional Ising-Like System in an External Field with the Use of the ρ6ρ^6ρ6 Model
The microscopic approach to calculating the free energy of a three-dimensional Ising-like system in a homogeneous external field is developed in the higher non-Gaussian approximation (the ρ6ρ^6ρ6 model) at temperatures above the critical value of TcT_cTc​ (TcT_cTc​ is the phase-transition temperature in the absence of an external field). The free energy of the system is found by separating the contributions from the short- and long-wave spin-density oscillation modes taking into account both the temperature and field fluctuations of the order parameter. Our analytical calculations do not involve power series in the scaling variable and are valid in the whole field-temperature plane near the critical point including the region in the vicinity of the limiting field h~c\tilde h_ch~c​, which divides external fields into the weak and strong ones (i.e., the crossover region). In this region, the temperature and field effects on the system are equivalent, the scaling variable is of the order of unity, and power series are not efficient. The obtained expression for the free energy contains the leading terms and terms determining the temperature and field confluent corrections.
A three-dimensional Ising-like system in a homogeneous external field is studied on the basis of the higher non-Gaussian measure density (the model). The presented solutions of recurrence relations for the coefficients of the effective measure densities and the generalized point of exit of the system from the critical regime are used for calculating the free energy of the system at temperatures ( is the phase transition temperature in the absence of an external field). A calculation technique is based on the first principles of statistical physics and is naturally realized without any general assumptions and without any adjustable parameters. The obtained expression for the free energy does not involve series expansions in the scaling variable and is valid near the critical point not only in the regions of the so-called weak and strong external fields, but also in the crossover region between these fields, where power series in the scaling variable are not effective.
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 analytical calculation of the free energy of a three‐dimensional Ising‐like system in a homogeneous external field is performed in the higher non‐Gaussian approximation (the ρ6 model) at temperatures above the critical value of Tc (⁠Tc is the phase‐transition temperature in the absence of an external field). The free energy of the system is found by separating the contributions from the short‐ and long‐wave spin‐density oscillation modes taking into account the generalized point of exit of the system from the critical regime as a function of both the temperature and field variables. A calculation technique is based on the first principles of statistical physics and is naturally realized without any general assumptions and without any adjustable parameters. The obtained expression for the free energy does not involve series expansions in the scaling variable and is valid near the critical point not only in the regions of the so‐called weak and strong external fields but also in the crossover region between these fields. In this region, the temperature and field effects on the system are equivalent, the scaling variable is of the order of unity and power series are not efficient. The free energy contains the leading terms and terms determining the temperature and field confluent corrections.
We study the behavior of a three-dimensional Ising-like system in an external field near the critical point by using the non-Gaussian spin-density fluctuations, namely, the quartic measure density with the even and odd powers of the variable (the asymmetric ${\ensuremath{\rho}}^{4}$ model). The basic idea of the analytic method for deriving complete expressions of the thermodynamic characteristics (including the scaling functions) is described. The proposed method allows us to perform the calculations on the microscopic level without any adjustable parameters. Explicit expressions for the total free energy, order parameter, susceptibility, entropy, and specific heat of the system are obtained as functions of the temperature and field. The regions of the so-called weak and strong fields are considered for temperatures above and below Tc{T}_{c}Tc​ (Tc{T}_{c}Tc​ is the phase-transition temperature in the absence of an external field). The average spin moment and susceptibility, depending on the field variation and the proximity to Tc{T}_{c}Tc​, are investigated. It is confirmed that the temperature and field fluctuations for the order parameter play the leading roles in the weak and strong fields, respectively.
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.