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AIP conference proceedings

Calculation of free energy of three-dimensional uniaxial magnet in external field based on the higher non-Gaussian distribution

Ihor V. Pylyuk, Mykhailo P. Kozlovskii, Yurij V. Holovatch, Bertrand Berche, N. N. Bogolyubov, R. Folk

Published Jan 1, 2009• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.3284408

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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.
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.
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.
The influence of a homogeneous external field on a three-dimensional uniaxial magnet behaviour near the critical point is investigated on the microscopic level by the nonperturbative collective variable (CV) method using the ρ4 model. The research is carried out in the high-temperature region. The critical exponents related with external field are calculated. The analytic explicit expressions for the free energy, average spin moment and susceptibility are obtained without using any phenomenological assumptions and adjusting parameters. The field dependences of the order parameter are plotted for the field region including the neighbourhood of the pseudocritical line. It is established that long-wave fluctuations of the order parameter are responsible for main features of the crossover between the temperature-dependence and field-dependence critical behaviour of the system. The maximum locations on the temperature scale for the long-wave part of the susceptibility are determined for different values of the field.
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.
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.