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arXiv:cond-mat/9812371 · Physical review. B, Condensed matter

Thermodynamic characteristics of the classicaln-vector magnetic model in three dimensions

Z. Usatenko, Mykhailo P. Kozlovskii

Published Oct 1, 2000• 9 citations
arXivHTMLPDF

Abstract

A method of calculating the free energy and thermodynamic characteristics of the classical n-vector three-dimensional magnetic model at the microscopic level without any adjustable parameters is proposed. Mathematical description is performed with the collective variables method used within the framework of the ${\ensuremath{\rho}}^{4}$ model approximation. The exponentially decreasing function of the distance between the particles situated at the N sites of a simple cubic lattice is used as the interaction potential. Explicit analytical expressions for entropy, internal energy and specific heat near the phase-transition point as functions of the temperature are obtained. The dependence of the amplitudes of the thermodynamic characteristics of the system for $T>{T}_{c}$ and $T<{T}_{c}$ on the microscopic parameters of the system are studied for the cases $n=1,2,3.$ The obtained results provide the basis for accurate analysis of the critical behavior in three dimensions including the nonuniversal characteristics of the system.

DOI
10.1103/physrevb.62.9599

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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.
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.
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 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 critical behavior of the three-dimensional n -vector model in the presence of an external field is investigated. The mathematical description is performed with the collective variables method in the framework of the ρ 4 model approximation at the microscopic level without any adjustable parameters. The recurrence relations of the renormalization group as functions of the external field and temperature were found. The analytical expression for the free energy of the system for the temperatures T > T c and different n was obtained. The equation of state of the n -vector model for the general case of small and large external fields was written. The explicit form of the correspondent scaling function for different values of the order parameter was derived. The obtained results are in qualitative agreement with the data of Monte Carlo simulations.
Abstract A method for the calculation of the thermodynamic functions in the three-dimensional using model near the phase transition is proposed. In the frame of a uniform approach, the critical exponents and the critical amplitudes of the main characteristics of the model (i.e. free energy, enthropy, specific heat, etc.) are obtained as functions of microscopic Hamiltonian parameters. It is shown that the correct sign and temperature behaviour of the entropy are provided due to the contribution of the long-wave spin moment modes. The short-wave ones are responsible for the values of the critical exponents.