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Journal of the Physical Society of Japan

Spin-One Ising Model with Bilinear and Biquadratic Interactions

Kazuhiro Takahashi, Motoyuki Tanaka

Published May 1, 1979• 45 citations
Publisher

Abstract

The spin-1 Ising model with bilinear and biquadratic interactions is investigated using a generalized constant-coupling approximation with two parameters. Equations to determine these parameters are derived by minimizing the variational free energy. The averages of the dipolar and quadrupolar moments expressed in terms of these parameters are first ascertained to give exact results for a linear chain, then investigated in detail for the quadrupolar-coupled system in comparison with the molecular-field results. The pair-correlation and the heat capacity of this system are also investigated in connection with the phase transition. For the system with the bilinear and biquadratic interactions the effect of the anisotropy constant to the phase transition is discussed.

DOI
10.1143/jpsj.46.1428

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The spin-1 Ising model with bilinear and biquadratic interactions is investigated using a generalized constant-coupling approximation with two parameters. Equations to determine these parameters are derived by minimizing the variational free energy. The averages of the dipolar and quadrupolar moments expressed in terms of these parameters are first ascertained to give exact results for a linear chain, then investigated in detail for the quadrupolar-coupled system in comparison with the molecular-field results. The pair-correlation and the heat capacity of this system are also investigated in connection with the phase transition. For the system with the bilinear and biquadratic interactions the effect of the anisotropy constant to the phase transition is discussed.
The equilibrium properties of the spin-1 Ising system Hamiltonian with arbitrary bilinear (J), biquadratic (K) and odd (L), which is also called dipolar-quadrupolar, interactions is studied for zero magnetic field in the lowest approximation of the cluster variation method. The odd interaction is combined with the bilinear (dipolar) and biquadratic (quadrupolar) exchange interactions by the geometric mean. In this system, phase transitions depend on the ratio of the coupling parameters, alpha = J/K; therefore, the dependence of the nature of the phase transition on alpha is investigated extensively and it is found that for alpha less than or equal to 1 and alpha greater than or equal to 2000 a second-order phase transition occurs, and for 1 < alpha < 2000 a first-order phase transition occurs. The critical temperatures in the case of a second-order phase transition and the upper and lower limits of stability temperature in the case of a first-order phase transition are obtained for different values of alpha calculated using the Hessian determinant. The first-order phase transition temperatures are found by using the free energy values while increasing and decreasing the temperature. Besides the stable branches of the order parameters, we establish also the metastable and unstable parts of these curves and the thermal variations of these solutions as a function of the reduced temperature are investigated. The unstable solutions for the first-order phase transitions are obtained by displaying the free energy surfaces in the form of a contour map. Results are compared with the spin-1 Ising system Hamiltonian with the bilinear and biquadratic interactions and it is found that the odd interaction greatly influences the phase transitions.
A statistical-mechanical study of a solvable spin-one system with bilinear and biquadratic interactions is presented for various values of α, the ratio of the bilinear to biquadratic interaction. First, equations to determine the order-parameters and the free energy in the equilibrium are derived from the partition function to investigate the phase transitions occurring the model. The magnetic susceptibility is then calculated, and an argument about the critical exponents at the second-order phase transition is given for the case of the critical ratio α=2/3, below which the first order phase transition takes place.
Analytical expressions of the order parameters near the transition temperatures in the spin-3/2 Ising system with bilinear (J) and biquadratic (K) interactions are presented for various values of J/K. First, we obtain the free energy expression and the equations to determine order parameters by using the mean-field approximation. Then, the order parameters are expressed in the vicinity of the transition temperatures in which these expressions are very important to study the dynamics of the system by means of Onsager's theory of irreversible thermodynamics. Hence, we investigate the phase transitions occurring in the system and also obtain two tricritical points analytically. Finally, the specific heat and magnetic susceptibility are calculated and an argument about the critical exponents at the second-order phase transitions and tricritical points is given.
The critical properties of a three-dimensional spin-1 system characterized by quadrupolar interactions are analyzed under the assumption of weakly coupled chains. The problem is approached introducing first a trial one-dimensional Hamiltonian whose free energy can be evaluated exactly. Then using a variational procedure the free energy of the model is determined. The phase transitions are examined and the phase diagram derived. In addition some characteristic thermodynamic quantities such as average quadrupolar momentum, quadrupolar correlation function and specific heat are evaluated.
The spin-1 Ising model with bilinear and biquadratic exchange interactions and single-ion crystal field is solved on the Bethe lattice using exact recursion equations. The general procedure of critical properties investigation is discussed and full set of phase diagrams are constructed for both positive and negative biquadratic couplings. In latter case we observe all remarkable features of the model, uncluding doubly-reentrant behavior and ferrimagnetic phase. A comparison with the results of other approximation schemes is done.