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arXiv:1905.07513 · The European Physical Journal B

Phase diagrams of spin-1 Ising model with bilinear and quadrupolar interactions under magnetic field. Two-particle cluster approximation

R. R. Levitskii, O. R. Baran, Bohdan Lisnyi

Published Apr 1, 2006• 9 citations
arXivHTMLPDF
DOI
10.1140/epjb/e2006-00162-5

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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.

•Oct 1, 1996•23 citations
We study the phase diagram and magnetic properties of a transverse spin-1 Ising model with random longitudinal crystal-field interactions using an expansion technique for cluster identities of spin-1 localized spin systems. Partially ordered phases appear for a particular two valued distribution P(D i )=1/2( delta (D i -(1+ alpha )D)+ delta (D i -(1- alpha )D)), and sufficiently small transverse field Omega . The longitudinal and transverse magnetizations and the quadrupolar moments are calculated. General formulae, applicable to structures with arbitrary coordination number N, are given.
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.