Stable, metastable and unstable solutions of the Blume-Emery-Griffiths model

The temperature dependence of the magnetization and quadrupole order parame ters of the Blume-Emery-Griffiths (BEG) model Hamiltonian with the nearest- neighbor ferromagnetic exchange interactions [both bilinear (J) and biquadr atic (K)] and crystal field interaction (D) is studied using the lowest app roximation of the cluster variation method. Besides the stable solutions, m etastable and unstable solutions of the order parameters are found for vari ous values of the two different coupling parameters, alpha = J/K and gamma = D/K. These solutions are classified using the free energy surfaces in the form of a contour map. The phase transitions of the stable, metastable and unstable branches of the order parameters are investigated extensively. Th e critical temperatures in the case of a second-order phase transition are obtained for different values of alpha and gamma calculated by the Hessian determinant. The first-order phase transition temperatures are found using the free energy values while increasing and decreasing the temperature. The temperature when both the free energies equal each other is the first-orde r phase transition temperature. Finally, the results are also discussed for the Blume-Capel model which is the special case of the BEG model. (C) 1999 Elsevier Science B.V. All rights reserved.