Thermodynamic states of the mixed spin 1/2 and spin 1 hexagonal nanotube system obtained from a eighteen-site cluster within an improved mean field approximation

The mean field approximation results in the mixed spin 1/2 Ising model and spin 1 Blume-Capel model, in the hexagonal nanotube two layers system, are obtained from the Bogoliubov inequality. The Gibbs free energy, magnetization and critical frontiers are obtained. Besides the stable branches of the order parameters, we obtain the metastable and unstable parts of these curves and also find phase transitions of the metastable and unstable branches of the order parameters. The classification of the stable, metastable and unstable states is made by comparing the free energy values of these states. An uncommon increase in the magnetization, already observed by Canko et al., is described by the presence of an unstable state with order parameter nonzero.