The magnetization plateaus of the ferro and anti-ferro spin-1 classical models with $S_z^2$ term

We study in detail the exact thermodynamics of the one-dimensional standard and staggered spin-1 Ising models with a single-ion anisotropy term in the presence of a longitudinal magnetic field at low temperatures. The results are valid for the ferromagnetic and anti-ferromagnetic (AF) models and for positive and negative values of the crystal field for $T>0$. Although the excited states of the ferro and anti-ferro models are highly degenerate, we show that the temperature required for reaching the first excited state in the classical spin-1 ferro model gives a scale of temperature that permits fitting the z-component of the magnetization only by the contribution of two ground states of the model. This approximation is not true for the equivalent AF function due to the fact that the AF model is gapless along the lines separating the phases in its phase diagram at T=0. We relate the number of plateaus in the magnetization of each model to their respective phase diagrams at T=0. The specific heat per site of the AF model distinguishes, at low temperature, the transitions $\mbox{A} \rightleftharpoons \mbox{E}$ and $\mbox{G} \rightleftharpoons \mbox{E}$ as the external magnetic field is varied. The exact Helmholtz free energy of the classical spin-1 model is mapped onto the equivalent function of the ionic limit of the 1D extended Hubbard model by proper transformations.