Spin conversion, (SC), compounds are composed of molecules organized around a transition metal ion. The ion spin value is smaller for the ion fundamental level than for its first excited one. So, increasing the temperature changes the spin mean value. This spin conversion can be continuous or can display a first order phase transition called spin transition. The atom phonon coupling model, introduced recently, allows to describe at least qualitatively different experimental results. Up to now, this model has been applied on a linear chain of atoms. In this paper we apply it on a square lattice. We study the thermal variations of different thermodynamic parameters and the metastable states which are present around the transition. In this study, it is expected that the critical point of some (SC) compounds can be reached by applying on them a small hydrostatic pressure; it is also expected that ultrasound pulses can induce, at a very low temperature, a conversion between the stable low spin state and the metastable high spin state and it is also predicted that the crystal sound velocity can display a discontinuity at the first order phase transition. © 2011 EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg.