In this paper we revisit the onset of the instability of the solid state in classical systems within self-consistent phonon theory (SCPT). Spanning the whole phase diagram versus volume and versus pressure, we identify two different kinds of mechanism: one mainly relevant at constant volume, associated with the vanishing of the SCPT solution; and one related to the disappearing at a spinodal temperature of the solid phase as a metastable energy minimum. We show how the first mechanism occurs at extremely high temperatures and it is not reflected in any singular behavior of the thermodynamical properties. In contrast, the second one appears at physical temperatures which correlate well with the melting temperature and it is signalized by the divergence of the thermal compressibility as well as of the the lattice expansion coefficient.

(submitted to PRB)