Study of the atom-phonon coupling model for (SC) partition function: first order phase transition for an infinite linear chain

In spin-conversion (SC) compounds containing molecules organized around an iron (II) ion the fundamental level of the ion is low spin (LS), S = 0, and its first excited one is high spin (HS), S = 2. This energy diagram is due to the ligands field interaction on 3 d electrons and to the spin pairing energy. Heating the compound increases the magnetic susceptibility which corresponds to a change of populations of both levels and consequently a change of spin value of the molecules. This mechanism, called spin conversion (SC), can be accompagnied by thermal hysteresis observed by studying magnetic susceptibility or high spin fraction. In that case one considers that the (SC) takes place through a first-order phase transition due to intermolecular interactions. In the atom-phonon coupling model the molecules are considered as two-level systems, or two-level atoms, and it is assumed that the elastic force constant value of the spring which links two atoms first neighbours is depending on the electronic states of both atoms. In this study we calculate the partition function of a linear chain of N atoms ( N ≤ 16) and we describe the role of phonons and that of the parameter Δ which corresponds to the distance in energy between both levels. The chain free-energy function is F atph . We introduce for the chain a free-energy function defined by the set ( F HS , F LS , F barr ) and we show that F atph tends towards the previous set when N → ∞. The previous set allows to describe a first order phase transition between a (LS) phase and a (HS) one. At the crossing point between the function F LS and F HS , and around this point, there is an intermediate free-energy barrier which prevents the chain to change phase which can lead to thermal hysteresis. The energy gap between the free-energy function F atph and that defined by the set ( F HS , F LS , F barr ) is small. So we can expect that a nanoparticule takes for free-energy function that defined by the set and then displays a thermal hysteresis.