Phase transition between quantum and classical regimes for the escape rate of dimeric molecular nanomagnets in a staggered magnetic field

We study the phase transition of the escape rate of exchange-coupled dimer of single-molecule magnets which are coupled either ferromagnetic ally or antiferromagnetically in a staggered magnetic field and an easy $z$-axis anisotropy. The Hamiltonian for this system has been used to study molecular dimer nanomagnets [Mn${}_4$]${}_2$. We generalize the method of mapping a single-molecule magnetic spin problem onto a quantum-mechanical particle to dimeric molecular nanomagnets. The problem is mapped to a single particle quantum-mechanical Hamiltonian in terms of the relative coordinate and a coordinate dependent reduced mass. It is shown that the presence of the external staggered magnetic field creates a phase boundary separating the first- from the second-order transition. With the set of parameters used by R. Tiron, $\mathit{\text{et}}\mathit{\text{al}}$, \prl {\bf 91}, 227203 (2003), and S. Hill, $\textit{et al}$science\mathbf{302},1015(2003)tofitexperimentaldatafor[Mn$_{4}$]$_2$dimerwefindthatthecriticaltemperatureatthephaseboundaryis$T^{©}_0 =0.29K$. Therefore, thermally activated transitions should occur for temperatures greater than $T_0^{(c)}$.