Magnetic Properties and Inelastic Neutron Scattering for a Spin Hexamer: Application to the V$_{6}$ Molecular Magnets

We present the study of the magnetic susceptibility and inelastic neutron scattering energies and intensities for a spin hexamer formed by two interacting spin S and S' trimers. Using an isotropic Heisenberg Hamiltonian, we conclude that, regardless of spin, the structure factors for the magnetic excitations will have a specific function form which is dependent on the symmetric parts of the hexamer being excited. This work is then compared to previous work performed on two vanadium compounds (CN$_3$H$_6$)$_4$Na$_2$[H$_4$V$_6$O$_8$(PO$_4$)$_4$((OCH$_2$)$_3$CCH$_2 $OH)$_2$]$\cdot 14$H$_2$O and Na$_6$[H$_4$V$_6$O$_8$(PO$_4$)$_4$((OCH$_2$)$_3$CCH$_2$OH)$_2$]$\cdot 18$H$_2$O, which are thought to be magnetically well described as trimers of S=1/2 V$^{4+}$ ions. We show that it is possible to that a very weak inter-trimer interaction exists and we predict the energy and momentum transfer dependence for these compounds that may be observable with inelastic neutron scattering.