Determination of spin Hamiltonian in the Ni$_4$ magnetic molecule

Magnetic excitations in a Ni$_4$ magnetic molecule were investigated by inelastic neutron scattering and bulk susceptibility ($\chi_\text{bulk}$) techniques. The magnetic excitation spectrum obtained from the inelastic neutron scattering experiments exhibits three modes at energy transfers of $\hbar\omega=0.5$, 1.35, and 1.6 meV. We show that the energy, momentum, and temperature dependences of the inelastic neutron scattering data and $\chi_\text{bulk}$ can be well reproduced by an effective spin Hamiltonian consisted of intra-molecule exchange interactions, a single-ionic anisotropy, biquadratic interactions, and Zeeman term. Under a hydrostatic pressure, the bulk magnetization decreases with increasing pressure, which along with the biquadratic term indicates spin-lattice coupling present in this system.