Quadrupole Excitation in Tunnel Splitting Oscillation in Nano-Particle   $Mn_{12}$
  We analyze the interference between tunneling paths that occurs for a spin system with special Hamiltonian both for dipole and quadrupole excitation. Using an instanton approach, we find that as the strength of the second-order transverse anisotropy is increased, the tunnel splitting for both excitations are modulated, with zeros occurring periodically and the number of quenching points for quadrupole excitation decreases. This effect results from the interference of four tunneling paths connecting easy-axis spin orientations and occurs in the absence of any magnetic field. 
Yousef Yousefi and Khikmat Kh. Muminov
cond-mat.other