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We analyze the interference between tunneling paths that occur for a spin system with special Hamiltonian both for dipole and quadrupole excitations. Using an instanton approach, we find that as the strength of the second-order transverse anisotropy is increased, the tunnel splitting for both excitations is modulated, with zeros occurring periodically and the number of quenching points for quadrupole excitation decreasing. This effect results from the interference of four tunneling paths connecting easy-axis spin orientations and occurs in the absence of any magnetic field.