Quantum Tunneling of Spin Particles in Periodic Potentials with   Asymmetric Twin Barriers
  The tunneling effect of a periodic potential with an asymmetric twin barrier per period is calculated using the instanton method. The model is derived from the Hamiltonian of a small ferromagnetic particle in an external magnetic field using the spin-coherent-state path integral. The instantons in two neighbouring barriers differ and lead to different level shifts $\triangle\epsilon_1, \triangle\epsilon_2$. We derive with Bloch theory the energy spectrum which has formally the structure of an energy band. The spectrum depends on both level shifts. The removal of Kramer's degeneracy by an external magnetic field is discussed. In addition we find a new kind of quenching of macroscopic quantum coherence which is irrelevant to Kramer's degeneracy. 
J.-Q. Liang, H.J.W. Mueller-Kirsten, Jian-Ge Zhou, F. Zimmerschied and   F.-C. Pu