Directionally Independent Energy Gap Formation Due to the Hyperfine Interaction

We study energy gap formation at the level-crossing point due to the hyperfine interac-tion. In contrast to the energy gap induced by the Dzyaloshinskii-Moriya interaction, the gap induced by the hyperfine interaction is independent of the direction of the magnetic field. We also study the dynamics of the magnetization under a time dependent field that changes at a constant rate. Recently, adiabatic magnetic transitions in nanoscale magnets have attracted a great deal of interest. Among many interesting problems in this context is the search for the origin of the transition or, in other words, to determine how an energy gap is formed at the level-crossing point. 1) The Dzyaloshinskii-Moriya interactio