Anisotropy Energy, Spin-Atomic Vibration Interaction, and Spin-Flip Hamiltonian of a Single Atomic Spin in a Crystal Field

We derive the anisotropy energy $V_{\rm A}$, the spin-atomic vibration interaction $V_{\rm SA}$, and the spin-flip Hamiltonian $V_{\rm SF}$ of a single atomic spin system, ``Fe$^ {2+}$ (3d$^6$) in a crystal field of tetragonal symmetry'' [1,2]. We here apply the perturbation theory to a model with the spin- orbit interaction and the kinetic and potential energies of electrons. The model also takes into account the difference in vibration displacement between an effective nucleus and electrons, $\Delta r$. We first find conditions to enhance a coefficient $|D|$ of $V_{\rm A}$=$-|D|S_Z^2$, where $D$ is an anisotropy constant and $S_Z$ is the $Z$ component of a spin operator. Second, we show that $V_{\rm SA}$ appears for $\Delta r\ne 0$, while $V_{\rm SF}$ is present independently of $\Delta r$. Also, the magnitudes of the coefficients of $V_{\rm SA}$ can be larger than those of the conventional spin-phonon interaction depending on vibration frequency. \\[4pt] [1] S. Kokado {\it et al}., J. Phys. Soc. Jpn. {\bf 79}, 114721 (2010).\\[0pt] [2] S. Kokado {\it et al}., phys. stat. solidi (c) {\bf 7}, 2612 (2010).