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

We derive the spin-atomic vibration interaction $V_{\rm SA}$ and the spin-flip Hamiltonian $V_{\rm SF}$ of a single atomic spin in a crystal field. 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 {{\boldmath r}}. Examining the coefficients of $V_{\rm SA}$ and $V_{\rm SF}$, we first show that $V_{\rm SA}$ appears for \Delta {{\boldmath r}}$\ne$0, while $V_{\rm SF}$ is present independently of \Delta {{\boldmath r}}. As an application, we next obtain $V_{\rm SA}$ and $V_{\rm SF}$ of an Fe ion in a crystal field of tetragonal symmetry. It is found that the magnitudes of the coefficients of $V_{\rm SA}$ can be larger than those of the conventional spin-phonon interaction depending on vibration frequency. In addition, transition probabilities per unit time due to $V_{\rm SA}$ and $V_{\rm SF}$ are investigated for the Fe ion with an anisotropy energy of $-|D|S_Z^2$, where $D$ is an anisotropy constant and $S_Z$ is the $Z$ component of a spin operator.