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

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We derive the spin–atomic vibration interaction $V_{\text{SA}}$ and the spin-flip Hamiltonian $V_{\text{SF}}$ 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, $\Delta\mathbf{r}$. Examining the coefficients of $V_{\text{SA}}$ and $V_{\text{SF}}$, we first show that $V_{\text{SA}}$ appears for $\Delta\mathbf{r}\neq 0$, while $V_{\text{SF}}$ is present independently of $\Delta\mathbf{r}$. As an application, we next obtain $V_{\text{SA}}$ and $V_{\text{SF}}$ of an Fe ion in a crystal field of tetragonal symmetry. It is found that the magnitudes of the coefficients of $V_{\text{SA}}$ can be larger than those of the conventional spin–phonon interaction depending on vibration frequency. In addition, transition probabilities per unit time due to $V_{\text{SA}}$ and $V_{\text{SF}}$ 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.
2010-11-15