Zero-Point Phonon Contribution to the Hyperfine Coupling ofS-State Ions

Abstract

The temperature-independent phonon zero-point motion is shown to decrease the hyperfine coupling constant $A$ for ${\mathrm{Mn}}^{2+}$ ions in a cubic environment. A spherical approximation is made for the phonon vibrational modes appropriate to a rocksalt structure, enabling the inclusion of contributions from both the acoustic and optical branches. Numerical calculations are performed for vibrational parameters appropriate to both MgO and SrO. The temperature-dependent part of the decrease in $A$ due to thermal phonon excitation is also evaluated for both hosts. Comparing the two calculations, the temperature can be determined at which the decrease in $A$ due to thermal excitations equals the decrease due to zero-point excitations. These temperatures are 383°K for MgO and 133°K for SrO. Use is then made of the observed decrease in $A$ with temperature to find the magnitude of the zero-point reduction in both hosts. We find $\Delta A_{\mathrm{Z}.\mathrm{P}.\mathrm{}}$ (MgO: ${\mathrm{Mn}}^{2+}$) ≅-0.8×${10}^{-4\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$ and $\Delta A_{\mathrm{Z}.\mathrm{P}.\mathrm{}}$ $(\mathrm{SrO}:{\mathrm{Mn}}^{2+})\cong -1.4\mathrm{}\times {10}^{-4\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$. The change in $A$, $\frac{\Delta A_{\mathrm{Z}.\mathrm{P}.\mathrm{}}}{\left[\Delta A\right(T=0\mathrm{})+\Delta A_{\mathrm{Z}.\mathrm{P}.\mathrm{}}]}$, due to phonon zero-point motion is therefore 0.97% for MgO:${\mathrm{Mn}}^{2+}$ and 1.70% for SrO: ${\mathrm{Mn}}^{2+}$, and the $T=0\mathrm{}$ "rigid-lattice" values are estimated to be -82.3×${10}^{-4\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$ for both hosts.