Thorium-229 solid-state nuclear clock prospects in MgF$_2$ and LiSAF

The ${}^{229}$Th isomer is thought to be a good candidate for a nuclear clock based on its relatively low-energy isomer excitation of $\approx\,7.8$\,eV. We report on the study of Th atoms embedded in two crystals, MgF$_2$ and LiSAF (LiSrAlF$_6$). For MgF$_2$ we perform an oxidation study to find the preferred ionization state of the Th atom in the crystal; Th$^{n+}$, where $n=2-4$. We find that the preferred state is $n=4$ which requires two interstitial Fluorine atoms to charge compensate. Using the results of MgF$_2$ we then search within LiSAF for suitable dopant sites (the Sr, Al, or Li can all serve). Employing a standard density functional package using a plane-wave basis and psuedopotentials, we optimize a doped cell of increasing particle number sizes and use this to estimate the dilute doped-limit band-gap of LiSAF. Placement of the dopant on the Sr and Al sites with accompanying double and single F interstitial atom placements is also studied to determine the ground state, and comparisons are made with previous calculations\,[1]. In both crystal ground states, we find that the band gap is large enough for the observation of the ${}^{229}$Th nuclear isomer transition; $> 9$\,eV. \\ \, [1] R. A. Jackson et al., J. Phys.: Condens. Matter 21, 325401 (2009).