Estimating Isotope Shifts for $^{227}$Th and $^{229}$Th in Th$^{3+}$: 5F$_{5/2} \rightarrow$ 6D$_{5/2}$ Transition

Abstract

\textbf{Motivation:} Accurate spectroscopic information for isotopes of $^{224-232}$Th are critical for experimental programs investigating such thorium isotopes as candidates for next-generation nuclear optical clocks and as platforms for searches of symmetry-violating effects. Direct experimental data on isotope shifts in the $\text{Th}^{3+}:5F_{5/2} \to 6D_{5/2}$ line at $690$ nm are sparse, with measurements available only for $^{229}\text{Th}$ and $^{230}\text{Th}$ relative to the reference isotope $^{232}\text{Th}$.\\

\textbf{Method:} To address this gap, we employed a King-plot analysis comparing the well-characterized isotope shifts of the $\text{Th}^{+}$ transition at $583.9$ nm to the limited data available for the $690$ nm transition of $\text{Th}^{3+}$. Using nuclear structure information on mean-square charge radii and nuclear quadrupole deformations, we extracted the field-shift constant $F_{690}$ and mass-shift constant $M_{690}$ for the $690$ nm transition. We subsequently calculated the missing isotope shifts by incorporating published values of $\delta\langle r^2\rangle$ where available and estimating $\delta\langle r^2\rangle$ for unmeasured isotopes using nuclear quadrupole deformation coefficients $\beta_2$ from the FRDM model.\\

\textbf{Key Results:} The calculated isotope shifts for the $\text{Th}^{3+}:5F_{5/2} \to 6D_{5/2}$ transition relative to $690~$nm transition of $^{232}\text{Th}$ are:

\begin{align}

\delta\nu^{224,232}_{690} &= -29296(5585) \text{MHz} \nonumber\\

\delta\nu^{225,232}_{690} &= -25840(4930) \text{MHz} \nonumber\\

\delta\nu^{226,232}_{690} &= -22113(4219) \text{MHz} \nonumber\\

\delta\nu^{227,232}_{690} &= -18631(6238) \text{MHz} \nonumber\\

\delta\nu^{228,232}_{690} &= -14970(6181) \text{MHz} \nonumber\\

\delta\nu^{231,232}_{690} &= -3742(715) \nonumber\text{MHz}

\end{align}

Our results for the isotopes of $^{227,228}$Th use experimental $\delta\langle r^2\rangle$ values, whereas the remaining ones use theoretical values of $\delta\langle r^2\rangle$ calculated from quadrupole deformation coefficients. These results provide essential spectroscopic data for future precision measurements with thorium isotopes in various ionized states.