Nuclear Charge Radii of 229Th from Isotope and Isomer Shifts

The isotope $229\mathrm{Th}$ is unique in that it possesses an isomeric state of only a few electron volts above the ground state, suitable for nuclear laser excitation. An optical clock based on this transition is expected to be a very sensitive probe for variations of fundamental constants, but the nuclear properties of both states have to be determined precisely to derive the actual sensitivity. We carry out isotope shift calculations in ${\mathrm{Th}}^{+}$ and ${\mathrm{Th}}^{2+}$ including the specific mass shift, using a combination of configuration interaction and all-order linearized coupled-cluster methods and estimate the uncertainty of this approach. We perform experimental measurements of the hyperfine structure of ${\mathrm{Th}}^{2+}$ and isotopic shift between $229{\mathrm{Th}}^{2+}$ and $232{\mathrm{Th}}^{2+}$ to extract the difference in root-mean-square radii as $\delta ⟨r^2{⟩}^{232,229}=0.299\left(15\right){\mathrm{fm}}^2$. Using the recently measured values of the isomer shift of lines of $229m\mathrm{Th}$, we derive the value for the mean-square radius change between $229\mathrm{Th}$ and its low-lying isomer $229m\mathrm{Th}$ to be $\delta ⟨r^2{⟩}^{229m,229}=0.0105\left(13\right){\mathrm{fm}}^2$.