Sensitivity factor $K$ of the $\mathrm{Th}-229$ nuclear transition to $\alpha$
In plain words
The thorium-229 nucleus has an unusually low-energy excited state that can be reached by a laser, and its frequency may react to changes in $\alpha$ thousands of times more than atomic clocks do. How large that factor is depends on nuclear-structure numbers still poorly known.
Precise statement
Define $K$ by $\delta \nu / \nu = K \delta \alpha / \alpha$ for the $8.36\,\mathrm{eV}$ Th-229 isomer transition. Combining the measured quadrupole-moment ratio ($\Delta Q_0/Q_0 = 1.791(2)\%$) with charge-radius data gives $K = 5900(2300)$ under a prolate-spheroid nuclear model (Beeks et al, arXiv 2407.17300, 2024). Determine $K$ with uncertainty below 10%, which requires the isomer-ground mean-square charge-radius difference to that precision and a model-independent treatment of the deformation.
What would settle it
A measurement of the isomer shift of an electronic transition in Th ions with independent atomic-theory field-shift factors, giving $\delta(r^{2})$ between isomer and ground state to 10%.
Status in the literature
Unverified note
The 2024-2025 laser spectroscopy of Th-229 in CaF2 established $K$ of order 1e3 to 1e4; the remaining uncertainty is about 40%.