Proposed Experimental Method to Determine $\alpha$ Sensitivity of Splitting between Ground and 7.6 eV Isomeric States in ${}^{229}\mathrm{Th}$

The 7.6 eV electromagnetic transition between the nearly degenerate ground state and first excited state in the ${}^{229}\mathrm{Th}$ nucleus may be very sensitive to potential changes in the fine-structure constant, $\alpha =e^2/\hslash c$. However, the sensitivity is not known, and nuclear calculations are currently unable to determine it. We propose measurements of the differences of atomic transition frequencies between thorium atoms (or ions) with the nucleus in the ground state and in the first excited (isomeric) state. This will enable extraction of the change in nuclear charge radius and electric-quadrupole moment between the isomers, and hence the $\alpha$ dependence of the isomeric transition frequency with reasonable accuracy.