  The 7.6 eV electromagnetic transition between the nearly degenerate ground
state and first excited state in the Th-229 nucleus may be very sensitive to
potential changes in the fine-structure constant, $\alpha = e^2/\hbar c$.
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.
