Nuclear structure of lowestTh229states and time-dependent fundamental constants

The electromagnetic transition between the almost degenerate $5/2^{+}$ and $3/2^{+}$ states in ${}^{229}\mathrm{Th}$ is deemed to be very sensitive to potential changes in the fine structure constant $\alpha$. State of the art Hartree-Fock and Hartree-Fock-Bogoliubov calculations are performed to compute the difference in Coulomb energies of the two states that determines the sensitivity of the transition frequency ν on variations in $\alpha$. The kinetic energies are also calculated that reflect a possible variation in the nucleon or quark masses. As the two states differ mainly in the orbit occupied by the last unpaired neutron the Coulomb energy difference results from a change in the nuclear polarization of the proton distribution. This effect turns out to be rather small and to depend on the nuclear model. The sensitivity $q_s$ of the frequency shift $\delta \nu$ on $\delta \alpha /\alpha (\delta \nu =q_s\delta \alpha /\alpha )$ varies for the different models between about $+{10}^{20}\hspace{0.5em}\mathrm{Hz}$ and $-{10}^{20}\hspace{0.5em}\mathrm{Hz}$. Therefore, much more effort must be put into the improvement of the nuclear models before one can draw conclusions from a measured drift in the transition frequency on the size of a temporal drift of $\alpha$.